Singular Kähler-Einstein metrics

Abstract

We study degenerate complex Monge-Ampère equations of the form (ω + dd c ϕ) n = e tϕ µ where ω is a big semi-positive form on a compact Kähler manifold X of dimension n, t ∈ R + , and µ = f ω n is a positive measure with density f ∈ 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 → V is a proper birational morphism to a normal projective variety, [ω ′ ] ∈ N S 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.

Existence of minimal models for varieties of log general type

“…Recently, Shokurov has shown the existence of flips in dimension 4 [34] and Hacon and M c Kernan [8] have shown that assuming the minimal model program in dimension n − 1 (or even better simply finiteness of minimal models in dimension n − 1), then flips exist in dimension n. Thus we get an inductive approach to finite generation.…”

The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation

“…Kolodziej ([29,30]) proved the existence and Hölder estimate of solution to the complex Monge-Ampère equation when the right hand side is a nonnegative L p function for p > 1. There are further existence and regularity results on the complex Monge-Ampère equation with right hand side less regular or degenerate, see references [3,13,26,4,52,43,27,20,19,22] for details.…”

Untitled

“…The long time existence of the Kähler-Ricci flow on a minimal projective manifold with any initial Kähler metric is established in [TiZha]. The regularity problem of the canonical singular Kähler-Einstein metrics on minimal projective manifolds of general type is intensively studied in [Zh,EyGuZe1].…”

“…The unique Kähler-Einstein metric with bounded local potential and the associated Kähler-Einstein measure on the canonical model X can are constructed in [EyGuZe1]. The Kähler-Einstein measure in Theorem C.1 is invariant under birational transformations and so it can be considered as a birational invariant.…”

“…Its generalization was independently carried out in [Zh] and [EyGuZe1]. They proved that there is a bounded solution when ω is semi-positive and big.…”