Singular Kähler-Einstein metrics

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.