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

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.