Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. (English) Zbl 0994.32021

Demailly, Jean-Pierre; Kollár, János

Ann. Sci. Éc. Norm. Supér. (4) 34, No. 4, 525-556 (2001).

Let \(\varphi\) be a plurisubharmonic function on a complex manifold \(X\). The complex singularity exponent \(c_K(\varphi)\) of \(\varphi\) on a compact set \(K\subset X\) is the supremum over \(c\geq 0\) such that \(\exp(-2c \varphi)\) is integrable on a neighborhood of \(K\). The notion plays an important role in complex analysis and algebraic geometry, and several other characteristics of singularities for analytic objects (holomorphic functions, coherent ideal sheaves, divisors, currents) are its particular cases.
The main results of the paper is lower semicontinuity of the map \(\varphi\mapsto c_K (\varphi)\), which means that if \(\varphi_j\to \varphi\) in \(L^1_{\text{loc}}(X)\) then \(\exp(-2 \subset\varphi_j) \to\exp(-2 \subset\varphi)\) in \(L^1\)-norm over a neighborhood of \(K\) for all positive \(c<c_K (\varphi)\).
As a consequence, a comparatively simple proof is given for the existence of Kähler-Einstein metrics on certain Fano orbifolds. In this way, the authors produce three new examples of rigid del Pezzo surfaces with quotient singularities which admit a Kähler-Einstein metric.

Reviewer: Alexandr Yu.Rashkovsky (Khar’kov)

MSC:

32S05	Local complex singularities
14B05	Singularities in algebraic geometry
14J45	Fano varieties
32U05	Plurisubharmonic functions and generalizations
32U25	Lelong numbers

Keywords:

Arnold multiplicity; multiplier ideal sheaf; Lelong number; complex singularity exponent; Kähler-Einstein metrics; Fano orbifolds; del Pezzo surfaces

References:

[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 · Zbl 0847.32035 · doi:10.1007/BF01231446
[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 · Zbl 0138.06604 · doi:10.1007/BF02684398
[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 · 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 · Zbl 0508.32003 · doi:10.1007/BF01394271
[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 · Zbl 0214.33702 · doi:10.1007/BF01418775
[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 · Zbl 0629.32011 · doi:10.1007/BF02392558
[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 · Zbl 1112.32303
[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 · 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 · 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 · 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 · Zbl 0783.32013
[14]	Demailly J.-P. , L 2 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 . arXiv | Zbl 0883.14005 · 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 · 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 · 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 · Zbl 0798.14009
[18]	Futaki A. , An obstruction to the existence of Einstein-Kähler metrics , Invent. Math. 73 ( 1983 ) 437 - 443 . Zbl 0506.53030 · Zbl 0506.53030 · doi:10.1007/BF01388438
[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 · Zbl 0122.38603 · doi:10.2307/1970486
[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 · 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 · Zbl 0974.14023 · doi:10.5802/aif.1815
[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 · Zbl 0672.14006
[23]	Kollár J. , (with 14 coauthors) , Flips and Abundance for Algebraic Threefolds , Astérisque , 211 , 1992 . Zbl 0814.14038 · Zbl 0814.14038
[24]	Kollár J. , Shafarevich Maps and Automorphic Forms , Princeton Univ. Press , 1995 . MR 1341589 | Zbl 0871.14015 · 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 · 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 · Zbl 0079.30901
[27]	Lelong P. , Plurisubharmonic Functions and Positive Differential Forms , Gordon and Breach, New York, and Dunod , Paris , 1969 . Zbl 0195.11604 · 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 · Zbl 0080.37501
[29]	Lichtin B. , An upper semicontinuity theorem for some leading poles of | f | 2 s , in: Complex Analytic Singularities , Adv. Stud. Pure Math. , 8 , North-Holland , Amsterdam , 1987 , pp. 241 - 272 . MR 894297 | Zbl 0615.32007 · Zbl 0615.32007
[30]	Lichtin B. , Poles of | f ( z , w )| 2 s and roots of the B -function , Ark. för Math. 27 ( 1989 ) 283 - 304 . MR 1022282 | Zbl 0719.32016 · Zbl 0719.32016 · doi:10.1007/BF02386377
[31]	Manivel L. , Un théorème de prolongement L 2 de sections holomorphes d’un fibré vectoriel , Math. Z. 212 ( 1993 ) 107 - 122 . Article | MR 1200166 | Zbl 0789.32015 · Zbl 0789.32015 · doi:10.1007/BF02571643
[32]	Matsushima Y. , Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne , Nagoya Math. J. 11 ( 1957 ) 145 - 150 . Article | MR 94478 | Zbl 0091.34803 · 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 · Zbl 0711.53056 · doi:10.1073/pnas.86.19.7299
[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 · Zbl 0731.53063 · doi:10.2307/1971429
[35]	Ohsawa T. , Takegoshi K. , On the extension of L 2 holomorphic functions , Math. Z. 195 ( 1987 ) 197 - 204 . MR 892051 | Zbl 0625.32011 · Zbl 0625.32011 · doi:10.1007/BF01166457
[36]	Ohsawa T. , On the extension of L 2 holomorphic functions, II , Publ. RIMS, Kyoto Univ. 24 ( 1988 ) 265 - 275 . Article | MR 944862 | Zbl 0653.32012 · Zbl 0653.32012 · doi:10.2977/prims/1195175200
[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. arXiv | MR 1792297 | Zbl 0995.11065 · Zbl 0995.11065 · doi:10.2307/2661384
[38]	Phong D.H. , Sturm J. , On a conjecture of Demailly and Kollár , preprint , April 2000 . MR 1803721 · Zbl 0978.32004
[39]	Shokurov V. , 3-fold log flips , Izv. Russ. Acad. Nauk Ser. Mat. 56 ( 1992 ) 105 - 203 . MR 1162635 · Zbl 0785.14023
[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 · Zbl 0289.32003 · doi:10.1007/BF01389965
[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 · 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 · Zbl 0651.53035 · doi:10.2307/2007006
[43]	Siu Y.T. , An effective Matsusaka big theorem , Ann. Inst. Fourier. 43 ( 1993 ) 1387 - 1405 . Numdam | MR 1275204 | Zbl 0803.32017 · Zbl 0803.32017 · doi:10.5802/aif.1378
[44]	Skoda H. , Sous-ensembles analytiques d’ordre fini ou infini dans C n , Bull. Soc. Math. France 100 ( 1972 ) 353 - 408 . Numdam | MR 352517 | Zbl 0246.32009 · Zbl 0246.32009
[45]	Skoda H. , Estimations L 2 pour l’opérateur \(\overline{\partial}\) 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 · Zbl 0363.32004
[46]	Tian G. , On Kähler-Einstein metrics on certain Kähler manifolds with c 1 ( M )>0 , Invent. Math. 89 ( 1987 ) 225 - 246 . Zbl 0599.53046 · Zbl 0599.53046 · doi:10.1007/BF01389077
[47]	Varchenko A.N. , Complex exponents of a singularity do not change along the stratum \mu =constant , Functional Anal. Appl. 16 ( 1982 ) 1 - 9 . Zbl 0498.32010 · Zbl 0498.32010 · doi:10.1007/BF01081801
[48]	Varchenko A.N. , Semi-continuity of the complex singularity index , Functional Anal. Appl. 17 ( 1983 ) 307 - 308 . Zbl 0536.32005 · Zbl 0536.32005 · doi:10.1007/BF01076724
[49]	Varchenko A.N. , Asymptotic Hodge structure \cdots , Math. USSR Izv. 18 ( 1992 ) 469 - 512 . Zbl 0489.14003 · Zbl 0489.14003 · doi:10.1070/IM1982v018n03ABEH001395
[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 · Zbl 0369.53059 · doi:10.1002/cpa.3160310304