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Affine hyperplane arrangements at finite distance

Anaëlle Pfister

TL;DR

The paper develops a concrete, Orlik–Solomon–type description of the cohomology at finite distance for the complement of an affine hyperplane arrangement, linking it to the kernel of the boundary map on the Orlik–Solomon algebra and to the kernel of residues at infinity via suitable compactifications. It introduces partial wonderful compactifications at infinity to bridge two kernel descriptions and proves their equivalence, with a Brieskorn-type decomposition guiding the structure. The authors further connect these cohomological structures to canonical forms in the sense of positive geometry, providing explicit formulas for canonical forms as sums over non-broken circuits and showing their logarithmic, residue-vanishing nature. Together, these results illuminate the interplay between combinatorial data of the arrangement, the geometry at infinity, and the differential-geometric realization of forms, with implications for cosmological correlators and related areas. The work thus extends the OS framework to non-compact ambient spaces, clarifying how finite-distance cohomology captures bounded-region combinatorics and residues, and offering practical tools for constructing canonical differential forms on affine arrangement complements.

Abstract

We study the relative homology group of an affine hyperplane arrangement and its Poincaré 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.

Affine hyperplane arrangements at finite distance

TL;DR

The paper develops a concrete, Orlik–Solomon–type description of the cohomology at finite distance for the complement of an affine hyperplane arrangement, linking it to the kernel of the boundary map on the Orlik–Solomon algebra and to the kernel of residues at infinity via suitable compactifications. It introduces partial wonderful compactifications at infinity to bridge two kernel descriptions and proves their equivalence, with a Brieskorn-type decomposition guiding the structure. The authors further connect these cohomological structures to canonical forms in the sense of positive geometry, providing explicit formulas for canonical forms as sums over non-broken circuits and showing their logarithmic, residue-vanishing nature. Together, these results illuminate the interplay between combinatorial data of the arrangement, the geometry at infinity, and the differential-geometric realization of forms, with implications for cosmological correlators and related areas. The work thus extends the OS framework to non-compact ambient spaces, clarifying how finite-distance cohomology captures bounded-region combinatorics and residues, and offering practical tools for constructing canonical differential forms on affine arrangement complements.

Abstract

We study the relative homology group of an affine hyperplane arrangement and its Poincaré 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.
Paper Structure (21 sections, 24 theorems, 131 equations, 5 figures)

This paper contains 21 sections, 24 theorems, 131 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Conventions
  4. Acknowledgements
  5. Preliminaries on Orlik--Solomon Algebras
  6. The case of a central arrangement
  7. The case of an affine arrangement
  8. The case of a projective arrangement
  9. Cohomology at finite distance
  10. Cohomology at finite distance of a complex arrangement
  11. Relative homology of a real arrangement
  12. Residues at infinity
  13. Cohomology at finite distance in terms of a compactification
  14. Vanishing of residue at infinity
  15. Partial wonderful compactifications at infinity
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let ${\mathscr{A}} \subseteq {\mathbb{C}}^n$ be an essential affine hyperplane arrangement. Then, the only non-vanishing cohomology group is the middle one, i.e., Furthermore,

Figures (5)

  • Figure 1: The arrangement ${\mathscr{A}}.$
  • Figure 2: An example of a partial wonderful compactification at infinity. From left to right, the arrangements ${\mathscr{A}}$ in ${\mathbb{C}}^2$, $\tilde{{\mathscr{A}}}$ in ${\mathbb{P}}_{\mathbb{C}}^2$ and ${\overline{\mathscr{A}}}$ in $X$.
  • Figure 4: The arrangement ${{\mathscr{B}}}$
  • Figure 5: The arrangement $\tilde{{\mathscr{B}}} \cap L_\infty$ in $L_\infty$
  • Figure 6: From left to right, the arrangements $\overline{{\mathscr{B}}} \cap E_{P_{123}}$, and $\overline{{\mathscr{B}}} \cap E_{P_{345}}.$

Theorems & Definitions (68)

  • Remark 1.1
  • Theorem 1.2: \ref{['thmcharacterization']} and \ref{['propchar']}
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • ...and 58 more