Topological Realizations of Line Arrangements
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 lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isoto...
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Veröffentlicht in: | International mathematics research notices 2019-04, Vol.2019 (8), p.2295-2331 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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 lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to $\mathbb R\rm{P}^1$) in the real projective plane. In this article we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is 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. |
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ISSN: | 1073-7928 1687-0247 |
DOI: | 10.1093/imrn/rnx190 |