Irregular perverse sheaves

TL;DR

This paper introduces irregular constructible sheaves with Novikov ring coefficients, establishing an equivalence with holonomic D-modules and generalizing the irregular Riemann-Hilbert correspondence, including algebraic aspects and connections to Fukaya categories.

Contribution

It develops the theory of irregular constructible sheaves, extending the irregular Riemann-Hilbert correspondence and introducing a new irregular perverse t-structure.

Findings

Equivalence between derived categories of irregular constructible sheaves and holonomic D-modules.

Introduction of irregular perverse t-structure and its abelian heart.

Discussion of Novikov ring's role via a conjectural Fukaya category reformulation.

Abstract

We introduce irregular constructible sheaves, which are $\mathbb{C}$-constructible with coefficients in a finite version of Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic $\mathcal{D}$-modules by a modification of D'Agnolo--Kashiwara's irregular Riemann--Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse t-structure, which is a straightforward generalization of usual perverse t-structure and we see its heart is equivalent to the abelian category of holonomic $\mathcal{D}$-modules. We also develop the algebraic version of the theory. Furthermore, we discuss the reason of the appearance of Novikov ring by using a conjectural reformulation of Riemann--Hilbert correspondence in terms of certain Fukaya category.