Boundary points, minimal $L^2$ integrals and concavity property II: on weakly pseudoconvex K\"ahler manifolds

TL;DR

This paper investigates the concavity of minimal $L^2$ integrals on weakly pseudoconvex K"ahler manifolds, providing new insights into their boundary behavior, openness properties, and related modules.

Contribution

It establishes a concavity property for minimal $L^2$ integrals on such manifolds and explores related modules, including conditions for linearity and openness properties.

Findings

Concavity property of minimal $L^2$ integrals established.

Necessary condition for degenerating concavity to linearity identified.

Strong openness property of modules proved and extended.

Abstract

In this article, we consider minimal $L^2$ integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K\"ahler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal $L^2$ integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to the modules, a strong openness property of the modules and a twisted version, an effectiveness result of the strong openness property of the modules.