Weighted extremal K\"ahler metrics on resolutions of singularities

TL;DR

This paper extends the invariance of weighted extremal K"ahler metrics under smooth blowups to a broader class of singularities, using a uniform coercivity estimate for the Mabuchi energy.

Contribution

It generalizes previous results by proving invariance under blowups for weighted extremal K"ahler metrics on resolutions of singularities, under certain conditions.

Findings

Invariance under smooth blowups for weighted extremal K"ahler metrics.

A uniform coercivity estimate for the Mabuchi energy on resolutions.

Applicability to equivariant resolutions of Fano type of K"ahler klt spaces.

Abstract

Generalizing previous results of Arezzo-Pacard-Singer, Seyyedali-Sz\'ekelyhidi and Hallam, we prove the invariance under smooth blowups of the class of weighted extremal K\"ahler manifolds, modulo a log-concavity assumption on the first weight. Through recent work of Di Nezza-Jubert-Lahdili and Han-Liu, this is obtained as a consequence of a general uniform coercivity estimate for the (relative, weighted) Mabuchi energy on the blowup, which applies more generally to any equivariant resolution of singularities of Fano type of a compact K\"ahler klt space whose Mabuchi energy is assumed to be coercive.