Families of compact complex tori

A compact complex torus of dimension g is a complex Lie group isomorphic to V/Λ, where V is a complex vector space of dimension g and Λ is a lattice in V. Although all compact complex tori of dimension g are isomorphic in the category of real differentiable manifolds, they can have non-isomorphic structures of complex manifolds. As is common in algebraic geometry, it is possible to simplify the study of compact complex tori using a categorical approach. One possibility is through a functor F yielding an equivalence of categories between the category of compact complex tori of dimension g, and the category of triples (Λ, V, γ), where Λ is a free abelian group of rank 2g, V is a complex vector space of dimension g and γ from Λ to V is a lattice inclusion. This is important for several reasons. The first reason is that the treatment of compact complex tori under the categorical point of view allows us to focus our attention more on the relations and morphisms between them, rather than on the objects themselves. The second reason is that, since F is an equivalence, the two categories satisfy the same properties. So, in order to understand compact complex tori, it is enough to understand the category of triples (Λ, V, γ), which is an easier category to handle. The aim of this thesis is to extend F to a functor yielding an equivalence of categories between the category of families of compact complex tori of dimension g over a fixed complex manifold B, and the category of triples (Λ, V, γ), where Λ is a locally constant B-Lie group with structural group a free abelian group of rank 2g, V is a holomorphic vector bundle of rank g over B and γ from Λ to V is a morphism of B-Lie groups, such that it yields a lattice inclusion fiberwise. If B is just a point, these two categories coincide with the ones previously defined.