Semihomogenous vector bundles, $\mathbb Q$-twisted sheaves, duality, and linear systems on abelian varieties

In this paper we point out the natural relation between $\mathbb Q$-twisted objects of the derived category of abelian varieties, cohomological rank functions, and semihomogeneous vector bundles. We apply this to two basic classes of objects, corresponding to each other via the Fourier-Mukai-Poincar\'e transform: positive twists of the ideal sheaf of one point and of the evaluation complexes of ample simple semihomogeneous vector bundles. This naturally leads to the introduction of $\mathbb Q^{\ge 0}$- graded section modules associated to line bundles on abelian varieties built by means of semihomogeneous vector bundles (containing the usual section rings). We prove a duality relation between such modules associated to dual polarizations, which is not visible at the level of the usual section rings. Other applications include formulas relating the thresholds of relevant cohomological rank functions appearing in this context. As a consequence we show a lower bound for the base point free threshold of a polarization in function of its type, and some obstructions to surjectivity of multiplication maps of global sections of certain line bundles.