Coupled equations for Kähler metrics and Yang–Mills connections

Abstract

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kähler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kähler 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.