Notes on regular holonomic $D$-modules for algebraic geometers

A corollary of the b-function lemma

Let $X$ be an algebraic variety, $f$ a regular function, $j:U\subset X$ the complement to the locus of vanishing of $f$, and $M$ a holonomic D-module on $U$. Consider the $D_U[s]$-module $M\otimes "f^s"$. The goal of this note is to describe all $D_X[s]$-submodules $N\subset j_*(M\otimes "f^s")$ such that $j^*(N)\simeq M\otimes "f^s"$.

通过实例介绍移位辛几何

Shifted Symplectic Geometry by Examples

本文通过实例向具有扎实同调代数背景的泊松几何学者介绍移位辛几何，涵盖核心方法与主要贡献。

On the Mellin transform of a D-module

Given a holonomic D-module M on a 1-dimensional torus, one can consider its Mellin transform, which is a difference system on the affine line. In this note we prove a formal stationary phase formula, which shows that the formal behavior at infinity of the Mellin transform of M is determined by the local germs defined by M at its singular points.

On involutivity of p-support

The p-support of a holonomic D-module was introduced by Kontsevich. Thomas Bitoun in his PhD thesis proved several properties of p-support conjectured by Kontsevich. In this note we give an alternative proof for involutivity by reducing it to a slight extension of Gabber's theorem on the integrability of the characteristic variety. For the benefit of the reader we review how this extension follows from Kaledin's proof of Gabber's theorem.

On partial differential operators which annihilate the roots of the universal equation of degree k

The aim of this paper is to study in details the regular holonomic $D-$module introduced in \cite{[B.19]} whose local solutions outside the polar hyper-surface $\{Δ(σ).σ_k = 0 \}$ are given by the local system generated by the local branches of the multivalued function which is the root of the universal degree $k$ equation $z^k + \sum_{h=1}^k (-1)^h.σ_h.z^{k-h} = 0 $. Note that it is surprising that this regular holonomic $D-$module is given by the quotient of $D$ by a left ideal which has very simple explicit generators despite the fact it necessary encodes the analogous systems for any root …

指数层的实现与傅里叶变换

Realizations of exponential sheaves and Fourier transform

本文构造了从指数层的相对范畴到全纯（不一定正则）D-模的实现函子，并证明了它们在心上的t-正合性和忠实性，同时发展了与经典傅里叶变换交换的“通用”傅里叶变换，并证明其保持纯度。

关于一类 G-表示的射影空间上 D-模的一个注记

A note on D-modules on the projective spaces of a class of G-representations

本文研究满足抽象 Capelli 条件的不可约无重数 $G$-表示，证明其射影空间上正则完整 D-模与一类分次 $\mathcal{A}$-模商范畴等价，并用于分类反对称矩阵情形。