Coupled equations for K\"ahler metrics and Yang-Mills connections (Thesis)

TL;DR

This thesis introduces coupled equations linking K"ahler metrics and Yang-Mills connections on principal bundles over complex manifolds, generalizing classical geometric conditions and exploring their moduli and obstructions.

Contribution

It formulates new coupled equations, provides a moment map interpretation, and studies obstructions and their relation to algebraic geometry moduli problems.

Findings

Generalization of constant scalar curvature and Hermite-Yang-Mills conditions

Moment map interpretation of the coupled equations

Analysis of obstructions via Futaki character and Mabuchi K-energy

Abstract

We study equations on a principal bundle over a compact complex manifold coupling connections on the bundle with K\"ahler structures in the base. These equations generalize the conditions of constant scalar curvature for a K\"ahler 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.