Fourier transforms, generic vanishing theorems and polarizations of abelian varieties

The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: - we give a cohomological characterization of principal polarizations - we prove that if $X$ an abelian variety and $\Theta $ a polarization of type $(1,...,1,2)$, then a general pair $(X,\Theta )$ is log canonical