Holonomy Lie algebras and the LCS formula for subarrangements of A_n

TL;DR

This paper extends the understanding of the lower central series ranks of fundamental groups of hyperplane arrangement complements, providing a new formula for subarrangements of A_n using holonomy Lie algebras.

Contribution

It introduces a formula for the lower central series ranks of subarrangements of A_n, expanding Kohno's results beyond hypersolvable and decomposable arrangements.

Findings

Provides LCS formula for subarrangements of A_n

Extends Kohno's results to non-decomposable arrangements

Offers new insights into the structure of holonomy Lie algebras

Abstract

If X is the complement of a hypersurface in P^n, then Kohno showed that the nilpotent completion of the fundamental group 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 phi_k of the lower central series quotients of the fundamental group of X are known for isolated examples, and for two special classes: 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 three as a direct product of local components. In this paper, we use the holonomy Lie algebra to obtain a formula for phi_k when A is a subarrangement of A_n. This extends Kohno's result for braid arrangements, and provides an instance of an LCS formula for arrangements which are not decomposable or hypersolvable.