For a certain class of real analytic varieties with Lie group actions we develop a theory of (free-monodromic) tilting sheaves, and apply it to flag varieties stratified by real group orbits. For quasi-split real groups, we construct a fully faithful embedding of the category of tilting sheaves to a real analog of the category of Soergel bimodules, establishing real group analogs of Soergel's Structure Theorem and Endomorphism Theorem. We apply these results to give a purely geometric proof of the theorem of Bezrukavnikov and Vilonen which proves Soergel's conjecture for quasi-split groups.
In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.
We are going to show that the sheafication of graded Koszul modules $% K_Γ$ over $Γ_{n}=K[ x_{0},x_{1}...x_{n}] $ form an important subcategory $\overset{\wedge}{K}_Γ$ of the coherents sheaves on projective space, $Coh(P^{n}).$ One reason is that any coherent sheave over $P^{n}$ belongs to $\overset{\wedge}{K}_Γ$up to shift. More importantly, the category $K_Γ$ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over $Γ$ and over the exterior algebra $Λ.$ This is then used to develop a kind of relative Auslander-Reiten theory for the ca…
We investigate the behaviour of tilting sheaves under pushforward by a finite Galois morphism. We determine conditions under which such a pushforward of a tilting sheaf is a tilting sheaf. We then produce some examples of Severi Brauer flag varieties and arithmetic toric varieties in which our method produces a tilting sheaf, adding to the list of positive results in the literature. We also produce some counterexamples to show that such a pushfoward need not be a tilting sheaf.
We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called "non-cancellation property" which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their weights effectively. When we have a certain Radon transform, we prove a geometric analogue of Ringel duality which sends tilting objects to projective objects. We apply these methods to (partial) flag varieties and affine (partial) flag varieties and show that the weight polynomials of mixed tilting sheaves on flag and affine flag varieties are essentially give…
Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves is a hereditary, locally noetherian Grothendieck category. We classify all tilting sheaves which have a non-coherent torsion subsheaf. In case of nonnegative orbifold Euler characteristic we classify all large (that is, non-coherent) tilting sheaves and the corresponding resolving classes. In particular we show that in the elliptic and in the tubular cases every large tilting sheaf has a well-defined slope.
A new section on projections of coherent sheaves from a projective space to a lower-dimensional projective space has been added. Also some of the notation has been altered to bring it into line with the joint paper with Eisenbud and Schreyer.
The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. We relate it to a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of these cohomology is described in terms of cells in affine Weyl groups, and the basis in the Grothendieck group they provide is related to the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik. The proof is based on earlier res…
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
Let $X$ be a finite connected simplicial complex, and let $δ$ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices ($δ$-sheaves). We interpret the categories (1) and (2) as categories of modules over certain quadratic (and even Koszul) algebras $A(X,δ)$ and $B(X,δ)$ respectively, and we prove that $A(X,δ)$ and $B(X,δ)$ are Koszul dual to each other. We define the $δ$-perverse topo…
For a stratified topological space we introduce the category of IC-modules, which are linear algebra devices with the relations described by the equation d^2=0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties. As an application, we describe an algorithm calculating the quiver underlying the BGG category O for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand.
Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$. The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even…
A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).
Recently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open GIT substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity.
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.
This is an application of the theory of tilting objects to the geometric setting of perverse sheaves. We show that this theory is a natural framework for Beilinson's gluing of perverse sheaves construction. In the special case of Schubert stratification of a flag variety we get a short proof of Soergel's "Struktursatz", and describe (following a conjecture of Kapranov) Serre functor for category O. Some of our results were obtained independently by Rouquier.
We show that certain categories of perverse sheaves on a pair of affine toric varieties defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel. The functor expressing this duality is constructed explicitly using a combinatorial model for mixed sheaves on toric varieties.
In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded d…
For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $B^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee/U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and certain "Whittaker model" category of mixed complexes on $G^\vee/B^\vee$. In …
We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.
A fundamental result of Beilinson-Ginzburg-Soergel states that on flag varieties and related spaces, a certain modified version of the category of l-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobe…
This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the…
We show that, with coefficients in a field or a complete local ring k, the Braden-MacPherson algorithm computes the stalks of parity sheaves with coefficients in k. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the p-smooth locus of (Kac-Moody) Schubert varieties agrees with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.
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.
The $\otimes^\star$-monoidal structure on the category of sheaves on the $\mathrm{Ran}$ space is not pro-nilpotent in the sense of Francis-Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the $\mathrm{Ran}$ space and integrating along the $\mathrm{Ran}$ space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah…
We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are Morita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) - c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3 surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non empty, is a proper algebraic space.
Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain r…
Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\rm H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\cite{BMR…
We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative two-torus with real multiplication. These results are based on the techniques of math.QA/0211262 allowing to interpret all the data in terms of autoequivalences of the deriv…
We apply Treumann's "Smith theory for sheaves" in the context of the Iwahori--Whittaker model of the Satake category. We deduce two results in the representation theory of reductive algebraic groups over fields of positive characteristic: (a) a geometric proof of the linkage principle; (b) a character formula for tilting modules in terms of the $\ell$-canonical basis, valid in all blocks and in all characteristics.
We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group G (equivalently, the category of Harish Chandra bimodules of trivial central character) and its monodromic variant. The Hecke category is a categorified analogue of the finite Hecke algebra, which is a finite-dimensional semi-simple symmetric Frobenius algebra. We establish parallel properties of the Hecke category, showing it is a two-dualizable Calabi-Yau …
In the setting of a variety $X$ admitting a tilting bundle $T$ we consider the problem of constructing $X$ as a quiver GIT quotient of the endomorphism algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting bundle. We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of $X$ and the closed points of a quiver GIT moduli functor for $A$ then $X$ is indeed a fine moduli space for this quiver GIT moduli functor, and we prove this result without any assumptions on the singularities of $X$. As an application we consider varieties which are…
We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.
A localization theorem for the cyclotomic rational Cherednik algebra $H_c=H_c((\mathbb{Z}/l)^n\rtimes \mathfrak{S}_n)$ over a field of positive characteristic has been proved by Bezrukavnikov, Finkelberg and Ginzburg. Localizations with different parameters give different $t$-structures on the derived category of coherent sheaves on the Hilbert scheme of points on a surface. In this short note, we concentrate on the comparison between different $t$-structures coming from different localizations. When $n=2$, we show an explicit construction of tilting bundles that generates these $t$-structures…
We propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X. Among the objects of F_X^m, in addition to the Tate motives Z/m(j), there are the cohomological relative motives with compact support M_cc^m(Y/X) of varieties Y quasi-finite over X. Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties Y\to …
We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra…
Let $\text{X}$ denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands' philosophy, postulates that the equivariant derived category of bounded complexes with constructible equivariant cohomology sheaves on $\text{X}$ is equivalent to a full subcategory of the derived category of modules over a graded ring defined as a suitable graded $Ext$. Only special cases of this conjecture have been proven so far. {\it The purpose of this paper is…
