Weighted extremal Kähler metrics on resolutions of singularities
Sébastien Boucksom, Mattias Jonsson, Antonio Trusiani
TL;DR
The paper extends the theory of weighted extremal Kähler metrics to resolutions and, more broadly, to compact Kähler spaces with klt singularities by proving a uniform coercivity estimate for the relative weighted Mabuchi energy on blowups. The approach is anchored in pluripotential theory, extending finite-energy spaces $\mathcal{E}^1$ and the weighted Monge–Ampère formalism to singular settings and equivariant contexts via moment maps and polytopes. A key result is that coercivity on a base Fano-type space transfers to large smooth blowups, yielding weighted extremal metrics on the perturbed classes, with corollaries for iterated blowups. This work synthesizes prior gluing techniques with an analytic framework and has implications for the study of klt varieties and Minimal Model Program moduli through canonical metrics.
Abstract
Generalizing previous results of Arezzo-Pacard-Singer, Seyyedali-Székelyhidi and Hallam, we prove the invariance under smooth blowups of the class of weighted extremal Kähler 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 Kähler klt space whose Mabuchi energy is assumed to be coercive.