Singular Kahler-Einstein metrics

Abstract

We study degenerate complex Monge-Ampère equations of the form $(ω+dd^c φ)^n = e^{t φ} μ$ where $ω$ is a big semi-positive form on a compact Kähler manifold $X$ of dimension $n$, $t \in \R^+$, and $μ=fω^n$ is a positive measure with density $f\in L^p(X,ω^n)$, $p>1$. We prove the existence and unicity of bounded $ω$-plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition. In case $X$ is projective and $ω=ψ^*ω'$, where $ψ:X\to V$ is a proper birational morphism to a normal projective variety, $[ω']\in NS_{\R} (V)$ is an ample class and $μ$ 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.