The Abel-Jacobi isomorphism for the cubic fivefold

Abstract

Let X be a general cubic fivefold, JX the associated intermediate Jacobian, F the Fano surface of the planes contained in X.We prove that the Abel-Jacobi map induces an isomorphism from the Albanese variety of F to JX. Introduction.It is a standard fact (see [7] Exp.XI (2.9)) that the only smooth hypersurfaces X in p 2J+1 9 d > 0, for which the intermediate Jacobian JX is an abelian variety are the quadrics, the cubic and the quartic threefolds in P 4 , and the cubic fivefold in P 6 .For a quadric JX = 0.In [5], Clemens and Griffiths proved that the Abel-Jacobi map ( + ) a: AlbF-+JX