  In the first part of this paper we prove a vanishing criterion for higher
direct images of projective families of line bundles on a Cohen-Macaulay
variety X. The result involves certain first-order deformations of certain
curves on X, and makes essential use of the notion of global co-gaussian maps,
a generalization of Wahl's gaussian maps. In the second part we apply the
criterion above, combined with Fourier-Mukai transform on abelian varieties, to
prove an algebraic version of Green-Lazarsfeld's Generic Vanishing Theorem. In
fact we prove a stronger result concerning higher direct images of Poincar\'e
line bundles, which -- in the compact K\"ahler setting -- was conjectured by
Green and Lazarsfeld and was recently proved, by completely different methods,
by Hacon (math.AG/0308198)
