Première classe de Chern et courbure de Ricci : preuve de la conjecture de Calabi

Helffer, B. ; Gallot, Sylvain ; Polombo, Albert ; Bérard Bergery, Lionel ; Averous, Geneviève ; Deschamps, Annie ; Calabi, Eugenio ; Bourguignon, J.-P. ; Yau, Shing Tung ; Ezin, J. P.

Notes prises par : Bourguignon, J.-P.

Astérisque, no. 58 (1978) , 176 p.

Résumé

These notes from a seminar held in the Spring 1978 at the Centre de Mathématiques de l'Ecole Polytechnique in Palaiseau detail the proof of the Calabi conjecture given by S. T. Yau in the Fall 1976. The conjecture made in 1954 asserts that on a compact Kähler manifold any closed form of type $(1, 1)$ whose cohomology class is a multiple of the first Chern class is the Ricci form of a Kähler metric.

The necessary background material, both in non linear analysis (Schauder estimates, the continuity method) and in Kähler geometry (the differential calculus, special properties of the curvature, the first Chern form), is presented at some length. On the other hand applications of the conjecture are not treated.

The proof given follows basically S. T. Yau's proof except for the uniform estimate, where an extension of an argument which was special to two complex dimensions is presented. Efforts have been made to give a more intrinsic treatment of the whole subject.

Zbl

@book{AST_1978__58__1_0,
     author = {Helffer, B. and Gallot, Sylvain and Polombo, Albert and B\'erard Bergery, Lionel and Averous, Genevi\`eve and Deschamps, Annie and Calabi, Eugenio and Bourguignon, J.-P. and Yau, Shing Tung and Ezin, J. P.},
     title = {Premi\`ere classe de {Chern} et courbure de {Ricci} : preuve de la conjecture de {Calabi}},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {58},
     year = {1978},
     zbl = {0397.35028},
     language = {fr},
     url = {http://www.numdam.org/item/AST_1978__58__1_0/}
}

TY  - BOOK
AU  - Helffer, B.
AU  - Gallot, Sylvain
AU  - Polombo, Albert
AU  - Bérard Bergery, Lionel
AU  - Averous, Geneviève
AU  - Deschamps, Annie
AU  - Calabi, Eugenio
AU  - Bourguignon, J.-P.
AU  - Yau, Shing Tung
AU  - Ezin, J. P.
TI  - Première classe de Chern et courbure de Ricci : preuve de la conjecture de Calabi
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%0 Book
%A Helffer, B.
%A Gallot, Sylvain
%A Polombo, Albert
%A Bérard Bergery, Lionel
%A Averous, Geneviève
%A Deschamps, Annie
%A Calabi, Eugenio
%A Bourguignon, J.-P.
%A Yau, Shing Tung
%A Ezin, J. P.
%T Première classe de Chern et courbure de Ricci : preuve de la conjecture de Calabi
%S Astérisque
%D 1978
%N 58
%I Société mathématique de France
%U http://www.numdam.org/item/AST_1978__58__1_0/
%G fr
%F AST_1978__58__1_0

Helffer, B.; Gallot, Sylvain; Polombo, Albert; Bérard Bergery, Lionel; Averous, Geneviève; Deschamps, Annie; Calabi, Eugenio; Bourguignon, J.-P.; Yau, Shing Tung; Ezin, J. P. Première classe de Chern et courbure de Ricci : preuve de la conjecture de Calabi. Astérisque, no. 58 (1978), Bourguignon, J.-P. (red.), 176 p. http://numdam.org/item/AST_1978__58__1_0/

Bibliographie
Cité par

[1] T. Aubin, Métriques riemanniennes et courbure, J. Diff. Geom., 4 (1970), 383-424. | MR | Zbl | DOI

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

[3] E. Calabi, The space of Kähler metrics, Proc. Internat. Congress Math. Amsterdam, Vol. 2 (1954), 206-207.

[4] E. Calabi, On Kähler manifolds with vanishing canonical class, Algebraic geometry and topology, A symposium in honor of Lefschetz, Princeton Univ. Press (1955), 78-89. | MR | Zbl

[5] J. Kazdan, A remark on the preceding paper of Yau, Comm. Pure and Appl. Math. XXXI (1978), 413-414. | MR | Zbl | DOI

[6] S. T. Yau, On Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. U.S.A. 74 (1977), 1798-1799. | MR | Zbl | DOI

[7] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure and Appl. Math. XXXI (1978), 339-411. | Zbl | MR | DOI

[1] R. A. Adams, Sobolev spaces, Acad. Press (1975). | MR | Zbl

[2] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598. | Zbl | MR | DOI

[3] T. Aubin, Espaces de Sobolev sur les variétés riemanniennes, Bulletin des Sciences mathématiques, 100 (1976), 149-173. | MR | Zbl

[4] J. L. Lions, Problèmes aux limites dans les équations aux dérivées partielles, Séminaire de Montréal (1965). | Zbl | MR

[5] J. L. Lions, J. Peetre, Sur une classe d'espaces d'interpolation, Publications de l'I.H.E.S., n° 19 (1969). | EuDML | Zbl | MR | Numdam

[1] L. Bers, F. John, J. Schechter, Partial differential equations, Lectures in Applied Mathematics, vol. 3, Interscience, ch.II.5, (1974). | Zbl | MR

[2] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Math. Wiss., vol. 224, Springer-Verlag, (1977). | MR | Zbl

[3] 0. A. Ladyzenskaya, N. U. Ural'Tseva, Equations aux dérivées partielles de type elliptique, Dunod, ch.III, 99-128, (1968). | MR | Zbl

[1] K. Kodaira, J. Morrow, Complex manifolds, Holt, Rinehart and Winston (1971). | MR | Zbl

[2] A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65 (1967), 391-404. | Zbl | MR | DOI

[3] R. C. Gunning, H. Rossi, Analytic functions of several complex variables, Prentice-Hall, (1965) . | MR | Zbl

[1] M. Berger, A. Lascoux, Variétés kählériennes compactes, Lecture Notes in Mathematics, Springer-Verlag, Vol. 154 (1970). | MR | Zbl

[2] R. Bott, S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Mathematica, Vol. 144, (1965) 71-112. | MR | Zbl | DOI

[3] J. P. Bourguignon, Géométrie riemannienne et opérateurs différentiels naturels, Cours de troisième cycle, (à paraître).

[4] S. S. Chern, Complex manifolds without potentiel theory, Van Nostrand Mathematical Studies n° 15 (1967). | MR | Zbl

[5] F. Hirzebruch, Topological methods in algebraic geometry, Springer Verlag (1966). | MR | Zbl

[6] J. Milnor, J. Stasheff, Characteristic classes, Annals of Mathematical studies n° 74 (1974), Princeton University Press. | MR | Zbl

[1] S. T. Yau, On the Ricci curvature of a complex Kähler manifold and the complex Monge-Ampère equation I , Comm. Pure and Appl. Math. XXXI (1978), 339-411. | MR | Zbl | DOI

[1] J. Kazdan, F. Warner, Curvature functions on $2$ -manifolds, Ann. of Math., 99 (1974), 14-47. | MR | Zbl | DOI

[2] G. De Rham, Variétés différentiables, Hermann (1960). | Zbl

[3] F. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co (1971). | MR | Zbl

[1] T. Aubin, Métriques riemanniennes et courbure, J. Diff. Geom., 4 (1970), 383-424. | MR | Zbl | DOI

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

[3] A. V. Pogorelov, Monge-Ampère equations of elliptic type, Noordhoff, Groningen (1964). | MR | Zbl

[4] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure and Appl. Math., XXXI (1978), 339-411. | MR | Zbl | DOI

[1] V. Belinskii, G. Gibbons, D. Page, C. Pope, Asymptotically euclidean Bianchi IX metrics in quantum gravity, preprint (1978). | MR | DOI

[2] E. Calabi, A construction of nonhomogeneous Einstein metrics, Proc. Symp. Pure Math. Stanford, Vol. 27 (1975), 17-24. | MR | Zbl | DOI

[3] E. Calabi, Métriques kählériennes et fibrés holomorphes, à paraître aux Ann. Sc. de l'E.N.S. | DOI | MR | Zbl | EuDML | Numdam

[4] S. Y. Cheng, S. T. Yau, On the regularity of the Monge-Ampère equation $det (\frac{\partial^{2} u}{\partial x^{i} \partial x^{j}}) = F (x, u)$ , Comm. Pure Appl. Math., XXX (1977), 47-68. | MR | Zbl

[5] T. Eguchi, A. Hanson, Asymptotically flat self-dual solutions to Euclidean gravity, à paraître aux Physics Letters B. | MR

[1] M. Demazure, Sous-groupes algébriques du groupe de Cremona, Ann. SCi. E.N.S. Paris, 3 (1970), 507-589. | MR | Zbl | EuDML | Numdam

[2] F. Hirzebruch, Über eine Klasse von einfach-zusammenhängenden komplexen Mannigfaltigkeiten, Math. Ann., 124 (1951), 77-86. | MR | Zbl | EuDML | DOI

[3] N. Hitchin, On the curvature of rational surfaces, Proc. of Symp. Pure Math., XXVII (1975), 65-80. | MR | Zbl | DOI

[4] S. Kobayashi, Transformation groups in differential geometry, Erg. der Math. 70, Springer (1970). | MR | Zbl

[5] A. Lichnerowicz, Géométrie des groupes de transformations, Dunod (1958). | MR | Zbl

[6] G. De Rham, Variétés différentiables, Hermann (1955). | MR | Zbl

[7] S. T. Yau, On the curvature of compact Hermitian manifolds, Inv. Math., 25 (1974), 213-239. | MR | Zbl | EuDML | DOI

[1] E. Calabi, Improper affine hyperspheres and a generalization of a theorem of K. Jörgens, Michigan Math. J. 5, (1958) 105-126. | MR | Zbl

[2] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure and App. Math. XXXI, (1978) 339-411. | MR | Zbl | DOI

[1] E. Calabi, Improper affine hyperspheres and a generalization of a theorem of K. Jörgens, Mich. J., 5 (1958), 105-126. | MR | Zbl | DOI

[2] S. Y. Cheng, S. T. Yau, On the regularity of the Monge-Ampère equation $det (\frac{\partial^{2} u}{\partial x^{i} \partial x^{j}}) = F (x, u)$ , Comm. Pure Appl. Math., XXX (1977). 47-68. | MR | Zbl