Weighted Extremal Metrics on Blowups

TY - JOUR

T1 - Weighted Extremal Metrics on Blowups

AU - Hallam, Michael

N1 - Publisher Copyright: © Mathematica Josephina, Inc. 2025.

PY - 2025/6

Y1 - 2025/6

N2 - 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 μ-cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.

AB - 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 μ-cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.

KW - Canonical metrics

KW - Geometric analysis

KW - Kahler manifolds

UR - https://www.scopus.com/pages/publications/105004352528

U2 - 10.1007/s12220-025-02004-5

DO - 10.1007/s12220-025-02004-5

M3 - Journal article

AN - SCOPUS:105004352528

SN - 1050-6926

VL - 35

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 6

M1 - 187

ER -