Extremal metrics on blow ups

Details Extremal metrics on blow ups Extremal metrics on blow ups

2006-12-31

arxiv.org/abs/math/0701028v1

Mathematics

Differential Geometry

39 pages

Scientific paper

Given a compact Kahler manifold with an extremal metric (M,\omega), 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 \pi^*[\omega] - \epsilon (a_1 PD[E_1] + .. + a_n PD[E_n]) for all \epsilon sufficiently small. In particular our result implies that if (M,\omega) 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.

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