The Goresky-MacPherson formula for toric arrangements

Abstract

A subspace arrangement is a finite collection of affine subspaces in $\mathbb{R}^n$. One of the main problems associated to arrangements asks up to what extent the topological invariants of the union of these spaces, and of their complement are determined by the combinatorics of their intersection. The most important result in this direction is due to Goresky and MacPherson. As an application of their stratified Morse theory they showed that the additive structure of the cohomology of the complement is determined by the underlying combinatorics. In this paper we consider toric arrangements; a finite collection of subtori in $(\mathbb{C}^{\ast})^l$. The aim of this paper is to prove an analogue of the Goresky-MacPherson's theorem in this context. When all the subtori in the arrangement are of codimension-$1$ we give an alternate proof of a theorem due to De Concini and Procesi.