On the topology and combinatorics of decomposable arrangements

A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the I-adic completion of the Alexander invariant $B(G)$ also decomposes as a direct sum of "local" invariants and the Chen ranks of G are the sums of the local contributions. Moreover, if $B(G)$ is separated, then the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.