Coupled equations for K\"ahler metrics and Yang-Mills connections

TL;DR

This paper introduces coupled equations linking Kähler metrics and Yang-Mills connections on complex manifolds, generalizing classical conditions and exploring obstructions to solutions with a geometric and stability perspective.

Contribution

It formulates new coupled equations, provides a moment map interpretation, and extends stability obstructions like Futaki invariant and Mabuchi K-energy to this setting.

Findings

Generalization of constant scalar curvature and Hermite-Yang-Mills conditions

Identification of obstructions via extended Futaki invariant and Mabuchi K-energy

Examples illustrating solutions to the coupled equations

Abstract

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kahler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kahler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K-energy and geodesic stability. We finish by giving some examples of solutions.