Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over ${\Bbb P}^2$ 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 ${\Bbb P}^2$. We further generalize the isomorphism to the primitive {\em cohomology} of a smooth cubic hypersurface in ${\Bbb P}^n$. 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.