Thèse de doctorat

Transcription

Thèse de doctorat

Université
Paris
VII - Denis Diderot
Université Pierre et
Marie Curie
École Doctorale de Sciences Mathématiques de Paris
Centre
Thèse de doctorat
Discipline : Mathématiques
présentée par
Jeremy Daniel
Variations de structures de Hodge lacées et
fibrés harmoniques
dirigée par Bruno Klingler
Soutenue le 24 septembre 2015 devant le jury composé de :
M. Olivier Biquard
M. François Charles
M. Philippe Eyssidieux
M. Bruno Klingler
M. Claude Sabbah
M. Carlos Simpson
Mme Claire Voisin
ENS de Paris
Université Paris-Sud
Université Joseph Fourier
Université Paris Diderot
École Polytechnique
Université Sophia Antipolis
Université Pierre et Marie Curie
Rapporteur absent lors de la soutenance :
M. Takuro Mochizuki Kyoto University
examinateur
examinateur
examinateur
directeur
examinateur
rapporteur
examinatrice
2
Institut de Mathématiques de
Jussieu-Paris Rive Gauche
Bâtiment Sophie Germain
Case 2047
75205 Paris CEDEX 13
UPMC
École Doctorale de Sciences
Mathématiques de Paris Centre
4 place Jussieu
75252 Paris Cedex 05
Boite courrier 290
À la mémoire de mon père.
Remerciements
Tout commença un matin d’hiver, il y a bientôt cinq ans. Je m’étais exilé
un semestre à Munich et, de retour à Paris pour un week-end, je profitais de
mon samedi matin pour rencontrer mon possible directeur de thèse. J’arrive
au bâtiment de mathématiques de Chevaleret et, après quelques minutes d’incompréhension avec le gardien du lieu, parviens à contacter Bruno Klingler.
Quelques semaines plus tard, il me proposait un sujet : c’était le début d’une
longue aventure.
Le temps a passé depuis. La nature mathématique, sauvage et implacable,
m’a contraint à plusieurs reprises à modifier drastiquement la direction de mes
recherches ; des semaines d’un travail aussi fastidieux qu’inutile s’envolaient aussitôt. On aurait tort de mésestimer les angoisses qu’éprouve un jeune chercheur
durant sa thèse.
À sa manière, singulière mais pleine d’humanité, Bruno m’a aidé à surmonter les difficultés, m’a guidé vers de nouvelles pistes de réflexion et a stimulé ma
curiosité mathématique quand je tournais au ralenti. Il m’a permis de franchir
une étape importante dans ma maturité mathématique – poser des questions
avant de résoudre des problèmes. En cela, j’ai fait mienne sa démarche intellectuelle, je cherche à isoler les atomes de la vérité mathématique et je fais en
sorte que mes démonstrations montrent effectivement quelque chose.
Je lui suis reconnaissant pour tout cela, ainsi que pour nos discussions vaguement philosophiques à l’heure du déjeuner ; et aussi pour cette belle intuition
qu’une inscription des fibrés harmoniques dans la théorie de Hodge était possible. Je lui dois aussi la dénomination de structure de Hodge lacée, qui a vite
supplanté mon peu inventif et sibyllin structure de Hodge non-abélienne.
Je remercie Carlos Simpson pour les discussions électroniques que nous
avons échangées durant ma thèse, à défaut de parvenir à nous rencontrer à
Paris ou Princeton. C’est un grand honneur pour moi qu’il ait accepté de rapporter ma thèse et ses encouragements enthousiastes me vont droit au cœur.
I am very grateful to Takuro Mochizuki who referred my manuscript. I have
been able to improve the quality and the rigor of my text, thanks to his many
remarks, and it is very precious to have his opinion on my work.
La constitution de mon jury de thèse n’a pas été sans heurts et je suis
très heureux de présenter le fruit de mes réflexions à Olivier Biquard, Philippe
Eyssidieux et Claude Sabbah – avec qui j’ai échangé des idées durant ma thèse –
ainsi qu’à François Charles et Claire Voisin. Merci également à Frédéric Helein
pour avoir été compréhensif devant l’embrouillamini administratif dont je lui ai
6
fait part.
Je remercie Xiaonan Ma, co-auteur de mon premier article. Sa grande rigueur et son aisance technique ont permis de clarifier les problèmes soulevés par
une approche harmonique à la cohomologie caractéristique, aboutissant aux résultats de la deuxième partie de mon manuscrit.
Yohan Brunebarbe fut mon aîné dans cette aventure doctorale. Nous avons
évolué en parallèle, partageant nos doutes et nos réflexions, à Paris et dans
les nombreuses conférences où nous allâmes ensemble. Je le remercie pour ces
moments et m’excuse de lui redemander les définitions des fibrés big et nef, à
chaque fois que nous nous voyons.
En plus des personnes déjà nommées, j’ai eu le plaisir d’échanger des idées
– ou simplement de discuter – avec de nombreux.ses mathématicien.nes. Par
ordre chronoalphabélogétique : Robert Bryant, Philip Griffiths, Madhav Nori,
Colleen Robles, Michel Rumin sur la cohomologie caractéristique des systèmes
différentiels extérieurs ; Nicolas Bergeron, Henri Carayol, Wushi Goldring, JeanLoup Waldspurger sur la correspondance thêta, malheureusement absente de
cette thèse ; Josef Dorfmeister, Walter Freyn sur les groupes de lacets ; Patrick
Brosnan, Serge Cantat, Pierre-Henri Chaudouard, Agnès Desolneux, Ania Otwinowska pour les déjeuners durant mon séjour à Princeton ; Gregory Pearlstein
pour une matinée très instructive lors de mon court voyage au Texas.
Je remercie mes anciens professeurs qui ont eu une influence décisive sur
ma vocation mathématique : Philippe Aubé, pour les nombreuses énigmes que
je m’échinais à résoudre au collège ; Simone Guérin, pour avoir supporté mes
bavardages incessants au lycée ; Serge Dupont, pour ses remarques historiques,
et pour sa conversation cinéphilique et philosophique ; Yves Duval, pour ses
néologismes remarquables, ses explications à base de steack et d’anti-steacks,
et pour l’existence et l’unicité de sa personnalité inimitable. Je remercie aussi
mes professeurs de l’ENS, en particulier Frédéric Paulin et Pierre Pansu pour
avoir confirmé mon goût pour la géométrie et la topologie.
Je ne peux concevoir mon activité de recherche mathématique sans l’enseignement qui l’accompagne. Aussi ai-je une pensée émue pour mes élèves de L1
de Physique de l’université Paris Diderot, premier semestre 2012-2013, à qui
Gauss doit encore donner des boutons, à force d’avoir ingurgité de son pivot.
Ces deux dernières années, j’ai eu la joie d’enseigner les travaux dirigés de
géométrie différentielle aux élèves de l’ENS Ulm. Leurs remarques (presque)
toujours pertinentes m’ont beaucoup appris sur le sujet ; j’espère leur avoir
transmis le goût de la géométrie et la beauté d’un raisonnement intrinsèque.
Je m’excuse par ailleurs auprès d’eux d’écrire comme un cochon. Merci aussi
à Claude Viterbo et Patrick Bernard avec qui j’ai eu plaisir à enseigner cette
discipline.
J’ai également eu la chance de donner un cours de vulgarisation mathématique à des élèves des départements littéraires de l’École. C’est un exercice délicat que de trouver des sujets abordables par des élèves sans bagage technique,
sans toutefois donner une image faussée de la réalité de l’activité scientifique ;
j’espère avoir contribué à l’acquisition par ces élèves de quelques monuments
du paysage mathématique.
7
Je remercie le personnel administratif, disponible et efficace, qui a répondu
à mes interrogations et m’a aidé à résoudre les difficultés que j’ai rencontrées
ces dernières années : Pascal Chiettini, irremplaçable responsable administratif
de l’UFR de mathématiques de Paris Diderot ; Bénédicte Auffray, Zaïna Elmir
et Lara Morise, administratives à l’ENS Ulm ; Laurence Vincent, que de nombreuses générations de mathématiciens ne sauront oublier.
Aux doctorants de Paris 7 ; à la génération qui m’accueillit à Chevaleret :
Alexandre B., Amaury, Dragos, Élodie, Hoël, Louis-Hadrien, Mathieu, Robert.
Et à Arnaud et Victoria pour un mois de juillet (d’août ?) passé à jouer à la
pétanque plutôt qu’à travailler ma thèse.
À la génération de Sophie Germain, avec qui j’aime fréquenter le séminairedont-on-ne-doit-pas-prononcer-le-nom et prendre un goûter à 16h qui s’éternise :
Alexandre, pour les anecdotes en tout genre, la conversation passionnante et
la voix apaisante ; Aurélien, pour qui le corps à un élément n’est pas juste
une blague de matheux ; Baptiste, pour son humeur toujours égale ; Charles,
pour les exposants de Lyapunov et sa manière décontractée de faire des maths ;
Julie, pour les discussions du goûter et son humour pince-sans-rire ; Kévin, pour
les parties de Bomberman, pour le ping-pong et pour ses préoccupations qui
intersectent les miennes ; Marco, pour ses goûts musicaux, son prosélytisme
artistique et son accent qui enchante la langue française ; Martin L., même s’il
veut me faire chanter ; Simon, pour les conversations sans fin et ses explications
autour du tiers-exclu.
À ceux de l’ENS : Bertrand, l’homme qui ne mange pas, pour sa simplicité
et pour les jolies maths en perspective ; Diego, celui qui arrive avant les autres,
pour sa conversation aussi enrichissante que reposante ; Florent, pour les discussions politiques au vitriol ; Omid, pour ses énigmes et son rire communicatif,
qui égaye les toits du DMA ; Tony, celui qui part après les autres, compagnon
de correction, pour les discussions sur le vin ou l’enseignement. À Antoine, Fu
Lie, Henri, Nicolas, Olivier T., Olivier W., Selim, Zoé pour les discussions du
déjeuner. Aux nouveaux venus, qui amènent avec eux une nouvelle dynamique
à notre étage isolé. À Irène et Gabriel, avec qui j’ai toujours plaisir à déjeuner,
prendre un café ou aller voir de vieux films au cinéma.
À ma promotion de l’ENS (et à leurs amours), compagnons de mathématiques et de moments de vie, amoureux de jeux de mots en tout genre, collectionneurs de magnets au C6, séminaristes de feu l’Abbé Mol, co-agrégatifs du crû
2010, commentateurs de l’actualité, Présidents de la République ou Premiers
Ministres, Miss France ou Miss Monde, buveurs d’eau municipale et amateurs de
sushi avec baguettes : Arnaud, vous êtes sérieux, pour l’arnaque du 21 octobre
2015 ; Bastien, pour les pauses-café quand il daigne monter sur les toits et pour
les longues discussions où sa mauvaise foi fait écho à la mienne ; Catherine, pour
sa conversation toujours agréable et pour avoir été au cœur d’une rencontre qui
a tout changé ; Fathi, pour son humour décalé et ses invitations à déguster la
papaye ; François, pour notre compréhension respective des nombres p-adiques
et des catégories aristotéliciennes ; Jean-François, pour le crumble lyophilisé et
les réponses physiques aux problèmes concrets ; Max, pour la bière et les blagues
juives ; Sham, pour les Pink Floyd et les thèses matérialistes ; Silvain, co-bureau
8
à l’ENS, pour une bouteille de vodka qui m’a fait le plus grand bien ; Vincent,
à jamais Doudou, amateur des géométries de Cartan, pour ses colles en théorie
de Lie ; Yasmine, pour une chorégraphie mémorable et pour sa bonne humeur.
À David G., Julia, Marie A., Marie D., Léa, Rémi B.
Aux gens du club de bridge, pour les contrats volés et les chelems non déclarés : Adeline et Olivier, mes ravissants partenaires de compétition ; Baptiste,
pour les soirées pizza ; Daphné et Rémi, qui m’initièrent au jeu ; et puis Axel,
Félix, Jean, Julien, Lionel et Quentin.
Aux anciens de la Prépa : Alexandre, pour avoir été le premier à franchir avec
moi la treizième dimension ; Alicia, amatrice de bandes-annonces nanardesques ;
Claire, pour les déprimes du McDonalds, les dimanches de prépa ; Martin, enfin,
je sais pas, peut-être ; Nicolas, pour les films avec ou sans Scarlett Johansson ;
Pavel, pour m’avoir initié aux joies de la boisson, et pour son imitation du
Parrain, avec l’accent russe...
À mon petit groupe d’amis du lycée, qui survit malgré les années et les
oracles pessimistes ; Adélaïde, pour son rire ; Éléonore, pour les vacances en
Lorraine ; Guillaume F., pour ses conseils avisés ; Guillaume Ma., pour ses déboires éducatifs ; Guillaume Mi., qui nous revient enfin d’outre-Atlantique ; Lucie, pour la joie qu’elle m’apporte quand j’arrive à en placer une, et qui nous
manque en Australie, même s’il ne faut pas le lui dire ; Sébastien, qui n’en a
rien à foutre ; Thibault, malgré ses prédictions, en m’excusant de n’avoir pas
tout à fait suivi le chemin d’Évariste Galois ; Thomas, qui sait rester cool, même
quand il passe à la télévision. À Alix, à Thomas et à leur bébé.
À Julien et Delphine, pour les balades, les repas, les films, les jeux et les
vacances. Aux problèmes de maths de Julien que je ne sais jamais résoudre et
à mon ironie qui ne passe jamais chez Delphine. À Éline, sans qui je n’aurais
pas connu les joies de la course, et dont la seule présence est si réconfortante.
À Nicolas, ex-coloc, finalement ami pour la vie, pour tous les moments passés
et ceux à venir, pour son amour de la bonne orthographe, son refus de l’amphigourisme, et pour les mets qu’il a gentiment préparés pour ma soutenance. À
Martin et Julie, pour les cacahuètes et pour le reste, ineffable. Et à Zoé, petite
chose naissante qui, bientôt, bouleversera la donne.
À mon frère, à Virginie et à Matteo, qui me rend un peu gaga. À ma mère,
pour ses combats et son courage. À Aurélie, qui vint à ma rencontre.
À la littérature, au cinéma, à la musique, à l’art en général, à tout ce qui
permet le dépassement de la réalité et permet de croire au dévoilement d’une
vérité authentique. À Marcel Proust, pour la justesse de sa plume, quand il dépeint la rupture amoureuse ; à François Truffaut, arrêtant Jean-Pierre Léaud,
sur une plage de noir et de blanc, le retrouvant quinze ans plus tard pour parler
de trains qui avancent dans la nuit ; à Nina Simone, pour sa crainte universelle
de rester incomprise ; au fœtus de l’espace, qui nous regarde et se demande quoi
faire à présent. Aux centaines d’autres morceaux d’œuvres, dont j’aurais aimé
parler ici ; et à ceux pour qui ces lignes ne paraîtront pas singulières.
À tous et à tout, merci. Et, pour ton honneur à ne paraître jamais à la
télévision, thank you, Satan !
Résumé
Résumé
Cette thèse étudie deux problèmes indépendants liés à la théorie de Hodge.
Dans le premier chapitre, on introduit une généralisation en dimension infinie des structures de Hodge : les structures de Hodge lacées. La donnée d’une
variation de structures de Hodge lacées est équivalente à celle d’un fibré harmonique, permettant l’étude des fibrés harmoniques via les outils classiques de
théorie de Hodge, notamment l’existence d’un domaine et d’une application de
périodes.
Dans le deuxième chapitre, on étudie la possibilité de développer une théorie
de formes harmoniques pour le calcul de la cohomologie caractéristique, attachée
à une variété différentielle munie d’un système différentiel extérieur. Ceci est
motivé par l’exemple des domaines de périodes qui portent un tel système,
provenant de la distribution horizontale.
Mots-clefs
Théorie de Hodge, fibrés de Higgs, fibrés harmoniques, groupes de lacets,
géométrie sous-riemannienne, cohomologie caractéristique.
10
Variations of loop Hodge structures and harmonic
bundles
Abstract
Two Hodge-theoretic independent problems are discussed in this thesis.
In the first chapter, we introduce an object that generalizes a Hodge structur: a loop Hodge structure. We prove that the datum of a variation of loop
Hodge structures is equivalent to the datum of a harmonic bundle, so that one
can study harmonic bundles using classical tools of Hodge theory, especially the
existence of a period map.
In the second chapter, we consider the problem of defining harmonic forms
computing the characteristic cohomology of a manifold endowed with an exterior differential system. This is motivated by the example of the period domains,
where the exterior differential system is induced by the horizontal distribution.
Keywords
Hodge theory, Higgs bundles, harmonic bundles, loop groups, sub-Riemannian
geometry, characteristic cohomology.
Sommaire
Introduction
0.1 Théorie de Hodge classique . . .
0.2 Théorie de Hodge non-abélienne
0.3 Espaces de lacets et twisteurs . .
0.4 Cohomologie caractéristique . . .
0.5 Présentation des résultats . . . .
0.6 Perspectives . . . . . . . . . . . .
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1 Hodge theory of harmonic bundles
1.1 Harmonic bundles as loop VHS . . . . .
1.2 Classical and loop Hodge structures . .
1.3 The Higgs field . . . . . . . . . . . . . .
1.4 The nilpotent orbit theorem . . . . . . .
1.5 The determinant line bundle . . . . . . .
1.6 Relation with the Shafarevich morphism
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83
2 Characteristic Laplacian
85
2.1 The characteristic Laplacian . . . . . . . . . . . . . . . . . . . . . 85
2.2 Answer to question 0.5.2 . . . . . . . . . . . . . . . . . . . . . . . 98
A Geometry of infinite-dimensional manifolds
111
B Moduli spaces
117
C SL(2)-orbit theorem
121
D Spectral sequences & characteristic cohomology
125
Bibliography
131
Introduction
Cette thèse traite de deux sujets indépendants, les questions étudiées prenant
leurs racines dans la théorie de Hodge.
Le chapitre 1 consiste en une réinterprétation de la théorie des fibrés harmoniques via le prisme de la théorie de Hodge. Si le lien entre variations de
structures de Hodge et fibrés harmoniques est bien connu dans la littérature,
un analogue du domaine de périodes de la théorie de Hodge manquait pour parfaire cette correspondance et pour asseoir encore davantage le méta-théorème
de C. Simpson, affirmant que les résultats de théorie de Hodge ont leur analogue
dans la théorie des fibrés harmoniques.
Le chapitre 2 est issu, à des détails mineurs près, de l’article publié [DM15]
écrit en collaboration avec X. Ma. La donnée d’une distribution de sous-espaces
du fibré tangent d’une variété confère à celle-ci la structure d’un système différentiel extérieur et on définit des espaces de cohomologie caractéristique pour
étudier ce système. Dans notre article, on étudie la possibilité de développer une
théorie de Hodge permettant de mieux comprendre ces espaces de cohomologie.
Malheureusement, nos résultats montrent qu’une approche directe ne permet
pas d’atteindre un tel objectif.
Dans cette introduction, les sections 0.1 à 0.4 précisent le contexte de mes
recherches et rappellent certains résultats, réutilisés plus tard dans le texte. La
section 0.5 détaille la progression de mon travail de thèse et en souligne les
théorèmes principaux. Enfin, la section 0.6 discute des pistes de réflexion qu’il
m’apparait naturel d’exploiter dans l’avenir.
0.1
Théorie de Hodge classique
Par le terme de théorie de Hodge, on peut désigner deux domaines mathématiques voisins mais distincts. Le premier consiste en l’introduction d’objets
harmoniques, solutions d’équations aux dérivées partielles, pour comprendre
des invariants topologiques ou géométriques attachés à une variété différentielle compacte. Le second domaine est plus spécifique : il désigne l’étude de
la correspondance, obtenue sur une variété kählérienne compacte via ces objets
harmoniques, entre invariants topologiques et géométriques. Nous rappelons ici
les résultats de la théorie de Hodge dite classique ou abélienne.
Seules la définition des variations de structures de Hodge et la structure des
domaines de période seront utilisées par la suite. On trouvera des preuves des
affirmations dans [GH78], [Voi02] et [CMSP03].
14
0.1.1
Introduction
Théorème de Hodge
Soit X une variété différentielle orientée. On note A• le faisceau gradué des
formes différentielles à valeurs réelles et d : A• → A•+1 la différentielle extérieure. Comme d2 = 0, le couple (A• , d) est un complexe de faisceaux. On
• (X, R) la cohomologie du comappelle cohomologie de de Rham et on note HdR
•
plexe (A (X), d). La cohomologie de de Rham s’identifie à la cohomologie du
faisceau localement constant R.
Soit g une métrique riemannienne sur X. L’opérateur d admet un adjoint
formel d∗ pour la métrique g et on considère le laplacien ∆d := dd∗ + d∗ d. Une
forme α est dite d-harmonique si ∆d (α) = 0. On note Hd• (X, R) l’espace des
formes harmoniques sur X. Il est équivalent d’être d-harmonique et d’être à la
fois d-fermé et d∗ -fermé. En particulier, on a une application linéaire naturelle
• (X, R).
Hd• (X, R) → HdR
Théorème 0.1.1 (de Hodge). Si X est compacte, H• (X, R) est de dimension
• (X, R) est un isomorphisme.
finie et l’application H• (X, R) → HdR
Un point crucial dans la preuve de ce théorème est l’utilisation du caractère
elliptique du laplacien.
Si X est une variété complexe, le faisceau A•,•
C des formes différentielles à
valeurs complexes est bigradué et la différentielle extérieure se décompose en
¯ le premier opérateur étant de type (1, 0), le second de type (0, 1).
d = ∂ + ∂,
¯
Pour tout p, on considère le complexe (Ap,•
C , ∂). On appelle cohomologie de
p,•
Dolbeault de bidegré (p, •) et on note HDol (X) la cohomologie du complexe
¯
(Ap,•
C (X), ∂). Cette cohomologie s’identifie à la cohomologie du faisceau des
formes holomorphes de degré p.
Étant donnée une métrique g compatible avec la structure complexe J de X,
¯ On note Hp,• (X) l’espace
on définit de même un laplacien ∆∂¯ correspondant à ∂.
∂¯
¯
des formes ∂-harmoniques.
Dans cette situation, l’analogue du théorème 0.1.1
affirme que cet espace est de dimension finie et que l’application H∂p,•
¯ (X) →
p,•
HDol
(X) est un isomorphisme d’espaces vectoriels complexes.
0.1.2
Décomposition de Hodge
Si (X, J) est une variété complexe, munie d’une métrique riemannienne g compatible avec J, on dit que X est kählérienne si J est parallèle pour la connexion
de Levi-Civita. Une classe importante d’exemples de variétés kählériennes compactes est donnée par les variétés projectives lisses définies sur le corps des
nombres complexes : les variétés analytiques sous-jacentes sont kählériennes via
la métrique de Fubini-Study. L’intérêt de cette notion provient en partie du
résultat suivant :
Théorème 0.1.2. Si X est une variété kählérienne, l’égalité ∆d = 2∆∂¯ est
satisfaite.
Le laplacien ∆∂¯ respecte le bidegré des formes différentielles. Ainsi, le théorème de Hodge 0.1.1 appliqué à d et ∂¯ et le théorème 0.1.2 impliquent une
égalité entre espaces de cohomologie :
15
0.1. Théorie de Hodge classique
Théorème 0.1.3 (Décomposition de Hodge). Si X est une variété kählérienne
compacte, on a pour tout k un isomorphisme
M
k
HdR
(X, C) =
p,q
HDol
(X).
p+q=k
De plus, cet isomorphisme est indépendant de la métrique kählérienne considérée.
Ce théorème est remarquable. Le terme de gauche est un invariant topologique de la variété différentielle X alors que celui de droite est défini à partir de
la structure complexe de X. On dira des espaces de cohomologie de Dolbeault
qu’ils constituent des invariants géométriques de X ; dans le cas particulier
où X est une variété projective lisse, ce sont même des invariants algébriques,
comme toute cohomologie d’un faisceau cohérent par le principe GAGA. Si on
déforme X convenablement, le terme de gauche sera constant tandis que la décomposition de droite évoluera. Cette évolution est très contrainte et est l’objet
d’étude de la sous-section 0.1.4.
Avant cela, nous devons introduire un ingrédient supplémentaire : une polarisation. On considère toujours une variété kählérienne compacte X, et on note
ω la forme de Kähler donnée par ω(Y, Z) = −g(Y, JZ). C’est une forme fermée
de bidegré (1, 1). Soit Q la forme bilinéaire réelle (−1)k -symétrique définie sur
k (X, R) par
HdR
Z
α ∧ β ∧ ω n−k .
Q(α, β) =
X
k (X, C) par
Elle induit une forme hermitienne h, dite de polarisation, sur HdR
h(α, β) = QC (α, β̄),
où QC est la forme C-bilinéaire complexifiée de Q et où = i si k est impair et
1 si k est pair.
k (X, R) → H k+2 (X, R) par
Enfin, on définit l’opérateur de Lefschetz L : HdR
dR
L(α) = α ∧ ω.
k (X, R) →
Le théorème de Lefschetz vache affirme que l’opérateur Ln−k : HdR
•
est un isomorphisme. La cohomologie primitive H0,dR est défini
k (X, R), noyau de l’opérateur Ln−k+1 .
en degré k comme le sous-espace de HdR
p,q
p+q
p,q
On note H0,Dol (X) := H0,dR (X, C) ∩ HDol
(X), l’intersection ayant lieu dans
k
HdR (X, C), grâce au théorème 0.1.3.
2n−k
HdR
(X, R)
Théorème 0.1.4 (Hodge-Riemann). La décomposition de Hodge induit une
décomposition
M
p,q
k
H0,dR
(X, C) =
H0,Dol
(X).
p+q=k
p,q
Cette décomposition est orthogonale pour h, h est définie sur chaque H0,Dol
(X)
et son signe alterne avec la parité de p.
16
0.1.3
Introduction
Domaine de périodes
D’après la sous-section précédente, si X est une variété kählérienne compacte,
k
l’espace vectoriel H0,dR
(X, C) admet une décomposition en somme directe de
sous-espaces, cette décomposition obéissant à certaines contraintes liées à la
forme hermitienne h. On abstrait cette situation via la notion de structure de
Hodge. Pour simplifier, nous ne parlerons que de structures de Hodge complexes
polarisées.
Définition 0.1.5. Soit (V, h) un espace vectoriel complexe de dimension finie
muni d’une forme hermitienne non-dégénérée. Une structure de Hodge (complexe polarisée de poids 0) sur V est une décomposition h-orthogonale
V =
M
Vp
telle que h est définie de signe (−1)p sur V p . La filtration de Hodge associée est
la filtration décroissante de V donnée par
F p V :=
M
V q.
q≥p
On fixe (V, h) comme précédemment et des entiers positifs ou nuls (np )p∈Z
P
tels que np = dim V .
Définition 0.1.6. Le domaine de périodes D associé à (V, h, (np )p∈Z ) est l’enL
semble des structures de Hodge V = V p sur V telles que dim V p = np pour
tout p. Son compact dual Ď est l’ensemble des filtrations décroissantes F • V de
P
V telles que dim F p V = q≥p nq , pour tout p.
Le groupe unitaire U (V, h) agit sur D et on montre que cette action est tranL
sitive. En un point o de D, correspondant à une décomposition V = V p , le staQ
bilisateur de cette action est donné par le groupe compact H = U (V p , h|V p ).
Ainsi, D s’identifie à l’espace homogène
D = U (V, h)/
Y
U (V p , h|V p ).
Le groupe linéaire GL(V ) agit transitivement sur Ď et son stabilisateur en un
point o de D, correspondant à une filtration F • V , est le sous-groupe parabolique
P de GL(V ) des éléments stabilisant cette filtration ; le groupe H est égal à
l’intersection U (V, h) ∩ P . On a donc un isomorphisme
Ď = GL(V )/P
et Ď est une variété complexe, en fait projective. Le domaine de périodes D
s’injecte dans son compact dual Ď en associant la filtration de Hodge à la
structure de Hodge. On voit aisément que cette injection identifie D à un ouvert
(pour la topologie usuelle) de Ď.
Q
Par définition, le compact dual Ď se plonge dans la variété Gr(n0p , V ),
où Gr(k, V ) désigne la variété grassmannienne des k-plans de V et où n0p =
P
q≥p nq . Si o est un point de Gr(k, V ), correspondant à un sous-espace W de V ,
0.1. Théorie de Hodge classique
17
l’espace tangent To Gr(k, V ) s’identifie canoniquement à Hom(W, V /W ). Ainsi,
si o est un point de Ď correspondant à la filtration F • V , on a une inclusion
To Ď ,→
Y
Hom(F p V, V /F p V ).
p
Le sous-espace horizontal est défini comme étant
To,h Ď := To Ď ∩
Y
Hom(F p V, F p−1 V /F p V ).
p
Ces sous-espaces définissent une distribution holomorphe dans Ď, équivariante
sous l’action du groupe linéaire. En conclusion :
Proposition 0.1.7. Le domaine de périodes D est une variété complexe, homogène sous l’action du groupe unitaire U (V, h) qui agit par biholomorphismes. De
plus, on peut définir une distribution holomorphe naturelle sur Ď, équivariante
sous l’action du groupe U (V, h).
0.1.4
Variations de structures de Hodge
Dans le langage de la sous-section 0.1.3, le théorème 0.1.4 affirme l’existence
k
d’une structure de Hodge (au signe près) sur l’espace vectoriel H0,dR
(X, C),
muni de sa forme de polarisation h. On s’intéresse à l’évolution de cette structure
de Hodge quand on déforme la variété X ; précisons d’abord cette notion de
déformation.
Définition 0.1.8. Soit S une variété complexe connexe. Une fibration kählérienne au-dessus de S est la donnée d’une application propre, submersive et
holomorphe π : Ξ → S et d’une classe de cohomologie ω dans H 2 (Ξ, R) telle
que, en restriction à chaque fibre Xs := π −1 (s), ωs soit la classe de cohomologie
d’une forme de Kähler sur Xs .
Par le théorème de fibration d’Ehresmann, toutes les fibres de π sont difféomorphes et on peut considérer la famille (Xs )s∈S comme la déformation d’une
fibre Xs0 . Localement, l’isomorphisme entre espaces de cohomologie de de Rham
des fibres est canonique. En effet, si R désigne le faisceau localement constant
sur Ξ, les faisceaux R• π∗ R sont dans ce cas des faisceaux localement constants
sur S et le fibré vectoriel attaché au faisceau localement libre Rk π∗ R ⊗R C ∞ (R)
k (X , R). Sur ce fibré, on
a pour fibre en un point s de S l’espace vectoriel HdR
s
a donc une connexion plate D, appelée connexion de Gauss-Manin.
k (X , C) vient d’une
Comme la forme de polarisation sur chaque fibre de HdR
∗
classe de cohomologie définie sur tout Ξ, on peut vérifier que la forme de polarisation est plate pour D. En particulier, le sous-fibré de cohomologie primitive
définit un sous-fibré plat, noté simplement V dans la suite.
Quitte à passer au revêtement universel de S, le fibré plat (V, D) est trivial,
de fibre notée V . La polarisation étant plate, l’espace vectoriel V est muni
d’une forme hermitienne non dégénérée notée h. La décomposition de Hodge sur
chaque fibre de V donne une structure de Hodge sur l’espace vectoriel hermitien
(V, h). En conclusion, on obtient une application f : S̃ → D où D est le domaine
18
Introduction
p,k−p
de périodes attaché à l’espace (V, h) et aux dimensions np = dim H0,Dol
(Xs )
(indépendantes de s). Le théorème suivant décrit deux aspects qualitatifs de
l’évolution de la structure de Hodge portée par la cohomologie primitive en
degré k des fibres.
Théorème 0.1.9 (Griffiths). L’application f est holomorphe et sa différentielle
est à valeurs dans le fibré horizontal Th D.
On peut donner une version plus intrinsèque de ce théorème et l’abstraire
dans la notion de variation de structures de Hodge.
Définition 0.1.10. Soit S une variété complexe. Une variation de structures de
Hodge (complexes polarisées) sur S est la donnée d’un fibré vectoriel complexe
¯ d’une forme hermitienne non dégénérée h
V, d’une connexion plate D = ∂ + ∂,
L
sur V et d’une décomposition lisse V = V p en sous-fibrés vectoriels tels que,
L
notant F p V = q≥p V q , on ait :
1. La forme hermitienne h est D-plate ;
2. Sur chaque fibre, la décomposition Vs =
Hodge complexe polarisée par hs ;
L
Vsp définit une structure de
3. ∂¯ : F p V → F p V ⊗ A0,1 ;
4. D : F p V → F p−1 V ⊗ A1 .
On notera souvent (V, h, D, F p V) une variation de structures de Hodge, les
décompositions de Hodge se déduisant des filtrations via la forme hermitienne
h. La condition 3. dit que les sous-fibrés F p V varient holomorphiquement, la
structure holomorphe du fibré V étant déduite de sa structure plate. De plus,
la condition 4., appelée condition de transversalité de Griffiths, demande que
la filtration soit presque plate, au sens où une section plate qui prend sa valeur
en s dans (F p V )s ne peut infinitésimalement que tomber dans le sous-espace
(F p−1 V )s , qui suit dans la filtration.
Le théorème 0.1.9 affirme qu’une famille de variétés kählériennes compactes
induit une variation de structures de Hodge portée par le fibré de cohomologie
primitive, en degré k, des fibres de la famille.
0.2
Théorie de Hodge non-abélienne
De même que la théorie de Hodge classique fournit un lien entre les invariants
topologiques et géométriques donnés par les espaces de cohomologie de de Rham
et de Dolbeault, de même la théorie de Hodge non-abélienne établit une correspondance entre un invariant topologique et un invariant géométrique. Ces
invariants sont plus compliqués : il s’agit d’une part de l’espace des représentations du groupe fondamental de la variété, et d’autre part, de l’espace des fibrés
de Higgs, fibrés holomorphes avec une structure additionnelle.
Pour ces deux objets, on montre que dans certaines conditions il existe
un analogue harmonique, donnant au final une bijection entre les invariants
discutés, de nature pourtant très différente.
19
0.2. Théorie de Hodge non-abélienne
0.2.1
Variations de structures de Hodge et fibrés harmoniques
Soit (V, h, D, F • V) une variation de structures de Hodge ; on peut définir deux
objets à partir de cette variation. Le premier est très simple : il s’agit d’une
représentation linéaire du groupe fondamental, obtenue en prenant la monodromie de la connexion plate D. Pour le second, on considère le fibré vectoriel
E = GrF • V =
M F p−1 V
F pV
.
Il a une structure holomorphe naturelle, puisque les sous-fibrés F p V sont holomorphes. Par la condition de transversalité de Griffiths, le gradué de la connexion
plate D définit un opérateur θ sur E, dont on vérifie facilement qu’il est tensoriel. Ainsi, θ est une forme différentielle de degré 1 à valeurs dans End(E) ; on
montre aisément qu’elle a bidegré (1, 0) et qu’elle est en fait holomorphe. De
plus, l’égalité D2 = 0 implique θ ∧ θ = 0 sur le gradué E.
Définition 0.2.1. Un fibré de Higgs sur X est la donnée d’un fibré holomorphe
E et d’une forme holomorphe θ de degré 1, à valeurs dans End(E), satisfaisant
θ ∧ θ = 0.
Ainsi, une variation de structures de Hodge sur X induit une représentation
linéaire de π1 (X) et un fibré de Higgs sur X.
Plus généralement, considérons (E, D) un fibré plat sur X et h une métrique
hermitienne sur E. Il existe une unique décomposition D = ∇ + α telle que ∇
est une connexion h-métrique et α une forme de degré 1 à valeurs dans les
endomorphismes h-hermitiens de E. On peut décomposer à nouveau en types :
∇ = ∇1,0 + ∇0,1 et α = α1,0 + α0,1 .
Définition 0.2.2. La métrique h est harmonique si l’opérateur différentiel
∇0,1 + α1,0 est de carré nul. On dit aussi que le fibré (E, D, h) est un fibré
harmonique.
Une définition équivalente des fibrés harmoniques est donnée dans la section
1.1.2. La condition donnée dans l’énoncé se décompose en fait en trois conditions
distinctes : (∇0,1 )2 = 0, qui affirme que ∇0,1 est l’opérateur ∂¯ d’une structure
holomorphe sur E (théorème d’intégrabilité de Koszul-Malgrange) ; ∇0,1 α1,0 =
0, qui affirme que α1,0 est holomorphe pour cette structure ; α1,0 ∧ α1,0 = 0 qui
est une condition algébrique supplémentaire.
En particulier, de même qu’une variation de structures de Hodge, un fibré
harmonique sur X induit à la fois une représentation linéaire de π1 (X) et un
fibré de Higgs sur X. En fait, les variations de structures de Hodge induisent
des fibrés harmoniques et les deux constructions évoquées sont les mêmes, voir
la proposition 1.2.5.
0.2.2
Du côté de Betti
On s’intéresse ici aux représentations linéaires du groupe fondamental d’une
variété complexe connexe X. Leur espace de module est traditionnellement
20
Introduction
dénommé espace de module de Betti (voir l’annexe B). Soit donc ρ : π1 (X) →
GL(n, C) une représentation du groupe fondamental. On considère (E, D) le
fibré plat associé et on se demande s’il admet une métrique harmonique h.
Définition 0.2.3. La représentation ρ est semi-simple si tout sous-espace vectoriel de Cn , invariant par la représentation, admet un supplémentaire invariant.
Le théorème suivant est dû à S. Donaldson [Don87], dans le cas où X est
une courbe et où n = 2, et à K. Corlette [Cor88] dans le cas général.
Théorème 0.2.4 (Corlette-Donaldson). Soit X une variété kählérienne compacte. Une représentation ρ : π1 (X) → GL(n, C) est la monodromie d’un fibré
harmonique si et seulement si elle est semi-simple. De plus, la métrique harmonique est unique à isomorphisme près.
0.2.3
Le côté de Higgs
Soit (E, θ) un fibré de Higgs sur X. On se pose la même question que précédemment : quand ce fibré de Higgs est-il obtenu à partir d’un fibré harmonique ? Si
h est une métrique sur E, on peut définir une connexion Dh de la façon suivante
[Sim92].
Notons ∂¯ l’opérateur différentiel donné par la structure holomorphe sur E.
Soit ∇h = ∂ + ∂¯ la connexion de Chern de (E, h). Notons de plus, θ∗ l’adjoint
de θ pour la métrique h. Alors, Dh est défini par
Dh := ∇h + θ + θ∗ .
C’est bien une connexion puisqu’elle diffère de ∇h par une forme de degré 1 à
valeurs dans End(E). De plus, on vérifie aisément que le fibré de Higgs (E, θ)
provient d’un fibré harmonique si et seulement si Dh est plat.
Définition 0.2.5. Soit X une variété kählérienne compacte de dimension n,
de forme de Kähler ω. Si E est un fibré vectoriel complexe sur X, on définit le
degré de E par
Z
c1 (E) ∧ ω n−1
deg(E) :=
X
et la pente de E comme le quotient
deg(E)
.
rang(E)
Définition 0.2.6. Soit (E, θ) un fibré de Higgs sur une variété kählérienne
compacte X de dimension n, de forme de Kähler ω. On dit que (E, θ) est stable
si tout sous-fibré holomorphe F de E, stable par θ, a une pente strictement
inférieure à celle de E.
On dit que (E, θ) est polystable s’il est somme directe de fibrés de Higgs
stables de même pente.
Le théorème suivant répond alors à la question posée ; il a été montré par
N. Hitchin [Hit87], dans le cas où X est une courbe et le fibré E de rang 2, et
par C. Simpson [Sim88] dans le cas général.
0.3. Espaces de lacets et twisteurs
21
Théorème 0.2.7 (Hitchin-Simpson). Il existe, sur un fibré de Higgs (E, θ), une
métrique h telle que Dh est plate si et seulement si (E, θ) est polystable et vérifie
c1 (E) = c2 (E) = 0. De plus, une telle métrique (dite harmonique) est unique à
isomorphisme près.
0.3
Espaces de lacets et twisteurs
Les applications harmoniques viennent en famille.
Cette idée fondamentale a germé à la fin du xixe siècle dans l’étude des surfaces minimales de l’espace euclidien. Une surface minimale est l’image dans R3 ,
par une application conforme harmonique, d’une surface de Riemann ; elle peut
toujours être déformée en une famille de surfaces minimales, l’espace des paramètres de la déformation étant un cercle. Un exemple célèbre de ce phénomène
est la déformation d’une caténoïde en hélicoïde.
Ceci a été généralisé dans plusieurs directions depuis les années 1980. D’une
part, par la considération d’espaces d’arrivée de plus en plus généraux : groupes
de Lie compacts [Uhl89], espaces symétriques de type compact [DPW98], espaces symétriques généraux [ET98] et [BD05]. D’autre part, en considérant des
variétés complexes de dimension plus grande au départ, et en remplaçant la
notion d’harmonicité par la notion de pluriharmonicité (cf. définition 1.1.14),
[OV90] et [ET98] notamment.
En changeant de point de vue, ce cercle d’applications harmoniques peut être
vu comme une application unique à valeurs dans un espace D construit à partir
d’un espace de lacets du groupe d’isométries de l’espace symétrique. Ce point de
vue est expliqué dans les références précitées, notamment [DPW98]. Cet espace
D est une variété complexe de dimension infinie et est muni d’une distribution
horizontale holomorphe. De plus, D fibre au-dessus de l’espace symétrique ;
le résultat fondamental de la théorie stipule alors qu’une application à valeurs
dans l’espace symétrique se relève en une application holomorphe et horizontale
à valeurs dans D si et seulement si elle est pluriharmonique.
Ce lien entre applications (pluri)harmoniques et holomorphes a également
des origines lointaines. Par exemple, si M est une surface de Riemann et si
φ : M → R3 est une immersion conforme, alors φ est harmonique (et donc
φ(M ) est une surface minimale) si et seulement si son application de Gauss
γ : M → P1 (C) est anti-holomorphe. La théorie des twisteurs généralise ce
genre de résultats : un espace de twisteurs T est une variété complexe qui fibre
au-dessus d’une variété riemannienne M donnée, induisant une correspondance
entre certaines applications holomorphes à valeurs dans l’espace de twisteurs T
et applications (pluri)harmoniques à valeurs dans M , voir [BR90] et [DPW98].
De ce point de vue, l’espace D évoqué ci-dessus devient un espace de twisteurs
universel, pour l’espace symétrique considéré.
Les relations entre ces théories et notre travail sont détaillées dans les soussections 1.1.5 et 1.2.2.
22
0.4
Introduction
Cohomologie caractéristique
Cette section motive le chapitre 2 de ma thèse.
Soit X une variété différentielle. Un système différentiel extérieur [BCG+ 90]
sur X est la donnée d’un idéal différentiel sur X, c’est-à-dire d’un sous-faisceau
J du faisceau A des formes différentielles, tel que J soit stable par produit
extérieur avec une forme quelconque de A et par l’opérateur d de différentiation
extérieure. Une solution d’un système différentiel extérieur est une immersion
f : Y → X tel que f ∗ (J ) = 0.
L’idée est la suivante : considérons par exemple une équation aux dérivées
partielles du premier ordre F (x, y, u, ux , uy ) = 0, où u est une fonction à valeurs
réelles de x et y, ux et uy sont les dérivées partielles de u par rapport à x et y
et F est une fonction réelle définie sur R5 . On se place dans l’espace des jets R5
des points (x, y, z, p, q) et on note α la forme de contact α = dz − pdx − qdy. Si
u(x, y) est une solution de l’équation différentielle de départ, son jet au premier
ordre est une sous-variété de dimension 2 de R5 et est une solution du système
différentiel extérieur sur R5 , engendré par la fonction F et la forme différentielle
α. Réciproquement, toute solution du système différentiel extérieur sur laquelle
la forme dx ∧ dy ne s’annule pas est le jet au premier ordre d’une solution de
l’équation initiale.
Ainsi, les systèmes différentiels extérieurs permettent une approche géométrique des équations aux dérivées partielles. Une classe importante de tels
systèmes est obtenu en considérant une distribution W (de rang constant pour
simplifier) sur X. L’annulateur de W dans A1 engendre en effet un idéal différentiel et une solution de ce système différentiel extérieur est simplement la
donnée d’une immersion f : Y → X telle que la différentielle de f est à valeurs
dans la distribution W. Ces systèmes sont appelés pfaffiens.
Remarquons en passant que l’idéal algébrique engendré par l’annulateur de
W est déjà stable par différentiation si et seulement si W vérifie la condition
d’intégrabilité de Frobenius. Dans ce cas, les solutions du système différentiel
extérieur sont simplement les immersions à valeurs dans les sous-variétés intégrant W.
Dans les articles [BG95a] et [BG95b], les auteurs définissent la cohomologie
caratéristique HJ• (X, R) d’un système différentiel extérieur J comme la cohomologie du complexe
A• (X) ¯
,d .
J • (X)
L’opérateur d passe en effet au quotient puisque J • est stable par d. Il est
immédiat qu’une solution f : Y → X du système J induit une application en
cohomologie
•
f ∗ : HJ• (X, R) → HdR
(Y, R).
Ainsi, une partie de la cohomologie de de Rham de la variété Y provient de
ce qu’elle est solution d’un certain système différentiel extérieur et la cohomologie caractéristique de (X, J ) peut être vue comme une sorte de cohomologie
universelle liée à ce système. Pour étudier cette cohomologie, on peut essayer
0.5. Présentation des résultats
23
de développer une théorie de Hodge dont les résultats imiteraient ceux de la
section 0.1.
Plus précisément, dans [Gri09, §III], P. Griffiths demande si la cohomologie
caractéristique attachée à l’idéal horizontal dans un quotient co-compact d’un
domaine de périodes D porte une structure de Hodge naturelle. Sauf dans de
rares cas, un quotient co-compact de D ne peut pas être muni d’une métrique
kählérienne ; en revanche, il porte une métrique qui est kählérienne dans les
directions horizontales. On peut donc espérer qu’une théorie de Hodge associée à cette métrique permettra de munir la cohomologie caractéristique d’une
structure de Hodge. De plus, dans cette situation, les applications de période
pourraient induire des morphismes de structures de Hodge en cohomologie.
Dans [GGK10], un laplacien caractéristique est défini. La question consiste
alors à déterminer si un analogue du théorème 0.1.1 et de la proposition 0.1.2
sont valides dans cette situation.
0.5
0.5.1
Présentation des résultats
Chapitre 1
C’est le chapitre principal de cette thèse. Nous avons décrit les structures de
Hodge, leur domaine de périodes et les variations associées. De plus, nous avons
expliqué que les variations de structures de Hodge pouvaient être considérées
comme un cas particulier de fibrés harmoniques. Le but de ce chapitre est la
compréhension de tout fibré harmonique du point de vue de la théorie de Hodge.
Le résultat principal est la définition de la notion de structure de Hodge
lacée, une généralisation naturelle en dimension infinie de la notion de structure
de Hodge classique. Dans ce cadre, tout fibré harmonique s’interprète comme
variation de structures de Hodge lacées (théorème 1.1.23) :
Théorème A. La catégorie des variations de structures de Hodge lacées est
équivalente à la catégorie des fibrés harmoniques.
Le formalisme des structure de Hodge lacées ne permet pas seulement d’incorporer naturellement la S 1 -action de Hitchin-Simpson – la théorie des twisteurs le fait aussi – il permet surtout d’associer une application de périodes à
tout fibré harmonique, réalisant ainsi un vœu de T. Mochizuki. On commence
par définir un domaine de période D de dimension infinie, homogène non plus
pour le groupe unitaire U (V, h) mais pour un groupe de lacets, dont la géométrie est tout à fait analogue à celle des domaines de périodes classique (distribution horizontale, courbure négative ou nulle dans les directions horizontales...).
Comme dans le cas des variations de structures de Hodge, on montre (théorème
1.1.49) :
Théorème B. La donnée d’un fibré harmonique sur X est équivalente à la
donnée d’une application holomorphe et horizontale à valeurs dans D, obéissant
à une certaine condition d’équivariance.
Si cette application existe déjà dans la littérature (voir les références dans
la section 0.3), son interprétation cruciale en termes d’application classifiante
24
Introduction
faisait défaut.
Avant d’utiliser cette approche pour étudier les fibrés harmoniques, nous
établissons le lien entre variations de structures de Hodge classiques et lacées
(théorème 1.2.3) :
Théorème C. Il existe une action naturelle du cercle S 1 sur l’ensemble des
variations de structures de Hodge lacées au-dessus d’une variété complexe X.
Les variations de structures de Hodge classiques peuvent être identifiées aux
variations de structures de Hodge lacées, isomorphes aux variations dans leur
S 1 -orbite.
De même que dans le cas classique, la différentielle de l’application des
périodes obtenue à partir d’un fibré harmonique s’identifie au champ de Higgs
du fibré. On en déduit le théorème 1.3.11, plus précis que ce qui est établi dans
la littérature :
Théorème D. Si (E, θ) est un fibré de Higgs provenant d’un fibré harmonique,
le feuilletage défini par le noyau du champ de Higgs θ est intégrable.
Dans l’article [Sch73], deux théorèmes fondamentaux sont établis concernant
l’asymptotique des variations de structures de Hodge, c’est-à-dire le comportement d’une variation de structures de Hodge à l’infini sur une variété ouverte :
il s’agit du théorème de l’orbite nilpotente et du théorème de l’orbite SL(2). La
lecture de [Moc07a] suggère de prouver des théorèmes analogues dans le cas des
fibrés harmoniques. Pour l’orbite nilpotente, il s’agit de notre théorème 1.4.5 :
Théorème E. Pour un fibré harmonique modéré nilpotent avec structure parabolique triviale, on peut établir un théorème de l’orbite nilpotente.
Pour le théorème de l’orbite SL(2) – plus technique – nous proposons dans
l’annexe C une discussion conjecturale des résultats que notre approche pourrait
permettre d’obtenir.
Enfin, on démontre un résultat lié à la conjecture de Carlson-Toledo – qui
affirme que le groupe fondamental Γ d’une variété kählérienne compacte admet
virtuellement de la cohomologie en degré 2 (théorème 1.5.17) :
Théorème F. Soit X une variété kählérienne compacte dont le groupe fondamental admet une représentation linéaire irréductible non-rigide. Alors, l’espace
de cohomologie H 2 (π1 (X), R) est non trivial.
Si ce résultat est déjà connu dans la littérature [KKM11], la preuve que nous
en donnons est assez différente. Elle utilise en particulier un résultat important
en soi et semble-t-il inédit (corollaire 1.5.13) :
Théorème G. Si (E, θ) est un fibré de Higgs sur X, provenant d’un fibré
harmonique, alors la classe de cohomologie de la forme de degré 2 définie par
1
Tr(θ ∧ θ∗ )
4iπ
est entière sur le revêtement universel de X.
βX :=
Ici, θ∗ désigne l’adjoint de θ pour la métrique harmonique.
0.5. Présentation des résultats
0.5.2
25
Chapitre 2
L’objet de ce chapitre est d’étudier la possibilité de développer une théorie de
Hodge pour la cohomologie caractéristique d’une variété munie d’un système
différentiel extérieur, plus précisément d’un système pfaffien. Malheureusement,
une approche directe ne fonctionne pas et nos résultats sont négatifs.
Dans l’article [GGK10], les auteurs suggèrent que le laplacien caractéristique
qu’ils définissent est hypoelliptique quand la distribution W du système pfaffien
engendre l’espace par crochets. Ceci est incorrect ; on donne le contre-exemple
2.1.10 :
Proposition 0.5.1. Il existe une variété complexe compacte de dimension 3,
avec un système pfaffien de contact, telle que le laplacien caractéristique ne soit
pas hypoelliptique en degré 2.
De plus, si la complexité de la situation rend un énoncé général assez hasardeux, nous estimons que cette situation est générale et que l’hypoellipticité
de ce laplacien en tel degré ne peut être que miraculeuse.
Nous étudions ensuite une question formulée dans [GGK10].
Question 0.5.2. Soit X une variété complexe, soit g une métrique sur X
compatible à sa structure complexe. Soit W une distribution holomorphe sur X,
de fibré réel sous-jacent WR . Quelles sont les conditions nécessaires et suffisantes
pour que le laplacien caractéristique correspondant au système pffafien défini
par WR respecte le bidegré des formes ?
Il s’agirait donc d’établir l’analogue de la proposition 0.1.2. La réponse est
inattendue et décevante (théorème 2.2.1) :
Théorème H. Le laplacien caractéristique ne respecte le bidegré des formes en
aucun degré supérieur à 1 quand la distribution W n’est pas involutive.
Dans la situation classique, i.e. quand la distribution W est égale à tout
l’espace, on a le théorème 2.2.2 :
Théorème I. Sur une variété complexe munie d’une métrique riemannienne
compatible à la structure complexe, le laplacien ∆d respecte le bidegré des formes
si et seulement si la métrique est kählérienne.
Le sens direct résulte de la proposition 0.1.2 mais nous n’avons pas trouvé
de référence dans la littérature pour la réciproque.
Mentionnons enfin l’annexe D de notre thèse. On y définit une suite spectrale, intéressante dans l’étude de la cohomologie caractéristique d’un système
pfaffien, en accord avec l’article [Rob14a].
26
0.6
0.6.1
Introduction
Perspectives
Chapitre 1
La motivation pour notre travail est proche de celle qui a poussé à introduire
la théorie des twisteurs de [Sim97] (sans rapport avec celle de la section 0.3).
Une compréhension plus fine des liens entre les deux approches reste à étudier ;
par exemple, la raison pour laquelle le point de vue des twisteurs ne permet pas
d’avoir l’analogue d’un domaine de périodes ne nous apparait pas clairement.
Deux questions plus précises sont liées à cette problématique : d’une part,
les résultats des articles [Moc07a] et [Moc07b] sont exprimés avec le point de
vue des twisteurs et il nous faut comprendre si ces résultats sont équivalents
à ceux que nous conjecturons dans l’annexe C. D’autre part, des domaines de
périodes pour des généralisations de variations de structures de Hodge, dénommées structures TERP, sont apparus dans des travaux comme [HS07] et [HS08].
Il serait intéressant de comprendre si ces domaines TERP peuvent être vus à
l’intérieur du domaine de périodes des structures de Hodge lacées, ou plus probablement dans un analogue de ce domaine de périodes.
Une autre source de questions est liée à l’action de C∗ sur l’espace de modules des fibrés de Higgs. L’action de S 1 que nous décrivons sur les variations
de structures de Hodge lacées correspond à l’action usuelle de S 1 sur l’espace
de modules des fibrés de Higgs. Au niveau du fibré harmonique, la métrique
ne change pas et donc l’action est facile à décrire. Si maintenant on cherche
à décrire l’action de C∗ , il faut comprendre comment évolue la métrique harmonique radialement et seul le théorème de Hitchin-Simpson prouve l’existence
d’une métrique harmonique pour toute l’orbite sous C∗ . Il s’agit d’un théorème
d’existence pour une équation aux dérivées partielles. Nous posons la question
de savoir s’il ne suffit pas en fait, connaissant une métrique harmonique, de
résoudre des équations différentielles ordinaires, pour déterminer les métriques
harmoniques dans l’orbite sous C∗ . Cette intuition est peut-être infondée ; disons
seulement qu’elle provient d’une interrogation plus générale sur le lien entre la
méthode DPW de l’article [DPW98], permettant de construire des applications
pluriharmoniques, et le théorème de Hitchin-Simpson.
0.6.2
Chapitre 2
Malgré les résultats négatifs dont nous avons discuté, une idée semble toutefois
intéressante à exploiter. Dans [BG95a], une définition plus compliquée de la
cohomologie caractéristique est donnée. L’idée est de considérer plutôt un système différentiel formel sur la limite projective formelle d’une tour d’extensions
de variétés. Ce système différentiel formel a l’avantage de satisfaire la condition
d’intégrabilité de Frobenius. Si l’on parvient à donner un sens à un laplacien
caractéristique sur cette limite de variétés, peut-être la condition d’avoir une
métrique kählérienne dans les directions de W , sur la variété de base, suffira-telle à assurer que le laplacien respecte le bidegré des formes différentielles ?
Quand bien même cette approche donnerait des résultats, il resterait encore
à obtenir un théorème de Hodge affirmant qu’on peut véritablement étudier la
cohomologie caractéristique via les formes harmoniques...
Chapter 1
Hodge theory of harmonic
bundles
This chapter is the main part of my thesis. We define a generalization of Hodge
structure, that we call loop Hodge structure. We show that any harmonic
bundle on a complex manifold can be reinterpreted as a variation of loop Hodge
structures and derive some consequences of this correspondence.
1.1
Harmonic bundles as variations of loop Hodge
structures
This section is the heart of chapter 1. We introduce the notion of loop Hodge
structure and show that their variations are in 1 − 1 correspondence with harmonic bundles. With this point of view, one can associate a holomorphic period
map to any harmonic bundle; its target is a complex Hilbert manifold whose
geometry mimics the geometry of finite dimensional classical period domains
discussed in the introduction.
1.1.1
Loop Hodge structures
We introduce the fundamental notion of loop Hodge structure. We will need
to consider infinite-dimensional complex vector spaces endowed with a nondegenerate indefinite metric. Such spaces can behave badly but Krein spaces
share a lot of properties with Hilbert spaces. The standard reference for this
topic is [Bog74].
Krein spaces
Let K be a (possibly infinite-dimensional) complex vector space and let B be a
Hermitian form on K (we do not assume that B is positive definite). A subspace
W of K is positive (resp. negative) if B|W ⊗W̄ is positive definite (resp. negative
definite). Such a subspace is endowed with a pre-Hilbert structure; we say that
it is intrinsically complete if this structure is Hilbert, that is if the induced
distance is complete.
28
Chapter 1. Hodge theory of harmonic bundles
We say that K is a Krein space if there exists a B-orthogonal decomposition
K = K+ ⊕⊥ K− such that K+ (resp. K− ) is an intrinsically complete positive
(resp. negative) subspace. Such a decomposition is called a fundamental decomposition of the Krein space K. This is not unique, even for finite-dimensional
spaces endowed with an indefinite metric.
From a fundamental decomposition of the Krein space K, one can define
a unique Hilbert inner product h on K such that K− and K+ are orthogonal
for h, h = B on H+ and h = −B on H− . This inner product depends on the
fundamental decomposition but we have the following proposition:
Proposition 1.1.1. (Corollary IV.6.3 in [Bog74]) Let K be a Krein space. The
Hilbert structures coming from two fundamental decompositions of K define the
same topology on K.
Hence, one can speak of the Hilbert topology of the Krein space K. Any
reference to the topology of K will be with respect to this one.
Let W be a non-degenerate closed subspace of K. Then W admits a decomposition W = W+ ⊕⊥ W− , with W+ (resp. W− ) a positive (resp. negative)
closed subspace of K (theorem V.3.1 in [Bog74]). However, W+ and W− are
not necessarily intrisically complete, that is W is not necessarily a Krein space
itself. The following proposition gives a characterization of closed subspaces of
a Krein space, that are Krein spaces. We sometimes call them Krein subspaces,
though it does not depend on the ambient space.
Proposition 1.1.2. (Theorems V.3.4 and V.3.5 in [Bog74]) Let W be a closed
subspace in a Krein space K. Let W = W+ ⊕⊥ W− be a decomposition of
W as above. Then W is a Krein subspace of K if and only if there exists a
fundamental decomposition K = K+ ⊕⊥ K− such that W+ ⊂ K+ and W− ⊂ K− .
Moreover, if W is a Krein subspace of K, the orthogonal W ⊥ of W in K is a
Krein subspace and satisfies W ⊕⊥ W ⊥ = K.
Loop Hodge structures
Let K be a Krein space and let T be an anti-isometric operator on K, that is
B(T u, T v) = −B(u, v) for u and v in K.
Definition 1.1.3. A closed subspace W of K is an outgoing subspace for T if
• W is a Krein subspace of K;
• TW ⊂ W;
•
T
n∈N T
nW
= 0;
•
S
n∈Z T
nW
is dense in K;
• the orthogonal of T W in W is positive definite.
This terminology comes from [LP90] and [PS88], where it is defined in a
Hilbert setting; in this case, T is an isometric operator. In a Krein space K,
an outgoing operator T is an anti-isometric operator that admits an outgoing
1.1. Harmonic bundles as loop VHS
29
subspace. An outgoing Krein space (K, B, T ) is a Krein space (K, B) endowed
with an outgoing operator T . It is easy to see that a non-trivial outgoing Krein
space is infinite-dimensional and that the positive and negative components in
any fundamental decomposition are infinite-dimensional too. In particular, the
Krein metric has to be indefinite. We will sometimes write (K, T ) or K for an
outgoing Krein space.
Definition 1.1.4. A loop Hodge structure (K, B, T, W ) is the datum of an
outgoing Krein space (K, B, T ) and an outgoing subspace W for T .
An isomorphism from a loop Hodge structure (K, B, T, W ) to a loop Hodge
structure (K 0 , B 0 , T 0 , W 0 ) is a Krein isometry φ : (K, B) → (K 0 , B 0 ) that intertwines the outgoing operators and such that φ(W ) = W 0 .
This terminology is justified by the fact that classical Hodge structures can
be realized as loop Hodge structures. This will be discussed in section 1.2.
Definition 1.1.5. The Hodge filtration of a loop Hodge structure (K, B, T, W )
is the decreasing filtration F • K of K, given by F i K := T i W . It is a complete
T
S
topologically separated filtration, meaning that n∈Z F n K = 0 and n∈Z F n K
is dense in K.
Canonical form
Let H be a Hilbert space, with scalar product denoted by h. Let K be the
space of square-integrable functions from the circle S 1 to H. We define a Krein
structure on K by
Z
B(f, g) :=
h(f (λ), g(−λ))dν(λ),
S1
where ν is the invariant volume form on S 1 with integral 1.
P
P
Using Fourier series, we write f (λ) = n∈Z fn λn and g(λ) = n∈Z gn λn ,
P
with (fn )n∈Z , (gn )n∈Z in l2 (Z, H). Then B(f, g) = n∈Z (−1)n h(fn , gn ). A
fundamental decomposition of K is obtained by taking the odd and even components of a function in K; for this fundamental decomposition, the induced
Hilbert structure is the usual Hilbert structure on L2 (S 1 , H).
The right-shift operator of K is the operator T , given by (T f )(λ) = λf (λ).
In terms of the Fourier series representation, T is defined by T ((an )n∈Z ) =
(an−1 )n∈Z . This is an anti-isometric operator and in fact an outgoing operator.
Indeed, the subspace L2+ (S 1 , H) of functions with Fourier series concentrated
in nonnegative degrees is easily seen to be an outgoing subspace for T . This
space will be called the Fourier-nonnegative subspace. One should notice that
functions in L2+ (S 1 , H) are holomorphic functions on the open unit disk in C
P
P
whose powers series representation n∈N an z n at 0 satisfy n∈N |an |2 < ∞.
This defines a canonical loop Hodge structure on L2 (S 1 , H). The next
proposition is an adaptation with Krein spaces of a well known-fact in abstract
scattering theory [PS88], [LP90], [HSW13].
Proposition 1.1.6. Let (K, B, T, W ) be a loop Hodge structure and let H be
the orthocomplement of T W in W . Then, H is a Hilbert space and there exists
a canonical isomorphism of loop Hodge structures from K to L2 (S 1 , H).
30
Proof. Since W is a Krein subspace of K and T is an anti-isometric operator,
T W is a Krein space too. In particular, one has an orthogonal direct sum W =
T W ⊕⊥ H by proposition 1.1.2. The subspaces T n H, n ∈ Z are in orthogonal
direct sum. Indeed, if i < j, then T i H is orthogonal to T i+1 W and T j H ⊂
T j W ⊂ T i+1 W . By induction, one gets an orthogonal direct sum
W =
⊥
M
T i H ⊕⊥ T n+1 W,
0≤i≤n
for all positive n. Since n∈N T n W = 0, this implies that W is the Hilbert sum
S
of the spaces T n H, for n ≥ 0. Moreover, n∈Z T n W is dense in K; hence K is
the Hilbert sum of the T n H, for n ∈ Z.
The Krein space K can thus be identified with L2 (S 1 , H), with its standard Krein metric (the Krein metric B is by assumption positive definite on
H) since T is an anti-isometric operator. By construction, this identification
intertwines T with the right-shift operator of L2 (S 1 , H) and sends W to the
Fourier-nonnegative subspace of L2 (S 1 , H).
T
Remark 1.1.7. With the notations of proposition 1.1.6, the isomorphism Φ
from L2 (S 1 , H) to K is given by
Φ:
X
n∈Z
an λn 7→
X
T n an .
n∈Z
Proposition 1.1.8. Let (K, B, T ) be an outgoing Krein space and let W1 and
W2 be two outgoing subspaces. We denote by Hi the orthocomplement of T Wi
in Wi . Then H1 and H2 are isometric Hilbert spaces.
Proof. By proposition 1.1.6, the outgoing Krein space (K, B, T ) is isomorphic
to L2 (S 1 , Hi ) (as outgoing Krein space). In particular, there is an isomorphism
L2 (S 1 , H1 ) ∼
= L2 (S 1 , H2 ),
that intertwines the outgoing operators. The proposition is then a consequence
of the following lemma.
Lemma 1.1.9. Let H1 and H2 be two Hilbert spaces and let φ : L2 (S 1 , H1 ) →
L2 (S 1 , H2 ) be an isomorphism that intertwines the outgoing operators. Then,
H1 and H2 are isometric.
Sketch of the proof. I only sketch the proof. This statement is implicit in proposition 8.12.4 of [PS88] and is a generalization of the beginning of the proof of
their theorem 8.3.2. For the sake of simplicity, we assume that H1 = Cn and
H2 = Cm are finite-dimensional, with n > m. Since φ intertwines the outgoing
operators, there exists M in L∞ (S 1 , Mm,n (C)) such that φ is matrix multiplication by M , that is:
φ(f )(λ) = M (λ).f (λ),
for f in L2 (S 1 , Cn ) and almost every λ in S 1 . This statement is an easy generalization of proposition A.3.4.
31
For almost every λ, the kernel of M (λ) is not reduced to 0 since n > m.
By using Gauss elimination, we can define a non-zero measurable vector-valued
function v(λ) in Cn such that v(λ) is in the kernel of M (λ) for almost every
λ. Moreover, up to a renormalization of v, we can assume that v lives in
L2 (S 1 , Cn ).
Then, v is not zero but φ(v) is zero, contradicting that φ is an isomorphism.
In particular, in a loop Hodge structure (K, B, T, W ), the dimension of H
is independent of the outgoing subspace W . This will be called the virtual
dimension of the outgoing Krein space (K, B, T ).
We will only consider loop Hodge structures of finite virtual dimension.
Families of loop Hodge structures
Let π : K → X be a Hilbert bundle over a differentiable manifold X (see
Appendix A). Let B be a fibrewise Hermitian form on K and let T be a section
of the Banach bundle End(K) → X. We say that (K, B, T ) is an outgoing Krein
bundle if for every x in X, there exists a neighbourhood U of x, a Hilbert space
K, a trivialization π −1 (U ) ∼
= U × K and an outgoing Krein structure on K
such that the restriction of the trivialization to each fiber is an isomorphism
of outgoing Krein spaces. In particular, each fiber of K has a structure of
outgoing Krein space and the intrinsic topology of this Krein space coincides
with its topology as a fiber of the Hilbert bundle K.
Let π : (K, B, T ) → X be an outgoing Krein bundle. A subbundle of outgoing subspaces W in K is a subset of K such that, for any x in X, there
exists a neighbourhood U of x, an outgoing Krein space K and a trivialization
∼ U × K of outgoing Krein bundles such that, in this trivialization,
π −1 (U ) =
−1
K ∩ π (U ) is identified with U × W , where W is a fixed outgoing subspace of
K.
Definition 1.1.10. A family of loop Hodge structures over a differentiable manifold X is the datum of an outgoing Krein bundle (of finite virtual dimension)
with a subbundle of outgoing subspaces. An isomorphism of such families is a
bundle map which is an isomorphism of loop Hodge structures on each fiber.
Let (E, h) be a finite-dimensional Hermitian bundle over X. By proposition
A.2.7, the union of the Hilbert spaces L2 (S 1 , Ex ), x ∈ X, has a natural structure
of Hilbert bundle. Moreover, this Hilbert bundle is in a natural way a family
of loop Hodge structures, by the discussion before proposition 1.1.6.
Proposition 1.1.11. Let X be a differentiable manifold and let (K, B, T , W)
be a family of loop Hodge structures of virtual dimension n. Then this family
is isomorphic to the family of loop Hodge structures on L2 (S 1 , E), where E is
the orthogonal complement of T W in W.
Proof. Since in a trivialization, W becomes a fixed outgoing subspace in a fixed
outgoing Krein space, the orthogonal complement E of T W in W is a smooth
bundle of rank n over X. Moreover, by the definition of outgoing subspaces,
32
the Hermitian form B restricts to a Hermitian inner product on E. Hence, one
can indeed consider the outgoing Krein bundle L2 (S 1 , E), and by proposition
1.1.6, one gets a vector bundle map K → L2 (S 1 , E) which is an isomorphism
of loop Hodge structures in each fiber. Since locally in some trivialization, this
isomorphism does not depend on the point in X, it is smooth. This concludes
the proof.
Corollary 1.1.12. Let X be a differential manifold. The category of families of loop Hodge structures over X (with morphisms being isomorphisms) is
equivalent to the category of finite-dimensional Hermitian bundles over X (with
morphisms being bundle isometries).
1.1.2
Variations of loop Hodge structures and harmonic bundles
In this subsection, we recall the definition of a harmonic bundle and we define
our fundamental object of study: variations of loop Hodge structures. We prove
that these two notions are equivalent; this is theorem A of the introduction.
Developing map of a metric
Let (X, x0 ) be a pointed connected differentiable manifold. We choose a basepoint x̃0 in the fiber above x0 of the universal cover π : X̃ → X. Let (E, D)
be a flat bundle of rank r over X, with a trivialization Ex0 ∼
= Cr of the fiber
above x0 . The pullback bundle π ∗ E can be trivialized to X̃ × Cr by parallel
transport with respect to the flat connection π ∗ D and base-point x̃0 . If h
is a Hermitian metric on E, the pullback metric π ∗ h can thus be seen as a
varying Hermitian inner product on the fixed space Cr . The space of such
inner products is homogeneous under the action of GL(r, C) by congruence,
with stabilizer equal to the unitary group U (r) at the standard inner product.
We thus get a map f : X̃ → GL(r, C)/U (r), which is equivariant under the
monodromy ρ : π1 (X, x0 ) → GL(r, C) of the flat bundle. This map is called the
developing map of the metric h. Conversely, such a map defines a Hermitian
metric on the flat bundle (E, D).
Harmonic bundles
Let (M, g M ) and (N, g N ) be two Riemannian manifolds. If f : M → N is
a smooth map, the bundle W := T ∗ M ⊗ f −1 T N on M is endowed with a
connection ∇W , induced by the Levi-Civita connections on M and N .
Definition 1.1.13. The tension field τ (f ) of a smooth map f : M → N is a
tensor in f −1 T N , obtained by taking the trace of the tensor ∇W df , that lives
in (T ∗ M )⊗2 ⊗ f −1 T N . We say that f is harmonic if its tension field vanishes.
We refer to [HW08] for a variational point of view. In the particular case
where M is a complex manifold of dimension 1, this notion of harmonic map
is independent of the (necessarily Kähler) metric on M (see [HW08], example
2.2.12). Hence, the following definition makes sense:
33
Definition 1.1.14. A smooth map f : X → N from a complex manifold X to
a Riemannian manifold N is pluriharmonic if its restriction to every complex
curve of X is harmonic.
It is known that a pluriharmonic map from a Kähler manifold X to a Riemannian manifold N is harmonic. The best result in the converse direction is
the Siu-Sampson’s theorem [Siu80],[Sam86]:
Theorem 1.1.15 (Siu-Sampson). Let X be a compact Kähler manifold, let
N be a Riemannian manifold of non-positive Hermitian curvature and let ρ :
π1 (X) → Isom(N ) be a representation of π1 (X) in the isometry group of N .
Then, any ρ-equivariant harmonic map from X̃ to N is pluriharmonic.
To be precise, this theorem follows from the results in the two quoted references. A self-contained proof of this theorem can be found in [ABC+ 96]: it is
given in a non-equivariant setting but it is asserted there that the proof works
also with equivariant maps.
There is an equivalent notion of pluriharmonicity. Let (E, D) be a flat bundle
over a complex manifold X and let h be a Hermitian metric on E. There is a
unique decomposition D = ∇ + α, where ∇ is a metric connection for h and
α is a h-Hermitian 1-form with values in End(E). We write ∇ = ∂ + ∂¯ and
α = θ + θ∗ , for their decompositions in types. Since α is Hermitian, θ∗ is the
adjoint of θ with respect to h.
Definition 1.1.16. The metric h on the flat bundle (E, D) is harmonic if the
differential operator ∂¯ + θ has vanishing square. We also say that (E, D, h) is a
harmonic bundle.
The vanishing of the square of ∂¯ + θ is equivalent to the vanishing of ∂¯2 ,
¯ and θ ∧ θ. By the Koszul-Malgrange integrability theorem [KM58], ∂¯ gives
∂θ
a structure of holomorphic bundle on E. For this holomorphic structure, θ is a
¯ θ) is
holomorphic 1-form satisfying the equation θ ∧ θ = 0. Such a datum (E, ∂,
called a Higgs bundle and θ is the Higgs field.
These two notions of harmonicity are related by the following proposition:
Proposition 1.1.17. A metric h on (E, D) is harmonic if and only if its developing map is pluriharmonic.
The following proof is due to T. Mochizuki; I thank him for his explanations
on the work of K. Corlette.
Proof. In page 376 of [Cor88], it is proved that, on a Kähler manifold X, the
¯ is primitive. Apdeveloping map is harmonic if and only if the (1, 1)-form ∂θ
plying this result to the germs of curves in a complex manifold X, this shows
¯ vanishes (on a curve,
that the developing map is pluriharmonic if and only if ∂θ
there is no non-trivial primitive (1, 1)-forms). This shows one implication: if
the metric h is harmonic, its developing map is pluriharmonic.
For the converse, we use a Bochner-type formula, given in the proof of
theorem 5.1 of [Cor88]. This is proposition 21.39 in [Moc07b]. By decomposing
34
the equation D2 = 0 and separating the bidegrees and the Hermitian/antiHermitian parts, we have the equations:
¯ ∗ = ∂θ∗ + ∂θ
¯ = 0,
∂θ = ∂θ
¯ + [θ, θ∗ ] = 0.
∂ 2 + θ ∧ θ = ∂¯2 + θ∗ ∧ θ∗ = ∂ ∂¯ + ∂∂
Since the result is local, we can assume that X is a Kähler manifold, with
Kähler form ω. Then, there exist positive constants C1 and C2 such that:
¯ 2 ).ω dim X .
∂ ∂¯ Tr(θ ∧ θ∗ ).ω dim X−2 = (C1 .|θ ∧ θ|2h + C2 .|∂θ|
h
¯ ∧ θ∗ − θ ∧ ∂θ
¯ ∗ ). The term ∂θ
¯ ∗ always vanishes
The left-hand side term is ∂ Tr(∂θ
¯
and ∂θ vanishes if the map is pluriharmonic. Hence, if the developing map
is pluriharmonic, we get θ ∧ θ = 0. On the other hand, a similar formula
gives θ∗ ∧ θ∗ = 0 since ∂θ∗ = 0 also vanishes when the developing map is
pluriharmonic. Since ∂¯2 + θ∗ ∧ θ∗ = 0, this gives ∂¯2 = 0 and concludes the
proof.
Remark 1.1.18. The terminology is a little ambiguous but well established and
we will not use the more precise term of pluriharmonic bundle. Anyway, in the
case where X is a compact Kähler manifold, harmonicity and pluriharmonicity
are equivalent by theorem 1.1.15 since the symmetric space GL(r, C)/U (r) has
non-positive Hermitian curvature.
Circle of connections
We refer to Appendix A for a general discussion about circle of connections.
We consider a harmonic bundle (E, D, h) and write D = ∇ + α, ∇ = ∂ + ∂¯ and
α = θ + θ∗ as before. For any λ in S 1 , we write αλ for the Hermitian 1-form
λ−1 θ + λθ∗ and Dλ for the connection
Dλ = ∇ + αλ .
The following lemma is well known.
Lemma 1.1.19. The bundle (E, Dλ , h) is harmonic.
Proof. We denote by F∇ the curvature of the connection ∇. The curvature of
D is D2 = F∇ + ∇α + α ∧ α. Since F∇ + α ∧ α is anti-Hermitian and ∇α is
Hermitian, the vanishing of D2 is equivalent to F∇ + α ∧ α = 0 and ∇α = 0.
The curvature of Dλ is Dλ2 = (F∇ + αλ ∧ αλ ) + ∇αλ . The decomposition by
¯ = 0 yield ∇θ = ∇θ∗ = 0.
types of the equality ∇α = 0 and the condition ∂θ
2
∗
∗
Hence, ∇αλ = 0. Moreover, αλ ∧αλ = λ θ ∧θ +(θ∧θ∗ +θ∗ ∧θ)+λ−2 θ∧θ. Since
θ∧θ = 0, only the central term doesn’t vanish. Hence, F∇ +α∧α = F∇ +αλ ∧αλ
and the vanishing of D2 is equivalent to the vanishing of Dλ2 .
It remains to show that the operator ∂¯ + λ−1 θ has vanishing square. This is
equivalent to λ−1 θ being a Higgs field for the ∂¯ holomorphic structure, which
is obvious.
35
The family (Dλ )λ∈S 1 is called the circle of flat connections of the harmonic
bundle (E, D, h). We give another lemma which relates the harmonicity of E
with the existence of such a circle of flat connections.
Lemma 1.1.20. Let X be a complex manifold and let (E, h) be a Hermitian
bundle over X. We assume that there exists a metric connection ∇ and a
(1, 0)-form θ with values in End(E) such that, for every λ in S 1 , the connection
Dλ := ∇ + λ−1 θ + λθ∗
is flat.
Then (E, Dλ , h) is a harmonic bundle, for every λ in S 1 .
Proof. By the previous lemma, it is sufficient to show that (E, D1 , h) is a harmonic bundle. The connection D1 has vanishing curvature by assumption. We
¯ The equality D2 = 0 gives
write ∇ = ∂ + ∂.
λ
0 = λ−2 θ ∧ θ + λ−1 ∇θ + (F∇ + θ ∧ θ∗ + θ∗ ∧ θ) + λ∇θ∗ + λ2 θ∗ ∧ θ∗ .
Since this is true for every λ, the terms with different exponents of λ vanish
separately. Hence, θ ∧ θ = 0 and ∇θ = 0, which implies that the (1, 1)-part
¯ = 0 vanishes. Moreover, the constant term in λ has (0, 2)-type ∂¯2 , hence ∂¯2
∂θ
vanishes too.
Variations of loop Hodge structures
The following definition mimics the one of variations of Hodge structures, given
in the introduction.
Definition 1.1.21. A variation of loop Hodge structures over a complex manifold X is the datum of a family of loop Hodge structures (K, B, T , W) and a
flat connection D = ∂ + ∂¯ on K such that:
1. the Krein metric B and the outgoing operator T are D-flat;
2. ∂¯ : W → W ⊗ A0,1 ;
3. D : W → T −1 W ⊗ A1 .
Remark 1.1.22. We recall that a family of loop Hodge structures (K, B, T , W)
carries a topological decreasing filtration given by F p K := T p W. Since in a
variation of loop Hodge structures T is D-flat, the two differential conditions
given above are equivalent to the a priori stronger ones:
1. ∂¯ : F p K → F p K ⊗ A0,1 ;
2. D : F p K → F p−1 K ⊗ A1 .
36
The equivalence of categories
We can now prove theorem A of the introduction. Let X be a complex manifold.
We write HX for the category of harmonic bundles over X, the morphisms
being the flat and isometric vector bundle maps. We write VX for the category
of variations of loop Hodge structures over X, the morphisms being the flat
isomorphisms of family of Hodge structures. We define a functor F : HX → VX
in the following way:
If (E, D, h) is a harmonic bundle. Let ∇ be an arbitrary connection on E and
˜
let ∇ be its naive extension to L2 (S 1 , E). The circle of flat connections (Dλ )λ∈S 1
˜ + Γ(X, T ∗ X ⊗ L∞ (S 1 , End(E))), hence as
can be thought as an element in ∇
a connection D̃ on the Hilbert bundle L2 (S 1 , End(E)) (see Corollary A.3.5).
The functor F is defined by F : (E, D, h) 7→ (L2 (S 1 , E), D̃), where L2 (S 1 , E) is
endowed with its canonical structure of family of loop Hodge structures.
Theorem 1.1.23. The functor F is well-defined on objects and establish an
equivalence of categories from HX to VX .
Proof. We first prove that F (E, D, h) is indeed a variation of loop Hodge structures. Then, we construct a quasi-inverse G : VX → HX . That F and G are
quasi-inverse is straightforward and left to the reader, as the definition of F
and G on morphisms.
Let (E, D, h) be a harmonic bundle and write
(L2 (S 1 , E), B, T , L2+ (S 1 , E), D̃) := F (E, D, h).
We recall that D̃ is defined by (D̃f )(λ, x) = Dλ f (λ, x), where f is a local section
of L2 (S 1 , E) and x is in E. The connection D̃ is flat since all Dλ are flat. By
Corollary A.3.5, T is D̃-flat. The Hermitian form B is D̃-flat too. Indeed, let
f and g be local sections of L2 (S 1 , End(E)). Then,
Z
dh(f (λ), g(−λ))dν(λ)
dB(f, g) =
ZS
1
=
S1
h(Dλ f (λ), g(−λ)) + h(f (λ), D−λ g(−λ) dν(λ)
Z
h((D̃f )(λ), g(−λ)) + h(f (λ), (D̃g)(−λ)) dν(λ)
=
S1
= B(D̃f, g) + B(f, D̃g).
For the second equality, we use that Dλ and D−λ are adjoint for the metric h.
Indeed, Dλ = ∇ + αλ and D−λ = ∇ − αλ , where ∇ is a metric connection and
αλ is a Hermitian 1-form. Notice that the flatness of the Hermitian form is the
main motivation for considering indefinite Hermitian forms.
We now check the differential constraints 2. and 3. of Definition 1.1.21 on
L2+ (S 1 , E). By definition, the Fourier series of elements in L2+ (S 1 , E) have only
nonnegative powers of λ. On the other hand, the flat connection D̃ is equal
to ∇ + λ−1 θ + λθ∗ , with the usual notations. Hence, its (0, 1)-part is concentrated in nonnegative Fourier coefficients and only powers of λ greater or equal
to −1 appear in the Fourier series of D. This shows that the two differential
37
constraints on L2+ (S 1 , E) are indeed satisfied.
Conversely, let (K, B, T , W, D) be a variation of loop Hodge structures. By
proposition 1.1.11, there is an isomorphism of families of loop Hodge structures
K ∼
= L2 (S 1 , E), where E is the orthogonal of T W in W. Let h denote the
restriction of B to E ⊗ Ē; this is a Hermitian inner product.
˜ the
Choose an arbitrary metric connection ∇ on (E, h) and denote by ∇
2
1
induced naive connection on L (S , E). Since T is D-flat, by Corollary A.3.5,
D can be written as
˜ + ω,
D=∇
where ω lives in Γ(X, T ∗ X ⊗ L∞ (S 1 , End(E))). Since ∇ is metric for h, one
˜ is metric for B. Moreover, B is D-flat; hence we get the
checks easily that ∇
equality:
B(ω.f, g) + B(f, ω.g) = 0,
where f and g are local sections of L2 (S 1 , E). Let u and v be local sections of E,
let i be in Z and consider the local sections f = λi u, g = v. Then the previous
equality gives:
h(ω−i .u, v) + (−1)i h(u, ωi .v) = 0,
(1.1)
where ω = n∈Z ωn λn , with ωj ∈ Γ(X, T ∗ X ⊗ End(E)). Considering the types
of the ωj , this equality proves that the (1, 0)-part of ωj vanishes if, and only if
the (0, 1)-part of ω−j vanishes.
The two differential constraints on W give the following information: ωj
0,1
vanishes too. Hence,
vanishes for j ≤ −2 (transversality condition) and ω−1
P
by equation (1.1), the only possible non-vanishing terms in ω = n∈Z ωn λn are
1,0
ω0 , ω−1
and ω10,1 . Thus, D can be written as
P
˜ + ω0 ) + λ−1 ω−1 + λω1 ,
D = (∇
and ω−1 (resp. ω1 ) is of type (1, 0) (resp. of type (0, 1)). The equations in
(1.1) for i = 0 and i = 1 prove that ω0 is an anti-Hermitian form and that ω1
and ω−1 are adjoint. The connection D comes from the circle of connections
˜ + ω0 ) + λ−1 ω−1 + λω1 . Hence,
(Dλ )λ∈S 1 defined, for every λ in S 1 , by Dλ := (∇
all Dλ are flat connections. We define the functor G by
G(K, B, T , W, D) = (E, D1 , h)
and we conclude by lemma 1.1.20.
1.1.3
Algebraic variation of loop Hodge structures and nonpolarized variations
Theorem 1.1.23 and its proof show that the Hilbert bundles K that we consider
in the definition of a variation of loop Hodge structures are obtained from a
finite-dimensional bundle E via the functor E 7→ L2 (S 1 , E) and that the flat
connection D on K involves only a finite number of exponents of λ, relatively
to the isomorphism K ∼
= L2 (S 1 , E). Hence, it is natural to give a more algebraic
definition of variations of loop Hodge structures.
38
Definition 1.1.24. Let E be a finite-dimensional complex vector bundle on
a differentiable manifold X. We define E[t, t−1 ] to be the infinite-dimensional
complex vector bundle on X whose sheaf of smooth sections F is defined for
S
any open set U by F(U ) = n∈N F n (U ), where
F n (U ) := {
X
ak tk | ak ∈ Γ(U, E)}.
|k|≤n
We define the right-shift operator T on E[t, t−1 ] to be the multiplication by
the formal variable t.
If (E, h) is a Hermitian vector bundle, we define the Krein metric B on
E[t, t−1 ] by
X
X
X
B(
fn tn ,
gn tn ) =
(−1)n h(fn , gn ),
n∈Z
n∈Z
n∈Z
where all the sums are in fact finite.
Definition 1.1.25. A connection D on E[t, t−1 ] is a C-linear map Γ(X, T X ⊗
E[t, t−1 ]) → Γ(X, E[t, t−1 ]) which is C ∞ (X)-linear in the T X direction and
which satisfies Leibniz’s rule. As usual, one can define the notion of flatness of
a connection or speak of flat tensors.
Proposition 1.1.26. Let D be a connection on E[t, t−1 ]. Then T is D-parallel
P
if and only if D can be written D = D0 + 1≤|k|≤n0 ak tk , where D0 is a connection on E and the ak are 1-forms with values in End(E).
Proof. This is analogous to proposition A.3.5 and is in fact easier to prove
since there are no topological issues. Let D0 be any connection on E and
consider it as a connection D̃0 on E[t, t−1 ]. Then, ω := D − D̃0 is a section of
T ∗ X ⊗ End(E[t, t−1 ]) that commutes with T . Considering the action of ω on
elements in E ⊂ E[t, t−1 ], one obtains a form A ∈ Γ(X, T ∗ X ⊗ End(E)[t, t−1 ])
such that, for every x in X, Yx in Tx X and vx in Ex , ωx (Yx )(vx ) = Ax (Yx )(vx ).
Since ω commutes with T , this equality is necessarily true for v in E[t, t−1 ]x .
Otherwise said, D lives in D̃0 +Γ(X, T ∗ X ⊗End(E)[t, t−1 ]) and this is exactly
the statement of the theorem.
Definition 1.1.27. An algebraic variation of (complex polarized) loop Hodge
structures over a complex manifold X is the datum of a finite-dimensional
Hermitian bundle (E, h) over X and a flat connection D = ∂ + ∂¯ on E[t, t−1 ]
such that
1. the Krein metric B and the right-shift operator T are D-flat;
2. ∂¯ : E[t] → E[t] ⊗ A0,1 ;
3. D : E[t] → t−1 E[t] ⊗ A1 .
Remark 1.1.28. By completing the vector bundle E[t, t−1 ] with respect to its
natural pre-Hilbert structure, one obtains a variation of loop Hodge structures
in the sense of definition 1.1.21. On the other hand, by the proof of theorem
39
1.1.23, every variation of loop Hodge structures is the completion of an algebraic
variation of loop Hodge structures.
This algebraic definition may seem simpler than the analytic one. If one
is only interested in the variational point of view, this is certainly true. But,
as soon as a classifying space for loop Hodge structures will be needed, we
will have to solve ordinary differential equations and we need the Hilbert setting to do that. Moreover, it seems preferable in the definition of loop Hodge
structures to emphasize the role of the outgoing subspace W rather than the
finite-dimensional subspace E which is the orthocomplement of T W in W. This
can be done in the Hilbert setting, thanks to proposition 1.1.6; but there is no
obvious analogue of this proposition in an algebraic setting. For these reasons,
apart from the following discussion of non-polarized variations, we will not use
this algebraic setting anymore.
If one is interested in variations of loop Hodge structures that are not polarized (that is, which carry no metric), the algebraic definition looks simpler
to generalize in the following way:
Definition 1.1.29. A non-polarized variation of (complex) loop Hodge structures over a complex manifold X is the datum of a finite-dimensional complex
vector bundle E over X and a flat connection D = ∂ + ∂¯ on E[t, t−1 ] such that
1. the right-shift operator T is D-flat;
2. ∂¯ : E[t] → E[t] ⊗ A0,1 ;
3. D : E[t] → t−1 E[t] ⊗ A1 ;
4. ∂ : E[t−1 ] → E[t−1 ] ⊗ A1,0 ;
5. D : E[t−1 ] → tE[t−1 ] ⊗ A1 .
Remark 1.1.30. In the polarized case, the two additionnal assumptions 4.
and 5. are satisfied, thanks to the polarization. The interest of this notion lies
in the fact that it is equivalent to a non-polarized harmonic bundle, as defined
in [Sim97].
Definition 1.1.31. A non-polarized harmonic bundle over a complex manifold X is a triple (E, D0 , D00 ), where E is a complex vector bundle over X, D0
(resp. D00 ) is a differential operator on E which satisfy the Leibniz’s rule for ∂X
(resp. ∂¯X ); moreover we ask that D0 and D00 satisfy the integrability conditions
(D0 )2 = (D00 )2 = D0 D00 + D00 D0 = 0.
Remark 1.1.32. In the polarized situation, where the flat connection D is
¯ + (θ + θ∗ ), one takes D0 = ∂ + θ∗ and D00 = ∂¯ + θ.
written D = ∇ + α = (∂ + ∂)
Theorem 1.1.33. The category of non-polarized variations of loop Hodge structures is equivalent to the category of non-polarized harmonic bundles.
40
Sketch of the proof. The proof is very similar to the proof of theorem 1.1.23; we
0 + D 0 and D 00 = D 00 + D 00 according
only sketch it. We decompose D0 = D1,0
0,1
1,0
0,1
1
to their types. For λ in S , a differential operator is defined by
0
00
00
0
Dλ := D1,0
+ D0,1
+ λ−1 D1,0
+ λD0,1
.
In the polarized case, this is the usual definition, thanks to remark 1.1.32. For
all λ, Dλ is a connection, and one checks that the integrability conditions imply
the flatness of Dλ . This circle of connections thus defines a flat connection in
E[t, t−1 ] (replacing λ by the formal variable t) and this defines a non-polarized
variation of loop Hodge structures.
Conversely, if D is the flat connection on E[t, t−1 ], one first uses proposition
1.1.26 to have a good representation of D. Then, the differential assumptions
on E[t] and E[t−1 ] force a lot of vanishings in the coefficients of D, as in the
proof of theorem 1.1.23. One obtains that D can be written
D = D0 + t−1 α−1 + tα1 ,
where D0 is a connection on E, α−1 is a (1, 0)-form with values in End(E) and
α1 is a (0, 1)-form with values in End(E).
The differential operators D0 and D00 are defined by D0 = D01,0 + α1 and
00
D = D00,1 + α−1 .
Remark 1.1.34. This jugglery with differential operators appears in particular in [Sim97], lemma 3.1, in the proof of the equivalence between (polarized)
harmonic bundles and polarizable pure variations of twistor structures.
1.1.4
Loop period domain
In section 1.1.2, we have proved that a harmonic bundle corresponds to an
object of variational type. We now construct a classifying space for loop Hodge
structures; this classifiying space is infinite-dimensional, has the same structure
than the period domains defined in classical Hodge theory and in fact contains
all classical period domains, as will be explained in section 1.2.
Another variational definition for harmonic bundles can be given via the
notion of twistor structures ([Sim97],[Moc07a]). But, in this point of view,
there is no classifying space, as remarked in the introduction of [Moc07a], 1.3.2.
Our hope is that the classifying space will permit a better Hodge understanding
of harmonic bundles.
In all this section, we use the following notations; n is a fixed positive integer,
G is the complex reductive Lie group GL(n, C), K is its maximal compact
subgroup U (n), σ is the Cartan involution of G relative to K, given by the
inverse of the transconjugate. The Lie algebras of these groups are denoted by
the same gothic letter, σ also denotes the differential of the Cartan involution,
acting on g, and the Cartan decomposition is given by g = k ⊕ p.
We refer to Appendix A for questions about Lie groups and Lie algebras of
infinite dimensions.
41
The loop algebra
On g, we define the usual Hermitian inner product:
(A, B)g := Tr(AB ∗ ).
The corresponding norm will be written |.|g or |.| if there is no possible confusion.
Let s be a nonnegative real number. The space of loops with regularity H s
and with values in g is by definition:
Λs g = {
X
an λn | an ∈ g,
n∈Z
X
(1 + |n|2 )s |an |2g < ∞}.
n∈Z
In particular, these loops are in L2 (S 1 , g). We define a Hermitian inner
product on Λs g by
(
X
an λn ,
X
bn λn )Λs g :=
X
(1 + |n|2 )s (an , bn )g .
This endows Λs g with the structure of a complex Hilbert space. Moreover,
a map f ∈ L2 (S 1 , g) is in Λs g if and only if all its matrix coefficients are in
H s (S 1 , C). We now assume that s > 12 and recall the two fundamental facts
([Tao01], proposition 1.1):
Proposition 1.1.35. If s > 12 , then H s (S 1 , C) injects continuously in C 0 (S 1 , C),
the space of continuous functions from the circle to C. Moreover, the product
of two functions in H s (S 1 , C) is still in H s (S 1 , C) and the product H s (S 1 , C) ⊗
H s (S 1 , C) → H s (S 1 , C) is continuous.
The second fact shows that Λs g is endowed with the structure of an associative algebra (coming from the associative algebra structure of g). In particular,
it is equipped with a complex Hilbert-Lie algebra structure, that is a Lie algebra on a complex Hilbert space with continuous bracket for the Hilbert norm.
Until Remark 1.1.45, the real number s > 1/2 will be fixed and omitted in the
notations of the different loop groups and loop algebras.
The loop group
Although this is quite unnatural, we see G as an open set in its Lie algebra g.
Since functions in Λg are continuous, the following definition makes sense:
ΛG := {f ∈ Λg | f (λ) ∈ G, ∀λ ∈ S 1 }.
This subset is an open set of Λg since the H s -topology is finer than the compactopen topology (and since S 1 is compact). Hence, ΛG is a complex Hilbert
manifold. We claim that ΛG is also a group. The stability by product works as
for the Lie algebra; for the inverse, it is sufficient by Cramer’s rule to remark
that the determinant is bounded away from zero since it is a continuous nonvanishing function. Moreover the group operations are smooth for the Hilbert
manifold structure. Indeed, since the product in ΛG is the restriction of the
product in Λg, which is a bilinear map, it is sufficient to show that the product
is continuous. This is the last part of proposition 1.1.35.
The Lie algebra of ΛG can be identified with Λg and one can show that the
exponential map is the pointwise exponential map from g to G. In summary:
42
Proposition 1.1.36. The space ΛG is a complex Hilbert-Lie group with complex Hilbert-Lie algebra Λg.
The twisted loop group
We recall that σ denotes the Cartan (anti-linear) involution of G relative to
K = U (n). In the loop Lie group or in the loop Lie algebra, one defines the
corresponding (anti-linear) involution:
σ̂ : γ(λ) 7→ σ(γ(−λ)).
The closed subgroup (resp. subalgebra) of fixed points of σ̂ is denoted by Λσ G
(resp. Λσ g). Using the Fourier series decomposition, one has
Λσ g = {
X
an λn | σ(an ) = (−1)n a−n }.
Denoting by g = k + p the Cartan decomposition of g, this means that the loops
in Λσ g are exactly the sums of an even loop with values in k and an odd loop
with values in p. In particular, Λσ g is a real form of Λg and the group Λσ G is
a real Hilbert-Lie group, which is a real form of ΛG.
The period domain and its compact dual
Let Λ+ g be the subalgebra of Λg of functions with only nonnegative Fourier
coefficients and let Λ+ G be the subgroup of ΛG of functions with only nonnegative Fourier coefficients. It is a complex Hilbert-Lie subgroup of ΛG, with Lie
algebra Λ+ g.
Proposition 1.1.37. The intersection Λσ G Λ+ G equals K, where an element
k of K is viewed as the constant map from S 1 to k.
T
Proof. Since a loop γ in Λ+ G has no negative Fourier coefficients, it defines a
holomorphic function f on the open unit disk that extends continuously on the
boundary. Let z be in CP1 with |z| > 1 and define f (z) = σ(f (− z̄1 )). Then f
is continuous on CP1 if γ is in Λσ G; moreover f is holomorphic in the outer
disk |z| > 1. In fact, this extension f is the canonical extension given by a
generalization of Schwarz reflection principle ([Ahl66], section 6.5, theorem 24),
which implies that f is holomorphic on the whole sphere.
In order to be more precise, consider the complex group G × G and the
anti-linear involution θ defined by
θ : G × G → G × G,
(A, B) 7→ (σ(B), σ(A)).
Embed ΛG in Λ(G × G) by the map ι : γ(λ) 7→ (γ(λ), γ(−λ)). Let γ be
in Λσ G ∩ Λ+ G. Then, ι(γ) has a holomorphic extension to the open unit disk,
extends continuously to the closed unit disk and, on the unit circle, it takes
values in the real form of G × G defined by θ. By Schwarz reflection principle,
such a map extends to a holomorphic map from CP1 to G × G, which has to
be constant. Hence, γ is constant and its value is necessarily in K; the other
inclusion is obvious.
43
Another self-contained proof will be given at then end of the proof of proposition 1.1.44.
Proposition 1.1.38. The multiplication map Λσ G × Λ+ G → ΛG is open.
Proof. It is enough to show that the equality Λg = Λσ g + Λ+ g holds at the
Lie algebra level since, in Banach-Lie groups, the exponential map is a local
P
diffeomorphism. Let γ = n∈Z an λn be a loop in Λg. Define
γσ :=
X
ãn λn ,
n∈Z
with ãn = an for n ≤ 0 and ãn = (−1)n σ(a−n ) for n ≥ 1 and
γ+ =
X
bn λn ,
n∈N
with bn = an − ãn , for n ≥ 1.
Then, by construction γ = γσ + γ+ , γσ is in Λσ g and γ+ is in Λ+ g.
Remark 1.1.39. About the connected components of these groups: let ΩG
be the based loop space of G and let LG be its free loop space. Both spaces
are endowed with the compact open topology. It is well known that πi (ΩG) =
πi+1 G and that, as topological spaces, LG ∼
= ΩG × G. In particular, π0 (LG) ∼
=
∼
π1 (G) × π0 (G), hence π0 (LG) = Z. By some approximation results, this can be
extended to the space ΛG, of H s -loops, so that π0 (ΛG) = Z.
Loops in Λ+ G extend continuously to the whole disk, so they are nullhomotopic. On the other hand, I claim that Λσ G encounters exactly the connected components containing the loops λ 7→ λk Id for even k. One inclusion
is clear since these loops are in Λσ G; for the other inclusion, one can use the
fact that the connected component in which a loop γ lives depends only on the
homotopy type of the loop
γ̃ : λ 7→
det(γ(λ))
,
| det(γ(λ))|
with values in S 1 . If γ is in Λσ G, then the loop γ̃ is even, which proves the
claim.
Definition 1.1.40. The homogeneous space D := Λσ G/K is the loop period
domain for the symmetric space G/K. The homogeneous space Ď := ΛG/Λ+ G
is its compact dual. The terminology comes from the corresponding spaces in
classical Hodge theory, as explained in the introduction and in section 1.2.
Remark 1.1.41. By propositions 1.1.37 and 1.1.38, the period domain D embeds as an open set in its compact dual Ď.
44
The complex structure on the period domain
Since the period domain D is open in its compact dual Ď which is a complex
Hilbert manifold, it inherits a complex structure. Let us see a more intrinsic
way of viewing this complex structure.
The group K, seen as a group of constant loops, acts on the Lie algebra Λσ g
by the adjoint action. It preserves its Lie algebra k and its orthogonal
Λ0σ g := {γ ∈ Λσ g | γ =
X
an λ n }
n6=0
for the (real) Hilbert structure on Λσ g coming from the complex Hilbert structure on Λg. Hence, as a K-module, the quotient Hilbert space Λσ g/k is isomorphic to Λ0σ g.
In the same way, Λg/Λ+ g is isomorphic as a K-module to
Λ− g := {γ ∈ Λg | γ =
X
an λn }.
n<0
The spaces Λ0σ g and Λ− g are the tangent spaces at the (same) base point
of D and Ď. Hence, by the proof of proposition 1.1.38, there is a canonical
isomorphism between these two spaces. In particular, the complex structure
on Λ− g gives a complex structure J on Λ0σ g. One computes that for γ =
P
n
0
n6=0 an λ ∈ Λσ g,
Jγ = i
X
n<0
an λn + (−i)
X
an λn .
(1.2)
n>0
It is of course K-invariant. Since the tangent space of D is isomorphic as a
Λσ G-bundle to
TD ∼
= Λσ G ×K Λ0σ g,
this provides an almost complex structure J on T D, which is necessarily integrable since it coincides with the complex structure coming from the inclusion
in the compact dual. The holomorphic tangent bundle is thus given by
Thol D = Λσ G ×K Λ− g.
The horizontal structure on the period domain
The complex subspace Λ≥−1 g = {γ ∈ Λg | γ = n≥−1 an λn } is stable by the
adjoint action of Λ+ G. The holomorphic horizontal bundle is defined to be
P
Th,hol Ď := ΛG ×Λ+ G Λ≥−1 g/Λ+ g ⊂ Thol Ď.
This is a finite-dimensional ΛG-equivariant holomorphic vector bundle over Ď.
In particular, it gives a Λσ G-equivariant holomorphic vector bundle over D.
A holomorphic map with target D or Ď will be called horizontal if its differential sends the holomorphic tangent bundle to the holomorphic horizontal
tangent bundle.
45
The Grassmannian of outgoing subspaces
In the following, we use the letter H for Krein spaces, in order to avoid confusion with the unitary group K.
Let (H, B, T ) be an outgoing Krein space of virtual dimension n and denote
by Gr(H) the outgoing Grassmannian of (H, B, T ), that is the set of all outgoing
subspaces. Let W be a base-point in Gr(H, T ). By proposition 1.1.6, one can
assume that H = L2 (S 1 , Cn ) with its natural outgoing Krein structure and W
is L2+ (S 1 , Cn ).
Consider the set of essentially bounded mesurable maps from S 1 to G, where
G is embedded in g. This set is stable by the pointwise product but the pointwise inverse of an essentially bounded map need not be essentially bounded
(compare with the discussion before proposition 1.1.36). Hence, we define Λ∞ G
to be the set of essentially bounded measurable maps whose pointwise inverse
are also essentially bounded, modulo the equivalence relation of equality almost everywhere. The requirement on the inverse is equivalent to asking that
det(g(λ))−1 is an essentially bounded function. As for the H s -loops, we can
give a structure of complex Banach-Lie group to Λ∞ G, with Lie algebra Λ∞ g.
We can also define the subgroup Λ∞
σ G, which is a real Banach-lie subgroup
∞
with Lie algebra Λσ g.
The group Λ∞ G acts on H by pointwise multiplication and gives an embedding Λ∞ G ,→ Aut(H) of complex Banach-Lie groups. The analogue of
proposition A.3.4 at the group level says that:
Proposition 1.1.42. The commutant of T in Aut(L2 (H)) is Λ∞ G.
Moreover, the group Λ∞
σ G is related to the Krein metric on H:
Lemma 1.1.43. The subgroup of Λ∞ G that acts isometrically on the Krein
space H is Λ∞
σ G.
Proof. Let f, g be in H and let γ in Λ∞ G. Then
Z
B(γ(λ)f (λ), γ(−λ)g(−λ))dν(λ)
B(γ.f, γ.g) =
ZS
1
=
B(f (λ), σ(γ(λ))−1 γ(−λ)g(−λ))dν(λ)
S1
= B(f, (σ̂(γ)−1 γ).g).
For γ to be an isometry, it thus has to satisfy B(f, g) = B(f, (σ̂(γ)−1 γ).g)
for any f and g and this is equivalent to σ̂(γ) = γ.
Proposition 1.1.44. The group Λ∞
σ G acts transitively on Gr(H), and the
stabilizer of W equals the subgroup K of constant loops in K.
Proof. By proposition 1.1.42 and lemma 1.1.43, Λ∞
σ G acts on H and preserves
the Krein metric and the right-shift operator T . By the definition 1.1.3 of an
outgoing subspace, Λ∞
σ G preserves the outgoing Grassmannian. Let W̃ be an
outgoing subspace. By lemma 1.1.9, the orthocomplement V of T W̃ in W̃ has
46
dimension n. Let φ : Cn → V be an isometry (with respect to the restriction
of the Krein metric). There is a unique way to extend φ to a map from H to
L2 (S 1 , V ) ∼
= H that commutes with the outgoing operator (see also the proof
of Theorem 8.3.2 in [PS88]). This extension, still denoted φ, commutes with
T by construction, and is a Krein isometry: indeed, it sends T i ej to T i φ(ej ),
both having square norm equal to (−1)i , the elements T i ej are B-orthogonal
and they give a Hilbert basis of H. Hence, by proposition 1.1.42 and lemma
n
1.1.43, φ lives in Λ∞
σ G and sends C on V . Hence, it sends W on W̃ .
The assertion about the stabilizer is essentially proposition 1.1.37 but we
give another self-contained proof. Let γ be in Λ∞
σ G, stabilizing the Fouriernonnegative subspace. Writing Fourier series representation, this implies that
γ has its Fourier series in nonnegative degrees. We claim that its Fourier series
also has to be concentrated in nonpositive degrees. Indeed, since γ −1 is also
holomorphic in the open unit disk, we can write
γ −1 =
X
An λn .
n∈N
Then σ̂(γ) is given by
σ̂(γ) =
X
A∗n (−λ−n ),
n∈N
S1
λ−1 .
since the conjugate of λ in
is
Since γ = σ̂(γ), this proves the claim: γ
has to be constant and necessarily in K.
Remark 1.1.45. This proposition gives a bijection between Gr(H) and the
homogenous space Λ∞
σ G/K. This enables us to give a Banach manifold structure to Gr(H). In order to distinguish the different regularities discussed, we
write again the superscript s for loops with regularity H s .
The complex structure defined above for Λsσ G/K is not well-defined on
Indeed, let f be in Λ0,s
σ g. By Equation 1.2, Jf is given by if− − if+ ,
where f− and f+ are the Fourier negative and positive parts of f . If we only
∞
assume that f is in Λ∞
σ g, then Jf is in Λσ g if and only if f− and f+ are in
∞
Λ g. We claim that this is not true in general.
Indeed, let M any matrix in p and consider the function f : S 1 → g, such
that f (λ) = M if Im λ > 0 and f (λ) = −M if Im λ ≤ 0. This is a square signal
function which belongs to Λ∞
σ g. Its Fourier series are given by
Λ∞
σ G/K.
f=
X 1
λn ,
n odd
n
up to some multiplicative factor. Then, f− and f+ are not essentially bounded.
∞
The issue here seems to be that Λ∞
+ G is not a closed Lie subgroup of Λ G,
∞
in the sense of [Lan96], section VI.5. Indeed, Λ+ g does not seem to admit a
complementary closed subspace in Λ∞ g.
However one can show that all this construction works in the slightly more
general regularity H 1/2 ∩ L∞ ([PS88], Chapter 8). Thanks to lemma 1.1.46, it
will be sufficient for our purpose to consider the H s -regularity.
47
The period map
Let (H, B, T , W, D) → X be a variation of loop Hodge structures of virtual
dimension n and denote by π : X̃ → X the universal cover of X. Let x̃0 be
a base-point in X̃ and consider the pullback of the variation of loop Hodge
structure to X̃. Its fiber over x̃0 can be identified with H := L2 (S 1 , Cn ) with
its canonical outgoing Krein structure such that (π ∗ W )x̃0 corresponds to the
Fourier-nonnegative subspace.
Since π ∗ D is a flat connection over a simply connected manifold, one can
trivialize the variation over X̃. In this way, the variation on X̃ is carried by
the trivial bundle X̃ × H, the Krein metric and outgoing operator are constant
and W is a subbundle of outgoing subspaces which is the Fourier-nonnegative
subspace over x̃0 . The datum of this subbundle is equivalent to a map f : X̃ →
Λ∞
σ G/K, such that f (x̃0 ) = eK by proposition 1.1.44.
We write ρtot : π1 (X) → Aut(H) for the monodromy of the flat connection
D. Since B and T are D-flat, ρtot takes its values in Λ∞
σ G. The map f is
equivariant with respect to the monodromy representation ρtot , since the data
come from X. Moreover, f is smooth by the very definition of a subbundle of
outgoing subspaces in an outgoing Krein bundle.
Lemma 1.1.46. Let s be any real number greater than 1/2. Then f takes its
values in Λsσ G/K and the monodromy ρtot takes its values in Λsσ G.
Proof. By theorem 1.1.23, the variation comes from a harmonic bundle and the
flat connection D comes from a circle of flat connections. This circle of flat
connections depends smoothly on the parameter λ in S 1 (even algebraically).
Hence, by the theorem about smooth dependence of solutions of ODE, the
parallel transport of D stabilizes the smooth loops with values in Cn . This
proves that a local lift of f to Λ∞
σ G takes its values in the smooth loop group.
By the Sobolev embedding theorem, a map is smooth if and only if it belongs
to all H s ; this concludes the proof for f . The proof for the monodromy is
analogous.
Thanks to this lemma, we will not need to consider the space Gr(H) =
anymore; we choose a number s > 1/2 and we write again Λσ G/K for
the period domain of loops with regularity H s .
Λ∞
σ G/K
Remark 1.1.47. The same proof shows that the map f takes its values in
Λhσ G/K, where Λhσ G is the subgroup of Λσ G of loops which are restrictions of
holomorphic maps from C∗ to G. This will be used in section 1.4.
Definition 1.1.48. The developing map f : X̃ → D of a variation of loop
Hodge structures is called the period map of the variation.
Theorem 1.1.49. The period map of a variation of loop Hodge structures is
holomorphic and horizontal. Conversely, let ρ : π1 (X) → Λσ G be a group morphism and let f : X̃ → Λσ G/K be a holomorphic, horizontal and ρ-equivariant
L
i ) defines a
map that satisfies f (x̃0 ) = eK. Then, the datum (H, d, ˆ i∈N HX̃
variation of loop Hodge structures on X̃ that descends to one on X.
48
i will be explained
This is theorem B of the Introduction. The notation HX̃
in subsection 1.3.1, where we postpone the proof of the theorem.
Remark 1.1.50. By theorems 1.1.23 and 1.1.49, we can associate to any harmonic bundle (E, D, h) on X a map f : X̃ → D, that is ρtot -equivariant, where
ρtot is the monodromy of the variation of loop Hodge structures. For λ in S 1 ,
we write ρλ : π1 (X) → G for the monodromy of the flat connection Dλ . Then,
by the proof of theorem 1.1.23, the monodromy ρtot is equal to the circle of
monodromies (ρλ )λ∈S 1 .
1.1.5
Relation with twistor theory
Given a harmonic bundle (E, D, h) over a complex manifold X, one can define
the developing map of its metric: it is a pluriharmonic ρ-equivariant map f from
X̃ to the symmetric space G/K. One can also consider the period map F : X̃ →
D of its associated variation of loop Hodge structures: it is a holomorphic,
horizontal and ρtot -equivariant map. This correspondence between harmonic
and holomorphic data already appeared in the literature ([DPW98], [DE03],
[ET98]) and can be proved without the geometric interpretation that we have
presented. In this subsection, we briefly explain this viewpoint and its relation
with our construction. We will come back again to this discussion in subsection
1.2.2, when dealing with classical variations of Hodge structures.
Let X be a simply connected complex manifold and let f : X → G/K be
a smooth map. Writing I for the identity and J for the complex structure on
T X, we can define rotation operators
Rλ := (Re λ)I + (Im λ)J,
on T X, for λ in S 1 . An associated family for f is a smooth family of maps fλ
from X to G/K such that
Φλ ◦ dfλ = df ◦ Rλ ,
where Φλ : fλ∗ T S → f ∗ T S is a parallel vector bundle isomorphism (parallel for
the pullback of the Levi-Civita connection on G/K) that respects the pullback
of the curvature tensor of G/K.
Proposition 1.1.51 ([DE03],[ET98]). The map f admits an associated family
if and only if it is pluriharmonic.
Thinking of f as giving a harmonic bundle (E, D, h) over X (with E a trivial
bundle and D the trivial flat connection), the maps fλ are obtained using the
flat connections Dλ , as defined in subsection 1.1.2.
Hence, we get a circle of pluriharmonic maps fλ : X → G/K. Moreover, if
˜
f : X → G is a local lift of f , then it is explained in the references given above
how to define canonically some local lifts f˜λ for fλ . One can consider the local
maps f˜λ as a single local map F̃ : X → ΛG and F̃ takes in fact its values in
Λσ G. Quotienting by the subgroup K of constant loops with values in K, we
get a canonical (global) map F : X → Λσ G/K associated to the pluriharmonic
map f .
49
Proposition 1.1.52 ([DE03],[DPW98]). The map F is holomorphic and horizontal. If ev1 : Λσ G/K → G/K is the evaluation map at λ = 1 then, by
construction, f = ev1 ◦ F
Remark 1.1.53. It is essentially written in the literature that these properties characterize the map F (this is the uniqueness statement in theorem 3 of
[DE03]). The map ev1 : Λσ G/K → G/K is a generalized twistor fibration and
the map F is the twistor lift of f .
Proposition 1.1.54. If (E, D, h) is a harmonic bundle over a complex manifold
X, let f : X̃ → G/K be the developing map of its metric and let F : X̃ →
Λσ G/K be the period map. Then, F is the twistor lift of f .
Proof. By the above remark, it is sufficient to show that ev1 ◦ F = f . Moreover,
we can assume that X = X̃. Let x0 be the base-point in X and identify the fiber
Ex0 to Cn with its canonical Hermitian structure and (L2 (S 1 , Ex0 ), h, T, Wx0 )
to L2 (S 1 , Cn ) with its canonical loop Hodge structure. We write γ (resp. γ̃) for
the parallel transport in (E, D) (resp. in (L2 (S 1 , E), D̃)) from x to x0 , where x
is some point in X.
The subspace γ̃.L2+ (S 1 , Ex ) of L2 (S 1 , Cn ) is an outgoing subspace since γ̃
respects both T and the Krein metric. Let g̃ be in Λσ GL(n, C) such that
γ̃.L2 (S 1 , Ex ) = g̃.L2+ (S 1 , Cn ). The metric hx on Ex induces a metric γ∗ hx on
Ex0 = Cn . Let g be in GL(n, C) such that γ∗ hx = g∗ h0 , where h0 is the
canonical metric on Cn . We claim that we can choose g to be ev1 ◦ g̃.
Indeed, consider the map g̃ −1 ◦ γ̃ from L2 (S 1 , Ex ) to L2 (S 1 , Ex0 ), which
is just L2 (S 1 , Cn ). By construction, it is an isometry, sending the outgoing
subspace L2+ (S 1 , Ex ) to the outgoing subspace L2+ (S 1 , Cn ). Hence, it sends Ex
to Cn . If v1 and v2 are in Ex , we can consider them in L2 (S 1 , Ex ) and
Bx (v1 , v2 ) = hx (v1 , v2 ).
On the other hand, Bx0 (g̃ −1 ◦ γ̃.v1 , g̃ −1 ◦ γ̃.v2 ) = hx0 (g̃ −1 ◦ γ̃.v1 , g̃ −1 ◦ γ̃.v2 ) since
the vectors g̃ −1 ◦ γ̃.vi are in Cn . Since g̃ −1 ◦ γ̃ is a Krein isometry, we thus get
hx0 (g̃ −1 ◦ γ̃.v1 , g̃ −1 ◦ γ̃.v2 ) = hx (v1 , v2 ).
The vectors g̃ −1 ◦ γ̃.vi are constant in λ, hence equal to their evaluation at 1,
which is g̃ −1 (1)◦γ.vi . Here, we use that ev1 is a flat map for the flat connections
D̃ and D, by construction of D̃. The last equality gives
g̃(1)∗ h0 = γ∗ hx ,
showing that one can indeed take g to be g̃(1).
By definition of F and f , F̃ (x) = g̃K in Λσ G/K and f (x) = gK in G/K.
This concludes the proof.
50
1.2
Classical and loop Hodge structures
It is well known that a variation of classical Hodge structures as defined in
the introduction underlies a harmonic bundle. This construction is recalled in
proposition 1.2.5. In this section, we explain further the relations between these
two notions, through the viewpoints of variations of loop Hodge structures and
twistor theory.
1.2.1
Intrinsic point of view
By theorem 1.1.23, a harmonic bundle can be interpreted as a variation of loop
Hodge structures. In this subsection, we explain how to realize a variation of
classical Hodge structures as a variation of loop Hodge structures, justifying
the chosen terminology.
The circle action
Let (K, B, T , W, D) be a variation of loop Hodge structures over a complex
manifold X. If µ is a complex number of modulus one, one checks easily that
(K, B, µT , W, D) is also such a variation. This is called the circle action on the
variations of loop Hodge structures. In the correspondence given by theorem
1.1.23, this corresponds to an operation on harmonic bundles: one keeps the
metric but changes the flat connection from D = D1 to Dµ , with the usual
notations.
A variation of loop Hodge structures is a fixed point of this circle action if
it is isomorphic to all variations in its orbit.
Classical Hodge structures interpreted as loop Hodge structures
Let (V, h, F • V ) be a classical Hodge structure. We write V = n V n for the
Hodge decomposition. Let K be the Hilbert space L2 (S 1 , V ) endowed with the
Hermitian form
Z
h(f (λ), g(−λ))dν(λ).
B(f, g) =
L
S1
Let T denote the right-shift operator. Let W be the subspace of K given by
W =
M
d
n∈Z
T −n F n V.
Proposition 1.2.1. The quadruple (K, B, T, W ) is a loop Hodge structure.
Proof. Since the Hermitian form h is non-degenerate, the Hermitian form B
defines a structure of Krein space on H. A fundamental decomposition is for
instance given by
K+ :=
M
d
K− :=
M
d
T nV
even T nV
odd n even
M M
d
n odd
T nV
odd and
n even
M M
d
n odd
T nV
even .
1.2. Classical and loop Hodge structures
51
The operator T is anti-isometric. Let us show that W is an outgoing Krein
subspace. All points in definition 1.1.6 are trivial, except that the orthogonal
E of T W in W is definite positive. This space is equal to
E=
M
d
n∈Z
T −n V n .
This concludes since h is (−1)n positive definite on V n and since T is an antiisometry.
Characterization of classical Hodge structures
Let (V, h, D, F • V) be a variation of classical Hodge structures over a complex
manifold X. By a family generalization of proposition 1.2.1, this gives a family
of loop Hodge structures (K, B, T , W). We write D̃ for the naive connection
induced by D on K = L2 (S 1 , V).
Proposition 1.2.2. The quintuple (K, B, T , W, D̃) is a variation of loop Hodge
structures over X, which is a fixed point of the circle action.
Proof. The D-flatness of h implies the D̃-flatness of B. Moreover, the right-shift
operator T is D̃-flat as usual. It is thus enough to show that the differential
constraints 2. and 3 of definition 1.1.21 are satisfied. Let f be a local section
of W. It can be written in Fourier series as
f=
X
an λ n ,
n∈Z
where an is a local section of F −n V. Then D̃f is given by
D̃f =
(Dan )λn = T −1
X
X
n
n
(Dan )λn+1 .
By the Griffiths transversality condition, Dan is a local section of T ∗ X ⊗
F −n−1 V, hence D̃f is a local section of T ∗ X ⊗ T −1 W. The differential constraint on the (0, 1)-part of D̃ is proved in the same way.
Let µ be in S 1 and consider the map φ from K to K given by
φ(f )(λ) = f (µλ).
Then, φ is a flat isometry of K satisfying µT ◦ φ = φ ◦ T . This shows that the
variation is a fixed point of the circle action.
This proposition has the following converse (compare with lemma 4.1 in
[Sim92]), which is theorem C of the introduction:
Theorem 1.2.3. Let (K, B, T , W, D) be a variation of loop Hodge structures,
which is a fixed point of the circle action. Then, it is canonically isomorphic to
a variation of loop Hodge structures coming from a variation of classical Hodge
structures, by the construction of proposition 1.2.2.
52
Proof. Let µ a complex number of modulus one, that is not a root of unity,
such that there exists an isomorphism
φ : (K, B, T , W, D) → (K, B, µT , W, D).
Since (µT )W = T W , and since φ preserves W and B, it also preserves R the
orthogonal of T W in W. Up to a canonical isomorphism, K is L2 (S 1 , R), T is
the right-shift operator and W is the Fourier-nonnegative subbundle L2+ (S 1 , R).
Denote by ψ : R → R the restriction of φ to R. It is an isometry of R and φ
is determined by ψ:
φ(f )(λ) = ψ(f (µλ)),
(1.3)
as comes easily from the condition φ ◦ T = µT ◦ φ.
If x is in X and ν is a complex number of modulus one, we denote by Rνx
the eigenspace of ψ in R for the eigenvalue ν. Since ψ is an isometry, one has
M
Rx =
Rνx .
ν∈S 1
Let Kxν be the (finite-dimensional) subspace of Kx given by
Kxν =
M
−n ν
T n Rµx
.
n∈Z
From the equation (1.3), Kxν is the eigenspace of φ for the eigenvalue ν and
M M
d
Kx =
ν∈S 1
n∈Z
T n Kxν .
If v is any vector in Kxν and ṽ denotes its local flat prolongation, ṽy is in Kyν
over any point y near x since φ is flat. This shows that Kν is a flat subbundle
−n
of K, in particular a smooth subbundle. Hence, all the components T n Rµ ν
are also smooth, and in particular Rν is a smooth subbundle of R for any ν.
Let Y denote the set of eigenvalues ν of ψ such that µν is not an eigenvalue
L
of ψ. Let V denote the (finite-dimensional) flat subbundle ν∈Y Kν and denote
L
−i
by V i the subbundle of V defined by V i := ν∈Y T i Rµ ν . One has the smooth
decomposition
M
V=
V i.
i∈Z
µ−i ν
Let v be a local section of T i R
and let Y be a local vector field on X. Then,
DY v is a local section of Kν . By the transversality condition for a variation of
L
loop Hodge structures, it is also a local section of c j≥i−1 T j R. Hence, DY v is a
L
local section of j≥i V j . This is the transversality condition for classical Hodge
structures. The holomorphicity condition is proved in the same way. Finally,
the metric B has sign (−1)i on V i : indeed, V i is contained in T i R, the metric
L
is positive on R and T is an anti-isometry. This shows that V = i V i defines
a variation of classical Hodge structures.
In order to conclude, we show that the variation of loop Hodge structures
(K0 , B 0 , T 0 , W 0 , D0 ) obtained from this variation of classical Hodge structures
53
i
V =
i V (by proposition 1.2.2) is the variation of loop Hodge structures
(K, B, T , W, D) that we started with. First, K0 = L2 (S 1 , V) = L2 (S 1 , R) = K.
Moreover, it is straightforward to show that the Krein metrics, the outgoing
operators and the flat connections are the same. It is thus enough to show that
W = W 0 . By definition, the decreasing filtration F • V is given by
L
F nV =
M
T n W ∩ Kν .
ν∈Y
Hence, W 0 is formally given by
W0
=
L
=
L
ν∈Y
ν∈Y
L
n
−n V)
c
n∈Z T (F
L
n ν
c
n∈Z W
∩T K .
Hence, W 0 is in W. For the reverse inclusion, it is sufficient to show that R is
in W 0 . If ω is any eigenvalue of ψ, then there exists n0 ∈ N such that µn0 ω =: ν
is in Y , by definition of Y . Hence, Rω = T −n0 (T n0 Rω ) is in T −n0 Kν , hence in
W 0 . This concludes the proof.
Remark 1.2.4. With the notations of the proof, the decomposition V =
L
ν
ν∈Y K is a decomposition of V in sub-variations of Hodge structures. In
particular, V is irreducible only if Y has cardinal 1.
1.2.2
Back to twistor theory
In this subsection, we give another point of view on theorem 1.2.3 in the framework of twistor theory. This discussion was begun in subsection 1.1.5.
Variations of Hodge structures and harmonic bundles
We consider a classical variation of Hodge structures (V, h, F • V, D) over a comL
plex manifold X. Let V = i V i be the (h-orthogonal) Hodge decomposition.
Let h0 be the Hermitian form on V such that this decomposition is also orthogonal for h0 and such that h0 = (−1)i h on V i ⊗ V i . The Hermitian form h0 is
thus positive definite.
Proposition 1.2.5. The bundle (V, h0 , D) is a harmonic bundle.
Sketch of the proof. By Griffiths transversality condition, the holomorphicity of
the filtration F • V and the orthogonality of the Hodge decomposition, the flat
connection D can be decomposed as D = ∇ + θ + µ, where
• ∇ is a connection on V which respects the Hodge decomposition;
• µ is a (0, 1)-form with values in
L
i , V i−1 );
• θ is a (1, 0)-form with values in
L
i , V i+1 ).
i Hom(V
i Hom(V
54
Using the D-flatness of h, the definition of h0 and the orthogonality of the
Hodge decomposition, one shows that ∇ is a metric connection for h0 and that
µ is the adjoint of θ for h0 . Hence, we write µ = θ∗ . The flatness of D gives
(∇2 + θ ∧ θ∗ + θ∗ ∧ θ) + ∇θ + ∇θ∗ + θ ∧ θ + θ∗ ∧ θ∗ = 0.
These five terms have to vanish separately because they act differently on
the Hodge decomposition of V . The vanishing of the first term implies that
(∇0,1 )2 = 0 for bidegree considerations. Finally, ∇θ = 0 also implies ∇0,1 θ = 0
for the same reason. Hence,
(∇0,1 + θ)2 = 0,
which is the harmonicity condition.
Let Dc be the classical period domain associated to the variation. If (p, q)
is the signature of h and if we write pi (resp. qj ) for the dimensions of the
even (resp. odd) subspaces in the Hodge decomposition, then we recall from
the introduction that
∼ U (p, q)/
Dc =
Y
i
U (pi ) ×
Y
U (qj ).
j
The subgroup i U (pi ) × j U (qj ) is compact and is contained in the maximal
compact subgroup U (p) × U (q). We write pr for the canonical projection from
Dc to the symmetric space U (p, q)/U (p)×U (q). We also write ι for the homogeneous embeddings of symmetric spaces U (p, q)/U (p) × U (q) ,→ GL(n, C)/U (n).
Finally, we write π : Dc → GL(n, C)/U (n) =: G/K for the composition ι ◦ pr.
At the level of homogeneous spaces, the passage from the variation of classical
Hodge structures to the harmonic metric is given by the following easy lemma:
Q
Q
Lemma 1.2.6. Let f : X̃ → Dc be the period map of the variation of loop
Hodge structures (V, h, F • V, D). Then π ◦ f : X̃ → G/K is the developing map
of the harmonic bundle (V, h0 , D).
Twistor theory and isotropic pluriharmonic maps
We have defined a projection map π from a classical period domain Dc to
the symmetric space G/K. This is called a twistor fibration. Its fundamental
property is the following: if f is a holomorphic and horizontal map from X to
Dc , then π ◦ f : X → G/K is pluriharmonic. Conversely, we want to explain
which pluriharmonic maps from X to G/K can be lifted to a holomorphic and
horizontal map to some period domain Dc .
We recall that a pluriharmonic map f : X → G/K admits an associated
family (fλ )λ∈S 1 , whose properties are defined in subsection 1.1.5. We say that
f is isotropic if the associated family can be chosen constant: fλ = f , for every
λ in S 1 .
Proposition 1.2.7. An isotropic pluriharmonic map f from a simply connected complex manifold X to G/K takes its values in a totally geodesic proper
55
subspace isomorphic to U (p, q)/U (p) × U (q) and can be lifted to a holomorphic
horizontal map with values in some classical period domain Dc .
Hence, the datum of an isotropic pluriharmonic map to G/K is equivalent
to the datum of a variation of classical Hodge structures.
Sketch of the proof. A pluriharmonic map corresponds to a variation of loop
Hodge structures, hence to a holomorphic and horizontal map φ : X → Λσ G/K.
Another description for this map is given for instance in [DE03]; that the two
descriptions coincide is the content of proposition 1.1.54. In the given reference,
paragraph 8, it is shown that there exists a lift Φ : X → Λσ G of φ and a group
morphism q : S 1 → U (n) such that Φ takes its values in the finite-dimensional
subgroup Λq of Λσ G given by
Λq := {qgq −1 | g ∈ G} ∩ Λσ G.
Here, (qgq −1 )(λ) = q(λ)gq −1 (λ). This loop is in Λσ G if and only if
q(−1)gq(−1) = σ(g)
or, otherwise said, if and only if g preserves the non-degenerate Hermitian form
given by q(−1). If this form has signature (p, q), Λq is thus a subgroup of Λσ G
which is isomorphism to U (p, q).
Remark 1.2.8. In [DE03], only full isotropic pluriharmonic maps are considered (that is, maps whose values are not contained in a proper totally geodesic
subspace) and in this setting, for every g in the structural group G, all loops
qgq −1 are in Λσ G. But, as explained for instance in [ET98], a full isotropic
pluriharmonic map can only exist if the symmetric space is inner, which is
not the case of GL(n, C). For people in Hodge theory, this is essentially the
statement that GL(n, C) is not a group of Hodge type.
We come back to the proof of proposition 1.2.7. Since Φ takes its values
in Λq , φ takes its values in the orbit of Λq in Λσ G/K. One checks that the
stabilizer subgroup is the group
Cq := {qgq −1 |g ∈ Z(q) ∩ U (q(−1))},
where Z(q) is the centralizer of the circle q. In order to understand this centralizer, we diagonalize q(θ). The diagonal terms are of the form einθ , with
some integers n. There are p even integers k and q odd integers k 0 . We can
decompose further: there are pi integers ki and qj integers kj0 where i and j are
P
P
some indices and i pi = p, j qj = q. This decomposition shows that Cq is
Q
Q
isomorphic to a product i U (pi ) × j U (qj ).
Hence, φ takes its values in a period domain
Dc = Λq /Cq ∼
= U (p, q)/
Y
i
U (pi ) ×
Y
U (qj ).
j
The embedding Dc ,→ Λσ G/K is holomorphic and respects the horizontal structure. Hence, φ is holomorphic and horizontal. That f takes its values in a totally
geodesic proper subspace isomorphic to U (p, q)/(U (p) × U (q)) follows from the
commutativivity of the diagram in the next paragraph.
56
Pluriharmonic maps and loop period domain
Since we consider both the classical and loop period domains, we will precise
the notations: Dc for the classical period domain, and Dl for the loop period
domain.
Proposition 1.2.9. Let q be a morphism from S 1 to K = U (n). Then h :=
q(−1) is a non-degenerate Hermitian form. Let Cq be the centralizer of q in
U (h) ∩ K. Then, we have a commutative diagram
U (h)
Dc = U (h)/Cq / Λσ G
/ Dl = Λσ G/K

U (h)/U (h) ∩ K
ev1
/ G/K
The vertical arrows are the canonical projections, ev1 , being induced by the
evaluation at 1 from Λσ G to G. The top horizontal arrow is U (h) 3 g 7→
qgq −1 ∈ Λσ G, the middle horizontal arrow is induced from the top one and the
bottom horizontal arrow is induced from the inclusion U (h) ,→ G.
Proof. The top horizontal arrow induces a middle horizontal arrow by the proof
of proposition 1.2.7. Hence, the only thing to check is the commutativity of
the big square. Since q : S 1 → K is a group morphism, q(1) = 1 and the
commutativity of this square is thus obvious.
In conclusion, the picture is as follows: a harmonic bundle corresponds to a
pluriharmonic map to G/K that can always be lifted to a holomorphic and horizontal map with values in the loop period domain Dl . When the pluriharmonic
map is isotropic, then the lift can be performed in a classical period domain Dc ,
that can be seen in Dl . In particular, Dl contains all classical period domains
and the map ev1 : Λσ G/K → G/K can be thought as a universal twistor map.
1.3
The Higgs field
Let (E, D, h) be a harmonic bundle. We recall that we write D = ∇ + α, with
∇ a metric connection and α a Hermitian 1-form with values in End(E) and
that the Higgs field θ is the (1, 0)-part of α, so that α = θ + θ∗ . In this section,
we prove that the Higgs field can be seen as the differential of the period map
to D = Λσ G/K and we discuss the consequences for the Higgs foliation.
1.3.1
Differential of the period map
The following notations will be used in the whole section. The space L2 (S 1 , Cn )
with its structure of loop Hodge structure is written H. The period domain is
D = Λσ G/K and the Lie algebra of Λσ G is Λσ g. When there is no possible
confusion, the trivial bundles Y × H and Y × Λσ g over a complex manifold Y
will be simply written H and Λσ g.
57
1.3. The Higgs field
The tangent bundle of the period domain
Definition 1.3.1. The tautological bundle over D is the complex Hermitian
bundle of rank n defined by (H 0,D )W ∈D = W ⊥ T W , where the orthogonal
is taken with respect to the Krein metric. The tautological decomposition of H
over D is the Hilbert sum decomposition
H=
M
d
i∈Z
H i,D ,
where H i,D := T i H 0,D . The tautological decomposition of Λσ g over D is the
Hilbert sum decomposition
Λσ g =
M
d
i∈Z
Λi,D
σ g,
0,D ) ⊂ H i,D } = {f ∈ Λ g | ∀j, f (H j,D ) ⊂
where Λi,D
σ g := {f ∈ Λσ g | f (H
σ
i+j,D
H
}.
Proposition 1.3.2. The tangent bundle of D is canonically isomorphic to the
bundle Λσ g/Λ0,D
σ g.
Proof. We recall that the tangent bundle of D is isomorphic to Λσ G×K Λσ g/Λ0σ g,
as a Λσ G-equivariant bundle. Here, Λ0σ g is the subspace of Λσ g of functions
with vanishing zero Fourier coefficient. In this representation, a point in T D
over g.o, where g is in Λσ G and o = eK is the base-point, is given by
[g : f + Λ0σ g],
where f is in Λσ g and the equivalence relation is given by [g : f + Λ0σ g] = [gk :
Ad(k)−1 f + Λ0σ g].
We define a vector bundle map from T D to Λσ g/Λσ0,D g by
[g : f + Λ0σ g] 7→ Ad(g)f + Λσ0,D g,
over g.o. This is well-defined. Indeed, on the one hand, if one changes g by gk
and f by Ad(k −1 f ), then gk.o = g.o and Ad(gk) Ad(k −1 )f = Ad(g)f . On the
0
other hand, one has the equality (Λ0,D
σ g)g.o = Ad(g)Λσ g. Since the inverse of
this vector bundle map is given by Ad(g −1 ) over g.o, this is an isomorphism.
L
Lemma 1.3.3. In the identification Λσ g/Λσ0,D g ∼
= c j6=0 Λj,D
σ g, the complex
L
j
c
structure on T D is given by multiplication by i on j<0 Λσ gD and by −i on
L
j
−1
1
c
j>0 Λσ gD . The horizontal subbundle is given by Λσ g ⊕ Λσ g.
Proof. This is true at the base-point and both the complex structure and the
horizontal subbundle are stable by the Λσ G action. By the explicit isomorphism
given in the proof of proposition 1.3.2, this concludes the proof.
58
The differential of a smooth map to the period domain
Let f : X → D be a smooth map. The trivial bundle H over X has a canonical
L
decomposition H = c i∈Z H i,X given by pullback of the decomposition of D×H.
We also have a decomposition of the trivial bundle X × Λσ g. Let d be the
trivial connection on the bundle H over X. We denote by π the vector bundle
projection π : H → H/H 0,X . Let
0,X
∼
ι : Λσ g/Λ0,X
, H/H 0,X )
σ g = Hom(H
be the canonical isomorphism of vector bundles, given by
ι : f + ∈ Λσ g 7→ (h ∈ H 0,X 7→ f (h) + H 0,X ).
0 to H/H 0 .
Finally, let β be the differential operator π ◦ d|H 0,X from HX
X
Proposition 1.3.4. The differential operator β is a tensor, that is a 1-form
with values in Hom(H 0,X , H/H 0,X ). Moreover, if one considers the differential
of f to be a 1-form with values in Λσ g/Λ0,X
σ g (by proposition 1.3.2), then
β = ι ◦ df.
Proof. Let g be a local lift of f to Λσ G. We write ω for the left Maurer-Cartan
form in Λσ G. In the homogeneous representation T D = Λσ G ×K Λσ g/Λ0σ g, df
is given by
dx f = [g(x) : (g ∗ ω)x + Λ0σ g].
By the proof of proposition 1.3.2, as a 1-form with values in Λσ g/Λ0,D
σ g, df
is thus given by
dx f = Ad(g(x))(g ∗ ω)x + (Λ0,D
σ g)x .
On the other hand, we need to compute β. Let s be a local smooth section
of H 0,X . It can be written s(x) = g(x)h(x), where h is a smooth map with
values in Cn ⊂ H. Then
dx s = (dx g)h(x) + g(x)(dx h).
Hence modulo (H 0,X )x , dx s = (dx g)h(x) = (dx g)g −1 (x)s(x). In particular, β
is a tensor, given by
βx (s(x)) = (dx g)g −1 (x)s(x) + (H 0,X )x ,
as a 1-form with values in Hom(H 0,X , H/H 0,X ).
Since ι ◦ df is given by
(ι ◦ df )x (s(x)) = Ad(g(x))(g ∗ ω)x (s(x)) + (H 0,X )x ,
it is sufficient to show that Ad(g(x))(g ∗ ω)x = (dx g)g −1 (x). This follows from
(g ∗ ω)x = g(x)−1 dx g and concludes the proof.
59
The differential of the period map
Let (E, D, h) be a harmonic bundle over a simply-connected manifold X and
let f : X → D be the period map. Let (L2 (S 1 , E), L2+ (S 1 , E), D̃) be the associated variation of loop Hodge structures. By parallel transport, there is an
isomorphism of variations of loop Hodge structures
γ : (L2 (S 1 , E), D̃) → (H, d)
i . The differential of f
where the outgoing subbundle in H is given by c i∈N HX̃
takes its values in the bundle Hom(H 0,X , H/H 0,X ) which, by ι, is isomorphic
to Λσ g/Λ0,X
σ g. We omit the isomorphism ι in what follows.
On the other hand, the 1-form α = λ−1 θ + λθ∗ takes its values in the
bundle Λσ End(E)/Λ0σ End(E). Here, Λσ End(E) is the Hilbert subbundle of
H s (S 1 , End(E)) whose elements are the loops γ with values in End(E), satisfying
σ(γ(λ)) = γ(−λ), where σ is the Cartan involution induced from the harmonic
metric h. The quotient Λσ End(E)/Λ0σ End(E) is a complex Hilbert bundle.
The isomorphism γ induces an isomorphism γ : Λσ End(E)/Λ0σ End(E) ∼
=
Λσ g/Λ0,X
g
of
complex
Hilbert
bundles.
σ
L
Proposition 1.3.5. The equality df = γ ◦ α holds.
Proof. We recall that β was defined by β := π ◦ d|H 0 and that β = ι ◦ df . Since
X
the flat connection d on H is sent by γ −1 to the flat connection D̃ = ∇ + α, the
form γ −1 ◦ β is equal to the projection of D̃ on Λσ End(E)/Λ0σ End(E), that is to
α. Hence, β = γ ◦ α, which concludes the proof, since we omit the isomorphim
ι.
We can now prove theorem 1.1.49, which we recall.
Theorem 1.3.6. The period map induced from a harmonic bundle is holomorphic and horizontal. Conversely, let ρ : π1 (X) → Λσ G and let f : X̃ → Λσ G/K
be a holomorphic, horizontal and ρ-equivariant map that satisfies f (x̃0 ) = eK.
L
i ) defines a variation of loop Hodge structures
Then, the datum (H, d, c i∈N HX̃
on X̃ that descends to one on X.
Proof. We can assume that X = X̃. Since the isomorphism γ respects all the
structures, it is sufficient to check that the 1-form α lives in the horizontal
subbundle of Λσ End(E)/Λ0σ End(E) and commutes with the complex structure.
As before, the horizontal subbundle is given by the loops in Λσ End(E) that
have only Fourier coefficients in degrees −1 and 1. Hence, α is horizontal.
Since θ (resp. θ∗ ) is a holomorphic (resp. anti-holomorphic) section of End(E),
the computation of Jα gives:
Jα = λ−1 JEnd(E) ◦ θ − λJEnd(E) ◦ θ∗ by equation (1.2)
= λ−1 θ ◦ JT X̃ + λθ∗ ◦ JT X̃
= α ◦ JT X̃ ,
which proves the claim.
60
i ) defines a
For the converse, the same analysis shows that (H, d, c i∈N HX̃
variation of loop Hodge structures on X̃ and it descends to X since the map f
is equivariant under ρ.
L
For future reference, we add a simple corollary to proposition 1.3.5. We
consider a harmonic bundle (E, D, h) over a complex manifold X and we endow
X with a Hermitian metric. The harmonic metric h induces a metric hEnd(E) on
End(E). Since θ is a 1-form with values in End(E), it makes sense to ask that
θ is bounded on X (meaning that its operator norm is bounded by a constant,
independent of the point in X).
0,X ) and H 0,X
On the other hand, the space Λ0,X
σ g is isomorphic to End(H
is endowed with a (positive) Hermitian metric which is the restriction of the
Krein metric of H; hence Λ0,X
σ g is also endowed with a natural metric. In this
way, we endow Λσ End(E) and Λσ g with a Hilbert metric and γ is an isometry
by construction. Beware that the Hilbert metric on the trivial bundle Λσ g is
not the trivial one.
Corollary 1.3.7. If θ is bounded, then the differential of the period map f :
X̃ → D is bounded, as an operator from T X̃ to Λσ g/Λ0,X
σ g. Here, the metric
on X̃ is induced from the metric on X.
Proof. If θ is bounded, its pullback to X̃ is also bounded so that we can assume that X = X̃. Since γ is an isometry of Hilbert bundles, it is sufficient to show that α = λ−1 θ + λθ∗ is bounded (as an operator with values in
Λσ End(E)/Λ0σ End(E)). The adjoint operator θ∗ will be bounded if θ is, so this
is obvious.
The holomorphic differential
We consider the trivial complex Hilbert bundle Λg over D. There is a tautological filtration on Λg given by (F i,D Λg)W ∈D = {f ∈ Λg | f (W ) ⊂ T i W }. This
is a holomorphic filtration since it is homogenous with respect to Λσ G: indeed,
F i,D Λg = Λσ G ×K F i Λg, where F i Λg is the subspace of Λg of loops having
Fourier coefficients in degrees ≥ i.
The holomorphic tangent space of D is canonically isomorphic to Λg/F 0,D Λg
and the horizontal subspace to F −1,D Λg/F 0,D Λg. If f : X̃ → D is the period
map of a harmonic bundle, the holomorphic differential of f thus takes its values
in F −1,X̃ Λg/F 0,X̃ Λg, with obvious notations. As before, the isomorphism γ
induces an isomorphism of holomorphic vector bundles
F −1,X̃ Λg/F 0,X̃ Λg ∼
= L2,≥−1 (S 1 , End(E))/L2,≥0 (S 1 , End(E)),
where L2,≥i (S 1 , End(E)) is the subbundle of L2 (S 1 , End(E)) of loops whose
Fourier coefficients are ≥ i. There is an isomorphism
j : End(E) → L2,≥−1 (S 1 , End(E))/L2,≥0 (S 1 , End(E))
given by
j : ν ∈ End(E)) 7→ λ−1 ν + L2,≥0 (S 1 , End(E)).
61
Lemma 1.3.8. The map j is an isomorphism of holomorphic vector bundles,
where End(E) is endowed with its Higgs bundle structure.
Proof. We consider an arbitrary harmonic bundle (F, D = ∇ + θ + θ∗ , h).
¯
The ∂-operator
of F, seen as a Higgs bundle, is ∇(0,1) . On the other hand,
L2 (S 1 , F) is endowed with a flat connection D̃ = ∇ + λ−1 θ + λθ∗ . The induced
¯
∂-operator
is its (0, 1)-part, that is ∇(0,1) + λθ∗ . When acting on the quotient
L2,≥−1 (S 1 , F)/L2,≥0 (S 1 , F), only the term ∇(0,1) remains.
We apply this to the harmonic bundle F = End(E).
Our last proposition easily follows from proposition 1.3.5.
Proposition 1.3.9. Under the isomorphism (of holomorphic vector bundles)
End(E) ∼
= F −1,X̃ Λg/F 0,X̃ Λg, the holomorphic differential of f is the Higgs field
θ.
1.3.2
Higgs foliation
If (E, θ) is a Higgs bundle over a complex manifold X, then the kernel of θ defines
a (possibly singular) holomorphic distribution in X. When (E, θ) comes from
a harmonic bundle, it is shown in [Mok92], theorem 3, that this distribution is
involutive. Here, we show that this distribution is in fact integrable.
Holomorphic distributions
First, we recall some facts about holomorphic distributions. A holomorphic
distribution D in X is a subset of the holomorphic tangent bundle T X which is
locally the vanishing set of a finite number of holomorphic 1-forms on X. Let x
be some point in X, and let Dx be the germ in x of a holomorphic distribution.
For any family ω = (ω1 , . . . , ωp ) of germs of holomorphic 1-forms defining Dx ,
we define qx (ω) to be the codimension in X of the vanishing set of the p-form
ω1 ∧· · ·∧ωp , with the convention qx (ω) = ∞ if this set is empty. We write q(Dx )
for the supremum of such qx (ω) where ω is any family of germs of holomorphic
1-forms defining Dx .
A distribution D is involutive if it can locally be defined by holomorphic
1-forms ω1 , . . . , ωp which satisfy dωi ∧ ω1 ∧ · · · ∧ ωp = 0, for all i in {1, . . . , p}.
Dually, this is equivalent to the usual definition: locally, for every holomorphic
vector fields X and Y , tangent to D, the bracket [X, Y ] is also tangent to D.
Equivalently, an involutive distribution is called a foliation.
A distribution D is integrable if it can locally be defined by exact holomorphic 1-forms df1 , . . . , dfp , where the fi are local holomorphic functions. An integrable distribution is involutive and, in the regular case, that is when q(Dx ) = ∞
for any x, then the Frobenius theorem asserts that the converse is true. In the
singular case, things are subtler but this still works if the singular locus is small
enough:
Theorem 1.3.10 (Malgrange, [Mal77]). Let D be an involutive distribution in
X and assume that q(Dx ) ≥ 3 for any x in X. Then, D is integrable.
62
Higgs foliation
Let (E, D, h) be a harmonic bundle over X and let θ denote the induced Higgs
field. The kernel of θ defines a holomorphic distribution in X since θ is a
holomorphic 1-form with values in End(E), for some holomorphic structure on
E. We can now state theorem D of the Introduction.
Theorem 1.3.11. The holomorphic distribution defined by the Higgs field is
integrable.
I thank P. Eyssidieux, who suggested this application to me.
Proof. Locally in X, one can define a holomorphic period map f : X → D =
Λσ G/K. By theorem 1.3.9, a holomorphic vector field is in the kernel of the
Higgs field if and only if it is in the kernel of df . Since we work locally, we
can assume that f takes its values in a Hilbert space H. We write (fi )i∈N for
the coordinates of f relative to some Hilbert basis of H. The distribution is
thus defined by the vanishing of an infinite number of holomorphic 1-forms.
We claim that, locally, we can consider only a finite number of holomorphic
1-forms. Indeed, the germ of the distribution at x is completely determined by
the germs at (x, 0) of the functions (dfi )i∈N that go from the total space T X to
C. Since, the local ring of the sheaf of holomorphic functions in any complex
manifold is Noetherian, we can consider only a finite number of dfi , proving the
claim.
Remark 1.3.12. If (E, θ) is a Higgs bundle over a complex manifold X, there
is a priori no reason that the distribution that it defines is involutive. Indeed,
there is no differential assumptions about the Higgs field in the definition of
a Higgs bundle. Nevertheless, if X is a compact Kähler manifold, if (E, θ)
is polystable with vanishing Chern classes, then the Hitchin-Simpson theorem
0.2.7 asserts that it comes from a harmonic bundle, so that one can apply
theorem 1.3.11.
1.4
The nilpotent orbit theorem
In the introduction, paragraph 1.3.2 of [Moc07a], the author says the following:
In the study of complex variation of polarized Hodge structure
(CVHS), the nilpotent orbit theorem due to W. Schmid is quite
important. It was the starting point of the later studies on CVHS.
However, we do not know even the formulation of nilpotent orbit
theorem for harmonic bundles, for we do not have the counterparts
of the classifying space.
In this section we use the loop period domain D to prove a nilpotent orbit
theorem. We show that the proof of [Sch73] can be adapted to the loop period
domain, confirming the interest of this construction.
1.4. The nilpotent orbit theorem
63
However we have to remark that, in [Moc07a] and [Moc07b], the author
studies the asymptotic behaviour of harmonic bundles without needing a nilpotent orbit theorem. Moreover, some new phenomena appear in the case of
harmonic bundles: for instance, the Higgs field is always nilpotent in harmonic
bundles coming from variations of Hodge structures, but it is not in general.
Hence it is reasonable to think that this section (and the conjectural results
of appendix C) will give results already contained in [Moc07a] and [Moc07b].
Since the viewpoint is different, we nevertheless believe that this work is worth
it.
1.4.1
General setting
Assumptions on harmonic bundles
Let ∆∗ denote the punctured disk, and let (E, D, h) be a harmonic bundle over
∆∗ . We denote by (Dλ )λ∈S 1 the circle of flat connections on E; in particular
D1 is D. The Higgs field of the harmonic bundle E is written θ.
We write H for the upper half-plane, which is the universal cover of ∆∗ via
the map z 7→ w = exp(2iπz). The fundamental group Z of ∆∗ acts on H by
the translation z 7→ z + 1. We endow H with the hyperbolic metric given by
gH :=
dz ⊗ dz̄
.
(Im z)2
Since this metric is invariant by the map z 7→ z + 1, it endows ∆∗ with a
metric, given by
dw ⊗ dw̄
g∆∗ :=
.
2
|w| (− log |w|)2
The Higgs field θ is a section of T ∗ (1,0) ∆∗ ⊗ End(E). From the hyperbolic
metric g∆∗ and the metric h on E, this vector bundle inherits a metric. We
make the following assumptions:
Assumption 1.4.1. The Higgs field θ is bounded.
Assumption 1.4.2. The flat connections Dλ , λ in S 1 , have unipotent monodromy.
Proposition 1.4.3. A harmonic bundle over ∆∗ satisfies assumption 1.4.1 if
and only if it is tame nilpotent.
∗
Proof. We write θ = f dw
w , where f is a holomorphic section of End(E), over ∆ .
The characteristic polynomial det(t − f ) has coefficients which are holomorphic
functions on ∆∗ . We recall (definition 4.4 in [Moc02]) that the harmonic bundle is tame if these coefficients can be extended to a holomorphic function on
∆. Moreover, a tame harmonic bundle is nilpotent if the extended coefficients
vanish at 0, that is if the extended characteristic polynomial is tr at 0, where r
is the rank of E.
That a tame nilpotent harmonic bundle satisfies assumption 1.4.1 is known
in the literature as the main estimate for tame nilpotent harmonic bundles.
64
It is proposition 4.1 in [Moc02] and essentially appeared in the previous work
[Sim90].
Conversely, we assume that (E, h) satisfies assumption 1.4.1 and we denote
by α1 (w), . . . , αr (w) the eigenvalues
of f (w). Then, the norm of f (w) with
pP
respect to h(w) is larger than
|ai (w)|2 , up to some multiplicative constant.
For the metric g∆∗ , the norm of dw is |w|(− log(|w|)). Hence, we get
||θ(w)|| ≥
qX
|ai (w)|2 (− log(|w|)).
Since θ is bounded, this implies that |ai (w)|2 tends to 0 as w goes to 0. Hence,
the characteristic polynomial det(t − f ) can be extended to a holomorphic
function on ∆ and its value at 0 is tr .
P
Remark 1.4.4. With the first assumption, the second assumption is equivalent to asking that the harmonic bundle is tame nilpotent with trivial parabolic
structure (section 4 of [Moc02] for the definition of this notion). Indeed, this
follows from the formulas giving the residues of the λ-connections of the harmonic bundle, cf. definition 2.1.7 and corollary 7.71 in [Moc07a].
These assumptions are satisfied if the harmonic bundle comes from a Zvariation of Hodge structures (possibly after passing to a finite cover of ∆∗ ).
Indeed, the monodromies of the flat connections are then conjugated, because
of the characterization of variations of Hodge structures as fixed points of the
S 1 -action, and they are (quasi-)unipotent by the monodromy theorem (see e.g.
[CMSP03]); moreover the description of the Higgs field θ in proposition 1.2.5
shows that it is everywhere nilpotent, hence the harmonic bundle is tame nilpotent and the Higgs field is bounded.
Statement of the theorem
Let (E, D, h) be a harmonic bundle satisfying assumptions 1.4.1 and 1.4.2 over
∆∗ and let φ̃ : H → D = Λσ G/K be its period map, given by theorems 1.1.23
∼ π1 (∆∗ ) → Λσ G is the
and 1.1.49; this map is ρ-equivariant, where ρ : Z =
monodromy of the connection D̃ in the Hilbert bundle L2 (S 1 , E). We denote
by N ∈ Λσ g the logarithm of ρ(1), which is well defined by assumption 1.4.2.
One thus has the following equivariance property:
φ̃(z + 1) = exp(N ) · φ̃(z).
We define the twisted period map ψ̃ : H → Ď = ΛG/Λ+ G by
ψ̃(z) := exp(−zN ) · φ̃(z).
By construction, this map is invariant by z 7→ z + 1. We denote by ψ the
1
corresponding map from ∆∗ to Ď, defined by ψ(w) = ψ̃( 2iπ
log w). We now
state the nilpotent orbit theorem; this is theorem E in the introduction.
65
Theorem 1.4.5. We make the assumptions 1.4.1 and 1.4.2. The map ψ extends holomorphically to ∆. Writing a = ψ(0) in Ď, there exist positive constants α, β such that, for Im z > α, the map exp(zN · a) takes its values in D
and the inequality
dD (exp(zN ) · a, φ̃(z)) ≤ (Im z)β exp(−2π Im z)
holds, for some Λσ G-invariant Riemannian metric on D, defined in subsection
1.4.2. Moreover, the map z 7→ exp(zN · a) is horizontal.
Our proof follows closely the proof of [Sch73], for period domains coming
from Z-variations of classical Hodge structures.
In order to obtain informations about the asymptotic of the harmonic bundle
we started with, we need to understand the evaluation at λ = 1 of the map
exp(zN ) · a, which is well-defined for Im(z) sufficiently large. This will certainly
need an analogue of the SL(2)-orbit theorem of [Sch73]. For the moment, we
just write exp(zN ) · a = γ(z)K, where γ(z) is defined for Im(z) > α and lies in
Λσ G. We define γ1 (z) in G by γ1 = ev1 ◦ γ. Then, writing φ1 : X̃ → G/K for
the developing map of the harmonic bundle (E, D, h):
Corollary 1.4.6. There exist positive constants α, β and an element g ∈ G
such that the map z 7→ γ1 (z)K, defined for Im z > α is pluriharmonic and
satisfies:
dG/K (γ1 (z)K, φ1 (z)) ≤ (Im z)β exp(−2π Im z).
In this statement, G/K is endowed with a G-invariant Riemannian metric.
Proof. By proposition 1.1.54, φ1 = ev1 ◦ φ̃, where ev1 : Λσ G/K → G/K is the
evaluation at 1. It is thus enough to show that ev1 decreases the distances,
up to some constant. Since the metrics on Λσ G/K and G/K are respectively
Λσ G and G-invariant and since ev1 (γ.x) = γ(1).ev1 (x), for γ in Λσ G and x in
Λσ G/K, we need an estimate on a single point x of the period domain. Since
ev1 is smooth, its differential dx ev1 is a bounded linear map; this concludes the
proof.
1.4.2
Preliminaries
As usual, we choose a regularity H s for loops in ΛG. We recall that the period
domain is the homogeneous space D = Λσ G/K and that it naturally embeds as
an open set in its compact dual, which is the homogeneous space Ď = ΛG/Λ+ G.
We define two Riemannian metrics on these spaces.
The metric on Ď.
Consider a point x in Ď, and write πx for the orbital map πx : ΛG → Ď, given
by πx (g) = g · x. We write sx for the Lie subalgebra, in Λg, of the stabilizer of
x for the ΛG-action. By the orbital map πx , one has a canonical identification:
∼ Λg/sx .
Tx Ď =
66
On Λg, we have defined a H s -metric. Hence, the quotient Λg/sx can be
identified with the orthogonal (sx )⊥ in Λg. Via the identification Tx Ď ∼
= (sx )⊥ ,
we obtain a Hermitian inner product on Tx Ď. By definition, the norm of a vector
X in Tx Ď is the minimal norm of Y in Λg = Te (ΛG) such that de πx (Y ) = X.
Proposition 1.4.7. This construction gives a structure of complex HilbertRiemannian manifold to Ď.
Proof. The only thing to check is that the metric depends smoothly on the
point in Ď. We refer to [Lan96], chapter VII, for details about Hilbert bundles.
Consider the trivial Hilbert vector bundle Ď × Λg on Ď. Let x be some point in
Ď. On a small neighborhood of x, there exists a smooth section g of the orbital
map πx , so that y = g(y)x. Moreover, the Lie subalgebras of stabilizers at the
point x and at the point y = g(y)x are related by
sg(y)x = g(y)sx g −1 (y).
Hence, we have a Hilbert subbundle s of Ď × Λg, given over a point y by
the Lie subalgebra of the stabilizer of y in ΛG. The orbital maps give an exact
sequence
0 → s → Λg → T Ď → 0
and the orthogonal of s for the metric on Λg gives a splitting of this exact
sequence (see [Lan96],VII,3.). Hence, we get an isomorphism of Hilbert bundles
(s)⊥ ∼
= T Ď and this shows that the metric defined above depends smoothly on
the point in Ď.
This metric will be simply referred as the metric on Ď. The norm relative
to it will be written k · kĎ . This metric is not invariant under the action of ΛG.
Nevertheless, the next proposition controls the distortion of the metric by its
action (compare with lemma 8.12 in [Sch73]).
Proposition 1.4.8. Let g be in ΛG; if x is in Ď, we write dx Lg for the induced
infinitesimal translation from Tx Ď to Tg·x Ď. Then, the operator norm of dx Lg ,
relative to the metric on Ď, is bounded by the operator norm of Ad(g) acting
on Λg.
Proof. Write y = g · x and ιg for the conjugation map from ΛG to ΛG given
by ιg (h) = ghg −1 . We also recall the notations πx and πy for the orbital maps
centered in x and y. Then, if h is in ΛG, we have the equality
g · (πx (h)) = πy ◦ ιg (h).
Infinitesimally at the identity of ΛG, this gives
dx Lg ◦ de πx (Y ) = de πy ◦ Ad(g)(Y ),
with Y in Λg.
Hence, if Z is in Tx Ď and Y in Λg lifts Z, then Ad(g)(Y ) lifts dx Lg (Z). In
particular, the norm of dx Lg (Z) is less than the operator norm of Ad(g) acting
on Λg, times the norm of Y . Since the norm of Z is by definition the minimal
norm of such Y , this concludes the proof.
67
Remark 1.4.9. This construction of a metric in a homogeneous space is completely general but I have not found any reference to it in the literature. One
should also notice that this is not the metric that is used in [Sch73]. Indeed, for
classical period domains, one has another homogeneous representation M/V of
the compact dual and since the stabilizer group V is compact, one can define
a M -invariant metric on the compact dual. Nevertheless, the M -invariance of
the metric is never used, except to establish the result analogous to proposition
1.4.8.
In our situation, we also have another homogeneous representation for the
compact dual. Namely, Ď ∼
= ΛK/K (cf. [PS88]); since K acts isometrically for
s
the H -inner product, we can define a corresponding ΛK-invariant metric on
Ď. But it is not at all clear whether proposition 1.4.8 is true or not for this
metric. The problem is that the group ΛK does act isometrically for a L2 -inner
product on Λg, but not for the H s -inner product.
The metric on D.
The holomorphic tangent space of D is isomorphic to Λσ G ×K Λ− g. The space
Λ− g inherits a Hermitian inner product from Λg, which is Ad(K)-invariant.
Hence, we get a Λσ G-invariant metric on D. This metric will be simply called
the metric on D; the norm will be denoted by k · kD .
Comparing the definitions on the metrics on Ď and D, we remark that they
coincide at the base-point eK of D, which is identified to the base-point eΛ+ G
of Ď. These base-points will be simply written o in the following.
Let g be an element in Λσ G. Then, infinitesimal translation on D by g is an
isometry for the metric on D and proposition 1.4.8 controls its operator norm
for the metric on Ď. Hence,
Proposition 1.4.10 (Corollary 8.13 in [Sch73]). Let x = g · o be some point in
D, with g in Λσ G. Let X be in Tx Ď. Then,
−1
k Ad(g)k−1
Λg kXkĎ ≤ kXkD ≤ k Ad(g )kΛg kXkĎ .
Riemannian distances on D and Ď.
As in a finite-dimensional setting, these Riemannian structures induce distances
on Ď and D. They will respectively be called the distance on Ď and the distance
on D and denoted by dĎ (·, ·) and dD (·, ·). The induced topologies coincide with
the manifold topologies (proposition 6.1 in [Lan96]). Since the manifolds are
modeled on Hilbert spaces, the metric spaces (Ď, dĎ ) and (D, dD ) are locally
complete. Since Λσ G acts transitively and isometrically on D, (D, dD ) is a
complete metric space. For Ď, I ask the following question:
Question 1.4.11. Is (Ď, dĎ ) a complete metric space?
Remark 1.4.12. If one uses the analogous metric in the compact dual of
classical Hodge theory, then the answer is affirmative since the compact dual is
compact! It is reasonable that the answer is still affirmative in this setting but
I do not know how to replace the compactness argument.
68
Proposition 1.4.8 has a global analogue, which I state for future reference.
Proposition 1.4.13. Let x and y be in Ď and let g be in ΛG. Then
dĎ (g · x, g · y) ≤ k Ad(g)kΛg dĎ (x, y).
Estimates with holomorphic functions
We will need to estimate the H s norms of some loops in the proof of the theorem.
Since our loops are restrictions of holomorphic maps from C∗ , this will be
facilitated by the following lemma.
Proposition 1.4.14. Let D be a compact subset of C which contains the circle
S 1 in its interior. Let f be a complex-valued continuous function on D, holomorphic in the interior of D. Writing kf kH s for the H s -norm of the restriction
of f to S 1 and kf k∞ for its supremum norm on D, one has:
kf kH s ≤ K0 kf k∞ ,
where K0 is a constant depending on D and s, but not on f .
Proof. Let be a positive number such that, for any w in S 1 , the disk of radius
centered in w lies in the interior of D. Such exists by compactness of S 1 .
By Cauchy’s differentiation formula, one has:
f
(n)
n!
(w) =
2πi
Z
w+S 1
f (z)
dz.
(z − w)n+1
Hence,
n!kf k∞
.
n
Since this is true for any n, one can in particular estimate the H s norm of f ,
proving the proposition.
|f (n) (w)| ≤
We will also use lemma (8.17) of [Sch73] in a Hilbert setting. The proof
is identical since Cauchy’s integral formula is valid for holomorphic functions
with values in a Hilbert space (see for instance theorem 2.6.3 in [Neu03]).
Lemma 1.4.15. (Lemma 8.17, [Sch73]) Let η be a positive real number. Let f
be a bounded holomorphic function on the strip | Im u| < η, periodic with period
1, with values in a Hilbert space H. Then,
kf 0 (0)kH ≤ π(sinh πη)−2 kf k∞,H .
Distance-decreasing properties
By corollary 1.3.7, if we have a bound on the Higgs field θ, then we have a
bound on the differential of f , as an operator from T X̃ to Λσ g/Λ0,X
σ g. The
0,X
Hilbert metric on Λσ g/Λσ g was defined before the corollary and it is easy to
show that it is Λσ G-invariant. Since at the base-point, it coincides with the
metric defined above in this section, they coincide everywhere. Hence,
69
Proposition 1.4.16. Under assumption 1.4.1 , the period map f : H → D
decreases the distances. More precisely, there exists a positive constant C such
that, for every x, y ∈ H, the inequality
dD (f (x), f (y)) ≤ CdH (x, y)
holds.
We will also need an analogous result for the map that goes to the symmetric
space G/K, which we endow with a G-invariant Riemmanian metric.
Proposition 1.4.17. Let (E, D, h) be a harmonic bundle. For any λ in C∗ , let
fλ : X̃ → G/K be the developing map of the metric h, on the flat bundle
(E, Dλ := ∇ + λ−1 θ + λθ∗ ).
Then, under assumption 1.4.1, the differential of dfλ is bounded. Moreover, the
bound is uniform if λ is contained in a compact subset of C∗ .
Proof. We can assume that X = X̃. We consider a flat vector bundle (E, D)
with a metric h. There is a unique decomposition D = ∇ + α, where ∇ is
a metric connection and α is a 1-form with values in Endh (E), the bundle of
h-hermitian endomorphisms of E. Under parallel transport relative to D, this
bundle is equal to the pullback of the tangent bundle of G/K. Moreover, we
can endow G/K with a G-invariant Riemannian metric. This induces a metric
on the bundle Endh (E). Up to some uniform bounds, this metric is simply the
restriction of the metric on End(E), induced by the metric h on E.
By the following lemma 1.4.18, the norm of the differential of f is equal to
the norm of α in Endh (E), up to some constant factor. In order to conclude,
we have to identify α for the various connections Dλ . We can write
Dλ = (∇ +
(λ−1 − λ)θ + (λ − λ−1 )θ∗
(λ−1 + λ)θ + (λ + λ−1 )θ∗
)+
.
2
2
The first term is a metric connection and the second term is a Hermitian
form. If θ is bounded, the second term is bounded, uniformly in λ, when λ lives
in a compact subset of C∗ . This concludes the proof.
Lemma 1.4.18. The notations are as in the proof of the proposition 1.4.17.
Then, the norm of the differential df is equal to a constant times the norm of
α, under the isomorphisms defined above.
Proof. We can assume that (E, D) = (Cn , d) the trivial flat bundle. We write
H for the matrix of h in the canonical basis of Cn , so that f = H when
G/K is seen as the set of positive definite Hermitian matrices. The equality d = (d + 21 H −1 dH) − 12 H −1 dH holds and one checks easily that the first
term is a metric connection and the second term is a h-Hermitian 1-form.
Hence, α = − 21 H −1 dH. The square norm of α in Endh (Cn ) is given by
1
−1 (dH)H −1 (dH)) since H −1 dH is a h-Hermitian matrix.
4 Tr(H
On the other hand, df = dH. An invariant metric on G/K is given by
< u, v >K = Tr(K −1 uK −1 v), where, K is a positive definite Hermitian matrix
70
and u and v are Hermitian matrices. Indeed, G acts on the space of positive
definite Hermitian matrices by g · K = g ∗ Kg so that
< g∗ u, g∗ v >g·K = Tr(g −1 K −1 g ∗ −1 g ∗ ugg −1 K −1 g ∗,−1 g ∗ vg)
= Tr(K −1 uK −1 v)
=< u, v >K
Hence, the square norm of df at the point H is given by Tr(H −1 (dH)H −1 (dH)).
This concludes the proof.
1.4.3
Proof of theorem 1.4.5
We consider the map ψ̃(z) = exp(−zN ) · φ̃(z). The idea of the proof is to show
that the behaviour of this map when Im z tends to infinity is nice, so that the
corresponding map ψ can be extended on the puncture of ∆. With this aim in
view, we estimate the derivative of this map, when Im z tends to infinity. We
define the vector Xz that lives in Tφ̃(z) D by
Xz := dz φ̃(∂z ) − N (φ̃(z)),
(1.4)
∂
where ∂z is the holomorphic vector field ∂z
over H and N denotes the vector
field on Ď obtained from infinitesimal translation by N . Then, we compute:
dz ψ̃(∂z ) = dφ̃(z) L(exp(−zN )).Xz .
(1.5)
We first need to control the vector field Xz . Let z be in H, let g(z) be an
element of Λσ G such that φ̃(z) = g(z) · o and consider the auxiliary function Fz
defined from H − z to Ď, by
Fz (u) = g(z)−1 exp(−uN ).φ̃(z + u)
−1
= exp(−u Ad g(z)
−1
N )g(z)
(1.6)
· φ̃(z + u).
(1.7)
Since the derivative of Fz at 0 is given by
d0 Fz (∂z ) = dφ̃(z) L(g(z)−1 ).Xz ,
(1.8)
estimates on Fz will give estimates on Xz , and finally on dz ψ̃(∂z ).
Study of Fz
First, we estimate the norm of Ad g(z)−1 N .
Proposition 1.4.19. (Lemma 8.14, [Sch73]) There exists positive constants
α, β such that, if Im z ≥ α, then
k Ad g(z)−1 N kΛg ≤ β(Im z)−1 .
71
Proof. Let D be any compact subset of C∗ , containing S 1 in its interior, for
instance an annulus thickening S 1 . Since g(z) and N are restriction of holomorphic maps defined on C∗ , we can evaluate these loops at any λ in D. For
any such λ, we write φ̃λ for the (non-necessarily pluriharmonic) developing map
from H to G/K, of the metric h with respect to the flat bundle (E, Dλ ), as in
subsection 1.1.2.
Writing evλ : Λhσ G/K → G/K for the evaluation map at λ, I recall from (a
generalization of) proposition 1.1.54 that φ̃λ = evλ ◦ φ̃. We endow the upperhalf plane with its hyperbolic distance and the symmetric space G/K with a
G-invariant distance dG/K . By assumption 1.4.1 and proposition 1.4.17, φ̃λ
decreases the distances up to some constant Cλ . Moreover, for λ varying in a
compact subset of C, the constant can be made uniform in λ; up to a rescaling
of the distance in G/K, we can assume that this constant is one. Hence,
(Im z)−1 ≥ dG/K (φ̃λ (z + 1), φ̃λ (z))
= dG/K (exp Nλ · φ̃λ (z), φ̃λ (z))
= dG/K (exp Nλ gλ (z)K, gλ (z)K)
= dG/K (exp(Ad gλ (z)−1 Nλ )K, eK).
Since Ad g(z)−1 Nλ is nilpotent, there exists a unitary matrix k = k(λ, z)
such that Ad(kg(z)−1 )Nλ lies in the fixed nilpotent subalgebra u of strictly
triangular matrices. Since
dG/K (exp(Ad gλ (z)−1 Nλ )K, eK) = dG/K (exp(Ad(kgλ (z)−1 )Nλ )K, eK),
this shows that
dG/K (exp(Ad(kgλ (z)−1 )Nλ )K, eK) ≤ (Im z)−1 .
(1.9)
Recall the Iwasawa decomposition of G: writing A for the subgroup of
diagonal matrices with real positive coefficients, the map
A × u → G/K, (a, X) 7→ a(exp X)K
is a diffeomorphism. In particular, it is locally bounded with respect to any
metric on A × u and G/K, in small neighborhoods of (e, 0) and eK. By inequality (1.9), for Im z sufficiently large, exp(Ad(kgλ (z)−1 )Nλ )K is in such
a neighborhood of eK. Hence, there exist constants C and α such that for
Im z > α,
k exp(Ad(kgλ (z)−1 )Nλ )kg ≤ C(Im z)−1 .
Since Ad(k) acts isometrically on g, this gives the same estimate for
k Ad gλ (z)−1 Nλ )kg .
By (a vector-valued version of) proposition 1.4.14, this gives the estimate
k Ad g(z)−1 N kΛg ≤ β(Im z)−1 ,
with β = K0 × C.
72
Let P be a neighborhood of o in D. By proposition 1.4.19, we obtain the
following lemma.
Lemma 1.4.20. (Lemmas (8.15) and (8.16), [Sch73]) There exist positive constants α and ζ (independent of z) such that Fz (u) belongs to P, whenever
Im z > α and | Im u| < ζ Im z.
I omit the proof of this lemma since it is identical to the proof in [Sch73],
with m = 1. Beware of the typo in the proof of Lemma (8.16): the first two
occurences of (Im z)−1 have to be replaced by Im z.
By shrinking P, we can assume that it corresponds to an open set contained
in the ball of radius R centered in 0, in a Hilbert space H, via some coordinate
chart. Moreover, we can also assume that the Hilbert metric on H and the
metric on Ď restricted to P are mutually bounded. Since Fz is periodic of
period 1, we can apply lemma 1.4.15 with η = ζ Im z. This gives:
kd0 Fz (∂z )kH ≤ π(sinh πζ Im z)−2 R.
Since the metrics on H and the metric on Ď are mutually bounded, we obtain
that there is a constant R0 such that
kd0 Fz (∂z )kĎ ≤ R0 exp(−2πζ Im z).
Since the constant R0 is proportional to R, we can shrink P again in order
that the constant R0 becomes 1. Hence,
Proposition 1.4.21. There exist positive constants α and such that, if Im z >
α, then
kd0 Fz (∂z )kĎ ≤ exp(− Im z).
Study of Xz
By equation (1.8) and propositions 1.4.21 and 1.4.8, an estimate of the operator
norm of Ad g(z) acting on Λg will give an estimate for the norm of Xz in Ď.
Proposition 1.4.22. (Lemma (8.19),[Sch73]) Let α > 1 be given. Then, there
exist positive constants C and β such that, for Im z > α and | Re z| ≤ 1, the
operator norms of Ad g(z) and Ad g(z)−1 are bounded by C(Im z)β .
Proof. As in the proof of proposition 1.4.19, let D denote a compact subset
of C containing the circle in its interior. For any λ in D, we write gλ (z) for
the evaluation of g(z) at λ. Using the KAK decomposition of G, we can
write gλ (z) = k1 exp(Y )k2 , with k1 , k2 are in K and Y is a matrix in the Lie
subalgebra a of diagonal matrices with real entries.
We recall that g is endowed with its natural Hermitian inner product. Moreover, G can be viewed as a subset of g and G acts naturally on g by left or right
multiplication. For these actions, K acts isometrically. Writing kgλ (z)kg for the
norm of gλ (z) via the inclusion G ⊂ g, one thus have kgλ (z)kg = k exp(Y )kg .
73
Moreover, the map a 3 D → exp(D)K ∈ G/K is known to be a global
isometric embedding, up to some normalization. Hence, there is a constant η
such that
kY kg = ηd(exp(Y )K, eK)
= ηd(gλ (z)K, eK)
≤ ηd(φλ (z), φλ (i)) + ηd(φλ (i), eK).
In the last sum, the right summand in bounded by a constant independent of λ
since λ lives in a compact set D. By proposition 1.4.17, the left summand is less
than the hyperbolic distance between z and i, up to some constant independent
of λ. Hence, we get an inequality of the kind
kY kg ≤ A(log Im z + | Re z| + A2 ).
With the restriction | Re z| ≤ 1, exponentiation gives
exp(kY kg ) ≤ A3 (Im z)β .
Moreover, k exp(Y )kg ≤ A4 exp(A5 kY kg ) (because one can take A4 = A5 =
1 for the corresponding inequalities with a norm algebra and all norms in g are
equivalent). Hence, we have an inequality
kgλ (z)kg ≤ A6 (Im z)β2 .
By proposition 1.4.14, we get a similar inequality kg(z)kΛg ≤ A7 (Im z)β2 ,
where the norm is taken with respect to the embedding ΛG ,→ Λg. Moreover,
we get the same inequality for kg(z)−1 k.
Our aim was to obtain a similar bound for the operator norm of Ad g(z)
and Ad g(z)−1 , but these operators are just given by multiplication with g(z)
and g(z)−1 . Hence the bounds on g(z) and g(z)−1 give that
k Ad g(z)kΛg ≤ C(Im z)β ,
for some constants C and β and the same bound holds for Ad g(z)−1 .
Corollary 1.4.23. There exist positive constants α, such that, if Im z > α,
| Re z| ≤ 1, then
kXz kĎ ≤ exp(− Im z).
Proof. We first take α and as in proposition 1.4.21, so that if Im z > α, then
kd0 Fz (∂z )kĎ ≤ exp(− Im z).
Since, Xz = do L(g(z)).d0 Fz (∂z ), by proposition 1.4.8 and proposition 1.4.22,
we get
kXz kĎ ≤ C(Im z)β exp(− Im z),
if Im z > max(α, 1) and | Re z| ≤ 1. By increasing α and decreasing , one can
absorb the polynomial factor, concluding the proof.
74
End of the proof
By equation (1.5), proposition 1.4.8 and corollary 1.4.23, we obtain that
kdz ψ̃(∂z )kĎ ≤ k Ad exp(−zN )kΛg exp(− Im z),
for Im z > α, | Re z| ≤ 1, where α and are some positive constants. Since N is
nilpotent, exp(zN ) is polynomial in z and in the range | Re z| ≤ 1, the norm of
Ad exp(−zN ) can be bounded by a polynomial in Im z. Hence, by increasing α
and decreasing , one can absorb this polynomial. This gives
Corollary 1.4.24. There exist positive constants α and such that, for Im z >
α,
kdz ψ̃(∂z )kĎ ≤ exp(− Im z).
Indeed, the restriction | Re z| ≤ 1 becomes unnecessary since both terms in
the inequality are invariant under the translation z 7→ z + 1.
Let β be greater than α and let z1 and z2 be such that Im zi ≥ β. By
integrating the estimate in corollary 1.4.24 and using again the invariance of ψ̃,
we obtain
dĎ (ψ̃(z1 ), ψ̃(z2 )) ≤ (1 + −1 ) exp(−β) =: C exp(−β).
(1.10)
We claim that this implies that ψ̃(z) has a limit in Ď when Im z tends to
infinity. Beware that one cannot directly conclude since we do not know if Ď
is complete for dĎ .
We write as before g(z) for an element in Λσ G such that φ̃(z) = g(z).o.
Then ψ̃(z) = exp(−zN )g(z).o. Using proposition 1.4.13 and the estimates
on Ad g(z)−1 and Ad(exp(zN )), the inequality (1.10) shows that there exist
positive constants α, C, A, such that
dĎ (g(z2 )−1 exp(z2 N )ψ̃(z1 ), o) ≤ C(Im z2 )A exp(−β)
holds if β ≥ α, Im z1 , Im z2 ≥ β and | Re z2 | ≤ 1. In particular, if Im z1 ≥ Im z2 ,
we can take β = Im z2 .
Let W a complete neighborhood of o in the metric space (Ď, dĎ ). Then, if
Im z2 is sufficiently large, the last inequality shows that the points
g(z2 )−1 exp(z2 N )ψ̃(z1 )
are contained in W for Im z1 ≥ Im z2 . Fixing such z2 and choosing a sequence zn that tends to infinity, the previous estimates show moreover that
g(z2 )−1 exp(z2 N )ψ̃(zn ) are Cauchy sequences. Hence, these sequences converge;
translating back by exp(−z2 N )g(z2 ), this shows that ψ̃(zn ) converges in Ď for
any sequence zn tending to infinity.
In term of the function ψ : ∆∗ → Ď, this gives:
Proposition 1.4.25. The map ψ extends continuously, hence holomorphically,
to ∆.
1.5. The determinant line bundle
75
Proof. For the holomorphic statement, one uses Riemann’s extension theorem
which is valid for holomorphic functions with values in a Hilbert space. Indeed,
the proof of this fact only uses that holomorphic functions are complex-analytic
(see for instance [Neu03]).
We write a := ψ(0). The fact that ψ extends holomorphically gives a better
estimate for the constant appearing in the equations above. Indeed, since
ψ is continuously differentiable, it is locally bounded in a neighborhood of 0.
Hence, there exist constants C, ρ, such that if |w| ≤ ρ, then dĎ (a, ψ(w)) ≤ C|w|.
Returning to the universal cover, this gives
dĎ (a, ψ̃(z)) ≤ C exp(−2π Im z),
if Im z ≥ α, where α is some constant. Translating things by g(z)−1 exp(−zN ),
we get
dĎ (g(z)−1 exp(−zN ) · a, o) ≤ C2 (Im z)β exp(−2π Im z),
assuming Im z ≥ α and | Re z| ≤ 1. This shows in particular that if Im z is
sufficiently large (and | Re z| ≤ 1), then g(z)−1 exp(−zN ) · a is in the period
domain D = Λσ G/K. We can moreover assume that for such z, the points
g(z)−1 exp(−zN ) · a are in a neighborhood of o where the distances dD and dĎ
are mutually bounded. Hence,
dD (g(z)−1 exp(zN ) · a, o) ≤ C3 (Im z)β exp(−2π Im z).
Since g(z) lives in Λσ G, it acts isometrically for dD . Hence,
Proposition 1.4.26. There exist positive constants α and β such that
˜ ≤ (Im z)β exp(−2π Im z).
dD (exp(zN ) · a, φ(z))
Indeed, by increasing α and β, the constant C2 can be absorbed and the
restriction | Re z| ≤ 1 becomes unnecessary by invariance of the inequality under
z 7→ z + 1.
In order to finish the proof of theorem 1.4.5, it remains to prove the horizontality of the map z 7→ exp(zN ) · a. The proof of this fact is identical to the
end of the proof in [Sch73] and I omit it.
1.5
The determinant line bundle
In this section, we define a holomorphic line bundle over the loop period domain
D. This bundle is not equivariant with respect to Λσ G but we can construct
a central extension of this group for which the bundle is equivariant. Via the
period map, this line bundle is related in a rather subtle way to some geometrical
objects that we can associate to any harmonic bundle. This will lead us to a
discussion of the Carlson-Toledo conjecture.
The results here are based on a joint work with Bruno Klingler.
76
1.5.1
Central extensions
We want to define a central extension by the circle S 1 of the group Λσ G. The
analogous problem for ΛK is discussed in detail in [PS88], chapters 4 et 6, from
which we collect some results.
Let H be a (Banach-)Lie group and let H̃ be a central extension of H by
S 1 . As a vector space, the Lie algebra h̃ can be written h̃ = h ⊕ iR and the Lie
bracket is given by
[(ξ, λ), (η, µ)] = ([ξ, η], ω(ξ, η)),
where ω is a skew-symmetric map from h × h to iR satisfying the cocycle condition
ω([ξ, η], ζ) + ω([ζ, η], ξ) + ω([η, ξ], ζ) = 0.
Conversely, let ω be such a map. Then, ω defines a H-invariant 2-form
on H (that we still denote ω), and the cocycle condition is equivalent to the
closedness of this form.
Proposition 1.5.1. (Corollary of proposition 4.4.2 in [PS88]) If H is simply
connected, then a cocycle ω is obtained from a central extension of H by S 1 if
i
and only if the cohomology class of the 2-form 2π
ω is integral.
In order to define a central extension by S 1 of Λσ G, a first possibility is thus
to define a cocycle on the Lie algebra Λσ g. We write < X, Y >= − Tr(XY ) in
g.
Definition 1.5.2. The fundamental cocycle on Λg is defined to be the skewsymmetric map with values in C defined by
ω(ξ, η) : =
i
2π
Z 2π
< ξ(θ), η 0 (θ) > dθ.
0
Remark 1.5.3. The skew-symmetry follows from an integration by parts. The
cocycle condition is proved in section 4.2 of [PS88], for the Lie algebra Λk and
we conclude by complex linearity.
Proposition 1.5.4. The restriction of ω to Λk and Λσ g takes its values in iR.
Proof. If ξ is in Λk, then ξ(θ)∗ = −ξ(θ) and if η is in Λσ g, then η(θ)∗ = −η(−θ).
For i = 1, 2, we take ξi in Λk. Then
−2iπω(ξ1 , ξ2 ) =
Z 2π
0
Z 2π
=
0
Z 2π
=
0
< ξ1 (θ), ξ20 (θ) > dθ
< t ξ1 (θ), t ξ20 (θ) > dθ
< ξ1 (θ), ξ20 (θ) > dθ
= −2iπω(ξ1 , ξ2 ).
The third line is obtained from the properties of the trace operator.
The computation is similar for Λσ g, since the measure dθ is invariant under
θ 7→ −θ.
77
The group SU (n) is simply-connected and this implies that ΛSU (n) is
simply-connected. By proposition 1.5.1, in order to define the central extension
^
ΛSU
(n), it is sufficient to show that the 2-form induced by the cocycle ω (restricted to Λsu(n)) has its cohomology class in H 2 (ΛSU (n), 2iπZ). This is done
with propositions 4.4.4 and 4.4.5. Then, one can define the central extension
g of ΛK; this is section 4.7 of [PS88].
^
ΛU
(n) = ΛK
For the group Λσ G, the construction of an analogue of the transgression
map used in propositions 4.4.4 and 4.4.5 is not clear and we will define the
central extension of Λσ G in another way.
A central extension by C∗ of ΛG is defined in section 6.7 of [PS88]. This
^
is done by pullback of a central extension C∗ → GL
res → GLres , where GLres
is some subgroup of the group of linear automorphisms of the Hilbert space
L2 (S 1 , Cn ), so that we have an embedding ΛG ,→ GLres . The construction of
the central extension of GLres is done in section 6.6. After proposition 6.6.5,
it is remarked that the same construction can be applied in order to obtain
a S 1 -extension of the group Ures which is the subgroup of GLres of isometries
of the Hilbert space L2 (S 1 , Cn ). Since, the group ΛK embeds in Ures , this
gives a central extension by S 1 of ΛK. In proposition 6.7.1, it is proved that
this extension has the cocycle ω defined above; hence this gives a concrete
g
description of ΛK.
Considering the space L2 (S 1 , Cn ) as a Krein space, we can do the same construction with the Krein isometry group instead of the Hilbert isometry group.
The only point to check, which is implicit in section 6.6, is that an operator of
determinant class which is unitary (for either the Hilbert or Krein metric) has
determinant of modulus one. This is proved as in the finite-dimensional setting,
using that if M has a determinant and if X is invertible, then XM X −1 has a
determinant too, equal to det(M ).
A third possibility to define the central extension of Λσ G is less natural but
easier, since it does not involve the description of the central extension by C∗
of ΛG. Let 1 → C∗ → E → Λσ G → 1 be the pullback of the central extension
by C∗ of ΛG. The group C∗ decomposes as C∗ = S 1 × R∗+ ; we can quotient the
extension E by R∗+ , giving a central extension
1 → S 1 → Λg
σ G → Λσ G → 1.
g → LG → 1 is the fundamental coThe cocycle of the extension 1 → C∗ → LG
cycle of definition 1.5.2. This is essentially proposition 6.7.1 of [PS88]. Hence,
the cocycle of Λg
σ G is the restriction to the fundamental cocycle to Λσ g, followed
by projection onto iR (with respect to the decomposition C = iR ⊕ R). From
proposition 1.5.4, the fundamental cocycle restricted to Λσ g already takes its
values in iR.
We sum up this discussion with the following proposition:
78
Proposition 1.5.5. There is a central extension
1 → S 1 → Λg
σ G → Λσ G → 1,
whose cocycle is given by definition 1.5.2.
Remark 1.5.6. The restriction of Λg
σ G over the subgroup K of Λσ G is trivial;
indeed, the fundamental cocycle vanishes on k ⊗ k. In the following, we write
K̃ for K × S 1 .
1.5.2
Curvature
Another homogeneous representation of the loop period domain is given by
0
g
D = Λg
σ G/K̃. At the level of Lie algebras, we can write Λσ g = k̃ ⊕ Λσ g, where
Λ0σ g is the subspace of Λσ g of functions whose zeroth Fourier coefficient vanishes.
A Λg
σ G-equivariant complex line bundle on D is obtained from a character of
1
S × K.
Definition 1.5.7. The determinant line bundle over D is the complex line
bundle given by the character (k, λ) 7→ λ of K̃.
As a homogeneous space under Λg
σ G, the loop period domain is a reductive
0
space, thanks to the decomposition Λg
σ g = k̃ ⊕ Λσ g of its Lie algebra. For
such spaces, one can define a canonical connection for the principal K̃-bundle
f G → D. Let ω
g
Λ
σ
M C be the Maurer-Cartan form of Λσ G. Then, the curvature
of the canonical connection is given by
1 Λ0
Λ0 g
Ω = − [ωMσC g, ωMσC ]k̃ ,
2
(1.11)
where the Lie algebras in exponents stand for the projection on these Lie algebras, relative to the reductive decomposition given above. (see [CMSP03],
equation 12.3.2).
The determinant line bundle over D (resp. Ď) is endowed with a Λg
σ Ginvariant metric since the character of K̃ which induces this bundle is unitary.
We write ∇ for the connection on the determinant line bundle induced by the
canonical connection. Then, by [CMSP03], theorem 12.3.5,
Proposition 1.5.8. The determinant line bundle over D admits a unique structure of holomorphic line bundle such that ∇ is the Chern connection.
We can compute the curvature of the connection ∇ and in particular prove
the following result:
Proposition 1.5.9. The connection ∇ has positive curvature in the horizontal
directions.
Proof. Since the determinant line bundle and the connection ∇ are invariant
under Λg
σ G, we only consider the base-point of D. Over this point, the curvature
79
of ∇D is a skew-symmetric map β from Λ0σ g⊗Λ0σ g with values in End(C). From
equation (1.11), it is given by
1
β(ξ, η)w = − [ξ, η]k̃ · w,
2
where w ∈ C, ξ, η ∈ Λ0σ g and the dot is the infinitesimal action of the character
of K̃ defined above. In the decomposition k̃ = k⊕iR, only the second component
acts non-trivially on C. From the description of the Lie algebra Λ̃k, this gives:
1
β(ξ, η)w = − ω(ξ, η)w.
2
(1.12)
We write I for the almost-complex structure on Λ0σ g. If f is in Λ0σ g, we
write
X
f (λ) =
an λ n .
n6=0
We recall that an = (−1)n+1 a∗n and that I.f is given by
(I.f )(λ) = i
X
an λ n − i
X
an λn .
n>0
n<0
Hence, as a function of θ, the derivative of I.f is given by
(I.f )0 (λ) = −
X
nan λn +
n<0
X
nan λn .
n>0
Since < X, Y >= − Tr(XY ), inverting the sum and the integral, we get:
iβ(f, I.f ) = −
=
X
2π X
− Tr(−nan a−n ))
− Tr(nan a−n ) +
(
4π n<0
n>0
X
n(−1)n+1 Tr(an a∗n )
n<0
Since a function in the horizontal distribution has only Fourier coefficients
in degrees 1 and −1, this proves the proposition.
1.5.3
Hitchin energy
Let (E, D, h) be a harmonic bundle over a complex manifold X whose associated
Higgs field is denoted by θ.
1
Definition 1.5.10. The Hitchin energy is the 2-form βX := 4iπ
Tr(θ ∧ θ∗ ).
∗
Here, Tr is the trace operator on End(E) and θ is the adjoint of θ for the
harmonic metric h.
Proposition 1.5.11. The Hitchin energy βX is closed.
80
Proof. We recall the following: if E is a vector bundle over X and α is a k-form
with values in End(E), then for any connection ∇ on E:
d Tr(α) = Tr(∇α).
Using this formula, the differential of the Hitchin energy is given by
4iπdβX = Tr(∇θ ∧ θ∗ − θ ∧ ∇θ∗ ),
where ∇ = D − (θ + θ∗ ) is the canonical metric connection of the harmonic
bundle E. By (the proof of) lemma 1.1.19, ∇θ = ∇θ∗ = 0. This concludes the
proof.
We write βD for the curvature of the determinant line bundle. Let βX̃ be
the pullback of the Hitchin energy to X̃ and let f : X̃ → D be the period map.
Proposition 1.5.12. The relation βX̃ =
i ∗
2π f (βD )
holds.
Proof. We use the notations of subsection 1.3.1. We have defined an isomorphism γx : Ex → Hf0(x) . This induces an isomorphism from Λσ−1,1 End(Ex ) to
−1
∗
Λ−1,1
σ,f (x) g. Under this isomorphism, the 1-form α = λ θ + λθ corresponds to
¯ . If X and Y are vector fields of type (1, 0) and (0, 1),
the 1-form λ−1 ∂f + λ∂f
we thus get
Tr(θ(X)θ∗ (Y ))x = Tr(dx f (X)dx f (Y ))f (x) ,
since the trace of two conjugated endomorphisms is the same. On the other
hand, by definition 1.5.2 and equation (1.12), the curvature β is given at the
base-point o of D by
1
βD,o (X, Y ) = − Tr(XY ),
2
where X is of type (1, 0) with values in Λ−1 g and Y of type (0, 1) with values in
Λ1 g. Here, X and Y are seen as endomorphisms on the vector space Cn , which
is isomorphic to Ho0 . Since the curvature is invariant under Λσ G, the formula
1
βD,y (X, Y ) = − Tr(XY )
2
1
holds, where X and Y are now with values in Λ−1
y g and Λy g and are seen as
0
endomorphism of Hy . We thus get that
Tr(θ(X)θ∗ (Y ))x = −2βD,f (x) (dx f (X), dx f (Y )).
In other terms, βX̃ =
i ∗
2π f (βD );
this concludes the proof.
An immediate consequence of this fact is the following corollary, which seems
to be not known in the literature.
Corollary 1.5.13. The cohomology class of βX̃ lives in H 2 (X̃, Z).
81
Remark 1.5.14. There is a priori no reason that the cohomology class of βX
lives in H 2 (X, Z). For instance, let X be an elliptic curve and let ω be a nonzero holomorphic 1-form on X. Let E be the trivial holomorphic line bundle
over X. Then (E, tω) is a Higgs bundle for any t ∈ C and we check easily that
the trivial Hermitian metric on E is the harmonic metric for any t ∈ C. The
|t|2
Hitchin energy of (E, tω) is given by 4iπ
ω ∧ ω̄; since its cohomology class does
not vanish, it is not integral for almost all t ∈ C.
Writing LX̃ for the pullback of the determinant line bundle to X̃, βX has
integral cohomology class if and only if LX̃ is the pullback of a holomorphic
line bundle LX . This holomorphic line bundle, having curvature βX , would
be positive when the period map is an immersion, thanks to propositions 1.5.9
and 1.5.12 and the horizontality of the period map. In particular, X will be a
projective variety when it is compact.
An integrality criterion is given in theorem 1.5.18.
1.5.4
Carlson-Toledo conjecture
An interesting conjecture concerning fundamental groups of compact Kähler
manifolds is the following:
Conjecture 1.5.15. Let Γ be an infinite group, which is the fundamental group
of a compact Kähler manifold. Then, virtually, H 2 (Γ, R) 6= 0.
The following theorem 1.5.17, which is theorem F in the introduction, is a
particular case of lemma 3.2 in [KKM11]. Our proof here is quite different and
relies on corollary 1.5.13. In both proofs, the assumption that the fundamental
group admits a non-rigid representation is crucial.
Definition 1.5.16. Let Γ be a finitely generated group. An irreducible representation ρ in Rs (Γ, G) is rigid if its connected component in Rs (Γ, G) coincides
with its conjugation class under G.
Theorem 1.5.17. Let X be a compact Kähler manifold such that there exists
a non-rigid irreducible representation of π1 (X) in G. Then, H 2 (π1 (X), R) is
non-trivial.
Proof. Let ρ be an irreducible representation of π1 (X). Let Λ be the connected
component of ρ in Rs (π1 (X), G). We write βX,λ for the Hitchin energy of the
representation λ in Λ. It varies continuously with λ. Moreover, by corollary
1.5.13, its pullback βX̃,λ to X̃ has an integral cohomology class. The cohomology
class of βX̃,λ is thus constant in λ.
Hence, if λ and λ0 are in Λ, the cohomology class of βX,λ − βX,λ0 is in the
kernel of the pullback map H 2 (X, R) → H 2 (X̃, R)π1 (X) . This map is part of
the following exact sequence
0 → H 2 (π1 (X), R) → H 2 (X, R) → H 2 (X̃, R)π1 (X) ,
and the cohomology class of βX,λ − βX,λ0 thus lives in H 2 (Γ, R). We can assume
that this cohomology class is always zero (or the proof is over). Then, the
82
function F defined before theorem B.2.7 is constant in Λ/G, which is an open
set in X s (π1 (X), G). All smooth points in Λ/G are thus critical points of F ,
hence come from variations of Hodge structures by theorem B.2.7. The set of
smooth points in Λ/G is thus contained in a totally real analytic submanifold
of X s (π1 (X), G) by lemma B.1.1. Since it is also open, it has to be a point;
which is possible only if Λ/G is itself a point. Hence, ρ is a rigid representation.
1.5.5
Criterion of integrality
Let (E, D, h) be a harmonic bundle over a complex manifold X and let f : X̃ →
D and ρtot : π1 (X) → Λσ G be the period map and total monodromy. Thanks
to the monodromy ρtot , we can define a central S 1 -extension π^
1 (X) of π1 (X)
from the central S 1 -extension Λg
σ G of Λσ G by the following formula:
g
π^
1 (X) = {(γ, g̃) ∈ π1 (X) × Λσ G | ρtot (γ) = p(g̃)},
where p : Λg
σ G → Λσ G is the projection map.
We recall that, if A is an abelian group and Γ a discrete group, then the
central extensions of Γ by A are classified up to isomorphism by the elements of
H 2 (Γ, A), where A is seen as a trivial Γ-module. We write e ∈ H 2 (π1 (X), S 1 )
for the cohomology class of π^
1 (X).
Theorem 1.5.18. The cohomology class of βX in H 2 (X, R) is integral if and
only if e = 0 in H 2 (π1 (X), S 1 ).
Proof. We write L → D for the determinant line bundle over D, f : X̃ → D for
the period map and π : X̃ → X for the canonical projection map.
Assume that e = 0. Then, as a S 1 -central extension, π^
1 (X) is isomorphic to
π1 (X) × S 1 . In particular, the bundle f ∗ L is then π1 (X)-equivariant. Hence, it
defines a holomorphic line bundle LX over X such that π ∗ LX ∼
= f ∗ L. The bundle f ∗ L is equipped with a connection whose Chern form is βX̃ by proposition
1.5.12. Since this connection is π1 (X)-invariant, the bundle LX is also equipped
with a connection and its Chern form ωX satisfies π ∗ (ωX ) = βX̃ = π ∗ (βX ).
Since, π is a local diffeomorphism, this implies that ωX = βX . This shows that
the cohomology class of βX is integral since ωX is the Chern form of a line
bundle.
Conversely, we assume that βX has integral cohomology class. Then, the
bundle f ∗ L → X̃ is isomorphic to the pullback by π of a bundle LX → X.
In particular, the bundle f ∗ L is then π1 (X)-equivariant and the trivial S 1 central extension π1 (X) × S 1 can thus be identified with the group of vector
bundle isometries of f ∗ L → X̃ that cover the action of an element of π1 (X).
On the other hand, an element of π^
1 (X) also acts on this bundle as a vector
bundle isometry covering the action of an element of π1 (X). Since both groups
1
π1 (X) × S 1 and π^
1 (X) are central S -extensions of π1 (X), they are isomorphic
as S 1 -extensions of π1 (X).
1.6. Relation with the Shafarevich morphism
83
Remark 1.5.19. The condition e = 0 means that the total monodromy ρtot :
π1 (X) → Λσ G can be lifted to Λg
σ G. I do not know if such a condition can be
studied in some cases.
1.6
Relation with the Shafarevich morphism
This section is a joint work with Yohan Brunebarbe.
Definition 1.6.1. Let X be a connected compact Kähler manifold and let
ρ : π1 (X) → GL(n, C) be a representation. A Shafarevich morphism for (X, ρ)
is the datum of a connected complex normal space Shρ (X) and a holomorphic
surjective map with connected fibers shρ : X → Shρ (X) such that for any
connected complex manifold Z and any holomorphic map f : Z → X, the map
shρ ◦f is constant if and only if the morphism ρ ◦ f∗ : π1 (Z) → GL(n, C) has
finite image.
Let X be a compact Kähler manifold and let ρ : π1 (X) → GL(n, C) be a
representation. In theorem 1 of [CCE15], it is shown that if the Zariski-closure
of the image of ρ is a semisimple group, then the Shafarevich morphism exists
and its image is a projective normal algebraic variety of general type if ρ(π1 (X))
has no torsion. In theorem 1.6.3, we show under simplifying hypotheses that
the Shafarevich morphism can be understood via the period map attached to
the harmonic bundle with monodromy ρ.
We first recall the notion of Stein factorization for a proper holomorphic
map between complex spaces:
Theorem 1.6.2 (Stein factorization, [GR84], page 213). Let X and Y be complex spaces and let f : X → Y be a proper holomorphic map. Then f admits a
unique Stein factorization
fˆ
g
X → Ŷ → Y
through a complex space Ŷ with the following properties:
• fˆ is a proper holomorphic surjection; g is finite and holomorphic; f =
g ◦ fˆ;
• fˆ∗ (OX ) = OŶ ; in particular all fibers of fˆ are connected.
If X is normal, then Ŷ is normal too.
Let ρ : π1 (X) → GL(n, C) be a semisimple representation of the fundamental group of a compact Kähler manifold X. By the Corlette-Donaldson theorem
0.2.4, there is a harmonic bundle (E, D, h) of rank n with monodromy ρ.
Theorem 1.6.3. Let X be a connected compact Kähler manifold and let ρ :
π1 (X) → GL(n, C) be a semisimple representation with discrete image. Let
f : X̃ → D and ρtot : π1 (X) → Λσ be the period map and total monodromy of
the harmonic bundle (E, D, h) with monodromy ρ. We assume that the image
of the total monodromy is without torsion. Then, the image of the map f¯ :
84
X → D/ρtot (π1 (X)) is a finite-dimensional complex space. Moreover, the map
fˆ : X → Ŷ in its Stein factorization
fˆ
X → Ŷ → Im(f¯)
is a Shafarevich morphism for (X, ρ).
Proof. Since ρ(π1 (X)) is discrete in G, ρtot (π1 (X)) is discrete in Λσ G. The
group ρtot (π1 (X)) acts on D with stabilizer a discrete subgroup of K, which
has to be finite, and thus trivial since we assume that ρtot (π1 (X)) has no torsion. Hence, the quotient D/ρtot (π1 (X)) has a structure of complex Hilbert
manifold. The map f¯ : X → D/ρtot (π1 (X)) is proper since X is compact;
by a generalization of Remmert proper mapping theorem, for maps whose target space is infinite-dimensional (corollary 3, page 180 in [Maz84]), the image
of f¯ is a finite-dimensional complex analytic subspace of D/ρtot (π1 (X)). Let
fˆ
X → Y → Im(f¯) be its Stein factorization. Then fˆ is a proper holomorphic
surjection with connected fibers and Ŷ is a normal space.
Let Z be a connected complex manifold and let g : Z → X be a holomorphic
map. If the map f¯◦ g : Z → Ŷ is constant, then the period map of the variation
of loop Hodge structures on Z (obtained via pullback from g) is constant. This
simply means that the pullback of the harmonic bundle on X is trivial on Z,
in the sense that the harmonic metric is flat. Hence, the image of π1 (Z) by
ρ ◦ g∗ is contained in a unitary group. Since it is also discrete, it has to be
finite. Conversely, it the image of π1 (Z) by ρ ◦ g∗ is finite, it is contained
in a unitary group. Hence, the flat bundle associated to the representation
π1 (Z) → GL(n, C) admits a flat metric, which has to be the harmonic metric.
This shows that the period map f¯ ◦ g : Z → D/ρ(π1 (X)) is constant. This
factorizes through the map fˆ ◦ g : Z → Ŷ , which has finite image since the
map Ŷ → D/ρ(π1 (X)) is finite. Since Z is connected, the map fˆ ◦ g is in fact
constant. This concludes the proof.
Remark 1.6.4. Following for instance proposition 3.5 in [CCE15], one can
try to deduce properties of Shρ (X). In the case where ρ is the monodromy
of a variation of Hodge structures, the period map takes values in a classical
period domain Dc . The invariant metric g on Dc is Kähler in the horizontal
directions. Moreover, the holomorphic bisectional curvature is nonpositive in
the horizontal directions and the holomorphic sectional curvature is negative in
the horizontal directions. This shows that (some desingularization of) Shρ (X)
has ample canonical bundle since its curvature form (with respect to the Kähler
metric g| Shρ (X) ) is positive.
On the loop period domain, we still have a Λσ G-invariant metric. It is also
Kähler in the horizontal directions and its holomorphic bisectional curvature
is still nonpositive in the horizontal directions. However, the holomorphic sectional curvature can vanish in the horizontal directions and we cannot conclude
as before. One should notice that this problem was already encountered in
[Mok92].
Chapter 2
Characteristic Laplacian in
sub-Riemannian geometry
This chapter is a published article [DM15], written with X. Ma. We study the
possibility of defining a theory of harmonic forms to compute the characteristic
cohomology of a differentiable manifold endowed with a distribution. In section
2.1, we define the characteristic Laplacian, study its properties and discuss
about its hypoellipticity. In section 2.2, we answer question 0.5.2, proving that
in a complex setting the characteristic laplacian never respects the bigrading
on differential forms.
2.1
The characteristic Laplacian
This section is organized as follows. In subsection 2.1.1, we define the characteristic Laplacian associated with a distribution for Riemannian manifolds. In
subsection 2.1.2, we explain Taylor’s counterexample for the hypoellipticity of
the characteristic Laplacian. In subsection 2.1.3, we explain why the hypoellipticity does not seem to hold in general by computing its principal symbol. In
subsection 2.1.4, we give a counterexample for the hypoellipticity of the characteristic Laplacian in the complex setting which is the context of the original
question.
2.1.1
Definitions and notations
Let (X, g T X ) be a smooth compact Riemannian manifold, endowed with a
(constant-rank) distribution W . We denote by F the annihilator of W in T ∗ X;
it is a vector subbundle of T ∗ X. We denote by Ω• (X) the graded algebra of
differential forms on X and we endow it with the natural metric g Ω(X) induced
by g T X . We consider
• I the algebraic ideal generated by the smooth sections IX = C ∞ (X, F );
• J the differential ideal generated by the smooth sections of F on X, that
is the minimal algebraic ideal containing the smooth sections of F and
stable by exterior differentiation.
86
Chapter 2. Characteristic Laplacian
Remark that J is the algebraic ideal generated by IX and dIX . If (θj ) is a
frame of F , the forms in I can locally be written as
X
θj ∧ φ j ,
j
where φj are arbitrary forms on X, and those in J are of the form
X
θj ∧ φj + dθj ∧ ψj ,
j
where φj , ψj are arbitrary forms on X.
Let Q be the orthogonal complement of J in Ω(X). Remark that Q is
naturally graded. We define a differential operator for forms in Q by
dQ := πQ ◦ d ◦ πQ : Q → Q,
(2.1)
where πQ is the orthogonal projection from Ω(X) onto Q. In (2.1), we use a
common abuse of notation: dQ should formally be defined by dQ = πQ ◦ d ◦ ιQ ,
where ιQ is the inclusion of Q in Ω(X). The right πQ in (2.1) is thus here to
emphasize that dQ is defined on the subspace Q of Ω(X).
Since J is stable by d, we have
πQ ◦ d ◦ πQ = πQ ◦ d : Ω• (X) → Ω• (X),
(2.2)
As a direct consequence of (2.2), we know
d2Q = πQ ◦ d ◦ πQ ◦ d ◦ πQ = πQ ◦ d2 ◦ πQ = 0.
(2.3)
Since Q is the orthogonal complement of J and J is stable by d, we know
that Q is stable by the adjoint d∗ of d and the restriction of d∗ to Q is d∗Q , the
adjoint of dQ for the natural L2 -structure on Q. Indeed, if α (resp. β) is in Q
(resp. J ), then dβ is in J and this implies
(d∗ α, β) = (α, dβ) = 0.
Since this is true for any β in J , d∗ α is in Q. Moreover, if α and β are in Q,
then
(d∗ α, β) = (α, dβ) = (α, dQ β)
(2.4)
and this shows that d∗Q α = d∗ α.
Definition 2.1.1. The characteristic Laplacian ∆Q on X with respect to W
is the differential operator on Q
∆Q = dQ d∗Q + d∗Q dQ : Q → Q.
(2.5)
Remark 2.1.2. In sub-Riemmanian geometry (where the distribution W is
bracket-generating), one defines a sub-Laplacian on functions (see [Mon02]).
This sub-Laplacian is hypoelliptic and coincides with the characteristic Laplacian in degree 0. In example 2.1.5, we will see that hypoellipticity can fail in
positive degrees.
2.1. The characteristic Laplacian
87
Remember that we defined the characteristic cohomology of X (associated
to the distribution W ) to be
HJ• (X) := H • (Ω• (X)/J • , d),
(2.6)
with the differential induced by exterior differentiation on Ω• (X).
The characteristic cohomology of an exterior differential system was deeply
studied in [BG95a] and [BG95b]. More recently, some aspects of the characteristic cohomology of period (or Mumford- Tate) domains have been treated
in [Rob14b] and [Rob14a]. In particular, it is explained in the last reference
that in low dimensions characteristic and de Rham cohomologies coincide. Although written in a quite specific setting in [Rob14a], this is true in general, as
explained in the Appendix D.
By definition of Q and dQ , this characteristic cohomology is naturally isomorphic to the cohomology of the complex (Q• , dQ ). An analogue of Hodge
theory would be the following conjecture.
• (X) the space of characteristic harmonic
Conjecture 2.1.3. We denote by HQ
forms, that is the kernel of ∆Q . Then
• (X) is of finite dimension.
• HQ
• There is an orthogonal decomposition
•
Q• (X) = HQ
(X) ⊕ dQ (Q•−1 (X)) ⊕ d∗Q (Q•+1 (X)).
(2.7)
• (X) → H • (X) is an isomorphism.
• The natural application HQ
J
In classical Hodge theory, one gets these results as consequences of the
ellipticity of the Laplacian. In the next section, we show that the characteristic
Laplacian in not even hypoelliptic in general. Hence we cannot really hope this
conjecture to be true.
2.1.2
The question of hypoellipticity
We recall that, if E and F are vector bundles over X and P : E → F is a
differential operator, then P is said to be hypoelliptic if the following condition
is satisfied: for every local distribution u with values in E, if P u is smooth on
an open set U ⊂ X, then the restriction of u to U is smooth. Elliptic operators,
like the usual Hodge Laplacian, are hypoelliptic. It is a natural question to ask
whether ∆Q is hypoelliptic since this would be the first step in order to prove
Conjecture 2.1.3.
The best known sufficient condition for a second order differential operator
to be hypoelliptic is due to Hörmander ([Hör67]).
Theorem 2.1.4 (Sum of squares condition of hypoellipticity). Let P be a
second order differential operator from a vector bundle E to itself. Suppose that
88
locally one can find smooth vector fields X0 , . . . , Xk and a smooth function c
such that in a local frame of E,
k
X
Pu = (
Xi2 + X0 + c)u
i=1
(in particular P acts componentwise). Then P is hypoelliptic if and only if
X0 , . . . , Xk generate T X by brackets.
In [GGK10], it is suggested that this theorem implies that the characteristic
Laplacian is hypoelliptic when the distribution W is bracket-generating. We will
first give a counterexample due to Michael Taylor [Tay10] and then compute the
principal symbol of ∆Q in order to understand why the hypoellipticity certainly
fails in general.
Example 2.1.5. A contact structure on a 3-manifold M is one of the simplest
examples of Pfaffian systems. It is the datum of a 2-rank distribution W on
M which is bracket-generating. For instance, one can take for W the kernel
of the 1-form θ = du − pdx in coordinates (x, u, p) in M = R3 . A natural
example in which M is compact is constructed as follows: we consider H3 the
3-dimension Heisenberg group, that is R3 with coordinates (p, q, t) and group
structure (p, q, t) · (p0 , q 0 , t0 ) = (p + p0 , q + q 0 , t + t0 + 21 (pq 0 − p0 q)). One checks
that the 1-form θ = dt − 12 qdp + 21 pdq is right-invariant and defines a contact
structure on H3 . Taking a cocompact discrete subgroup Γ of G, M = G/Γ still
carries the contact structure.
Locally, all contact structures on 3-manifolds are the same (however, the
characteristic Laplacian also needs a Riemannian metric to be defined; see also
example 2.1.10 for this important issue). Let M be a 3-dimensional Riemannian
manifold with a contact form θ and corresponding 2-rank distribution W . Let
U be an open set in M . Then
• J 0 (U ) = 0,
• J 1 (U ) = C ∞ (U, Rθ),
• J 2 (U ) = Ω2 (U ), J 3 (U ) = Ω3 (U ),
and
• Q0 (U ) = C ∞ (U ),
• Q1 (U ) = C ∞ (U, E),
• Q2 (U ) = 0, Q3 (U ) = 0,
where E → U is the real vector bundle of rank 2, which is the orthogonal of
Rθ in T ∗ R3 . In degree 1, the characteristic Laplacian is well-defined on forms
with compact support. We have
∆1Q = dQ d∗Q : Cc∞ (U, E) → Cc∞ (U, E)
89
and one can consider it as a second order differential operator on E. Let us
denote by (X1 , X2 ) a (smooth) orthonormal frame of W over U and by (α1 , α2 )
the dual frame of W ∗ . Using the metric, W ∗ can be seen as a subbundle of
T ∗ M and E can be identified with W ∗ (see also the following subsection 2.1.3).
A form µ ∈ L2 (U, E) can thus be written µ = µ1 α1 + µ2 α2 , with µi ∈
2
L (U, R), and d∗Q µ is given by
d∗Q µ = Y1 µ1 + Y2 µ2 ,
(2.8)
where Y1 , Y2 are first order scalar differential operators on U .
In particular
d∗Q (µ1 α1 ) = Y1 µ1 .
We claim that Y1 µ1 can be smooth (even zero) without µ1 being smooth. Indeed, as any first-order scalar differential operator, Y1 can be written Y1 =
X + f , with X a non-vanishing vector field and f a smooth function. Let
(x1 , x2 , x3 ) be some local system of centered coordinates, star-shaped in 0, such
that X = ∂x∂ 1 , then Y1 has the form
Y1 =
∂
+ f (x1 , x2 , x3 ).
∂x1
Consider the function
v(x1 , x2 , x3 ) = exp
Z x1
f (t, x2 , x3 )dt µ1 (x1 , x2 , x3 )
0
which is smooth if and only if µ1 is smooth. We compute that
∂
v(x1 , x2 , x3 ) = exp
∂x1
Z x1
f (t, x2 , x3 )dt Y1 µ1 (x1 , x2 , x3 ).
0
Choosing v independent of x1 but not smooth, one has Y1 µ1 = 0, proving the
claim.
Remark 2.1.6. In an analogous example occurring in the complex situation
(example 2.1.10), we will need to be more precise. In particular, formula (2.8)
can be made explicit:
d∗Q µ = − div(µ1 X1 + µ2 X2 ),
(2.9)
where div(X) is the divergence of the vector field X.
In order to better understand why the hypoellipticity seems to fail, we
will compute the principal symbol of ∆Q . It will not show that ∆Q is not
hypoelliptic but it will at least show that the sum of squares condition cannot
be applied, at least if one only considers the second order terms.
90
2.1.3
The principal symbol of the characteristic Laplacian
Recall that we denote by F the annihilator of W in T ∗ X. Let N be the
orthogonal complement of W in (T X, g T X ). Then as a smooth vector bundle,
we have
T X = W ⊕ N,
and
T ∗X = W ∗ ⊕ N ∗.
(2.10)
We can identify N ∗ and F as C ∞ vector bundles, and
∗
b
X)).
I = C ∞ (X, N ∗ ⊗Λ(T
(2.11)
The quotient Ω(X)/I can be identified with the orthogonal complement of
I, i.e.,
ΩW (X) := C ∞ (X, ΛW ∗ ).
(2.12)
In what follows, we do these identifications without further notice.
Since I ⊂ J , Q = J ⊥ can be viewed as a subspace of ΩW (X). The
orthogonal complement of Q in ΩW (X) will be identified with J /I. We thus
have the following decompositions:
Ω(X) = I ⊕ ΩW (X)
(2.13)
ΩW (X) = J /I ⊕ Q.
(2.14)
and
All of these spaces are naturally graded. We define a map ϕ : F → Λ2 (W ∗ )
by: for θ ∈ C ∞ (X, F ), v, w ∈ C ∞ (X, W ), set
ϕ(θ)(v, w) := (dθ)(v, w) = −θ([v, w]).
(2.15)
We check that for any x ∈ X, ϕ(θ)(v, w)x depends only on θx , vx and wx .
b k W ∗ → Λk+2 W ∗ for any
The map ϕ : F → Λ2 W ∗ induces a map ϕ : F ⊗Λ
k. We will assume that the rank of these maps is constant on X, for any k.
b k W ∗ ) forms a vector subbundle of Λk+2 W ∗ on X for any
In particular, ϕ(F ⊗Λ
∗ ) and let F
b
k. Set Fϕ := ϕ(F ⊗ΛW
ϕ,⊥ be the orthogonal complement of Fϕ in
∗
ΛW over X.
By construction, we thus have an orthogonal decomposition
ΛW ∗ = Fϕ ⊕ Fϕ,⊥
(2.16)
and by (2.12) and (2.16), this decomposition induces (2.14), that is
J /I = C ∞ (X, Fϕ ),
Q = C ∞ (X, Fϕ,⊥ ).
(2.17)
We denote by πFϕ and πFϕ,⊥ the orthogonal projections from ΛW ∗ onto Fϕ and
Fϕ,⊥ . In order to make computations with the operators on Q, we construct
intermediate operators on ΛW ∗ . First we define
dW := πW ◦ d ◦ πW : ΛW ∗ → ΛW ∗ ,
(2.18)
91
where πW is the projection from Λ(T ∗ X) on ΛW ∗ in the decomposition (2.13).
Beware that there is no reason for d2W to be 0. The adjoint d∗W of dW satisfies
d∗W = πW ◦ d∗ ◦ πW .
(2.19)
From (2.1), (2.18) and (2.19), we have
dQ = πQ ◦ dW ◦ πQ ,
d∗Q = πQ ◦ d∗W ◦ πQ .
(2.20)
By (2.18) and (2.19), dW and d∗W are first order differential operators on
ΛW ∗ and their principal symbols are, for ξ ∈ T ∗ X
√
σ1 (dW , ξ) = −1 ξW ∧,
√
(2.21)
∗ ,
σ1 (d∗W , ξ) = − −1 iξW
∗ ∈ W is the metric
where ξW is the orthogonal projection of ξ on W ∗ and ξW
dual of ξW .
By (2.17) and (2.20), dQ and d∗Q are first order differential operators on
Fϕ,⊥ and their principal symbols are
√
σ1 (dQ , ξ) = −1 πFϕ,⊥ ξW ∧ πFϕ,⊥ ,
√
(2.22)
∗ πF
σ1 (d∗Q , ξ) = − −1 πFϕ,⊥ iξW
.
ϕ,⊥
One also gets the adjoint formula of (2.2)
πQ ◦ d∗ ◦ πQ = d∗ ◦ πQ .
(2.23)
Taking the principal symbols of (2.2) and (2.23), we have
πFϕ,⊥ ξW ∧ πFϕ,⊥ = πFϕ,⊥ ξW ∧,
∗ πF
∗ πF
πFϕ,⊥ iξW
= iξW
.
ϕ,⊥
ϕ,⊥
(2.24)
Proposition 2.1.7. The characteristic Laplacian is a second order differential
operator on Fϕ,⊥ and its principal symbol is
∗ πF ξW ∧)πF
σ2 (∆Q , ξ) = πFϕ,⊥ (|ξW |2 − iξW
.
ϕ
ϕ,⊥
(2.25)
Remark 2.1.8. The first term in (2.25) is the term suggested in [GGK10] but
the second term was forgotten. Because of this second term, one cannot apply
Hörmander’s condition of hypoellipticity; see example 2.1.9.
Proof of proposition 2.1.7. By (2.5), (2.22) and (2.24), ∆Q is a second order
differential operator on Fϕ,⊥ and its principal symbol is
∗ πF
∗ πF
σ2 (∆Q , ξ) = πFϕ,⊥ ξW ∧ πFϕ,⊥ iξW
+ πFϕ,⊥ iξW
ξ ∧ πFϕ,⊥
ϕ,⊥ W
ϕ,⊥
∗ π F ξW ∧ π F
∗ πF
∗ ξW ∧ πF
= πFϕ,⊥ ξW ∧ iξW
+ πFϕ,⊥ iξW
− πFϕ,⊥ iξW
ϕ
ϕ,⊥
ϕ,⊥
ϕ,⊥
∗ πF ξW ∧)πF
= πFϕ,⊥ (|ξW |2 − iξW
.
ϕ
ϕ,⊥
In the second equality, we use (2.24) and in the third, we use the identity
2
∗ ξW ∧ +ξW ∧ iξ ∗ = |ξW | .
iξW
W
The proof of proposition 2.1.7 is completed.
92
Example 2.1.9. We review example 2.1.5 and compute the principal symbol
of the characteristic Laplacian in degree one. With the identifications at the
beginning of the paragraph, Q1 is the space of sections of W ∗ and Q2 is zero.
Let η be in W ∗ . Since in (2.25), only the projection ξW of ξ is involved, we can
restrict the symbol to the ξ belonging to W ∗ . We have
σ2 (∆Q , ξ)η = |ξ|2 η − πFϕ,⊥ iξ∗ πFϕ (ξ ∧ η)
= |ξ|2 η − iξ∗ (ξ ∧ η)
= |ξ|2 η − (|ξ|2 η − η(ξ ∗ )ξ)
= η(ξ ∗ )ξ.
In the second equality, we use that Fϕ,⊥ (resp. Fϕ ) equals ΛW ∗ in degree 1
(resp. 2).
If ξ = (ξ1 , ξ2 )t and η = (η1 , η2 )t in an orthonormal frame for W ∗ , we get
η1
σ2 (∆Q , ξ)
η2
!
ξ1
= (ξ1 η1 + ξ2 η2 )
ξ2
!
!
=
ξ12 η1 + ξ1 ξ2 η2
.
ξ1 ξ2 η1 + ξ22 η2
!
!
Otherwise said, we have the equality
−∆Q
η1
η2
!
=
X12 η1 + X1 X2 η2
η1
+ X0
,
X1 X2 η1 + X22 η2
η2
where (X1 , X2 ) is a local frame of W and X0 is a first order differential operator,
which does not necessarily act componentwise. Since the second order does not
act componentwise, one cannot apply Hörmander’s condition of theorem 2.1.4.
2.1.4
The complex situation
In the remainder of the article, we will be interested in Pfaffian systems over a
complex manifold. More precisely, let (X, J) be a compact complex manifold;
(0,1)
J induces a splitting T X ⊗R C = T (1,0) X ⊕ T (0,1) X, where T (1,0)
√ X and T √ X
are the eigenbundles of J corresponding to the eigenvalues −1 and − −1,
respectively. Let T ∗(1,0) X and T ∗(0,1) X be the corresponding dual bundles.
We still denote by Ωk (X) the space of smooth k-forms on X with values in
C. Let
Λp,q (T ∗ X) = Λp (T ∗(1,0) X) ⊗ Λq (T ∗(0,1) X),
Ωp,q (X) := C ∞ (X, Λp,q (T ∗ X)).
(2.26)
Then Ωp,q (X) is the space of smooth (p, q)-forms on X, and
Ωk (X) =
M
Ωp,q (X).
p+q=k
Let Θ be a real (1, 1)-form such that
g T X (·, ·) = Θ(·, J·)
(2.27)
93
defines a Riemannian metric on T X. The triple (X, J, Θ) is called a complex
Hermitian manifold. If Θ is a closed form, then the form Θ is called a Kähler
form on X.
We denote the holomorphic tangent bundle by Th X. Let W ⊂ Th X be
a (constant-rank) holomorphic distribution. We consider the Pfaffian system
associated to the distribution WR . Otherwise said, if we denote by F ⊂ Th∗ X
the holomorphic annihilator of W , then the exterior differential system J we
consider is generated by FR ⊂ T ∗ X. Beware of the notations that differ from
¯
the real case. Remark that J is not only d-stable: it is also ∂ and ∂-stable.
∞
Indeed, if IX = C (X, F ) is the space of smooth sections of F on X then
I X = C ∞ (X, F ) and d acts on IX (resp. I X ) as ∂ modulo IX · Ω0,1 (X) (resp.
∂¯ modulo I X · Ω1,0 (X)).
Locally, if (θj ) is a holomorphic frame of F , the forms in I can be written
as
X
θj ∧ ψj + θj ∧ φj ,
j
where ψj , φj are arbitrary smooth forms, and those in J are of the form
X
θj ∧ ψj + θj ∧ φj + dθj ∧ ωj + dθj ∧ χj ,
j
where ψj , φj , ωj , χj are arbitrary smooth forms.
We still denote by Q the orthogonal of J . Besides the operator dQ , we also
define ∂Q and ∂¯Q by
∂Q := πQ ◦ ∂ ◦ πQ ,
∂¯Q := πQ ◦ ∂¯ ◦ πQ .
(2.28)
Moreover, as in (2.4), the adjoints of ∂Q and ∂¯Q (for the natural L2 -structure
¯
on Q) are the restrictions to Q of ∂ ∗ , ∂¯∗ the adjoints of ∂ and ∂.
If we denote by N the orthogonal complement of W in (T X, g T X ), one
obtains decompositions analogous to those in the previous subsection. In particular, denoting by I the algebraic ideal generated by FR , one has the analogues
of (2.11), (2.12), (2.13) and (2.14):
∗
b
I = C ∞ (X, NR∗ ⊗Λ(T
X)),
ΩW (X) := C ∞ (X, ΛWR∗ ),
Ω(X) = I ⊕ ΩW (X),
(2.29)
ΩW (X) = J /I ⊕ Q.
All of these spaces carry a natural bigrading. Denoting by πW the orthogonal projection from Λ(T ∗ X) onto ΛWR∗ , we define
∂W := πW ◦ ∂ ◦ πW ,
∗
∂W
= πW ◦ ∂ ∗ ◦ πW ,
∂¯W := πW ◦ ∂¯ ◦ πW ,
∂¯∗ = πW ◦ ∂¯∗ ◦ πW .
(2.30)
W
Then, we have, besides equations (2.18), (2.19) and (2.20),
∂Q = πQ ◦ ∂W ◦ πQ ,
∗
∗
∂Q
= πQ ◦ ∂W
◦ πQ ,
∂¯Q = πQ ◦ ∂¯W ◦ πQ ,
∗
∗
∂¯Q
= πQ ◦ ∂¯W
◦ πQ .
(2.31)
94
We still define a map ϕ : F → Λ2 W ∗ as in (2.15). Since in the definition
(2.15), we can take θ, v, w to be holomorphic sections, it proves that ϕ is a
∗
∗
b
holomorphic map. We still assume that the induced map ϕ : F ⊗ΛW
R → ΛWR
∗ ) and F
b
has constant rank. Set Fϕ := ϕ(F ⊗ΛW
ϕ,⊥ the orthogonal complement
∗
of Fϕ in ΛW and let πFϕ , πFϕ,⊥ be the orthogonal projections from ΛW ∗ onto
Fϕ , Fϕ,⊥ .
∗
b
b ϕ and Fϕ,⊥ ⊗F
b ϕ,⊥ are vector subbundles of ΛW ∗ ⊗R
Then Fϕ ⊗ΛW
+ΛW ∗ ⊗F
R
C over X, and
∗
b ϕ ⊕ Fϕ,⊥ ⊗F
b ϕ,⊥ .
b
+ ΛW ∗ ⊗F
ΛWR∗ ⊗R C = Fϕ ⊗ΛW
(2.32)
In this decomposition, we have, as in (2.17),
∗
b
b ϕ ).
J /I = C ∞ (X, Fϕ ⊗ΛW
+ ΛW ∗ ⊗F
b ϕ,⊥ ).
Q = C ∞ (X, Fϕ,⊥ ⊗F
(2.33)
Example 2.1.10. Example 2.1.5 can also be seen in the complex situation
but an interesting phenomenon appears in degree 1. Consider the complex
manifold M = C3 with complex coordinates (x, u, p) and the holomorphic 1form θ = du − pdx. We denote by (·, ·) a Hermitian metric on M and by
| · | the corresponding norm. Using the notations of this subsection, W is a
holomorphic vector subbundle of Th X of rank 2. Hence Λ2 W ∗ is of rank 1 and
we have Fϕ = Λ2 W ∗ . We thus obtain the following bidegree decomposition of
Q over an open set U :
• Q0,0 (U ) = C ∞ (U ),
• Q1,0 (U ) = C ∞ (U, W ∗ ),
∗
• Q0,1 (U ) = C ∞ (U, W ),
∗
b
• Q1,1 (U ) = C ∞ (U, W ∗ ⊗W
).
Take (X1 , X2 ) a holomorphic frame of W on U and (α1 , α2 ) its dual frame.
We study the smoothness of harmonic forms for ∆Q in degrees 0, 1 and 2.
For f ∈ L2 (U, C), one has ∆Q f = 0 if and only if dQ f = 0. Using the
frames, one obtains
dQ f = X1 (f )α1 + X2 (f )α2 + X 1 (f )α1 + X 2 (f )α2 .
Since X1 , X2 , X 1 and X 2 generate by brackets the tangent bundle of U , a
function f such that dQ f = 0 is in fact locally constant. Remark that the
same argument works for any distribution W which is bracket-generating. This
proves that the harmonic functions are smooth.
In degree 1, let µ = µ1 α1 + µ2 α2 be in L2 (U, W ∗ ) (here µ1 , µ2 ∈ L2 (U, C)).
Such a form is killed by ∆1Q if and only if it is killed by both dQ and d∗Q .
We denote by div(Y ) the divergence of a vector field Y and by dvX the
Riemannian volume form on (X, g T X ).
95
Lemma 2.1.11. The following identities hold:
dQ µ = ∂¯Q µ =
d∗Q µ =
P2
j
i
i,j=1 X j (µi )α ∧ α ,
P
− div( 2i=1 (µ, αi )Xi ).
(2.34)
(2.35)
Proof. Since Q2,0 (U ) = 0, we know dQ µ = ∂¯Q µ. The 2-form dαk satisfies
dαk (Y1 , Y2 ) = Y1 (αk (Y2 )) − Y2 (αk (Y1 )) − αk ([Y1 , Y2 ]).
Since (α1 , α2 ) is the dual basis of (X1 , X2 ), this simplifies to dαk (Y1 , Y2 ) =
−αk ([Y1 , Y2 ]) when Yl is either a Xi or a X j . Since the Xi are holomorphic, the
brackets [Xi , X j ] vanish and dαk (Xi , X j ) = 0. This implies that
dQ µ =
i =
¯
i=1 ∂Q µi α
P2
2
X
X j (µi )αj ∧ αi .
i,j=1
For the second equality, one has for every smooth function f ,
∗
(d∗Q µ, f ) = (∂Q
µ, f )
= (µ, ∂Q f )
=
2 Z
X
Xi (f )(µ, αi )dvX
i=1 M
2 Z
X
=−
f¯ div((µ, αi )Xi )dvX .
i=1 M
Hence d∗Q µ = − div(
P2
i=1 (µ, α
i )X ).
i
Using lemma 2.1.11, if ∆1Q µ = 0 then X j (µi ) = 0 for all i, j from 1 to 2.
Since the X j generate by brackets the anti-holomorphic tangent bundle, this
is equivalent to µi being holomorphic in the weak sense. Since ∂¯ is an elliptic
operator on functions, the µi are holomorphic. This shows that the harmonic
forms of bidegree (1, 0) (resp. (0, 1)) are holomorphic (resp. anti-holomorphic)
∗
sections of W ∗ (resp. W ). In particular, they are smooth, contrary to what
happened in the real setting.
P
b ∗ ). It is
In degree 2, consider a form ν = 2i,j=1 νij αj ∧ αi in L2 (U, W ∗ ⊗W
annihilated by ∆2Q if and only if d∗Q ν = 0.
Lemma 2.1.12. The 2-form ν is harmonic if and only if for k = 1, 2,
div
div
2
X
l
k
l=1
2
X
k
l
!
(ν, α ∧ α )Xl
!
(ν, α ∧ α )X l
l=1
= 0,
= 0.
96
Proof. We just show that the first equation is equivalent to the vanishing of
∗ ν. For every (1, 0)-form µ = µ α1 + µ α2 , we get
∂¯Q
1
2
∗
(∂¯Q
ν, µ) = (ν, ∂¯Q µ)
2
X
= (ν,
X l (µk )αl ∧ αk ) by equation (2.34) in lemma 2.1.11
k,l=1
=
2
X
Z
k,l=1 M
=−
Xl (µk )(ν, αl ∧ αk )dvX
2 Z
X
k,l=1 M
µk div((ν, αl ∧ αk )Xl )dvX .
P2
This is zero for all µ if and only if div(
l=1 (ν, α
l
∧ αk )Xl ) is 0 for k = 1, 2.
It seems difficult to unravel these equations in general. We will only consider
two different choices for the metric.
Standard metric In the particular case where the metric on C3 is the standard one, one can choose for X1 , X2 orthogonal holomorphic vectors with zero
∂
∂
∂
and X2 = p ∂u
+ ∂x
. Moreover, X1 has norm 1. Then
divergence (take X1 = ∂p
the equations of lemma 2.1.12 become
X
Xl (νlk |αl ∧ αk |2 ) = 0,
l
X
X l (νkl |αk ∧ αl |2 ) = 0,
l
where ν = νlk αl ∧ αk . One can take ν12 = ν21 = ν22 = 0. Then the equations
are simply X1 ν11 = X 1 ν11 = 0. Since X1 is holomorphic, [X1 , X 1 ] = 0 and, by
Frobenius theorem, one can choose ν11 constant in the directions of X1 and X 1
but ν11 not smooth. This shows the existence of non-smooth harmonic 2-forms.
Heisenberg metric Consider the case where we see C3 as the complex
Heisenberg group (see example 2.1.5), endowed with a right-invariant Hermitian metric and with a right-invariant contact form. Choose a basis (X10 , X20 )
of W at the identity e of H3 . Consider the corresponding right-invariant vector
f0 . Since these vector fields and the volume
f0 and X
fields on H3 , denoted by X
1
2
f0 and X
f0 are constant. Hence,
form are right-invariant, the divergences of X
2
1
to certain linear combination X1 of X10 and X20 corresponds a right-invariant
vector field with zero divergence. We can moreover assume that X1 is a unit
vector and complete it to an orthonormal basis (X1 , X2 ) of We . Thus we get
f1 , X
f2 ) of W and X
f1 has zero divergence.
a holomorphic orthonormal frame (X
Then, the same argument as above shows the existence of non-smooth harmonic
2-forms.
97
Remark 2.1.13. Still considering a contact system on a 3-dimensional complex manifold, there is another natural example to study. For more precisions,
one can look at [Car98] or at the general reference [CMSP03] on period domains
of (real) Hodge structures.
This example is one of the simplest situations encountered in Hodge theory.
Let h be the non-degenerate Hermitian form of signature (2, 1) on the complex
vector space C3 , given by the diagonal matrix with diagonal entries 1, −1, 1.
We define a complex Hodge structure (of type (1, 1, 1)) to be an h-orthogonal
decomposition
C3 = H 2,0 ⊕ H 1,1 ⊕ H 0,2
such that H i,2−i is a complex line on which h is (−1)i positive definite.
Writing (e1 , e2 , e3 ) for the canonical basis of C3 , then H 2,0 = Ce1 , H 1,1 =
Ce2 , H 0,2 = Ce3 is such a complex Hodge structure, that we call the standard one. Around the standard one, the complex Hodge structures can be
parametrized by three complex coordinates (x, u, p) in the following way: let
F 2 (x, u, p) be the line generated by the vector with coordinates (1, x, u) and let
F 1 (x, u, p) be the plane generated by F 2 (x, u, p) and the vector with coordinates (0, 1, p). Then, if x, u, p are sufficiently small, H 2,0 (x, u, p) := F 2 (x, u, p),
1
3
H 1,1 := F 2 (x, u, p)⊥F (x,u,p) and H 0,2 := F 1 (x, u, p)⊥C define a complex Hodge
structure, the orthogonals being taken with respect to h. In this way, the period
domain D, that is the set of all complex Hodge structures, is endowed with the
structure of a complex manifold of dimension 3.
∼ U (2, 1) naturally acts on D and it
Moreover, the real Lie group U (h) =
can be shown that this action is transitive, with stabilizer V ∼
= U (1)×3 at the
×3
∼
standard complex Hodge structure. Hence, D = U (2, 1)/U (1) . Since, U (1)×3
is compact, one can endow D with a U (2, 1)-invariant structure of Hermitian
manifold.
Finally, we can also define a holomorphic distribution in the (holomorphic)
tangent space of D. Geometrically, this distribution gives constraints to the
line H 2,0 : more precisely, we want the so-called infinitesimal period relation
dH 2,0 ⊂ H 2,0 ⊕ H 1,1 = F 1 to be satisfied. In the coordinates (x, u, p), this
simply means that the form (0, dx, du) has to be in the vector space generated
by (1, x, u) and (0, 1, p), which is equivalent to the vanishing of the 1-form
pdx − du. Hence, we find again a contact system, which is U (2, 1)-invariant.
In conclusion, we have defined a 3-dimensional complex manifold, with a
Hermitian structure and a contact distribution. As before, we can ask whether
the characteristic Laplacian is hypoelliptic in degree 2 in this situation. But the
computations seem difficult to handle and we were not able to perform them.
Our intuition is that there is no reason for the characteristic Laplacian to be
hypoelliptic; but if it were, it would be very interesting to understand is this is
a general phenomenon coming from the Hodge-theoretic setting.
98
2.2
Answer to question 0.5.2
The aim of this section is to prove the following theorem, which is an answer
to question 0.5.2:
Theorem 2.2.1. In the notations of question 0.5.2 and section 2.1, the characteristic Laplacian never respects the bigrading on Q• when the distribution
W is not involutive.
This section is organized as follows. In subsection 2.2.1, we show that
a complex Hermitian manifold is Kähler if and only if the Hodge Laplacian
preserves the bigrading on Ω(X). In subsection 2.2.2, we establish a generalized
sub-Kähler identity. In subsection 2.2.3, we establish theorem 2.2.1.
2.2.1
The classical case
First we study the case where the distribution W is the whole tangent space T X,
which is interesting for itself. We thus have Q = Ω(X) and the characteristic
Laplacian is the usual Hodge Laplacian, which we simply denote by ∆. Remark
that theorem 2.2.1 says nothing in this case.
It is well known that for a Kähler manifold, its Hodge Laplacian preserves
the bigrading of the differential forms (cf. [GH78, §0.7], [MM07, Corollary
1.4.13]). This implies the decomposition of the complex valued de-Rham cohomology in bidegree type for a compact Kähler manifold; this was in fact
the initial interest of the authors for the general question 0.5.2. In [GGK10,
§III. A], Green, Griffiths and Kerr claimed that Chern [Che57] proved that
for Hermitian manifolds, if its Hodge Laplacian preserves the bigrading of the
differential forms, then the Hermitian metric is Kähler. After communications
with Professors Bryant and Griffiths, we realized that Chern did not claim this
result in his paper [Che57], and it seems that one could not find a proof in the
literature.
Theorem 2.2.2. The complex Hermitian manifold (X, J, Θ) is Kähler if and
only if ∆ preserves the bigrading on Ω(X), i.e., ∆ sends (p, q)-forms to (p, q)forms.
We first introduce some notations from [MM07].
For any Z2 -graded vector space V = V + ⊕ V − , the natural Z2 -grading on
End(V ) is defined by
End(V )+ = End(V + )⊕End(V − ),
End(V )− = Hom(V + , V − )⊕Hom(V − , V + ),
and we define deg B = 0 for B ∈ End(V )+ , and deg B = 1 for B ∈ End(V )− .
For B, C ∈ End(V ), we define their supercommutator (or graded Lie bracket)
by
[B, C] = BC − (−1)deg B·deg C CB.
(2.36)
Then for B, B 0 , C ∈ End(V ), the Jacobi identity holds:
0
0
(−1)deg C·deg B B 0 , [B, C] + (−1)deg B ·deg B B, [C, B 0 ]
+ (−1)deg B·deg C C, [B 0 , B] = 0. (2.37)
99
2.2. Answer to question 0.5.2
We will apply the above notation for Ω• (X) with natural Z2 -grading induced
by the parity of the degree, (cf. [MM07, (1.3.31)]).
We define the Lefschetz operator L = Θ ∧ on Λ•,• (T ∗ X) and its adjoint
Λ = i(Θ) with respect to the Hermitian product h·, ·iΛ•,• induced by g T X . For
(1,0) X, we have
{wj }m
j=1 a local orthonormal frame of T
L=
√
−1
m
X
m
√ X
Λ = − −1
iwj iwj ,
wj ∧ wj ∧ ,
j=1
(2.38)
j=1
where ∧ and i denote the exterior and interior product, respectively. The
Hermitian torsion operator is defined by
T := [Λ, ∂Θ] = [i(Θ), ∂Θ] .
(2.39)
Proof of theorem 2.2.2. If (X, J, Θ) is Kähler, then it is a classical result that
∆ preserves the bigrading on Ω• (X), cf. for example [MM07, Corollary 1.4.13]
for a proof.
We assume now that ∆ preserves the bigrading on Ω• (X).
∗
∗
Let := ∂∂ ∗ + ∂ ∗ ∂; := ∂ ∂ + ∂ ∂ be the usual ∂-Laplacian and ∂Laplacian. Then as d = ∂ + ∂ and d2 = 0, we have (cf. [MM07, 1.4.50)])
∗
∗
∆ = [d, d∗ ] = [∂ + ∂, ∂ ∗ + ∂ ] = + + [∂, ∂ ] + [∂, ∂ ∗ ].
(2.40)
∗
As , preserve the bigrading on Ω• (X), and [∂, ∂ ] : Ω•,• (X) → Ω•+1,•−1 (X),
we know that ∆ preserves the bigrading on Ω• (X) if and only if
∗
[∂, ∂ ] = 0.
(2.41)
By the generalized Kähler identities [MM07, (1.4.38d)] (cf. [Dem86]) for
E = C therein, we get
√
∗
∗
Λ, ∂ = −1 ∂ + T .
(2.42)
From (2.42), we get
h
∂, ∂
∗
i
i
√ h i h
∗
= − −1 ∂, Λ, ∂ − ∂, T .
(2.43)
But by (2.37), we get
h
∂, Λ, ∂
i
h
= Λ, ∂, ∂
i
h
i
+ ∂, ∂, Λ .
(2.44)
As ∂, ∂ = 2∂ 2 = 0 and ∂, Λ = − Λ, ∂ , we get from (2.44) that
h
i
∂, Λ, ∂
= 0.
(2.45)
From (2.43) and (2.45), we know that (2.41) is equivalent to
∗
∂ , T = 0.
(2.46)
100
∗
∗
By [MM07, (1.4.9)], the operator ∂ has the form ∂ = −
e TX
j iwj ∇wj ,
P
e T X is certain connection on Λ(T ∗ X), thus
some 0-order terms; here ∇
plus
∗
∂ ,T
is a first order differential operator, and its principal symbol σ is: for ξ ∈ T ∗ X,
√ X
σ(ξ) = − −1 (ξ, wi ) · iwi , T .
(2.47)
j
By [MM07, Lemma 1.4.10, (1.2.48), (1.2.54)],
√
i
h
−1 X
(∂Θ)(wj , wk , wl ) 2 wk ∧wl ∧iwj −2 δjl wk −wj ∧wk ∧iwl . (2.48)
T =−
2 jkl
From (2.48), we get
iw i , T =
√
−1
X
(∂Θ)(wj , wk , wl )wk ∧ iwi , wl ∧ iwj
jkl
=
√
−1
X
(∂Θ)(wj , wk , wi )wk ∧ iwj . (2.49)
jk
By (2.47) and (2.49), the equation (2.46) implies that
∂Θ = 0.
(2.50)
Thus ∂Θ = ∂Θ = 0 and dΘ = 0. This means that if ∆ preserves the bigrading
on Ω• (X), then (X, J, Θ) is Kähler.
Remark 2.2.3. After we sent our preliminary version to Professor Bryant, he
sent us an easier proof which works also in the almost-complex case. Here is
the argument:
Let (X, J, Θ) be an almost complex manifold with almost complex structure
J and Θ a real (1, 1)-form as in (2.27). We suppose that the Hodge Laplacian
∆ preserves the bigrading. In fact, we may only suppose that ∆ sends (0, 1)forms to (0, 1)-forms. In particular, ∆ commutes with J : T X → T X. Using
the following lemma, this implies that J is parallel with respect to the LeviCivita connection ∇T X on (T X, g T X ). It is well-known (cf. [MM07]) that this
condition is equivalent to the metric being Kähler.
Lemma 2.2.4. Let (X, g T X ) be a Riemannian manifold and L ∈ C ∞ (X,
End(T ∗ X)). If L commutes with the Hodge Laplacian ∆ on 1-forms, then
L is parallel with respect to the Levi-Civita connection ∇T X .
∗
∗
Proof. Let ∇Λ(T X) (resp. ∇End(T X) ) be the connection on Λ(T ∗ X) (resp.
End(T ∗ X)) induced by the Levi-Civita connection ∇T X on (T X, g T X ).
For ∇F a connection on a vector bundle F , let ∆F be the Bochner Laplacian on F associated to ∇F . By definition, for {ej }j an orthonormal frame of
(T X, g T X ), we have
F
∆ =−
X j
∇Fej
2
−
∇F∇Te X ej
j
.
(2.51)
101
∗
As ∇Λ(T X) preserves the Z-grading on Λ(T ∗ X), we know that the Bochner
∗
∗
Laplacian ∆Λ(T X) on Λ(T ∗ X) associated to ∇Λ(T X) , also preserves the Z∗
grading on Ω(X). One can relate ∆ and ∆Λ(T X) by the Weitzenböck formula
(cf. [BGV92, §3.6]). In particular, if α ∈ Ω1 (X), one has the equality
∆α = ∆T
∗X
α + Ric α,
(2.52)
where the Ricci curvature Ric is identified with a section of the bundle End(T ∗ X)
by means of g T X .
By (2.51) and (2.52), the principal symbol σ2 (∆) of ∆ is σ2 (∆)(ξ) =
2
|ξ| IdΛ(T ∗ X) for ξ ∈ T ∗ X. Thus σ2 (∆L − L∆) = 0 and ∆L − L∆ is a first order
differential operator. We now compute the principal symbol σ1 (∆L − L∆) by
computing limt→∞ t−1 e−itf (∆L − L∆)eitf when t → +∞ for any f ∈ C ∞ (X).
By (2.51) and (2.52), we know for any s ∈ C ∞ (X, T ∗ X)
σ1 (∆L − L∆)(df )s = lim t−1 e−itf (∆L − L∆)eitf s = −2i(∇End(T
ej
t→∞
∗ X)
L)ej (f )s.
(2.53)
By assumption, one has
∆L − L∆ = 0.
(2.54)
This implies σ1 (∆L − L∆) = 0. Thus from (2.53), we know (2.54) implies
∇End(T
∗ X)
L = 0.
(2.55)
The proof of lemma 2.2.4 is completed.
Remark 2.2.5. In the first proof of theorem 2.2.2, we use the generalized
Kähler identity. When we began to clarify the situation of question 0.5.2, when
there is a distribution, we computed an analogue of the generalized Kähler
identity in this case. This is the object of the following subsection, which is
independent from subsection 2.2.3.
2.2.2
A generalized sub-Kähler identity
The Chern connection ∇Th X on Th X induces a connection on T X and on the
e T X . In
bundle Λ•,• (T ∗ X) ([MM07, §1.2.2]). This connection is denoted by ∇
(1,0)
X the
what follows, we identify Th X with T (1,0) X and thus we denote by ∇T
(0,1) X
T
X
(1,0)
∞
(0,1)
T
h
connection ∇
on T
X. For v ∈ C (X, T
X), we define ∇
v=
(1,0) X
(0,1) X
(1,0) X
T
X
T
T
e
T
∇
v. Then ∇
=∇
⊕∇
. Moreover, we denote by
T ∈ Λ2 (T ∗ X) ⊗ T X
e TX.
the torsion of ∇
By identifying N in (2.10) to Th X/W , N gets a holomorphic structure from
Th X/W . Let πN be the orthogonal projection from Th X onto N . We denote
by h, i the C-bilinear form on T X ⊗R C induced by g T X .
102
∇N
Let hW , hN be the Hermitian metrics on W, N induced by hTh X . Let ∇W ,
be the Chern connections on (W, hW ), (N, hN ). Then we have
∇W = πW ∇Th X πW ,
∇N = πN ∇Th X πN .
(2.56)
As W is a holomorphic subbundle, we know
00
00
00
A = ∇Th X − (∇W ⊕ ∇N ) ∈ T ∗(0,1) X ⊗ Hom(N, W ),
(2.57)
00
where ∇ denotes the (0, 1)-part of a connection ∇. The adjoint A∗ of A takes
values in T ∗(1,0) X ⊗ Hom(W, N ). Note that for w ∈ W, v ∈ N, U ∈ T X ⊗R C,
we have
D
E
hA∗ (U )w, vi = w, A(U )(v) .
(2.58)
Then, under the decomposition Th X = W ⊕ N , we have
∇
Th X
=
∇W
A
∗
−A ∇N
!
.
(2.59)
eW, ∇
e N be the connection on WR , NR induced by ∇W , ∇N as above
Let ∇
or as in [MM07, (1.2.35)]. Set
⊕
∇
TX
eW ⊕ ∇
e N.
=∇
(2.60)
e W , ⊕∇
e T X be the connections on Λ(W ∗ ), Λ(T ∗ X) induced by ∇W , ⊕ ∇T X
Let ∇
R
as in [MM07, §1.2.2], respectively.
(1,0) X such that {w }n
Let {wj }m
j j=1 is an
j=1 be an orthonormal frame of T
orthonormal frame of W . Then by [MM07, Lemma 1.4.4], we have
∂=
m
X
e TX +
wj ∧ ∇
wj
j=1
m
1 X
hT (wj , wk ), wl iwj ∧ wk ∧ iwl ,
2 j,k,l=1
(2.61)
and
∗
∂ =−
m
X
m
X
e TX −
iwj ∇
wj
j=1
hT (wj , wk ), wk iiwj
j,k=1
m
1 X
+
hT (wj , wk ), wl iwl ∧ iwk ∧ iwj .
2 j,k,l=1
(2.62)
Note that for any 1 ≤ j, k ≤ m, U ∈ T X, we have
e T X wk , wj ) = −(wk , ∇
e T X wj ) = −hwk , ∇
e T X wj i = h ∇
e T X wk , wj i,
(∇
U
U
U
U
e T X wk , wj ) = h∇
e T X wk , wj i.
(∇
U
U
(2.63)
From (2.59), for 1 ≤ k ≤ m, n + 1 ≤ γ ≤ m, we get
TX
e T X wγ =⊕ ∇ wγ +
∇
wk
wk
n D
X
E
A(wk )wγ , wj wj .
j=1
(2.64)
103
From (2.59), (2.63) and (2.64), we know that on Λ•,• (T ∗ X),
TX
e T X =⊕ ∇
e
∇
wk
wk
+
m
n h
X
X
D
E
− hA∗ (wk )wj , wβ i wβ ∧ iwj + A(wk )wβ , wj wj ∧ iwβ
β=n+1 j=1
m
n D
E
X
X
e TX +
=⊕ ∇
A(w
)w
,
w
j
k
β
wk
β=n+1 j=1
i
− wβ ∧ iwj + wj ∧ iwβ .
(2.65)
By (2.30), (2.60), (2.61) and (2.65), we know that
∂W =
n
X


eW
wj ∧ ∇
wj
j=1
n
1 X
+
hT (wj , wk ), wl iwk ∧ iwl  .
2 k,l=1
(2.66)
From (2.65), we get for n + 1 ≤ α ≤ m,
e T X πW = −
πW iwα ∇
wα
n D
X
E
wj , A(wα )wα iwj .
(2.67)
j=1
From (2.30), (2.62), (2.65) and (2.67), we get
∗
∂¯W
=
n X
j=1
+
=
m
X
eW −
− iwj ∇
wj
hT (wj , wk ), wk iiwj
k=1
n
1 X
n D
m
X
X
hT (wj , wk ), wl iwl ∧ iwk ∧ iwj +
2 j,k,l=1
n
X
E
wj , A(wα )wα iwj
α=n+1 j=1
n
eW +
iwj − ∇
wj
j=1
n
1 X
hT (wj , wk ), wl iiwk ∧ wl
2 k,l=1
m
D
X
o
E
wj , A(wα )wα − hT (wj , wα ), wα i
+
.
α=n+1
(2.68)
We also generalize the definition of the operators L = Θ∧ and Λ its adjoint,
by defining ΘW ∈ Λ1,1 (WR∗ ) as the restriction to Λ1,1 (WR∗ ) of Θ. We thus get
operators LW and ΛW on Λ•,• (WR∗ ). By (2.38), we have
LW =
√
−1
n
X
wj ∧ wj ∧ ,
n
√ X
ΛW = − −1
iwj iwj .
j=1
(2.69)
j=1
The Hermitian torsion operator is defined as in (2.39) and [MM07, (1.4.34)] by
TW := [ΛW , ∂W ΘW ].
We have the analogue of [MM07, theorem 1.4.11].
(2.70)
104
Proposition 2.2.6. Generalized sub-Kähler identity
[ΛW , ∂W ] =
−
√
√ ∗
∗
−1 ∂¯W + T W
−1
m
n D
X
X
E
wj , A(wα )wα − hT (wj , wα ), wα i iwj πW . (2.71)
α=n+1 j=1
Proof. Set πW,⊥ = Id −πW . By (2.42), we have
√
∗
∗
πW Λ, ∂ πW = −1 ∂ W + πW T πW .
(2.72)
πW ΛπW = ΛπW .
(2.73)
Note that
From (2.73), we know
πW Λ, ∂ πW = πW ΛπW ∂πW + πW ΛπW,⊥ ∂πW − πW ∂πW ΛπW
= ΛW , ∂W + πW ΛπW,⊥ ∂πW . (2.74)
By (2.61) and (2.65), we have
m
X
√
πW ΛπW,⊥ ∂πW = − −1πW
iwγ iwγ πW,⊥ ∂πW
γ=n+1
=
√
n D
m
X
X
−1
E
wj , A(wα )wα iwj πW . (2.75)
α=n+1 j=1
By (2.38), (2.61) and (2.65), we have ∂W ΘW = πW ∂ΘπW . Thus similarly
to (2.74), we have
πW [Λ, ∂Θ]πW = [ΛW , ∂W ΘW ] + πW ΛπW,⊥ ∂ΘπW .
(2.76)
By [MM07, (1.2.48), (1.2.54)], we have
√
m
−1 X
∂Θ =
hT (wi , wj ), wk iwi ∧ wj ∧ wk .
2 i,j,k=1
(2.77)
From (2.38) and (2.77), we know
πW ΛπW,⊥ ∂ΘπW = −
m
n
X
X
hT (wα , wj ), wα i wj πW .
(2.78)
α=n+1 j=1
Taking the adjoint of (2.76), from (2.70) and (2.78), we know
πW T ∗ πW = TW∗ −
m
n
X
X
α=n+1 j=1
hT (wα , wj ), wα i iwj πW .
(2.79)
105
Thus
∗
m
n
X
X
∗
πW T πW = T W −
hT (wα , wj ), wα i iwj πW .
(2.80)
α=n+1 j=1
Finally from (2.72), (2.74), (2.75) and (2.80), we get
[ΛW , ∂W ] =
−
√
√
∗
∗
−1 ∂¯W
+TW
−1
m
n D
X
X
E
wj , A(wα )wα + hT (wα , wj ), wα i iwj πW . (2.81)
α=n+1 j=1
From (2.81), we get (2.71).
Remark 2.2.7. Because of the presence of the double sum in (2.71), we do not
know if there is a nice geometric interpretation for the vanishing of TW , except
in the case where W = T X, when the vanishing is equivalent to the metric
being Kähler, as explained in subsection 2.2.1.
2.2.3
The proof of theorem 2.2.1
b • W ∗ → Λ•+2 W ∗
Remember the construction of the holomorphic map ϕ : F ⊗Λ
in subsection 2.1.4. There we assumed for simplicity that this map has constant
rank. For the purpose of theorem 2.2.1, one can easily reduce to this case.
b kW ∗ →
Indeed, there exists an analytic subset V of X such that ϕ : F ⊗Λ
k+2
∗
b k W ∗)
Λ W has maximum rank on X \ V for any k. In particular, ϕ(F ⊗Λ
forms a vector subbundle of Λk+2 W ∗ on X \ V for any k. We can define the
vector bundles Fϕ and Fϕ,⊥ on X \ V as before. On X \ V , we have the
decompositions (2.32) and (2.33) for forms with compact support; in particular
b ϕ,⊥ ).
Q ∩ Ω•c (X \ V ) = Cc∞ (X \ V, Fϕ,⊥ ⊗F
(2.82)
As X\V is an open connected dense subset of X, the characteristic Laplacian
∆Q preserves the bigrading on Q if and only if it preserves the bigrading on
Q ∩ Ω•c (X \ V ). Thus we can work on X \ V instead of X.
b k W ∗)
From the above discussion, in the rest, we will assume that ϕ(F ⊗Λ
k+2
∗
forms a vector subbundle of Λ W on X for any k. Then we can use the
formalism developed in subsection 2.1.4.
By (2.1), (2.28), as in (2.40), we have
h
∗
i
h
∗
i
h
i
∗
∗
∆Q = ∂Q + ∂ Q , ∂Q
+ ∂ Q = Q + Q + ∂Q , ∂ Q + ∂ Q , ∂Q
.
(2.83)
∗
As Q , Q preserve the bigrading on Q, and [∂Q , ∂ Q ] : Q•,• → Q•+1,•−1 , we
know that ∆Q preserves the bigrading on Q if and only if
h
∗
i
∂Q , ∂ Q = 0.
(2.84)
∗
We would like to understand the operator [∂Q , ∂ Q ].
106
For f ∈ C ∞ (X), by (2.30), we have
∂W f =
n
X
wj (f )wj ∈ W ∗ ,
(2.85)
j=1
(1,0) X such that {w }n
where {wj }m
j j=1 is an
j=1 is an orthonormal frame of T
∗
∗
orthonormal frame of W . For ξ ∈ TR X, let ξ ∈ TR X be the metric dual of ξ.
In particular, if ξ ∈ W ∗ , then ξ ∗ ∈ W .
¯ as in (2.2), we have as maps on Ω(X),
Since J is stable by d, ∂, ∂,
πQ ◦ ∂ ◦ πQ = πQ ◦ ∂,
¯
πQ ◦ ∂¯ ◦ πQ = πQ ◦ ∂,
πQ ◦ ∂¯∗ ◦ πQ = ∂¯∗ ◦ πQ .
(2.86)
∗
Let hFϕ , hFϕ,⊥ be the Hermitian metrics on Fϕ , Fϕ,⊥ induced by hΛW on
ΛW ∗ , which is induced by hW . We recall that Fϕ is a holomorphic vector
subbundle of ΛW ∗ . As in (2.56), let ∇Fϕ , ∇Fϕ,⊥ be the Chern connections on
∗
(Fϕ , hFϕ ), (Fϕ,⊥ , hFϕ,⊥ ). Let ∇ΛW be the connection on ΛW ∗ induced by ∇W ,
∗
∗
then ∇ΛW is the Chern connection on (ΛW ∗ , hΛW ). Set
B = ∇ΛW
∗ 00
00
00
− (∇Fϕ ⊕ ∇Fϕ,⊥ ) ∈ T ∗(0,1) X ⊗ Hom(Fϕ,⊥ , Fϕ ).
(2.87)
The adjoint B ∗ of B takes values in T ∗(1,0) X ⊗ Hom(Fϕ , Fϕ,⊥ ). Then under the
decomposition ΛW ∗ = Fϕ ⊕ Fϕ,⊥ , we have
∇
ΛW ∗
!
∇Fϕ
B
.
−B ∗ ∇Fϕ,⊥
=
(2.88)
∗
b
b ϕ,⊥ ,
We denote also πQ the orthogonal projection from ΛW ∗ ⊗ΛW
onto Fϕ,⊥ ⊗F
⊥
and πQ = IdΛW ∗ ⊗
b ΛW ∗ −πQ . Then
πQ = πFϕ,⊥ ⊗ π Fϕ,⊥ ,
⊥
πQ
= πFϕ ⊗ π Fϕ,⊥ + πFϕ,⊥ ⊗ π Fϕ + πFϕ ⊗ π Fϕ .
(2.89)
∗
operator acting
Lemma 2.2.8. The operator [∂Q , ∂ Q ] is a first order differential √
b ϕ,⊥ . Its principal symbol, evaluated on ξ ∈ T ∗ X, is −1 times
on Fϕ,⊥ ⊗F
πQ
n
X
h
i
iwj wk ∧ (ξW , T (wj , wk )) − (ξ, πN [wj , wk ]) πQ
j,k=1
+
n
X
∗
b wj − πQ
(B (wj )πFϕ ξW )⊗i
j=1
n
X
b (ξ )∗ B(wj ), (2.90)
wj ⊗i
W
j=1
where ξW is the orthogonal projection of ξ on W ∗
Remark 2.2.9. Theorem 2.2.1 is an easy corollary of (2.90). Indeed, if we take
ξ a holomorphic one-form which is orthogonal to W ∗ , the principal symbol is
n
X
√
− −1πQ
iwj wk ∧ (ξ, πN [wj , wk ])πQ .
j,k=1
107
Evaluating at wj , which belongs to Fϕ,⊥ , one gets
√
−1
n
X
(ξ, πN [wj , wk ])wk .
k=1
This term vanishes for any j and ξ if and only if the distribution is involutive,
which shows theorem 2.2.1.
b ϕ,⊥ , by (2.16), (2.24)
Proof of lemma 2.2.8. Note that for ξ ∈ W ∗ , ψ ∈ Fϕ,⊥ ⊗F
⊥
b ϕ,⊥ . Thus by (2.24), iξ ∗ π ⊥ (ξ ∧ ψ) ∈
and (2.89), we know that πQ (ξ ∧ ψ) ∈ Fϕ ⊗F
Q
b ϕ,⊥ , as ξ ∗ ∈ W , and this implies
Fϕ ⊗F
⊥
πQ iξ∗ πQ
ξ ∧ πQ = 0
for any ξ ∈ W ∗ .
(2.91)
∗
We compute now the principal symbol of [∂Q , ∂ Q ] by computing the asymp∗
totics of e−itf [∂Q , ∂ Q ]eitf when t → +∞ for f ∈ C ∞ (X). By (2.31), (2.66) and
(2.68), we have first
e−itf ∂Q eitf = it πQ ∂W f ∧ πQ + ∂Q ,
∗
(2.92)
∗
e−itf ∂ Q eitf = −it πQ i(∂W f )∗ πQ + ∂ Q .
∗
Thus from (2.24), (2.92), the principal symbol of [∂Q , ∂ Q ] as a second order
∗
differential operator is limt→∞ t−2 e−itf [∂Q , ∂ Q ]eitf , that is
h
iπQ ∂W f ∧ πQ , −iπQ i(∂W f )∗ πQ
i
= πQ ∂W f ∧ i(∂W f )∗ πQ + πQ i(∂W f )∗ πQ ∂W f ∧ πQ
⊥
= −πQ i(∂W f )∗ πQ
∂W f ∧ πQ = 0, (2.93)
∗
here we use (2.91) in the last equality. The equation (2.93) means that [∂Q , ∂ Q ]
is a first order differential operator.
∗
By (2.92), the principal symbol of [∂Q , ∂ Q ] as a first order differential oper∗
ator is limt→∞ t−1 e−itf [∂Q , ∂ Q ]eitf , that is
∗
h
i
h
i
i πQ ∂W f ∧ πQ , ∂ Q − i ∂Q , πQ i(∂W f )∗ πQ .
(2.94)
By (2.24) and (2.31), we get
∗
∗
∗
πQ + πQ ∂¯W
πQ ∂W f ∧ πQ
πQ ∂W f ∧ πQ , ∂ Q = πQ ∂W f ∧ ∂¯W
i
h
∗
∗ ⊥
= πQ ∂¯W
(∂W f )πQ − πQ ∂¯W
πQ ∂W f ∧ πQ . (2.95)
Again by (2.24) and (2.31), we get
h
− ∂Q , πQ i(∂W f )∗ ∧ πQ
i
= −πQ i(∂W f )∗ πQ ∂W πQ − πQ ∂W i(∂W f )∗ πQ
⊥
= −πQ ∂W (i(∂W f )∗ ) ∧ πQ + πQ i(∂W f )∗ πQ
∂W πQ . (2.96)
108
By (2.68) and (2.85), we get

∗
∗
∂¯W
(∂W f ) = [∂¯W
, ∂W f ] = − 
n
X

e W , ∂W f 
iwj ∇
wj
j=1
=−
n
X
h
e W wk
iwj wj (wk (f ))wk + wk (f )∇
wj
i
j,k=1
n
X
=−
h
i
iwj wk wj (wk (f )) − (∂W f, ∇W
wj wk ) . (2.97)
j,k=1
By (2.66) and (2.85), we get
− ∂W (i(∂W f )∗ ) = −[∂W , i(∂W f )∗ ] = −
" n
X
#
eW ,i
wk ∇
wk (∂W f )∗
k=1
=−
n
X
i
h
wk ∧ wk (wj (f ))iwj + wj (f )i∇W
w
k
j,k=1
n
X
=
wj
h
i
iwj wk wk (wj (f )) − (∂W f, ∇W
wk wj ) . (2.98)
j,k=1
By (2.59), (2.97) and (2.98), we get
∗
∂¯W
(∂W f ) − ∂W (i(∂W f )∗ )
n
X
=
h
W
iwj wk (∂f, −[wj , wk ]) + (∂W f, ∇W
wj wk − ∇wk wj )
i
j,k=1
=
n
X
h
i
iwj wk (∂W f, T (wj , wk )) − (∂f, πN [wj , wk ]) . (2.99)
j,k=1
From (2.94)–(2.99), we know
√ that the principal symbol of the first order differ∗
ential operator [∂Q , ∂ Q ] is −1 times
πQ
n
X
iwj wk [(∂W f, T (wj , wk )) − (∂f, πN [wj , wk ])] πQ
j,k=1
∗ ⊥
⊥
− πQ ∂¯W
πQ ∂W f ∧ πQ + πQ i(∂W f )∗ πQ
∂W πQ . (2.100)
⊥∂ f ∧ π ⊂ F ⊗ F
Now by (2.89), πQ
ϕ
W
Q
ϕ,⊥ . By (2.24), (2.68) and (2.88), we
know that
∗ ⊥
πQ ∂¯W
πQ ∂W f ∧ πQ = πQ
n X
e W π ⊥ ∂W f ∧ πQ
− iwj ∇
wj
Q
j=1
= πQ
n X
j=1
⊥
iwj B ∗ (wj ) ⊗ 1 πQ
∂W f ∧ π Q = −
n
X
b wj .
(B ∗ (wj )πFϕ ∂W f ∧)⊗i
j=1
(2.101)
109
Let P be the orthogonal projection from Λ•,• (WR∗ ) onto Fϕ,⊥ ⊗ F ϕ . Note that
∗
∗
⊥ ∂ π ⊂ F ⊗ ΛW ∗ ⊕ F
πQ
ϕ
W Q
ϕ,⊥ ⊗ F ϕ , as i(∂W f )∗ Fϕ ⊗ ΛW ⊂ Fϕ ⊗ ΛW , from
(2.66), (2.88) and (2.89), we have also
⊥
πQ i(∂W f )∗ πQ
∂W πQ = πQ i(∂W f )∗ P ∂W πQ
= πQ i(∂W f )∗ P
n
X
e W πQ = −πQ
wj ∧ ∇
wj
j=1
n
X
b (∂ f )∗ B(wj ). (2.102)
wj ⊗i
W
j=1
The proof of lemma 2.2.8 is completed.
We finally give an explicit computation of this phenomenon, in the setting
of example 2.1.10, for the choice of the standard metric on C3 .
We start with a form µ = µ1 α1 + µ2 α2 in C ∞ (U, W ∗ ) = Q1,0 (U ).
∗ ]µ is given by
Lemma 2.2.10. The 1-form [∂¯Q , ∂Q
∗
[∂¯Q , ∂Q
]µ =
2 X
X̄j . (X̄i .µj )(αj ∧ ᾱi , αj ∧ ᾱi )
(αi , αi )
i,j=1
− X̄i . X̄j .(µj (αj , αj )) ᾱi
Proof. By the equation (2.34), we have
2
X
∂¯Q µ =
(X̄j .µi )ᾱj ∧ αi =:
2
X
βij ᾱj ∧ αi = β.
i,j=1
i,j=1
∗
∗ β. Let ν = ν ᾱ1 + ν ᾱ2 be in C ∞ (U, W ) = Q0,1 (U ).
Let us compute ∂Q
2
1
Then,
∗
(∂Q
β, ν) = (β, ∂Q ν)
2
X
= (β,
(Xj .νi )αj ∧ ᾱi )
i,j=1
=
2
X
Z
(X̄j .ν¯i )(β, αj ∧ ᾱi )dνX
i,j=1 M
=−
2 Z
X
ν¯i X̄j .(β, αj ∧ ᾱi )dνX since X̄j has zero divergence.
i,j=1 M
∗ β = γ ᾱ1 + γ ᾱ2 , where γ satisfies
This shows that ∂Q
1
2
i
P2
γi = −
=
j=1 X̄j .(β, α
(ᾱi , ᾱi )
j
∧ ᾱi )
2
X
X̄j .(βji (αj ∧ ᾱi , αj ∧ ᾱi ))
j=1
(αi , αi )
.
110
Hence,
∗ ¯
∂Q
∂Q µ
2
X
X̄j . (X̄i .µj )(αj ∧ ᾱi , αj ∧ ᾱi ) i
=
ᾱ .
i
i
(α , α )
i,j=1
On the other hand, since the Xi have zero divergence and by equation (2.35),
one has
∗
∂Q
µ=−
2
X
X̄j .(µj (αj , αj ))
j=1
Hence,
∗
∂¯Q ∂Q
µ=−
2
X
X̄i . X̄j .(µj (αj , αj )) ᾱi .
i,j=1
(αj
In order to simplify the notations, we write Ni := (αi , αi ) and Nij :=
∧ ᾱi , αj ∧ ᾱi ). In particular, Nij = Ni .Nj . Then,
∗
[∂¯Q , ∂Q
]µ =
2 X
X̄j . (X̄i .µj )Nij
i,j=1
=
2 X
i,j=1
Ni
− X̄i .X̄j .(µj Nj ) ᾱi
[X̄j , X̄i ].µj × Nj + (X̄i .µj )(
X̄j .Nji
− X̄j .Nj )
Ni
− (X̄j .µj )(X̄i .Nj ) − µj X̄i .X̄j .Nj ᾱi .
Now, we take µ1 = 0 and consider the term before ᾱ1 . We consider a
function µ2 that, at some fixed point z0 ∈ U satisfies: µ2 (z0 ) = 0, (X̄1 .µ2 )(z0 ) =
(X̄2 .µ2 )(z0 ) = 0 and ([X̄2 , X̄1 ].µ2 )(z0 ) 6= 0. Such a function exists since the
bracket [X̄1 , X̄2 ] is not contained in the vector subspace generated by X̄1 and
X̄2 .
∗ ]µ does not vanish at z , which proves that in this particular
Then, [∂¯Q , ∂Q
0
example the characteristic Laplacian cannot respect the bigrading.
Appendix A
Differential geometry of
infinite-dimensional manifolds
In this appendix, we collect some definitions and results of infinite-dimensional
differential geometry. A reference for this subject is the book [Lan96].
A.1
Banach manifolds
Definition A.1.1. Let X be a set. An atlas on X is a collection of pairs
(Ui , φi )i∈I such that
• the collection (Ui )i∈I is a covering of X;
• each φi is a bijection from Ui to an open subset of a Banach space Ei ;
• for any i, j ∈ I, the set φi (Ui ∩ Uj ) is open in Ei and the map
φi ◦ φ−1
j : φj (Ui ∩ Uj ) → φi (Ui ∩ Uj )
is a diffeomorphism.
Remark A.1.2. There is a unique topology on X such that the φi are homeomorphisms. This topology will always be metrizable and separable in the cases
we are concerned with.
Definition A.1.3. Two atlases on a set X are compatible if their union is still
an atlas. A structure of Banach manifold on X is an equivalence class of atlases.
Remark A.1.4. A Banach manifold X is thus locally modeled on some Banach space E. The isomorphism class of E (in the category of topological vector
spaces) is well determined in a connected component of X. All Banach manifolds that we will consider – though not necessarily connected – will be modeled
on a unique Banach space.
We can define in the same way Hilbert manifolds, which are locally modeled
on Hilbert spaces. They are in particular Banach manifolds. We can speak of
smooth maps between Banach manifolds, as in the finite-dimensional setting.
If the Banach space E is a complex Banach space and the maps φi ◦ φ−1
are
j
holomorphic, we say that X has a structure of complex Banach manifold, or
complex Hilbert manifold if E is a complex Hilbert manifold.
112
A.2
Appendix A. Geometry of infinite-dimensional manifolds
Banach bundles
If E1 and E2 are Banach spaces, we write Hom(E1 , E2 ) for the space of bounded
linear morphisms from E1 to E2 . We recall that Hom(E1 , E2 ) is itself a Banach
space for the operator norm, hence a Banach manifold. If E is a Banach space,
we write End(E) for Hom(E, E) and Aut(E) for its set of invertible elements.
It is an open set in End(E), hence it inherits a structure of Banach manifold
too.
Definition A.2.1. ([Lan96], section III.1) Let X be a Banach manifold, E a
topological space and π : E → X a surjective continuous map. We assume that
each fiber Ex := π −1 (x) has been given the topology of a Banach space. A
trivializing cover of π : E → X is a collection of pairs (Ui , τi )i∈I such that
• the collection (Ui )i∈I is an open covering of X;
• each τi is a homeomorphism from π −1 (Ui ) to Ui ×Ei , where Ei is a Banach
space;
• writing pr1 : Ui × Ei → Ui for the first projection, τi satisfies pr1 ◦τi =
π|π−1 (Ui ) ;
• for every x in Ui the induced map τi,x : Ex → Ei is an isomorphism in
the category of topological vector spaces;
• for every i, j ∈ I, the natural map τi ◦ τj−1 : Ui ∩ Uj → Hom(Ej , Ei ) is
smooth.
Remark A.2.2. We emphasize that there is no natural norm on a fiber Ex :
only its topology of Banach space is well defined.
Definition A.2.3. Two trivializing covers of π : E → X are compatible if their
union is still a trivializing cover. A structure of Banach bundle on π : E → X
is an equivalence class of trivializing covers.
Remark A.2.4. In each Banach bundle that we will consider, all fibers will be
isomorphic, as topological vector spaces.
We can define Hilbert bundles in the same way and speak of morphism
between Banach bundles. A holomorphic Banach bundle is a complex Banach
bundle such that the transition maps τi ◦ τj−1 are holomorphic. Given a smooth
map f : X → Y and a Banach bundle π : E → Y , we define the pullback bundle
π : f ∗ E → X as in the finite-dimensional setting. Finally, we define the tangent
space of a Banach manifold X in the usual way; it has a natural structure of
Banach bundle over X.
Definition A.2.5. Let X be a Banach manifold and let Y be a subset of
X. We assume that Y has a structure of Banach manifold whose underlying
topology is the topology induced by X. We say that Y is a Banach submanifold
of X if the inclusion i of Y in X is an immersion, that is if for every y in Y ,
dy i : Ty Y → Ty X is injective and if dy i(Ty T ) is a closed complemented Banach
subspace of Ty X.
113
A.2. Banach bundles
We recall that a closed subspace F of a Banach subspace E is called complemented if there exists a closed subspace G such that F = E ⊕ G. This is
automatic in Hilbert spaces.
The following lemma is useful to define a structure of Banach bundle.
Lemma A.2.6. ([Lan96], III, Proposition 1.2) Let X be a Banach manifold,
let E be a set and let π : E → X be a map. Suppose that we are given a Banach
space E, an open covering (Ui )i∈I of X, bijections τi : π −1 (Ui ) → Ui × E
commuting with the first projection such that
• the maps τi ◦ τj−1 : Ui ∩ Uj × E → Ui ∩ Uj × E are isomorphisms of
topological vector spaces in the second variable;
• the maps Ui ∩ Uj → Aut(E) are smooth;
• the τi ’s satisfy the cocycle condition.
Then, there is a unique structure of Banach bundle on π : E → X such that
the collection (Ui , τi ) is a trivializing cover.
Let π : E → X be a complex vector bundle of rank r over X. The next
proposition endows the union of the Hilbertizable spaces L2 (S 1 , Ex ) with the
structure of a Hilbert bundle over X. We denote by L2 (S 1 , E) this union and
by p the natural projection L2 (S 1 , E) → X.
Proposition A.2.7. Let (Ui , τi : π −1 (Ui ) → Ui × Cr ) be a trivializing cover of
π : E → X. Write H for the Hilbert space L2 (S 1 , Cr ), τ˜i : p−1 (Ui ) → Ui × H
for the map given by τ˜i (x, f ) = (x, τi ◦ f ). Then, (Ui , τ˜i ) satisfy the conditions
of lemma A.2.6 and endow L2 (S 1 , E) with the structure of a Hilbert bundle.
Moreover, this structure is independent of the chosen trivializing cover.
Proof. The vector space L2 (S 1 , Ex ) has the topology of a Hilbert space, by
choosing an arbitrary norm on Ex . It is clear that the τ˜i are well-defined
bijections, commuting with the first projection and isomorphisms of topological
vector spaces in the second variable. They satisfy the cocycle condition since
the τi satisfy it. The only point to check is that the maps Ui ∩ Uj → Aut(H)
are smooth. Indeed, since the union of two trivializing covers of X is still a
trivializing cover, the two Hilbert bundle structures defined by two different
trivializing covers will automatically be the same.
Let gij : Uij → GL(r, C) be a transition function for the vector bundle E.
The map Ui ∩ Uj → Aut(H) is given by
g̃ij : x 7→ (f 7→ gij (x)f (·)).
The next lemma and an induction argument conclude the proof.
Lemma A.2.8. Let U be an open set in Rn , A : U → M (r, C) be a C 1 map.
Then the map Ã : U → End(L2 (S 1 , Cr )) given by
Ã : x 7→ (f 7→ A(x)f (·))
is C 1 with directional derivative in the direction h ∈ Rn given by
Dh Ã : x 7→ (f 7→ Dh A(x)f (·)).
114
Proof. If M is any element in M (r, C) and one defines M̃ in the same way as an
operator in L2 (S 1 , Cr ), then the operator norm of M̃ (for the canonical Hilbert
structure on L2 (S 1 , Cr )) is the operator norm of M for the canonical Hermitian
structure on Cr . Applying this remark to 1t (A(x+th)−A(x)−tDh A(x)) proves
the formula for Dh Ã. The same remark implies that Dh Ã is continuous since
Dh A is.
A.3
Connections on Banach bundles
Definition A.3.1. Let π : E → X be a Banach bundle over a Banach manifold
X. A connection D on E is a local operator, from the sections of T X ⊗ E to
the sections of E such that, in a local trivialization EU ∼
= U × E, D can be
written D = d + A, where A : T U × E → E is smooth as a map from T U to
End(E). As usual, a connection D is C ∞ (X)-linear in the T X-direction and
satisfies Leibniz’s rule.
Remark A.3.2. The space of connections on a Banach bundle π : E → X
is thus an affine space, modeled on the space of global sections of the Banach
bundle Hom(T X, End(E)).
Let π : E → X be a Banach manifold over a finite-dimensional connected
differentiable manifold X. A connection D on E induces differential operators
from Λi T ∗ X ⊗ E to Λi+1 T ∗ ⊗ E. The curvature of D is the differential operator
D2 : E → Λ2 T ∗ X ⊗ E. It is in fact tensorial: that is given by a section FD of
the Banach bundle Λ2 T ∗ X ⊗ End(E). We say that the connection D is flat if its
curvature vanishes. The following correspondence between representations of
the fundamental group and flat connections is well-known for finite-dimensional
flat bundles and the same proof works in the Banach setting. This will be used
without mention in subsection 1.1.4.
Proposition A.3.3. Let x0 be a base-point in X and let p : X̃ → X be the
universal cover of X. If ρ : π1 (X, x0 ) → Aut(E) is a representation of π1 (X, x0 )
in the space of bounded automorphisms of E, one defines a Banach bundle on
X by
Eρ := X̃ ×ρ E.
It is endowed with a flat connection D, which comes from the canonical connection d on the trivial bundle X̃ × E → X̃.
Conversely, if (E, D) is a Hilbert bundle with flat connection over X, then
the parallel transport defines a monodromy representation ρ : π1 (X, x0 ) →
Aut(Ex0 ).
These two constructions give an equivalence of categories between the category of representations of π1 (X, x0 ) in the bounded automorphisms of a Banach
space and the category of Banach bundles endowed with a flat connection over
X.
Let us specialize the discussion to the Hilbert bundle H = L2 (S 1 , E), where
E is some complex finite-dimensional bundle over X. We want to understand
when a connection on H comes from connections in E.
A.4. Banach-Lie groups
115
Let H be the Hilbert space L2 (S 1 , Cn ) and let T denote the right-shift
operator. There is an embedding of Banach spaces
L∞ (S 1 , M (n, C)) ,→ End(H)
given by A 7→ (f (λ) 7→ A(λ)f (λ)). We recall the following:
Proposition A.3.4 ([PS88], Theorem 6.1.1). The commutant of T in End(H)
is L∞ (S 1 , M (n, C)).
Let D be an arbitrary connection on the vector bundle E; it induces a
connection D̃ on H = L2 (S 1 , E) by the formula
(D̃Y f )(λ, v) = (DY f (λ, ·))(v),
where f is a local section of H, Y is a local vector field on X, λ is in S 1 and
v is in E. The connections D and D̃ give an origin to the space of connections
of E and L2 (S 1 , E). Hence, this identifies these spaces to Γ(X, T ∗ X ⊗ End(E))
and to Γ(X, T ∗ X ⊗ End(L2 (S 1 , E))). There is a natural embedding of Banach
bundles L∞ (S 1 , End(E)) → End(L2 (S 1 , E)). We can now state the corollary of
proposition A.3.4 in a geometrical setting:
˜ on L2 (S 1 , E) lives in the affine subspace
Corollary A.3.5. A connection ∇
∗
∞
1
D̃ + Γ(X, T X ⊗ L (S , End(E))) if and only if the right-shift operator T is
˜
parallel for ∇.
Hence, if (∇λ )λ∈S 1 is a circle of connections on E (with weak conditions on
˜ on H by the
the regularity with respect to λ), one can define a connection ∇
formula
˜ Y f )(λ, v) = ((∇λ )Y f (λ, ·))(v),
(∇
where f is a local section of H, Y is a local vector field on X, λ is in S 1 and v
˜ on H is obtained in this way if and only if
is in E. Conversely, a connection ∇
˜
the right-shift operator T is parallel for ∇.
A.4
Banach-Lie groups
A Banach-Lie group is a Banach manifold G, with a structure of group, such
that the multiplication map m : G × G → G and the inverse map ι : G → G are
smooth. If G is a complex Banach manifold, we ask that the group operations
are holomorphic and speak of a complex Banach-Lie group. The tangent space
of G at the identity element is the Lie algebra of G. It is denoted by g and is
a Banach-Lie algebra, meaning that the bracket is continuous for the Banach
space topology.
Since the general theory of ordinary differential equations works in a Banach setting, one can define the exponential map exp : g → G. It is a local
diffeomorphism in a neighborhood of 0 ∈ g. Indeed, its differential at 0 is the
identity of g by definition and the local inversion theorem is true for Banach
spaces.
116
A Banach-Lie subgroup H of a Banach-Lie group G is a subgroup of G
which is also a Banach submanifold of G. In particular, the Lie algebra of H is
complemented in the Lie algebra of G. The following proposition is called the
homogeneous space construction theorem ([Bou72], Proposition III.1.6.11).
Proposition A.4.1. If H is a Banach-Lie subgroup of a Banach-Lie group G,
then there exists a unique structure of Banach manifold on the quotient space
G/H such that the projection G → G/H is a smooth submersion.
Appendix B
Moduli spaces
In this appendix, we collect some facts about the various moduli spaces that
one can attach to a compact Kähler manifold. This is only used in the proof of
theorem 1.5.17.
B.1
B.1.1
Betti moduli space
Construction of the moduli space
Let Γ be a finitely presented group and let G = GL(n, C). We briefly review
the construction of the moduli space of representations X(Γ, G), which is a
categorical quotient (in the sense of geometric invariant theory) for the action
of G by conjugation on the space of representations R(Γ, G). All this material is
contained in [JM87], (though in the case G semisimple rather than reductive).
Let γ1 , . . . , γN be generators of Γ and let r1 , . . . , rk be relators that define a presentation of G. We define R(Γ, G) to be the subset of GN of elements
(M1 , . . . , MN ) satisfying the relations ri (M1 , . . . , MN ) = 0. In this way, R(Γ, G)
is realized as an affine algebraic variety and this affine algebraic structure is independent of the presentation of G. The quotient variety X(Γ, G) is by definition
the affine variety corresponding to the algebra of G-invariant polynomials on
R(Γ, G). We write π : R(Γ, G) → X(Γ, G). The map π does not in general
induce a bijection R(Γ, G)/G ∼
= X(Γ, G).
If U is a real form of G, both R(Γ, G) and X(Γ, G) are defined over the real
numbers. Moreover, one can define in the same way a space X(Γ, U ) and this
space is contained in the real points of X(Γ, G).
A representation ρ in R(Γ, G) is stable if its orbit under conjugation is closed
in R(Γ, G). It is equivalent to the condition that Im ρ is not contained in any
proper parabolic subgroup of G; since G = GL(n, C), this simply means that
the representation is irreducible.
We write Rs (Γ, G) for the set of stable representations. It is Zariski-open
in R(Γ, G). Its image by π is written X s (Γ, G); it is Zariski-open in X(Γ, G).
If Γ is the fundamental group of some topological space X, we write MB (X)
for X s (Γ, G) and call it the Betti moduli space of X. The map π : Rs (Γ, G) →
X s (Γ, G) induces a homeomorphism Rs (Γ, G)/G ∼
= X s (Γ, G). If U is a real form
118
Appendix B. Moduli spaces
of G, we define in the same way X s (Γ, U ). It is homeomorphic to (Rs (Γ, G) ∩
R(Γ, U ))/U and is a subset of the real points of X s (Γ, G).
B.1.2
Variations of Hodge structures
We write Z for the equivalence classes in X s (Γ, G) of representations whose
associated flat bundles admit a complex variation of Hodge structures. We will
need the following lemma:
Lemma B.1.1. The subset Z is contained in a totally real analytic subspace
of the complex-analytic space X s (Γ, G)
Proof. If ρ is a representation coming from a complex variation of Hodge structures, then it is conjugated to a representation with values in a real form U (p, q)
of G. Hence, Z is contained in the union of the sets X s (Γ, U (p, q)). Each of
these sets is included in the set of real points (for different real structures) of
X s (Γ, G); in particular, each one is included in a totally real analytic subset
of the complex-analytic space X s (Γ, G). The union being finite, this concludes
the proof.
B.2
B.2.1
Hyperkähler structure
Space of connections
Let M be a compact Kähler manifold, with Kähler form ω. Let (E, h) be a
Hermitian vector bundle on X. We choose an integer k ≥ dim M + 2 and
consider the spaces A (resp AR ) of connections with Sobolev regularity Hk on
E (resp. metric connections on (E, h)). The space A has a natural structure of
a hyperkähler affine Hilbert space defined in the following way:
We choose a metric connection D0 so that we identifify A (resp. AR ) to the
Hilbert spaces Hk1 (M, End(E)) (resp. Hk1 (M, u(E, h))): these notations stand
for the space of global 1-forms with regularity Hk in the bundles End(E) and
u(E, h). From the complex structure on M , the space AR = Hk1 (M, u(E, h))
inherits a complex structure C. It is also endowed with a C-invariant inner
product < ·, · > defined by
Z
< φ, ψ >=
< φ(x), ψ(x) >x
M
ωn
,
n!
where the inner product < ·, · >x on Tx∗ M ⊗u(Ex , hx ) is induced from the Kähler
metric on Tx M and from minus the Killing form on u(Ex , hx ).
We consider A as the complexification of AR and we write J for the corresponding complex structure. We extend C and < ·, · > by complex linearity
to A. Then h(x, y) :=< x, ȳ > (the bar is relative to the real form AR ) is a
Hermitian metric on A; we write g for its real part. Finally, we write I for the
unique antilinear extension of C to A. Then, ([Fuj91], (4.1)):
Lemma B.2.1. The space (A, g, I, J) is hyperkähler.
119
B.2. Hyperkähler structure
There is also a natural S 1 -action on A ([Fuj91], (4.2)). Let λ be a unit
number in C∗ and let D be an element in A. As usual, we write D = ∇ + α,
where ∇ is a metric connection and α is a Hermitian 1-form. We decompose α
in types : α = α1,0 + α0,1 . The action of S 1 on A is given by
λ.D = ∇ + λ−2 α1,0 + λ2 α0,1 .
B.2.2
(B.1)
Hyperkählerian reduction
A connection D in A is Einstein if its curvature FD satisfies
i Tr(FD ) = λ IdE ,
where λ is some constant depending only on M , ω and E and where the trace
operator is defined from Hk1,1 (M, End(E)) to Hk (M, End(E)) and is induced by
the Kähler metric.
Let D be a connection in A and write D = ∇ + α. We write ∇∗ for the
formal adjoint of ∇ and we say that D is weakly harmonic if the equation
∇∗ α = 0
is satisfied.
A connection D is irreducible if there is no non-trivial subbundle F of E,
which is stable by the connection.
A remarkable point is that both the Einstein and weakly harmonic conditions can be interpreted as asking that the connections are in the zero sets
of moment maps, relatively to the hyperkähler structure on A and the action of the complex gauge group G := Hk+1 (M, GL(E)) and its real form
K := Hk+1 (M, U (E, h)) ([Fuj91], (6.2) and (6.3)). We write F̃ for the subset of
A of irreducible, Einstein and weakly harmonic connections. Then hyperkählerian reduction leads to:
Theorem B.2.2 ([Fuj91], (6.6.1)). We write F = F̃/K for the set of equivalence classes of irreducible, Einstein and weakly harmonic connections on (E, h).
Then, F inherits from A a structure of hyperkähler Hilbert orbifold and a S 1 action.
B.2.3
Hyperkähler structure on the moduli space
We now assume that the first and second Chern classes of E vanish. We write
N for the set of isomorphism classes of connections D that are irreducible, flat
and admit an harmonic metric. Then, ([Fuj91], (8.1.2)):
Lemma B.2.3. The space N can be naturally realized as a finite-dimensional
locally closed complex analytic subspace of F.
In subsection B.1, we have defined the Betti moduli space MB (X) of irreducible representations from π1 (X) to GL(n, C). We consider its subset
MB,E (X) of representations whose associated bundle is isomorphic to E as
120
Appendix B. Moduli spaces
a smooth vector bundle. It is a union of connected components of MB . By
the Corlette-Donaldson theorem 0.2.4, the spaces N and MB,E are in bijection.
Moreover,
Lemma B.2.4 ([Fuj91], (8.2.3)). The complex analytic structures of N and
MB,E are the same.
We finally write M0 for the set of smooth points in the underlying reduced
space of MB,E .
Theorem B.2.5 ([Fuj91], (8.3.1)). By the inclusion M0 ,→ F, M0 inherits a
hyperkähler structure. Moreover, M0 is stable by the action of S 1 .
Remark B.2.6. Much more is said in [Fuj91]. Our aim here is only to define
the hyperkähler structure on the Betti moduli space of X. If one is interested
in Higgs bundles, a moduli space of stable Higgs bundles can be defined (for X
projective, this is done in [Sim94a] and [Sim94b]) and its set of smooth points
will be in bijection with M0 . Moreover, its complex structure will be one of
the complex structures of the hyperkähler manifold M0 . We omit the details
and refer the interested reader to [Fuj91].
B.2.4
Variations of Hodge structures
The fixed points of the circle action on M0 are the irreducible representations
whose associated flat bundle admits a complex variation of Hodge structures.
This comes from equation (B.1) and the well-known corollary 4.2 of [Sim92],
which is our theorem 1.2.3, in the context of variations of loop Hodge structures.
In the space A, we consider the functional
Z
F (D) = i
Tr(φ1,0 ∧ φ0,1 ) ∧ ω n−1 ,
M
where as usual, we write D = ∇ + φ. There exists a real constant C such that
F (D) = C.g(φ, φ), where g is the hyperkähler metric of A.
The metric is invariant under the unitary gauge group K, hence F defines
a function on F, and in particular on M0 . We still denote this function by
F . The tangent space of A at the point D can be canonically identified with
Hk1 (M, u(E, h)) × Hk1 (M, iu(E, h)). Hence, the differential of F is given by
dD F (χ, ψ) = 2Cg(φ, ψ).
On the other hand, the infinitesimal vector field Y generating the circle action
on A is given by YD = −2iφ1,0 + 2iφ0,1 = 2Iφ, where the complex structure I
was defined before lemma B.2.1. Hence, the gradient of F is given by C.IY .
Since the hyperkähler structure of A induces a hyperkähler structure on
M0 , the complex structure I is in particular well-defined on M0 and we get
the following theorem:
Theorem B.2.7. The critical values of F on M0 are the irreducible representations whose associated flat bundle admits a complex variation of Hodge
structures.
Appendix C
SL(2)-orbit theorem
Two crucial results for the understanding of the asymptotic of variations of
Hodge structures are proved in [Sch73]: the nilpotent orbit theorem, that we
generalized to variations of loop Hodge structures in section 1.4, and the SL(2)orbit theorem. The nilpotent orbit theorem roughly says that the behaviour of a
variation of Hodge structures at infinity looks like the behaviour of a variation
obtained from some nilpotent orbit. The SL(2)-orbit theorem says that any
nilpotent orbit can be itself approached – in a rather subtle way – by an orbit
constructed from a representation of SL(2, R), reducing the understanding of
the asymptotic to Lie group-theoretic questions.
In this section, we present a possible generalization of the SL(2)-orbit theorem to variations of loop Hodge structures and some of its corollaries. We
follow closely the presentation of [Sch73]. The author adresses two warnings
to the reader at this point: the first is that we give (and have) no proof of
the asserted facts, though we prefer to write Proposition or Theorem rather
than Conjecture everywhere. The proof of the SL(2)-orbit theorem is intricate
and it is not clear – but quite plausible – whether one can generalize it to the
infinite-dimensional loop period domain, as in section 1.4.
The second is that the author does not have a good knowledge of the literature on the asymptotic of harmonic bundles; hence we will not try to distinguish
well-known facts from conjectured new ones and we will not discuss the relation
with the approach presented here and the one used in [Moc07a] and [Moc07b]
for instance.
The whole section is thus conjectural and should be read as the sketch of a
future work.
C.1
Statement of the theorem
We consider the upper half-plane H and the period domain D = Λσ G/K, which
is open in its compact dual Ď = ΛG/Λ+ G.
Definition C.1.1. A nilpotent orbit is a horizontal and holomorphic map f :
H → Ď such that
122
Appendix C. SL(2)-orbit theorem
• f (z) = exp(zN ) · a for some nilpotent element N in Λσ g and some point
a in Ď;
• f (z) belongs to D if z has sufficiently large imaginary part.
The nilpotent orbit theorem 1.4.5 asserts that any period map coming from
a harmonic bundle (E, D, h) on the pointed disk ∆∗ can be approached by a
nilpotent orbit.
The group SL(2, C) acts transitively by homographies on the Riemann
sphere CP1 . The orbit of i of its real form SL(2, R) is the upper half-plane
H. We thus get homogeneous representations CP1 ∼
= SL(2, C)/L and H ∼
=
SL(2, R)/SO(2), where L is the stabilizer of i in the SL(2, C)-action.
A map from CP1 to Ď is said to be equivariant if it comes from a group
morphism φ : SL(2, C) → ΛG, that sends L to Λ+ G. It is said to be real
equivariant if the group morphisms φ sends SL(2, R) to Λσ G. In particular, a
real equivariant map from CP1 to Ď sends H to D.
Theorem C.1.2 (Theorem (5.13) of [Sch73]). Let exp(zN ) · a be a nilpotent
orbit. Up to a change of the base-point in D, it is possible to choose
1. a holomorphic, horizontal, real equivariant embedding ψ̃ : CP1 → Ď, coming from a group morphism ψ : SL(2, C) → ΛG;
2. a holomorphic mapping W 3 z 7→ g(z) ∈ ΛG, with W a neighborhood of
∞ in CP1 ;
such that:
1. exp(zN ) · a = g(−iz) · ψ̃(z), for z ∈ W − {∞};
2. g(y) ∈ Λσ G, for iy ∈ W ∩ iR;
3. N is the image under ψ∗ of ( 00 10 ) in sl2 (R);
4. g(∞) = 1 and the coefficients of the power series of g and g −1 at infinity
can be controlled (see the following remark).
Remark C.1.3. The theorem thus says that a nilpotent orbit can be approached by a map coming from a group morphism, which we call an SL(2)orbit. Assertion 3. implies that ψ̃ has the equivariance property ψ̃(z + 1) =
exp(N ).ψ̃(z). Indeed, ψ̃ is equivariant and the exponential of the matrix ( 00 10 )
acts as translation by 1 on H.
Assertion 4. should be an analogue of the assertion g) in theorem (5.13) of
[Sch73]. Since it makes references to eigenvalues and eigenspaces, it might be a
good idea to evaluate the coefficients in the power series at any point λ in S 1 .
In this way, we obtain loop of coefficients in GL(n, C) and we can work with
finite-dimensional spaces rather than Hilbert spaces. We omit the details of a
precise statement.
Definition C.1.4. A harmonic bundle over ∆∗ is said to be obtained from
SL(2) if its developing map f : H → D is the restriction of a real equivariant
holomorphic map f : CP1 → Ď.
C.2. Weight filtration
123
Proposition C.1.5. Let (E, D, h) be a harmonic bundle on ∆∗ satisfying the
assumptions 1.4.1 and 1.4.2 of section 1.4. Then, it can be approached by a
harmonic bundle obtained from SL(2), in the sense that, on a vertical strip, the
distance between their developing maps tend to 0, as the imaginary part of the
variable goes to infinity.
Indeed, by the nilpotent orbit theorem and the SL(2)-orbit theorem, one
can approach the variation of loop Hodge structures by a variation obtained
from a group morphism ψ : SL(2, C) → ΛG as in theorem C.1.2. The projection
D = Λσ G/K → G/K decreases the distances by the proof of corollary 1.4.6 and
the map obtained after this projection is the developing map of the harmonic
bundle by proposition 1.1.54.
C.2
Weight filtration
If V is a complex vector space and N a nilpotent endomorphism, a canonical
increasing filtration of V can be constructed. In our situation, the nilpotent
element N lives in Λσ g and acts on the Krein space L2 (S 1 , Cn ). Algebraically,
there is no choice to define the filtration; we think that the subspaces in the
filtration are Krein subspaces, although this does not seem obvious.
Proposition C.2.1. Let k be an integer such that N k+1 = 0 on the Krein space
H = L2 (S 1 , Cn ). Then, there exists a unique increasing filtration
0 ⊂ W0 ⊂ W1 ⊂ · · · ⊂ W2k−1 ⊂ W2k = H
by Krein subspaces such that N (Wi ) ⊂ Wi−2 and such that
N̄ i : Grk+i (W• ) → Grk−i (W• )
is an isomorphism.
Moreover, Wi is the orthogonal of W2k−i−1 .
Remark C.2.2. Another approach would be to define a filtration for each value
of λ and glue them together.
One of the consequences of the SL(2)-orbit theorem is that the weight filtration can be understood via the harmonic metric.
Theorem C.2.3 (Theorem (6.6) of [Sch73]). Let (E, D, h) be a harmonic bundle on ∆∗ satisfying the assumptions 1.4.1 and 1.4.2 of section 1.4 and let
(K, B, T , W, D) be its corresponding variation of loop Hodge structures. Let h̃
be the Hilbert inner product on K, obtained from the inner product h. Over a
radial ray L, the Krein bundle can be trivialized to L × H by parallel transport
and H is endowed with a monodromy operator N , which is nilpotent. Let W•
be its filtration by Krein subspaces.
Then, an element v in H belongs to Wi if, and only if,
h̃(v, v)z = O((− log |t|)i−k
as z tends to 0 in L.
This has to be compared with the estimates of page 756 in [Sim90].
124
C.3
Appendix C. SL(2)-orbit theorem
Mixed loop Hodge structures
We can generalize the notion of mixed Hodge structure to our infinite-dimensional
setting.
Definition C.3.1. A (complex polarized) mixed loop Hodge structure is the
datum of a Hilbertizable vector space H, a finite increasing weight filtration
W• of H by closed subspaces, an invertible bounded operator T of H, a closed
subspace V of H and non-degenerate Hermitian forms Bi on the quotient spaces
GriW• := Wi /Wi−1 such that
• T preserves the filtration W• ;
• Each quotient space GriW• carries a loop Hodge structure, with Krein
∩Wi
metric Bi , outgoing operator T̄ and outgoing subspace V̄ := VV∩W
.
i−1
The relation between this notion and the notion of mixed twistor structure
[Sim97] has to be clarified.
Remark C.3.2. One can be more precise and add the datum of a nilpotent
operator N in H. The weight filtration has to be the one associated to N and
the Hermitians forms Bi are obtained from a Hermitian form B defined on H,
as in lemma (6.6) of [Sch73].
We consider again a harmonic bundle (E, D, h) on ∆∗ satisfying the assumptions 1.4.1 and 1.4.2 of section 1.4 and we write (K, B, T , W, D) for its variation
of loop Hodge structures and N for its monodromy operator. We trivialize the
variation over H and write Vz for the outgoing subspace of H = L2 (S 1 , Cn )
at the point z ∈ H. By the nilpotent orbit theorem, one can define a limit
subspace
V∞ := lim exp(−zN )Vz .
Im z→∞
Then,
Theorem C.3.3 (Theorem (6.16) of [Sch73]). The Krein space H carries a
mixed loop Hodge structure. The weight filtration is constructed from N by
proposition C.2.1, T is the right-shift operator, the subspace V of definition
C.3.1 is V∞ and the Hermitian forms Bi are obtained from the Krein metric on
H and the nilpotent operator N (see remark C.3.2).
Appendix D
Spectral sequences and
characteristic cohomology
This appendix is based on a correspondence written with Colleen Robles, after
her preprint [Rob14a]. In her paper, isomorphisms between characteristic, de
Rham and other cohomologies on a Mumford-Tate domain – a generalization of
a period domain – are obtained in low degrees (Theorem 5.3 for instance), using
spectral sequences and Lie algebra cohomology. My goal here is to highlight the
fact that the isomorphism in low degrees between characteristic and de Rham
cohomologies is true in a much more general setting.
Let M be a differentiable manifold and let W be a bracket-generating
constant-rank distribution on M . We write A• for the sheaf of differential
forms. If X is a vector field and f is a function, we write X.f for the function
df (X).
D.1
Filtrations
First, we construct from the distribution W a filtration on T X.
Lemma D.1.1. Let U and V be two vector fields on M and let x be in M . Let
Px be the linear span of Ux and Vx . Then [U, V ]x mod Px only depends only
on Ux and Vx .
Proof. The formula
[f U, gV ] = f (U.g)V − g(V.f )U + f g[U, V ],
where f and g are smooth functions, shows that [f U, gV ]x = f (x)g(x)[U, V ]x
mod Px . It is well-known that this tensoriality property implies the lemma.
Using the lemma, one can define an increasing filtration on each Tx M by
F0 Tx M = 0, F1 Tx M = Wx and Fi+1 Tx M = Fi Tx M + [Wx , Fi Tx M ]. We assume
that this a constant-rank filtration so that it defines an increasing filtration
F• T M by subbundles on T M . It also induces an increasing filtration by subbundles on the exterior algebra of T M : Fp Λq T M is the linear span of the
wedges w1 ∧ · · · ∧ wq , such that there exist indices ji that satisfy wi ∈ Fji T M
P
and qi=1 ji ≤ p.
126Appendix D. Spectral sequences & characteristic cohomology
Lemma D.1.2. [Fi T M, Fj T M ] ⊂ Fi+j T M .
Proof. This in an induction argument, using that the bracket of vector fields
satisfies Jacobi formula.
The increasing filtration on Λ• T M induces a decreasing filtration by subbundles on Λ• T ∗ M : F p Λq T ∗ X is by definition the annihilator in Λq T ∗ X of
Fp−1 Λq T M . We also denote by F • Aq the corresponding filtration on the sheaf
Aq .
Lemma D.1.3. The differential operator d : Aq → Aq+1 is compatible with the
filtration.
Proof. Let ω be in F p Aq (U ), where U is some open set in M and let X1 ∧ · · · ∧
Xq+1 be a multi-vector field in C ∞ (Fp−1 Λq+1 T M, U ). We recall that
dω(X1 ∧ · · · ∧ Xq+1 ) =
X
(−1)i Xi .ω(X1 ∧ . . . X̂i ∧ · · · ∧ Xq+1 )
i
+
X
i+j
(−1)
ω([Xi , Xj ] ∧ X1 ∧ · · · ∧ X̂i ∧ · · · ∧ X̂j ∧ · · · ∧ Xq+1 ).
i<j
All the multi-vectors are in C ∞ (Fp−1 Λq T M, U ); we use lemma D.1.2 for the
second sum.
We thus get a filtered complex (A• (M ), d, F • ) and we consider the spectral
sequence (Eip,q ) attached to it. We recall that the zeroth page is given by
E0p,q =
F p Ap+q (M )
F p+1 Ap+q (M )
and the differential ∂0 goes from E0p,q to E0p,q+1 . Since sheaves of smooth sections
of a vector bundle are acyclic for the functor of global sections, E0p,q can also
be described as the space of global sections of the quotient bundle
Dp,q =
F p Λp+q (T ∗ M )
.
F p+1 Λp+q (T ∗ M )
The bundle Dp,q can equivalently be described as the annihilator of the bundle Fp−1 Λp+q T M in (Fp Λp+q T M )∗ . This is the description we will generally use.
A little reflection shows that E0p,q is not zero only if p ≥ 0, q ≤ 0 and
q ≥ 1 − p unless (p, q) = (0, 0). The picture is thus as follows:
q=0
q = −1
q = −2
q = −3
E00,0 E01,0
E02,0
E03,0
0
0
E02,−1 E03,−1
0
0
0
E03,−2
0
0
0
0
p=0 p=1 p=2 p=3
(D.1)
D.2. Study of the spectral sequence
D.2
127
Study of the spectral sequence
We want to have a better understanding of (E0•,• , ∂0 ). For q = 0, we have
Fp Λp T M = Λp W and Fp−1 Λp T M = 0 so that Dp,0 = Λp W ∗ .
Proposition D.2.1. Let α be an element of E0p,q , seen as a global linear form
on Fp Λp+q T M , vanishing on Fp−1 Λp+q T M . Then as a global linear form on
Fp Λp+q+1 T M vanishing on Fp−1 Λp+q+1 T M , ∂0 α is given by
∂0 α(X1 ∧· · ·∧Xp+q+1 ) =
X
(−1)i+j α([Xi , Xj ]∧X1 ∧· · ·∧X̂i ∧· · ·∧X̂j ∧· · ·∧Xp+q+1 ),
i<j
where X1 ∧ · · · ∧ Xp+q+1 is in C ∞ (Fp Λp+q+1 T M, M ).
Proof. We know that α can be represented by a differential form ω of degree
p+q vanishing on Fp−1 Λp+q T M , the restriction of ω to Fp Λp+q T M being α. We
want to compute the restriction of dω to Fp Λp+q+1 T M , so let X1 ∧ · · · ∧ Xp+q+1
be in C ∞ (Fp Λp+q+1 T M, M ). One has
dω(X1 ∧ · · · ∧ Xp+q+1 ) =
X
(−1)i ω(X1 ∧ · · · ∧ X̂i ∧ · · · ∧ Xp+q+1 )
i
+
X
i+j
(−1)
ω([Xi , Xj ] ∧ X1 ∧ · · · ∧ X̂i ∧ · · · ∧ X̂j ∧ · · · ∧ Xp+q+1 )
i<j
The multi-vectors in the first sum are in Fp−1 Λp+q T M , thus the first sum
vanishes. In the second sum, one can replace ω by α, since the multi-vectors
are in Fp Λp+q T M . This proves the proposition.
Remark D.2.2. The proposition shows in particular that the map ∂0 is defined
pointwise. We will assume that it is of constant rank, so that it is induced by
a vector bundle map, still denoted ∂0 from D•,• to D•,•+1 .
We say that the diagonal l vanishes in the first page of the spectral sequence
if E1p,l−p = 0 for every p ≥ l + 1. We write l0 for the maximal number l such
that all diagonals below l vanishes. The corresponding page is thus of the form
q=0
q = −1
q = −2
q = −3
E10,0
0
0
0
p=0
E1l0 +1,0
E1l0 +2,0
E1l0 +3,0
. . . E1l0 ,0
...
0
0
E1l0 +2,−1
E1l0 +3,−1
...
0
0
0
E1l0 +3,−2
...
0
0
0
0
. . . p = l0 p = l0 + 1 p = l0 + 2 p = l0 + 3
(D.2)
Proposition D.2.3. The number l0 is at least 1.
Proof. We have to show the following: for every p ≥ 2, the map ∂0 : Dp,1−p →
Dp,2−p is injective. The spaces Dp,1−p and Dp,2−p have been described as respectively the bundle of linear forms on Fp T M vanishing on Fp−1 T M and the
bundle of linear forms on Fp Λ2 T M vanishing on Fp−1 Λ2 T M . Moreover, ∂0 is
simply given by
∂0 α(X ∧ Y ) = −α([X, Y ]),
128Appendix D. Spectral sequences & characteristic cohomology
where X ∧Y is a multi-vector field in Fp Λ2 T M . Thus, if ∂0 α = 0, α([X, Y ]) = 0
for any X in F1 T M and Y in Fp−1 T M . Thus, α vanishes on [F1 T M, Fp−1 T M ]
and on Fp−1 T M . By the definition of F• T M by induction, it follows that
α = 0.
D.3
Characteristic cohomology
We write I for the annihilator of W in T ∗ M and J • ⊂ A• for the graded
differential ideal sheaf generated by I.
Lemma D.3.1. With the assumption of constant rank of the filtration F• T M ,
the sheaf J • is locally free.
Proof. Let α be a section of I and let f be a smooth function. Then d(f α) =
f dα + df ∧ α so that the map θ : I 3 α 7→ [dα] ∈ Λ2 T ∗ M/I ∧ T ∗ M is welldefined as a vector bundle map. The bundle I ∧ T ∗ M is easily seen to be the
annihilator of Λ2 W in Λ2 T ∗ M . Hence, θ(α) = 0 if and only if dα vanishes on
Λ2 W, which is equivalent to the vanishing of α on [W, W].
Hence, the image K2 of θ is a subbundle of Λ2 T ∗ M/I ∧ T ∗ M , isomorphic to
I/Ann([W, W]). Writing π : Λ2 T ∗ M → Λ2 T ∗ M/I ∧ T ∗ M for the projection,
the sheaf J 2 is the sheaf of sections of the subbundle π −1 (K2 ) of Λ2 T ∗ M .
More generally, in any degree q ≥ 2, J q is the q-th degree of the algebraic ideal
generated by π −1 (K2 ) in Λ• T ∗ M ; hence it is a vector bundle.
We write Q• for the graded sheaf A• /J • . By the previous lemma, it is
locally free. The differential operator d defines a differential operator dQ from
• (X) of (X, W ) is
Q• to Q•+1 ; we recall that the characteristic cohomology HW
defined to be the cohomology of the complex of global sections (Q• (X), dQ ).
Theorem D.3.2. The characteristic cohomology appears as the zeroth line of
the second page of the spectral sequence.
Proof. This is a consequence of the following two lemmas.
Lemma D.3.3. The space E1p,0 can be identified with Qp (X).
Lemma D.3.4. Under the identification of lemma D.3.3, the differential dQ
of Q• (X) corresponds to the differential ∂1 of the complex E1•,0
Proof of lemma D.3.3. The graded space J • contains the algebraic ideal generated by I, which is I ∧ Λ•−1 T ∗ M . Since, this ideal is the annihilator of Λ• W
in Λ• T ∗ M , this shows that Q• can be seen as a quotient bundle of Λ• W ∗ . With
the notations of lemma D.3.1, we have the equality
Q• = Λ• W ∗ /K2 ∧ Λ•−2 W ∗ ,
which we simply write Q• =: Λ• W ∗ /K• . An element in K• is the restriction to
Λ• W of a form dα, with α in the algebraic ideal generated by I. Hence, Kp is
generated by elements of the form
w1 ∧ · · · ∧ wp 7→
X
(−1)i+j α([wi , wj ] ∧ w1 ∧ · · · ∧ ŵi ∧ · · · ∧ ŵj ∧ · · · ∧ wp ),
i<j
D.3. Characteristic cohomology
129
where α is in the algebraic ideal generated by I. Notice that the formula only
depends on the values of α on the space Fp Λp−1 T M .
On the other hand, E1p,0 is the space of global sections of the cokernel of
the map ∂0 : Dp,−1 → Dp,0 . The bundle Dp,−1 is the bundle of linear forms
on Fp Λp−1 T M vanishing on Fp−1 Λp−1 T M = Λp−1 W and Dp,0 is the bundle
Λp W ∗ . By proposition D.2.1, the image of the map ∂0 : Dp,−1 → Dp,0 is thus
generated by the elements of the form
w1 ∧ · · · ∧ wp 7→
X
(−1)i+j α([wi , wj ] ∧ w1 ∧ · · · ∧ ŵi ∧ · · · ∧ ŵj ∧ · · · ∧ wp ),
i<j
where α is in (Fp Λp−1 T M )∗ and vanishes on Λp−1 W. Since the vanishing
condition means that α is in the algebraic ideal generated by I, this concludes
the proof.
Proof of lemma D.3.4. The map ∂1 : E1p,0 → E1p+1,0 is constructed as follows:
E1p,0 is a quotient of C ∞ (Λp W ∗ , X); if α is in E1p,0 , take a representative in
C ∞ (Λp W ∗ , X), lift it to a global differential form of degree p, take its differential, restrict it to Λp+1 W ∗ and consider its image in the quotient E1p+1,0 .
When Q• is considered as a quotient of Λ• W ∗ , as in the proof of lemma
D.3.3, the map dQ : Qp (X) → Qp+1 (X) is constructed in the same way.
Our main corollary is the following.
p
(X).
Theorem D.3.5. For all p < l0 , there is an equality H p (X) = HW
Proof. Since the spectral sequence was obtained by filtering (A• (X), d), it abuts
to the cohomology H • (A• (X), d), which is simply H • (X) by de Rham theorem.
p =
A look at the spectral sequence (D.2) should convince the reader that E∞
p,0
p,0 = E
E∞
2 for all p < l0 , which proves the theorem.
l0
Remark D.3.6. One can also describe an inclusion H l0 (X, C) ,→ HW
(X).
It goes without saying that this theorem is useful only if one can evaluate
the number l0 . Some computations are done for Mumford-Tate domains in
Appendix A of [Rob14a].
Remark D.3.7. If X is a complex manifold and the distribution W is a holomorphic subbundle of the holomorphic tangent bundle, then one can obtain
other comparison theorems, analogous to theorem D.3.5, that involve the complex structure. This is the situation considered in [Rob14a].
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Cours spécialisés. SMF, 2002.

Thèse de doctorat

Transcription

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