Extremal K\"ahler metrics on blowups

TL;DR

This paper proves that the existence of extremal K"ahler metrics on blowups of certain manifolds is equivalent to a stability condition called relative K-stability, confirming a conjecture and providing a geometric interpretation.

Contribution

It establishes the equivalence between extremal K"ahler metrics on blowups and relative K-stability, extending previous results to a broader class of manifolds.

Findings

Extremal K"ahler metrics exist on blowups if and only if the manifold is relatively K-stable.

Provides a geometric interpretation of relative K-stability via geometric invariant theory.

Completes the solution to a problem on constant scalar curvature K"ahler metrics in higher dimensions.

Abstract

Consider a compact K\"ahler manifold which either admits an extremal K\"ahler metric, or is a small deformation of such a manifold. We show that the blowup of the manifold at a point admits an extremal K\"ahler metric in K\"ahler 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 Sz\'ekelyhidi for constant scalar curvature K\"ahler metrics in dimension at least three.