Gauss-Manin connections for arrangements, III Formal connections

Abstract. We study the Gauss-Manin connection for the moduli space of anarrangement of complex hyperplanes in the cohomology of a complex rank onelocal system. We define formal Gauss-Manin connection matrices in the Ao-moto complex and prove that, for all arrangements and all local systems, theseformal connection matrices specialize to Gauss-Manin connection matrices. 1. IntroductionLet A = {H 1 ,...,H n } be an arrangement of nordered hyperplanes in C l , andlet L be a local system of coefficients on M = M(A) = C l \S nj=1 H j , the comple-ment of A. The need to calculate the local system cohomology H • (M;L) arises invarious contexts. For instance, local systems may be used to study the Milnor fiberof the non-isolated hypersurface singularity at the origin obtained by coning thearrangement, see [8, 5]. In mathematical physics, local systems on complements ofarrangements arise in the Aomoto-Gelfand theory of multivariable hypergeometricintegrals [2, 12, 18] and the representation theory of Lie algebras and quantumgroups. These considerations lead to solutions of the Knizhnik-Zamolodchikov dif-ferential equation from conformal field theory, see [21, 23]. Here a central problemis the determination of the Gauss-Manin connection on H

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