Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

Details Deformation of extremal metrics, complex manifolds and the relative Futaki invariant Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

2011-07-03

arxiv.org/abs/1107.0456v4

Mathematics

Differential Geometry

22 pages. A mistake concerning the application of our main result to the Mukai-Umemura 3-fold and its deformations has been co

Scientific paper

Let (X,\Omega) be a closed polarized complex manifold, g be an extremal metric on X that represents the K\"ahler class \Omega, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deformations of (X,\Omega)\simeq (M_0,\Theta_0) provided with a holomorphic action of G with trivial action on B. Then for every t\in B sufficiently small, there exists an h^{1,1}(X)-dimensional family of extremal Kaehler metrics on M_t whose K\"ahler classes are arbitrarily close to \Theta_t. We apply this deformation theory to show that certain complex deformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.

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