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

TL;DR

This work investigates how Q-twisted objects, cohomological rank functions, and semihomogeneous vector bundles interact on polarized abelian varieties through the Fourier-Mukai transform. It introduces $S^Q_A(L,\alpha)$-type modules built from semihomogeneous bundles, establishes duality relations between thresholds for a polarization and its dual, and links fractional polarizations to graded section modules. The authors derive lower bounds for base point freeness thresholds via dual rank data and expose obstructions to the surjectivity of multiplication maps, yielding insights into projective normality and syzygies. The framework also outlines avenues for upper bounds, jump loci analysis, jets-separation, and higher Gaussian maps, highlighting a rich connection between FM transform, cohomology, and geometric properties of polarization on abelian varieties.

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-Poincaré 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.

Theorems & Definitions (43)

• Remark
• Theorem A
• Proposition B
• Corollary C
• Theorem D
• Corollary E
• Proposition 2.1.1
• proof
• Proposition 2.1.2
• proof