Mikhail Kapranov

Koszul duality for operads

(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] and Kontsevich [Kon 1] on the cell decomposition and intersection theory on the moduli spaces of curves. The other is the theory of Koszul duality for quadratic associative algebras which was introduced by Priddy [Pr] and has found many applications in homological algebra, algebraic geometry and representation theory (see e.g., [Be] [BGG] [BGS] [Ka 1] [Man]). The unifying concept here is that of an operad. This paper can be divided into two parts consisting of chapters 1, 3 and 2, 4, respectively. The purpose of the first part is to establish a relationship between operads, moduli spaces of stable curves and graph complexes. To each operad we associate a collection of sheaves on moduli spaces. We introduce, in a natural way, the cobar complex of an operad and show that it is nothing but a (special case of the) graph complex, and that both constructions can be interpreted as the Verdier duality functor on sheaves. In the second part we introduce a class of operads, called quadratic, and introduce a distinguished subclass of Koszul operads. The main reason for introducing Koszul operads (and in fact for writing this paper) is that most of the operads ”arising from nature” are Koszul, cf. (0.8) below. We define a natural duality on quadratic operads (which is

Derived quot schemes

Abstract We construct a “derived” version of Grothendiecks Quot scheme which is a dg-scheme, i.e., an object RQuot of a certain nonabelian right derived category of schemes. It has the property of being manifestly smooth in an appropriate sense (whereas the usual Quot scheme is often singular). The usual scheme Quot is obtained from RQuot by degree 0 truncation. The construction of RQuot can be seen as realization of a part of the Derived Deformation Theory program, which proposes to replace all the moduli spaces arising in geometry by their derived versions by retaining the information about all the higher cohomology instead of H 1 in the classical theory.

Cellular strings on polytopes

The complex of cellular strings with respect to a generic linear functional on a d-dimensional convex polytope has the homotopy type of the (d 2)-sphere. This result was conjectured in a special case by H.-J. Baues.

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.

Braided monoidal 2-categories and Manin-Schechtman higher braid groups

Abstract We study a certain coherence problem for braided monoidal 2-categories. For ordinary braided monoidal categories such a problem is well known to lead to braid groups: If we denote by T(n) the pure braid group on n strands then this group acts naturally on each product A1 ⊗ · ⊗ An. It turns out that in the 2-categorical case we have to consider the so-called higher braid group T(2,n) introduced by Manin and Schechtman. The main result is that T(2,n) naturally acts by 2-automorphisms on the canonical 1-morphism A1 ⊗ · ⊗ An → An⊗ · ⊗ A1 for any objects A1,…, An.

Double affine Hecke algebras and 2-dimensional local fields

The concept of an n-dimensional local field was introduced by A.N. Parshin [Pal] with the aim of generalizing the classical adelic formalism to (absolutely) ndimensional schemes. By definition, a 0-dimensional local field is just a finite field, and an n-dimensional local field, n > 0, is a complete discrete valued field whose residue field is (n 1)-dimensional local. Thus for n = 1 we get locally compact fields such as Qp, Fq((t)), and for n = 2 we get fields such as Qp((t)), Fq((tI))((t2)), etc. In representation theory, harmonic analysis on reductive groups over 0and 1dimensional local fields leads, in particular, to consideration of the finite and affine Hecke algebras Hq, Hq associated to aniy finite root system R and anly q E C*. These algebras can be defined in several ways, one being by generators and relations, another as the convolution algebra, with respect to the Haar measure, of functions on the group bi-invariant with respect to an appropriate subgroup (i.e., as the algebra of double cosets). Harmonic analysis on groups over 2-dimensional local fields has not been developed, the main difficulty being the infinite dimensionality (absense of local compactness) of such fields. However, the double affine Hecke

HYPERGEOMETRIC FUNCTIONS, TORIC VARIETIES AND NEWTON POLYHEDRA

In this talk we give a survey of our recent results on multidimensional hypergeometric functions [GZK 1,2,7], Before developing the general theory we briefly discuss main features of the classical Gauss function F(x)= 2F1 (a,b;c;x). By definition, F(x) is the solution of the hypergeometric equation

Discriminants and Resultants for Forms in Several Variables

x\left( {1 - x} \right)\frac{{{d^{2}}F}}{{d{x^{2}}}} + \left[ {c - \left( {a + b + 1} \right)x} \right]\frac{{dF}}{{dx}} - abF = 0