Extremal metrics on blow ups

Abstract

Given a compact Kahler manifold with an extremal metric (M,ω), we give sufficient conditions on finite sets points p_1,...,p_n and weights a_1,...a_n for which the blow up of M at p_1,...,p_n has an extremal metric in the Kahler class π^*[ω] - ε(a_1 PD[E_1] + .. + a_n PD[E_n]) for all εsufficiently small. In particular our result implies that if (M,ω) is a toric manifold and p_1,...,p_n is any subset of the fixed locus of the torus action, then such metrics exist for any choice of the weights. The relationship with previous constructions of the first two authors for Kahler constant scalar curvature metrics is discussed.

39 pages

References in corpus (3)

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• Asymptotic properties of extremal Kähler metrics of Poincaré type
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• Torus equivariant K-stability
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• Gluing in geometric analysis via maps of Banach manifolds with corners and applications to gauge theory
• Deformation theory of scalar-flat Kähler ALE surfaces
• Remarks on the existence of bilaterally symmetric extremal Kähler metrics on ${\mathbb{CP}}^2♯2\overline{{\mathbb{CP}}^2}$
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• On complete constant scalar curvature Kähler metrics with Poincaré-Mok-Yau asymptotic property
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• A note on blow-ups of toric surfaces and cscK metrics
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