The Kodaira Dimension of Contact 3-Manifolds and Geography of Symplectic Fillings

Li, Tian Jun; Mak, Cheuk Yu

doi:10.1093/imrn/rny166

Cited by 4 publications

(3 citation statements)

References 47 publications

(44 reference statements)

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“…Along the way, we will give some alternative proofs for these known results and strengthen them. Further study of related questions along this line can be found in . All contact manifolds in this paper are assumed to be closed, three‐dimensional and have co‐oriented contact structures.…”

Section: Introductionmentioning

confidence: 99%

Calabi–Yau caps, uniruled caps and symplectic fillings

Mak

Yasui

2017

Proc. London Math. Soc.

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We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, they imply that any exact filling of the standard contact structure on the unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinite family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large b 2 .Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness properties and extend Wand's obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero.

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Section: Introductionmentioning

confidence: 99%

Calabi–Yau caps, uniruled caps and symplectic fillings

Mak

Yasui

2017

Proc. London Math. Soc.

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“…Since then, the geography for symplectic fillings of many contact 3-manifolds have been figured out, even though their symplectic fillings have not been classified yet. For example some contact Brieskorn spheres [42,43,12], the contact 3-manifolds supported by planar (spinal) open book decompositions [10,21,32], the contact 3-manifolds satisfying certain Floer homology conditions [37,29], and the contact 3-manifolds wearing certain symplectic caps [27,28], etc.…”

Section: Introductionmentioning

confidence: 99%

On geography of symplectic fillings of contact branched covers

Li¹,

Zhang²

2022

Preprint

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In this paper, we determine the Euler characteristics and signatures of the exact symplectic fillings of the contact double, 3-fold or 4-fold cyclic covers of the standard contact 3-sphere branched over certain transverse quasi-positive links. These links include all quasipositive knots with crossing numbers smaller than 11 and all quasi-positive links with crossing numbers smaller than 12 and nonzero nullity.

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“…From the contact point of view, symplectic log Calabi-Yau pairs are separated into 3 groups, as stated in the following theorem. Here, Kod(Y, ξ) is the contact Kodaira dimension introduced in [13].…”

Section: Introductionmentioning

confidence: 99%