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 $\mathrm{\Theta }$ a polarization of type $(1,\mathrm{.}\mathrm{.}\mathrm{.},1,2)$, then a general pair $(X,\mathrm{\Theta })$ is log canonical