Pierre Schapira
We introduce the category of b-analytic manifolds, a natural tool to define constructible sheaves and functions up to infinity. We study with some details the operations on these objects and also recall the Radon transform for constructible functions.
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Categories of Constructible Sheaves
本文研究了可构造sheaf范畴的等价性问题，并探讨了局部常数sheaf和可构造sheaf的基本性质。
Constructible sheaves on schemes
本文在任意概形上以拓扑环或凝聚环为系数建立可构造层的统一理论，将光滑层定义为预平展导出无穷范畴中的可对偶对象，证明其满足预平展下降并构造 t-结构，同时提供系数环操作工具以统一并推广文献中的可构造 ℓ-adic 层理论。
This article contains a proof of the basic lemma. This lemma, discovered by Beilinson, yields a motivic proof of the Andreotti-Frankel theorem for affine varieties. Next, it is shown that the category of Cohomologically Constructible Sheaves (as it is referred to in the Riemann-Hilbert correspondence) coincides with the derived category of bounded complexes of constructible sheaves. It is also shown that higher direct images and the sheaf-Ext groups are effaceable in the category of constructible sheaves.
Good moduli space for constructible sheaves and Stokes functors
本文为可构造层与Stokes函子构造良好模空间，并考虑其导出增强。
Unusual functorialities for weakly constructible sheaves
该研究针对层上六个Grothendieck运算相关态射的性质问题，通过分析弱上同调可构造层的特性，证明了这类态射在限制于弱可构造层时为同构，并推导了其在弱$\boldsymbol{\rm R}$-可构造层下的运算相容性，推进了代数几何中层论的相关研究。
We explain a method for calculating the cohomology of line bundles on a toric variety in terms of the cohomology of certain constructible sheaves on the polytope. We show its effective use by means of some examples.
The derived category of the abelian category of constructible sheaves
本文证明了代数簇上有界可构造复形三角范畴与可构造层阿贝尔范畴的有界导出范畴等价，并将 Nori 定理推广至正特征。
By using sheaf-theoretical methods such as constructible sheaves, we generalize the formula of Libgober-Sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. In particular, some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained.
Holonomic étale sheaves are constructible
基于贝林森工作，引入平展层全纯性概念，在完美基域上建立贝林森结果逆命题，证全纯层可构造，是柏原定理平展类似物。
We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for constructible sheaves and a relative vanishing theorem for a homomorphism between abelian varieties. Our proof relies on a Tannakian description for convolution products of perverse sheaves, and with future applications in mind we discuss the basic properties of the arising Tannaka groups.
A. D'Agnolo and M. Kashiwara proved that their enhanced solution functor induces a fully faithful embedding of the triangulated category of holonomic D-modules into the one of R-constructible enhanced ind-sheaves. In this paper, we define C-constructible enhanced ind-sheaves and show that the triangulated category of them is equivalent to its essential image. Moreover we show that there exists a t-structure on it whose heart is equivalent to the abelian category of holonomic D-modules.
A Microlocal Theory for Zariski-Constructible Sheaves on Rigid Analytic Varieties
本文在 Kashiwara–Schapira 框架下发展刚性解析簇上 Zariski 可构造层的微局部理论，引入单值层、Fourier 变换、特化、微局部化、micro-hom 与奇异支集等工具。
Constructible sheaves on toric varieties
本文针对环面簇上的可构造层范畴给出显式计算，既简化了复数域上Braden与Lunts的相关描述，又首次在正特征下对任意域分裂环面簇的驯分歧可构造平展层完成了平展外定理的显式计算。
Characteristic cycles of constructible sheaves and microlocalization
研究正特征域上光滑概型可构造层的奇异支撑和特征圈，探讨类似超越情形的微局部化构造在代数情形的可行性，取得部分肯定成果。
On relative constructible sheaves and integral transforms
本文刻画了射影参数空间上的相对可构造层，研究了相对 Fourier-Mukai 变换诱导的范畴等价，并引入相对 Euler-Poincaré 指标，证明其给出 Grothendieck 群与相对可构函数环的同构。
Vanishing Cycles for Zariski-Constructible Sheaves on Rigid Analytic Varieties
本文在非阿基米德域上建立了Zariski可构造层的近环与消失环理论，证明其保持可构造性、具有Milnor纤维解释、满足Beilinson粘合、perverse t-精确且与Verdier对偶交换。
The convolution algebra of constructible sheaves
研究可构造sheaves的卷积乘法下的可逆对象，证明可逆sheaf的逆为其反向变换的对偶，并探讨支撑凸的紧支撑常sheaf的可逆性条件。
We show that the characteristic cycle of the exterior product of constructible complexes is the exterior product of the characteristic cycles of factors. This implies the compatibility of characteristic cycles with smooth pull-back which is a first step in the proof of the index formula.
We give bounds for the cohomology of constructible sheaves on the moduli stacks M_{g,n} over the complex field. This enables us recover Harer's bound for the virtual cohomological dimension of the associated mapping class groups as well the theorem of Diaz on complete subvarieties of M_g. We also obtain such bounds for any open subset of the Deligne-Mumford compactification of M_{g,n} that is a union of strata. Our proof yields a template for obtaining similar bounds for the cohomological dimension for quasicoherent sheaves on M_{g,n}.
We show that for any constructible sheaf F on a smooth algebraic variety X over a field of arbitrary characteristic its singular support SS(F) is equidimensional of dimension dim X. Here SS(F) is the minimal closed subset of the cotangent bundle of X such that every (local) function on X with df(X) disjoint from SS(F) is locally acyclic relative to F.
We obtain a linear algebra data presentation of the category of constructible with respect to perverse triangulation sheaves on a finite simplicial complex. We also establish Koszul duality between the above mentioned category and the category of perverse sheaves constructible with respect to the triangulation
For an irreducible subvariety Z in an algebraic group G we define a nonnegative integer gdeg(Z) as the degree, in a certain sense, of the Gauss map of Z. It can be regarded as a substitution for the intersection index of the conormal bundle to Z with the zero section of T^*G, even though G may be non-compact. For G a semiabelian variety (in particular, an algebraic torus (C^*)^n) we prove a Riemann-Roch-type formula for constructible sheaves on G, which involves our substitutions for the intersection indices. As a corollary, we get that a perverse sheaf on such a G has nonnegative Euler charac…
Characteristic cycles of real and complex constructible sheaves, revisited
本文重新定义了实与复构造sheaf的相对特征环，并推导了实近邻sheaf的特征环公式，统一处理了多种构造sheaf的特征环计算。
A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian
本文将复简约群球面Hecke范畴的卷积乘积提升为∞范畴中的左t正合E₃-幺半结构，利用Beilinson–Drinfeld格拉斯曼流形等工具完成内蕴自守侧的构造。
Local Monodromy of Constructible Sheaves
研究复代数簇间态射下可构造层的局部单值性，推广了拟幂单层情形下的经典结果，并应用于阿贝尔覆盖和广义Alexander模。
We prove an analog of the Gauss-Bonnet formula for constructible sheaves on reductive groups. As a corollary from this formula we get that if a perverse sheaf on a reductive group is equivariant under the adjoint action, then its Euler characteristic is nonnegative.
Cartier crystals and dual-constructible $p$-torsion complexes
本文证明在Noetherian且$F$-有限概形上，Cartier晶体的不动点函子全忠实嵌入Kato定义的平坦相对完美景的层范畴，其本质像为对偶可构造层。
Given a smooth projective toric variety $X_Σ$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_Σ)$ into the dg derived category of constructible sheaves on a torus $Sh(T^n, Λ_Σ)$. Recently, Kuwagaki \cite{Ku2} proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent …
We prove a generic smoothness result in rigid analytic geometry over a characteristic zero nonarchimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well-adapted to "spreading out" arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.
In this note we relate three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other papers of the authors ([JS], [Sa], [SaH] in the list of references).
We prove that intermediate extensions of perverse constructible $\mathbb{F}_p$-sheaves commute with smooth pullbacks for schemes admitting a closed embedding into a smooth scheme over a field of characteristic $p$ (embeddable schemes for short). Along the way we also prove that the equivalence of categories of Cartier crystals with unit $R[F]$-modules commutes with $f^!$ for a smooth morphism $f: X \to Y$ of embeddable schemes.
Morelli's computation of the K-theory of a toric variety X associates a polyhedrally constructible function on a real vector space to every equivariant vector bundle E on X. The coherent-constructible correspondence lifts Morelli's constructible function to a complex of constructible sheaves kappa(E). We show that certain filtrations of the cohomology of kappa(E) coming from Morse theory coincide with the Klyachko filtrations of the generic stalk of E. We give Morse-theoretic (i.e. microlocal) conditions for a complex of constructible sheaves to correspond to a vector bundle, and to a nef vect…
Lagrangian structures on the derived moduli of constructible sheaves
本文研究了紧致定向流形上 $\mathcal{D}(k)$-valued 构造sheaves 和 perverse sheaves 的模形，证明其具有 $(2-n)$-移位的拉格朗日结构，并通过立方体的松散粘合结果实现。
We give a simple proof for the rigidity of a complex in the bounded derived category of sheaves with constructible cohomology on an abelian variety.
We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible étale sheaf around a point is determined by wild ramification at the point. The key ingredient is to prove that wild ramification of the stalk of the nearby cycle complex of a constructible étale s…
Computing the cohomology of constructible étale sheaves on curves
本文提出了一种计算不可约曲线上的构造层上同调复形的算法，该算法仅涉及有限群，并研究了其复杂性。
Additivity of non-acyclicity classes for constructible étale sheaves
本文利用双变上同调对应为非反圈类建立范畴迹类公式，并以此证明其可加性。
We explore the relation between the positive dimensional irreducible components of the characteristic varieties of rank one local systems on a smooth surface and the associated (rational or irrational) pencils. Our study, which may viewed as a continuation of D. Arapura's work, yields new geometric insight into the translated components relating them to the multiplicities of curves in the associated pencil, in a close analogy to the compact situation treated by A. Beauville. The new point of view is the key role played by the constructible sheaves naturally arising from local systems.
A categorical Künneth formula for constructible Weil sheaves
本文证明了特征 $p>0$ 概型上 lisse 与可构造 Weil 层导出范畴的 Künneth 型等价，涵盖多种系数，并提出了适用于全局函数域上 shtuka 模叠上同调的 ind-可构造层变体。
In this paper we develop a theory of Grothendieck's six operations for adic constructible sheaves on Artin stacks continuing the study of the finite coefficients case in math.AG/0512097.
Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of real vector spaces that play well with sheaves having microsupport in a given cone or, equivalently, sheaves in the corresponding conic topology.
Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone
本文证明几何分次 Hecke 代数可实现为幂零锥上等变半单纯层复形的自同态代数，并将相应等变可构造层导出范畴等价描述为分次 Hecke 代数直和上的 DG 模导出范畴，从而给出其范畴化。
For an $F$-finite scheme $X$ separated over a perfect field $k$ of characteristic $p>0$ which admits an embedding into a smooth $k$-scheme, we establish an equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb{Z}/p\mathbb{Z}$-sheaves on the étale site $X_{\text{ét}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal{O}_{F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by usi…
The coherent-constructible correspondence is a relationship between coherent sheaves on a toric variety X, and constructible sheaves on a real torus T. This was discovered by Bondal, and explored in the equivariant setting by Fang, Liu, Treumann and Zaslow. In this paper we collect partial results towards a proof of the non equivariant coherent-constructible correspondence. Also, we give applications to the construction of tilting complexes in the derived category of toric DM stacks.
Under some assumptions we prove that the Deligne tensor product of categories of constructible perverse sheaves on pseudomanifolds X and Y is the category of constructible perverse sheaves on X\times Y. The Deligne external tensor product functor is identified with the geometrical external tensor product.
In this paper we introduce constructible analogs of the discrete complexity classes $\mathbf{VP}$ and $\mathbf{VNP}$ of sequences of functions. The functions in the new definitions are constructible functions on $\mathbb{R}^n$ or $\mathbb{C}^n$. We define a class of sequences of constructible functions that play a role analogous to that of $\mathbf{VP}$ in the more classical theory. The class analogous to $\mathbf{VNP}$ is defined using Euler integration. We discuss several examples, develop a theory of completeness, and pose a conjecture analogous to the $\mathbf{VP}$ vs. $\mathbf{VNP}$ conje…
Let S be a Noetherian scheme and f:X -> S a proper morphism. By SGA 4 XIV, for any constructible sheaf F of Z/nZ-modules on X, the sheaves of Z/nZ-modules R^if_*F obtained by direct image (for the etale topology) are also constructible: there is a stratification of S on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which S is generically of characteristic zero or the sheaves F are constant (with invertible torsion on S), here we study the dependency of the stratification on F. We show that a na…
We prove that the length function for perverse sheaves and algebraic regular holonomic D-modules on a smooth complex algebraic variety Y is an absolute Q-constructible function. One consequence is: for "any" fixed natural (derived) functor F between constructible complexes or perverse sheaves on two smooth varieties X and Y, the loci of rank one local systems L on X whose image F(L) has prescribed length are Zariski constructible subsets defined over Q, obtained from finitely many torsion-translated complex affine algebraic subtori of the moduli of rank one local systems via a finite sequence …
We give a complete quiver description of the category of perverse sheaves on Hermitian symmetric spaces in types A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety.
This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the étale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or noda…
Let $X$ be a real analytic manifold endowed with a distance satisfying suitable properties and let $\mathbf{k}$ be a field. In [PS20], the authors construct a pseudo-distance on the derived category of sheaves of $\mathbf{k}$-modules on $X$, generalizing a previous construction of [KS18]. Here, we prove that if the distance between two constructible sheaves with compact support (or more generally, constructible sheaves up to infinity) on $X$ is zero, then these two sheaves are isomorphic. This answers in particular a question of [KS18].
We prove the following result of Bondal's: that there is a fully faithful embedding $κ$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $κ$ is an open problem.
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of…
In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks which are locally of finite type over suitable regular basis of dimension at most 1.
In arXiv:math/0311139, as evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In arXiv:0911.4711, we showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on R^n. In this paper we investigate and reprove some of Kawamata's results from this perspective.
We introduce irregular constructible sheaves, which are $\mathbb{C}$-constructible with coefficients in a finite version of Novikov ring $Λ$ 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 generalizatio…
For a pair of affine toric varieties X and Y defined by dual cones, we define an equivalence between two triangulated categories. The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel. A similar duality was constructed in math.AG/0308216; this new approach is more canonical.
We present a detailed introduction of the theory of constructible sheaf complexes in the complex algebraic and analytic setting. All concepts are illustrated by many interesting examples and relevant applications, while some important results are presented with complete proofs. This paper is intended as a broadly accessible user's guide to these topics, providing the readers with a taste of the subject, reflected by concrete examples and applications that motivate the general theory. We discuss the stability of constructible sheaf complexes under the standard functors, and explain the relation…
We extend our previous work arXiv:1007.0053 on coherent-constructible correspondence for toric varieties to include toric Deligne-Mumford (DM) stacks. Following Borisov-Chen-Smith, a toric DM stack $\cX_\bSi$ is described by a "stacky fan" $\bSi=(N,\Si,β)$, where $N$ is a finitely generated abelian group and $\Si$ is a simplicial fan in $N_\bR=N\otimes_{\bZ}\bR$. From $\bSi$ we define a conical Lagrangian $Λ_\bSi$ inside the cotangent $T^*M_\bR$ of the dual vector space $M_\bR$ of $N_\bR$, such that torus-equivariant, coherent sheaves on $\cX_\bSi$ are equivalent to constructible sheaves on $M…
Rouquier dimension is Krull dimension for normal toric varieties
本文借助凝聚—可构造对应，证明任意正规环面簇的凝聚层有界导出范畴的 Rouquier 维数与其 Krull 维数相等。
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