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Astérisque - Titles - 390 (2017) 245-275

Titles < 2017 < 390

Séminaire Bourbaki, volume 2015/2016, exposés 1104-1119
Astérisque 390 (2017), xi+533 pages
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Presentation, Summary

Exposé 1112 : Variational approach for complex Monge-Ampère equations and geometric applications
Jean-Pierre DEMAILLY
Astérisque 390 (2017), 245-275
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Résumé :
Exposé 1112 : Approche variationnelle pour les équations de Monge-Ampère complexes et applications géométriques
Solutions of Monge-Ampère equations on compact Kähler manifolds can be obtained by a variational method independent of Yau's theorem. The technique relies on the study of certain functionals (Ding-Tian, Mabuchi) on the space of Kähler metrics, and on their geodesic convexity, due to Berndtsson-Berman in its general form. Applications include the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities and a new proof of a uniform version of the Yau-Tian-Donaldson conjecture.

Mots-clefs : Variational method, Monge-Ampère equation, Kähler-Einstein metric, plurisubharmonic potential, Fano variety, Tian-Yau-Donaldson conjecture, Donaldson-Futaki invariant, K-stability, space of Kähler potentials, Aubin functional, Mabuchi functional, Ding functional.

Abstract:
Les équations de Monge-Ampère sur les variétés kählériennes compactes peuvent être résolues par une méthode variationnelle indépendante du théorème de Yau. La technique repose sur l'étude de certaines fonctionnelles (Ding-Tian, Mabuchi) sur l'espace des métriques de Kähler, et sur leur convexité géodésique, due à Berndtsson-Berman dans sa forme générale. Les applications incluent l'existence et l'unicité de métriques de Kähler-Einstein sur les variétés Q-Fano à singularités terminales, et une nouvelle preuve d'une version uniforme de la conjecture de Yau-Tian-Donaldson.

Keywords: Méthode variationnelle, Monge-Ampère equation, potentiel plurisousharmonique, métrique de Kähler-Einstein, variété de Fano, conjecture de Tian-Yau-Donaldson, invariant de Futaki-Donaldson, K-stabilité, espaces des métriques de Kähler, fonctionnelle d'Aubin, fonctionnelle de Mabuchi, fonctionnelle de Ding.

Class. math. : 53C55, 32J27, 32P05.


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Arezzo, Claudio and Pacard, Frank
Blowing up and desingularizing constant scalar curvature Kähler manifolds
Acta Math. 196 (2006) 179–228
Math Reviews MR2275832
2
Aubin, Thierry
Équations du type Monge-Ampère sur les variétés kählériennes compactes
Bull. Sci. Math. 102 (1978) 63–95
Math Reviews MR494932
3
Aubin, Thierry
Réduction du cas positif de l'équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d'une inégalité
J. Funct. Anal. 57 (1984) 143–153
Math Reviews MR749521
4
Bando, Shigetoshi and Mabuchi, Toshiki
Uniqueness of Einstein Kähler metrics modulo connected group actions
in Algebraic geometry, Sendai, 1985
Adv. Stud. Pure Math. 10 (1987) 11–40
Math Reviews MR946233
5
Beauville, Arnaud
Variétés Kähleriennes dont la première classe de Chern est nulle
J. Differential Geom. 18 (1983) 755–782
Math Reviews MR730926
6
Bedford, Eric and Taylor, B. A.
The Dirichlet problem for a complex Monge-Ampère equation
Invent. math. 37 (1976) 1–44
Math Reviews MR0445006
7
Bedford, Eric and Taylor, B. A.
A new capacity for plurisubharmonic functions
Acta Math. 149 (1982) 1–40
Math Reviews MR674165
8
Bedford, Eric and Taylor, B. A.
Fine topology, Šilov boundary, and (ddc)n
J. Funct. Anal. 72 (1987) 225–251
Math Reviews MR886812
9
10
Berman, Robert and Boucksom, Sébastien
Growth of balls of holomorphic sections and energy at equilibrium
Invent. math. 181 (2010) 337–394
Math Reviews MR2657428
11
12
Berman, Robert and Demailly, Jean-Pierre
Regularity of plurisubharmonic upper envelopes in big cohomology classes
in Perspectives in analysis, geometry, and topology
Progr. Math. 296 (2012) 39–66
Math Reviews MR2884031
13
Berman, Robert J.
A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics
Adv. Math. 248 (2013) 1254–1297
Math Reviews MR3107540
14
Berman, Robert J.
K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics
Invent. math. 203 (2016) 973–1025
Math Reviews MR3461370
15
16
Berman, Robert J. and Boucksom, Sébastien and Guedj, Vincent and Zeriahi, Ahmed
A variational approach to complex Monge-Ampère equations
Publ. Math. IHÉS 117 (2013) 179–245
Math Reviews MR3090260
17
18
Berndtsson, Bo
Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
Ann. Inst. Fourier (Grenoble) 56 (2006) 1633–1662
Math Reviews MR2282671
19
Berndtsson, Bo
A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
Invent. math. 200 (2015) 149–200
Math Reviews MR3323577
20
Boucksom, Sébastien
Divisorial Zariski decompositions on compact complex manifolds
Ann. Sci. École Norm. Sup. 37 (2004) 45–76
Math Reviews MR2050205
21
Boucksom, Sébastien and Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed
Monge-Ampère equations in big cohomology classes
Acta Math. 205 (2010) 199–262
Math Reviews MR2746347
22
Boucksom, Sébastien and Favre, Charles and Jonsson, Mattias
Valuations and plurisubharmonic singularities
Publ. Res. Inst. Math. Sci. 44 (2008) 449–494
Math Reviews MR2426355
23
Boucksom, Sébastien and Favre, Charles and Jonsson, Mattias
Singular semipositive metrics in non-Archimedean geometry
J. Algebraic Geom. 25 (2016) 77–139
Math Reviews MR3419957
24
Boucksom, Sébastien and Hisamoto, T. and Jonsson, Mattias
Uniform K-stability and coercivity of the K-energy
in preparation
25
26
Bourguignon, Jean-Pierre
Premières formes de Chern des variétés kählériennes compactes [d'après E. Calabi, T. Aubin et S. T. Yau]
Séminaire Bourbaki, vol. 1977/1978, exposé no 507, Lecture Notes in Math. 710 (1979) 1–21
Math Reviews MR554212
27
Bourguignon, Jean Pierre
Métriques d'Einstein-Kähler sur les variétés de Fano: obstructions et existence (d'après Y. Matsushima, A. Futaki, S. T. Yau, A. Nadel et G. Tian)
Séminaire Bourbaki, vol. 1996/1997, exposé no 830, Astérisque (1997) 277–305
Math Reviews MR1627115
28
Calabi, Eugenio
The Space of Kähler Metrics
in Proc.International Congress of Mathematicians, Amsterdam, II
(1954) 206–207
29
Calabi, Eugenio
On Kähler manifolds with vanishing canonical class
in Algebraic geometry and topology. A symposium in honor of S. Lefschetz
(1957) 78–89
Math Reviews MR0085583
30
Calabi, Eugenio and Chen, Xiuxiong
The space of Kähler metrics. II
J. Differential Geom. 61 (2002) 173–193
Math Reviews MR1969662
31
Campana, F. and Demailly, Jean-Pierre and Peternell, Th.
Rationally connected manifolds and semipositivity of the Ricci curvature
in Recent advances in algebraic geometry
London Math. Soc. Lecture Note Ser. 417 (2015) 71–91
Math Reviews MR3380444
32
Cao, Junyan
Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact Kähler manifolds
Compos. Math. 150 (2014) 1869–1902
Math Reviews MR3279260
33
Cegrell, Urban
Pluricomplex energy
Acta Math. 180 (1998) 187–217
Math Reviews MR1638768
34
Chen, Xiuxiong
The space of Kähler metrics
J. Differential Geom. 56 (2000) 189–234
Math Reviews MR1863016
35
Chen, Xiuxiong
Space of Kähler metrics. III. On the lower bound of the Calabi energy and geodesic distance
Invent. math. 175 (2009) 453–503
Math Reviews MR2471594
36
Chen, Xiuxiong and Donaldson, S. K. and Sun, Song
Kähler-Einstein metrics on Fano manifolds, I-III
J. Amer. Math. Soc. 28 (2015) 183–197, 199–234, 235–278
37
38
Chen, Xiuxiong and Sun, Song
Space of Kähler metrics (V)—Kähler quantization
in Metric and differential geometry
Progr. Math. 297 (2012) 19–41
Math Reviews MR3220438
39
40
Chen, Xiuxiong and Tian, Gang
Geometry of Kähler metrics and foliations by holomorphic discs
Publ. Math. IHÉS (2008) 1–107
Math Reviews MR2434691
41
42
Darvas, Tamás
The Mabuchi geometry of finite energy classes
Adv. Math. 285 (2015) 182–219
Math Reviews MR3406499
43
44
45
46
Datar, Ved and Székelyhidi, Gábor
Kähler–Einstein metrics along the smooth continuity method
Geom. Funct. Anal. 26 (2016) 975–1010
Math Reviews MR3558304
47
Demailly, Jean-Pierre
Regularization of closed positive currents and intersection theory
J. Algebraic Geom. 1 (1992) 361–409
Math Reviews MR1158622
48
Demailly, Jean-Pierre
On the cohomology of pseudoeffective line bundles
in Complex Geometry and Dynamics
Abel Symposia 10 (2015) 51–99
49
Demailly, Jean-Pierre and Peternell, Thomas and Schneider, Michael
Pseudo-effective line bundles on compact Kähler manifolds
Internat. J. Math. 12 (2001) 689–741
Math Reviews MR1875649
50
Dembo, Amir and Zeitouni, Ofer
Large deviations techniques and applications
Springer, Berlin, 2010
Math Reviews MR2571413
51
52
Dinew, Sławomir
Uniqueness in E(X,)
J. Funct. Anal. 256 (2009) 2113–2122
Math Reviews MR2498760
53
Ding, Wei Yue
Remarks on the existence problem of positive Kähler-Einstein metrics
Math. Ann. 282 (1988) 463–471
Math Reviews MR967024
54
Ding, Wei Yue and Tian, Gang
Kähler-Einstein metrics and the generalized Futaki invariant
Invent. math. 110 (1992) 315–335
Math Reviews MR1185586
55
Donaldson, S. K.
Scalar curvature and projective embeddings. I
J. Differential Geom. 59 (2001) 479–522
Math Reviews MR1916953
56
Donaldson, S. K.
Scalar curvature and stability of toric varieties
J. Differential Geom. 62 (2002) 289–349
Math Reviews MR1988506
57
Donaldson, S. K.
Lower bounds on the Calabi functional
J. Differential Geom. 70 (2005) 453–472
Math Reviews MR2192937
58
El Mir, Hassine
Sur le prolongement des courants positifs fermés
Acta Math. 153 (1984) 1–45
Math Reviews MR744998
59
Eyssidieux, Philippe
Métriques de Kähler-Einstein sur les variétés de Fano [d'après Chen-Donaldson-Sun et Tian]
Séminaire Bourbaki, exp. no 1095, vol. 2014/2015, Astérisque 380 (2016) 207–229
Math Reviews MR3522175
60
Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed
Singular Kähler-Einstein metrics
J. Amer. Math. Soc. 22 (2009) 607–639
Math Reviews MR2505296
61
Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed
Continuous approximation of quasiplurisubharmonic functions
in Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong
Contemp. Math. 644 (2015) 67–78
Math Reviews MR3372461
62
63
Guedj, Vincent and Zeriahi, Ahmed
The weighted Monge-Ampère energy of quasiplurisubharmonic functions
J. Funct. Anal. 250 (2007) 442–482
Math Reviews MR2352488
64
Jeffres, Thalia and Mazzeo, Rafe and Rubinstein, Yanir A.
Kähler-Einstein metrics with edge singularities
Ann. of Math. 183 (2016) 95–176
Math Reviews MR3432582
65
Keller, Julien
Ricci iterations on Kähler classes
J. Inst. Math. Jussieu 8 (2009) 743–768
Math Reviews MR2540879
66
Kołodziej, Sławomir
The complex Monge-Ampère equation and pluripotential theory
Mem. Amer. Math. Soc. 178 (2005)
Math Reviews MR2172891
67
68
Li, Chi and Xu, Chenyang
Special test configuration and K-stability of Fano varieties
Ann. of Math. 180 (2014) 197–232
Math Reviews MR3194814
69
Mabuchi, Toshiki
A functional integrating Futaki's invariant
Proc. Japan Acad. Ser. A Math. Sci. 61 (1985) 119–120
Math Reviews MR796483
70
Nadel, Alan M.
On the absence of periodic points for the Ricci curvature operator acting on the space of Kähler metrics
in Modern methods in complex analysis (Princeton, NJ, 1992)
Ann. of Math. Stud. 137 (1995) 277–281
Math Reviews MR1369142
71
Phong, D. H. and Song, Jian and Sturm, Jacob and Weinkove, Ben
The Moser-Trudinger inequality on Kähler-Einstein manifolds
Amer. J. Math. 130 (2008) 1067–1085
Math Reviews MR2427008
72
Rubinstein, Yanir A.
Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics
Adv. Math. 218 (2008) 1526–1565
Math Reviews MR2419932
73
Sesum, Natasa and Tian, Gang
Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman)
J. Inst. Math. Jussieu 7 (2008) 575–587
Math Reviews MR2427424
74
Sibony, Nessim
Quelques problèmes de prolongement de courants en analyse complexe
Duke Math. J. 52 (1985) 157–197
Math Reviews MR791297
75
Siu, Yum Tong
Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics
Birkhäuser, 1987
Math Reviews MR904673
76
Skoda, Henri
Prolongement des courants, positifs, fermés de masse finie
Invent. math. 66 (1982) 361–376
Math Reviews MR662596
77
78
Stoppa, Jacopo
K-stability of constant scalar curvature Kähler manifolds
Adv. Math. 221 (2009) 1397–1408
Math Reviews MR2518643
79
Székelyhidi, Gábor
Greatest lower bounds on the Ricci curvature of Fano manifolds
Compos. Math. 147 (2011) 319–331
Math Reviews MR2771134
80
Székelyhidi, Gábor
Filtrations and test-configurations
Math. Ann. 362 (2015) 451–484
Math Reviews MR3343885
81
Tian, Gang
Kähler-Einstein metrics with positive scalar curvature
Invent. math. 130 (1997) 1–37
Math Reviews MR1471884
82
Tian, Gang
Canonical metrics in Kähler geometry
Birkhäuser, 2000
Math Reviews MR1787650
83
Tian, Gang
K-stability and Kähler-Einstein metrics
Comm. Pure Appl. Math. 68 (2015) 1085–1156
Math Reviews MR3352459
84
Wang, Xiaowei
Height and GIT weight
Math. Res. Lett. 19 (2012) 909–926
Math Reviews MR3008424
85
Yau, Shing Tung
On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I
Comm. Pure Appl. Math. 31 (1978) 339–411
Math Reviews MR480350
86
Yau, Shing Tung
Non-Linear Analysis in Geometry
Enseignement Math. 33 (1986) 1–54