Sergey Yuzvinsky

Cohomology of the Orlik–Solomon Algebras and Local Systems

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.

Cohomology of the brieskorn-orlik-solomon algebras

Let V be an affine space of dimension ` over a field F and let A = {H1, H2, . . . , Hn} be a non-empty arrangement of hyperplanes of V . For each H ∈ A fix an affine functional αH such that kerαH = H and put αi = αHi . The main character of the paper is the graded F -algebra A = A(A) = ⊕p=0Ap generated by the differential forms ωi = dαi/αi ∈ A1. If F = C then, according to Brieskorn’s theorem [2], this algebra is isomorphic under the de Rham map to the cohomology algebra of M = V \ ⋃ni=1Hi. Explicit and pure combinatorial description of this algebra has been given by Orlik and Solomon [6] and is presented in detail in Section 3 of [7]. For every λ = (λ1, . . . , λn) ∈ F n the left multiplication dλ by ωλ = ∑n i=1 λiωi defines a cochain complex (A, dλ) 0 → A0 dλ → A1 dλ → · · · dλ → A` → 0. The goal of this paper is to study the cohomology H = H(A, dλ) of this complex. The study of H is motivated by [4] and [5]. These papers are concerned with H∗(M,L) for F = C where L is a local system on M . The cohomology is used in theory of hypergeometric functions and Knizhnik-Zamolodchikov equations. Kohno [5] proved that if L is the local system of flat sections of the trivial bundle with respect to the connection d+ωλ, then under a certain genericity condition on λ, H(M,L) = 0 for p < `. Also if A is real and transverse to the hyperplane at infinity, he found a basis of H(M,L) that does not depend on λ. Then Esnault, Schechtman, and Viehweg [4] proved that under a weaker genericity condition on λ

KOSZUL ALGEBRAS FROM GRAPHS AND HYPERPLANE ARRANGEMENTS

This work was started as an attempt to apply theory from noncommutative graded algebra to questions about the holonomy algebra of a hyperplane arrangement. We soon realized that these algebras and their deformations, which form a class of quadratic graded algebras, have not been studied much and yet are interesting to algebra, arrangement theory and combinatorics.

Hyperplane arrangement cohomology and monomials in the exterior algebra

We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.

Completely reducible hypersurfaces in a pencil

Abstract We study completely reducible fibers of pencils of hypersurfaces on P n and associated codimension one foliations of P n . Using methods from theory of foliations we obtain certain upper bounds for the number of these fibers as functions only of n . Equivalently this gives upper bounds for the dimensions of resonance varieties of hyperplane arrangements. We obtain similar bounds for the dimensions of the characteristic varieties of the arrangement complements.

Chow rings of toric varieties defined by atomic lattices

We study a graded algebra

Small rational model of subspace complement

D=D(\mathcal{L},\mathcal{G})

Derivations of an effective divisor on the complex projective line

over ℤ defined by a finite lattice ℒ and a subset

De Rham cohomology of logarithmic forms on arrangements of hyperplanes

\mathcal{G}

A free resolution of the module of derivations for generic arrangements

in ℒ, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice ℒ, as the Chow ring of a smooth toric variety that we construct from ℒ and