Graham Denham

Critical points and resonance of hyperplane arrangements

If �� is a master function corresponding to a hyperplane arrangement A and a collection of weights �, we investigate the relationship between the critical set of ��, the variety defined by the vanishing of the one-form !� = dlog ��, and the resonance of �. For arrangements satisfying certain conditions, we show that if � is resonant in dimension p, then the critical set of �� has codimension at most p. These include all free arrangements and all rank 3 arrangements.

Torsion in Milnor fiber homology

In a recent paper, Dimca and N emethi pose the problem of nding a homogeneous polynomial f such that the homology of the complement of the hypersurface dened by f is torsion-free, but the homology of the Milnor ber of f has torsion. We prove that this is indeed possible, and show by construction that for each prime p, there is a polynomial with p-torsion in the homology of the Milnor ber. The techniques make use of properties of characteristic varieties of hyperplane arrangements.

The Orlik–Solomon complex and Milnor fibre homology

Abstract Fix a real hyperplane arrangement, and let F be the Milnor fibre of its complexified defining polynomial. We consider a filtration of the homology of F that arises from the algebraic monodromy, using integer or Z /p Z coefficients. We compare it with the cohomology of the Orlik–Solomon algebra, over Z or Z /p Z , respectively, with respect to a suitable “Koszul” boundary map, and find isomorphisms in certain cases. This continues work by various authors in comparing the cohomology of certain local systems on a hyperplane complement with that of the Orlik–Solomon algebra.

Combinatorial covers and vanishing of cohomology

We use a Mayer–Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen–Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients and obtain new generalizations.

A note on de Concini and Procesi's curious identity

We give a short, case-free and combinatorial proof of de Concini and Procesis formula for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.

Local cohomology of logarithmic forms

Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms along Y, and their relationship with the property that Y is a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

Mini-Workshop: Cohomology Rings and Fundamental Groups of Hyperplane Arrangements, Wonderful Compactifications, and Real Toric Varieties

Toric manifolds. In a seminal paper [7] that appeared some twenty years ago, Michael Davis and Tadeusz Januszkiewicz introduced a topological version of smooth toric varieties, and showed that many properties previously discovered by means of algebro-geometric techniques are, in fact, topological in nature. Let P be an n-dimensional simple polytope with facets F1, . . . , Fm, and let χ be an integral n×m matrix such that, for each vertex v = Fi1 ∩ · · · ∩ Fin , the minor of columns i1, . . . , in has determinant ±1. To such data, there is associated a 2n-dimensional toric manifold, MP (χ) = T ×P/ ∼, where (t, p) ∼ (u, q) if p = q, and tu belongs to the image under χ : T → T n of the coordinate subtorus corresponding to the smallest face of P containing q in its interior. Here is an alternate description, using the moment-angle complex construction (see for instance [10] and references therein). Given a simplicial complex K on vertex set [n] = {1, . . . , n}, and a pair of spaces (X,A), let ZK(X,A) be the subspace of the cartesian product X, defined as the union ⋃ σ∈K(X,A) σ , where (X,A) is the set of points for which the i-th coordinate belongs to A, whenever i / ∈ σ. It turns out that the quasi-toric manifold MP (χ) is obtained from the moment angle manifold ZK(D , S), where K is the dual to ∂P , by taking the quotient by the relevant free action of the torus T = ker(χ).The purpose of this workshop was to bring together researchers with a common interest in the objects mentioned in the title from, respec- tively, the points of view of toric and tropical geometry, arrangement theory, and geometric group theory.

Abelian duality and propagation of resonance

We explore the relationship between a certain “abelian duality” property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the “EPY property”. The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups, and Bestvina–Brady groups. Our approach brings to the fore the relevance of the Cohen–Macaulay condition in this combinatorial context.

Homological Aspects of Hyperplane Arrangements

The purpose of this expository article is to survey some results and applications of free resolutions related to hyperplane arrangements. We include some computational examples and open problems.

Equivariant Euler characteristics of discriminants of reflection groups

Abstract Let G be a finite, complex reflection group acting on a complex vector space V, and δ its disciminant polynomial. The fibres of δ admit commuting actions of G and a cyclic group. The virtual G × Cm character given by the Euler characteristics of a fibre is a refinement of the zeta function of the geometric monodromy, calculated in [8]. We show that this virtual character is unchanged by replacing δ by a slightly more general class of polynomials. We compute it explicitly, by studying the poset of normalizers of centralizers of regular elements in G, and the subspace arrangement given by the proper eigenspaces of elements of G. As a consequence we also compute orbifold Euler characteristics and find some new ‘case-free’ information about the discriminant.