[Springer Monographs in Mathematics] Theory of Hypergeometric Functions Volume 3841 || Representation of Complex Integrals and Twisted de Rham Cohomologies

✍ Scribed by Aomoto, Kazuhiko; Kita, Michitake

• Book ID: 111872571
• Publisher: Springer Japan
• Year: 2011
• Tongue: Japanese
• Weight: 893 KB
• Edition: 2011
• Category: Article
• ISBN: 4431539387
• DOI: 10.1007/978-4-431-53938-4_2

✦ Synopsis

This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne’s rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff’s classical theory on analytic difference equations on the other.