Homogeneous line bundles over a toroidal group

Abstract

A connected complex Lie group without non-constant holomorphic functions is called a toroidal group ([5]) or an (H, C)-group ([9]). Let X be an n-dimensional toroidal group. Since a toroidal group is commutative ([5], [9] and [10]), X is isomorphic to the quotient group Cn/Γ by a lattice of Cn. A complex torus is a compact toroidal group. Cousin first studied a non-compact toroidal group ([2]).Let L be a holomorphic line bundle over X. L is said to be homogeneous if is isomorphic to L for all x ε X, where Tx is the translation defined by x ε X. It is well-known that if X is a complex torus, then the following assertions are equivalent: