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.
General
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