Projective algebraic varieties whose universal covering spaces are biholomorphic to \(\mathbb{C}^n\)
In: Number Theory, Algebraic Geometry, Commucative Algebra, in Honor Y. Akizuki, 147-167 (1973; Zbl 0271.14015), \textit{S. Iitaka} made the conjecture \(U_n\): If \(V\) is a smooth projective variety with \({\mathbb{C}}^n\) as its universal cover, then a finite unramified cover of \(V\) is an abelian variety (such \(V\) will be called para-abelian variety). In an early version of this paper, the author proved that \(U_3\) is true using \(\partial\)-étale cohomology theory developed in his early paper [``Global structure of an elliptic fibration'' (Kyoto Univ. 1996)]. Here the author generalizes and proves, among other things, the following: (1) If \(V\) is smooth projective of type \(U\) (this is satisfied if \(V\) has \({\mathbb{C}}^n\) as its universal cover) and if the canonical divisor is semi-ample (i.e., a positive multiple of it is base point free), then \(V\) is a para-abelian variety. (2) If \(V\) is smooth projective of type \(U\) and if the canonical divisor \(K_F\) of a general fibre \(F\) of the Albanese mapping \(\alpha : V \rightarrow \text{Alb}(V)\) is semi-ample, then \(V\) is a para-abelian variety and \(\alpha\) is an étale fibre bundle.
scientific article; zbMATH DE number 1347252