Non trivial examples of coupled equations for K\"ahler metrics and Yang-Mills connections

TL;DR

This paper constructs non-trivial solutions to coupled equations that unify constant scalar curvature K"ahler metrics and Hermitian-Einstein metrics, advancing the understanding of geometric structures on complex manifolds.

Contribution

It provides explicit examples of solutions to coupled equations linking K"ahler metrics and Yang-Mills connections, extending previous theoretical frameworks.

Findings

Explicit non-trivial solutions to coupled equations

Generalization of uniformization problem for complex manifolds

Bridging K"ahler geometry and gauge theory

Abstract

We provide non trivial examples of solutions to the system of coupled equations introduced by M. Garc\'ia-Fern\'andez for the uniformization problem of a triple $(M,L,E)$ where $E$ is a holomorphic vector bundle over a polarized complex manifold $(M,L)$, generalizing the notions of both constant scalar curvature K\"ahler metric and Hermitian-Einstein metric.