On blowing up extremal Kähler manifolds

By G'abor Sz'ekelyhidi. 2010

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We show that the blowup of an extremal Kahler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kahler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo-Pacard-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 Kahler-Einstein manifold at a point admits a constant scalar curvature Kahler metric in classes that make the exceptional divisor small, if it is K-polystable with respect to these classes.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4ff5cd41c1e5087998301d5c2195b8d5aef759d3