On the K\"ahler-Yang-Mills-Higgs equations

In this paper we introduce a set of equations on a principal bundle over a compact complex manifold coupling a connection on the principal bundle, a section of an associated bundle with K"ahler fibre, and a K"ahler structure on the base. These equations are a generalization of the K"ahler-Yang-Mills equations introduced by the authors. They also generalize the constant scalar curvature for a K"ahler metric studied by Donaldson and others, as well as the Yang-Mills-Higgs equations studied by Mundet i Riera. We provide a moment map interpretation of the equations, construct some first examples, and study obstructions to the existence of solutions.