## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1421-1432 |

Number of pages | 12 |

Journal | International Mathematics Research Notices |

Volume | 2009 |

Issue number | 8 |

DOIs | |

State | Published - 2009 |

## ASJC Scopus subject areas

- Mathematics(all)

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