Perverse Sheaves Research Articles

Spherical adjunctions of stable ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}-categories and the relative S-construction

We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\infty }$$\\end{document}-categories. We study the relative Waldhausen S-construction S∙(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_\\bullet (F)$$\\end{document} of the spherical functor F and show that it has a natural paracyclic structure (“rotation symmetry”). This fulfills a part of the general program of perverse schobers which are conjectural categorical upgrades of perverse sheaves. If we view a spherical functor as defining a schober on a disk, then each component Sn(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n(F)$$\\end{document} of the S-construction gives a categorification of the cohomology of a perverse sheaf on a disk with support in a union of (n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+1)$$\\end{document} closed arcs in the boundary. In other words, Sn(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n(F)$$\\end{document} can be interpreted as the Fukaya category of the disk with coefficients in the schober and with support (“stops”) at the boundary arcs. The importance of the paracyclic structure is that it allows us to naturally associate the above data to disks on oriented surfaces. The action of the paracyclic rotation is a categorical analog of the monodromy of a perverse sheaf.

General multiple Dirichlet series from perverse sheaves

We give an axiomatic characterization of multiple Dirichlet series over the function field Fq(T), generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.

Symplectic Fourier–Deligne Transforms on G/U and the Algebra of Braids and Ties

Abstract We explicitly identify the algebra generated by symplectic Fourier–Deligne transforms (i.e., convolution with Kazhdan–Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type $A$ by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya’s generators of the Yokonuma–Hecke algebra (previously proved case-by-case in types $A, D, E$ by Juyumaya and separately for types $B, C, F_{4}, G_{2}$ by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan–Lusztig basis, and finally an explicit formula for the dimension of Marin’s algebra in Type $A_{n}$ (previously only known for $n \leq 4$).

Constant term functors with $\mathbb{F}_{p}$-coefficients

We study the constant term functor for \mathbb{F}_{p} -sheaves on the affine Grassmannian in characteristic p with respect to a Levi subgroup. Our main result is that the constant term functor induces a tensor functor between categories of equivariant perverse \mathbb{F}_{p} -sheaves. We apply this fact to get information about the Tannakian monoids of the corresponding categories of perverse sheaves. As a byproduct we also obtain geometric proofs of several results due to Herzig on the mod p Satake transform and the structure of the space of mod p Satake parameters.

The four operations on perverse motives

Let k be a field of characteristic zero with a fixed embedding \sigma:k\hookrightarrow \mathbf{C} into the field of complex numbers. Given a k -variety X , we use the triangulated category of étale motives with rational coefficients on X to construct an abelian category \mathscr{M}(X) of perverse mixed motives. We show that over \mathop{\mathrm{Spec}}(k) the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories \mathrm{D}^{\mathrm{b}}(\mathscr{M}(X)) are equipped with the four operations of Grothendieck (for morphisms of quasi-projective k -varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo–Migliorini and provides an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.

Twistor D-Modules and the Decomposition Theorem

The purpose of this Arbeitsgemeinschaft is to introduce the notion of twistor \mathcal{D} -modules and their main properties. The guiding principle leading this discussion is Simpson’s “meta-theorem”, which gives a heuristic for generalizing (mixed) Hodge-theoretic results into (mixed) twistor-theoretic results. The strength of the twistor approach is that it enables to enlarge the scope of Hodge theory not only to arbitrary semi-simple perverse sheaves, equivalently semi-simple regular holonomic \mathcal{D} -modules via the Riemann-Hilbert correspondence, but also to possibly semi-simple irregular holonomic \mathcal{D} -modules. An overarching goal for this session is Mochizuki’s proof of the decomposition theorem for semi-simple holonomic \mathcal{D} -modules on a smooth complex projective variety, first conjectured by Kashiwara in 1996.

Artin perverse sheaves

We show that the perverse t-structure induces a t-structure on the category DA(S,Zℓ) of Artin ℓ-adic complexes when S is an excellent scheme of dimension less than 2 and provide a counter-example in dimension 3. The heart PervA(S,Zℓ) of this t-structure can be described explicitly in terms of representations in the case of 1-dimensional schemes.When S is of finite type over a finite field, we also construct a perverse homotopy t-structure over DA(S,Qℓ) and show that it is the best possible approximation of the perverse t-structure. We describe the simple objects of its heart PervA(S,Qℓ)# and show that the weightless truncation functor ω0 is t-exact. We also show that the weightless intersection complex ECS=ω0ICS is a simple Artin homotopy perverse sheaf. If S is a surface, it is also a perverse sheaf but it need not be simple in the category of perverse sheaves.

Relative perversity

We define and study a relative perverse t t -structure associated with any finitely presented morphism of schemes f : X → S f: X\to S , with relative perversity equivalent to perversity of the restrictions to all geometric fibres of f f . The existence of this t t -structure is closely related to perverse t t -exactness properties of nearby cycles. This t t -structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category P e r v U L A ( X / S ) \mathrm {Perv}^{\mathrm {ULA}}(X/S) with many of the same properties familiar in the absolute setting (e.g., noetherian, artinian, compatible with Verdier duality). For S S connected and geometrically unibranch with generic point η \eta , the functor P e r v U L A ( X / S ) → P e r v ( X η ) \mathrm {Perv}^{\mathrm {ULA}}(X/S)\to \mathrm {Perv}(X_\eta ) is exact and fully faithful, and its essential image is stable under passage to subquotients. This yields a notion of “good reduction” for perverse sheaves.

Proof of Vogan’s conjecture on Arthur packets: irreducible parameters of p-adic general linear groups

In this paper we prove Vogan’s conjecture on local Arthur packets, as recalled in Cunningham et al. [Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples, Memoirs of the American Mathematical Society, Boston, 2022, Section 8.3, Conjecture 1(a)], for irreducible Arthur parameters of p-adic general linear groups. This result shows that these Arthur packets may be characterized by properties of simple perverse sheaves on a moduli space of Langlands parameters.

Tame quivers and affine bases I: A Hall algebra approach to the canonical bases

In [17], Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztig's canonical basis and obtains a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang ([14]) and closely related with the work of Beck-Nakajima ([2]). A crucial method in our construction is a generalization of that in [8].

A new approach to the generalized Springer correspondence

The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decomposition theorem to this map yields all simple perverse sheaves on the nilpotent cone with multiplicity as predicted by Lusztig’s generalized Springer correspondence.

Perverse sheaves on Riemann surfaces as Milnor sheaves

Abstract Constructible sheaves of abelian groups on a stratified space can be equivalently described in terms of representations of the exit-path category. In this work, we provide a similar presentation of the abelian category of perverse sheaves on a stratified surface in terms of representations of the so-called paracyclic category of the surface. The category models a hybrid exit–entrance behaviour with respect to chosen sectors of direction, placing it ‘in between’ exit and entrance path categories. In particular, this perspective yields an intrinsic definition of perverse sheaves as an abelian category without reference to derived categories and t-structures.

The parity of Lusztig's restriction functor and Green's formula

Our investigation in the present paper is based on three important results. (1) In [14], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [6], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [11], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula.

Perverse Sheaves over Real Hyperplane Arrangements II

Let \mathcal{H} be an arrangement of hyperplanes in \mathbb{R}^n and \operatorname{Perv}(\mathbb{C}Cn,\mathcal{H}) be the category of perverse sheaves on \mathbb{C}^n smooth with respect to the stratification given by complexified flats of \mathcal{H} . We give a description of \operatorname{Perv}(\mathbb{C}^n, \mathcal{H}) in terms of “matrix diagrams”, i.e., diagrams formed by vector spaces E_{A,B} labeled by pairs (A,B) of real faces of \mathcal{H} (of all dimensions) or, equivalently, by the cells iA+B of a natural cell decomposition of \mathbb{C} . A matrix diagram is formally similar to a datum describing a constructible (nonperverse) sheaf but with the direction of one-half of the arrows reversed.

Perverse sheaves on $\mathbf{C}^{2}$ without vanishing cycles at the origin along a general plane curve with singularities

Generalizing MacPherson-Vilonen’s method [2] to arbitrary plane curve singularities, we provide a classification of perverse sheaves on the neighborhood of the origin in the complex plane, which are adapted to a germ of a complex analytic plane curve. We rely on the presentation of the fundamental group of the complement of the curve as obtained by Neto and Silva [5]. The main result is an equivalence of categories between the category of perverse sheaves on $ \mathbf{C}^{2}$ stratified with respect to a singular plane curve and the category of $n$-tuples of finite dimensional vector spaces and linear maps satisfying a finite number of suitable relations. As an application, we classify perverse sheaves with no vanishing cycles at the origin for a special case.

Vanishing cohomology and Betti bounds for complex projective hypersurfaces

We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tib\u{a}r in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.

Perverse Sheaves and the Cohomology of Regular Hessenberg Varieties

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 Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kähler package.”

A GEOMETRIC STEINBERG FORMULA

We prove an isomorphism for simple perverse sheaves on the affine Grassmannian of a connected reductive algebraic group that is a geometric counterpart (in light of the Finkelberg–Mirković conjecture) of the Steinberg tensor product formula for simple representations of reductive groups over fields of positive characteristic.

Perverse Sheaves with Nilpotent Singular Support on the Stack of Coherent Sheaves on an Elliptic Curve

We define a stratification of the moduli stack of coherent sheaves on an elliptic curve which allows us (1) to give an explicit description of the irreducible components of the global nilpotent cone of elliptic curves, (2) to establish an explicit bijection between the simple objects of the category of perverse sheaves defined by Schiffmann to categorify the elliptic Hall algebra (the so-called spherical Eisenstein sheaves) and the irreducible components of the global nilpotent cone and (3) to give an explicit description and parametrization of the perverse sheaves on the moduli stack of coherent sheaves on an elliptic curve having nilpotent singular support. Along the way, we find a combinatorial parametrization of the irreducible components of the semistable locus of the elliptic global nilpotent cone. The comparison with Bozec’s parametrization leads to an interesting combinatorial problem.

Equivariant K-Theory Approach to $\imath$-Quantum Groups

Various constructions for quantum groups have been generalized to \imath -quantum groups. Such a generalization is called an \imath -program. In this paper, we fill one of the parts in the \imath -program. Namely, we provide an equivariant K-theory approach to \imath -quantum groups, which is the Langlands dual picture of that constructed in Bao et al. (Transform. Groups 23 (2018), 329–389), where a geometric realization of \imath -quantum groups is provided by using perverse sheaves. As an application of the main results, we prove Li’s conjecture (Li, Represent. Theory 23 (2019), 1–56) for special cases.