Nero Budur
In this mostly expository note we give a down-to-earth introduction to the V-filtration of M. Kashiwara and B. Malgrange on D-modules. We survey some applications to generalized Bernstein-Sato polynomials, multiplier ideals, and monodromy of vanishing cycles.
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继续探索
In this note we prove a conjecture of Kashiwara, which states that the Euler class of a coherent analytic sheaf F on a complex manifold X is the product of the Chern character of F with the Todd class of X. As a corollary, we obtain a functorial proof of the Grothendieck-Riemann-Roch theorem in Hodge cohomology for complex manifolds.
The Legend of Masaki Kashiwara and Algebraic Analysis
本文概述了马崎正彦在D-模、微局分析及相关领域的工作，阐述其在代数分析发展中的贡献。
An introduction to $V$-filtrations
本文介绍了 Malgrange 和 Kashiwara 的 $V$-滤过理论，探讨其基本性质、与 $b$-函数的联系，并展示了其在加权齐次孤立奇点、真前推与对偶相容性及奇点不变量描述中的应用。
We study Fourier transforms of regular holonomic D-modules. By using the theory of Fourier-Sato transforms of enhanced ind-sheaves developed by Kashiwara-Schapira and D'Agnolo-Kashiwara, a formula for their enhanced solution complexes will be obtained. Moreover we show that some parts of their characteristic cycles and irregularities are expressed by the geometries of the original D-modules.
In D-module theory, we have the notion of the restriction of a module along a smooth variety. T. Oaku and N. Takayama have described a process to compute the restriction, which starts from a free resolution adapted to the V-filtration of Kashiwara-Malgrange. By using the theory of free resolutions adapted to the (order and V)-bifiltration (from T. Oaku , N. Takayama and M. Granger), we endow the restriction with an order filtration and we show that the process of Oaku and Takayama respects the order filtration. In consequence, we compute a minimal presentation of the algebraic local cohomology…
We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for $\mathrm{U}(p,q)$ by analyzing a four-dimensional singular subvariety of $\mathbb{C}^8$. We relate this singularity to the Kashiwara-Saito singularity arising for Schubert varieties for ${\mathrm{GL}}(8,\mathbb{C})$ with reducible characteristic cycles, as well as recent related examples of Williamson. In particular, this gives another explanation of the simple highest weight modules for ${\mathfrak{g}}{\mathfrak{l}}(12,\mathbb{C})$ with reducible associated varieties that Williamson discovered.
In this article we are interested in morphisms without slope for mixed Hodge modules. We first show the commutativity of iterated nearby cycles and vanishing cycles applied to a mixed Hodge module in the case of a morphism without slope. Then we define the notion "strictly without slope" for a mixed Hodge module and we show the preservation of this condition under the direct image by a proper morphism. As an application we prove the compatibility of the Hodge filtration and Kashiwara-Malgrange filtrations for some pure Hodge modules with support an hypersurface with quasi-ordinary singularitie…
In this paper, we reprove the Riemann-Hilbert correspondence for regular holonomic D-modules of [M. Kashiwara, Publ. Res. Inst. Math. Sci., 1984] (see also [Z. Mebkhout, Compositio Math., 1984.]) by using the irregular Riemann-Hilbert correspondence of [A. D'Agnolo and M. Kashiwara, Publ. Math. Inst. Hautes Etudes Sci., 2016]. Moreover, we also prove the algebraic one by the same argument. For this purpose, we study C-constructible enhanced ind-sheaves of [Y. Ito, Tsukuba journal of Mathematics, 2020, Rend. Sem. Mat. Univ. Padova., 2021.] in more detail.
For an algebraic vector bundle $E$ over a smooth algebraic variety $X$, a monodromic $D$-module on $E$ is decomposed into a direct sum of some $O$-modules on $X$. We show that the Hodge filtration of a monodromic mixed Hodge module is decomposed with respect to the decomposition of the underlying $D$-module. By using this result, we endow the Fourier-Laplace transform $M^{\wedge}$ of the underlying $D$-module $M$ of a monodromic mixed Hodge module with a mixed Hodge module structure. Moreover, we describe the irregular Hodge filtration on $M^{\wedge}$ concretely and show that it coincides with…
Geometric construction of Kashiwara crystals on multiparameter persistence
本研究构建了多参数持续性模簇不可约分支上Kashiwara晶体的几何构造，以交换关系为核心区别于既有研究，并明确了1、2参数情形下的Kashiwara算子，为相关晶体结构提供具体认知。
We give explicit formulas for the Hodge filtration on mixed Hodge modules associated with certain hypersurfaces.
On the Hodge and V-filtrations of mixed Hodge modules
本文通过Hodge滤波证明了V滤波的Beilinson型公式，并利用该公式研究了更高乘数理想与Hodge理想的关系，改进了相关结果并回答了若干开放问题。
Relaxed Highest Weight Modules from $\mathcal{D}$-Modules on the Kashiwara Flag Scheme
本文将松弛最高权表示推广到任意仿射 Kac-Moody 代数，并在 Kashiwara 旗簇上通过扭曲 D-模的整体截面实现此类模，给出上同调完整描述，推广了 Kashiwara-Tanisaki 的结果。
In this article we describe three constructions of complex variations of Hodge structure, proving the existence of interesting opposite filtrations that generalize a construction of Deligne. We also analyze the relation between deformations of Frobenius modules and certain maximally degenerate variations of Hodge structures. Finally, under a certain generation hypothesis, we show how to construct a Frobenius manifold starting from a deformation of a Frobenius module.
The notions Hodge-Newton decomposition and Hodge-Newton filtration for F-crystals are due to Katz and generalize Messing's result on the existence of the local-étale filtration for p-divisible groups. Recently, some of Katz's classical results have been generalized by Kottwitz to the context of F-crystals with additional structures and by Moonen to $μ$-ordinary p-divisible groups. In this paper, we discuss further generalizations to the situation of crystals in characteristic p and of p-divisible groups with additional structure by endomorphisms.
Restrictions of mixed Hodge modules using generalized V-filtrations
本文利用Sabbah定义的广义V-滤子，通过循环覆盖与通常V-滤子的比较，计算了零截面包含映射的限制函子，并应用于研究完全交子簇的奇点，分类了加权齐次孤立完全交奇点的k-Du Bois和k-理性性质。
On $V$-filtration, Hodge filtration and Fourier transform
本文研究光滑簇闭浸入下$V$-滤过与混合Hodge模Hodge滤过的相互作用，给出相关函子表达式，得到monodromic混合Hodge模滤过结果、Skoda定理的Hodge证明，并研究其Fourier-Laplace变换。
We extend the notion of test module filtration introduced by Blickle for Cartier modules. We then show that this naturally defines a filtration on unit $F$-modules and prove that this filtration coincides with the notion of $V$-filtration introduced by Stadnik in the cases where he proved existence of his filtration. We also show that these filtrations do not coincide in general. Moreover, we show that for a smooth morphism $f: X \to Y$ test modules are preserved under $f^!$. We also give examples to show that this is not the case if $f$ is finite flat and tamely ramified along a smooth diviso…
Note on Relation between Enhanced Ind-Sheaves and Enhanced Subanalytic Sheaves
本文通过阐明增强归纳层与增强亚解析层之间的关系，解释了D'Agnolo–Kashiwara与Kashiwara关于全纯D-模及不规则全纯D-模的Riemann-Hilbert对应定理之间的联系。
We define and study Harder-Narasimhan filtrations on Breuil-Kisin-Fargues modules and related objects relevant to p-adic Hodge theory.
Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties
本文将乘子理想与$V$-滤子的关系推广至奇异不可约簇，引入乘子模与相交复Hodge模，应用于Skoda定理与Ajit公式证明，并定义簇与理想层对的Bernstein-Sato多项式，建立其根与跳跃数联系。
Arithmetic Kashiwara Regularity and Orbit Classification for Filtered Strongly Equivariant $\mathcal{D}^{\dagger}$-Modules
本文证明形式旗流形上过滤强等变 Berthelot 算术 $\mathscr D^\dagger$-模的柏原正则性的算术类比，并通过轨道和等变 $F$-等晶体分类简单 Frobenius 过滤强等变相干算术 $\mathscr D^\dagger$-模。
We show that the Hodge filtration of a tempered Hodge module is generated by the lowest piece of its Hodge filtration. As a consequence, we prove the main conjecture of [SV] in the special case of tempered representations of real reductive Lie groups.
Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N < 6 or p > 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a res…
Kodaira-Saito vanishing for the irregular Hodge filtration
本文借助 Mochizuki 的最新成果修正并改进了非正则混合 Hodge 模范畴的定义，进而为该范畴的非正则 Hodge 滤过导出了 Kodaira-Saito 型消没定理。
We consider mixed Hodge module structures on GKZ-hypergeometric differential systems. We show that the Hodge filtration on these D-modules is given by the order filtration, up to suitable shift. As an application, we prove a conjecture on the existence of non-commutative Hodge structures on the reduced quantum D-module of a nef complete intersection inside a toric variety.
Verdier specialization and restrictions of Hodge modules
本文利用 Verdier 特化与 $V$-过滤给出光滑簇闭浸入下 Hodge 模块上同调拉回函子的显式公式，并通过对余维数归纳提供了该定理的简化证明。
We give a Hodge-theoretic interpretation of the multiplier ideal of an effective divisor on a smooth complex variety. More precisely, we show that the associated graded coherent sheaf with respect to the jumping-number filtration can be recovered from the smallest piece of M. Saito's Hodge filtration of the D-module of vanishing cycles.
An example of the Jantzen filtration of a D-module
本文计算SL₂(ℂ)旗簇上D-模的Jantzen滤过，结合整体截面的𝔰𝔩₂(ℂ)-模结构给出代数图景，验证了Beilinson–Bernstein定理。
Given a mixed Hodge module and a meromorphic function f on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module, which extends the construction of arXiv:1302.4537. We show the strictness of the push-forward filtered D-module through any projective morphism, by using the theory of mixed twistor D-modules of T. Mochizuki. We consider the example of the rescaling of a regular function f, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss-Manin systems of f in ter…
We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis. In the favourable case a basis of the module is parameterized by the lattice points of a normal polytope. The filtrations induce flat degenerations of the corresponding flag variety to its abelianized version and to a toric variety, the specia…
We characterise the symplectic Weyl group elements such that the FFLV basis is compatible with the PBW filtration on symplectic Demazure modules, extending type A results by the second author. Surprisingly, the number of such elements depends not on the type A or C of the Lie algebra but on the rank only.
We develop a Gröbner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight vectors. The approach is tailored for bifiltered D-modules satisfying properties of mixed Hodge modules. As a key ingredient in functors of such modules our theory applies to compute the order filtration on pieces of a V-filtration.
We show that each direct summand of the associated graded module of the test module filtration $τ(M, f^λ)_{λ\geq 0}$ admits a natural Cartier structure. If $λ$ is an $F$-jumping number, then this Cartier structure is nilpotent on $τ(M, f^{λ-\varepsilon})/τ(M, f^λ)$ if and only if the denominator of $λ$ is divisible by $p$. We also show that these Cartier structures coincide with certain Cartier structures that are obtained by considering certain $\mathcal{D}$-modules associated to $M$ that were used to construct Bernstein-Sato polynomials. Moreover, we point out that the zeros of the Bernstein…
Let $R$ be a ring essentially of finite type over an $F$-finite field. Given an ideal $\mathfrak{a}$ and a principal Cartier module $M$ we introduce the notion of a $V$-filtration of $M$ along $\mathfrak{a}$. If $M$ is $F$-regular then this coincides with the test module filtration. We also show that the associated graded induces a functor $Gr^{[0,1]}$ from Cartier crystals to Cartier crystals supported on $V(\mathfrak{a})$. This functor commutes with finite pushforwards for principal ideals and with pullbacks along essentially étale morphisms. We also derive corresponding transformation rules…
Let G be a semi simple linear algebraic group over a field of characteristic zero and let V be a finite dimensional irreducible G-module with highest weight vector v. Let P in G be the parabolic subgroup fixing v and let g=Lie(G). We get a canonical filtration of V by P-modules U^k(g)v where U^k(g) is the filtration of the universal enveloping algebra U(g). This filtration was in a previous paper studied in the case where P in G=SL(E) is the subgroup fixing an m-dimensional subspace. The aim of this paper is to use higher direct images of G-linearized sheaves, filtrations of generalized Verma …
Filtrations on the globalization of twisted D-modules over Dedekind schemes
本文在 Dedekind 概形上证明，由 K-等变闭浸入正像得到的扭曲 D-模整体化具有 K-不变正穷举滤过，且各相伴分次为有限秩局部自由模。
In this paper, we consider $\mathbb{Z}^{r}-$graded modules on the $\mathrm{Cl}(X)$ $-$graded Cox ring $\mathbb{C}[x_{1},\dotsc,x_{r}]$ of a smooth complete toric variety $X$. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive $\mathbb{Z}^{s+r+2}$-graded modules over non-standard bigraded polynomial rings $\mathbb{C}[x_{0},\dotsc,x_{s},y_{0},\dotsc,y_{r}]$. In this case, we give sharp bounds for the multigraded regularity index of their multig…
Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic p. Generalizing the construction of the PBW filtration on Weyl modules for G we construct a G-stable filtration on tensor products of Weyl modules which we call the induced PBW filtration. We use this filtration to give some purely representation-theoretic conditions which are equivalent to the existence of a Frobenius splitting of the double flag variety $G/B \times G/B$ that maximally compatibly splits the diagonal. In particular, this gives a sufficient condition for Wahl's conjecture to hold…
Filtrations of D-modules along semi-invariant functions
研究连通可约线性代数群G作用下光滑代数簇X上的D-模M的G分解，探讨V-、Hodge和权重滤子。通过半不变函数局部化纯twistor D-模，确定不可约同构分量中元素的权重级别与根函数的乘数有关。
Let $(A, \mathfrak{m})$ be a Noetherian local ring and $\mathfrak{F}=(I_{n})_{n\geq 0}$ a filtration. In this paper, we study the Gorenstein properties of the fiber cone $F(\mathfrak{F})$, where $\mathfrak{F}$ is a Hilbert filtration. Suppose that $F(\mathfrak{F})$ and $G(\mathfrak{F})$ are Cohen-Macaulay. If in addition, the associated graded ring $G(\mathfrak{F})$ is Gorenstein; similarly to the $I$-adic case, we obtain a necessary and sufficient condition, in terms of lengths and minimal number of generators of ideals, for Gorensteiness of the fiber cone. Moreover, we find a description of …
Harder-Narasimhan filtrations of persistence modules: metric stability
本文将skyscraper不变量从有限持久模推广到$\mathbb Z^n$和$\mathbb R^n$上的多参数持久模，并建立了该不变量的侵蚀型度量稳定性结果。
Local cohomology with support in Schubert varieties
本文研究广义旗形簇上局部上同调sheaves在闭Schubert簇的支持结构，利用Kazhdan-Lusztig理论和Saito的混合Hodge模理论，通过Kempf引入的Grothendieck-Cousin复形计算简单分解因子、Hodge滤和权重滤。
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