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

TL;DR

This paper introduces a new set of coupled equations on principal bundles over complex manifolds, generalizing several known equations in K"ahler geometry and gauge theory, with analysis of solutions and obstructions.

Contribution

It formulates a unified framework for K"ahler-Yang-Mills-Higgs equations, extending previous models and providing geometric and analytical insights.

Findings

Provided a moment map interpretation of the equations

Constructed initial examples of solutions

Analyzed obstructions to existence of solutions

Abstract

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.