Michitake Kita

On vanishing of the twisted rational de Rham cohomology associated with hypergeometric functions

Recent development in hypergeometric functions in several variables has made the importance of studying twisted rational de Rham cohomology clear to many specialists. Roughly speaking, a hypergeometric function in our sense is the integral of a product of complex powers of polynomials P j (u 1 , . . . . , u n ) : ∫ U du 1 ∧ · · · ∧ du n , U = Π , integration being taken over some cycle. So we are led naturally to consider the twisted rational de Rham cohomology, which is a direct generalization of the usual de Rham cohomology to multivalued case.

Introduction: the Euler−Gauss Hypergeometric Function

The binomial series \({(1 + x)}^{\alpha } ={ \sum \nolimits }_{n=0}^{\infty }\frac{\alpha (\alpha - 1)\cdots (\alpha - n + 1)} {n!} {x}^{n},\quad \vert x\vert < 1\) is the generating function of binomial coefficients \(\left (\begin{array}{*{10}c} \alpha \\ n \end{array} \right )=\frac{\alpha (\alpha - 1)\cdots (\alpha - n + 1)} {n!}.\)A hypergeometric function can be regarded as a generating analytic function of more complicated combinatorial numbers which generalizes the binomial series. By studying its analytic structure, it provides us with information such as relations among combinatorial numbers and their growth. The aim of this book is to treat hypergeometric functions of several variables as complex analytic functions. Hence, we assume that the reader is familiar with basic facts about complex functions.

Representation of Complex Integrals and Twisted de Rham Cohomologies

In integral representations of Euler type of classical hypergeometric functions of several variables or of hypergeometric functions which are studied these days, integrals of the product of powers of polynomials appear. We will establish a framework to treat such integrals, and after that, we will study hypergeometric functions of several variables as an application of the theory. Since ordinary theory of integrals of single-valued functions is formalized under the name of the de Rham theory, by modifying this theory, we will constuct a theory suitable for our purpose in this chapter. As the key to the de Rham theory is Stokes theorem, we will start by posing the question how to formulate Stokes theorem for integrals of multi-valued functions.

Holonomic Difference Equations and Asymptotic Expansion

As we have seen in Chapter 1, the Γ-function is a solution of a first-order difference equation which can be uniquely determined by its asymptotic behavior at infinity. This fact can be generalized to the cases of several variables that contain a finite number of unknown meromorphic functions which satisfy a holonomic system of difference equations. The hypergeometric functions discussed in Chapters 2 and 3 satisfy holonomic systems of difference equations with respect to the parameters α = (α1, …, α m ). Their asymptotic structure at infinity strongly reflects topological aspects of their twisted de Rham (co)homology.