Complete Kähler–Einstein metrics under certain holomorphic covering and Examples  [ Métriques complètes de Kähler–Einstein sous certains revêtements holomorphes et exemples ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2901-2921.

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.

Publié le : 2019-05-24
DOI : https://doi.org/10.5802/aif.3230
Classification : 32Q15,  32Q20,  53C55,  32H02
Mots 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--Tung},
     title = {Complete K\"ahler--Einstein metrics under certain holomorphic covering and Examples},
     journal = {Annales de l'Institut Fourier},
     pages = {2901--2921},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     doi = {10.5802/aif.3230},
     language = {en},
     url = {archive.numdam.org/item/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/item/AIF_2018__68_7_2901_0/

[1] Cheng, Shiu-Yuen; Yau, Shing-Tung Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., Volume 28 (1975) no. 3, pp. 333-354 | MR 0385749

[2] Cheng, Shiu-Yuen; Yau, Shing-Tung 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 | Article | MR 575736

[3] Cheng, Shiu-Yuen; Yau, Shing-Tung Inequality between Chern numbers of singular Kähler surfaces and characterization of orbit space of discrete group of SU (2,1), Complex differential geometry and nonlinear differential equations (Brunswick, 1984) (Contemporary Mathematics) Volume 49, American Mathematical Society, 1986, pp. 31-44 | Article | MR 833802

[4] Demailly, Jean-Pierre L 2 estimates for the ¯-operator on complex manifolds (1996) (Notes de cours, Ecole d’été de Mathématiques (Analyse Complexe))

[5] Diverio, Simone; Trapani, Stefano 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] Gao, Peng; Yau, Shing-Tung; Zhou, Wubin Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds, Math. Ann., Volume 369 (2017) no. 3-4, pp. 1271-1282 | Article | MR 3713541

[7] Griffiths, Phillip A. Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. Math., Volume 94 (1971), pp. 21-51 | Article | MR 0310284

[8] Heier, Gordon; Lu, Steven S. Y.; Wong, Bun On the canonical line bundle and negative holomorphic sectional curvature, Math. Res. Lett., Volume 17 (2010) no. 6, pp. 1101-1110 | Article | MR 2729634 | Zbl 1233.14005

[9] Heier, Gordon; Lu, Steven S. Y.; Wong, Bun Kähler manifolds of semi-negative holomorphic sectional curvature, J. Differ. Geom., Volume 104 (2016) no. 3, pp. 419-441 | MR 3568627

[10] Kikuta, Shin Carathéodory measure hyperbolicity and positivity of canonical bundles, Proc. Am. Math. Soc., Volume 139 (2011) no. 4, pp. 1411-1420 | Article | MR 2748434

[11] Klembeck, Paul F. 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 0463506

[12] Kobayashi, Shoshichi Intrinsic distances, measures and geometric function theory, Bull. Am. Math. Soc., Volume 82 (1976) no. 3, pp. 357-416 | MR 0414940

[13] Kobayashi, Shoshichi; Nomizu, Katsumi Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, 1996, xii+329 pages | MR 1393940

[14] Liu, Kefeng; Sun, Xiaofeng; Yau, Shing-Tung Canonical metrics on the moduli space of Riemann surfaces. I, J. Differ. Geom., Volume 68 (2004) no. 3, pp. 571-637 | MR 2144543

[15] Mok, Ngaiming; Yau, Shing-Tung 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 720056

[16] Royden, H. L. The Ahlfors–Schwarz lemma in several complex variables, Comment. Math. Helv., Volume 55 (1980) no. 4, pp. 547-558 | Article | MR 604712 | Zbl 0484.53053

[17] Schumacher, Georg Asymptotics of complete Kähler–Einstein metrics – negativity of the holomorphic sectional curvature, Doc. Math., Volume 7 (2002), pp. 653-658 | MR 2015056 | Zbl 1022.32008

[18] Shi, Wan-Xiong Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differ. Geom., Volume 45 (1997) no. 1, pp. 94-220 | MR 1443333

[19] Tian, Gang; Yau, Shing-Tung 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 915840 | Zbl 0682.53064

[20] Tosatti, Valentino; Yang, Xiaokui An extension of a theorem of Wu–Yau, J. Differ. Geom., Volume 107 (2017) no. 3, pp. 573-579

[21] Wu, Damin Higher canonical asymptotics of Kähler–Einstein metrics on quasi-projective manifolds, Commun. Anal. Geom., Volume 14 (2006) no. 4, pp. 795-845 | MR 2273294

[22] Wu, Damin Kähler–Einstein metrics of negative Ricci curvature on general quasi-projective manifolds, Commun. Anal. Geom., Volume 16 (2008) no. 2, pp. 395-435 | MR 2425471

[23] Wu, Damin; Yau, Shing-Tung Negative holomorphic curvature and positive canonical bundle, Invent. Math., Volume 204 (2016) no. 2, pp. 595-604 | Article | MR 3489705

[24] Wu, Damin; Yau, Shing-Tung A remark on our paper “Negative holomorphic curvature and positive canonical bundle”, Commun. Anal. Geom., Volume 24 (2016) no. 4, pp. 901-912

[25] Wu, Damin; Yau, Shing-Tung Invariant metrics on negatively pinched complete Kähler manifolds (2017) (https://arxiv.org/abs/1711.09475, submitted)

[26] Wu, Hung 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 1207887 | Zbl 0773.32017

[27] Yau, Shing-Tung A general Schwarz lemma for Kähler manifolds, Am. J. Math., Volume 100 (1978) no. 1, pp. 197-203 | MR 0486659

[28] Yau, Shing-Tung 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