Toric and tropical compactifications of hyperplane complements

These lecture notes are based on lectures given by the author at the summer school "Arrangements in Pyr\'en\'ees" in June 2012. We survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel'fand-MacPherson construction, Kapranov's visible contours compactification, and De Concini and Procesi's wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry. The paper contains a few new arguments intended to make the presentation as self-contained as possible.Comment: 26 page

Equivariant Euler characteristics of discriminants of reflection groups

Let G be a finite, complex reflection group and f its discriminant polynomial. The fibers of f admit commuting actions of G and a cyclic group. The virtual $G\times C_m$ character given by the Euler characteristic of the fiber is a refinement of the zeta function of the geometric monodromy, calculated in a paper of Denef and Loeser. We compute the virtual character explicitly, in terms of the poset of normalizers of centralizers of regular elements of G, and of the subspace arrangement given by proper eigenspaces of elements of G. As a consequence, we compute orbifold Euler characteristics and find some new "case-free" information about the discriminant.Comment: 18 page

Moment-angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio

On the homotopy Lie algebra of an arrangement

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincar\'e polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras.Comment: 20 pages; accepted for publication by the Michigan Math. Journa