Modulus of continuity of Monge--Ampère potentials in big cohomology classes

TL;DR

The paper addresses the modulus of continuity of solutions to the degenerate complex Monge–Ampère equation in big cohomology classes, where the right-hand side is not integrable. It develops a framework combining Demailly regularization, Kiselman–Legendre transforms, and capacity estimates to obtain a uniform modulus of continuity: for any $U\Subset \mathrm{Amp}(\theta)\setminus\{\psi=-\infty\}$, a function $F_U$ with $F_U(0)=0$ bounds $|u(z_1)-u(z_2)|$ in terms of $\mathrm{dist}(z_1,z_2)$, with $F_U$ depending on the data and an exponential integrability bound of $e^{\frac{2(V_\theta-u)}{a}}$ against $d\mu$. A corollary in the special case $\theta=\omega_X$ yields equicontinuity for the family of solutions under a growth condition on $\psi$ and the Hölder control of $\mu$. The results advance the understanding of regularity for Monge–Ampère potentials in big classes beyond the integrable-right-hand-side setting, with implications for geometric properties and convergence in Kähler geometry.

Abstract

In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Ampère equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.

Theorems & Definitions (39)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Definition 2.5