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Jian XIAO Positivité En Géométrie Kählérienne Positivity in Kähler

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Jian XIAO Positivité En Géométrie Kählérienne Positivity in Kähler THÈSE Pour obtenir le grade de DOCTEUR DE LA COMMUNAUTÉ UNIVERSITÉ GRENOBLE ALPES Spécialité : Mathématiques Arrêté ministériel : 25 mai 2016 Présentée par Jian XIAO Thèse dirigée par Jean-pierre DEMAILLY, préparée au sein du Laboratoire Institut Fourier dans l'École Doctorale Mathématiques, Sciences et technologies de l'information, Informatique Positivité en géométrie kählérienne Positivity in Kähler geometry Thèse soutenue publiquement le 23 mai 2016, devant le jury composé de : Monsieur JEAN-PIERRE DEMAILLY PROFESSEUR, UNIVERSITE GRENOBLE ALPES, Directeur de thèse Monsieur JIXIANG FU PROFESSEUR, UNIVERSITE DE FUDAN (SHANGHAI) - CHINE, Examinateur Monsieur SEBASTIEN BOUCKSOM DIRECTEUR DE RECHERCHE, CNRS DELEGATION ILE-DE-FRANCE SUD, Rapporteur Monsieur VALENTINO TOSATTI PROFESSEUR, UNIVERSITE DE NORTHWESTERN - ETATS-UNIS, Rapporteur Monsieur PHILIPPE EYSSIDIEUX PROFESSEUR, UNIVERSITE GRENOBLE ALPES, Examinateur Monsieur MATEI TOMA PROFESSEUR, UNIVERSITE DE LORRAINE, Président Positivité en géométrie kählerienne Jian Xiao 5 mai 2016 Acknowledgements First of all I would like to express my sincere gratitude to my advisors Professor Jean-Pierre Demailly (Université Grenoble Alpes) and Professor Jixiang Fu (Fudan University) for their invaluable guidance and support in the past years. My warmest thanks go to them. I would also like to thank Sébastien Boucksom, Michel Brion, Junyan Cao, Ya Deng, Philippe Eys- sidieux, Zhizhong Huang, Qingchun Ji, Mehdi Lejmi, Xiaonan Ma, Wenhao Ou, Matei Toma, Zhizhang Wang and Can Zhang for interesting and useful conversations, and thank Tristan C. Collins, Mattias Jonsson and Valentino Tosatti for suggestions and comments. Special acknowledgments are due to Brian Lehmann. A large part of this thesis is based on our joint work. I learnt a lot through the discussions with him. It is my great honor that Sébastien Boucksom, Stéphane Druel, Philippe Eyssidieux, Matei Toma and Valentino Tosatti are my committee members. I have been benefiting a lot by studying their works. In particular, I would like to thank Sébastien and Valentino for their helpful comments and corrections on this memoir. My PhD work has been conducted successively at Fudan University and Université Grenoble Alpes : this is a joint PhD thesis defended under the auspices of both math departments, with support from the China Scholarship Council and Institut Fourier. I am very grateful for their support. I would like to thank my lovely friends at Fudan University and Université Grenoble Alpes who have contributed to make my life interesting, and express my gratitude to the secretaries at both places who have made much of my work easier. Lastly I would like to thank my family for their love and support, this thesis is dedicated to them. 3 Table des matières 1 Introduction (Français)7 2 Introduction 20 3 Teissier’s proportionality problem over compact Kähler manifolds 33 3.1 Introduction.......................................... 34 3.2 Proof of the main theorem.................................. 35 3.3 Applications to the balanced cone.............................. 41 3.3.1 Background and main applications......................... 41 3.3.2 Injectivity : application of Theorem 3.1.1..................... 42 3.3.3 Surjectivity : the image of the boundary of Kähler cone............. 42 3.3.4 Characterization on a nef class being Kähler.................... 46 3.3.5 Appendix....................................... 48 4 Transcendental Morse inequalities over compact Kähler manifolds 50 4.1 Introduction.......................................... 51 4.2 Technical preliminaries.................................... 54 4.2.1 Monge-Ampère equation on compact Hermitian manifolds............ 54 4.2.2 Lamari’s lemma.................................... 54 4.2.3 Resolution of singularities of positive currents................... 54 4.2.4 Movable cohomology classes............................. 55 4.2.5 Movable intersections................................ 56 4.3 Proof of the main results................................... 56 4.3.1 Theorem 4.1.4.................................... 56 4.3.2 Theorem 4.1.10 and Theorem 4.1.13........................ 59 4.3.3 Theorem 4.1.15.................................... 64 4.4 Applications to a conjecture of Lejmi and Székelyhidi................... 65 4.4.1 Background and main applications......................... 65 4.4.2 Proof of the main applications........................... 66 4.4.3 Further discussions.................................. 68 4.5 Appendix........................................... 69 5 Characterizing volume via cone duality 72 5.1 Introduction.......................................... 73 5.2 Characterizing volume for divisors............................. 75 5.2.1 Technical preliminaries................................ 75 5.2.2 Volume characterization............................... 79 5.3 Volume functional for 1-cycles................................ 81 5.3.1 Definition and properties.............................. 81 5.3.2 Relation with mobility................................ 83 5.3.3 Towards Fujita approximation for 1-cycles..................... 85 5.4 Further discussions...................................... 90 5 6 Positivité en géométrie kählerienne 5.4.1 Another invariant of movable class......................... 90 5.4.2 A general approach.................................. 91 6 Zariski decomposition of curves on algebraic varieties 92 6.1 Introduction.......................................... 93 6.1.1 Examples....................................... 94 6.1.2 Volume and mobility................................. 95 6.1.3 Formal Zariski decompositions........................... 96 6.1.4 Other applications.................................. 97 6.1.5 Outline........................................ 98 6.2 Preliminaries......................................... 99 6.2.1 Khovanskii-Teissier inequalities........................... 100 6.2.2 Complete intersection cone............................. 101 6.2.3 Fields of characteristic p ............................... 102 6.2.4 Compact Kähler manifolds............................. 103 6.3 Polar transforms....................................... 103 6.3.1 Duality transforms.................................. 103 6.3.2 Differentiability.................................... 106 6.4 Formal Zariski decompositions................................ 109 6.4.1 Teissier proportionality............................... 112 6.4.2 Morse-type inequality................................ 114 6.4.3 Boundary conditions................................. 116 6.5 Positivity for curves..................................... 117 6.5.1 Basic properties.................................... 117 6.5.2 Zariski decompositions for curves.......................... 118 6.5.3 Strict log concavity.................................. 119 6.5.4 Differentiability.................................... 120 6.5.5 Negative parts.................................... 121 6.5.6 Birational behavior.................................. 122 6.5.7 Morse-type inequality for curves.......................... 123 6.6 Positive products and movable curves............................ 124 6.6.1 The volume function on big and movable divisors................. 125 6.6.2 The function M .................................... 127 6.7 Comparing the complete intersection cone and the movable cone............ 131 6.8 Toric varieties......................................... 134 6.8.1 Positive product on toric varieties.......................... 134 6.8.2 The function M .................................... 134 6.8.3 Zariski decompositions................................ 136 6.9 Hyperkähler manifolds.................................... 137 6.10 Comparison with mobility.................................. 139 6.10.1 Weighted mobility.................................. 142 6.11 Applications to birational geometry............................. 143 6.11.1 Seshadri constants.................................. 143 6.11.2 Rationally connected varieties............................ 145 6.12 Appendix A.......................................... 146 6.12.1 Reverse Khovanskii-Teissier inequalities...................... 146 6.12.2 Towards the transcendental holomorphic Morse inequality............ 147 6.13 Appendix B.......................................... 148 6.13.1 Non-convexity of the complete intersection cone.................. 148 Chapitre 1 Introduction (Français) L’objectif de cette thèse est d’étudier divers concepts de positivité en géométrie kählerienne. En par- ticulier, pour une variété kählerienne compacte de dimension n, nous étudions la positivité des classes transcendantes de type (1; 1) et (n − 1; n − 1) – ces classes comprennent donc en particulier les classes de diviseurs et les classes de courbes. Les résultats principaux présentés dans ce mémoire sont basés sur les articles suivants que j’ai rédigés au cours de mes études de doctorat : 1. (avec Brian Lehmann) Zariski decomposition of curves on algebraic varieties. arXiv preprint 2015, arXiv : 1507.04316, soumis. 2. A remark on the convergence of inverse σk-flow, arXiv preprint 2015, arXiv : 1505.04999, Comptes Rendus Mathématique 354 (2016) 395-399. 3. Characterizing volume via cone duality, arXiv preprint 2015, arXiv : 1502.06450, soumis. 4. (avec Jixiang Fu) Teissier’s problem on proportionality of nef and big classes over a compact Kähler manifold, arXiv preprint 2014, arXiv : 1410.4878, soumis. 5. Movable intersections and bigness criterion, arXiv preprint 2014, arXiv : 1405.1582. 6. Weak transcendental
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