Neron-Severi group for nonalgebraic elliptic surfaces. II: Non-Kählerian case
The Neron-Severi group of a non-Kählerian elliptic surface \(\varphi : X \to B\) is studied. The paper is a continuation of part I: The elliptic bundle case, ibid. 79, No. 2, 187-195 (1993; Zbl 0847.32032) which deals with the Neron-Severi group of elliptic bundles. First the author shows that a non-Kählerian elliptic surface \(\varphi : X \to B\) has the structure of a quasi-bundle, which means that it is an elliptic bundle, except for some singular fibers which are multiples of smooth elliptic curves. Then he proves that given a quasi-bundle \(\varphi : X \to B\) with odd Betti numbers, there exists an elliptic bundle which is a cyclic cover of \(\varphi\). This enables him to use the above quoted part I and so to get the following isomorphism: \(NS (X) \times \mathbb{Q} \cong \Hom (J_B,E) \times \mathbb{Q}\), where \(J_B\) denotes the Jacobian variety of \(B,E\) the general fiber of \(\varphi\) and \(\Hom (J_B,E)\) denotes the group of the morphisms of abelian varieties.
scientific article; zbMATH DE number 703893