Adjoint linear series on weakly 1-complete Kähler manifolds. II: Lefschetz type theorem on quasi-Abelian varieties (Q1271235)

The author proves the following two beautiful results on a holomorphic line bundle \(L\) on a quasi-torus. 1. A Kodaira-type theorem: Let \(L\) be a holomorphic line bundle over a quasi-torus \(\mathbb{C}^n/Gamma\). Then the following four conditions are equivalent: (a) \(L\) is ample; (b) \(L\) is positive; (c) The first Chern class \(c_1(L)\) contains a Kähler form; (d) The alternating form \(c_1(L):\Gamma\times\Gamma\to {\mathbb Z}\) given by the first Chern class is obtained as the imaginary part of a Hermitian form on \(\mathbb{C}^n\) which is positive on \(F=:{\mathbb R}\cap \sqrt{-1}{\mathbb R}\Gamma\). 2. A Lefschetz-type theorem: If \(L\) satisfies these equivalent conditions, then \(L\) has a nontrivial section, \(L^{\otimes^2}\) is generated by its global sections, and \(L^{\otimes^2}\) is very ample. The main ingredients in the proofs are the general results obtained earlier by the author in part I of this series of papers [Math. Ann. 311, No. 3, 501-531 (1998; Zbl 0912.32021)].