Quasi-regular Sasakian and K-contact structures on Smale–Barden manifolds

• Alejandro Cañas ^[1] ; Vicente Muñoz ^[1] ; Matthias Schütt ^[2] ; Aleksy Tralle ^[3]
  1. [1] Universidad de Málaga

     Universidad de Málaga

     Málaga, España

     
  2. [2] University of Hannover

     University of Hannover

     Region Hannover, Alemania

     
  3. [3] University of Warmia and Mazury in Olsztyn

     University of Warmia and Mazury in Olsztyn

     Olsztyn, Polonia

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 3, 2022, págs. 1029-1050
• Idioma: inglés
• DOI: 10.4171/RMI/1335
• Texto completo no disponible (Saber más ...)
• Resumen

  • Smale–Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale–Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Kollár). Recently, the second named author of this article started the systematic development of quasi-regular Seifert fibrations, that is, over orbifolds which are not necessarily smooth. The present work is devoted to several applications of this theory. First, we develop constructions of a Smale–Barden manifold admitting a quasi-regular Sasakian structure but not a semi-regular K-contact structure. Second, we determine all Smale–Barden manifolds that admit a null Sasakian structure. Finally, we show a counterexample in the realm of cyclic Kähler orbifolds to the algebro-geometric conjecture by Muñoz, Rojo and Tralle that claims that for an algebraic surface with b1=0 and b2>1 there cannot be b2 smooth disjoint complex curves of genus g>0 spanning the (rational) homology.