Neron-Severi group for nonalgebraic elliptic surfaces. I: Elliptic bundle case (Q1313591)

The Néron-Severi group of elliptic bundles over a curve is studied. An elliptic bundle \(X \to B\) is a principal fiber bundle over a complex, compact, connected smooth curve \(B\) whose typical fiber and structure group are an elliptic curve \(E\). Denote by \(E_B\) the sheaf of germs of local holomorphic maps from \(B\) to \(E\) and consider the cohomology class \(\xi\) in \(H^1 (E_B)\) which defines the elliptic bundle \(X \to B\). The author proves that, if \(c(\xi) \neq 0\), the group \(NS(X)/ \text{Tors} NS (X)\) is isomorphic to the group of morphisms of abelian varieties \(\Hom (J_B,E)\), where \(J_B\) denotes the Jacobian variety of \(B\). Moreover, also in case \(c(\xi) = 0\), the author determines the rank of the Néron-Severi group in terms of the genus and of the period matrix of the curve \(B\).