The Varchenko matrix for topoplane arrangements

Abstract

A topoplane is a mild deformation of a linear hyperplane contained in a given smooth manifold that is homeomorphic to a Euclidean space. We consider solidly transsective topoplane arrangements. These collections generalize pseudohyperplane arrangements. Even though the topoplane arrangements locally look like hyperplane arrangements, the global coning procedure is absent here. The main aim of the paper is to introduce the Varchenko matrix in this context and show that the determinant has a similar factorization as in the case of hyperplane arrangements. We achieve this by suitably generalizing the strategy of Aguiar and Mahajan. We also study a system of linear equations introduced by them and describe its solution space in the context of topoplane arrangements.