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

Anjos, Sílvia; Pinsonnault, Martin

doi:10.1007/s00209-012-1134-5

Cited by 19 publications

(63 citation statements)

References 23 publications

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“…Note that the above result is very similar but still different from [AP13] Proposition B.1. The only difference is that we consider space A u instead of J ω here.…”

Section: Clearly We Have the Decompositionmentioning

confidence: 52%

Stability of the symplectomorphism group of rational surfaces

Anjos¹,

Li²,

Li³

et al. 2019

Preprint

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“…Note that the above result is very similar but still different from [AP13] Proposition B.1. The only difference is that we consider space A u instead of J ω here.…”

Section: Clearly We Have the Decompositionmentioning

confidence: 52%

Stability of the symplectomorphism group of rational surfaces

Anjos¹,

Li²,

Li³

et al. 2019

Preprint

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“…Proof of claim: The second arrow of the first row is simply the restriction of an element φ to φ (2). The continuity of the vertical maps follows from the continuous dependence of solutions of an ODE on initial conditions when applying Moser's method.…”

Section: 1mentioning

confidence: 98%

“…Recall that the Hirzebruch surface F n is the projectivation P(O(n) ⊕ C). Under the action of its Kähler isometry group K n ≃ U (2)/Z n , the complex surface F n is partitionned into three orbits: the zero section C n , the section at infinity C ∞ n and their open complement F n \ {C n ∪ C ∞ n }, see Appendix B in [2]. Since the K n action preserves the ruling F n → CP 1 , every element in K n acts as an isometry of CP 1 and K n acts faithfully on the normal bundle ν on C n via derivatives.…”

Section: 1mentioning

confidence: 99%

“…In particular, we obtain the following result: Theorem 1.1. The group Symp c (sL(n, 1)), endowed with the C ∞ -topology, has countably many components, each being weakly homotopy equivalent to the based loop space of SU (2). There is a natural map from L (C Iso n ), the based loop group of contact isometry group of L(n, 1), to Symp c (sL(n, 1)) which induces the weak homotopy equivalence.…”

Section: Introductionmentioning

confidence: 99%

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Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians

Hind

Pinsonnault

2016

Journal of Symplectic Geometry

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We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we also prove the longstanding expected result that the space of Lagrangian RP 2 in T * RP 2 is weakly contractible.MSC classes: 53Dxx, 53D35, 53D12

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“…Thanks to the works by Abreu [5], Abreu-McDuff [6], Lalonde-Pinsonnault [7], as well as many other authors, much is known when χ(X) < 8. In one of the recent works, Anjos-Pinsonnault [8] computed the homotopy Lie algebra of Symp(X, ω) when ω is non-monotone.…”

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confidence: 99%

Symplectic (−2)-spheres and the symplectomorphism group of small rational 4-manifolds

2020

Pacific J. Math.

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For (CP 2 #5CP 2 , ω), let Nω be the number of (−2)-symplectic spherical homology classes. We completely determine the Torelli symplectic mapping class group (Torelli SMCG): Torelli SMCG is trivial if Nω > 8; it is π 0 (Diff + (S 2 , 5)) if Nω = 0 (by [1],[2]); it is π 0 (Diff + (S 2 , 4)) in the remaining case. Further, we completely determine the rank of π 1 (Symp(CP 2 #5CP 2 ) for any given symplectic form. Our results can be uniformly presented regarding Dynkin diagrams of type A and type D Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds.Compared to the monotone case when C 0 is contractible (where the form is of type D 5 ), we fall short of computing the homotopy type of it directly: indeed, the topology of the open strata of almost complex structure can be very complicated even in much simpler manifolds, see [6].We took a new approach here. Starting from a class of standard RP 2 packing symplectic forms (RP 2 forms for short, see Definition 3.16), we show that the map φ is indeed surjective when there is an RP 2 packing in X. This surjectivity is in turn related to another relative ball-packing problem and makes use of the ball-swapping symplectomorphism constructed in [20]. We then use a stability argument inspired by [21], paired with a Cremona equivalence computation, to relate a type A form with a RP 2 form.The forms of type D 4 is more complicated. We will construct a key commutative diagram (28) (compare [1]). The punchline is to remove those strata of almost complex structures which allows more than one (−2)-sphere, or spheres with self-intersection no greater than (−3) from the space of ω-compatible almost complex structure. This yields a 2-connected space. Such a space is not homeomorphic to C 0 , but captures π i (C 0 ) for i = 0, 1, 2, which suffices for the study of π 0 and π 1 of Symp(X, ω). An extensive study of diagram (28) enables one to compare the induced homotopy sequence in the lowest degrees with the strand-forgetting sequencewhich eventually deduces our main theorem for D 4 using the Hopfian property of braid groups.Remark 1.5. After the first draft of this manuscript was posted, Silvia Anjos informed us about her work with Sinan Eden ([22], [23] ), in which they independently obtain similar results in some toric cases for the 4-fold blow-up of CP 2 , including the generic case and the case where λ = 1 in the Table 1. Moreover, they have a result to show that the generators of π 1 (Ham(X, ω)) also generate the homotopy Lie algebra of Ham(X, ω), using similar ideas from [8]. K c J := {[ω] ∈ H 2 (M ; R)|ω is compatible with J}. K t J and K c J are both convex cones in the positive cone P = {c ∈ H 2 (M ; R)|c · c > 0}. Note that we have the tamed Nakai-Moishezon theorem for rational surfaces when Euler number is small: Theorem 2.19 (Theorem 1.6 in [33]). Suppose M = S 2 × S 2 or CP 2 #kCP 2 , k ≤ 9, and let C >0 J := {c ∈ H 2 (M ; R)|[Σ] = c for some some J-holomorphic subvariety Σ} be the curve cone of J. For an almost Kähler J on M , C ∨,>0...

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

Cited by 19 publications

References 23 publications

Stability of the symplectomorphism group of rational surfaces

Stability of the symplectomorphism group of rational surfaces

Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians

Symplectic (−2)-spheres and the symplectomorphism group of small rational 4-manifolds

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