In this thesis we study some questions on abelian varieties by means of the Fourier-Mukai-Poincaré (FMP) transform and generic vanishing. Specifically, we study cohomological rank functions, and give a reinterpretation of them in terms of Mukai's semi-homogeneous vector bundles. On the other hand, we introduce the theory of vanishing thresholds of coherent sheaves, extending the notion of basepoint-freeness threshold. In particular, we obtain a sensible notion of jets-separation thresholds of a polarization and relate them to a more classical invariant, namely the Seshadri constant of the polarization. In addition, we establish certain dualities, coming from the FMP transform, which give relations between thresholds on an abelian variety and thresholds on its dual. As concrete applications, we describe obstructions for a polarization to be normally generated and prove a result about the surjectivity of certain higher Gauss-Wahl maps, which allow us to show that the failure of the surjectivity of some of these maps characterize special kinds of abelian varieties.