Theorem 1.1 [Zaslavsky 1975]. Let A be a hyperplane arrangement in Rn . The number of regions into which A dissects Rn is equal to (−1)χA(−1). The number of regions which are relatively bounded is equal to (−1)χA(1). Theorem 1.2 [Orlik and Solomon 1980]. Let A be a hyperplane arrangement in Cn , and let MA =Cn− ⋃ H∈A H be its complement. Then the Poincaré polynomial of the cohomology ring of MA...

A classical result of Arnold and Brieskorn [Bk], [Bk2] states that the complement of the discriminant of the versal unfolding of a simple hypersurface singularity is a K(π, 1). Deligne [Dg] showed this result could be placed in the general framework by proving that the complement of an arrangement of reflecting hyperplanes for a Coxeter group is again a K(π, 1) (and more generally for simplicia...

We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of n hyperplanes in an r-dimensional linear space is min{n + 1, 2r}.

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 dim...

— There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the h...

Let A = {H1, ..., Hd} be an affine essential hyperplane arrangement in C , see [OT1], [OT2] for general facts on arrangements. We set as usual M = M(A) = C\X, X being the union of all the hyperplanes in A. One of the main problems now in hyperplane arrangement theory is to study the cohomology of the complement M with coefficients in some local system L on M , see for instance the introduction ...

The cohomology ring of the complement of a central complex hyperplane arrangement is the well-studied Orlik-Solomon algebra. The homotopy group of the complement is interesting, complicated, and few results are known about it. We study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field C over the Orlik-Solomon algebra.

R. Thomas proved that the Hodge conjecture is essentially equivalent to the existence of a Thomas hyperplane section having only ordinary double points as singularities and such that the restriction of a given primitive Hodge class to it does not vanish. We show that the relations between the vanishing cycles associated to the ordinary double points of a Thomas hyperplane section have the same ...

In this article there are two main results. The first result gives a formula, in terms of a log resolution, for the graded pieces of the Hodge filtration on the cohomology of a unitary local system of rank one on the complement of an arbitrary divisor in a smooth projective complex variety. The second result is an application of the first. We give a combinatorial formula for the spectrum of a h...