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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
DOI: https://doi.org/10.1090/S0894-0347-09-00629-8
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
MR Author ID: 602577
Email: eyssi@fourier.ujf-grenoble.fr

Vincent Guedj
Affiliation: LATP, UMR 6632, CMI, Université de Provence, 39 Rue Joliot-Curie, 13453 Marseille cedex 13, France
Email: guedj@cmi.univ-mrs.fr

Ahmed Zeriahi
Affiliation: Laboratoire Emile Picard, UMR 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France
Email: zeriahi@math.ups-tlse.fr

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.