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Extremal Kähler metrics on blowups

Ruadhaí Dervan, Lars Martin Sektnan

Abstract

Consider a compact Kähler manifold which either admits an extremal Kähler metric, or is a small deformation of such a manifold. We show that the blowup of the manifold at a point admits an extremal Kähler metric in Kähler classes making the exceptional divisor sufficiently small if and only if it is relatively K-stable, as predicted by the Yau-Tian-Donaldson conjecture. We also give a geometric interpretation of what relative K-stability means in this case in terms of finite dimensional geometric invariant theory. This gives a complete solution to a problem introduced and solved by Arezzo, Pacard, Singer and Székelyhidi for constant scalar curvature Kähler metrics in dimension at least three.

Extremal Kähler metrics on blowups

Abstract

Consider a compact Kähler manifold which either admits an extremal Kähler metric, or is a small deformation of such a manifold. We show that the blowup of the manifold at a point admits an extremal Kähler metric in Kähler classes making the exceptional divisor sufficiently small if and only if it is relatively K-stable, as predicted by the Yau-Tian-Donaldson conjecture. We also give a geometric interpretation of what relative K-stability means in this case in terms of finite dimensional geometric invariant theory. This gives a complete solution to a problem introduced and solved by Arezzo, Pacard, Singer and Székelyhidi for constant scalar curvature Kähler metrics in dimension at least three.

Paper Structure

This paper contains 25 sections, 36 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. The main argument
  3. Preliminaries
  4. Extremal metrics on blowups
  5. Step 1: the main gluing argument
  6. Step 2: moment map geometry
  7. Step 3: K-stability and GIT
  8. The semistable case
  9. The main gluing argument
  10. Constructing an initial approximate solution
  11. Weighted Hölder spaces
  12. Weighted spaces on punctured manifolds
  13. Weighted spaces on the blowup
  14. The linearisation
  15. Improving the approximate solution
  16. ...and 10 more sections

Key Result

Theorem 1.1

There is an $\varepsilon_0>0$ such that for all $\varepsilon \in (0, \varepsilon_0)$ the following are equivalent:

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • ...and 52 more