Log continuity of solutions of complex Monge-Ampère equations

TL;DR

This work establishes quantitative log-continuity for solutions to complex Monge-Ampère equations in semi-positive big classes on compact Kähler manifolds, under the assumption that $-c_1(K_X)$ is semi-positive and the associated algebraic model has isolated singularities. It develops a robust framework for log-continuity of pseudometrics, proves its stability under blowups, and employs Hölder-measure techniques and Kołodziej-type capacity arguments to obtain global regularity. A novel analytic regularization scheme for psh potentials on line bundles is constructed, with precise $L^p$ proximity, near-non-Kähler-locus control, and gradient bounds, which is then lifted to a desingularized setting. As an application, singular Ricci-flat metrics in semi-ample integral classes on projective Calabi-Yau surfaces are shown to have $\log^M$-continuous potentials, yielding explicit global distance estimates away from the non-Kähler locus. Overall, the paper delivers a quantitative, global regularity theory for Monge-Ampère metrics in semi-positive contexts, with implications for the study of singular Kähler-Einstein metrics and their moduli.

Abstract

Let $X$ be a compact Kähler manifold whose anticanonical cohomology class is semipositive. Let $L$ be a big and semi-ample line bundle on $X$ and $α$ be the Chern class of $L$. We give a sufficient condition ensuring that the solution of the complex Monge-Ampère equations in $α$ with $L^p$ right-hand side ($p>1$) is $\log^M$-continuous for every constant $M>0$. As an application, we show that every singular Ricci-flat metric in a semi-ample integral class in a projective Calabi-Yau surface $X$ is globally $\log^M$-continuous with respect to a smooth metric on $X$.

Theorems & Definitions (35)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4