Stability of measures on K\"ahler manifolds

TL;DR

This paper investigates the stability properties of measures on Kähler manifolds under complexified group actions, providing criteria for stability and a surjectivity result related to geometric analysis.

Contribution

It develops an abstract framework for momentum maps and stability criteria for measures on Kähler manifolds, extending classical results to a broader setting.

Findings

Numerical criteria for stability, semi-stability, and polystability of measures.

Application of stability criteria to actions of complexified groups on measures.

A general surjectivity result for a map studied by Hersch and Bourguignon-Li-Yau.

Abstract

Let $(M,\omega)$ be a K\"ahler manifold and let $K$ be a compact group that acts on $M$ in a Hamiltonian fashion. We study the action of $K^\mathbb{C}$ on probability measures on $M$. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystability. Next we apply this setting to the action of $K^\mathbb{C}$ on measures. We get various stability criteria for measures on K\"ahler manifolds. The same circle of ideas gives a very general surjectivity result for a map originally studied by Hersch and Bourguignon-Li-Yau.