Coupled K\"ahler-Einstein and Hermitian-Yang-Mills equations

TL;DR

This paper introduces a new coupled system of equations combining K"ahler-Einstein and Hermitian-Yang-Mills equations, providing geometric interpretations, obstructions, and examples, advancing the understanding of complex geometric structures.

Contribution

It formulates a novel coupled system of equations, offers a moment map perspective, identifies obstructions, and constructs explicit solutions on projective bundles.

Findings

Identified a Futaki type invariant as an obstruction.

Proved a Matsushima-Lichnerowicz type theorem.

Constructed explicit solutions on projective bundles.

Abstract

We introduce a new system of equations coupling K\"ahler-Einstein and Hermitian-Yang-Mills equations. We provide a moment map interpretation of these equations. We identify a Futaki type invariant as an obstruction to the existence of solutions to these equations. We also prove a Matsushima-Lichnerowicz type theorem. We prove a deformation result that produces nontrivial solutions of these equations under some conditions. We produce examples on some projective bundles using Calabi ansatz.