On toroidal groups (Q1825330)

A toroidal group is a quotient of \({\mathbb{C}}^ n\) by a lattice \(\Lambda\), which does not admit any non-constant holomorphic function. This is a generalization of the notion of a complex torus. For a complex torus it is well-known that the following statements are equivalent: (1) T is an Abelian variety, (2) T admits a positive line bundle and (3) T is projectively algebraic. The present paper gives a generalization of this result: For a toroidal group X the following statements are equivalent: (1) X is an quasi-Abelian variety, (2) X admits a positive line bundle, (3) X is quasi-projective, (4) X is meromorphically separable and (5) X is a covering space of an Abelian variety.