Symplectic surgeries and normal surface singularities

Stipsicz, András I.

doi:10.2140/agt.2009.9.2203

Cited by 27 publications

(51 citation statements)

References 19 publications

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“…In turn, the fact that the two contact structures ξ C and ξ M are contactomorphic will be proved by relating two compatible open book decompositions. The existence of this contactomrphism was verified in [1,11] for specific families of configurations of C; in this paper we extend the result of [11] by constructing an appropriate horizontal open book decomposition on ∂U C compatible with ξ C (cd. Theorem 3.2.…”

Section: Introductionmentioning

confidence: 67%

“…This symplectic feature of the construction has been extended to further configurations of symplectic surfaces in symplectic 4-manifolds and further smoothings of singularities in [1,9,10,11]. The general case, however, stayed open and was formulated as Conjecture 1.4 in [11]. The aim of the present paper is to prove this conjecture.…”

Section: Introductionmentioning

confidence: 92%

“…. , C m ) be the given set of symplectic surfaces in (X, ω), and W C a smoothing of a singularity with resolution graph Γ C given by the configuration C. Let U C be an ω-convex neighbourhood of C in X (the existence of which is proved in [11,Theorem 1.2]). According to Corollary 3.3 the contact structure ξ C induced on ∂U C is contactomorphic to the Milnor fillable contact structure ξ M on ∂W C , hence by the symplectic gluing theorem described in [7] (see also [18,Theorem 7.1.9]), the manifold X C = (X − intU C ) ∪ φ W C admits a symplectic structure ω C which on X − U C conicides with the given symplectic structure ω.…”

Section: Horizontal Open Book Decompositionsmentioning

confidence: 99%

“…In specific cases the nonvanishing of the Seiberg-Witten invariant of X S can be explained using symplectic topology: according to a result of Symington [22,23], if (X, ω) is a symplectic 4-manifold and the spheres in the configuration S are symplectic submanifolds (intersecting ω-orthogonally), then X S admits a symplectic structure (hence by Taubes' theorem [24] it has nontrivial Seiberg-Witten invariants). This symplectic feature of the construction has been extended to further configurations of symplectic surfaces in symplectic 4-manifolds and further smoothings of singularities in [1,9,10,11]. The general case, however, stayed open and was formulated as Conjecture 1.4 in [11].…”

Section: Introductionmentioning

confidence: 99%

“…The main idea of the proof of the above result is the following: by [11] the configuration C = ∪ i C i admits an ω-convex neighbourhood U C , with boundary ∂U C supporting a compatible contact structure ξ C . The smoothing W C , on the other hand, admits the structure of a Stein domain, inducing the so-called Milnor fillable contact structure ξ M on ∂W C .…”

Section: Introductionmentioning

confidence: 99%

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Smoothings of singularities and symplectic surgery

Park

Stipsicz

2014

Journal of Symplectic Geometry

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Abstract. Suppose that C = (C 1 , . . . , Cm) is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold (X, ω) with connected, negative definite intersection graph Γ C . We show that by replacing an appropriate neighborhood of ∪C i with a smoothing W S of a normal surface singularity (S, 0) with resolution graph Γ C , the resulting 4-manifold admits a symplectic structure. This operation generalizes the rational blow-down operation of Fintushel-Stern for other configurations, and therefore extends Symington's result about symplectic rational blow-downs.

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Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 92%

Section: Horizontal Open Book Decompositionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Smoothings of singularities and symplectic surgery

Park

Stipsicz

2014

Journal of Symplectic Geometry

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On Symplectic Caps

T.¹,

Stipsicz²

2011

Progress in Mathematics

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An important class of contact 3-manifolds are those that arise as links of rational surface singularities with reduced fundamental cycle. We explicitly describe symplectic caps (concave fillings) of such contact 3-manifolds. As an application, we present a new obstruction for such singularities to admit rational homology disk smoothings.We dedicate this paper to Oleg Viro on the occasion of his 60th birthday.

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Milnor fibers and symplectic fillings of quotient surface singularities

Park

Shin

et al. 2018

Advances in Mathematics

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We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that:-Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism.-Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called P -resolutions of a quotient surface singularity. We provide an explicit algorithm for identifying a given Milnor fiber as a minimal symplectic filling, that is, as complements Z − E ∞ . For this, we apply some techniques from the minimal model program for 3-folds such as divisorial contractions and flips.We first compactify a given smoothing of (X, 0). We briefly sketch the idea: Let M be the Milnor fiber of a smoothing π : X → ∆ of a quotient surface singularity X. Let Y → X be the P -resolution corresponding to π. According to Behnke-Christophersen [3], there is a special partial resolution, so called, M -resolution Y → X dominating Y (See Definition 6.14), and a Q-Gorenstein smoothing φ : Y → ∆ such that the smoothing φ blows down to π. We have a commutative diagram as described in Figure 1. It is easy to show that a general fiber Y t = φ −1 (t) is isomorphic to X t = π −1 (t). Therefore we haveWe then compactify X and Y to compact complex surfaces, following Lisca [26] and Pinkham [41], so that the two smoothings Y → ∆ and X → ∆ can be extended to the deformations Y → ∆ and X → ∆ of the natural compactifications X and Y of X and Y , where X and Y are obtained, roughly speaking, by pasting a regular neighborhood ν(E ∞ ) of the compactifying divisor E ∞ of X (See Section 7 for the definition of the natural compactification). We again have a commutative diagram, as described in Figure 1.The deformations Y → ∆ and X → ∆ are locally trivial along E ∞ . So the Milnor fiber Y t is given as the complement of the compactifying divisor E ∞ in a general fiber Y t (which is called a compactified Milnor fiber ; See Definition 7.6) of the deformation Y → ∆; See Proposition 7.5. So we haveWe need to recognize ( Y t , E ∞ ) as (Z, E ∞ ) in the lists of Lisca [26] and . For this, we show that Main Theorem 2 (Theorem 9.4). By applying specific divisorial contractions and flips in a controlled and explicit manner to the deformation Y → ∆, we obtain a new deformation W → ∆ such that all of its fibers are smooth.During this minimal model program (in short, MMP ) process, we ...

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Symplectic surgeries and normal surface singularities

Cited by 27 publications

References 19 publications

Smoothings of singularities and symplectic surgery

Smoothings of singularities and symplectic surgery

On Symplectic Caps

Milnor fibers and symplectic fillings of quotient surface singularities

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