Mahler measures, Stable Pairs, and the Global coercive estimate for the Mabuchi Functional

TL;DR

This paper establishes a deep connection between the properness of the Mabuchi energy on polarized manifolds and their asymptotic stability, linking geometric stability to the existence of constant scalar curvature Kähler metrics.

Contribution

It proves that the Mabuchi energy is bounded below if and only if the manifold is asymptotically (semi)stable, extending the understanding of stability conditions for Kähler metrics.

Findings

Mabuchi energy properness characterizes asymptotic stability.

Existence of cscK metrics is equivalent to asymptotic stability under certain automorphism conditions.

The work connects algebraic stability with differential geometric energy functionals.

Abstract

We show that the Mabuchi energy of any polarized manifold (X,L) is (bounded below) proper on the full space of Kahler metrics in the first Chern class of L if and only if (X,L) is asymptotically (semi)stable. In particular it now follows from work of Xiuxiong Chen and Jinguri Cheng that there exists a cscK metric in the first Chern class of L if and only if (X,L) is asymptotically stable, provided the reduced automorphism group of (X,L) is finite.