Quasi-regular sasakian and K-contact structures on smale-barden manifolds

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  • Universidad de Malaga
  • University of Warmia and Mazury
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Original languageEnglish
Pages (from-to)1029-1050
Number of pages22
JournalRevista Matematica Iberoamericana
Volume38
Issue number3
Publication statusPublished - 26 Jan 2022

Abstract

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 (Koll\'ar). 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 K\"ahler orbifolds to the algebro-geometric conjecture 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.

Keywords

    math.DG, math.AG, math.SG, 53C25, 53D35, 14J28, 14J17, cyclic orbifold, K-contact, K3 surface, Sasakian, Smale–Barden manifold

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Quasi-regular sasakian and K-contact structures on smale-barden manifolds. / Cañas, Alejandro; Muñoz, Vicente; Schütt, Matthias et al.
In: Revista Matematica Iberoamericana, Vol. 38, No. 3, 26.01.2022, p. 1029-1050.

Research output: Contribution to journalArticleResearchpeer review

Cañas A, Muñoz V, Schütt M, Tralle A. Quasi-regular sasakian and K-contact structures on smale-barden manifolds. Revista Matematica Iberoamericana. 2022 Jan 26;38(3):1029-1050. doi: 10.4171/RMI/1335
Cañas, Alejandro ; Muñoz, Vicente ; Schütt, Matthias et al. / Quasi-regular sasakian and K-contact structures on smale-barden manifolds. In: Revista Matematica Iberoamericana. 2022 ; Vol. 38, No. 3. pp. 1029-1050.
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abstract = "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 (Koll\'ar). 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 K\{"}ahler orbifolds to the algebro-geometric conjecture 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. ",
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N1 - Funding Information: Funding. The first author is supported by a PhD grant from Universidad de Málaga. The second author was partially supported by Project MINECO (Spain) PGC2018-095448-BI00. The fourth author was supported by the National Science Center (Poland), grant NCN no. 2018/31/B/ST1/00053.

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