Vadim Schechtman

Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors

In this paper we strenghten a theorem by Esnault-Schechtman-Viehweg, [3], which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain ”Aomoto non-resonance conditions” for monodromies are fulfilled at some ”edges” (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges, see Theorem 4.1. We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations, cf. [8]), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras, see Theorem 7.1.

On algebraic equations satisfied by hypergeometric correlators in WZW models. I.

The paper contains an explicit description of genus 0 conformal block bundles for Wess-Zumino-Witten models of Conformal field theory. We prove that an earlier construction due to the second and the third authors gives a map of these bundles to certain de Rham cohomology bundles.

Quantum Groups and Homology of Local Systems

In this paper we “quantize” results of [SV2], Part II. We establish and study the connection between (co)homology of local systems introduced in [SV1], [SV2] and homology of nilpotent subalgebras of certain Hopf algebras very close to Drinfeld-Jimbo q-analogues of Kac-Moody algebras.

Aomoto Dilogarithms, Mixed Hodge Structures and Motivic Cohomology of Pairs of Triangles on the Plane

It is known that a group of linear combinations of polytopes in R3 considered up to movements with respect to cutting of polytopes may be embedded into ℝ ⊗ ℝ/2πℤ ⊕ ℝ; this embedding assigns to a polytope its Dehn invariant and volume [C]. The study of motivic cohomology of a projective plane with two distinguished families of projective lines leads to an analogous problem: to describe a group of linear combinations of pairs of triangles on a plane considered up to the action of PGL(3), with respect to a cutting of any triangle of a pair. It turns out that this group is isomorphic up to 12—torsion to B2 ⊕ S2B1, where S 2 B1 is the symmetric square of the multiplicative group of a ground field, and B2 — the Bloch group of this field. This is the first main result of the paper (see Theorems 2.12, 3.8 and 3.6.2).

On algebraic equations satisfied by correlators in Wess-Zumino-Witten models

Apart from the Knizhnik-Zamolodchikov differential equations, correlation functions in Wess-Zumino-Witten models of conformal field theory satisfy a certain system of algebraic equations. We show that hypergeometric solutions of KZ equations constructed in [3] do satisfy these algebraic equations.

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

Une Algèbre Quadratique Liée à la Suite de Sturm

\mathfrak {g}

Arrangements of hyperplanes and Lie algebra homology

g and study the corresponding schober-type diagram. For