Topological realizations of line arrangements

TL;DR

This paper explores the realizability of combinatorial line arrangements as embedded 2-spheres in complex projective space, identifying topological obstructions and linking to symplectic fillability.

Contribution

It introduces new topological obstructions to realizing certain line arrangements as embedded spheres in the complex projective plane, especially over finite fields.

Findings

Obstructions to realizability in the topological category.

Certain contact graph manifolds are not symplectically fillable.

Pseudoline configurations can be complexified into symplectic spheres.

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.