Three results on holonomic D-modules

TL;DR

This paper explores local methods in the theory of irregular holonomic D-modules, establishing invariance properties, vanishing theorems, and a new perspective on Laplace transforms, advancing understanding in algebraic analysis.

Contribution

It introduces new invariance results, local vanishing theorems, and a novel construction of the Laplace transform for Stokes-filtered sheaves, enriching the theory of holonomic D-modules.

Findings

Euler characteristic invariance under localization and tensoring

Isomorphism of pushforward morphisms after twisting by differential forms

New construction of Laplace transform matching D-module correspondence

Abstract

In this text, we illustrate the use of local methods in the theory of (irregular) holonomic D-modules. I. (The Euler characteristic of the de~Rham complex) We show the invariance of the global or local Euler characteristic of the de~Rham complex after localization and dual localization of a holonomic D-module along a hypersurface, as well as after tensoring with a rank one meromorphic connection with regular singularities. II. (Local generic vanishing theorems for holonomic D-modules) We prove that the natural morphism from the proper pushforward to the total pushforward of an algebraic holonomic D-module by an open inclusion is an isomorphism if we first twist the D-module structure by suitable closed algebraic differential forms. III. (Laplace transform of a Stokes-filtered constructible sheaf of exponential type) Motivated by the construction in [YZ24], we~propose a slightly different construction of the Laplace transform of a Stokes-perverse sheaf on the projective line and show directly that it corresponds to the Laplace transform of the corresponding holonomic D-module via the Riemann-Hilbert-Birkhoff-Deligne-Malgrange correspondence. This completes the presentation given in [Sab13, Chap. 7]}, where only the other direction of the Laplace transformation is analyzed. We~also compare our approach with the construction made previously in [YZ24].