Boundary points, minimal $L^2$ integrals and concavity property V -- vector bundles

TL;DR

This paper investigates minimal L^2 integrals for singular hermitian metrics on vector bundles over pseudoconvex Kähler manifolds, establishing a concavity property and exploring its implications for module openness and support functions.

Contribution

It introduces a new concavity property of minimal L^2 integrals in the context of vector bundles and singular metrics, with several applications to module theory and support functions.

Findings

Concavity of minimal L^2 integrals established

Necessary conditions for linearity in concavity degeneracy

Strong openness and effectiveness properties of modules

Abstract

In this article, for singular hermitian metrics on holomorphic vector bundles, we consider minimal $L^2$ integrals on sublevel sets of plurisubharmonic functions on weakly pseudoconvex K\"ahler manifolds 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 strong openness property of the modules and a twisted version, an effectiveness result of the strong openness property of the modules, and an optimal support function related to the modules.