The diffeomorphic types of the complements of arrangements in CP 3 I: Point arrangements

Abstract

For any arrangement of hyperplanes in ${\mathbf{CP}}^3$, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. If the soul consists of a set of points (0-simplices) and a set of planes (2-simplices), then the arrangement is called point arrangement. In this paper, we give a sufficient combinatoric condition for two point arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient condition on combinatorics for the point arrangement of hyperplanes whose moduli space is connected.