A remark on virtual orientations for complete intersections

The aim of this note is to give a simple definition of genus zero virtual orientation classes (or fundamental classes) for projective complete intersections or, more generally, for complete intersections in convex varieties, and to prove a push forward formula for them.Comment: 3 pages, Late

On hypergeometric functions connected with quantum cohomology of flag spaces

Givental's recursion relations for the flag varieties $G/B$ are established.Comment: 26 pages, Te

De Rham complex of a Gerstenhaber algebra

We introduce a notion of the De Rham complex of a Gerstenhaber algebra which produces a notion of a "quasi-BV structure", and allows to classify these structures, generalizing the classical results for polyvector fields.Comment: 17 page

Local structure of moduli spaces

We describe the algebra of a universal formal deformation as the zeroth cohomology of the dg Lie algebra corresponding to this deformation problem. A report at Arbeitstagung 1997 on the joint work with V.Hinich.Comment: 3 pages, Tex. A slight modification in Tex is mad

Remarks on formal deformations and Batalin-Vilkovisky algebras

This note consists of two parts. Part I is an exposition of (a part of) the V.Drinfeld's letter, [D]. The sheaf of algebras of polyvector fields on a Calabi-Yau manifold, equipped with the Schouten bracket, admits a structure of a Batalin-Vilkovisky algebra. This fact was probably first noticed by Z.Ran, [R]. Part II is devoted to some generalizations of this remark.Comment: 29 pages, TeX. Minor changes made, and a reference adde

Screenings and a universal Lie-de Rham cocycle

Feigin and Fuchs have given a well-known construction of intertwining operators between "Fock-type" modules over the Virasoro algebra. The intertwiners are obtained via contour integration of certain "screening operators" over top homology classes of a configuration space. The main observation of the present paper is that the screening operators contain more information. Specifically, at the chain level, the screening operators provide a certain canonical cocycle of the Virasoro (resp. affine Kac-Moody) algebra with coefficients in the de Rham complex of an operator-valued local system on the configuration space. This way we obtain canonical morphisms from higher homology groups of the above local systems to appropriate higher Ext-groups between the Fock space representations. Our construction is motivated by, and in a special case reduces to the construction of Bowknegt et al, see [BMP1], [BMP2].Comment: Amstex, 38pp. Minor improvements made and some references adde

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.Comment: 19 pages, 1 figur

Chiral Poincar\'e duality

The aim of this note is to prove the analogue of Poincar\'e duality in the chiral Hodge cohomology.Comment: 14 pages, TeX. A remark adde

Rational differential forms on line and singular vectors in Verma modules over sl^2\widehat {sl}_2

We construct a monomorphism of the De Rham complex of scalar multivalued meromorphic forms on the projective line, holomorphic on the complement to a finite set of points, to the chain complex of the Lie algebra of $sl_2$-valued algebraic functions on the same complement with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{sl}_2$. We show that the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the new relations between the cohomology classes of logarithmic differential forms.Comment: Latex 16 pages, new abstract, proof of Theorem 5.12 extended, misprints correcte

Chiral de Rham complex. II

This paper is a sequel to math.AG/9803041. It consists of three parts. In the first part we give certain construction of vertex algebras which includes in particular the ones appearing in op. cit. In the second part we show how the cohomology ring $H^*(X)$ of a smooth complex variety $X$ could be restored from the correlation functions of the vertex algebra $R\Gamma(X;\Omega^{ch}_X)$. In the third part, we prove first a useful general statement that the sheaf of loop algebras over the tangent sheaf \Cal{T}_X acts naturally on $\Omega^{ch}_X$ for every smooth $X$ (see \S 1). The Z-graded vertex algebra $H^*(X;\Omega^{ch}_X)$ seems to be a quite interesting object (especially for compact $X$). In \S 2, we compute $H^0(CP^N;\Omega^{ch}_{CP^N})$ as a module over $\hat{sl}(N+1)$.Comment: 48 pages, TeX. Some typos are corrected and a remark in I.2.3 adde