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

TL;DR

This paper develops a deformation theory for extremal metrics on complex manifolds, showing existence of nearby extremal Kähler metrics under certain conditions, and applies it to find Kähler-Einstein metrics on deformations of the Mukai-Umemura 3-fold.

Contribution

It introduces a new deformation approach for extremal metrics with nondegenerate Futaki invariant, enabling the construction of extremal and Kähler-Einstein metrics on deformed complex manifolds.

Findings

Existence of extremal Kähler metrics on small deformations of (X, abla)

Application to Kähler-Einstein metrics on Mukai-Umemura 3-fold deformations

Nondegeneracy condition of the Futaki invariant ensures stability of extremal metrics

Abstract

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.