Cohomology of local systems on complex hyperplane complement
by
Sergey Yuzvinsky
University of Oregon

Cohomology of local systems on complex hyperplane complement

Sergey Yuzvinsky

University of Oregon

Eugene, OR 97403

yuz@math.uoregon.edu

Dedicated to the memory of V.A.Rokhlin

In the intersection of topology and algebraic geometry, a class of interesting examples is formed by the complements to divisors in complex projective spaces. In this talk we will focus on divisors that are unions of finite sets of hyperplanes (arrangement of hyperlanes). Topology of the complement M of such a union has been studied at least from 1969 when Arnold computed the integer cohomology ring of M for the braid arrangements given by equations x_i=x_j, 1 <= i < j <= l. In this case (as in many other cases) M is the K[\pi, 1]-space for \pi being a (colored) Braid group. Soon after that Brieskorn generalized this result to arbitrary arrangements. In particular he proved that the complex algebra of rational forms on the whole space generated by d(\alpha_i)/\alpha_i, where \alpha_i are linear polynomials whose 0-loci are the hyperplanes, is isomorphic under de Rham homomorphism to H^*(M;C). Then Orlik and Solomon gave an explicit presentation of this algebra proving in particular that it is defined by the combinatorics of the arrangement, more explicitly by the lattice consisting of all the intersections of hyperplanes ordered opposite to inclusion. This algebra became known as the Orlik-Solomon algebra.

A later development of the theme brought up the problems about the space H^*(M, L) where L is a rank one linear system of coefficients on M. There are several motivations for studying these spaces. For instance they define invariants of the fundamental group of M, namely its characteristic varieties. Another motivation comes from theory of multivariable hypergeometric functions developed at about the same time by two groups, led by Aomoto and Gelfand respectively. The knowledge when these cohomologies vanish and what they are generated by otherwise is very important for understanding hypergeometric integrals.

In the talk we give a systematic survey of recent results about these cohomology spaces and related questions. Most of unattributed results come from a join paper of the speaker with A.Libgober (to appear in Compositio Mathematica).

The starting point in the study of cohomology of local systems is the connection between it and the cohomology H^*(A, a) of the Orlik- Solomon algebra provided with the differential of multiplication by an element a in A of degree 1. Notice that the latter cohomology is defined by the lattice. Using a celebrated result by Deligne, the natural isomorphism

Hp(M, L(a))=Hp(A*, a)

(*)

was proved by Esnault, Schechtman, and Viehweg under certain general position conditions on a. Here L(a) is defined by the differential form represented by a. For an arbitrary a the dimension of the left hand side of (*) is not smaller than that of the right hand side which leads to Conjecture

dimHp(M, L(a))= sup N  dim Hp(A*, a+N)

(**)

where N runs through linear combinations of d(\alpha_i)/\alpha_i with integer coefficients. This conjecture though false in general has a promising version (see 5 below).

It was proved by Kohno and the speaker that both sides of (*) vanish for a general position a and for all but maximal dimensions p. More precisely the set of a for which the left (resp. right) side of (*) does not vanish for a given p (or have a large enough dimension) is for most p a proper algebraic subvariety of A₁ called characteristic (resp. resonance) variety. Most recent work consists of studying these varieties and comparing them to each other.

The deepest results in this direction were obtained for p=1 that can be viewed as cohomology of local systems on the complement of a complex plane projective curve consisting of several lines. The results about the characteristic varieties are based on theory of Alexander invariants for plane projective curves developed in several papers by Libgober and recent very general and deep results by Arapura. In the studies of the resonance varieties, quite surprisingly Vinberg's classification of Cartan-type matrices plays a very significant role. Combining these techniques one gets an interesting interplay between algebraic geometry and combinatorics. The following lists sums up the main known results (for p=1).

1. A resonance variety is the union of linear subspaces where each two intersect at 0. The space of 1-cocycles for a non-zero a from each linear space coincides with this space.

2. A resonance variety is the tangent cone at 1 to the respective characteristic variety (Cohen and Suciu).

3. For a positive dimensional irreducible component V of a characteristic variety that contains 1, the exponentiation defines the universal covering of V by a component of the respective resonance variety.

4. The equality (**) holds (for p=1) for all but finitely many local systems. The exceptional ones lie in finite subgroups of a torus that form discrete components of characteristic varieties.

5. The existence of a non-local characteristic variety of a positive dimension gives much information about the curve (arrangement of lines); for some cases even a classification is possible. By non-local variety we mean here a variety having components not supported on lines passing all through one point (the latter always form a so called local component). For instance if each line contains not more than 3 points of multiple intersection then the lines form either a braid arrangement or a subarrangement of the Hasse arrangement consisting of 12 lines passing each through 3 of 9 inflection points of a smooth cubic.

Date received: May 5, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-05.