Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds
Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 34 (2001) no. 4, p. 525-556
@article{ASENS_2001_4_34_4_525_0,
     author = {Demailly, Jean-Pierre and Koll\'ar, J\'anos},
     title = {Semi-continuity of complex singularity exponents and K\"ahler-Einstein metrics on Fano orbifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Elsevier},
     volume = {Ser. 4, 34},
     number = {4},
     year = {2001},
     pages = {525-556},
     doi = {10.1016/s0012-9593(01)01069-2},
     zbl = {0994.32021},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2001_4_34_4_525_0}
}
Demailly, Jean-Pierre; Kollár, János. Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 34 (2001) no. 4, pp. 525-556. doi : 10.1016/s0012-9593(01)01069-2. http://www.numdam.org/item/ASENS_2001_4_34_4_525_0/

[1] Angehrn U., Siu Y.-T., Effective freeness and point separation for adjoint bundles, Invent. Math. 122 (1995) 291-308. | MR 1358978 | Zbl 0847.32035

[2] Andreotti A., Vesentini E., Carleman estimates for the Laplace-Beltrami equation in complex manifolds, Publ. Math. I.H.E.S. 25 (1965) 81-130. | Numdam | Zbl 0138.06604

[3] Arnold V.I., Gusein-Zade S.M., Varchenko A.N., Singularities of Differentiable Maps, Progress in Math., Birkhäuser, 1985. | MR 777682

[4] Aubin T., Équations du type Monge-Ampère sur les variétés kählériennes compactes, C. R. Acad. Sci. Paris Ser. A 283 (1976) 119-121, Bull. Sci. Math. 102 (1978) 63-95. | Zbl 0333.53040

[5] Barlet D., Développements asymptotiques des fonctions obtenues par intégration sur les fibres, Invent. Math. 68 (1982) 129-174. | MR 666639 | Zbl 0508.32003

[6] Bombieri E., Algebraic values of meromorphic maps, Invent. Math. 10 (1970) 267-287, Addendum, Invent. Math. 11 (1970) 163-166. | MR 306201 | Zbl 0214.33702

[7] Boyer C., Galicki K., New Sasakian-Einstein 5-manifolds as links of isolated hypersurface singularities, Manuscript, February 2000.

[8] Demailly J.-P., Nombres de Lelong généralisés, théorèmes d'intégralité et d'analyticité, Acta Math. 159 (1987) 153-169. | MR 908144 | Zbl 0629.32011

[9] Demailly J.-P., Transcendental proof of a generalized Kawamata-Viehweg vanishing theorem, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989) 123-126, in: Berenstein C.A., Struppa D.C. (Eds.), Proceedings of the Conference “Geometrical and Algebraical Aspects in Several Complex Variables” held at Cetraro (Italy), 1989, pp. 81-94. | Zbl 01648048

[10] Demailly J.-P., Singular hermitian metrics on positive line bundles, in: Hulek K., Peternell T., Schneider M., Schreyer F. (Eds.), Proc. Conf. Complex algebraic varieties (Bayreuth, April 2-6, 1990), Lecture Notes in Math., 1507, Springer-Verlag, Berlin, 1992. | Zbl 0784.32024

[11] Demailly J.-P., Regularization of closed positive currents and intersection theory, J. Alg. Geom. 1 (1992) 361-409. | MR 1158622 | Zbl 0777.32016

[12] Demailly J.-P., Monge-Ampère operators, Lelong numbers and intersection theory, in: Ancona V., Silva A. (Eds.), Complex Analysis and Geometry, Univ. Series in Math., Plenum Press, New York, 1993. | Zbl 0792.32006

[13] Demailly J.-P., A numerical criterion for very ample line bundles, J. Differential Geom. 37 (1993) 323-374. | MR 1205448 | Zbl 0783.32013

[14] Demailly J.-P., L2 vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of the CIME Session, Transcendental Methods in Algebraic Geometry, Cetraro, Italy, July 1994, 96 p, Duke e-prints alg-geom/9410022. | Zbl 0883.14005

[15] Dolgachev I., Weighted projective varieties, in: Group Actions and Vector Fields, Proc. Polish-North Am. Semin., Vancouver 1981, Lect. Notes in Math., 956, Springer-Verlag, 1982, pp. 34-71. | Zbl 0516.14014

[16] Fletcher A.R., Working with weighted complete intersections, Preprint MPI/89-35, Max-Planck Institut für Mathematik, Bonn, 1989, Revised version: Iano-Fletcher A.R., in: Corti A., Reid M. (Eds.), Explicit Birational Geometry of 3-folds, Cambridge Univ. Press, 2000, pp. 101-173. | Zbl 0960.14027

[17] Fujiki A., Kobayashi R., Lu S.S.Y., On the fundamental group of certain open normal surfaces, Saitama Math. J. 11 (1993) 15-20. | MR 1259272 | Zbl 0798.14009

[18] Futaki A., An obstruction to the existence of Einstein-Kähler metrics, Invent. Math. 73 (1983) 437-443. | Zbl 0506.53030

[19] Hironaka H., Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. Math. 79 (1964) 109-326. | MR 199184 | Zbl 0122.38603

[20] Hörmander L., An Introduction to Complex Analysis in Several Variables, North-Holland Math. Libr., 7, North-Holland, Amsterdam, 1973. | MR 1045639 | Zbl 0271.32001

[21] Johnson J.M., Kollár J., Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces, Ann. Inst. Fourier 51 (2001) 69-79. | Numdam | Zbl 0974.14023

[22] Kawamata Y., Matsuda K., Matsuki K., Introduction to the minimal model problem, Adv. Stud. Pure Math. 10 (1987) 283-360. | MR 946243 | Zbl 0672.14006

[23] Kollár J., (with 14 coauthors) , Flips and Abundance for Algebraic Threefolds, Astérisque, 211, 1992. | Zbl 0814.14038

[24] Kollár J., Shafarevich Maps and Automorphic Forms, Princeton Univ. Press, 1995. | MR 1341589 | Zbl 0871.14015

[25] Kollár J., Singularities of pairs, Algebraic Geometry, Santa Cruz, 1995, in: Proceedings of Symposia in Pure Math. Vol. 62, AMS, 1997, pp. 221-287. | MR 1492525 | Zbl 0905.14002

[26] Lelong P., Intégration sur un ensemble analytique complexe, Bull. Soc. Math. France 85 (1957) 239-262. | Numdam | MR 95967 | Zbl 0079.30901

[27] Lelong P., Plurisubharmonic Functions and Positive Differential Forms, Gordon and Breach, New York, and Dunod, Paris, 1969. | Zbl 0195.11604

[28] Lichnerowicz A., Sur les transformations analytiques des variétés kählériennes, C. R. Acad. Sci. Paris 244 (1957) 3011-3014. | MR 94479 | Zbl 0080.37501

[29] Lichtin B., An upper semicontinuity theorem for some leading poles of |f|2s, in: Complex Analytic Singularities, Adv. Stud. Pure Math., 8, North-Holland, Amsterdam, 1987, pp. 241-272. | MR 894297 | Zbl 0615.32007

[30] Lichtin B., Poles of |f(z,w)|2s and roots of the B-function, Ark. för Math. 27 (1989) 283-304. | MR 1022282 | Zbl 0719.32016

[31] Manivel L., Un théorème de prolongement L2 de sections holomorphes d'un fibré vectoriel, Math. Z. 212 (1993) 107-122. | MR 1200166 | Zbl 0789.32015

[32] Matsushima Y., Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne, Nagoya Math. J. 11 (1957) 145-150. | MR 94478 | Zbl 0091.34803

[33] Nadel A.M., Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Proc. Nat. Acad. Sci. USA 86 (1989) 7299-7300. | Zbl 0711.53056

[34] Nadel A.M., Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Annals of Math. 132 (1990) 549-596. | Zbl 0731.53063

[35] Ohsawa T., Takegoshi K., On the extension of L2 holomorphic functions, Math. Z. 195 (1987) 197-204. | MR 892051 | Zbl 0625.32011

[36] Ohsawa T., On the extension of L2 holomorphic functions, II, Publ. RIMS, Kyoto Univ. 24 (1988) 265-275. | MR 944862 | Zbl 0653.32012

[37] Phong D.H., Sturm J., Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions, preprint, January 1999, to appear in Ann. of Math. | MR 1792297 | Zbl 0995.11065

[38] Phong D.H., Sturm J., On a conjecture of Demailly and Kollár, preprint, April 2000. | MR 1803721

[39] Shokurov V., 3-fold log flips, Izv. Russ. Acad. Nauk Ser. Mat. 56 (1992) 105-203. | MR 1162635

[40] Siu Y.T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974) 53-156. | MR 352516 | Zbl 0289.32003

[41] Siu Y.T., Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV Seminar (Band 8), Birkhäuser-Verlag, Basel, 1987. | Zbl 0631.53004

[42] Siu Y.T., The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group, Ann. of Math. 127 (1988) 585-627. | Zbl 0651.53035

[43] Siu Y.T., An effective Matsusaka big theorem, Ann. Inst. Fourier. 43 (1993) 1387-1405. | Numdam | MR 1275204 | Zbl 0803.32017

[44] Skoda H., Sous-ensembles analytiques d'ordre fini ou infini dans Cn, Bull. Soc. Math. France 100 (1972) 353-408. | Numdam | MR 352517 | Zbl 0246.32009

[45] Skoda H., Estimations L2 pour l’opérateur ¯ et applications arithmétiques, in: Séminaire P. Lelong (Analyse), année 1975/76, Lecture Notes in Math., 538, Springer-Verlag, Berlin, 1977, pp. 314-323. | Zbl 0363.32004

[46] Tian G., On Kähler-Einstein metrics on certain Kähler manifolds with c1(M)>0, Invent. Math. 89 (1987) 225-246. | Zbl 0599.53046

[47] Varchenko A.N., Complex exponents of a singularity do not change along the stratum μ=constant, Functional Anal. Appl. 16 (1982) 1-9. | Zbl 0498.32010

[48] Varchenko A.N., Semi-continuity of the complex singularity index, Functional Anal. Appl. 17 (1983) 307-308. | Zbl 0536.32005

[49] Varchenko A.N., Asymptotic Hodge structure …, Math. USSR Izv. 18 (1992) 469-512. | Zbl 0489.14003

[50] Yau S.T., On the Ricci curvature of a complex Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure and Appl. Math. 31 (1978) 339-411. | Zbl 0369.53059