Singular Kahler-Einstein metrics

Abstract

We study degenerate complex Monge-Ampère equations of the form $(\omega +d{d}^c\phi {)}^n=e^{t\phi }\mu$ 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 $\omega ={\psi }^{\ast }{\omega }^{\prime }$, where $\psi :X\to V$ is a proper birational morphism to a normal projective variety, $[{\omega }^{\prime }]\in N{S}_{\mathbb{R}}(V)$ 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.

To appear in Journal of A.M.S

References in corpus (3)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• Numerical characterization of the Kähler cone of a compact Kähler manifold
• Kähler-Ricci Flow and the Minimal Model Program for Projective Varieties

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