# Tagged Questions

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### Nearby cycles and specialisation - properties

I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
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### Sabbah b-functions factoring

Sabbah defined a version of b-functions for multiple functions in his 1987 paper here: http://archive.numdam.org/ARCHIVE/CM/CM_1987__62_3/CM_1987__62_3_283_0/CM_1987__62_3_283_0.pdf According to ...
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### Smooth mixed hodge modules - representations of fundamental group?

I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...
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### Algebraic methods in pde [duplicate]

I'm finding myself with a linear, very symetric system of first orders pde with polynomial coefficients. Wandering on the web, i learnt there is some nice alebraic way to deal with it involving ...
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### Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
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### Most general “finiteness of de Rham cohomology” statement for holonomic $D$-modules in the algebraic case?

Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) ...
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Consider an holonomic D-module on a smooth algebraic variety $X$ over a field $k$ of caracteristic 0. Let $i: Y \hookrightarrow X$ be a regular embedding. $Li^* M = \mathcal{D}_{Y\to X} ... 0answers 176 views ### singular support of D-module smooth w.r.t. a stratification (1) Suppose that$X$is a smooth complex algebraic variety, stratified by some nice smooth stratification$S$. Let$M$be a$D$-module on$X$, s.t. its shriek-pullback (or star... whatever is ... 2answers 287 views ### About “de-Rham” and “l-adic” local systems - comparison Hello, Suppose that$k$is an algebraically closed field of char. 0. Let$X$be a smooth connected variety over$k$. Then I have the category$A$of Regular Singular smooth$D$-modules on$X$(i.e. ... 1answer 240 views ### D-affine morphisms and composition I am just a beginner in$D$-module, so this could be a stupid question, but I can't find an easy reference for it. I would like to define the notion of$D$-affine morphism. The most obvious way would ... 0answers 221 views ### Weight decomposition and eigenspaces Euler vector field Let$V$be a finite dimensional vector space over$\mathbb{C}$, denote$V^{\times} = V -0$and let$\pi : V^{\times} \rightarrow \mathbb{P}(V)$be the quotient by the natural action of ... 1answer 437 views ### Analogues of D-modules and constructible sheaves For a smooth complex variety, one can consider the category of say holonomic$\mathcal D$modules on it. It is equipped with the deRham functor, which turns a$\cal D$-module into a constructible ... 2answers 337 views ### D-module that is coherent as O-module Suppose that$X$is an algebraic variety over$\mathbb C$, not necessarily smooth. Is it still true that each$\mathcal D_X$-module ($\mathcal D_X$is of course the sheaf of differential operators) ... 0answers 166 views ### Non-characteristic is to pullback as (blank) is to pushforward. Suppose$f:X\to Y$is a map of smooth complex algebraic varieties. There is a pushforward functor$f_\ast : D(X) \to D(Y)$on the derived category of$D$-modules. This certainly does not preserve ... 1answer 285 views ### Tensor product of$\mathcal{D}$-modules and constructible sheaves The Riemann-Hilbert correspondence, as proved by Kashiwara and Mebkhout, says that for X a smooth algebraic variety over$\mathbb{C}$there is an equivalence of triangulated categories ... 0answers 358 views ### An equivalence between$(\mathcal{D}_X^m)-\text{mod}$and$(\mathcal{D}_X^{m+1})-\text{mod}$This question is related to my other question. Consider a scheme$X$over$S=\text{Spec}(\mathbb{k})$where$\mathbb{k}=\overline{\mathbb{F}_p})$; let$F: X \rightarrow X$be the Frobenius$p$-th ... 1answer 605 views ###$\mathcal{D}$-modules of level m My question is regarding the definition of$\mathcal{D}$-modules of level$m$given in this paper. As an example, let$X=\mathbb{A}^1$over$S=\text{Spec }\overline{\mathbb{F}_p}$; I was told that a ... 0answers 210 views ###$D$-modules and quasi-projective varieties? Until recently I was under the impression, that for any morphism$f:X\rightarrow Y$of smooth complex varieties there exist functors six functors$f^*,f_{*},...$between the derived categories of ... 1answer 599 views ###$\mathcal{D}$-quasi-isomorphisms and coherent$\Omega$-modules Let$X$be a smooth$\mathbf{C}$-scheme of finite type. In section 7.2 of their preprint "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld define an ... 1answer 223 views ### Distinguished triangle of closed - open partition, for D-modules Hello, I am sorry if this question is not appropriate for MO. Suppose$X$is the affine line,$i:Z\to X$is the origin, and$j: U \to X$is the complement to$Z$in$X$. I then have a distinguished ... 0answers 164 views ### Computing the constant D-module on an intersection Let$M$be a smooth variety, say over the complex numbers, and let$i:W \hookrightarrow M, j: Z \hookrightarrow M$be smooth subvarieties. Let$i_{+},j_{+}$denote (derived) pushforward of D-modules, ... 1answer 427 views ### Localizability of differential operators a la Grothendieck Hello, Maybe this question is trivial, so sorry Let$A$be a (comm. with 1)$k$-algebra, where$k$is a ring (comm. with 1). Then we can define the module of differential operators$D^{\leq n} ...
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What would be an l-adic analogue of the Turrittin-Levelt decomposition theorem? Turrittin-Levelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a ...
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### Is base affine space a trivial fibration?

I am very confused about the base affine space. Let $G$ be a complex reductive algebraic group and $H=B/U$ the abstract torus. If I understand it correctly (Edit: which turns out not to be the case! ...
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### Examples for D-affine varieties?

A variety is called $\mathcal D$- affine if the global section functor induces an equivalence between quasi-coherent $\mathcal D$-modules and modules over $\Gamma(X,\mathcal D)$. It is easy to see ...
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### D-modules and Algebraic Solutions of PDEs

I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When ...
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### Base change for the Gauss-Manin sheaf

I want to see the following thing: $\ \$If $X$ is a smooth geometrically connected scheme over a field $k$ of characteristic 0, $U\subseteq X$ is a non-empty open, $(E,\nabla)$ is an integrable ...
Here arXiv:math/0510287, Golishev proposed the following conjecture: The hypergeometric pullback conjecture. Let $X$ be a Fano variety. Then,for any constituent $C$ of the quantum D-module $Q$ there ...