Constructions of symplectic surfaces in symplectic 4-manifolds with transversal intersections
Vicente Muñoz, Juan Rojo
TL;DR
The paper develops a toolkit for constructing large families of symplectic surfaces in a symplectic 4-manifold with transversal intersections, aimed at producing ramification loci for Seifert fibrations that yield K-contact 5-manifolds (and, in particular, examples with K-contact but non-Sasakian structures). It introduces a complex-like local model for multiple intersections, a perturbation-based procedure to achieve nice, pairwise intersections, and then builds Kähler tubular neighborhoods to glue local models into global, holomorphic-compatible structures. It also provides concrete methods to realize symplectic divisors as smooth symplectic surfaces, including explicit genus and self-intersection formulas for multiples of a given curve and general divisors, via line-bundle sections and resolution of intersections. Together these tools refine and expand the constructions used to address the geography of K-contact and Sasakian Smale-Barden manifolds, underpinning new examples and structural insights.
Abstract
In the breakthrough paper [V. Muñoz, A Smale-Barden manifold admitting K-contact but not Sasakian structure, 2024, 10.4171/JEMS/1496], it is constructed the first example of a simply connected compact 5-manifold (aka.\ Smale-Barden manifold) which admits a K-contact structure but does not carry a Sasakian structure, thus settling the question raised as Open Problem 10.2.1 in n [C. Boyer and K. Galicki, Sasakian Geometry, OUP, 2007]. In this paper we revise, refine and generalize the constructions of symplectic surfaces in a symplectic 4-manifold with transversal intersections. These are needed to produce the ramification locus of Seifert bundles over symplectic 4-orbifolds that serve to produce K-contact 5-manifolds.