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Singular Kähler-Einstein metrics. (English) Zbl 1215.32017
The paper deals with the complex Monge-Ampère equation on compact Kähler manifolds, but with the background form which is not Kähler. Let $$X$$ be an $$n$$-dimensional compact Kähler manifold with a semi-positive big $$(1,1)$$ form $$\omega$$ such that $$\int _X \omega^n >0$$. For a suitably normalized $$f\geq0 , \;f\in L^p (X), \;p>1$$, there exists a bounded (and, under an additional technical assumption, continuous) solution $$u$$ of the complex Monge-Ampère equation
$(\omega +dd^c u)^n =f\omega ^n,$
for which $$\omega +dd^c u$$ is a non-negative current. For $$\omega$$ a Kähler form, the result is due to the reviewer [Acta Math. 180, No. 1, 69–117 (1998; Zbl 0913.35043)].
The statement is then used for $$X$$ projective and $$\omega$$ which is Kähler outside an analytic set $$S$$ and satisfies $$\omega ^n = D\Omega ^n$$ for a Kähler form $$\Omega$$ and $$D^{-a} \in L^1 (\Omega ),\;a>0,$$ $$[\omega ], [\Omega ] \in NS_{\mathbb R } (X).$$ If $$\sigma _j , \tau _j$$ are collections of holomorphic sections of two line bundles with
$\int_X \frac{1}{\sum |\tau _j |^{2l} }\Omega ^n <\infty$
and
$\int _X \frac{\sum |\sigma _j |^{2k}}{\sum |\tau _j |^{2l} } e^F \Omega ^n=\int_X \omega^n,$
for nonnegative numbers $$k,l$$ and smooth $$F$$, then the Monge-Ampère equation with right hand side
$\frac{\sum |\sigma _j |^{2k}}{\sum |\tau _j |^{2l} } e^F \Omega ^n$
has a bounded solution which is smooth away from $$S$$ and zero sets of the sections. This is a version of Theorem 8 from S.-T. Yau’s paper [Commun. Pure Appl. Math. 31, 339–411 (1978; Zbl 0369.53059)] with a stronger hypothesis on the manifold but weaker on the sections. The above solutions are then used to construct Kähler-Einstein metrics with singularities and with negative curvature on projective klt pairs. In particular, such metrics are obtained on canonical models of algebraic varieties of general type. This result can be viewed as a generalization of the constructions given by H. Tsuji [Math. Ann. 281, No. 1, 123–133 (1988; Zbl 0631.53051)] and G. Tian and Z. Zhang [Chin. Ann. Math., Ser. B 27, No. 2, 179–192 (2006; Zbl 1102.53047)].

##### MSC:
 32W20 Complex Monge-Ampère operators 32Q20 Kähler-Einstein manifolds 32J27 Compact Kähler manifolds: generalizations, classification 14J17 Singularities of surfaces or higher-dimensional varieties
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##### References:
  Thierry Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63 – 95 (French, with English summary). · Zbl 0374.53022  Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1 – 40. · Zbl 0547.32012  Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207 – 302. · Zbl 0896.14006  C. Birkar, P. Cascini, C. D. Hacon, J. Mckernan: Existence of minimal models for varieties of log general type, arXiv:math/0610203. · Zbl 1210.14019  Zbigniew Błocki, Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 6, 1697 – 1701. · Zbl 1054.32024  Z. Blocki: The Monge-Ampère equation on compact Kähler manifolds. Lecture Notes of a course given at the Winter School in Complex Analysis, Toulouse, 24-328/02/2005.  Zbigniew Błocki and Sławomir Kołodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2089 – 2093. · Zbl 1116.32024  Charles P. Boyer, Krzysztof Galicki, and János Kollár, Einstein metrics on spheres, Ann. of Math. (2) 162 (2005), no. 1, 557 – 580. · Zbl 1093.53044  Eugenio Calabi, On Kähler manifolds with vanishing canonical class, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 78 – 89. · Zbl 0080.15002  P. Cascini & G. La Nave: Kähler-Ricci Flow and the Minimal Model Program for Projective Varieties. Preprint arXiv math. AG/0603064.  Urban Cegrell, Pluricomplex energy, Acta Math. 180 (1998), no. 2, 187 – 217. · Zbl 0926.32042  Alessio Corti , Flips for 3-folds and 4-folds, Oxford Lecture Series in Mathematics and its Applications, vol. 35, Oxford University Press, Oxford, 2007. · Zbl 05175029  Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. · Zbl 0978.14001  Jean-Pierre Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, Complex analysis and geometry, Univ. Ser. Math., Plenum, New York, 1993, pp. 115 – 193. · Zbl 0792.32006  Jean-Pierre Demailly, A numerical criterion for very ample line bundles, J. Differential Geom. 37 (1993), no. 2, 323 – 374. · Zbl 0783.32013  Jean-Pierre Demailly, Cohomology of \?-convex spaces in top degrees, Math. Z. 204 (1990), no. 2, 283 – 295. · Zbl 0682.32017  Jean-Pierre Demailly and Mihai Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247 – 1274. · Zbl 1064.32019  John Erik Fornæss and Raghavan Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), no. 1, 47 – 72. · Zbl 0411.32011  Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. · Zbl 0537.32001  Vincent Guedj, Approximation of currents on complex manifolds, Math. Ann. 313 (1999), no. 3, 437 – 474. · Zbl 0924.32014  Vincent Guedj and Ahmed Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607 – 639. · Zbl 1087.32020  Vincent Guedj and Ahmed Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), no. 2, 442 – 482. · Zbl 1143.32022  Christopher D. Hacon and James McKernan, Extension theorems and the existence of flips, Flips for 3-folds and 4-folds, Oxford Lecture Ser. Math. Appl., vol. 35, Oxford Univ. Press, Oxford, 2007, pp. 76 – 110. · Zbl 1286.14026  Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001  Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121 – 176. · Zbl 0431.14004  Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109 – 203; ibid. (2) 79 (1964), 205 – 326. · Zbl 0122.38603  Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. · Zbl 0835.32001  Alan Huckleberry, Subvarieties of homogeneous and almost homogeneous manifolds, Contributions to complex analysis and analytic geometry, Aspects Math., E26, Friedr. Vieweg, Braunschweig, 1994, pp. 189 – 232. · Zbl 0823.32016  Yujiro Kawamata, On the finiteness of generators of a pluricanonical ring for a 3-fold of general type, Amer. J. Math. 106 (1984), no. 6, 1503 – 1512. · Zbl 0587.14027  Ryoichi Kobayashi, Einstein-Kähler \?-metrics on open Satake \?-surfaces with isolated quotient singularities, Math. Ann. 272 (1985), no. 3, 385 – 398. · Zbl 0556.14019  János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, 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. · Zbl 0926.14003  Sławomir Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1, 69 – 117. · Zbl 0913.35043  Sławomir Kołodziej, The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 3, 667 – 686. · Zbl 1039.32050  Sławomir Kołodziej, The complex Monge-Ampère equation and pluripotential theory, Mem. Amer. Math. Soc. 178 (2005), no. 840, x+64. · Zbl 1084.32027  Kenji Matsuki and Martin Olsson, Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett. 12 (2005), no. 2-3, 207 – 217. · Zbl 1080.14023  Shigefumi Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117 – 253. · Zbl 0649.14023  Michael Nakamaye, Stable base loci of linear series, Math. Ann. 318 (2000), no. 4, 837 – 847. · Zbl 1063.14008  Mihai Paun, Sur l’effectivité numérique des images inverses de fibrés en droites, Math. Ann. 310 (1998), no. 3, 411 – 421 (French). · Zbl 1023.32014  Miles Reid, Canonical 3-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn — Germantown, Md., 1980, pp. 273 – 310.  Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345 – 414. · Zbl 0634.14003  Rolf Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257 – 286 (German). · Zbl 0153.15401  V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 82 – 219; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 75 – 213. · Zbl 1082.14019  Nessim Sibony, Dynamique des applications rationnelles de \?^{\?}, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix – x, xi – xii, 97 – 185 (French, with English and French summaries). · Zbl 1020.37026  Yum Tong Siu, A vanishing theorem for semipositive line bundles over non-Kähler manifolds, J. Differential Geom. 19 (1984), no. 2, 431 – 452. · Zbl 0577.32031  Yum-Tong Siu, Multiplier ideal sheaves in complex and algebraic geometry, Sci. China Ser. A 48 (2005), no. suppl., 1 – 31. · Zbl 1131.32010  Jian Song and Gang Tian, The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), no. 3, 609 – 653. · Zbl 1134.53040  Kenichi Sugiyama, Einstein-Kähler metrics on minimal varieties of general type and an inequality between Chern numbers, Recent topics in differential and analytic geometry, Adv. Stud. Pure Math., vol. 18, Academic Press, Boston, MA, 1990, pp. 417 – 433. · Zbl 0770.53021  Gang Tian, Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000. Notes taken by Meike Akveld. · Zbl 0978.53002  Gang Tian and Zhou Zhang, On the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179 – 192. · Zbl 1102.53047  Hajime Tsuji, Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), no. 1, 123 – 133. · Zbl 0631.53051  Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339 – 411. · Zbl 0369.53059  Ahmed Zeriahi, Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J. 50 (2001), no. 1, 671 – 703. · Zbl 1138.31302  Zhou Zhang, On degenerate Monge-Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. , posted on (2006), Art. ID 63640, 18. · Zbl 1112.32021
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