On blowing up extremal Kähler manifolds

Abstract

We show that the blowup of an extremal Kähler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kähler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo, Pacard, and Singer. We also study the K-polystability of these blowups, sharpening a result of Stoppa in this case. As an application we show that the blowup of a Kähler–Einstein manifold at a point admits a constant scalar curvature Kähler metric in classes that make the exceptional divisor small, if it is K-polystable with respect to these classes.