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.


Ruadhaí Dervan

University of Cambridge

Lars Martin Sektnan

University of Gothenburg

Chalmers, Mathematical Sciences, Algebra and geometry

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9/9/2022 7