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 -