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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Singular Kähler-Einstein metrics
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by Philippe Eyssidieux, Vincent Guedj and Ahmed Zeriahi
J. Amer. Math. Soc. 22 (2009), 607-639
Published electronically: February 6, 2009


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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Bibliographic Information
  • Philippe Eyssidieux
  • Affiliation: Institut Fourier - UMR5582, 100 rue des Maths, BP 74, 38402 St Martin d’Heres, France
  • MR Author ID: 602577
  • Email:
  • Vincent Guedj
  • Affiliation: LATP, UMR 6632, CMI, Université de Provence, 39 Rue Joliot-Curie, 13453 Marseille cedex 13, France
  • Email:
  • Ahmed Zeriahi
  • Affiliation: Laboratoire Emile Picard, UMR 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France
  • Email:
  • Received by editor(s): March 17, 2006
  • Published electronically: February 6, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 607-639
  • MSC (2000): Primary 32W20, 32Q20, 32J27, 14J17
  • DOI:
  • MathSciNet review: 2505296