Orlik-Solomon Algebras of Hyperplane Arrangements

Let V be a finite-dimensional vector space over a field k. A hyperplane arrangement in V is a collection A = (H1, . . . , Hn) of codimension one affine subspaces of V . The arrangement A is called central if the intersection ⋂ Hi is nonempty; without loss of generality the intersection contains the origin. We will always denote by n the number of hyperplanes in the arrangement, and by l the dimension of the ambient space V . The bulk of this paper is devoted to proving the theorem of Orlik-Solomon and Brieskorn, here Theorem 4.4, which gives a presentation in terms of generators and relations for the cohomology ring of the complement of a complex hyperplane arrangement. Before tackling the proof of Theorem 4.4, however, it may be instructive to study a much simpler topological invariant of a real hyperplane arrangement, the number γ(A) of connected components of the complement. The components of the complement are convex subsets of R, hence contractible, so γ is the only interesting topological invariant of a real arrangement. It turns out that the number γ depends only on certain combinatorial data associated to the arrangement. The intersection poset L(A) is the set of all nonempty subspaces of V that arise as intersections of hyperplanes in A, partially ordered by inclusion. The ambient space V is included as the intersection the empty set of hyperplanes. In general, L does not have a unique minimal element, but if A is a central arrangement, then the intersection of all hyperplanes in A is the unique minimal element of L, and in this case L has the structure of a lattice (i.e. any two elements have a least upper bound and greatest lower bound). To show that γ depends only on the intersection poset L, and to see how to compute γ given L, it is useful to consider the analogous problem over finite fields. Given a hyperplane arrangement defined over Fq , how many points in F l q lie in its complement? We might näıvely begin counting such points as follows. Beginning with all q points in Fq , subtract q l−1 points for each hyperplane in the arrangement. To add back the points we’ve double-counted, we need to know, for each codimension-two subspace X, the number of hyperplanes Hi containing X. To go further, we need to keep track of not just the number of