[Paper Review] Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds

Alejandro Cañas, Vicente Muñoz+2 more|arXiv (Cornell University)|Apr 27, 2020

Geometric and Algebraic Topology24 references4 citations

TL;DR

This paper develops quasi-regular Seifert fibrations over singular orbifolds to construct Smale-Barden 5-manifolds with quasi-regular Sasakian structures that do not admit semi-regular K-contact structures. It provides the first example of a manifold with a quasi-regular Sasakian structure but no semi-regular K-contact structure, and classifies null Sasakian structures on 5-manifolds via orbifold K3 surfaces with cyclic singularities and trivial orbifold fundamental group.

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 (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 that claims that for an algebraic surface with $b_1=0$ and $b_2>1$ there cannot be $b_2$ smooth disjoint complex curves of genus g>0 spanning the (rational) homology.

Motivation & Objective

• To address the open question of whether there exist Smale-Barden manifolds with quasi-regular Sasakian structures but no semi-regular K-contact structures.
• To extend the classification of Sasakian and K-contact structures on 5-manifolds beyond the semi-regular case by developing a theory of quasi-regular Seifert fibrations over singular orbifolds.
• To classify null Sasakian structures on simply connected 5-manifolds by analyzing orbifold K3 surfaces with cyclic singularities and trivial orbifold fundamental group.
• To provide a counterexample to an algebro-geometric conjecture regarding the maximum number of disjoint genus-g curves on a surface with b1=0 and b2>1, using cyclic Kähler orbifolds.

Proposed method

• Utilizes quasi-regular Seifert fibrations over Kähler orbifolds with non-smooth singularities, generalizing the semi-regular case studied by Kollár.
• Constructs a 5-manifold M as a Seifert bundle over a K3 surface X with an A19 singularity, obtained by contracting a configuration of 19 (−2)-curves from a K3 surface X′.
• Ensures the orbifold fundamental group π₁^orb(X) = 1 by verifying topological conditions (Z1) and (Z2) on the fiber configuration, using Theorem 41.
• Applies Theorem 13 to construct a Sasakian structure on M using a primitive Kähler class in H²(X−P, ℤ), ensuring the Reeb flow is quasi-regular.
• Employs the fact that H₁(M, ℤ) = 0 and π₁^orb(X) = 1 to conclude that M is simply connected and diffeomorphic to #₂(S²×S³).
• Analyzes extremal elliptic K3 fibrations with A-type configurations to classify null Sasakian structures via orbifold invariants and Mordell-Weil group vanishing.

Experimental results

Research questions

• RQ1 Are there Smale-Barden manifolds that admit a quasi-regular Sasakian structure but no semi-regular K-contact structure?
• RQ2 Can the classification of null Sasakian structures on 5-manifolds be achieved via orbifold K3 surfaces with cyclic singularities and trivial orbifold fundamental group?
• RQ3 Does there exist a counterexample to the algebro-geometric conjecture that for a surface with b₁=0 and b₂>1, there cannot be b₂ disjoint smooth genus-g curves (g>0) spanning rational homology?
• RQ4 Can the theory of quasi-regular Seifert fibrations over singular orbifolds yield new examples of Sasakian 5-manifolds not accessible via semi-regular fibrations?
• RQ5 What are the topological and geometric constraints on K3 surfaces with cyclic orbifold singularities that admit a Seifert bundle with trivial fundamental group and a Sasakian structure?

Key findings

• The paper constructs a Smale-Barden manifold with a quasi-regular Sasakian structure that does not admit any semi-regular K-contact structure, answering Question 1 in the affirmative.
• The constructed manifold is diffeomorphic to #₂(S²×S³), with H₁(M, ℤ) = 0 and π₁(M) = 1, arising as a Seifert bundle over a K3 surface with an A₁₉ singularity.
• The orbifold base X has b₂(X) = 3, π₁^orb(X) = 1, and is obtained by contracting a configuration of 19 (−2)-curves from a K3 surface X′ with a Kodaira type I₁₉ fiber.
• The existence of a primitive Kähler class in H²(X−P, ℤ) ensures the applicability of Theorem 13, yielding a Sasakian structure on M.
• The paper provides a counterexample to the conjecture in [23] by constructing a cyclic Kähler orbifold K3 surface with b₂ = 3 and 20 disjoint (−2)-curves, violating the expected bound.
• Null Sasakian structures on #₂(S²×S³) are classified via orbifold K3 surfaces with cyclic singularities and trivial orbifold fundamental group, with the classification reduced to extremal elliptic fibrations with A-type configurations.