Symplectic Blow-up Research Articles

Symplectic mapping class groups of blowups of tori

AbstractLet be a Kähler form on the real 4‐torus . Suppose that satisfies an irrationality condition that can be achieved by an arbitrarily small perturbation of . This note shows that the smoothly trivial symplectic mapping class group of the one‐point symplectic blowup of is infinitely generated.

Lagrangian Floer homology on symplectic blow ups

We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian quantum homology in the one-point blow up of (CP^2,\omega) of the proper transform of the Clifford torus.

Deformation of singular fibers of genus two fibrations and small exotic symplectic 4-manifolds

We introduce the [Formula: see text]-nodal spherical deformation of certain singular fibers of genus two fibrations, and use such deformations to construct various examples of simply connected minimal symplectic [Formula: see text]-manifolds with small topology. More specifically, we construct new exotic minimal symplectic [Formula: see text]-manifolds homeomorphic but not diffeomorphic to [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using [Formula: see text]-nodal spherical deformations of certain singular fibers of genus [Formula: see text] fibrations.

Kähler packings and Seshadri constants on projective complex surfaces

In analogy to the relation between symplectic packings and symplectic blow ups we show that multiple point Seshadri constants on projective complex surfaces can be calculated as the supremum of radii of multiple Kähler ball embeddings. We exemplify this connection on toric surfaces, also discussing how toric moment maps reflect the packing.

Distinguishing symplectic blowups of the complex projective plane

A symplectic manifold that is obtained from CP by k blowups is encoded by k + 1 parameters: the size of the initial CP, and the sizes of the blowups. We determine which values of these parameters yield symplectomorphic manifolds.

Locally conformal symplectic blow-ups

In this paper, we study the blow-up of a locally conformal symplectic manifold. We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.

Stein fillings of homology 3-spheres and mapping class groups

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology 3-sphere supported by an open book with page a 4-holed sphere admits a unique Stein filling up to symplectic deformation. Furthermore, according to a property of deforming symplectic fillings of rational homology 3-spheres into strong symplectic fillings, we also show that a symplectically fillable integral homology 3-sphere supported by an open book with page a 4-holed sphere admits a unique symplectic filling up to symplectic deformation and blow-up.

Lagrangian blow-ups, blow-downs, and applications to real packing

Given a symplectic manifold (M, {\omega}) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, we show that if M admits an anti-symplectic involution {\phi} and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an anti-symplectic involution on the blow-up as well. We derive a homological condition which determines when the topology of a real Lagrangian surface L = Fix({\phi}) changes after a blow down, and we use these constructions to study the real packing numbers and packing stability for real, rank-1 symplectic four manifolds which are non-Seiberg-Witten simple.

On blow-ups and cohomology of almost complex manifolds

We study the behavior of a special type of almost complex structures, called C∞pure and full and introduced by T.-J. Li, W. Zhang (2009) in [10], in relation to the complex blow-up and the symplectic blow-up.

Torus actions on small blowups of ℂℙ2

A manifold obtained by k simultaneous symplectic blowups of CP 2 of equal sizes ∈ (where the size of CP 1 C CP 2 is one) admits an effective two dimensional torus action if k ≤ 3. We show that it does not admit such an action if k ≥ 4 and ∈ ≤ 1/(3k2 2k ). For the proof, we show a correspondence between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves.

Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds

AbstractLet Mμ0 denote S2×S2 endowed with a split symplectic form $\mu \sigma \oplus \sigma $ normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding $\iota :B_{c}\hookrightarrow M^0_{\mu }$ of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up $\widetilde {M}^0_{\mu ,c}$. In this paper, we study the homotopy type of the symplectomorphism group ${\mathrm {Symp}}(\widetilde {M}^0_{\mu ,c})$ and that of the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ of unparametrized symplectic embeddings of Bc into Mμ0. Writing ℓ for the largest integer strictly smaller than μ, and λ∈(0,1] for the difference μ−ℓ, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λ is homotopically equivalent to the stabilizer of a point in Symp(Mμ0), while that of a blow-up of ‘large’ capacity c≥λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a non-trivial bundle $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$ obtained by blowing down $\widetilde {M}^0_{\mu ,c}$. It follows that, for c<λ, the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$. By contrast, we show that the embedding spaces $\Im {\mathrm {Emb}}(B_{c},\mathbb {C}P^{2})$ and $\Im {\mathrm {Emb}}(B_{c_{1}}\sqcup B_{c_{2}},\mathbb {C}P^{2})$, if non-empty, are always homotopy equivalent to the spaces of ordered configurations $F(\mathbb {C}P^{2},1)\simeq \mathbb {C}P^{2}$ and $F(\mathbb {C}P^{2},2)$. Our method relies on the theory of pseudo-holomorphic curves in 4 -manifolds, on the computation of Gromov invariants in rational 4 -manifolds, and on the inflation technique of Lalonde and McDuff.

Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$ denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. Our results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable.

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold ( M , ω ) into a smooth symplectic manifold ( M ˜ , ω ˜ ) . Then we study how the formality and the Lefschetz property of ( M ˜ , ω ˜ ) are compared with that of ( M , ω ) . We also study the formality of the symplectic blow-up of ( M , ω ) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.

Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$ denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. Our results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable.

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.

A formula for the Chern classes of symplectic blow-ups

It is shown that the formula for the Chern classes (in the Chow ring) of blow-ups of algebraic varieties, due to Porteous and Lascu-Scott, also holds (in the singular cohomology ring) for blow-ups of symplectic and complex manifolds. This was used by the second author in her solution of the geography problem for 8-dimensional symplectic manifolds. The proof equally applies to real blow-ups of arbitrary manifolds and yields the corresponding blow-up formula for the Stiefel-Whitney classes. In the course of the argument, the topological analogue of Grothendieck's formule clef in intersection theory is proved. © 2007 London Mathematical Society.

Circle and torus actions on equal symplectic blow-ups of $\mathbf{\text{CP}^2}$

A manifold obtained by $k$ simultaneous symplectic blow-ups of $\CP^2$ of equal sizes $\epsilon$ (where the size of $\CP^1\subset\CP^2$ is one) admits an effective two dimensional torus action if $k \leq 3$ and admits an effective circle action if $\epsilon < 1/(k-1)$. We show that these bounds are sharp if $\epsilon = 1/n$ where $n$ is a natural number. Our proof combines ``soft equivariant techniques with ``hard holomorphic techniques.

Symplectic Capacities of Toric Manifolds and Related Results

AbstractIn this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds withS1-action.

THE KIRWAN MAP FOR SINGULAR SYMPLECTIC QUOTIENTS

Let M be a Hamiltonian K -space with proper moment map $\mu$ . The symplectic quotient $X=\mu^{-1}(0)/K$ is a singular stratified space with a symplectic structure on the strata. In this paper we generalise the Kirwan map, which maps the K equivariant cohomology of $\mu^{-1}(0)$ to the middle perversity intersection cohomology of X , to this symplectic setting. The key technical results which allow us to do this are Meinrenken's and Sjamaar's partial desingularisation of singular symplectic quotients and a decomposition theorem, proved in Section 2 of this paper, exhibiting the intersection cohomology of a ‘symplectic blowup’ of the singular quotient X along a maximal depth stratum as a direct sum of terms including the intersection cohomology of X .

Symplectic submanifolds in special position

By working on the symplectic blow-up, we show that the symplectic divisors produced by Donaldson in [D] may be chosen so that they contain a fixed symplectic submanifold or, in a complementary direction, so that they cut it transversally with a symplectic intersection.