Sheaves of AV-modules on quasi-projective varieties

TL;DR

This paper investigates sheaves of modules for the Lie algebra of vector fields on quasi-projective varieties, introducing the concept of virtual jets and characterizing sheaves of AV-modules through modules for a specific Lie algebra and D-modules.

Contribution

It introduces the notion of virtual jets of vector fields and characterizes sheaves of AV-modules via modules for a Lie algebra and D-modules, providing new constructions and realizations.

Findings

Characterization of sheaves of AV-modules using $ ext{L}_+$ and D-modules

Construction of jet AV-modules from rational finite-dimensional representations

Realization of Rudakov modules as tensor products of jet modules and delta function D-modules

Abstract

We study sheaves of modules for the Lie algebra of vector fields with the action of the algebra of functions, compatible via the Leibniz rule. A crucial role in this theory is played by the virtual jets of vector fields - jets that evaluate to a zero vector field under the anchor map. Virtual jets of vector fields form a vector bundle $\mathcal{L}_+$ whose fiber is Lie algebra $\widehat{L}_+$ of vanishing at zero derivations of power series. We show that a sheaf of $AV$-modules is characterized by two ingredients - it is a module for $\mathcal{L}_+$ and an $\mathcal{L}_+$-charged $D$-module. For each rational finite-dimensional representation of $\widehat{L}_+$, we construct a bundle of jet $AV$-modules. We also show that Rudakov modules may be realized as tensor products of jet modules with a $D$-module of delta functions.