Kähler–Yang–Mills Equations and Vortices

The Kähler–Yang–Mills equations are coupled equations for a Kähler metric on a compact complex manifold and a connection on a complex vector bundle over it. After briefly reviewing the main aspects of the geometry of the Kähler–Yang–Mills equations, we consider dimensional reductions of the equations related to vortices — solutions to certain Yang–Mills–Higgs equations.