Deformations of complex structures and the coupled K\"ahler-Yang-Mills equations

TL;DR

This paper develops a deformation theory for the coupled Kähler-Yang-Mills equations, enabling the discovery of new solutions through complex structure deformations and analyzing recent examples in the field.

Contribution

It generalizes Székelyhidi's work on scalar curvature metrics to the coupled equations and introduces methods to find new solutions via complex structure deformations.

Findings

New solutions of coupled Kähler-Yang-Mills equations found through deformation.

Extended deformation theory applicable to complex structures and vector bundles.

Analysis of recent examples by Keller and Tønnesen-Friedman.

Abstract

In this work we define a deformation theory for the Coupled K\"ahler-Yang-Mills equations in arXiv:1102.0991, generalizing work of Sz\'ekelyhidi on constant scalar curvature K\"ahler metrics. We use the theory to find new solutions of the equations via deformation of the complex structure of a polarised manifold endowed with a holomorphic vector bundle. We also study the deformations of the recent examples of Keller and T{\o}nnesen-Friedman.