Irrationality of the general smooth quartic 3-fold using intermediate Jacobians

Abstract

We prove that the intermediate Jacobian of the Klein quartic 3-fold X is not isomorphic, as a principally polarized abelian variety, to a product of Jacobians of curves. As corollaries we deduce (using a criterion of Clemens-Griffiths) that X, as well as the general smooth quartic 3-fold, is irrational. These corollaries were known: Iskovskih-Manin [14] proved that every smooth quartic 3-fold is irrational. However, the method of proof here is different than that of [14], is significantly simpler, and produces an explicit example.