Equivariant CM minimization for extremal manifolds

TL;DR

This paper proves an equivariant version of the CM minimization conjecture for extremal Kähler manifolds, extending previous results to include automorphisms and extremal metrics, with implications for stability and geometric analysis.

Contribution

It generalizes the CM minimization conjecture to the extremal setting, incorporating automorphisms and extremal metrics, and extends stability results to this broader context.

Findings

Proves the equivariant CM minimization conjecture for extremal Kähler manifolds.

Extends Székelyhidi's asymptotic stability results to extremal manifolds.

Shows that extremal fillings minimize the relative CM degree in equivariant families.

Abstract

We prove an equivariant version of the CM minimization conjecture for extremal K\"ahler manifolds. This involves proving that, given an equivariant punctured family of polarized varieties, a relative version of the CM degree is strictly minimized by an extremal filling. This generalizes a result by Hattori for cscK manifolds with discrete automorphism group by allowing automorphisms and extremal metrics. As a main tool, we extend results by Sz\'ekelyhidi on asymptotic filtration Chow stability of cscK manifolds with discrete automorphism group to the extremal setting.