On the difference between real and complex arrangements

Abstract

If B is an arrangement of linear complex hyperplanes in C d , then the following can be constructed from knowledge of its intersection lattice: (a) the cohomology groups of the complement (Br), (b) the cohomology algebra of the complement (OS), (c) the fundamental group of the complement, if d ≤ 2, (d) the singularity link up to homeomorphism, if d ≤ 3, (e) the singularity link up to homotopy type (ZZ). If Bis, more generally, a 2-arrangement in IR 2d (an arrangement of real subspaces of codimension 2 with even-dimensional intersections), then the intersection lattice still determines (a) the cohomology groups of the complement (GM) and (e) the homotopy type of the singularity link (ZZ). We show, however, that for 2-arrangements the data (b), (c) and (d) are not deter- mined by the intersection lattice. They require the knowledge of extra information on sign patterns, which can be computed as determinants of linear relations, or (equiva- lently) as linking coefficients in the sense of knot theory.