Holonomic and perverse logarithmic D-modules

TL;DR

This paper extends the theory of holonomic D-modules to logarithmic schemes, introduces a perverse t-structure, and explores duality and extension properties in this new setting.

Contribution

It develops a framework for holonomic D-modules on logarithmic schemes, including duality, t-structures, and filtrations, which was not previously established.

Findings

Verdier duality extends to logarithmic D-modules.

The perverse t-structure makes duality t-exact for holonomic modules.

Holonomic modules can be extended from open subschemes to entire spaces.

Abstract

We introduce the notion of a holonomic D-module on a smooth (idealized) logarithmic scheme and show that Verdier duality can be extended to this context. In contrast to the classical case, the pushforward of a holonomic module along an open immersion is in general not holonomic. We introduce a "perverse" t-structure on the category of coherent logarithmic D-modules which makes the dualizing functor t-exact on holonomic modules. This allows us to transfer some of the formalism from the classical setting and in particular show that every holonomic module on an open subscheme can be extended to a holonomic module on the whole space. Conversely this t-exactness characterizes holonomic modules among all coherent logarithmic D-modules. We also introduce logarithmic versions of the Gabber and Kashiwara-Malgrange filtrations.