On blowing up extremal Kähler manifolds

Details On blowing up extremal Kähler manifolds On blowing up extremal Kähler manifolds

2010-10-25

arxiv.org/abs/1010.5130v2

Mathematics

Differential Geometry

29 pages; v2 30 pages, clarified exposition, fixed typos

Scientific paper

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.

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