Relative twisted homology and cohomology groups associated with Lauricella's $F_D$

Abstract

We introduce relative twisted homology and cohomology groups associated with Euler type integrals of solutions to Lauricella's system $F_D(a,b,c)$ of hypergeometric differential equations. We define an intersection form between relative twisted homology groups and that between relative twisted cohomology groups, and show their compatibility. We prove that the relative twisted homology group is canonically isomorphic to the space of local solutions to $F_D(a,b,c)$ for any parameters $a,b,c$. Through this isomorphism, we study $\cF_D(a,b,c)$ by the relative twisted homology and cohomology groups and the intersection forms without any conditions on $a,b,c$.