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arXiv · 2110.13579

Extremal K\"ahler metrics on blowups

Ruadhaí Dervan·Lars Martin Sektnan
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Abstract

Consider a compact K\"ahler manifold which either admits an extremal K\"ahler metric, or is a small deformation of such a manifold. We show that the blowup of the manifold at a point admits an extremal K\"ahler metric in K\"ahler classes making the exceptional divisor sufficiently small if and only if it is relatively K-stable, as predicted by 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 introduced and solved by Arezzo, Pacard, Singer and Sz\'ekelyhidi for constant scalar curvature K\"ahler metrics in dimension at least three.

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BibTeXRIS

Ruadhaí Dervan, Lars Martin Sektnan. 2021-10-26. Extremal K\"ahler metrics on blowups. https://arxiv.org/abs/2110.13579

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