We construct a co-$t$-structure on the derived category of coherent sheaves on the nilpotent cone $\mathcal{N}$ of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the "exotic parity objects" along the (classical) Springer resolution give indecomposable objects inside the coheart of the co-$t$-structure on $\mathcal{N}$. We also demonstrate how the various parabolic co-$t$-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type $A$ for $p>h$.
We show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.
In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.
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 tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.
We introduce Morse branes in the Fukaya category of a holomorphic symplectic manifold, with the goal of constructing tilting objects in the category. We give a construction of a class of Morse branes in the cotangent bundles, and apply it to give the holomorphic branes that represent the big tilting sheaves on flag varieties.
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.
Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good chara…
In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and …
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).
Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. The goal of this article is to generalize these results to positive characteristics.
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…
This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type $(2, 2, n)$, and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.
It is a well established fact that the notions of quasi-abelian categories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of $t$-structures. Firstly, we extend this picture into a hierarchy of $n$-quasi-abelian categories and $n$-tilting torsion classes. We prove that any $n$-quasi-abelian category admits a derived category endowed with a $n$-tilting pair of $t$-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these $t$-structures as quotient categories of coherent functors, genera…
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…
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.
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and each indecomposable tilting vector bundle $T$ on $C$ a certain complex $S(C,T)$ of $G \times \mathbb{G}_m$-equivariant coherent sheaves on $\mathcal{N}$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-$t$-structure. We then show that if $p$ is larger than the Coxeter number, then the hypercoho…
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.
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…
Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.
We introduce a class of orders on $P^d$ called Geigle-Lenzing orders and show that they have tilting bundles. Moreover we show that their module categories are equivalent to the categories of coherent sheaves on Geigle-Lenzing spaces introduced in Herschend, Iyama, Minamoto and Oppermann.
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close connection of this theory to approximation theory and cotorsion pairs. We also discuss the motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme.
Given certain intersection cohomology sheaves on a projective variety with a torus action, we relate the cohomology groups of their tensor product to the cohomology groups of the individual sheaves. We also prove a similar result in the case of equivariant cohomology.
Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.
We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot-action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Li…
We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is ju…
Let $\mathcal{H}$ be a connected hereditary abelian category with tilting objects. It is proved that the cluster-tilting graph associated with $\mathcal{H}$ is always connected. As a consequence, we establish the connectedness of the tilting graph for the category $\operatorname{coh}\mathbb{X}$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of wild type. The connectedness of tilting graphs for such categories was conjectured by Happel-Unger, which has immediately applications in cluster algebras. For instance, we deduce that there is a bijection between the set of indecomposa…
We introduce, on a topological space X, a class of stacks of abelian categories we call "stacks of type P." This class of stacks includes the stack of perverse sheaves (of any perversity, constructible with respect to a fixed stratification), and is singled out by fairly innocuous axioms. We show that some basic structure theory for perverse sheaves holds for a general stack of type P: such a stack is locally equivalent to a MacPherson-Vilonen construction, and under certain connectedness conditions its category of global objects is equivalent to the category of modules over a finite-dimension…
These notes aim to give a first introduction to intersection cohomology and perverse sheaves with applications to representation theory or quantum groups in mind.
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a…
We construct geometric realizations of the r-colored bosonic and fermionic Fock space on the equivariant cohomology of the moduli space of framed rank r torsion-free sheaves on CP^2. Using these constructions, we realize geometrically all level one irreducible representations of the affine Lie algebra \hat{gl(r)}. The cyclic group Z_k acts naturally on the moduli space of sheaves, and the fixed-point components of this action are cyclic Nakajima quiver varieties . We realize level k irreducible representations of \hat{gl(r)} on the equivariant cohomology of these quiver varieties.
In this paper, we carry out several computations involving graded (or $\mathbb{G}_{\mathrm{m}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the $\mathbb{G}_{\mathrm{m}}$-action on certain normalized (or "canonical") simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the coho…
We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on the complex projective plane. The top non-vanishing Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators.
These notes provide a description of the abelian categories that arise as categories of coherent sheaves on weighted projective lines. Two different approaches are presented: one is based on a list of axioms and the other yields a description in terms of expansions of abelian categories. A weighted projective line is obtained from a projective line by inserting finitely many weights. So we describe the category of coherent sheaves on a projective line in some detail, and the insertion of weights amounts to adding simple objects. We call this process `expansion' and treat it axiomatically. Thus…
We consider a hyperplane arrangement in $\mathbb{C}^n$ defined over $\mathbb{R}$, and the associated natural stratification of $\mathbb{C}^n$. The category of perverse sheaves smooth with respect to this stratification was described by Kapranov and Schechtman in terms of quiver representations. Using work of Weissman, we reinterpret this category as the category of finite-dimensional modules over an explicit algebra. We also describe recollement (open-closed decomposition) of perverse sheaves in terms of this module category. As an application, we identify the modules associated to all interse…
