Singular Kahler-Einstein metrics

TL;DR

This paper establishes existence, uniqueness, and regularity of solutions to degenerate complex Monge-Ampère equations on compact Kähler manifolds, leading to the construction of singular Kähler-Einstein metrics on algebraic varieties.

Contribution

It proves the existence and uniqueness of bounded solutions to degenerate Monge-Ampère equations and constructs singular Kähler-Einstein metrics on projective varieties with mild singularities.

Findings

Solutions are bounded and unique under certain conditions.

Solutions are continuous with additional technical assumptions.

Constructs singular Kähler-Einstein metrics on algebraic varieties of general type.

Abstract

We study degenerate complex Monge-Amp\`ere equations of the form $(\omega+dd^c \varphi)^n = e^{t \varphi} \mu$ where $\omega$ is a big semi-positive form on a compact K\"ahler manifold $X$ of dimension $n$, $t \in \R^+$, 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, $[\omega']\in NS_{\R} (V)$ 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 K\"ahler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.