Stability and Existence of Surfaces in Symplectic 4-Manifolds with $b^+=1$

Dorfmeister, Josef G.; Li, Tianjun; Wu, Weiwei

doi:10.48550/arxiv.1407.1089

Cited by 5 publications

(11 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, when e is J-nef and J-effective with g J (e) = 0, we have the following strong bound for the expected dimension of curve configuration for Θ ∈ M red,e (Lemma 4.10 in [17]) (5)…”

Section: J-holomorphic Subvarietiesmentioning

confidence: 91%

“…It is direct to see from the adjunction formula that there are finitely many negative J-holomorphic spheres on an irrational ruled surface. For a complete classification of symplectic spheres on irrational ruled surfaces, see [5] section 6.…”

Section: Now We Look At the Class −[C] By The Above Calculation We Ha...mentioning

confidence: 99%

“…We might apply the classifications of negative rational curves, e.g. [30,5], and calculate the intersection numbers with A directly. Now, we will investigate the moduli space of the subvarieties in class T .…”

Section: Now We Look At the Class −[C] By The Above Calculation We Ha...mentioning

confidence: 99%

“…Hence, for the convenience of notation, we will write e = 2H in the following. When k < 9, we can apply the classification of negative rational curves in [30] to identify all possible subvarieties in class 2H and show the moduli space is CP 5 .…”

Section: J-holomorphic Torimentioning

confidence: 99%

See 3 more Smart Citations

Moduli space of $J$-holomorphic subvarieties

Zhang

2016

Preprint

View full text Add to dashboard Cite

We study the moduli space of J-holomorphic subvarieties in a 4-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear system structures as in algebraic geometry. Among the applications, we show various uniqueness results of J-holomorphic subvarieties, e.g. for the fiber and exceptional classes in irrational ruled surfaces. On the other hand, non-uniqueness and other exotic phenomena of subvarieties in complex rational surfaces are explored. In particular, connected subvarieties in an exceptional class with higher genus components are constructed. The moduli space of tori is also discussed, and leads to an extension of the elliptic curve theory.

show abstract

Section: J-holomorphic Subvarietiesmentioning

confidence: 91%

Section: Now We Look At the Class −[C] By The Above Calculation We Ha...mentioning

confidence: 99%

Section: Now We Look At the Class −[C] By The Above Calculation We Ha...mentioning

confidence: 99%

Section: J-holomorphic Torimentioning

confidence: 99%

See 2 more Smart Citations

Moduli space of $J$-holomorphic subvarieties

Zhang

2016

Preprint

View full text Add to dashboard Cite

show abstract

“…By identifying X 1 b and X 2 b with a natural choice of diffeomorphism Φ b , the homology classes of the components of D 1 b and D 2 b are the same. By Proposition 1.2.15 of [20] or Theorem 2.9 of [3], up to a D-symplectic homotopy (ie.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Symplectic log Calabi–Yau surface: deformation class

Mak

2016

Advances in Theoretical and Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We study the symplectic analogue of log Calabi-Yau surfaces and show that the symplectic deformation classes of these surfaces are completely determined by the homological information.Theorem 1.4. Let (X i , D i , ω i ) be symplectic log Calabi-Yau surfaces for i = 1, 2. Then (X 1 , D 1 , ω 1 ) is (resp. strictly) symplectic deformation equivalent to (X 2 , D 2 , ω 2 ) if and only if they are (resp. strictly) homological equivalent.Moreover, the symplectomorphism in the (resp. strict) symplectic deformation equivalence has same homological effect as the (resp. strict) homological equivalence.We remark that when D is a smooth divisor, the relative Kodaira dimension κ(X, D, ω) was introduced in [14] and it was noted there that this notion could be extended to nodal divisors. With this extension understood, symplectic Calabi-Yau surfaces have relative Kodaira dimension κ = 0 (cf. Theorem 3.28 in [14]). Moreover, Theorem 1.4 is also valid when κ(X, D, ω) = −∞. This will be treated in the sequel. Coupled with the techniques developed in [11], [12], some applications to Stein fillings will also be treated in the sequel.The paper is organized as follows. In Section 2 we introduce marked divisors and establish the invariance of their deformation class under blow-up/down in Proposition 2.10. This reduces Theorem 1.4 to the minimal cases. In Section 3, we classify the deformation classes of minimal models and finish the proof of Theorem 1.4.The authors benefit from discussions with Mark Gross, Paul Hacking and Sean Keel.

show abstract

Symplectic log Calabi-Yau surface---deformation class

Mak

2015

Preprint

View full text Add to dashboard Cite

Stability and Existence of Surfaces in Symplectic 4-Manifolds with $b^+=1$

Cited by 5 publications

References 24 publications

Moduli space of $J$-holomorphic subvarieties

Moduli space of $J$-holomorphic subvarieties

Symplectic log Calabi–Yau surface: deformation class

Symplectic log Calabi-Yau surface---deformation class

Contact Info

Product

Resources

About