Singular Kahler-Einstein metrics

Philippe Eyssidieux (1) , Vincent Guedj (2) , Ahmed Zeriahi (2)

1 IF - Institut Fourier
2 LEP - Laboratoire Émile Picard

We study degenerate complex Monge-Ampère equations of the form $(\omega+dd^c\f)^n = e^{t \f}\mu$ where $\omega$ is a big semi-Kähler form on a compact Kähler 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 continuous $\o$-plurisubharmonic solutions. 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 Kähler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

Domaines: Géométrie algébrique [math.AG]; Variables complexes [math.CV]; Géométrie différentielle [math.DG]

Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi. Singular Kahler-Einstein metrics. 2006. ⟨hal-00021177⟩