Weighted extremal metrics on blowups

Abstract

We show that if a compact Kähler manifold admits a weighted extremal metric for the action of a torus, so too does its blowup at a relatively stable point that is fixed by both the torus action and the extremal field. This generalises previous results on extremal metrics by Arezzo--Pacard--Singer and Székelyhidi to many other canonical metrics, including extremal Sasaki metrics, deformations of Kähler--Ricci solitons and $\hat{I}¼$-cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.

50 pages, minor corrections, final version