Very ample line bundles on quasi-abelian varieties (Q5928278)

This article is a continuation of the author's previous works concerning adjoint bundles on weakly 1-complete Kähler manifolds [Math. Ann. 311, 501-531 (1998; Zbl 0912.32021); ibid. 312, 363-385 (1998; Zbl 0952.32013)]. The main result is the following Lefschetz-type theorem. Let \(X\) be a quasi-torus with a positive line bundle \(L\). (1) Assume that \(X\) has no positive dimensional compact subtorus. Then \(L\) is very ample. (2) \(L^{\otimes^2}\) is very ample if and only if there is no positive dimensional compact subtorus \(A\) of \(X\) such that \((A,L|_A)\) is a principally polarized abelian variety. (3) \(L^{\otimes^3}\) is very ample. The author also gives two extreme examples of the pairs \(X\) and \(L\) as in the theorem.