Gaussian maps and generic vanishing I: subvarieties of abelian varieties

We present an approach to Green-Lazarsfeld's generic vanishing combining gaussian maps and the Fourier-Mukai transform associated to the Poincar\`e line bundle. As an application we prove the Generic Vanishing Theorem for all normal Cohen-Macaulay subvarieties of abelian varieties over an algebraically closed field. This paper is a reworking of part of arXiv:math/0310026v2 [math.AG], which contained a mistake.