Intersection subgroups of complex hyperplane arrangements

Let A be a central arrangement of hyperplanes in Cn , 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 AX={H∈A: H⫆X} , and A/X={H/X: H∈AX} , and AX={H∩X: H∈A\AX} . We exhibit natural embeddings of M(AX) in M(A) that give rise to monomorphisms from π1(M(AX)) to π1(M(A)) . We call the images of these monomorphisms intersection subgroups of type X and prove that they form a conjugacy class of subgroups of π1(M(A)) . that X in L(A) is modular if X+Y is an element of L(A) for all Y in L(A) . We call X in L(A) supersolvable if there exists a chain 0⫅X1⫅⋯⫅Xd=X in L(A) such that Xμ is modular and dim Xμ=μ for all μ=1,…,d . Assume that X is supersolvable and view π1(M(AX)) as an intersection subgroup of type X of π1(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 π1(M(AX)) in π1(M(A)) . More precisely, we exhibit an embedding of π1(M(AX)) in π1(M(A)) and prove: (M(AX))∩π1(M(AX))={1} and π1(M(AX)) is included in the centralizer of π1(M(AX)) in π1(M(A)) ; e normalizer is equal to the commensurator and is equal to the direct product of π1(M(AX)) and π1(M(AX)) ; e centralizer is equal to the direct product of π1(M(AX)) and the center of π1(M(AX)) . udy starts with an investigation of the projection p :M(A)→M(A/X) induced by the projection Cn→Cn/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 Az0X .