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K-Stability and Kähler Manifolds with Transcendental Cohomology Class Zakarias Sjöström Dyrefelt

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K-Stability and Kähler Manifolds with Transcendental Cohomology Class Zakarias Sjöström Dyrefelt K-stability and Kähler manifolds with transcendental cohomology class Zakarias Sjöström Dyrefelt To cite this version: Zakarias Sjöström Dyrefelt. K-stability and Kähler manifolds with transcendental cohomology class. General Mathematics [math.GM]. Université Paul Sabatier - Toulouse III, 2017. English. NNT : 2017TOU30126. tel-01899546 HAL Id: tel-01899546 https://tel.archives-ouvertes.fr/tel-01899546 Submitted on 19 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSETHÈSE En vue de l’obtention du DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE Délivré par : l’Université Toulouse 3 Paul Sabatier (UT3 Paul Sabatier) Présentée et soutenue le 15/09/2017 par : Zakarias SJÖSTRÖM DYREFELT K-stabilité et variétés kähleriennes avec classe transcendante JURY Jacopo STOPPA SISSA Trieste Rapporteur Sébastien BOUCKSOM École Polytechnique Directeur de thèse Vincent GUEDJ Université Paul Sabatier Directeur de thèse Paul GAUDUCHON École Polytechnique Examinateur Éveline LEGENDRE Université Paul Sabatier Examinateur Julien KELLER Université Aix-Marseille Examinateur École doctorale et spécialité : MITT : Domaine Mathématiques : Mathématiques fondamentales Unité de Recherche : Institut de Mathématiques de Toulouse (UMR 5219) Directeur(s) de Thèse : Sébastien BOUCKSOM et Vincent GUEDJ Rapporteurs : Robert BERMAN et Jacopo STOPPA iii Remerciements La présente thèse a été réalisée en co-direction entre l’Université Paul Sabatier à Toulouse et l’École Polytechnique à Paris. Cela a entrainé un grand nombre d’aller-retours notamment en- tre Paris et Toulouse. Je tiens à remercier tous mes collègues dans les deux labos concernés pour leur accueil chaleureux. Ces voyages ont de plus été l’occasion de faire de nouvelles dé- couvertes (mathématiques et autres) et rencontrer un grand nombre de personnages fascinants et généreux, qui m’ont souvent beaucoup aidé, d’une manière directe ou indirecte, à réaliser le travail qui a aboutit a cette thèse. Tout d’abord, je tiens plus spécialement à remercier mes directeurs de thèse, les professeurs Sébastien Boucksom et Vincent Guedj, pour leur générosité, et leur volonté de me guider lors de mes premiers pas dans le monde de la recherche mathématique, de m’avoir montré la richesse de la tradition scientifique française, et de m’avoir permis de passer trois ans a decouvrir ce pays incroyable qui est la France. En particulier, je leur exprime ma sincère gratitude de m’avoir dirigé dans des projets influencé par plusieurs domaines mathématiques passionnants, et d’avoir partagé leurs idées avec moi. Je tiens également a remercier les membres du jury de m’avoir fait l’honneur d’être venu à ma soutenance. Je remercie les rapporteurs Jacopo Stoppa et Robert Berman pour un travail méticuleux de relecture du manuscrit de thèse. Des remerciements particuliers sont ici adressés au professeur Robert Berman, qui a joué un rôle déterminant pour mon parcours, avec son soutien constant, et en partageant des idées et des projets avec moi. Il a aussi beaucoup contribué a mon développement dans mes études de doctorat, ce dont je lui suis fortement reconnaissant. Pendant ces années j’ai eu la chance de rencontrer de nombreux collègues du monde entier avec qui j’ai pu échanger des idées mathématiques, qui ont joué des rôles importants pour mon parcours, et qui sont devenus des amis. Je remercie ainsi Ruadhai Dervan, Julius Ross, Robert Berman, Bo Berndtsson, David Witt-Nyström, Tamás Darvas, Yanir A. Rubinstein, Gabor Székelyhidi, Ahmed Zeriahi, Carl Tipler, Éveline Legendre et Paul Gauduchon. Je remercie tous les membres du projet ANR EMARKS pour l’interêt porté à mon travail. Un merci tout particulier est destiné ici à mes frères et sœurs de thèse: Chinh H. Lu, Eleonora Di Nezza, Tat Dat Tô et Henri Guenancia. Merci pour la disponibilité, l’amitié et le soutien dont vous m’avez fait part. Je remercie également toutes les collegues du CMLS et de l’IMT, le personnel administratif inclus, notamment Agnès, Jocelyne et Pascale pour leur grande disponibilité et générosité, ainsi que pour leur aide avec des procédures parfois compliqués a comprendre pour un étranger ! iv Je tiens a remercier mes collègues doctorant/post-doc à Toulouse et à Paris: André, Jordi, Suzanna, Fabrizio, Danny, Dat, Anne, Damien, Laura, Jorge, Jules, Ibrahim, Daniel, Sergio, Anton, Bac, Rita, Jacek, René... Sur une note plus personnelle je tiens, bien sur, à remercier ma famille et tous mes amis, qui sont restés toujours disponibles, souvent malgré une distance géographique considérable. Je suis reconnaissant à Bogdan, qui a toujours cru en moi. Je remercie également Cat d’avoir bien voulu partager avec moi l’aventure de venir poursuivre des études de doctorat en France. Finalement, je remercie en particulier mes parents pour leur soutien constant et inconditionel. Sans vous cette thèse ne serait pas devenue ce qu’elle est aujourd’hui. Merci à vous tous ! Zakarias Sjöström Dyrefelt Toulouse, le 15 Septembre 2017 R´esum´e Dans cette th`esenous ´etudionsdes questions de stabilit´eg´eom´etriquepour des vari´et´es k¨ahleriennes`acourbure scalaire constante (cscK) avec classe de cohomologie transcen- dante. En tant que point de d´epart,nous introduisons des notions g´en´eralis´ees de K- stabilit´e,´etendant une image classique introduite par G. Tian et S. Donaldson dans le cadre des vari´et´espolaris´ees. Contrairement `ala th´eorie classique, ce formalisme nous permet de traiter des questions de stabilit´epour des vari´et´esk¨ahleriennescompactes non projectives ainsi que des vari´et´esprojectives munis de polarisations non rationnelles. Dans une premi`erepartie, nous ´etudionsles rayons sous-g´eod´esiquesassoci´esaux configurations tests dites cohomologiques, objets introduitent dans cette th`ese. Nous ´etablissonsainsi des formules fondamentales pour la pente asymptotique d’une famille de fonctionnelles d’´energie,le long de ces rayons g´eod´esiques. Ceci est li´eau couplage de Deligne en g´eom´etriealg´ebrique,et ce formalise permet en particulier de comprendre le comportement asymptotique d’un grand nombre de fonctionelles d’´energieclassiques en g´eom´etriek¨ahlerienne,y compris la fonctionnelle d’Aubin-Mabuchi et la K-´energie. En particulier, ceci fournit une approche pluripotentielle naturelle pour ´etudierle com- portement asymptotique des fonctionelles d’´energiedans la th´eoriede K-stabilit´e. En s’appuyant sur cette premi`erepartie, nous d´emontrons ensuite un certain nombre de r´esultatsde stabilit´epour les vari´et´escscK. Tout d’abord, nous prouvons que les vari´et´escscK sont K-semistables dans notre sens g´en´eralis´e,prolongeant ainsi un r´esultat dˆu`aDonaldson dans le cadre projectif. En supposant que le groupe d’automorphisme est discret, nous montrons en outre que la K-stabilit´eest une condition n´ecessaire pour l’existence des m´etriquescscK sur des vari´et´esk¨ahleriennescompactes. Plus pr´ecis´ement, nous prouvons que la coercivit´ede la K-´energieimplique la K-stabilit´euniforme, ainsi g´en´eralisant des r´esultatsde Mabuchi, Stoppa, Berman, Dervan et Boucksom-Hisamoto- Jonsson pour des vari´et´espolaris´ees. Cela donne une preuve nouvelle et plus g´en´erale d’une direction de la conjecture Yau-Tian-Donaldson dans ce contexte. L’autre direction (suffisance de K-stabilit´e)est consid´er´eecomme l’un des probl`emesouverts les plus importants en g´eom´etriek¨ahlerienne. Nous donnons enfin des r´esultatspartiels dans le cas des vari´et´esk¨ahleriennescom- pactes qui admettent des champs de vecteurs holomorphes non triviaux. Nous discutons ´egalement autour des perspectives et applications de notre th´eoriede K-stabilit´epour les vari´et´esk¨ahleriennes avec classe transcendante, notamment `al’´etudedes lieux de stabilit´edans le cˆonede K¨ahler. 1 Abstract In this thesis we are interested in questions of geometric stability for constant scalar curvature K¨ahler(cscK) manifolds with transcendental cohomology class. As a starting point we develop generalized notions of K-stability, extending a classical picture for polarized manifolds due to G. Tian, S. Donaldson, and others, to the setting of arbitrary compact K¨ahler manifolds. We refer to these notions as cohomological K-stability. By contrast to the classical theory, this formalism allows us to treat stability questions for non-projective compact K¨ahlermanifolds as well as projective manifolds endowed with non-rational polarizations. As a first main result and a fundamental tool in this thesis, we study subgeodesic rays associated to test configurations in our generalized sense, and establish formulas for the asymptotic slope of a certain family of energy functionals along these rays. This is related to the Deligne pairing construction in algebraic geometry, and covers many of the classical energy functionals in K¨ahlergeometry (including Aubin’s J-functional and the Mabuchi K-energy functional). In particular, this yields a natural potential-theoretic aproach to energy functional asymptotics in the theory of K-stability. Building on this foundation we establish a number of stability results for cscK man- ifolds: First, we show that cscK manifolds are K-semistable in our generalized sense, extending a result due to S. Donaldson in the projective setting. Assuming that the au- tomorphism group is discrete we further show that K-stability is a necessary condition for existence of constant scalar curvature K¨ahlermetrics on compact K¨ahlermanifolds. More precisely, we prove that coercivity of the Mabuchi functional implies uniform K- stability, generalizing results of T. Mabuchi, J. Stoppa, R. Berman, R. Dervan as well as S. Boucksom, T. Hisamoto and M.
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