Let $G$ be a complex reductive group. The spherical Hecke category of $G$ can be presented as the category of $G_{\mathcal O}$-equivariant constructible sheaves on the affine Grassmannian $\mathrm{Gr}_G$. This category admits a convolution product, extending the convolution product of equivariant perverse sheaves. In this paper, we upgrade the mentioned convolution product to a left t-exact $\mathbb E_3$-monoidal structure in $\infty$-categories. The construction is intrinsic to the automorphic side. Our main tools are the Beilinson--Drinfeld Grassmannian, Lurie's characterization of $\mathbb E_k$-algebras via the topological Ran space, the homotopy theory of stratified spaces, and the formalism of correspondences.
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.
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.
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.
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.
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.
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…
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.
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.
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.
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…
It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf.
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…
We prove a theorem classifying the equivariant $K$-theoretic pushforwards of the product of arbitrary Schur functors applied to the tautological bundle on the moduli space of framed rank $r$ torsion-free sheaves on $\mathbb{P}^2$, and its dual. This is done by deriving a formula for similar coefficients on Grassmannian varieties, and by thinking of the moduli space as a class in the $K$-theory of the Grassmannian, in analogy with the construction of the Hilbert scheme when the rank is one. Our motivations stem from some vertex operator calculus studied recently by Nekrasov, Okounkov, and the a…
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.
For a reductive group over an algebraically closed field of characteristic $p > 0$ we construct the abelian category of perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod $p$ Satake isomorphism. Along the way we prove that affine Schubert varie…
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.
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 …
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 construct a full exceptional collection of vector bundles in the derived category of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in a symplectic vector space of dimension $2n$ and in the derived category of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in an orthogonal vector space of dimension $2n + 1$ for all $n$.
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.
