Extremal Kähler metrics on blowups

Preprint, 2021

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, thus proving a special case of 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 addressed in special cases by Arezzo, Pacard, Singer and Székelyhidi. In addition, the case of a deformation of an extremal manifold proves the first non-trivial case of a general conjecture of Donaldson.