Intersection numbers for twisted homology (Q1882768)

Gel'fand-Aomoto hypergeometric functions are defined as integrals of products of complex powers of linear functions \(U=\prod_{j=1}^m f_j^{\lambda_j}\) on \(\mathbb C^n\). These integrals have an interpretation as pairings between twisted (loaded) cycles and twisted differential forms. Let \(M:=\mathbb C^n \backslash \bigcup_j\{f_j=0\}\) be the complement of the associated hyperplane arrangement. A twisted cycle is a topological cycle in \(M\) with a branch of the multivalued function \(U\) on it, and the homology of the associated complex is called the twisted homology. Twisted cohomology is the cohomology of the local system associated to \(U\) and is dual to the twisted homology group. Study of twisted homology and cohomology groups gives information about the associated hypergeometric function. In this article, the author proves some conjectures posed by Aomoto on intersection numbers oftwisted cycles in \(M\). The proof is by reduction to an intersection number formula due to Kita and Yoshida.