Singular Kähler-Einstein metrics

Eyssidieux, Philippe¹ ; Guedj, Vincent² ; Zeriahi, Ahmed³

¹ Institut Fourier - UMR5582, 100 rue des Maths, BP 74, 38402 St Martin d’Heres, France
² LATP, UMR 6632, CMI, Université de Provence, 39 Rue Joliot-Curie, 13453 Marseille cedex 13, France
³ Laboratoire Emile Picard, UMR 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France

Journal of the American Mathematical Society, Tome 22 (2009) no. 3, pp. 607-639

Résumé

We study degenerate complex Monge-Ampère equations of the form $(\omega +dd^c\varphi )^n = e^{t \varphi }\mu$ where $\omega$ is a big semi-positive form on a compact Kähler manifold $X$ of dimension $n$, $t \in \mathbb {R}^+$, and $\mu =f\omega ^n$ is a positive measure with density $f\in L^p(X,\omega ^n)$, $p>1$. We prove the existence and unicity of bounded $\omega$-plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition. In case $X$ is projective and $\omega =\psi ^*\omega ’$, where $\psi :X\to V$ is a proper birational morphism to a normal projective variety, $[\omega ’]\in NS_{\mathbb {R}} (V)$ is an ample class and $\mu$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kähler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

Détail

DOI : 10.1090/S0894-0347-09-00629-8

Bibliographie

[1] Aubin, Thierry Équations du type Monge-Ampère sur les variétés kählériennes compactes Bull. Sci. Math. (2) 1978 63 95

[2] Bedford, Eric, Taylor, B. A. A new capacity for plurisubharmonic functions Acta Math. 1982 1 40

[3] Bierstone, Edward, Milman, Pierre D. Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant Invent. Math. 1997 207 302

[4] Bå‚Ocki, Zbigniew Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifolds Indiana Univ. Math. J. 2003 1697 1701

[5] Bå‚Ocki, Zbigniew, Koå‚Odziej, Så‚Awomir On regularization of plurisubharmonic functions on manifolds Proc. Amer. Math. Soc. 2007 2089 2093

[6] Boyer, Charles P., Galicki, Krzysztof, Kollã¡R, Jã¡Nos Einstein metrics on spheres Ann. of Math. (2) 2005 557 580

[7] Calabi, Eugenio On Kähler manifolds with vanishing canonical class 1957 78 89

[8] Cegrell, Urban Pluricomplex energy Acta Math. 1998 187 217

[9] Flips for 3-folds and 4-folds 2007

[10] Debarre, Olivier Higher-dimensional algebraic geometry 2001

[11] Demailly, Jean-Pierre Monge-Ampère operators, Lelong numbers and intersection theory 1993 115 193

[12] Demailly, Jean-Pierre A numerical criterion for very ample line bundles J. Differential Geom. 1993 323 374

[13] Demailly, Jean-Pierre Cohomology of 𝑞-convex spaces in top degrees Math. Z. 1990 283 295

[14] Demailly, Jean-Pierre, Paun, Mihai Numerical characterization of the Kähler cone of a compact Kähler manifold Ann. of Math. (2) 2004 1247 1274

[15] Fornã¦Ss, John Erik, Narasimhan, Raghavan The Levi problem on complex spaces with singularities Math. Ann. 1980 47 72

[16] Grauert, Hans, Remmert, Reinhold Coherent analytic sheaves 1984

[17] Guedj, Vincent Approximation of currents on complex manifolds Math. Ann. 1999 437 474

[18] Guedj, Vincent, Zeriahi, Ahmed Intrinsic capacities on compact Kähler manifolds J. Geom. Anal. 2005 607 639

[19] Guedj, Vincent, Zeriahi, Ahmed The weighted Monge-Ampère energy of quasiplurisubharmonic functions J. Funct. Anal. 2007 442 482

[20] Hacon, Christopher D., Mckernan, James Extension theorems and the existence of flips 2007 76 110

[21] Hartshorne, Robin Algebraic geometry 1977

[22] Hartshorne, Robin Stable reflexive sheaves Math. Ann. 1980 121 176

[23] Hironaka, Heisuke Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II Ann. of Math. (2) 1964

[24] Hã¶Rmander, Lars Notions of convexity 1994

[25] Huckleberry, Alan Subvarieties of homogeneous and almost homogeneous manifolds 1994 189 232

[26] Kawamata, Yujiro On the finiteness of generators of a pluricanonical ring for a 3-fold of general type Amer. J. Math. 1984 1503 1512

[27] Kobayashi, Ryoichi Einstein-Kähler 𝑉-metrics on open Satake 𝑉-surfaces with isolated quotient singularities Math. Ann. 1985 385 398

[28] Kollã¡R, Jã¡Nos, Mori, Shigefumi Birational geometry of algebraic varieties 1998

[29] Koå‚Odziej, Så‚Awomir The complex Monge-Ampère equation Acta Math. 1998 69 117

[30] Koå‚Odziej, Så‚Awomir The Monge-Ampère equation on compact Kähler manifolds Indiana Univ. Math. J. 2003 667 686

[31] Koå‚Odziej, Så‚Awomir The complex Monge-Ampère equation and pluripotential theory Mem. Amer. Math. Soc. 2005

[32] Matsuki, Kenji, Olsson, Martin Kawamata-Viehweg vanishing as Kodaira vanishing for stacks Math. Res. Lett. 2005 207 217

[33] Mori, Shigefumi Flip theorem and the existence of minimal models for 3-folds J. Amer. Math. Soc. 1988 117 253

[34] Nakamaye, Michael Stable base loci of linear series Math. Ann. 2000 837 847

[35] Paun, Mihai Sur l’effectivité numérique des images inverses de fibrés en droites Math. Ann. 1998 411 421

[36] Reid, Miles Canonical 3-folds 1980 273 310

[37] Reid, Miles Young person’s guide to canonical singularities 1987 345 414

[38] Richberg, Rolf Stetige streng pseudokonvexe Funktionen Math. Ann. 1968 257 286

[39] Shokurov, V. V. Prelimiting flips Tr. Mat. Inst. Steklova 2003 82 219

[40] Sibony, Nessim Dynamique des applications rationnelles de 𝐏^{𝐤} 1999

[41] Siu, Yum Tong A vanishing theorem for semipositive line bundles over non-Kähler manifolds J. Differential Geom. 1984 431 452

[42] Siu, Yum-Tong Multiplier ideal sheaves in complex and algebraic geometry Sci. China Ser. A 2005 1 31

[43] Song, Jian, Tian, Gang The Kähler-Ricci flow on surfaces of positive Kodaira dimension Invent. Math. 2007 609 653

[44] Sugiyama, Kenichi Einstein-Kähler metrics on minimal varieties of general type and an inequality between Chern numbers 1990 417 433

[45] Tian, Gang Canonical metrics in Kähler geometry 2000

[46] Tian, Gang, Zhang, Zhou On the Kähler-Ricci flow on projective manifolds of general type Chinese Ann. Math. Ser. B 2006 179 192

[47] Tsuji, Hajime Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type Math. Ann. 1988 123 133

[48] Yau, Shing Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I Comm. Pure Appl. Math. 1978 339 411

[49] Zeriahi, Ahmed Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions Indiana Univ. Math. J. 2001 671 703

[50] Zhang, Zhou On degenerate Monge-Ampère equations over closed Kähler manifolds Int. Math. Res. Not. 2006