The cohomology ring of the boundary manifold of a combinatorial line arrangement

TL;DR

This paper generalizes the understanding of the cohomology ring of boundary manifolds from complex projective line arrangements to all combinatorial line arrangements, including non-realizable ones, by explicitly constructing homology cycles.

Contribution

It extends Cohen--Suciu's result to arbitrary combinatorial line arrangements and provides explicit homology cycles for computing the cohomology ring.

Findings

Cohomology ring of boundary manifold is isomorphic to the double of the Orlik-Solomon algebra.

Constructed explicit homology cycles for arbitrary arrangements.

Derived results on the resonance variety of the boundary manifold.

Abstract

Cohen--Suciu proved that the cohomology ring of the boundary manifold of a complex projective line arrangement is isomorphic to the double of the cohomology ring of the complement. In this paper, we generalize this result to arbitrary combinatorial line arrangements, including non-realizable ones. The notion of the boundary manifold for combinatorial line arrangements was introduced by Ruberman--Starkston. To handle arbitrary combinatorial line arrangements, we construct explicit homology cycles following the method by Doig--Horn. Using these cycles, we compute the cohomology ring of the boundary manifold and prove that it is isomorphic to the double of the Orlik-Solomon algebra. As an application, we derive several results on the resonance variety of the boundary manifold.