arXiv subjects

M. Kapranov

Publications and source records attributed to M. Kapranov.

At least 19 recordsLinked to original sources

arXiv2006

Riemann-Roch for real varieties

If E is a C^\infty complex vector bundle on an oriented C^\infty manifold Σ, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:Σ-->B is a smooth family of circles as above and E is a vector bundle on Σ, then the smooth direct image q_*(E) is an infinite-dimensional bundle with fibers as above and so we have its determinantal gerbe on B with lien being the sheaf of invertible complex valued C^\infty functions, it gives a class in H^3(B, Z). In this paper we consider a family q:Σ-->B as above but with fibers being compact oriented C^\infty manifolds of dimension d. For a bundle E on Σone expects q_*(E) to possess a determinantal d-gerbe and hence to give a class in H^{d+2}(B, Z). We construct directly, by means of a version of the Chern-Weil theory, the real version of this would be class. We further prove a real version of the Grothendieck-Riemann-Roch theorem describing this class as a direct image of a certain characteristic class of E.

math.DG↗

arXiv2006

Noncommutative geometry and path integrals

We argue that there should exist a "noncommutative Fourier transform" which should identify functions of noncommutative variables (say, of matrices of indeterminate size) and ordinary functions or measures on the space of paths. Some examples are considered.

math.QA↗

arXiv2006

Formal loops IV: Chiral differential operators

We relate the gerbe of sheaves of chiral differential operators (CDO) on a algebraic variety X, studied by Gorbounov, Malikov and Schechtman, to the determinantal gerbe of the formal loop space LX introduced in our earlier paper. The liens of the two gerbes are related by a version of the symplectic action homomorphism. The determinantal gerbe of LX has a factorization structure in the spirit of Beilinson and Drinfeld. In our identification, sheaves of CDO correspond to factorizing objects of this factorization gerbe.

math.AG↗

arXiv2005

Formal loops III: Factorizing functions and the Radon transform

To any algebraic variety X and and closed 2-form ωon X, we associate the "symplectic action functional" T(ω) which is a function on the formal loop space LX introduced by the authors in math.AG/0107143. The correspondence ω--> T(ω) can be seen as a version of the Radon transform. We give a characterization of the functions of the form T(ω) in terms of factorizability (infinitesimal analog of additivity in holomorphic pairs of pants) as well as in terms of vertex operator algebras. These results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on X (whose lien is the sheaf of closed 2-forms) and the determinantal gerbe of the tangent bundle of LX (whose lien is the sheaf of invertible functions on LX). On the level of liens this relation associates to a closed 2-form ωthe invertible function exp T(ω).

math.AG↗

arXiv2005

Formal loops II: A local Riemann-Roch theorem for determinantal gerbes

If V is a bundle of Tate vector spaces over a base B, its determinantal gerbe has a class C_1(V) in the second cohomology group of the sheaf of invertible functions which can be seen as the Deligne cohomology H^3(B, Z(2)). An example of such a "Tate bundle" can be obtained from a finite rank vector bundle E on the product of B and a punctured formal disk. Our main result identifies the corresponding C_1(V) with the cohomological direct image of ch_2(E), the second Chern character of E. It can be seen as a "local" version of the Riemann-Roch-Grothendieck theorem for a family of curves. This theorem explains the results of Gorbounov, Malikov and Schechtman relating ch_2 of the tangent bundle of an algebraic variety X to the existence of a sheaf of chiral differential operators. To be precise, it implies that the determinantal anomaly of the formal loop space of X is the transgression of ch_2(TX).

math.AG↗

arXiv2004

Vertex algebras and the formal loop space

We introduce an algebro-geometric version of the free loop space for any scheme X of finite type. This is an ind-scheme of ind-infinite type containing the scheme of formal germs of curves on X. Then, we give a direct geometric construction, for smooth X, of the chiral de Rham complex of X (introduced by Malikov, Schechtman and Vaintrob).

math.AG↗

arXiv2001

Semiinfinite symmetric powers

We develop a theory of measures, differential forms and Fourier tramsforms on some infinite-dimensional real vector spaces by generalizing the following two constructions: (a) The construction of the semiinfinite wedge power of a Tate vector space V. Recall that it is obtained as a certain double inductive limit of the exterior algebras of finite-dimensional subquotients of V. (b) The construction of the space of measures on a nonarchimedean local field K with maximal ideal M as a double projective limit of the spaces of measures (=functions) on finite subquotients M^i/M^j of K.

math.QA↗

arXiv2001

Modules and Morita theorem for operads

Associative rings A, B are called Morita equivalent when the categories of left modules over them are equivalent. We call two classical linear operads P, Q Morita equivalent if the categories of algebras over them are equivalent. We transport a part of Morita theory to the operadic context by studying modules over operads. As an application of this philosophy, we consider an operadic version of the sheaf of linear differential operators ona a (super) manifold M and give a comparison theorem between algebras over this sheaf on M and M_{red}. The paper is dedicated to A.N.Tyurin on the occasion of his 60th birthday.

math.QA↗

arXiv2000

Derived Hilbert schemes

We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X) which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is just given by truncations of the homogeneous coordinate rings of subschemes in X. In particular, RHilb_h(X) differs from RQuot_h(O_X), the derived Quot scheme constructed in our previous paper (math.AG/9905174) which carries only a family of A-infinity modules over the coordinate algebra of X. As an application, we construct the derived version of the moduli stack of stable maps of (variable) algebraic curves to a given projective variety Y, thus realizing the original suggestion of M. Kontsevich.

math.AG↗

arXiv2000

Derived Quot schemes

Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a differential graded manifold RQuot_h(F) (so it is always smooth in an appropriate sense) whose tangent space at a point represented by a subsheaf K in F, is a cochain complex quasiisomorphic to RHom(K, F/K).

math.AG↗

arXiv2000

The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups

We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for Kac-Moody groups. For a pair consisting of a surface and a curve on it, we consider a refined geometric function E (involving G-bundles with parabolic structures along the curve) which depends both on elliptic and modular variables. We prove a functional equation for E with respect to the affine Weyl group, thus establishing the elliptic behavior. When the curve is P^1, we calculate the Eisenstein-Kac-Moody series explicitly and it turns out to be a certain deformation of an irreducible Kac-Moody character, more precisely, an analog of the Hall-Littlewood polynomial for the affine root system. We also get an explicit formula for the universal blowup function for any simply connected structure group.

math.AG↗

arXiv1999

The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties

For an irreducible subvariety Z in an algebraic group G we define a nonnegative integer gdeg(Z) as the degree, in a certain sense, of the Gauss map of Z. It can be regarded as a substitution for the intersection index of the conormal bundle to Z with the zero section of T^*G, even though G may be non-compact. For G a semiabelian variety (in particular, an algebraic torus (C^*)^n) we prove a Riemann-Roch-type formula for constructible sheaves on G, which involves our substitutions for the intersection indices. As a corollary, we get that a perverse sheaf on such a G has nonnegative Euler characteristic, generalizing a theorem of Loeser-Sabbah.

math.AG↗

arXiv1999

Kleinian singularities, derived categories and Hall algebras

We describe the derived category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2). Then, we give an application to the Euler-characteristic version of the Hall algebra of the category of coherent sheaves on an algebraic surface.

math.AG↗

arXiv1998

Double affine Hecke algebras and 2-dimensional local fields

We give an interpretation of the double affine Hecke algebra of Cherednik as the (suitably regularized) algebra of double cosets of a group G by a subgroup J, extending the well known interpretations of finite and affine Hecke algebras. In this interpretation, G consists of K-points of a split reductive group where K is a 2-dimensional local field such as Q_p((t)) or F_q((t_1))((t_2)), and J is a certain analog of the Iwahori subgroup.

math.AG↗

arXiv1998

Hypergeometric functions on reductive groups

The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced by an arbitrary reductive group H and A is a set of irreducible representations of H. The functions are thus defined on the space M_A of functions on H spanned by the matrix elements of representations from A. The properties of the system are related to the geometry of a certain algebraic variety X_A, which belongs to the class of group compactifications studied by De Concini and Procesi. We develop the theory of Euler integral representations for these generalized hypergeometric functions (with integrals taken over cycles in H). We also construct the analogs of hypergeometric series, by expanding the delta-function along a subgroup into a power series and taking the termwise Fourier transform.

alg-geom↗

arXiv1997

Injective resolutions of BG and derived moduli spaces of local systems

It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural "derived" moduli spaces which are always smooth in an appropriate sense and whose tangent spaces involve the entire cohomology of the sheaf of infinitesimal automorphisms, not just H^1. In this note we give an algebraic construction of such an extension for the simplest class of moduli spaces, namely for moduli of local systems (representations of the fundamental group).

alg-geom↗

arXiv1997

Rozansky-Witten invariants via Atiyah classes

Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a very simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyperKaehler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work on the level of cohomology and for any Kaehler manifold, if we work on the level of Dolbeault forms. In particular, for any sheaf A of commutative algebras on any complex manifold the shifted cohomology of the tangent sheaf tensored with A is a graded Lie algebra. We also show that a certain system of Gilkey-type complexes of "natural tensors" on Kaehler manifolds is, up to suspension (accounting for various changes of signs), identified with the PROP (in the sense of Adams and MacLane) describing Lie algebras "up to higher homotopies". As an outcome of our considerations, we give a formula for the Rozansky-Witten classes using any Kaehler metric on a holomorphic symplectic manifold.

alg-geom↗

arXiv1997

Heisenberg doubles and derived categories

Let A be an abelian category of finite type and homological dimension 1. Then by results of Green R(A), the extended Hall-Ringel algebra of A, has a natural Hopf algebra structure. We consider its Heisenberg double Heis(A) and study its relation with D(A), the derived category of A. We show that Heis(A) can be viewed as a "Hall algebra" of D^{0,1}(A), the subcategory of complexes situated in degrees 0 and 1, in the following sense: if B is the heart of a t-structure on D(A) lying in D^{0,1}(A), then R(B) is naturally a subalgebra in Heis(A). Further, we define a new algebra L(A) called the lattice algebra of A, obtained by taking infinitely many copies of R(A), one for each site of an infinite 1-dimensional lattice and imposing Heisenberg double-type relations between copies at adjacent sites and oscillator-type relations between copies at non-adjacent sites. This algebra serves as the "Hall algebra" of the full derived category D(A) in the following sense: any derived equivalence D(A)-->D(B) induces an isomorphism of lattice algebras L(A)-->L(B).

q-alg↗