On the Néron-Severi group of abelian complex Lie groups

Abstract

In this paper we want to discuss some phenomena concerning Chern classes of holomorphic line bundles on not necessarily compact quotients X of a complex vector space by a lattice. We give explicit equations describing the Neron-Severi group NS(X), from which we can deduce some facts on the behaviour of Picard numbers in complete families of abelian complex Lie groups. The equations enable us to calculate long lists of Picard numbers of abelian Lie groups of given dimension and rank. In contrary to the compact case the Chern classes of holomorphic line bundles on X are in general not of type (1, 1). We point out a representation of the (1, 1) -part of the Neron-Severi group, which is analoguous to the description of NS(X) as a subgroup of Hom(X, X), when X is an abelian variety. The projection of NS(X) to the space of cohomology classes of type (1, 1) is in general not infective; however if X is a toroidal group this is the case, but in this situation the image of NS(X) in-H1,1(X) is in general not a discrete subgroup.