Nori's connectivity theorem from the perspective of $D$-modules

TL;DR

This paper explores Nori's connectivity theorem within the framework of $D$-modules, providing an explicit description of cohomology sheaves related to hyperplane sections and vanishing cohomology in algebraic geometry.

Contribution

It offers a novel perspective on Nori's connectivity theorem by analyzing the $D$-module structure of the variation of Hodge structure associated with hyperplane sections.

Findings

Explicit description of cohomology sheaves in terms of vanishing cohomology

Connection between $D$-modules and variation of Hodge structure

Application of Nori's connectivity theorem to hyperplane sections

Abstract

Given a very ample line bundle on a smooth projective variety, the variation of Hodge structure associated to the universal family of hyperplane sections can be thought of as a $D$-module with action generated by the Gauss-Manin connection. The decomposition theorem for a projective morphism and results similar to the Lefschetz hyperplane theorem tell us that the only nontrivial part of this VHS occurs in the middle degree corresponding to the vanishing cohomology of the hyperplane sections. We use Nori's connectivity theorem to give an explicit description of the cohomology sheaves of the de Rham complex of this $D$-module in terms of the vanishing cohomology of the original variety.