Gauss-Manin connections of Schläfli type for hypersphere arrangements
[Connexions de Gauss-Manin de type Schläfli pour des arrangements hypersphériques]
Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 977-995.

Nous donnons la structure cohomologique des arrangements hypersphériques. Les connexions de Gauss-Manin pour les intégrales hypergéométriques associées sont données en termes de formes invariantes. Elles sont utilisées pour obtenir la formule explicite du volume d'un simplexe dont les faces sont hypersphériques.

The cohomological structure of hypersphere arragnements is given. The Gauss-Manin connections for related hypergeometrtic integrals are given in terms of invariant forms. They are used to get the explicit differential formula for the volume of a simplex whose faces are hyperspheres.

DOI : 10.5802/aif.1970
Classification : 32S22, 32S40, 33C70, 39B32
Keywords: hypersphere arrangements, twisted cohomology, Gauss-Manin connection
Mot clés : arrangements hypersphériques, cohomologie des n{\oe}uds, connexion de Gauss-Manin
Aomoto, Kazuhiko 1

1 University of Nagoya, Department of Mathematics, Nagoya (Japon)
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Aomoto, Kazuhiko. Gauss-Manin connections of Schläfli type for hypersphere arrangements. Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 977-995. doi : 10.5802/aif.1970. http://archive.numdam.org/articles/10.5802/aif.1970/

[1] K. Aomoto Configurations and invariant Gauss-Manin connections of integrals I, Tokyo J. of Math., Volume 5 (1982), pp. 249-287 | DOI | MR | Zbl

[1] K. Aomoto Configurations and invariant Gauss-Manin connections of integrals. II., Tokyo J. of Math., Volume 6 (1983), pp. 1-24 | DOI | MR | Zbl

[2] K. Aomoto Vanishing of certain 1-form attached to a configuration, Tokyo J. of Math, Volume 9 (1986), pp. 453-455 | DOI | MR

[3] K. Aomoto Errata to "Configurations and invariant Gauss-Manin connections of integrals. I", Tokyo J. of Math, Volume 22 (1999), pp. 511-512 | MR | Zbl

[4] K. Aomoto Hyperlogarithmic expansion and the volume of a hyperbolic Simplex, Partial Differential Equations, Volume 27 (1992), pp. 9-21 | Zbl

[5] K. Aomoto; M. Kita; P. Orlik; H. Terao Twisted de Rham cohomology groups of logarithmic forms, Adv. in Math, Volume 128 (1997), pp. 119-152 | DOI | MR | Zbl

[6] D.C. Cohen; P. Orlik Gauss-Manin connections for arrangements, II nonresonant weights (2002) (preprint) | MR | Zbl

[7] H. Horiuchi; H. Terao The Poincaré series of the algebra of rational functions which are regular outside hyperplanes (2002) (preprint, March) | MR | Zbl

[8] Yi Hu On the homology of complements of arrangements of subspaces and spheres, Proc. Amer. Math. Soc, Volume 122 (1994), pp. 285-290 | DOI | MR | Zbl

[9] P. Orlik; H. Terao Arrangements and Hypergeometric Integrals, MSJ memoirs (2001) no. 9 | MR | Zbl

[10] P. Orlik; H. Terao Commutative algebras for arrangements, Nagoya Math. J, Volume 134 (1994), pp. 65-73 | MR | Zbl

[11] F. Pham Introduction à l'Etude Topologique des Singularités de Landau, Gauthiers Villars, 1967 | MR | Zbl

[12] C. Sabbah On the comparison theorem for elementary irregular 𝒟-modules, Nagoya Math. J, Volume 141 (1996), pp. 107-124 | MR | Zbl

[13] H. Terao Moduli space of combinatorially equivalent arrangements of hyperplanes and logarithmic Gauss-Manin connections, Topology and its Appl., Volume 118 (2002), pp. 255-274 | DOI | MR | Zbl

[14] H. Terao Algebras generated by reciprocals of linear forms, J. of Algebra, Volume 250 (2002), pp. 549-558 | DOI | MR | Zbl

[15] K. Yoshida Hypergeometric Functions, My Love, Vieweg Verlag, 1997 | Zbl

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