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Gauss-Manin connections of Schläfli type for hypersphere arrangements

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  • Earth Science
  • Law
  • Mathematics

Abstract

Gauss-Manin connections of Schläfli type for hypersphere arrangements AN N A L E S D E L’INSTI T U T F O U R IE R ANNALES DE L’INSTITUT FOURIER Kazuhiko AOMOTO Gauss-Manin connections of Schläfli type for hypersphere arrangements Tome 53, no 4 (2003), p. 977-995. <http://aif.cedram.org/item?id=AIF_2003__53_4_977_0> © Association des Annales de l’institut Fourier, 2003, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ 977- GAUSS-MANIN CONNECTIONS OF SCHLÄFLI TYPE FOR HYPERSPHERE ARRANGEMENTS by Kazuhiko AOMOTO 1. Introduction. The theory of the hypergeometric integrals associated with hyper- plane arrangements has been developed by many authors, like P. Orlik, H. Terao , A. Varchenko, M. Yoshida, etc (see [9], [15]). An arrangement of one hypersphere and many hyperplanes is also interesting from geometric and combinatorial point of view. If we restrict this to the hypersphere, we have a general hypersphere arrangement. The purpose of this note is to present the Gauss-Manin connection of the hypergeometric integrals asso- ciated with a "generic" hypersphere arrangement in the unit hypersphere in an invariant form. This expression can be regarded as a natural exten- sion of the classical Schlafli formula for a geodesic spherical simplex. The author has given in [1] various formulae about the hypergeometric integrals involved in quadratic exponentials. This note heavily de

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