Stability of weighted extremal manifolds through blowups

TL;DR

This paper proves that weighted extremal manifolds are relatively weighted K-polystable, extending stability results to include possibly singular degenerations, and demonstrates the existence of weighted extremal metrics after blowups at stable points.

Contribution

It establishes the stability of weighted extremal manifolds under blowups and extends weighted K-polystability to include singular degenerations.

Findings

Weighted extremal manifolds are relatively weighted K-polystable.

Blowups at stable points admit weighted extremal metrics.

Weighted K-polystability holds even with singular degenerations.

Abstract

In a previous paper, we showed that the blowup of a weighted extremal K\"ahler manifold at a relatively stable fixed point admits a weighted extremal metric. Using this result, we prove that a weighted extremal manifold is relatively weighted K-polystable. In particular, a weighted cscK manifold is weighted K-polystable. This strengthens both the weighted K-semistability proved by Lahdili and Inoue, and the weighted K-polystability with respect to smooth degenerations by Apostolov--Jubert--Lahdili, allowing for possibly singular degenerations.