On the Serre-Swan theorem, and on vector bundles over real Abelian varieties

Abstract

This thesis is divided into two parts. In Chapter 1 we prove a generalization of two classical results of Serre and Swan on the relation between locally free sheaves and projective modules, by emphasizing the axiomatic aspect of the problem. We determine a class of ringed spaces (X,OX) for which the category of locally free sheaves of bounded rank over X is equivalent to the category of finitely generated projective _(X,OX) modules. The well-known Serre-Swan theorems for affine schemes, differentiable manifolds, Stein spaces, etc., are then derived. In Chapter 2 we study real algebraic vector bundles over a real abelian variety. The main theorem in this part gives various equivalent criteria for a real algebraic vector bundle over a real abelian variety to admit a flat holomorphic connection. In the course of the proof of the main theorem we also derive a version of a result of Simpson for real abelian varieties.