Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 4

Special Issue in Honor of Simon Donaldson: Part 2 of 2

Guest Editor: Richard Thomas (Imperial College London)

On the Kähler–Yang–Mills–Higgs equations

Pages: 1181 – 1217

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n4.a6


Luis Álvarez-Cónsul (Instituto de Ciencias Matemáticas, Madrid, Spain)

Mario Garcia-Fernandez (Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain; and Instituto de Ciencias Matemáticas, Madrid, Spain)

Oscar García-Prada (Instituto de Ciencias Matemáticas, Madrid, Spain)


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.

The authors were partially supported by the Spanish MINECO under ICMAT Severo Ochoa project No. SEV-2015-0554, and under grant No. MTM2016-81048-P.

Received 22 June 2018

Published 20 March 2020