The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements
Abstract
We construct an Orlik--Solomon algebra for any locally geometric poset as a natural generalization of the one for geometric lattices. This algebra has several interesting features, including a combinatorial no-broken-circuit basis that we obtain through Gröbner basis theory. When the poset captures the intersection data of an arrangement of certain subgroups in a complex abelian Lie group, we infuse the Orlik--Solomon algebra with topological information to compute the cohomology of the arrangement complement. In particular, when the Lie group is compact, we present an explicit differential graded algebra whose cohomology is the rational cohomology of the arrangement complement. We further exhibit a family of abelian arrangements whose integer cohomology has torsion.
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BibTeX
@misc{bibby2026orliksolomon,
title = {{The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements}},
author = {Christin Bibby and Peter Ramsey},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.23384},
eprint = {2608.23384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23384},
}
RIS
TY - UNPB
TI - The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements
AU - Christin Bibby
AU - Peter Ramsey
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.23384
UR - https://arxiv.org/abs/2608.23384
AB - We construct an Orlik--Solomon algebra for any locally geometric poset as a natural generalization of the one for geometric lattices. This algebra has several interesting features, including a combinatorial no-broken-circuit basis that we obtain through Gröbner basis theory. When the poset captures the intersection data of an arrangement of certain subgroups in a complex abelian Lie group, we infuse the Orlik--Solomon algebra with topological information to compute the cohomology of the arrangement complement. In particular, when the Lie group is compact, we present an explicit differential graded algebra whose cohomology is the rational cohomology of the arrangement complement. We further exhibit a family of abelian arrangements whose integer cohomology has torsion.
ER -