Topological realizations of line arrangements

Abstract

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In this paper we investigate whether a configuration is realized by a collection of $2$-spheres embedded, in the symplectic, smooth, or topological (locally flat) categories, in the complex projective plane. We find obstructions to realizability in the topological category, which apply to configurations specified by all projective planes over a finite field. Such obstructions are used to show that certain contact graph manifolds are not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, $2$-spheres.