D-modules and complex foliations

TL;DR

This paper explores the relationship between D-modules and complex foliations on analytic manifolds, introducing a new way to measure foliation irregularity using derived categories and associated modules.

Contribution

It develops a novel framework linking D-modules to complex foliations and defines irregularity measures through derived Hom functors and associated modules.

Findings

Introduces a natural $ ext{D}_X$-module $ ext{M}_ ext{shi}$ for a Lie subalgebra $ ext{shi}$.

Defines integers measuring foliation irregularity via derived Hom functors.

Establishes a new approach to analyze complex foliations using D-module theory.

Abstract

Consider a complex analytic manifold $X$ and a coherent Lie subalgebra $\shi$ of the Lie algebra of complex vector fields on $X$. By using a natural $\shd_X$-module $\shm_\shi$ naturally associated to $\shi$ and the ring (in the derived sense) $\rhom[\shd_X](\shm_\shi,\shm_\shi)$, we associate integers which measure the irregularity of the foliation associated with $\shi$.