A Prym Construction for the Cohomology of a Cubic Hypersurface

Mumford defined a natural isomorphism between the intermediate jacobian of a conic‐bundle over P 2 and the Prym variety of a naturally defined étale double cover of the discriminant curve of the conic‐bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold, and Beauville later generalized the isomorphism to intermediate jacobians of odd‐dimensional quadric‐bundles over P 2 . We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in P n . We give two applications of our construction: one is a special case of the generalized Hodge conjectures and the other is an Abel‐Jacobi isomorphism. 1991 Mathematics Subject Classification : primary 14J70; secondary 14J45.