Intersection subgroups of complex hyperplane arrangements

✍ Scribed by Luis Paris

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 252 KB
• Volume: 105
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00068-1

✦ Synopsis

Let A be a central arrangement of hyperplanes in C n , let M(A) be the complement of A, and let L(A) be the intersection lattice of A. For X in L(A) we set A X = {H ∈ A: H ⊇ X}, and

We call the images of these monomorphisms intersection subgroups of type X and prove that they form a conjugacy class of subgroups of π 1 (M(A)).

Recall that

such that X µ is modular and dim X µ = µ for all µ = 1, . . . , d. Assume that X is supersolvable and view π 1 (M(A X )) as an intersection subgroup of type X of π 1 (M(A)). Recall that the commensurator of a subgroup S in a group G is the set of a in G such that S ∩ aSa -1 has finite index in both S and aSa -1 . The main result of this paper is the characterization of the centralizer, the normalizer, and the commensurator of π 1 (M(A X )) in π 1 (M(A)). More precisely, we exhibit an embedding of π 1 (M(A X )) in π 1 (M(A)) and prove:

(1)

(2) the normalizer is equal to the commensurator and is equal to the direct product of π 1 (M(A X )) and π 1 (M(A X ));

(3) the centralizer is equal to the direct product of π 1 (M(A X )) and the center of π 1 (M(A X )).

Our study starts with an investigation of the projection p : M(A) → M(A/X) induced by the projection C n → C n /X. We prove in particular that this projection is a locally trivial C ∞ fibration if X is modular, and deduce some exact sequences involving the fundamental groups of the complements of A, of A/X, and of some (affine) arrangement A X z 0 .