Métriques d'Einstein-Kähler sur les variétés de Fano : obstructions et existence
Séminaire Bourbaki : volume 1996/97, exposés 820-834, Astérisque, no. 245 (1997), Exposé no. 830, 29 p.
@incollection{SB_1996-1997__39__277_0,
     author = {Bourguignon, Jean-Pierre},
     title = {M\'etriques {d'Einstein-K\"ahler} sur les vari\'et\'es de {Fano} : obstructions et existence},
     booktitle = {S\'eminaire Bourbaki : volume 1996/97, expos\'es 820-834},
     series = {Ast\'erisque},
     note = {talk:830},
     pages = {277--305},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {245},
     year = {1997},
     mrnumber = {1627115},
     zbl = {0935.32019},
     language = {fr},
     url = {http://www.numdam.org/item/SB_1996-1997__39__277_0/}
}
TY  - CHAP
AU  - Bourguignon, Jean-Pierre
TI  - Métriques d'Einstein-Kähler sur les variétés de Fano : obstructions et existence
BT  - Séminaire Bourbaki : volume 1996/97, exposés 820-834
AU  - Collectif
T3  - Astérisque
N1  - talk:830
PY  - 1997
SP  - 277
EP  - 305
IS  - 245
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/SB_1996-1997__39__277_0/
LA  - fr
ID  - SB_1996-1997__39__277_0
ER  - 
%0 Book Section
%A Bourguignon, Jean-Pierre
%T Métriques d'Einstein-Kähler sur les variétés de Fano : obstructions et existence
%B Séminaire Bourbaki : volume 1996/97, exposés 820-834
%A Collectif
%S Astérisque
%Z talk:830
%D 1997
%P 277-305
%N 245
%I Société mathématique de France
%U http://www.numdam.org/item/SB_1996-1997__39__277_0/
%G fr
%F SB_1996-1997__39__277_0
Bourguignon, Jean-Pierre. Métriques d'Einstein-Kähler sur les variétés de Fano : obstructions et existence, dans Séminaire Bourbaki : volume 1996/97, exposés 820-834, Astérisque, no. 245 (1997), Exposé no. 830, 29 p. http://www.numdam.org/item/SB_1996-1997__39__277_0/

[1] Atiyah, M. F., Bott, R., The Yang-Mills Equations over Riemann Surfaces, Phil. Trans. R. Soc. London A 308 (1983), 523-615. | DOI | MR | Zbl

[2] Aubin, T., Équations du type de Monge-Ampère sur les variétés kählériennes compactes, C.R. Acad. Sci. Paris 283 (1976), 119-121. | MR | Zbl

[3] Aubin, T., Réduction du cas positif de l'équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d'une inégalité, J. Functional Anal. 57 (1984), 143-153. | DOI | MR | Zbl

[4] Bahri, A., Coron, J.-M., The Scalar Curvature Problem on the Standard 3-Dimensional Sphere, J. Functional Anal. 95 (1991), 106-172. | DOI | MR | Zbl

[5] Bando, S., The K-Energy Map, Almost Einstein-Kähler Metrics and an Inequality of the Miyaoka-Yau Type, Tôhoku Math. J. 39 (1987), 231-235. | DOI | MR | Zbl

[6] Bando, S., Mabuchi, T., Uniqueness of Kähler-Einstein Metrics Modulo Connected Group Actions, in Algebraic Geometry, Sendai 1985, Adv. Studies in Pure Math. 10, Kinokuniya, Tokyo, 1987, 11-40. | MR | Zbl

[7] Bando, S., Mabuchi, T., On some Integral Invariants on Compact Complex Manifolds, Proc. Japan Acad. Sci. 62 (1986), 197-200. | DOI | MR | Zbl

[8] Berger, M., Sur les variétés d'Einstein compactes, C.R. IIIème Réunion Math. Expression Latine, Namur (1965), 35-55. | MR | Zbl

[9] Besse, A. L., Einstein Manifolds, Ergeb. Math. 10, Springer-Verlag, Berlin- Heidelberg, 1987. | MR | Zbl

[10] Bott, R., A Residue Formula for Holomorphic Vector Fields, J. Differential Geom. 1 (1967), 311-330. | DOI | MR | Zbl

[11] Bourguignon, J. P., Premières formes de Chern des variétés kählériennes compactes, in Séminaire Bourbaki 1977-78, Exposé n°507, Lect. Notes in Math. 710, Springer-Verlag, Berlin-Heidelberg, 1978, 1-21. | EuDML | Numdam | MR | Zbl

[12] Bourguignon, J. P., Ricci Curvature and Einstein Metrics, in Differentiel Geometry and Global Analysis, Berlin 1979, U. Simon and D. Ferus Ed., Lect. Notes in Math. 838, Springer, Berlin-Heidelberg-New York, 1981, 42-63. | DOI | MR | Zbl

[13] Bourguignon, J. P., Invariants intégraux fonctionnels pour des équations aux dérivées partielles d'origine géométrique, in Differential Geometry, Peñiscola, A.M. Naveira Ed., Lect. Notes in Math. 1209, Springer-Verlag, Berlin-Heidelberg, 1987, 100-108. | MR | Zbl

[14] Bourguignon, J. P., L'équation de la chaleur associée à la courbure de Ricci, in Séminaire Bourbaki 1985-86, Exposé n°653, Astérisque 145-146 (1987), 45-61. | EuDML | Numdam | MR | Zbl

[15] Burns, D., De Bartolomeis, P., Stability of Vector Bundles and Extremal Metrics, Inventiones Math. 92 (1988), 403-407. | DOI | EuDML | MR | Zbl

[16] Calabi, E., The Space of Kähler Metrics, in Proc. International Congress of Mathematicians, Amsterdam, II (1954), 206-207.

[17] Calabi, E., Improper Affine Hyperspheres and a Generalization of a Theorem of K. Jörgens, Michigan Math. J. 5 (1958), 105-126. | DOI | MR | Zbl

[18] Cao, H. D., Deformation of Kähler metrics to Kähler-Einstein metrics on Compact Kähler Manifolds, Inventiones Math. 81 (1985), 359-372. | DOI | EuDML | MR | Zbl

[19] Catanese, F., Lebrun, C., On the Scalar Curvature of Einstein Manifolds, Prépublication, Universität Göttingen. | DOI | Zbl

[20] Chang, S. Y. A., Yang, P. C., Prescribing Gaussian Curvature on S2, Acta Math. 159 (1987), 215-259. | DOI | MR | Zbl

[21] Chrusciel, P., Semi-Global Existence and Convergence of Solutions of the Robinson-Trautman (2-dimensional Calabi) Equation, Commun. Math. Phys. 137 (1991), 289-313. | DOI | MR | Zbl

[22] Debarre, O., Variétés de Fano, in Séminaire Bourbaki 1996-97, Exposé n° 827, 1-25. | EuDML | Numdam | MR | Zbl

[23] Demailly, J.-P., Kollár, J., Semi-continuity of Complex Singularity Exponents and Kähler-Einstein Metrics on Fano Orbifolds, Prépublication Institut Fourier, Grenoble, à paraître. | Numdam | Zbl

[24] Demazure, M., Surfaces de Del Pezzo, in Séminaire sur les singularités des surfaces 1976-1977, Palaiseau, Lect. Notes in Math. 777, Springer-Verlag, Berlin- Heidelberg-New York, 1980, 23-69. | EuDML | Numdam | Zbl

[25] Ding, W., Remarks on the Existence Problem of Positive Kähler-Einstein Met- rics, Math. Ann. 282 (1988), 463-471. | DOI | EuDML | MR | Zbl

[26] Ding, W., Tian, G., Kähler-Einstein Metrics and the Generalized Futaki Invariant, Inventiones Math. 110 (1992), 315-335. | DOI | EuDML | MR | Zbl

[27] Ding, W., Tian, G., The Generalized Moser-Trudinger Inequality, in Proc. Int. Conf. on Non-Linear Analysis, Tianjin, K.C. Chang et al. Ed., World Scientific, Singapore, 1992, 57-70. | Zbl

[28] Donaldson, S. K., Anti-Self-Dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundles, Proc. London Math. Soc. 50 (1985), 1-26. | DOI | MR | Zbl

[29] Donaldson, S. K., Infinite Determinants, Stable Bundles and Curvature, Duke Math. J. 54 (1987), 231-247. | DOI | MR | Zbl

[30] Donaldson, S. K., Remarks on Gauge Theory, Complex Geometry and 4-Manifold Topology, The Fields Medal Volume, M.F. Atiyah and D. Iagolnitzer Ed., World Scientific, 1997. | MR

[31] Donaldson, S. K., Symmetric Spaces, Kähler Geometry and Hamiltonian Dynamics, Preprint, Oxford Univ., Oxford, 1997. | MR | Zbl

[32] Futaki, A., On Compact Kähler Manifolds of Constant Scalar Curvatures, Proc. Japan Acad. Sci. 59 (1983), 401-402. | DOI | MR | Zbl

[33] Futaki, A., An Obstruction to the Existence of Kähler-Einstein Metrics, Inventiones Math. 73 (1983), 437-443. | DOI | EuDML | MR | Zbl

[34] Futaki, A., The Ricci Curvature of Symplectic Quotients of Fano Manifolds, Tôhoku Math. J. 39 (1987), 329-339. | DOI | MR | Zbl

[35] Futaki, A., Kähler-Einstein Metrics and Integral Invariants, Lect. Notes in Math. 1314, Springer, Berlin-Heidelberg-New York, 1988. | MR | Zbl

[36] Futaki, A., Mabuchi, T., An Obstruction Class and a Representation of Holomorphic Automorphisms, in Geometry and Analysis on Manifolds, Lect. Notes in Math. 1339, Springer, Berlin-Heidelberg-New York-Tokyo, (1988), 127-141. | MR | Zbl

[37] Futaki, A., Mabuchi, T., Sakane, Y., Einstein-Kähler Metrics with Positive Ricci Curvature, in Kähler Metrics and Moduli Spaces, Adv. Stud. Pure Math. 18 (1990), 11-83. | MR | Zbl

[38] Kazdan, J. L., Warner, F. W., Curvature Functions for Compact 2-manifolds, Ann. Math. 99 (1974), 14-47. | DOI | MR | Zbl

[39] Kobayashi, S., On Compact Kähler Manifolds with Positive Definite Ricci Tensor, Ann. Math. 74 (1961), 570-574. | DOI | MR | Zbl

[40] Kobayashi, S., Curvature and Stability of Vector Bundles, Proc. Japan Acad. Sci. 58 (1982), 158-162. | DOI | MR | Zbl

[41] Kobayashi, S., Differential Geometry of Complex Vector Bundles, Publ. Math. Soc. Japan 15, Princeton Univ. Press, Princeton, and Iwanami Shoten, Tokyo, 1987. | MR | Zbl

[42] Kohn, J. J., Subellipticity of the ∂-Neumann Problem on Pseudo-Convex Domains: Sufficient Conditions, Acta Math. 142 (1979), 79-122. | DOI | Zbl

[43] Koiso, N., Sakane, Y., Non-Homogeneous Kähler-Einstein Metrics on Compact Complex Manifolds, in Curvature and Topology of Riemannian Manifolds, Lect. Notes in Math. 1201, Springer, Berlin-Heidelberg-New York-Tokyo, 1986, 165-179. | DOI | MR | Zbl

[44] Lebrun, C., Polarized 4-Manifolds, Extremal Kähler Metrics and Seiberg-Witten Theory, Math. Res. Lett. 2 (1995), 653-662. | DOI | MR | Zbl

[45] Lebrun, C., Simanca, S., Extremal Kähler Metrics and Complex Deformation Theory, Geom. Funct. Anal. 4 (1994), 298-336. | DOI | EuDML | MR | Zbl

[46] Lichnerowicz, A., Sur les transformations analytiques des variétés kählériennes, C.R. Acad. Sci. Paris 244 (1957), 3011-3014. | MR | Zbl

[47] Lübke, M., Stability of Einstein-Kähler Vector Bundles, Manuscripta Math. 42 (1983), 245-257. | DOI | EuDML | MR | Zbl

[48] Mabuchi, T., Some Symplectic Geometry on Compact Kähler Manifolds, Osaka Math. J. 24 (1987), 227-252. | MR | Zbl

[49] Mabuchi, T., Einstein-Kähler Forms, Futaki Invariants and Convex Geometry on Toric Fano Manifolds, Osaka Math. J. 24 (1987), 705-737. | MR | Zbl

[50] Margerin, C., Fibrés stables et métriques d'Hermite-Einstein, in Séminaire Bourbaki 1986-87, Exposé n°683, Astérisque 152-153 (1987), 263-283. | EuDML | Numdam | MR | Zbl

[51] Matsushima, Y., Sur les espaces homogènes kählériens d'un groupe réductif, Nagoya Math. J. 11 (1957), 53-60. | DOI | MR | Zbl

[52] Matsushima, Y., Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne, Nagoya Math. J. 11 (1957), 145-150. | DOI | MR | Zbl

[53] Matsushima, Y., Remarks on Kähler-Einstein Manifolds, Nagoya Math. J. 46 (1972), 161-173. | DOI | MR | Zbl

[54] Moser, J., A Sharp Form of an Inequality by N. Trudinger, Indiana Math. J. 20 (1971), 1077-1091. | DOI | MR | Zbl

[55] Mumford, D., Stability of Projective Varieties, Enseignement Math. 23 (1977), fasc. 1-2, 39-110. | MR | Zbl

[56] Nadel, A. M., Multiplier Ideal Sheaves and Existence of Kähler-Einstein Metrics of Positive Scalar Curvature, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 7299- 7300. | DOI | MR | Zbl

[57] Nadel, A. M., Multiplier Ideal Sheaves and Kähler-Einstein Metrics of Positive Scalar Curvature, Ann. Math. 132 (1990), 549-596. | DOI | MR | Zbl

[58] Nadel, A. M., Multiplier Ideal Sheaves and Futaki's Invariant, Preprint, Univ. Southern California, Los Angeles, 1997. | MR | Zbl

[59] Nakagawa, Y., Einstein-Kähler toric Fano fourfolds, Tôhoku Math. J. 45 (1993), 297-310. | DOI | MR | Zbl

[60] Nakagawa, Y., Combinatorial Formulae for Futaki Characters and Generalized Killing Forms on Toric Fano Manifolds, Prépublication, Tôhoku Univ., Sendai. | Zbl

[61] Onofri, E., On the Positivity of the Effective Action in a Theory of Random Surfaces, Commun. Math. Phys. 86 (1982), 321-326. | DOI | MR | Zbl

[62] Sakane, Y., Examples of Compact Kähler-Einstein Manifolds with Positive Ricci Curvature, Osaka J. Math. 31 (1986), 585-617. | MR | Zbl

[63] Semmes, S., Complex Monge-Ampère Equations and Symplectic Manifolds, Amer. J. Math. 114 (1992), 495-550. | DOI | MR | Zbl

[64] Première classe de Chern et courbure de Ricci : preuve de la conjecture de Calabi, Séminaire Palaiseau, Astérisque 58 (1978). | MR | Zbl

[65] Siu, Y. T., Lectures on Hermite-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics, Deutscher Math. Ver. Seminar 8, Birkhäuser, Basel, 1987. | MR | Zbl

[66] Siu, Y. T., Kähler-Einstein Metrics for the Case of Positive First Chern Class, in Complex Analysis III, C.A. Berenstein Ed., Lect. Notes in Math. 1277, Springer-Verlag, Berlin-Heidelberg-New York, (1987), 120-130. | MR | Zbl

[67] Siu, Y. T., The Existence of Kähler-Einstein Metrics on Manifolds with Positive Anticanonical Line Bundle and a Suitable Finite Symmetry Group, Ann. Math. 127 (1988), 585-627. | DOI | MR | Zbl

[68] Skoda, H., Sous-ensembles analytiques d'ordre fini ou infini dans C n , Bull. Soc. Math. France 100 (1972), 353-408. | DOI | EuDML | Numdam | MR | Zbl

[69] Subramanian, S., Stability of the Tangent Bundle and Existence of a Kähler-Einstein Metric, Math. Ann. 291 (1991), 573-577. | DOI | EuDML | MR | Zbl

[70] Tian, G., On Kähler-Einstein Metrics on Certain Manifolds with c1(M) > 0, Inventiones Math. 89 (1987), 225-246. | DOI | EuDML | MR | Zbl

[71] Tian, G., On Calabi's Conjecture for Complex Surfaces with Positive First Chern Class, Inventiones Math. 101 (1990), 101-172. | DOI | EuDML | MR | Zbl

[72] Tian, G., A Harnack Inequality for some Complex Monge-Ampère Equations, J. Differential Geom. 29 (1989), 481-488. | DOI | MR | Zbl

[73] Tian, G., On Stability of the Tangent Bundles of Fano Varieties, Intern. J. Math. 3 (1992), 401-413. | DOI | MR | Zbl

[74] Tian, G., Kähler-Einstein Metrics on Algebraic Manifolds, in C.I.M.E. Conf. Transcendental Methods in Algebraic Geom., F. Catanese, C. Ciliberto Ed., 1994. | MR | Zbl

[75] Tian, G., The K-Energy on Hypersurfaces and Stability, Commun. Geom. Anal. 2 (1994), 239-265. | DOI | MR | Zbl

[76] Tian, G., Kähler-Einstein Metrics with Positive Scalar Curvature, Inventiones Math., à paraître. | MR | Zbl

[77] Tian, G., Yau, S. T., Kähler-Einstein Metrics on Complex Surfaces with c1 (M) positive, Commun. Math. Phys. 112 (1987), 175-203. | DOI | MR | Zbl

[78] Tian, G., Zhu, X., A Non-Linear Inequality of Moser-Trudinger Type, Preprint Mass. Inst. Technology, Cambridge. | MR | Zbl

[79] Trudinger, N., On Imbeddings into Orlicz Spaces and some Applications, J. Math. Phys. 17 (1967), 473-483. | MR | Zbl

[80] Uhlenbeck, K. K., Yau, S. T., On the Existence of Hermite-Yang-Mills Connections in Stable Vector Bundles, Commun. Pure Appl. Math. 39 (1986), 257-293. | DOI | MR | Zbl

[81] Yau, S. T., On the Curvature of Compact Hermitian Manifolds, Inventiones Math. 25 (1974), 213-239. | DOI | EuDML | MR | Zbl

[82] Yau, S. T., On Calabi's Conjecture and some New Results in Algebraic Geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798-1799. | DOI | MR | Zbl

[83] Yau, S. T., On the Ricci-Curvature of a Complex Kähler Manifold and the Complex Monge-Ampère Equation, I, Commun. Pure Appl. Math. 31 (1978), 339- 411. | DOI | MR | Zbl

[84] Yau, S. T., Non-Linear Analysis in Geometry, Enseignement Math. 33 (1986), 1-54. | MR | Zbl

[85] Yau, S. T., Open Problems in Geometry, in Differential Geometry, Part I : Partial Differential Equations on Manifolds, Proc. Symp. Pure Math. 54, (1993), 1-28. | MR | Zbl

[86] Yotov, M., Nadel's Subschemes of Fano Manifolds with a Picard Group Isomorphic to Z, Preprint, Humboldt Univ., 1996. | MR | Zbl