On Multivariate Picard–Fuchs Systems and Equations

Abstract

In this paper, we studied the Picard–Fuchs systems and equations which appear in the theory of Gauss–Manin systems and connections associated with deformations of isolated singularities. Among other things, we describe some interesting properties of such systems and relationships between them. Then we show how to calculate the fundamental solutions to the Gauss–Manin system for Aμ-singularities and to the corresponding generalized Legendre equations in terms of the multidimensional Horn’s hypergeometric functions. In conclusion, some important questions concerning basic properties of the local and global Picard–Fuchs systems of Pfaffian type, involving integrability conditions and commuting relations, are discussed in some detail.

Keywords

• Gauss–Manin systems and connections
• Legendre equation
• versal integrals
• period integrals
• deformations
• logarithmic differential forms