Category Of Equivariant Coherent Sheaves Research Articles

Equivariant Hodge theory and noncommutative geometry

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models.

Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

The first author is supported in part by the NSF grant DMS-1400390 and by the Russian Academic Excellence Project '5-100'. The second author is a senior researcher at the FWO and was supported by the FWO grant 1503512N.

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

Let G be a connected reductive group over an algebraically closed field 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 G, and Iwahori-constructible 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 characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence.

Reductive groups, the loop Grassmannian, and the Springer resolution

In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a graded version of a conjecture of Finkelberg-Mirkovic describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian, and deduce a new proof of Lusztig's conjecture in large characteristic.

Equivalences of derived factorization categories of gauged Landau–Ginzburg models

For a given Fourier–Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier–Mukai equivalences of derived factorization categories of gauged Landau–Ginzburg (LG) models.As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier–Mukai equivalence is linearizable with respect to a reductive affine algebraic group action, then the derived categories of equivariant coherent sheaves on the varieties are equivalent. This result is shown by Ploog for finite groups case.

On the Exotic t-Structure in Positive Characteristic

In this paper we study Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of a connected reductive algebraic group defined over a field of positive characteristic with simply-connected derived subgroup. In particular, we show that the heart of the exotic t-structure is a graded highest weight category, and we study the tilting objects in this heart. Our main tool is the geometric braid group action studied by Bezrukavnikov and the second author.

Exact results for boundaries and domain walls in 2d supersymmetric theories

We apply supersymmetric localization to $$ \mathcal{N}=\left(2,2\right) $$ gauged linear sigma models on a hemisphere, with boundary conditions, i.e., D-branes, preserving B-type supersymmetries. We explain how to compute the hemisphere partition function for each object in the derived category of equivariant coherent sheaves, and argue that it depends only on its K theory class. The hemisphere partition function computes exactly the central charge of the D-brane, completing the well-known formula obtained by an anomaly inflow argument. We also formulate supersymmetric domain walls as D-branes in the product of two theories. In particular 4d line operators bound to a surface operator, corresponding via the AGT relation to certain defects in Toda CFT’s, are constructed as domain walls. Moreover we exhibit domain walls that realize the sl(2) affine Hecke algebra.

Purity and decomposition theorems for staggered sheaves

AbstractTwo major results in the theory of ℓ-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specifically, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.

Staggered t-structures on toric varieties

Achar has recently introduced a family of t-structures on the derived category of equivariant coherent sheaves on a G-scheme, generalizing the perverse coherent t-structures of Bezrukavnikov and Deligne. They are called staggered t-structures, and one of their points of interest is that they are more often self-dual. In this paper we investigate these t-structures on the T-equivariant derived category of a toric variety.

Semiorthogonal decompositions of derived categories of equivariant coherent sheaves

Let be an algebraic variety with an action of an algebraic group . Suppose that has a full exceptional collection of sheaves and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of the bounded derived category of -equivariant coherent sheaves on into components that are equivalent to the derived categories of twisted representations of . If the group is finite or reductive over an algebraically closed field of characteristic 0, this gives a full exceptional collection in the derived equivariant category. We apply our results to particular varieties such as projective spaces, quadrics, Grassmannians and del Pezzo surfaces.

Staggered t-Structures on Derived Categories of Equivariant Coherent Sheaves

Let X be a scheme, and let G be an affine group scheme acting on X. Under reasonable hypotheses on X and G, we construct a t-structure on the derived category of G-equivariant coherent sheaves that in many ways resembles the perverse coherent t-structure, but which incorporates additional information from the G-action. Under certain circumstances, the heart of this t-structure, called the “staggered t-structure,” is a finite-length category, and its simple objects are particularly easy to describe. We also exhibit two small examples in which the staggered t-structure is better-behaved than the perverse coherent t-structure.

Equivariant derived category of bundles of projective spaces

We give an analog of D.O. Orlov’s theorem on semiorthogonal decompositions of the derived category of projective bundles for the case of equivariant derived categories. Under the condition that the action of a finite group on the projectivization X of a vector bundle E is compatible with the twisted action of the group on the bundle E, we construct a semiorthogonal decomposition of the derived category of equivariant coherent sheaves on X into subcategories equivalent to the derived categories of twisted sheaves on the base scheme.

Perverse sheaves on affine flags and nilpotent cone of the langlands dual group

This paper is a continuation of [2]. In [2], we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) a category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence related to cells in the affineWeyl group; provide a similar “Langlands dual” description for the category of equivariant coherent sheaves on the nilpotent cone, and link it to perverse coherent sheaves; and deduce some conjectures by Lusztig and Ostrik.

Staggered sheaves on partial flag varieties

Staggered t-structures are a class of t-structures on derived categories of equivariant coherent sheaves. In this Note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits a staggered t-structure with the property that all objects in its heart have finite length. As a consequence, we obtain a basis for its equivariant K-theory consisting of simple staggered sheaves. To cite this article: P.N. Achar, D.S. Sage, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.

Knot homology via derived categories of coherent sheaves, I: The sl(2)-case

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to that of Seidel and Smith [SS] by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology (see [Kh1])

Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves

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. It turns out to be related to irreducible objects in the heart of 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 the cohomology is described in terms of cells in affine Weyl groups. The basis in the Grothendieck group provided by the cohomology modules is shown to coincide with the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik. The proof is based on the results of [ABG ], [AB] and [B], which allow us to reduce the question to purity of IC sheaves on affine flag varieties.

Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone

A certain t t -structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group is introduced by the author in the paper Perverse coherent sheaves (the so-called perverse t t -structure corresponding to the middle perversity). In the present note we show that the same t t -structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means by the author in the paper On tensor categories attached to cells in affine Weyl groups, to be published).