arXiv subjects

Mikhail Kapranov

Publications and source records attributed to Mikhail Kapranov.

At least 37 records · Page 2Linked to original sources

arXiv2019

Perverse sheaves over real hyperplane arrangements II

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

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arXiv2019

Higher Kac-Moody algebras and moduli spaces of G-bundles

We provide a generalization to the higher dimensional case of the construction of the current algebra g((z)), of its Kac-Moody extension and of the classical results relating them to the theory of G-bundles over a curve. For a reductive algebraic group G with Lie algebra g, we define a dg-Lie algebra g_n of n-dimensional currents in g. We show that any symmetric G-invariant polynomial P on g of degree n+1 determines a central extension of g_n by the base field k that we call higher Kac-Moody algebra g_{n,P} associated to P. Further, for a smooth, projective variety X of dimension n>1, we show that g_n acts infinitesimally on the derived moduli space RBun_G(X,x) of G-bundles over X trivialized at the formal neighborhood of a point x of X. Finally, for a representation ϕ: G-->GL_r, we construct an associated determinantal line bundle on RBun_G(X,x) and prove that the action of g_n extends to an action of g_{n,P_ϕ} on such bundle for P_ϕthe (n+1)-st Chern character of ϕ.

math.AG↗

arXiv2018

Complete complexes and spectral sequences

By analogy with the classical (Chasles-Schubert-Semple-Tyrell) spaces of complete quadrics and complete collineations, we introduce the variety of complete complexes. Its points can be seen as equivalence classes of spectral sequences of a certain type. We prove that the set of such equivalence classes has a structure of a smooth projective variety. We show that it provides a desingularization, with normal crossings boundary, of the Buchsbaum-Eisenbud variety of complexes, i.e., a compactification of the union of its maximal strata.

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arXiv2018

Supergeometry in mathematics and physics

This is a chapter for a planned collective volume entitled "New spaces in mathematics and physics" (M. Anel, G. Catren Eds.). The first part contains a short formal exposition of supergeometry as it is understood by mathematicians. The second part discusses aspects of supergeometry that are used by physicists in relation to supersymmetry. Finally, the third part is an attempt to uncover homotopy-theoretic roots of the super formalism.

math.AG↗

arXiv2018

Perverse schobers and birational geometry

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a "web of flops" provided by the Grothendieck resolution associated to a reductive Lie algebra g and study the corresponding schober-type diagram. For g=sl(3) we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.

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arXiv2016

Perverse sheaves and graphs on surfaces

We give an explicit combinatorial description of the category Perv(S,N) of perverse sheaves on an oriented surface S (with boundary) with singularities at a given finite set N. The description is given in terms of any spanning graph K in S with the set of vertices N, so the category is defined entirely in terms of a ribbon graph. This description can be seen as an application, in the theory of perverse sheaves, of the idea of localization on a Lagrangian skeleton.

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arXiv2015

Perverse Schobers

We suggest a possibility for a categorical generalization of the concept of a perverse sheaf, in which vector spaces are replaced by triangulated categories. We call such hypothetical objects perverse Schobers and consider several examples, giving a natural but ad hoc definition in each case. In the simplest case (perverse sheaves on a disk with one possible singular point) we propose, as a categorical analog, the data of a spherical functor.

math.AG↗

arXiv2015

Microlocal sheaves and quiver varieties

We relate Nakajima Quiver Varieties (or, rather, their multiplicative version) with moduli spaces of perverse sheaves. More precisely, we consider a generalization of the concept of perverse sheaves: microlocal sheaves on a nodal curve X. They are defined as perverse sheaves on normalization of X with a Fourier transform condition near each node and form an abelian category M(X). One has a similar triangulated category DM(X) of microlocal complexes. For a compact X we show that DM(X) is Calabi-Yau of dimension 2. In the case when all components of X are rational, M(X) is equivalent to the category of representations of the multiplicative pre-projective algebra associated to the intersection graph of X. Quiver varieties in the proper sense are obtained as moduli spaces of microlocal sheaves with a framing of vanishing cycles at singular points. The case when components of X have higher genus, leads to interesting generalizations of preprojective algebras and quiver varieties. We analyze them from the point of view of pseudo-Hamiltonian reduction and group-valued moment maps.

math.SG↗

arXiv2015

Crossed simplicial groups and structured surfaces

We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to Ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orbicell decompositions of the respective G-structured moduli spaces. As an application, we show how, building on our theory of 2-Segal spaces, the resulting theory can be used to construct categorified state sum invariants of G-structured surfaces.

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arXiv2015

Derived varieties of complexes and Kostant's theorem for gl(m|n)

Given a graded vector space V, the variety of complexes Com(V) consists of all differentials making V into a cochain complex. This variety was first introduced by Buchsbaum and Eisenbud and later studied by Kempf, De Concini, Strickland and many other people. It is highly singular and can be seen as a proto-typical singular moduli space in algebraic geometry. We introduce a natural derived analog of Com(V) which is a smooth derived scheme RCom(V). It can be seen as classifying twisted complexes. We study the cohomology of the dg-algebra of regular functions om RCom(V). It turns out that the natural action of the group GL(V) (automorphisms of V as a graded space) on the cohomology has simple spectrum. This generalizes the known properties of Com(V) and the classical theorem of Kostant on the Lie algebra cohomology of upper triangular matrices.

math.AG↗

arXiv2015

Membranes and higher groupoids

We study the dg-Lie algebra f_n generated by the coefficients of the universal translation invariant flat dg-connection on the n-dimensional affine space. We describe its "semiabelianization" (in particular, the universal quotient which is a crossed module of Lie algebras) in terms of closed differential forms of arbitrary order. This generalizes the theorem of Reutenauer on the abelianization of the commutant of a free Lie algebra. Semiabelian dg-Lie algebras, i.e., non-poisitively graded dg-Lie algebras with brackets of any elements of strictly negative degree being 0, are Lie algebraic analogs of crossed complexes of Brown-Higgins. We show that dg-connections with values in such dg-Lie algebras give rise to higher-dimensional holonomy in the strict sense, i.e., to a system of data associating an object of a strict higher category to a geometric membrane (a class of parametrized membranes up to thin homotopy). Applied to canonical connection with coefficients in the semiabelianization of f_n, this gives a way to assiociate a power-series-type object to a geometric membrane in R^n.

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arXiv2015

Perverse sheaves over real hyperplane arrangements

Let H be an arrangement of real hyperplanes in R^n. The complexification of H defines a natural stratification of C^n. We denote by Perv(C^n, H) the category of perverse sheaves on C^n smooth with respect to this stratification. We give a description of Perv(C^n, H) as the category of representations of an explicit quiver with relations, whose vertices correspond to real faces of H (of all dimensions). The relations are of monomial nature: they identify some pairs of paths in the quiver. They can be formulated in terms of the oriented matroid associated to H.

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arXiv2014

Algebra of the infrared and secondary polytopes

We study algebraic structures ($L_\infty$ and $A_\infty$-algebras) introduced by Gaiotto, Moore and Witten in their recent work devoted to certain supersymmetric 2-dimensional massive field theories. We show that such structures can be systematically produced in any number of dimensions by using the geometry of secondary polytopes, esp. their factorization properties. In particular, in 2 dimensions, we produce, out of a polyhedral "coefficient system", a dg-category $R$ with a semi-orthogonal decomposition and an $L_\infty$-algebra $\mathfrak g$. We show that $\mathfrak g$ is quasi-isomorphic to the ordered Hochschild complex of $R$, governing deformations preserving the semi-orthogonal decomposition. This allows us to give a more precise mathematical formulation of the (conjectural) alternative description of the Fukaya-Seidel category of a Kahler manifold endowed with a holomorphic Morse function.

math.SG↗

arXiv2013

Triangulated surfaces in triangulated categories

For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F(S,A) is independent on the choice of a triangulation of S up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces. In the simplest case, where A is the category of 2-periodic complexes of vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya category of the surface S. Therefore, our construction can be seen as implementing a 2-dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine.

math.AG↗

arXiv2012

Higher Segal spaces I

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d=1, we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-Segal space. This 2-Segal space is given by Waldhausen's S-construction, a simplicial space familiar in algebraic K-theory. Other examples of 2-Segal spaces arise naturally in classical topics such as Hecke algebras, cyclic bar constructions, configuration spaces of flags, solutions of the pentagon equation, and mapping class groups.

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arXiv2012

The Hall algebra of a curve

Let X be a smooth projective curve over a finite field. We describe H, the full Hall algebra of vector bundles X as a Feigin-Odesskii shuffle algebra. This shuffle algebra corresponds to the scheme S of all cusp eigenforms and to the rational function of two variables on S coming from the Rankin-Selberg L-functions. This means that the zeroes of these L-functions control all the relations in H. The scheme S is a disjoint union of countably many G_m-orbits. In the case when X has a theta-characteristic defined over the base field, we embed H into the space of regular functions on the symmetric powers of S.

math.AG↗

arXiv2012

The spherical Hall algebra of Spec(Z)

We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices with quadratic forms. This algebra H consists of automorphic forms with respect to GL_n(Z), n>0, with multiplication given by the parabolic pseudo-Eisenstein series map. We concentrate on the subalgebra SH in H generated by functions on the Arakelov Picard group of Spec(Z). We identify H with a Feigin-Odesskii type shuffle algebra, with the function defining the shuffle algebra expressed through the Riemann zeta function. As an application we study relations in H. Quadratic relations express the functional equation for the Eisenstein-Maass series. We show that the space of additional cubic relations (lying an an appropriate completion of H and considered modulo rescaling), is identified with the space spanned by nontrivial zeroes of the zeta function.

math.AG↗

arXiv2011

Symmetric and exterior powers of categories

We define symmetric and exterior powers of categories, fitting into categorified Koszul complexes. We discuss examples and calculate the effect of these power operations on the categorical characters of matrix 2-representations.

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