Linear systems on quasi-abelian varieties
The study of periodic meromorphic functions by the authors [Acta Math. 166, No. 1/2, 27-68 (1991; Zbl 0719.32005)] led to the definition of Cousin quasi-tori. The Cousin quasi-tori admitting a positive line bundle (or equivalently admitting the structure of a quasi-projective algebraic variety) are called quasi-abelian varieties. In these terms abelian varieties are exactly the compact quasi-abelian varieties. Suppose \(X\) is a quasi-abelian variety and \(L\) is a positive line bundle on \(X\). It is shown that for non-compact \(X\) the vector space \(H^0 (X,L)\) is infinite dimensional. Moreover the following Lefschetz type results are proven: \(H^0 (X, L^n)\) is base point free if \(n \geq 2\), and \(H^0 (X, L^n)\) gives a projective embedding if \(n \geq 3\). Also the theorem of Ohbuchi generalizes to quasi-abelian varieties: If \((X,L)\) admits no principally polarized abelian variety as factor, then \(H^0 (X,L^2)\) gives a projective embedding of \(X\).
scientific article; zbMATH DE number 734585
scientific article