Picard-Vessiot groups of Lauricella's hypergeometric systems $E_C$ and Calabi-Yau varieties arising integral representations

Abstract

We study the Zariski closure of the monodromy group $\mathbf{Mon}$ of Lauricella's hypergeometric function $F_C$. If the identity component ${\mathbf{Mon}}^0$ acts irreducibly, then $\overline{\mathbf{Mon}}\cap {\mathbf{SL}}_{2^n}(\mathbb{C})$ must be one of classical groups ${\mathbf{SL}}_{2^n}(\mathbb{C}),{\mathbf{SO}}_{2^n}(\mathbb{C})$ and ${\mathbf{Sp}}_{2^n}(\mathbb{C})$. We also study Calabi-Yau varieties arising from integral representations of $F_C$.

18 pages, 3 figures