Sheaves of AV-modules on quasi-projective varieties

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 L₊ whose fiber is Lie algebra 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 L₊ and an L₊-charged D-module. For each rational finite-dimensional representation of 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.