A priori $L^{\infty}$-estimates for degenerate complex Monge-Amp\`ere equations

TL;DR

This paper establishes uniform $L^{ ablafty}$-estimates for solutions to degenerate complex Monge-Ampère equations, extending previous results and impacting the understanding of the Kähler-Ricci flow.

Contribution

It generalizes recent $L^{ ablafty}$-estimates to cases where cohomology classes degenerate, providing new tools for complex geometric analysis.

Findings

Uniform $L^{ ablafty}$-estimates for degenerate equations

Extension of Kolodziej and Tian's work to non-big classes

Implications for Kähler-Ricci flow stability

Abstract

We study families of complex Monge-Amp\`ere equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\infty}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the study of the K\"ahler-Ricci flow.