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

TL;DR

This paper explores the deep connections between $Q$-twisted sheaves, semihomogeneous vector bundles, and duality on abelian varieties, introducing new graded modules and revealing duality relations that impact the understanding of linear systems and cohomological thresholds.

Contribution

It introduces $Q^{ ext{≥}0}$-graded section modules linked to semihomogeneous bundles and establishes duality relations between these modules for dual polarizations.

Findings

Established a duality relation between graded modules for dual polarizations.

Derived formulas relating cohomological rank function thresholds.

Provided lower bounds for base point free thresholds based on polarization type.

Abstract

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.