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Geometry of Fano Varieties : Subsheaves of the Tangent Bundle and Fundamental Divisor Jie Liu

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Geometry of Fano Varieties : Subsheaves of the Tangent Bundle and Fundamental Divisor Jie Liu Geometry of Fano varieties : subsheaves of the tangent bundle and fundamental divisor Jie Liu To cite this version: Jie Liu. Geometry of Fano varieties : subsheaves of the tangent bundle and fundamental divisor. Algebraic Geometry [math.AG]. Université Côte d’Azur, 2018. English. NNT : 2018AZUR4038. tel-02000801 HAL Id: tel-02000801 https://tel.archives-ouvertes.fr/tel-02000801 Submitted on 1 Feb 2019 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. G F : Jie L Laboratoire de Mathématiques J. A. Dieudonné Présentée en vue de l’obtention Devant le jury, composé de : du grade de docteur en Mathématiques Cinzia Casagrande PR Examinatrice de l'Université Côte d'Azur Pierre-Emmanuel Chaput PR Examinateur Stéphane Druel CR Rapporteur Dirigée par : Andreas Höring et Christophe Mourougane Andreas Höring PR Co-directeur Christophe Mourougane PR Co-directeur Soutenue le : juin Christian Pauly PR Examinateur Laboratoire de Mathématiques J. A. Dieudonné UMR n° CNRS UCA Université Côte d'Azur Parc Valrose Nice Cedex France Résumé Cee thèse est consacrée à l'étude de la géométrie des variétés de Fano complexes en utilisant les propriétés des sous-faisceaux du fibré tangent et la géométrie du diviseur fondamental. Les résultats principaux compris dans ce texte sont : (i) Une généralisation de la conjecture de Hartshorne : une variété lisse projective est isomorphe à un espace projectif si et seulement si son fibré tangent contient un sous-faisceau ample. (ii) Stabilité du fibré tangent des variétés de Fano lisses de nombre de Picard un : à l'aide de het- héorèmes d'annulation sur les espaces hermitiens symétriques irréductibles de type compact M, nous montrons que pour presque toute intersection complète générale dans M, le fibré tangent est stable. La même méthode nous permet de donner une réponse sur la stabilité de la restriction du fibré tangent de l'intersection complète à une hypersurface générale. (iii) Non-annulation effective pour des variétés de Fano et ses applications : nous étudions la positivité de la seconde classe de Chern des variétés de Fano lisses de nombre de Picard un. Ceci nous permet de montrer un théorème de non-annulation pour les variétés de Fano lisses de dimension n et d'indice n − 3. Comme application, nous étudions la géométrie anticanonique des variétés de Fano et nous calculons les constantes de Seshadri des diviseurs anticanoniques des variétés de Fano d'indice grand. (iv) Diviseurs fondamentaux des variétés de Moishezon lisses de dimension trois et de nombre de Picard un : nous montrons l'existence d'un diviseur lisse dans le système fondamental dans certain cas particulier. Mots clés : Variétés de Fano, espaces projectifs, faisceaux amples, feuilletages, stabilité, espaces her- mitiens symétriques, théorèmes d'annulation, intersections complètes, propriétés de Lefschetz, non- annulation, seconde classe de Chern, birationalité, diviseurs fondamentaux, constante de Seshadri, var- iétés de Moishezon, singularités, courbes rationnelles, théorie de Mori Abstract is thesis is devoted to the study of complex Fano varieties via the properties of subsheaves of the tangent bundle and the geometry of the fundamental divisor. e main results contained in this text are : (i) A generalization of Hartshorne's conjecture : a projective manifold is isomorphic to a projective space if and only if its tangent bundle contains an ample subsheaf. (ii) Stability of tangent bundles of Fano manifolds with Picard number one : by proving vanishing theorems on the irreducible Hermitian symmetric spaces of compact type M, we establish that the tangent bundles of almost all general complete intersections in M are stable. Moreover, the same method also gives an answer to the problem of stability of the restriction of the tangent bundle of a complete intersection on a general hypersurface. (iii) Effective non-vanishing for Fano varieties and its applications : we study the positivity of the second Chern class of Fano manifolds with Picard number one, this permits us to prove a non- vanishing result for n-dimensional Fano manifolds with index n − 3. As an application, we study the anticanonical geometry of Fano varieties and calculate the Seshadri constants of anticanonical divisors of Fano manifolds with large index. (iv) Fundamental divisors of smooth Moishezon threefolds with Picard number one : we prove the existence of a smooth divisor in the fundamental linear system in some special cases. Key words : Fano varieties, projective spaces, ample sheaves, foliations, stability, Hermitian symmetric spaces, vanishing theorems, complete intersects, Lefschetz properties, non-vanishing, second Chern class, birationality, fundamental divisors, Seshadri constants, Moishezon manifolds, singularities, ra- tional curves, Mori theory MSC classifications : B, C, E, E, F, J, J, J, J, J, J, J, J, M, C, G, J, J, L, L, M, M, Q, Q Remerciements En premier lieu, un grand merci à mes directeurs, Christophe Mourougane et Andreas Höring, pour leur grande patience, pour leur aide, leur générosité, leur soutien, leurs explications méticuleuses, leurs exemples et leurs idées. Leurs contributions à ce manuscrit sont aussi profondes que nombreuses. Tout au long de ces années, j'ai été très sensible à la grande liberté qu'ils m'ont octroyée pour mener à bien mon travail. Ils ont écouté mes idées avec aention et sans complaisance et ont né mon goût pour les mathématiques. En particulier, nos conversations enrichissantes et divertissantes ont largement influencé le présent mémoire. Je leur en serai toujours reconnaissante. Les travaux de Stéphane Druel et Jarosław A. Wiśniewski ont été une grande source d'inspiration pour ma thèse. J'aimerais les remercier d'avoir accepté de rapporter celle-ci, c'est un honneur et un plaisir pour moi qu'ils aient pris le temps de lire mon manuscrit. Merci en particulier à Stéphane Druel pour m'avoir permis, via ses remarques et suggestions, d'améliorer le manuscrit. Il a fait son travail de rap- porteur de façon remarquable et a consacré énormément de temps à cee thèse. C'est une grande joie pour moi que de compter Cinzia Casagrande et Pierre-Emmanuel Chaput parmi mon jury de thèse. Les conversations que j'ai pu avoir avec l'un et l'autre m'ont beaucoup apporté. Merci également à Christian Pauly d'avoir bien voulu faire partie du jury. Tout au long de ma scolarité, j'ai été disciple d'enseignants merveilleux auxquels je dois beaucoup. A l'université de Wuhan, les conseils de Yuping Tu, Zhenhan Tu, anhua Xu et Wanke Yin m'ont été pré- cieux. En particulier, je tiens à exprimer ma reconnaissance à Yuping Tu et Qi Zhang pour leurs cours et conseils qui ont suscité mon intérêt pour la géométrie birationnelle. De mes études à l'université de Rennes , j'ai eu la chance de trouver la beauté du calcul de Schubert sous la direction de Michele Bolog- nesi. Sébastien Boucksom m'a initié au monde de la recherche et à la géométrie complexe lors de mon Master. Il m'a enseigné avec beaucoup de sympathie de très belles mathématiques, et les discussions que j’ai eues avec lui ont toujours été des plus enrichissantes. J'ai eu la chance de discuter avec nombreux autres chercheurs durant mon doctorat. J'ai déjà mentionné certains plus haut auxquels j'ajoute volontier Arnaud Beauville, Amaël Broustet, Serge Cantat, Junyan Cao, Benoît Claudon, Alexandru Dimca, Sorin Dumitrescu, Enrica Floris, Jun-Muk Hwang, Frank Loray, Daniel Li, Steven Lu, Adam Parusiński, Boris Pasquier, omas Peternell, Zhiyu Tian, Frédéric Touzet, Junyi Xie, Chenyang Xu, et bien d'autres. Je remercie aussi les autres doctorants avec qui j'ai parlé de géométrie, et ce fut très intéressant de les croiser au fil des évènements mathématiques : Vladimiro Benedei, Benoît Cadorel, Jean-Baptiste Campesato, Jiaming Chen, Lionel Darondeau, ibaut Delcroix, Ya Deng, Yanbo Fang, Wei-Guo Foo, Alexis Garcia, Zhizhong Huang, Louis-Clément Leèvre, Hsueh-Yung Lin, Binxiao Liu, Federico Lo Bianco, Chen Jiang, Pedro Montero, Olivier om, Nestor Fernandez Vargas, Xiaozong Wang, Zhenjian Wang, Jian Xiao, Songyan Xie, Chenglong Yu, Shengyuan Zhao, et bien d'atures. En particulier, je tiens à remercier ma grande sœur Liana Heuberger, mon petit frère Yanis Mabed et mon grand frère Basile Pillet pour toutes ces discussions mathématiques et aussi les pausses très agréables que nous avons partagés, et j’ai eu la chance d'accueillir ma petite sœur Zhixin Xie pendant la dernière année. Et maintenant arrive le moment de remercier tous mes collègues doctorants et ex-doctorants. L'am- biance entre doctorants au laboratoire fait ma vie à Nice très agréable. En particulier, je remercie Alexis Gilles et Mehdi Zaidi d'avoir aidé à corriger les fautes de français. Un merci particulier est réservé aux habitants du bureau (à Nice) avec Ayoub Badia, Eduard Balzin, Marcella Bonazzoli, Charles Col- lot, Alexis Gilles et Pengfei Huang. Merci aussi à mes cobureaux du (à Rennes) Yuliang Huang et Andres Sarrazola Alzate. Un autre merci groupé pour mes amis à Nice, même si ce groupe a une forte intersection avec le groupe précédent. Vous avez eu, avez et aurez toujours une place importante dans ma vie. Je remercie le LJAD pour la bonne qualité des conditions de la recherche. Je tiens à remercier aussi l'IRMAR qui m'ont toujours très bien accueillie après je suis arrivé en France. Le personnel admin- istratif m'a rendu la vie à le LJAD et l'IRMAR très aisée en faisant un travail remarquable et je les en remercie.
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