Perverse Sheaves Research Articles

Теория гомотопов в применении к несмещенным базисам, гармоническому анализу на графах и превратным пучкам

В статье дается обзор современных результатов и приложений теории гомотопов. В работе введено понятие хорошо темперированного элемента ассоциативной алгебры и доказано, что категория представлений гомотопа, построенного с помощью хорошо темперированного элемента, является сердцевиной подходящим образом склеенной $t$-структуры. Посчитаны глобальная и хохшильдова размерность гомотопа в хорошо темперированном случае. Рассматривается случай гомотопа, построенного с помощью обобщенного оператора Лапласа группоида Пуанкаре графа. Показано, что такой гомотоп является фактором алгебры Темперли-Либа графа. Показано, что превратные пучки на проколотом диске и на двумерной сфере с двойной точкой отождествляются с представлениями соответствующего гомотопа. Также в статье обсуждается связь гомотопов с теорией ортогональных разложений алгебры Ли $\operatorname{sl}(n,\mathbb{C})$ в сумму картановских подалгебр, с классификацией конфигураций прямых и взаимно несмещенных базисов, с квантовыми протоколами и обобщенными адамаровыми матрицами. Библиография: 56 названий.

Log -modules and index theorems

Abstract We study log $\mathscr {D}$ -modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.

Mirror Symmetry for Perverse Schobers from Birational Geometry

Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent–constructible correspondence and a recent result of Ganatra et al. (Microlocal morse theory of wrapped fukaya categories. arXiv:1809.08807) to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of Bondal et al. (Selecta Math 24(1):85–143, 2018) on our mirror schobers.

Recollements of derived categories I: Construction from exact contexts

Recollements were introduced originally by Beilinson, Bernstein and Deligne to study the derived categories of perverse sheaves, and nowadays become very powerful in understanding relationship among three algebraic, geometric or topological objects. The purpose of this series of papers is to study recollements in terms of derived module categories and homological ring epimorphisms, and then to apply our results to both representation theory and algebraic K-theory. In this paper we present a new and systematic method to construct recollements of derived module categories. For this aim, we introduce a new ring structure, called the noncommutative tensor product, and give necessary and sufficient conditions for noncommutative localizations which appears often in representation theory, topology and K-theory, to be homological. The input of our machinery is an exact context which can be easily obtained from a rigid morphism that exists in very general circumstances. The output is a recollement of derived module categories of rings in which the noncommutative tensor product of an exact context plays a crucial role. Thus we obtain a large variety of new recollements from commutative and noncommutative localizations, ring epimorphisms and extensions.

Recollement for perverse sheaves on real hyperplane arrangements

Consider the category of perverse sheaves on Cn smooth with respect to the stratification arising from a real hyperplane arrangement. We construct an algebra whose module category is equivalent to this category, building on work of Kapranov–Schechtman. We prove that the standard recollement of perverse sheaves is equivalent to a well-known recollement for algebras.As an application of our results, we give a new description of the category of representations of the fundamental group of the complement of such a hyperplane arrangement. We also identify the modules associated to all simple perverse sheaves, that is, the intersection cohomology complexes. Finally, we generalise our results to W-equivariant perverse sheaves for the reflection arrangement of a finite Coxeter group W, extending work of Weissman. As an application, we identify the modules associated to the equivariant simple perverse sheaves.

Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and Tsimerman

We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel–Venkatesh mixing conjecture about the equidistribution of CM points on a product of two modular curves.

𝐙/𝐦-graded Lie algebras and perverse sheaves, IV

Let G G be a reductive group over C \mathbf {C} . Assume that the Lie algebra g \frak g of G G has a given grading ( g j ) (\frak g_j) indexed by a cyclic group Z / m \mathbf {Z}/m such that g 0 \frak g_0 contains a Cartan subalgebra of g \frak g . The subgroup G 0 G_0 of G G corresponding to g 0 \frak g_0 acts on the variety of nilpotent elements in g 1 \frak g_1 with finitely many orbits. We are interested in computing the local intersection cohomology of closures of these orbits with coefficients in irreducible G 0 G_0 -equivariant local systems in the case of the principal block. We show that these can be computed by a purely combinatorial algorithm.

Perverse Sheaves and Finite Dimensional Algebras

Let $X$ be a topologically stratified space, $p$ be any perversity on $X$, and $k$ be a field. We show that the category of $p$-perverse sheaves on $X$, constructible with respect to the stratification and with coefficients in $k$, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if $X$ has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

Summands of theta divisors on Jacobians

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

Graham's variety and perverse sheaves on the nilpotent cone

Graham has constructed a variety with a map to the nilpotent cone which is similar in some ways to the Springer resolution. One aspect in which Graham's map differs is that it is not in general an isomorphism over the principal orbit, but rather the universal covering map. This map gives rise to a certain semisimple perverse sheaf on the nilpotent cone, and we discuss here the problem of describing the summands of this perverse sheaf. For type An, a key tool is a known description of an affine paving of Springer fibers.

A support theorem for nested Hilbert schemes of planar curves

Consider a family of integral complex locally planar curves. We show that under some assumptions on the basis, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension.

Nearby cycles of parahoric shtukas, and a fundamental lemma for base change

Using the Langlands-Kottwitz paradigm, we compute the trace of Frobenius composed with Hecke operators on the cohomology of nearby cycles, at places of parahoric reduction, of perverse sheaves on certain moduli stacks of shtukas. Following an argument of Ngo, we then use this to give a geometric proof of a base change fundamental lemma for parahoric Hecke algebras for GL_n over local function fields. This generalizes a theorem of Ngo, who proved the base change fundamental lemma for spherical Hecke algebras for GL_n over local function fields, and extends to positive characteristic (for GL_n) a fundamental lemma originally introduced and proved by Haines for p-adic local fields.

Composition series for GKZ-systems

In this paper, we find a composition series of GKZ-systems with semisimple successive quotients. We also study the composition series of the corresponding perverse sheaves and compare these two composition series under the Riemann-Hilbert correspondence.

Lyubeznik numbers of irreducible projective varieties depend on the embedding

We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.

Shuffle algebras and perverse sheaves

We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct an equivalence between: (i) Braided Hopf algebras of a certain type. (ii) Factorizable collections of perverse sheaves on all the symmetric products. Under this eqiuvalence, the Nichols algebra associated to an object V corresponds to the collection of the intersection cohomology extensions of the local systems on the open configuration spaces associated to the tensor powers of V. Our approach is based on using real skeleta of complex configuration spaces.

Bifurcation values of polynomial functions and perverse sheaves

We characterize bifurcation values of polynomial functions by using the theory of perverse sheaves and their vanishing cycles. In particular, by introducing a method to compute the jumps of the Euler characteristics with compact support of their fibers, we confirm the conjecture of Némethi–Zaharia in many cases.

Absolute sets and the decomposition theorem

We give a framework to produce constructible functions from natural functors between categories, without need of a morphism of moduli spaces to model the functor. We show using the Riemann-Hilbert correspondence that any natural (derived) functor on constructible sheaves on smooth complex algebraic varieties can be used to construct a kind of constructible sets, called absolute sets, generalizing a notion introduced by Simpson in presence of moduli. We conjecture that the absolute sets of local systems satisfy a special varieties package, among which is an analog of the Manin-Mumford, Mordell-Lang, and Andre-Oort conjectures. The conjecture gives a simple proof of the Decomposition Theorem for all semi-simple perverse sheaves, assuming the Decomposition Theorem for the geometric ones. We prove the conjecture in the rank one case by showing that the closed absolute sets in this case are finite unions of torsion-translated affine tori. This extends a structure result of the authors for cohomology jump loci to any other natural jump loci. For example, to jump loci of intersection cohomology and Leray filtrations. We also show that the Leray spectral sequence for the open embedding in a good compactification degenerates for all rank one local systems at the usual page, not just for unitary local systems.

Witt groups of abelian categories and perverse sheaves

In this paper we study the Witt groups of symmetric and anti-symmetric forms on perverse sheaves on a finite-dimensional topologically stratified space with even dimensional strata. We show that the Witt group has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another `splitting decomposition' for Witt classes of perverse sheaves obtained inductively from our main new tool, a `splitting relation' which is a generalisation of isotropic reduction. The Witt groups we study are identified with the (non-trivial) Balmer-Witt groups of the constructible derived category of sheaves on the stratified space, and also with the corresponding cobordism groups defined by Youssin. Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.

Generalizations of Intersection Homology and Perverse Sheaves with Duality over the Integers

We provide a generalization of the Deligne sheaf construction of intersection homology theory and a corresponding generalization of Poincaré duality on pseudomanifolds such that the Goresky–MacPherson, Goresky–Siegel, and Cappell–Shaneson duality theorems all arise as particular cases. Unlike classical intersection homology theory, our duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. Self-duality does require local conditions, but our perspective leads to a new class of spaces more general than the Goresky–Siegel IP spaces on which upper-middle perversity intersection homology is self-dual. We also examine torsion-sensitive t-structures and categories of perverse sheaves that contain our torsion-sensitive Deligne sheaves as intermediate extensions.

On the length of perverse sheaves on hyperplane arrangements

In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we provide combinatorial formulas for these lengths. As by-products, we also obtain in this case combinatorial formulas for the intersection cohomology Betti numbers of rank one local systems on the complement with same monodromy around the planes.