Abstract

Speaker: Paulo Lima-Filho, Texas A&M University

Title: On the holonomy Lie algebra of graphic arrangements

Abstract:

If X is the complement of a projective hypersurface, then the nilpotent completion of its fundamental group is isomorphic to the nilpotent completion of the holonomy Lie algebra of X, as shown by Kohno. In the particular case where X is the complement of a hyperplane arrangement A, the ranks \phi_k of the lower central series quotients of \pi_1(X) are well-known in only two very special cases: if the arrangement A is hypersolvable (a linear slice of an arrangement which admits a sequence of linear fibrations), or if the the holonomy Lie algebra decomposes in degree greater or equal to 2 as a direct product of local components.

The talk will describe jont work with Hal Schenck, where we use the holonomy Lie algebra to obtain an explicit combinatorial formula for the ranks \phi_k, whenever A is a graphic arrangement. This formula generalizes Kohno's result for braid arrangements, and provides the first instance of a lower central series formula for a large class of arrangements which are not decomposable or fiber-type.

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