Symplectic fillings of Seifert fibered spaces

Starkston, Laura

doi:10.1090/s0002-9947-2014-06420-9

Cited by 21 publications

(35 citation statements)

References 35 publications

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“…It follows that D blows down to a generic configuration C of three generically embedded symplectic spheres that are pseudo-holomorphic with respect to an almost complex structure tamed by the standard Kähler form on CP 2 . By a theorem of Gromov [9] (see also [23,Lemma 2.7]), the embedding of such three symplectic spheres in general position is unique up to isotopy. Therefore, up to isotopy we may assume that C coincides with a basic configuration (3 ) of three complex lines.…”

Section: Fillingsmentioning

confidence: 99%

On Stein fillings of contact torus bundles

Golla

Lisca

2015

Bulletin of London Math Soc

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We consider a large family MJX-tex-caligraphicscriptF of torus bundles over the circle, and we use recent work of Li–Mak to construct, on each Y∈F, a Stein fillable contact structure ξY. We prove that (i) each Stein filling of (Y,ξY) has vanishing first Chern class and first Betti number, (ii) if Y∈F is elliptic, then all Stein fillings of (Y,ξY) are pairwise diffeomorphic and (iii) if Y∈F is parabolic or hyperbolic, then all Stein fillings of (Y,ξY) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y∈F we exhibit non‐homotopy equivalent Stein fillings of (Y,ξY).

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Section: Fillingsmentioning

confidence: 99%

On Stein fillings of contact torus bundles

Golla

Lisca

2015

Bulletin of London Math Soc

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“…The first step toward this goal is to understand whether the given

(Y, ξ)

has finitely many or infinitely many fillings. Some families of contact 3‐manifolds that admit finitely many Stein fillings are found (, etc.). For minimal strong fillings, Ohta, Ono and others have systematically investigated the links of isolated singularities (, etc.…”

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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“…Therefore, the fillable contact strucutre on ∂P (D) is not the standard one. Applying the method in [26], [4] and [40], one can obtain a finiteness result on the number of minimal symplectic manifolds that can be compactified by D, up to diffeomorphism (See Proposition 4.37). •…”

Section: Example 222mentioning

confidence: 99%

“…Stetch of proof. We follow the strategy in [26], [4] and [40]. We remark that this question is answered in [4] for graphs in type (P1), (P2), (P3).…”

Section: Finitenessmentioning

confidence: 99%

Symplectic divisorial capping in dimension $4$

Mak

2019

Journal of Symplectic Geometry

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We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an ω-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.

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Symplectic fillings of Seifert fibered spaces

Cited by 21 publications

References 35 publications

On Stein fillings of contact torus bundles

On Stein fillings of contact torus bundles

Calabi–Yau caps, uniruled caps and symplectic fillings

Symplectic divisorial capping in dimension $4$

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