Holonomy lie algebras and the LCS formula for subarrangements of A _n

If X is the complement of a hypersurface in ℙ ⁿ, then Kohno showed in [11] that the nilpotent completion of π ₁(X) is isomorphic to the nilpotent completion of the holonomy Lie algebra of X. When X is the complement of a hyperplane arrangement A, the ranks φ κ of the lower central series quotients of π ₁(X) are known in only two very special cases: if X is hypersolvable (in which case the quadratic closure of the cohomology ring is Koszul), or if the holonomy Lie algebra decomposes in degree 3 as a direct product of local components. In this paper, we use the holonomy Lie algebra to obtain a formula for φκ when A is a subarrangement of A _n. This extends Kohno's result [12] for braid arrangements, and provides the first instance of an LCS formula for arrangements that are not decomposable or hypersolvable.