Compatible complex structures on symplectic rational ruled surfaces

Abreu, Miguel; Granja, Gustavo; Kitchloo, Nitu

doi:10.1215/00127094-2009-033

Cited by 21 publications

(85 citation statements)

References 33 publications

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“…J i ;j is a homotopy equivalence, that the equivariant diffeomorphism type of a normal neighborhood of the j -th stratum is the same in both stratifications and that this neighborhood does not depend on the parameter as long as > .j i /=2. These facts, together with the results of Appendix C in [2], imply that the action of G i on the normal bundle N .J All of Abreu, Granja and Kitchloo arguments on M 0 apply as well for the group of symplectomorphisms of the blow-up M 0 ;c if one has in mind the following geometric facts and observations. When passing to the blow-up, the spaces of compatible (almost) complex structures J 0 ;c and z I 0 ;c are now partitioned according to the degeneracy type of exceptional curves in class B E (using the notation introduced in Section 1.3).…”

Section: The Stratification Of J I As An Orbit Decompositionmentioning

confidence: 83%

“…Using this corollary and the Serre spectral sequence of fibration (12), we get: In their paper [2], Abreu, Granja and Kitchloo prove that, under some cohomological conditions, I ! is a genuine Fréchet submanifold of J !…”

Section: The Split Case With 1 < ämentioning

confidence: 92%

“…The actual computations for the groups Symp. M i ;c / are carried out in Appendices A and B, following the method introduced in Abreu, Granja and Kitchloo [2]. In Section 3, we express rationally the space =Emb i !…”

Section: Plan Of the Papermentioning

confidence: 99%

“…For in that range, the space J 0 is made of an open stratum J 2 by defining, for any indices i and j , an…”

Section: The Stratification Of J I As An Orbit Decompositionmentioning

confidence: 99%

“…A solution to this problem, found by Abreu, Granja and Kitchloo in [2], is to look at the restriction of the action G i J i ! J i to the subspace I i J i of compatible integrable complex structures.…”

Section: The Stratification Of J I As An Orbit Decompositionmentioning

confidence: 99%

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The homotopy type of the space of symplectic balls in rational ruled 4–manifolds

Anjos¹,

Lalonde

Pinsonnault

2009

Geom. Topol.

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We compute in this note the full homotopy type of the space of symplectic embeddings of the standard ball B 4 (c) ⊂ R 4 (where c = πr 2 is the capacity of the standard ball of radius r) into the 4-dimensional rational symplectic manifold Mµ = (S 2 × S 2 , µω 0 ⊕ ω 0 ) where ω 0 is the area form on the sphere with total area 1 and µ belongs to the interval (1, 2]. We know, by the work of Lalonde-Pinsonnault [8], that this space retracts to the space of symplectic frames of Mµ for any value of c smaller than the critical value µ − 1, and that its homotopy type does change when c crosses that value. In this paper, we compute the homotopy type for the case c ≥ µ − 1 and prove the claim announced in [8] that it does not have the type of a finite CW-complex.

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Section: The Stratification Of J I As An Orbit Decompositionmentioning

confidence: 83%

Section: The Split Case With 1 < ämentioning

confidence: 92%

Section: Plan Of the Papermentioning

confidence: 99%

“…For in that range, the space J 0 is made of an open stratum J 2 by defining, for any indices i and j , an…”

Section: The Stratification Of J I As An Orbit Decompositionmentioning

confidence: 99%

Section: The Stratification Of J I As An Orbit Decompositionmentioning

confidence: 99%

See 3 more Smart Citations

The homotopy type of the space of symplectic balls in rational ruled 4–manifolds

Anjos¹,

Lalonde

Pinsonnault

2009

Geom. Topol.

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The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

Anjos¹,

Pinsonnault

2013

Math. Z.

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Abstract. By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by the toric structures one can put on the manifold. Our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of Lalonde-McDuff.

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Symplectic isotopy on non-minimal ruled surfaces

Buse

2023

Math. Z.

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Compatible complex structures on symplectic rational ruled surfaces

Cited by 21 publications

References 33 publications

The homotopy type of the space of symplectic balls in rational ruled 4–manifolds

The homotopy type of the space of symplectic balls in rational ruled 4–manifolds

The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

Symplectic isotopy on non-minimal ruled surfaces

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