On Formality and Combinatorial Formality for Hyperplane Arrangements

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autoren

  • Tilman Möller
  • Paul Mücksch
  • Gerhard Röhrle

Externe Organisationen

  • Ruhr-Universität Bochum
Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Seiten (von - bis)73-90
Seitenumfang18
FachzeitschriftDiscrete & computational geometry
Jahrgang72
Ausgabenummer1
Frühes Online-Datum17 Feb. 2023
PublikationsstatusVeröffentlicht - Juli 2024
Extern publiziertJa

Abstract

A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e., is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e., asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.

ASJC Scopus Sachgebiete

Zitieren

On Formality and Combinatorial Formality for Hyperplane Arrangements. / Möller, Tilman; Mücksch, Paul; Röhrle, Gerhard.
in: Discrete & computational geometry, Jahrgang 72, Nr. 1, 07.2024, S. 73-90.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Möller T, Mücksch P, Röhrle G. On Formality and Combinatorial Formality for Hyperplane Arrangements. Discrete & computational geometry. 2024 Jul;72(1):73-90. Epub 2023 Feb 17. doi: 10.1007/s00454-022-00479-5
Möller, Tilman ; Mücksch, Paul ; Röhrle, Gerhard. / On Formality and Combinatorial Formality for Hyperplane Arrangements. in: Discrete & computational geometry. 2024 ; Jahrgang 72, Nr. 1. S. 73-90.
Download
@article{0561fd338d5d4114a4d75d013fd14a86,
title = "On Formality and Combinatorial Formality for Hyperplane Arrangements",
abstract = "A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e., is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e., asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.",
keywords = "Combinatorial formality, Factored arrangements, Formality, Free arrangements, Hyperplane arrangements, K(π, 1)-Arrangements, k-Formality",
author = "Tilman M{\"o}ller and Paul M{\"u}cksch and Gerhard R{\"o}hrle",
note = "Publisher Copyright: {\textcopyright} 2023, The Author(s).",
year = "2024",
month = jul,
doi = "10.1007/s00454-022-00479-5",
language = "English",
volume = "72",
pages = "73--90",
journal = "Discrete & computational geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "1",

}

Download

TY - JOUR

T1 - On Formality and Combinatorial Formality for Hyperplane Arrangements

AU - Möller, Tilman

AU - Mücksch, Paul

AU - Röhrle, Gerhard

N1 - Publisher Copyright: © 2023, The Author(s).

PY - 2024/7

Y1 - 2024/7

N2 - A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e., is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e., asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.

AB - A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e., is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e., asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.

KW - Combinatorial formality

KW - Factored arrangements

KW - Formality

KW - Free arrangements

KW - Hyperplane arrangements

KW - K(π, 1)-Arrangements

KW - k-Formality

UR - http://www.scopus.com/inward/record.url?scp=85148370940&partnerID=8YFLogxK

U2 - 10.1007/s00454-022-00479-5

DO - 10.1007/s00454-022-00479-5

M3 - Article

VL - 72

SP - 73

EP - 90

JO - Discrete & computational geometry

JF - Discrete & computational geometry

SN - 0179-5376

IS - 1

ER -