Existence of coupled K\"ahler-Einstein metrics using the continuity method

TL;DR

This paper establishes the existence of coupled Kähler-Einstein metrics on complex manifolds with ample canonical bundle using the continuity method, providing an alternative to variational approaches.

Contribution

It introduces a new proof technique for coupled Kähler-Einstein metrics on manifolds with ample canonical bundle via the continuity method.

Findings

Existence of coupled Kähler-Einstein metrics proven for manifolds with ample canonical bundle.

Reduction of existence proof in the Fano case to a C^0 estimate.

Method offers an alternative to variational techniques for these metrics.

Abstract

In this paper we prove the existence of coupled K\"ahler-Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nystr\"om using calculus of variations. We prove the result using the method of continuity. In the process of proving estimates, akin to the usual K\"ahler-Einstein metrics, we reduce existence in the Fano case to a C^0 estimate.