The Regularity of Envelopes

TL;DR

This paper proves that on a compact complex manifold, the envelope of a bounded function with bounded Laplacian, relative to a big cohomology class, is also locally bounded with bounded Laplacian on the ample locus.

Contribution

It establishes regularity properties of envelopes of bounded functions with bounded Laplacian in the setting of big cohomology classes on compact complex manifolds.

Findings

The $ ext{α}$-psh envelope $P(f)$ is locally bounded.

$P(f)$ has locally bounded distributional Laplacian.

Regularity holds on the ample locus of the cohomology class.

Abstract

Let $X$ be a compact complex manifold of complex dimension $n$ and $\alpha$ be a smooth closed real form on $X$ such that its cohomology class $\{ \alpha \}\in H^{1,1}(X, \mathbb{R})$ is big. In this paper we prove that, given a bounded function $f$ with bounded distributional laplacian in $X,$ the $\alpha$-psh envelope $P(f)$ is also locally bounded with locally bounded distributional laplacian on the ample locus of $\{\alpha\}.$