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

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 Z.The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the arrangement.

Homotopy transfer and formality

“…-This result about formality of complements of hyperplane arrangements and toric arrangements with coefficients in Z is new as far as the authors know. Let us mention however, that the paper [5] contains a result called integral formality for toric arrangements (see [5,Theorem 7.4]). We believe that our result and the result of [5] are independent.…”

“…Let us mention however, that the paper [5] contains a result called integral formality for toric arrangements (see [5,Theorem 7.4]). We believe that our result and the result of [5] are independent. If we denote by X the toric arrangement, our result is about formality of the singular cohomology algebra C * (X) whereas the result of [5] identifies a certain Z-subalgebra of the de Rham algebra of X that is isomorphic to the integral cohomology of X.…”

Two examples of toric arrangements

“…For each ordered subset S ⊆ A and each connected component W of ∩ H∈S H such that |S| = rk W , there exist a differential form ω W,S of degree |S| defined in [CDDMP18]. If S = {H i } then the form ω Hi,S coincides with 2ω i − ψ i .…”

On projective wonderful models for toric arrangements and their cohomology

“…Toric arrangements have been studied since the early 1990s, and over the last two decades several aspects have been investigated: in particular, as far as the topology of the complement is concerned, De Concini and Procesi [12] determined the generators of the cohomology modules over C in the divisorial case, as well as the ring structure in the case of totally unimodular arrangements; d'Antonio and Delucchi, generalizing an algebraic complex first introduced by Moci and Settepanella [25], provided a presentation of the fundamental group for the complement of a divisorial complexified arrangement [5,6]; Callegaro, Delucchi and Pagaria computed the graded cohomology ring with integer coefficients (see [3,4,26]); the cohomology ring itself was computed by Callegaro et al [2].…”

“…The first part (Sects. [2][3][4][5] of this paper is a short survey, enriched with examples, on the main results of the papers [8] and [9]: the construction of a projective wonderful model Y A for a toric arrangement A and the presentation of its cohomology ring by generators and relations.…”