The fundamental group of the complement of the singular locus of Lauricella's $F_C$

Abstract

We study the fundamental group of the complement of the singular locus of Lauricella's hypergeometric function $F_C$ of $n$ variables. The singular locus consists of $n$ hyperplanes and a hypersurface of degree $2^{n-1}$ in the complex $n$-space. We derive some relations that holds for general $n\geq 3$. We give an explicit presentation of the fundamental groupin the three-dimensional case. We also consider a presentation of the fundamental group of $2^3$-covering of this space. In the version 2, we omit some of the calculations. For all the calculations, refer to the version 1 (arXiv:1710.09594v1) of this article.