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

Abstract

Let (X,Ω) be a closed polarized complex manifold, g be an extremal metric on X that represents the Kähler class Ω, 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,Ω)\simeq (M_0,Θ_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 Kähler classes are arbitrarily close to Θ_t. We apply this deformation theory to show that certain complex deformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.