Weighted extremal K\"ahler metrics on resolutions of singularities

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.