Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this
paper we consider cohomological rank functions of Q-twisted
(complexes of) coherent sheaves on abelian varieties. They satisfy a natural
transformation formula with respect to the Fourier-Mukai-Poincar\'e transform,
which has several consequences. In many concrete geometric contexts these
functions provide useful invariants. We illustrate this with two different
applications, the first one to GV-subschemes and the second one to
multiplication maps of global sections of ample line bundles on abelian
varieties.Comment: 28 pages, minor changes. Final version to appear on Annales Scient.
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