Surface singularities and planar contact structures
Résumé
We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of An-singularities. In general, we characterize completely planar links of normal surface singularities (in terms of their resolution graphs); these singularities are precisely rational singularities with reduced fundamental cycle (also known as minimal singularities). We also establish non-planarity of tight contact structures on certain small Seifert fibered L-spaces and of contact structures arising from the Boothby--Wang construction applied to surfaces of positive genus. Additionally, we prove that every finitely presented group is the fundamental group of a Lefschetz fibration with planar fibers.