[Métriques complètes de Kähler–Einstein sous certains revêtements holomorphes et exemples]
Nous établissons l’unique métrique complète de Kähler–Einstein avec courbure scalaire négative sur une large classe de variétés de Kähler complètes, y compris les variétés dont l’espace de recouvrement peut être biholomorphiquement plongé dans une variété de Kähler à courbure sectionnelle holomorphe limitée au-dessus par une constante négative. Nous présentons en outre plusieurs nouveaux exemples de variétés complètes de Kähler–Einstein non compactes, générés par les résultats.
We establish the unique complete Kähler–Einstein metric with negative scalar curvature on a broad class of complete Kähler manifolds, including those manifolds whose covering space can be biholomorphically embedded into a Kähler manifold with holomorphic sectional curvature bounded above by a negative constant. We further present several new examples of complete noncompact Kähler–Einstein manifolds, generated by the results.
Keywords: Kähler–Einstein metric, holomorphic covering, complete Kähler manifold, examples
Mot clés : Métrique de Kähler–Einstein, revêtements holomorphes, variétés complètes de Kähler, exemples
@article{AIF_2018__68_7_2901_0,
author = {Wu, Damin and Yau, Shing{\textendash}Tung},
title = {Complete {K\"ahler{\textendash}Einstein} metrics under certain holomorphic covering and {Examples}},
journal = {Annales de l'Institut Fourier},
pages = {2901--2921},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {68},
number = {7},
year = {2018},
doi = {10.5802/aif.3230},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.3230/}
}
TY - JOUR AU - Wu, Damin AU - Yau, Shing–Tung TI - Complete Kähler–Einstein metrics under certain holomorphic covering and Examples JO - Annales de l'Institut Fourier PY - 2018 SP - 2901 EP - 2921 VL - 68 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3230/ DO - 10.5802/aif.3230 LA - en ID - AIF_2018__68_7_2901_0 ER -
%0 Journal Article %A Wu, Damin %A Yau, Shing–Tung %T Complete Kähler–Einstein metrics under certain holomorphic covering and Examples %J Annales de l'Institut Fourier %D 2018 %P 2901-2921 %V 68 %N 7 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3230/ %R 10.5802/aif.3230 %G en %F AIF_2018__68_7_2901_0
Wu, Damin; Yau, Shing–Tung. Complete Kähler–Einstein metrics under certain holomorphic covering and Examples. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2901-2921. doi : 10.5802/aif.3230. http://archive.numdam.org/articles/10.5802/aif.3230/
[1] Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., Volume 28 (1975) no. 3, pp. 333-354 | MR
[2] On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Commun. Pure Appl. Math., Volume 33 (1980) no. 4, pp. 507-544 | DOI | MR
[3] Inequality between Chern numbers of singular Kähler surfaces and characterization of orbit space of discrete group of , Complex differential geometry and nonlinear differential equations (Brunswick, 1984) (Contemporary Mathematics), Volume 49, American Mathematical Society, 1986, pp. 31-44 | DOI | MR
[4] estimates for the -operator on complex manifolds (1996) Notes de cours, Ecole d’été de Mathématiques (Analyse Complexe)
[5] Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle (2016) (https://arxiv.org/abs/1606.01381, to appear in J. Differ. Geom.)
[6] Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds, Math. Ann., Volume 369 (2017) no. 3-4, pp. 1271-1282 | DOI | MR
[7] Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. Math., Volume 94 (1971), pp. 21-51 | DOI | MR
[8] On the canonical line bundle and negative holomorphic sectional curvature, Math. Res. Lett., Volume 17 (2010) no. 6, pp. 1101-1110 | DOI | MR | Zbl
[9] Kähler manifolds of semi-negative holomorphic sectional curvature, J. Differ. Geom., Volume 104 (2016) no. 3, pp. 419-441 | MR
[10] Carathéodory measure hyperbolicity and positivity of canonical bundles, Proc. Am. Math. Soc., Volume 139 (2011) no. 4, pp. 1411-1420 | DOI | MR
[11] Kähler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana Univ. Math. J., Volume 27 (1978) no. 2, pp. 275-282 | MR
[12] Intrinsic distances, measures and geometric function theory, Bull. Am. Math. Soc., Volume 82 (1976) no. 3, pp. 357-416 | MR
[13] Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, 1996, xii+329 pages | MR
[14] Canonical metrics on the moduli space of Riemann surfaces. I, J. Differ. Geom., Volume 68 (2004) no. 3, pp. 571-637 | MR
[15] Completeness of the Kähler–Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, The mathematical heritage of Henri Poincaré, Part 1 (Bloomington, 1980) (Proceedings of Symposia in Pure Mathematics), Volume 39, American Mathematical Society, 1983, pp. 41-59 | MR
[16] The Ahlfors–Schwarz lemma in several complex variables, Comment. Math. Helv., Volume 55 (1980) no. 4, pp. 547-558 | DOI | MR | Zbl
[17] Asymptotics of complete Kähler–Einstein metrics – negativity of the holomorphic sectional curvature, Doc. Math., Volume 7 (2002), pp. 653-658 | MR | Zbl
[18] Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differ. Geom., Volume 45 (1997) no. 1, pp. 94-220 | MR
[19] Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, 1986) (Advanced Series in Mathematical Physics), Volume 1, World Scientific, 1987, pp. 574-628 | MR | Zbl
[20] An extension of a theorem of Wu–Yau, J. Differ. Geom., Volume 107 (2017) no. 3, pp. 573-579
[21] Higher canonical asymptotics of Kähler–Einstein metrics on quasi-projective manifolds, Commun. Anal. Geom., Volume 14 (2006) no. 4, pp. 795-845 | MR
[22] Kähler–Einstein metrics of negative Ricci curvature on general quasi-projective manifolds, Commun. Anal. Geom., Volume 16 (2008) no. 2, pp. 395-435 | MR
[23] Negative holomorphic curvature and positive canonical bundle, Invent. Math., Volume 204 (2016) no. 2, pp. 595-604 | DOI | MR
[24] A remark on our paper “Negative holomorphic curvature and positive canonical bundle”, Commun. Anal. Geom., Volume 24 (2016) no. 4, pp. 901-912
[25] Invariant metrics on negatively pinched complete Kähler manifolds (2017) (https://arxiv.org/abs/1711.09475, submitted)
[26] Old and new invariant metrics on complex manifolds, Several complex variables (Stockholm, 1987/1988) (Mathematical Notes), Volume 38, Princeton University Press, 1993, pp. 640-682 | MR | Zbl
[27] A general Schwarz lemma for Kähler manifolds, Am. J. Math., Volume 100 (1978) no. 1, pp. 197-203 | MR
[28] Métriques de Kähler–Einstein sur les variétés ouvertes, Première Classe de Chern et courbure de Ricci: Preuve de la conjecture de Calabi (Palaiseau, 1978) (Astérisque), Volume 58, Société Mathématique de France, 1978, pp. 163-167
Cité par Sources :
