Coupled K\"ahler-Einstein metrics

TL;DR

This paper introduces coupled Kähler-Einstein metrics as a new generalization of classical Kähler-Einstein metrics, establishing existence, uniqueness, and stability conditions on certain Kähler manifolds.

Contribution

It defines coupled Kähler-Einstein metrics and proves their existence, uniqueness, and stability implications on Kähler manifolds with specific properties.

Findings

Existence and uniqueness of coupled Kähler-Einstein metrics on ample canonical bundles.

Existence results for Kähler-Einstein Fano manifolds.

Coupled Kähler-Einstein metrics imply a generalized algebraic stability condition.

Abstract

We propose new types of canonical metrics on K\"ahler manifolds, called coupled K\"ahler-Einstein metrics, generalizing K\"ahler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is K\"ahler-Einstein Fano. In the Fano case we also prove that existence of coupled K\"ahler-Einstein metrics imply a certain algebraic stability condition, generalizing $K$-polystability.