Extremal K\"ahler metrics on blowups

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.