Abstract
Let X be a canonically polarized variety, i.e. a complex projective variety such that its canonical class K X defines an ample \({\mathbb{Q}}\) -line bundle, and satisfying the conditions G 1 and S 2. Our main result says that X admits a Kähler–Einstein metric iff X has semi-log canonical singularities i.e. iff X is a stable variety in the sense of Kollár–Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kähler–Einstein metric in this singular context simply means a Kähler–Einstein on the regular locus of X with volume equal to the algebraic volume of K X , i.e. the top intersection number of K X . We also show that such a metric is uniquely determined and extends to define a canonical positive current in c 1(K X ). Combined with recent results of Odaka our main result shows that X admits a Kähler–Einstein metric iff X is K-stable, which thus confirms the Yau–Tian–Donaldson conjecture in this general setting of (possibly singular) canonically polarized varieties. More generally, our results are shown to hold in the setting of log minimal varieties and they also generalize some prior results concerning Kähler–Einstein metrics on quasi-projective varieties.
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V. Alexeev. Log Canonical Singularities and Complete Moduli of Stable Pairs. arXiv:alg-geom/9608013 (1996).
T. Aubin. Équations du type Monge–Ampère sur les variétés Kählériennes compactes. Bull. Sci. Math., 102 (1978)
R. J. Berman and B. Berndtsson. Real Monge–Ampère Equations and Kähler–Ricci Solitons on Toric Log Fano Varieties. arXiv:1207.6128 (2012).
R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi. Kähler–Einstein Metrics and the Kähler–Ricci Flow on Log-Fano Varieties. arXiv:1111.7158v2 (2011).
R. J. Berman, S. Boucksom, V. Guedj and A. Zeriahi. A variational approach to complex Monge–Ampère equations (2009) (to appear in Publ. IHES, arXiv:0907.4490).
S. Boucksom, A. Broustet and G. Pacienza. Uniruledness of stable base loci of adjoint linear systems with and without Mori theory (2010) (to appear in Math. Zeit., arXiv:0902.1142).
C. Birkar, P. Cascini, C. Hacon and J. McKernan. Existence of minimal models for varieties of log general type. J. Am. Math. Soc., 23 (2010), 405468
R. J. Berman and J.-P. Demailly. Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in analysis, geometry, and topology. In: Progress in Mathematics, Vol. 296. Birkhäuser/Springer, New York (2012), p. 3966.
S. Boucksom, P. Eyssidieux and V. Guedj. Introduction to the Kähler–Ricci flow. In: Lecture Notes in Mathematics (to appear). Springer, New York (2013).
S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi. Monge–Ampère equations in big cohomology classes. Acta Math., (2)205 (2010), 199262
R. J. Berman. Bergman kernels and equilibrium measures for line bundles over projective manifolds. Am. J. Math. (5)131 (2009), 14851524
R. J. Berman. A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics. arXiv:1011.3976 (2011).
R. J. Berman. K-polystability of \({\mathbb{Q}}\) -Fano varieties admitting Kähler–Einstein metrics. arXiv:1205.6214 (2012).
B. Bhatt, W. Ho, Z. Patakfalvi and C. Schnell. Moduli of products of stable varieties. arXiv:1206.0438 (2012).
S. Boucksom. On the volume of a line bundle. Int. J. Math., (10)13 (2002), 10431063
S. Boucksom. Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4), (1)37 (2004), 4576
S. Brendle. Ricci flat Kähler metrics with edge singularities (2011) (to appear in IMRN, arXiv:1103.5454)
E. Bedford and B. Taylor. Fine topology, Silov boundary, and (dd c)n. J. Funct. Anal., (2)72 (1987), 225251
X. Chen, S. Donaldson and S. Sun. Kähler–Einstein Metrics on Fano Manifolds, I: Approximation of Metrics with Cone Singularities. arXiv:1211.4566 (2012).
X. Chen, S. Donaldson and S. Sun. Kähler–Einstein Metrics on Fano Manifolds, II: Limits with Cone Angle Less Than 2 π. arXiv:1212.4714 (2012).
X. Chen, S. Donaldson and S. Sun. Kähler–Einstein Metrics on Fano Manifolds, III: Limits as Cone Angle Approaches 2π and Completion of the Main Proof. arXiv:1302.0282 (2013).
J. Carlson and P. Griffiths. A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. Math. 95 (1972), 557584
F. Campana, H. Guenancia and M. Păun. Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. Ann. Sci. École Norm. Sup., 46 (2013), 879916
S.-Y. Cheng and S.-T. Yau. On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math., 33 (1980), 507544
J.-P. Demailly. Estimations L 2 pour l’opérateur \({\bar \partial }\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. Sup. (4), (3)15 (1982), 457511
J.-P. Demailly. Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. France (N.S.), 19 (1985), 124
J.-P. Demailly. Regularization of closed positive currents and intersection theory. J. Algebraic Geom., (3)1 (1992), 361409
P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math., 36 (1969), 75109
J.-P. Demailly and N. Pali. Degenerate complex Monge–Ampère equations over compact Kähler manifolds. Int. J. Math., 21 (2010)
P. Eyssidieux, V. Guedj and A. Zeriahi. A priori \({L^\infty}\)-estimates for degenerate complex Monge–Ampère equations. Int. Math. Res. Notes (2008), Art. ID rnn 070, 8
P. Eyssidieux, V. Guedj and A. Zeriahi. Singular Kähler–Einstein metrics. J. Am. Math. Soc., 22 (2009), 607639
R. Elkik. Fibrés d’intersections et intégrales de classes de Chern. Ann. Sci. École Norm. Sup. (4), (2)22 (1989), 195226
R. Elkik. Métriques sur les fibrés d’intersection. Duke Math. J., (1)61 (1990), 303328
O. Fujino and Y. Gongyo. Log Pluricanonical Representations and Abundance Conjecture. arXiv:1104.0361 (2012).
J. E. Fornæss and R. Narasimhan. The Levi problem on complex spaces with singularities. Math. Ann., (1)248 (1980), 4772
G. Freixas i Montplet. An arithmetic Hilbert–Samuel theorem for pointed stable curves. J. Eur. Math. Soc. (JEMS), (2)14 (2012), 321351
A. Fujiki and G. Schumacher. The moduli space of extremal compact Kähler manifolds and generalized Weil–Petersson metrics. Publ. Res. Inst. Math. Sci., (1)26 (1990), 101183
O. Fujino. Finite generation of the log canonical ring in dimension four. Kyoto J. Math., (4)50 (2010), 671684
D. Greb, S. Kebekus, S. J. Kovács and T. Peternell. Differential forms on log canonical spaces. Publ. Math. Inst. Hautes Études Sci. 114 (2011), 87169
H. Guenancia and M. Păun. Conic Singularities Metrics with Prescribed Ricci Curvature: The Case of General Cone Angles Along Normal Crossing Divisors. arXiv:1307.6375 (2013).
H. Grauert and R. Remmert. Plurisubharmonische Funktionen in komplexen Räumen. Math. Z., 65 (1956), 175194
P. A. Griffiths. Entire Holomorphic Mappings in One and Several Complex Variables (A. of Mathematics Studies, ed.). Princeton University Press, Princeton (1976).
S. Greco and C. Traverso. On seminormal schemes. Compos. Math., (3)40 (1980), 325365
H. Guenancia. Kähler–Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor (2012) (to appear in Ann. Inst. Fourier, arXiv:1201.0952)
H. Guenancia. Kähler–Einstein metrics with cone singularities on klt pairs. Int. J. Math., 24 (2013)
H. Guenancia and D. Wu. On the boundary behaviour of Kähler–Einstein metrics of log canonical pairs (in preparation)
V. Guedj and A. Zeriahi. Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal., (4)15 (2005), 607639
V. Guedj and A. Zeriahi. The weighted Monge–Ampère energy of quasi plurisubharmonic functions. J. Funct. Anal., 250 (2007), 442482
J. H. Hubbard and S. Koch. An analytic construction of the Deligne–Mumford compactification of the moduli space of curves. arXiv:1301.0062 (2013).
L. Hörmander. Notions of Convexity. Birkhäuser, Basel (1994).
T. Jeffres, R. Mazzeo and Y. Rubinstein. Kähler–Einstein metrics with edge singularities (2011) (to appear in Ann. Math., arXiv:1105.5216, with an appendix by C. Li and Y. Rubinstein)
J. Kollár et al. Flips and Abundance for Algebraic Threefolds. Société Mathématique de France, Paris. In: Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, August 1991, Astérisque No. 211 (1992).
K. Karu. Minimal models and boundedness of stable varieties. J. Algebraic Geom., (1)9 (2000), 93109
J. Kollár and S. Mori. Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, Vol. 134. Cambridge University Press, Cambridge (1998) (with the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original).
R. Kobayashi. Kähler–Einstein metric on an open algebraic manifolds. Osaka 1. Math., 21 (1984), 399418
J. Kollár. Book on Moduli of Surfaces (ongoing project, avalaible at the author’s webpage https://web.math.princeton.edu/kollar/book/chap3.pdf).
J. Kollár. Moduli of Varieties of General Type. arXiv:1008.0621 (2010).
S. J. Kovács. Singularities of stable varieties (2012) (to appear in Handbook of Moduli, arXiv:1102.1240).
J. Kollár and N. I. Shepherd-Barron. Threefolds and deformations of surface singularities. Invent. Math., (2)91 (1988), 299338
S. J. Kovács, K. Schwede and K. E. Smith. The canonical sheaf of Du Bois singularities. Adv. Math., (4)224 (2010), 16181640
A. Langer. Logarithmic Orbifold Euler Numbers of Surfaces with Applications. arXiv:0012180 (2000).
R. Lazarsfeld. Positivity in Algebraic Geometry II. Springer, New York (2004).
M. Miyanishi. On the affine-ruledness of algebraic varieties. Algebraic geometry. Proceedings of the Japan–France Conference, Tokyo and Kyoto 1982. In: Lecture Notes in Mathematics, Vol. 1016 (1983), pp. 449–485.
N. Mok and S.-T. Yau. Completeness of the Kähler–Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions. In: The Mathematical Heritage of Henri Poincaré, Part 1 (Bloomington, Ind., 1980). Proceedings of the Symposium on Pure Mathematics, Vol. 39. American Mathematical Society, Providence, RI (1983), p. 4159.
Y. Odaka. The Calabi Conjecture and K-stability. arXiv:1010.3597 (2011).
Y. Odaka. On the Moduli of Kahler–Einstein Fano Manifolds. arXiv:1211.4833 (2013).
Y. Odaka. The GIT-stability of polarised varieties via discrepancy. Ann. Math., (2)177 (2013), 645661
Y. Odaka, C. Spotti and S. Sun. Compact Moduli Spaces of Del Pezzo Surfaces and Kähler–Einstein Metrics. arXiv:1210.0858 (2012).
M. Păun. Regularity properties of the degenerate Monge–Ampère equations on compact Kähler manifolds. Chin. Ann. Math. Ser. B, (6)29 (2008), 623630
D. H. Phong and J. Sturm. Scalar curvature, moment maps, and the Deligne pairing. Am. J. Math., (3)126 (2004), 693712
G. Schumacher. Positivity of Relative Canonical Bundles and Applications. arXiv:1201.2930 (2012).
Y.-T. Siu. Lectures on Hermitian–Einstein Metrics for Stable Bundles and Kähler–Einstein Metrics. Birkhäuser, Basel (1987).
J. Song. Riemannian Geometry of Kähler–Einstein currents. arXiv:1404.0445 (2014).
G. Tian. K-stability and Kähler–Einstein Metrics. arXiv:1211.4669 (2013).
C. Traverso. Seminormality and Picard group. Ann. Scuola Norm. Sup. Pisa (3), 24 (1970), 585595
H. Tsuji. A characterization of ball quotients with smooth boundary. Duke Math. J., (2)57 (1988), 537553
H. Tsuji. Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type. Math. Ann., (1)281 (1988), 123133
G. Tian and S.-T. Yau. Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry. Adv. Ser. Math. Phys. 1, 1 (1987), 574628 (mathematical aspects of string theory, San Diego, California, 1986)
G. Tian and S.-T. Yau. Complete Kähler manifolds with zero Ricci curvature. I. J. Am. Math. Soc., (3)3 (1990), 579609
J. Varouchas. Kähler spaces and proper open morphisms. Math. Ann., (1)283 (1989), 1352
E. Viehweg. Quasi-projective moduli for polarized manifolds. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 30. Springer-Verlag, Berlin (1995).
D. Wu. Kähler–Einstein metrics of negative Ricci curvature on general quasi-projective manifolds. Commun. Anal. Geom., (2)16 (2008), 395435
D. Wu. Good Kähler metrics with prescribed singularities. Asian J. Math., (1)13 (2009), 131150
S.-T. Yau. Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math., 28 (1975), 201228
S.-T. Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I. Commun. Pure Appl. Math., 31 (1978), 339411
S.-T. Yau. A splitting theorem and an algebraic geometric characterization of locally Hermitian symmetric spaces. Commun. Anal. Geom., (3–4)1 (1993), 473486
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Berman, R.J., Guenancia, H. Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs. Geom. Funct. Anal. 24, 1683–1730 (2014). https://doi.org/10.1007/s00039-014-0301-8
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DOI: https://doi.org/10.1007/s00039-014-0301-8