Singular Kähler-Einstein metrics

Source abstract

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