D-modules and finite maps

TL;DR

This paper investigates how semisimplicity of holonomic D-modules is preserved under finite maps between smooth varieties, introducing filtrations and exploring algebraic analogs of topological theorems.

Contribution

It introduces a filtration of direct images of D-modules via local cohomology and proves algebraic versions of classical theorems for connections and fundamental groups.

Findings

Filtration of direct images using local cohomology.

Algebraic proof of a Grothendieck-Lefschetz type theorem for connections.

Generalization of rationally connected varieties being simply connected.

Abstract

We study the preservation of semisimplicity for holonomic D-modules with respect to the direct and inverse image of mainly finite maps $\pi : X \to Y$ of smooth varieties. A natural filtration of the direct image $\pi_+({\mathcal O}_X)$ is defined by the vanishing of local cohomology along a natural stratification of $\pi$. The notions are exemplified with the invariant map $X\to X^G$, where $G$ is a complex reflection group. Simply connected varieties are treated algebraically by considering connections instead of fundamental groups. For example, a "Grothendieck-Lefschetz" theorem for connections is proven and also a generalized version of the assertion that rationally connected varieties be simply connected, entirely by algebraic means, using the idea of a "differential covering".