Abstract

We introduce a system of partial differential equations coupling a Khler metric on a compact complex manifold X and a connection on a principal bundle over X. These equations generalize the constant scalar curvature equation for a Khler metric on a complex manifold and the Hermitian-Yang-Mills equations for a connection on a bundle. They have an interpretation in terms of a moment map, where the group of symmetries is an extension of the gauge group of the bundle that moves the base X. We develop a natural “formal picture” for the problem which leads to analytic obstructions for the existence of solutions. When the structure group of the bundle is trivial we recover known obstructions for the theory of cscK metrics, in particular the Futaki invariant and the notion of geodesic stability. This is joint work with Luis Alvarez Consul and Oscar Garcia Prada.