Some topics on Higgs bundles over projective varieties and their moduli spaces

In this thesis we study vector bundles on projective varieties and their moduli spaces. In Chapters 2, 3 and 4 we recall some basic notions as Higgs bundles, decorated bundles and generalized parabolic sheaves and introduce the problem we want to study. In chapter 5, we study Higgs bundles on nodal curves. After moving the problem on the normalization of the curve, starting from a Higgs bundle we obtain a generalized parabolic Higgs bundle. Using decorated bundles we are able to construct a projective moduli space which parametrizes equivalence classes of Higgs bundles on a nodal curve X. This chapter is an extract of a joint work with Andrea Pustetto Later on Chapter 6 is devoted to the study of holomorphic pairs (or twisted Higgs bundles) on elliptic curve. Holomorphic pairs were introduced by Nitsure and they are a natural generalization of the concept of Higgs bundles. In this Chapter we extend a result of E. Franco, O. Garc\'ia-Prada And P.E. Newstead valid for Higgs bundles to holomorphic pairs. Finally the last Chapter describes a joint work with Professor Ugo Bruzzo. We study Higgs bundles over varieties with nef tangent bundle. In particular generalizing a result of Nitsure we prove that if a Higgs bundle $(E,\phi)$ over the variety X with nef tangent remains semisatble when pulled-back to any smooth curve then it discrimiant vanishes.