Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Singular Kähler-Einstein metrics

Authors: Philippe Eyssidieux, Vincent Guedj and Ahmed Zeriahi
Journal: J. Amer. Math. Soc. 22 (2009), 607-639
MSC (2000): Primary 32W20, 32Q20, 32J27, 14J17
Published electronically: February 6, 2009
MathSciNet review: 2505296
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Abstract: 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.

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Additional Information

Philippe Eyssidieux
Affiliation: Institut Fourier - UMR5582, 100 rue des Maths, BP 74, 38402 St Martin d’Heres, France

Vincent Guedj
Affiliation: LATP, UMR 6632, CMI, Université de Provence, 39 Rue Joliot-Curie, 13453 Marseille cedex 13, France

Ahmed Zeriahi
Affiliation: Laboratoire Emile Picard, UMR 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France

Received by editor(s): March 17, 2006
Published electronically: February 6, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.