Existence problem of extremal Kaehler metrics

TL;DR

This paper addresses the existence of extremal Kähler metrics on polarized algebraic manifolds, providing affirmative results under conditions of strong K-stability relative to a maximal torus, thus advancing the understanding of the Yau-Tian-Donaldson conjecture.

Contribution

It establishes that strongly K-stable polarized manifolds relative to a maximal torus admit extremal Kähler metrics, offering a partial resolution to the extremal Kähler version of the Yau-Tian-Donaldson conjecture.

Findings

Existence of extremal Kähler metrics under strong K-stability.

Extension of the Yau-Tian-Donaldson conjecture to extremal metrics.

Relation between algebraic stability and differential geometric properties.

Abstract

In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold $(X,L)$, we choose a maximal algebraic torus $T$ in the group of holomorphic automorphisms of $X$. Then the polarization class $c_1(L)$ will be shown to admit an extremal Kaehler metric if $(X,L)$ is strongly K-stable relative to $T$.