On homothetic balanced metrics

Arezzo C;LOI, ANDREA;ZUDDAS, FABIO

2012-01-01

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Abstract

In this article, we study the set of balanced metrics given in Donaldson's terminology (J. Diff. Geometry 59:479-522, 2001) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kähler metrics. We prove that this set is finite when M admits a non-positive Kähler-Einstein metric, in the case of non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179-228, 2006; Ann. Math. 170(2):685-738, 2009).