Coupled equations for Kähler metrics and Yang-Mills connections (Thesis)

We study equations on a principal bundle over a compact complex manifold coupling connections on the bundle with Kähler structures in the base. These equations generalize the conditions of constant scalar curvature for a Kähler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki character and the Mabuchi K-energy. We explain their relationship to the algebro-geometric moduli problem for pairs consisting of a polarized variety and a holomorphic vector bundle.

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