Affine hyperplane arrangements at finite distance

TL;DR

This paper explores the cohomology of affine hyperplane arrangements at finite distance, providing a new algebraic description and linking it to canonical forms in positive geometry.

Contribution

It introduces an Orlik--Solomon-type description for the cohomology at finite distance and a partial compactification approach applicable in broader contexts.

Findings

Cohomology at finite distance is identified with logarithmic forms with vanishing residues.

A partial version of wonderful compactifications is introduced.

Cohomology coincides with the space of canonical forms in positive geometry.

Abstract

We study the relative homology group of an affine hyperplane arrangement and its Poincar\'e dual, the cohomology at finite distance of the complement. We give an Orlik--Solomon-type description of the latter, and identify it with the vector space of logarithmic forms having vanishing residues at infinity. To this end, we introduce a partial version of wonderful compactifications, which could be relevant in other contexts where blow-ups only occur at infinity. Finally, we show that the cohomology at finite distance coincides with the vector space of canonical forms in the sense of positive geometry.