On periodic functions of several complex variables. II: Reduction to the positive definite case
The quotient complex manifold \(X=\mathbb{C}^ n/ \Gamma\), where \(\Gamma\) denotes a discrete subgroup, is called a toroidal group, if \(H^ 0(X,{\mathcal O}_ X)=\mathbb{C}\). The aim of the paper is to characterize those line bundles \(L\) on \(X\) which admit nontrivial holomorphic sections. The author gives some necessary conditions and finally reduces the question to the case of a positive line bundle on \(X\). Remark: Recently the problem has been solved by Copocasa and Catanese, who showed (preprint 1993) that every positive line bundle on a toroidal group admits nonzero holomorphic sections. [For part I of the paper see the author, Nagoya Math. J. 122, 83-114 (1991; Zbl 0749.32015)].
scientific article; zbMATH DE number 469050