Residues and filtered D-modules

TL;DR

This paper constructs a filtered holonomic D-module using residues for high-degree embeddings of smooth projective varieties, providing insights into Hodge modules and their properties.

Contribution

It introduces a new residue-based method to define filtered D-modules on dual projective space, linking geometric embeddings to Hodge theory.

Findings

Concrete description of intermediate extensions to Hodge modules

Positivity and vanishing theorems for sheaves F_k M

Reflexivity properties of the sheaves

Abstract

For an embedding of sufficiently high degree of a smooth projective variety X into projective space, we use residues to define a filtered holonomic D-module (M, F) on the dual projective space. This gives a concrete description of the intermediate extension to a Hodge module on P of the variation of Hodge structure on the middle-dimensional cohomology of the hyperplane sections of X. We also establish many results about the sheaves F_k M, such as positivity, vanishing theorems, or reflexivity.