Proceedings of the Japan Academy, Series A, Mathematical Sciences

Some problems of hypergeometric integrals associated with hypersphere arrangement

Kazuhiko Aomoto and Yoshinori Machida

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The $n$ dimensional hypergeometric integrals associated with a hypersphere arrangement $S$ are formulated by the pairing of $n$ dimensional twisted cohomology $H_{\nabla}^{n} (X, \Omega^{\cdot} (*S))$ and its dual. Under the condition of general position there are stated some results and conjectures which concern a representation of the standard form by a special basis of the twisted cohomology, the variational formula of the corresponding integral in terms of special invariant 1-forms using Calyley-Menger minor determinants, a connection relation of the unique twisted $n$-cycle identified with the unbounded chamber to a special basis of twisted $n$-cycles identified with bounded chambers. General conjectures are presented under a much weaker assumption.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 6 (2015), 77-81.

First available in Project Euclid: 3 June 2015

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Zentralblatt MATH identifier

Primary: 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10] 33C70: Other hypergeometric functions and integrals in several variables
Secondary: 14H70: Relationships with integrable systems

Hypergeometric integral hypersphere arrangement twisted rational de Rham cohomology Cayley-Menger determinant contiguity relation Gauss-Manin connection


Aomoto, Kazuhiko; Machida, Yoshinori. Some problems of hypergeometric integrals associated with hypersphere arrangement. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 6, 77--81. doi:10.3792/pjaa.91.77.

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