Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles

The notion of holomorphic vector bundle is common to some branches of mathematics and theoretical physics. In particular, such a notion seems to play a fundamental role in complex differential geometry, algebraic geometry and Yang-Mills theory. In this thesis we study a kind of holomorphic vector bundles over complex manifolds, known as Higgs bundles, and some of their main properties. We restrict such objects to the case when the complex manifold is compact Kähler. On one hand, complex manifolds provide a rich class of geometric objects, which behave rather differently than real smooth manifolds. For instance, one of the main characteristic of a compact complex manifold is that its group of biholomorphisms is always finite dimensional. On the other hand, since the manifolds in which we are interested are compact Kähler, we have that certain invariants associated with the holomorphic bundle can be defined using cohomology classes...