Deletion-restriction in toric arrangements

Abstract

Deletion-restriction is a fundamental tool in the theory of hyperplane arrangements. Various important results in this field have been proved using deletion-restriction. In this paper we use deletion-restriction to identify a class of toric arrangements for which the cohomology algebra of the complement is generated in degree 1. We also show that for these arrangements the complement is formal in the sense of Sullivan.

Purity, Formality, and Arrangement Complements

“…Their proof uses differential forms and goes along the same lines as the one for hyperplane arrangements that we have described in §3.3. This was later generalized by Deshpande and Sutar [DS14] to a broader class of toric arrangements, called deletion-restriction type. It is natural to ask if these proofs can be extended to more general toric arrangements.…”

“…De Concini and Procesi [DCP05] already proved a special case of Theorem 1.3, namely the case of unimodular toric arrangements; this was later generalized to deletion-restriction type toric arrangements by Deshpande and Sutar [DS14]. In both cases, formality is proved by exhibiting an algebra of closed differential forms that maps bijectively to the cohomology of the complement of the arrangement, exactly as in the case of hyperplane arrangements.…”

The integer cohomology algebra of toric arrangements

“…We thus start with a brief discussion of the effect on the cohomology of removing an hypertorus from the arrangement and of restricting the arrangement to an hypertorus. This type of operation has been investigated by Bibby in [2] and by Deshpande and Sutar in [15]. Here we discuss how some of their results generalize to cohomology with integer coefficients, and start with a remark on degeneration of spectral sequences.…”

“…The question of whether a cohomology ring is generated in degree one is natural and well-studied. For toric arrangements, this question has been addressed also in [11,15].…”

“…Lately, Deshpande and Sutar [15], by an explicit study of the Gysin sequence, gave a sufficient criterion for the complex cohomology algebra of a toric arrangement to be generated in first degree and to be formal.…”