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Contact structures on AR-singularity links. (English) Zbl 1386.57029

A contact structure \(\xi\) on a 3-manifold \(M\) is compatible with an open book in \(M\) if a contact form of \(\xi\) on an oriented binding of the open book is a positive volume form and away from the binding \(\xi\) can be isotoped through contact structures to be arbitrarily close to the tangent planes of the pages of the open book. This compatibility relation is a one-to-one correspondence between contact 3-manifolds up to contact isotopy and open books up to positive stabilizations. The distribution of complex tangencies on \(M\) is a contact distribution. If a 3-manifold is realized as a link of an isolated singularity, then the associated contact structure is called the canonical contact structure on \(M\). A complex surface singularity is called rational if its geometric genus is zero. A rational singularity is an almost rational singularity (AR-singularity) if it admits a good resolution whose dual graph is a negative definite connected tree and the dual graph of a rational singularity is obtained by reducing the weight on a vertex. An almost rational singularity is called proper AR if it is not rational. If \((W,J)\) is a compact complex surface with oriented boundary \(-M_1\cup M_2\) that admits a strictly plurisubharmonic Morse function \(\varphi:W\to[t_1,t_2]\) such that \(M_i=\varphi^{-1}(t_i)\) and \(-dJ^*d\varphi\) is a symplectic form, then the set of complex tangencies on \(M_i\) constitutes contact structures \(\xi_i\), \(i=1,2\). It is said that \(W\) is a Stein cobordism from \((M_1,\xi_1)\) to \((M_2,\xi_2)\). If \(M_1=\varnothing\), then \(W\) is a Stein filling for \((M_2,\xi_2)\).
In this paper, the authors consider the existence problem of Stein cobordisms between canonical contact structures on the link manifolds of various classes of singularities. This is also related to the problem of symplectically embedding one Milnor fiber into another. They prove that every closed, contact 3-manifold is Stein cobordant to the canonical contact structure of a proper AR-singularity. To any co-oriented contact structure \(\xi\) on a 3-manifold \(M\) the authors associate an element \(c^+(\xi)\in\text{HF}^+(-M)\) which is a contactomorphism invariant of the contact structure and is natural under Stein cobordisms. If \(\xi\) is Stein fillable, then \(c^+(\xi)\) does not vanish since the tight structure on \(S^3\) has non-vanishing \(c^+\) and \(U(c^+(\xi))=0\). Then, the authors define a numerical invariant of contact structures as \(\sigma(\xi)=-\sup\{d\in\mathbb N\cup\{0\}\}\), for \(c^+(\xi)\in U^d\cdot\text{HF}^+(-M)\). The computation of \(\sigma(\xi)\) is not easy to make, nevertheless the authors show that it is zero for the canonical contact structures of proper AR-singularities. They prove that if \((M,\xi)\) is the canonical contact link manifold of a proper AR-singularity, then \(\sigma(\xi)=0\), and hence, there is no Stein cobordism from \(\xi\) to (i)any contact structure supported by a planar open book, (ii) any contact structure on the link of a rational singularity, or (iii) any contact structure with vanishing Ozsváth-Szabó invariant.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57R58 Floer homology
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
32S25 Complex surface and hypersurface singularities
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

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