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 Kähler fibre, and a Kähler structure on the base. These equations are a generalization of the Kähler–Yang–Mills equations introduced by the authors. They also generalize the constant scalar curvature for a Kähler 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.

Partially supported by the Spanish MINECO under ICMAT Severo Ochoaproject No. SEV-2015-0554, and under grant No. MTM2016-81048-P