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Weighted extremal metrics on blowups

Michael Hallam

TL;DR

This work extends the theory of canonical Kähler metrics under blowups to the weighted extremal setting. By gluing the Burns–Simanca model near the exceptional divisor and solving a weighted deformation problem with a moment-map obstruction, the authors show that if a compact weighted extremal manifold $(M,\\omega)$ with torus action has a relatively stable fixed point, then the blowup Bl_p M carries a $(v,w)$-weighted extremal metric in the class $[\\pi^*\\omega-\\epsilon^2 E]$ for all small $\\epsilon$. The analysis hinges on precise moment-map estimates, a carefully designed family of weighted Hölder norms, a weighted Lichnerowicz-type linearisation, and a right-inverse constructed by gluing inverses from model pieces; a subsequent paper will address relative weighted K-stability. The framework unifies and extends several canonical metrics, including extremal Sasaki metrics, deformations of Kähler–Ricci solitons, and $\\mu$-cscK metrics, demonstrating the robustness of the blowup technique in the weighted setting.

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

Weighted extremal metrics on blowups

TL;DR

This work extends the theory of canonical Kähler metrics under blowups to the weighted extremal setting. By gluing the Burns–Simanca model near the exceptional divisor and solving a weighted deformation problem with a moment-map obstruction, the authors show that if a compact weighted extremal manifold with torus action has a relatively stable fixed point, then the blowup Bl_p M carries a -weighted extremal metric in the class for all small . The analysis hinges on precise moment-map estimates, a carefully designed family of weighted Hölder norms, a weighted Lichnerowicz-type linearisation, and a right-inverse constructed by gluing inverses from model pieces; a subsequent paper will address relative weighted K-stability. The framework unifies and extends several canonical metrics, including extremal Sasaki metrics, deformations of Kähler–Ricci solitons, and -cscK metrics, demonstrating the robustness of the blowup technique in the weighted setting.

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 -cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.
Paper Structure (14 sections, 31 theorems, 252 equations)

This paper contains 14 sections, 31 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Kähler geometry
  4. Weighted extremal metrics
  5. Setting up the problem
  6. Burns--Simanca metric
  7. The approximate solution
  8. The deformation problem
  9. Weighted norms
  10. Moment map estimates
  11. Estimates for the weighted linearisation
  12. A right-inverse of the linearised operator
  13. Completing the proof
  14. Examples

Key Result

Theorem 1.1

Let $(M,\omega)$ be a $(v,w)$-weighted extremal manifold, and let $p\in M$ be a relatively stable point that is fixed by both the $T$-action and the extremal field. Denote by $\pi:\mathrm{Bl}_pM\to M$ the blowup of $M$ at $p$ with exceptional divisor $E\subset\mathrm{Bl}_pM$. Then for all $\epsilon>

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Lah19
  • Remark 2.6
  • ...and 50 more