| According to recent work of Barja, Pardini and Stoppino, on an abelian variety one can naturally define the ranks of the cohomology groups of a coherent sheaf (or a finite complex of coherent sheaves) twisted with a rational multiple of a polarization. This gives rise to cohomological rank functions defined on the rational numbers, which can be in turn extended to the real numbers. These functions seem to encode interesting geometric informations, as I will try to show by computing some simple examples. Finally, I will show an application to the so-called GV-subvarieties of principally polarized abelian varieties. The main tool is a transformation formula for cohomological rank functions with respect to the Fourier-Mukai equivalence associated to the Poincaré line bundle. This is a report of a joint work in progress with Zhi Jiang (Fudan University, Shanghai) |