Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements

TL;DR

This paper provides an explicit algebraic presentation for the integral cohomology of toric arrangement complements, extending classical hyperplane arrangement results to the toric setting and over integers.

Contribution

It introduces a new presentation for the integral cohomology ring of toric arrangements, generalizing Orlik-Solomon's hyperplane arrangement results and extending formality to integer coefficients.

Findings

Explicit integral cohomology presentation for toric arrangements

Extension of formality results over $\\mathbb{Z}$

Connection between cohomology and intersection poset

Abstract

We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over $\mathbb Z$. The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the arrangement.