Geometric relationship between cohomology of the complement of real and complexified arrangements

Let AR be a real hyperplane arrangement and let AC be its complexification. Let MR and MC be the respective complements. Then MR is the disjoint union of convex chambers whose number is given by its only Betti number, b0(MR). A real arrangement and its complexification satisfy the M-property: b0(MR)=∑qbq(MC), the number of chambers in MR equals the sum of the Betti numbers of MC. The no-broken-circuit set, nbc, is a field independent combinatorial object. It has been used to label a basis for H∗(MC) but not to label the chambers of MR in a way that makes the M-property explicit. In this paper we use the nbc set to label a combinatorial object in the nerve of the arrangement, which is field independent. This allows for simultaneous choices of nbc bases in H∗(MR) and H∗(MC). We also explore the geometrical connections between these bases.