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Spherical Lagrangians via ball packings and symplectic cutting

Abstract

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, \(S^{2}\) or \(\mathbb{RP }^{2}\), in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction, this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.

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Acknowledgments

The authors warmly thank Selman Akbulut, Josef Dorfmeister, Ronald Fintushel, Robert Gompf, Dusa McDuff, and Leonid Polterovich for their interest in this work and many helpful correspondences. Particular thanks are due to Dusa McDuff for generously sharing early versions of her paper [36] with us, which plays a key role in our arguments. We would also like to thank the anonymous referee for valuable comments, suggestions, clarifications, and pointing us to the fact that Corollary 1.2(1) leads to a description of the Hamiltonian isotopy classes of Lagrangian spheres in a compact symplectic manifold where there are Hamiltonian knotted Lagrangian spheres. Matthew Strom Borman was partially supported by NSF-grant DMS 1006610; Tian-Jun Li and Weiwei Wu were supported by NSF-grant DMS 0244663.

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Appendices

Appendix A: Lagrangian spheres in rational manifolds

A.1 Characteristic Lagrangian spheres

We first recall from [28, Definition 3.3] that a stable spherical symplectic configuration is an ordered configuration of symplectic spheres with the following properties: (1) \(c_1\ge 1\) for all irreducible components, (2) the intersection numbers between two different components are 0 or 1, (3) they are simultaneously holomorphic with respect to some almost complex structure \(J\) tamed by the symplectic form. We will call them stable configurations for brevity. In the proof of [28, Theorem 1.5], the following intermediate result is reached.

Lemma 6.1

In \(\mathbb{CP }^2\#4\overline{\mathbb{CP }}^2\), let \(L_1\) and \(L_2\) be Lagrangian spheres in the homology class \(E_{1} - E_{2}\) and suppose they are disjoint from a stable configuration with irreducible components in classes \(\{H-E_1-E_2, H-E_3-E_4, E_3, E_4\}\), then \(L_{1}\) and \(L_{2}\) are Hamiltonian isotopic in the complement of the stable configuration.

In particular in the proof of [28, Theorem 1.5], one uses [28, Proposition 6.8] to show that \(L_{1}\) and \(L_{2}\) are Hamiltonian isotopic in the complement of the stable configuration. The same holds true for \(\mathbb{CP }^2\#(k+1)\overline{\mathbb{CP }}^2\) as well for \(k=1,2\) with the stable configurations specified in [28].

Theorem 6.2

Lagrangian \(S^2\)’s in a symplectic rational manifold with \(\chi \le 7\) are unique up to Hamiltonian isotopy.

Proof

By [28, Theorem 1.5 and Proposition 4.10], we only need to deal with the case where \(M=\mathbb{CP }^2\#3\overline{\mathbb{CP }}^2\) and \([L_i]=H-E_1-E_2-E_3\) for \(i=1,2\).

Fix a Darboux chart \(U_p\subset M\) that is disjoint from \(L_{1} \cup L_{2}\) and centered at the point \(p \in M\). By blowing up a symplectically embedded ball \(B_p\subset U_p\), we can build a symplectic manifold \((M^{\prime }=\mathbb{CP }^2\#4\overline{\mathbb{CP }}^2,\omega ^{\prime })\) with a exceptional sphere \(C\) such that \(H_2(M^{\prime };\mathbb{Z })\) has a basis identified with the union of a basis of \(H_2(M,\mathbb{Z })\) and \([C]\), the intersection product \([L_{i}] \cap [C] = 0\).

From the Gromov–Taubes invariant theory, for generic compatible almost complex structure \(J\) the classes \(H-E_1-[C]\), \(H-E_2-[C]\), and \(H-E_3-[C]\) have unique representatives as \(J\)-holomorphic exceptional spheres \(C_1\), \(C_2\) and \(C_3\), respectively, which are disjoint. Since \([C_{i}] \cap [L_{j}] = 0\), [28, Corollary 3.13] builds Hamiltonian isotopies \(\psi _{j}\) so that \(\psi _{j}(L_{j})\) is disjoint from \(C_{1} \cup C_{2} \cup C_{3}\cup C\).

Notice that the set of classes \(\{H-E_1-[C], H-E_2-[C], H-E_3-[C], [C]\}\) are Cremona equivalent to \(\{H-E_1-E_2, H-E_3-E_4, E_3, E_4\}\), Lemma 5.2 applies. It follows that \(L_1\) and \(L_2\) are Lagrangian isotopic in the complement of a neighborhood of \(C\cup \bigcup C_i\) in \((M^{\prime },\omega ^{\prime })\), in particular the complement of \(C\) which is symplectomorphic to an open set of \(M\).\(\square \)

A.2 Proof of Corollary 1.2

Proof

Part (2) follows from Theorem 5.2 and [28, Theorem 1.6]. When \(\chi (M) = 6\) and the homology class of the Lagrangians is characteristic, then Theorem 5.2 covers part (1). In all the rest of cases, we assume that \([L_i]=E_1-E_2\) without loss of generality by [28, Proposition 4.10]. Our proof follows the steps sketched in [28].

For each pair \((M, L_i)\) by [28, Theorem 1.1], away from \(L_i\), there is a set of disjoint \((-1)\) symplectic spheres \(C^l_i\) for \(l=3,\ldots ,k+1,\) with

$$\begin{aligned} \left[ C_i^l\right] =E_l\; \mathrm{for}\; l=3,\ldots ,k \;\mathrm{and}\; \left[ C^{k+1}_i\right] = H-E_1-E_2. \end{aligned}$$

Blowing down the collections \(\mathcal C _{i} = (C^{3}_{i},\ldots , C^{k+1}_{i})\) separately, results in \((\tilde{M_i}, \tilde{L}_i, \mathcal B _{i})\) where \(\tilde{M_i}\) is a symplectic \(S^2\times S^2\) with equal symplectic areas in each factor, \(\tilde{L}_i\) a Lagrangian sphere, and \(\mathcal B _i = (B_{i}^3,\ldots , B_{i}^{k+1})\) is a symplectic ball packing in \(\tilde{M}_{i} \backslash \tilde{L}_{i}\) corresponding to \(\mathcal C _{i}\).

By Lalonde–McDuff [22] and Hind [16], there is a symplectomorphism between the pairs \(\Psi : (\tilde{M_1}, \tilde{L}_1) \rightarrow (\tilde{M_2}, \tilde{L}_2)\). For fixed \(l\), the symplectic balls \(\Psi (B_{1}^{l})\) and \(B_{2}^{l}\) have the same volume since they come from blowing down the same class. Hence, by Theorem 1.1, there is a compactly supported Hamiltonian isotopy \(\Phi \) of \(\tilde{M}_{2}{\setminus }L_{2}\) connecting the symplectic ball packing \(\Psi (\mathcal B _1) = \{\Psi (B_{1}^{l})\}_{l}\) and \(\mathcal B _{2}\) in \(\tilde{M}_{2}{\setminus }L_{2}\). Therefore, \(\Phi \circ \Psi \) is a symplectomorphism between the tuples \((\tilde{M_i}, \tilde{L}_i, \mathcal B _i)\) and hence upon blowing up induces a symplectomorphism

$$\begin{aligned} \psi : (M, L_{1}, \mathcal C _{1}) \rightarrow (M, L_{2}, \mathcal C _{2}). \end{aligned}$$

By design \(\psi \) preserves the homology classes \(E_{1} - E_{2}\), \(H-E_{1}-E_{2}\), \(E_{3},\ldots ,E_{k}\) as well as the class \([\omega ]\), from which it follows that \(\psi \), it also preserves \(H\) and hence \(\psi \in \text{ Symp}_{h}(M)\).\(\square \)

Appendix B: \(\mathbb Z _2\)-orthogonal systems and push-off systems

Recall that in the proof of Lemma 4.8 for \(1 \le k \le 8\), we need to compare the counts for the following two sets:

  • \(\mathbb{Z }_{2}\)-orthogonal systems for \(H\), i.e. sets of \(k\) exceptional classes that are \(\mathbb{Z }_2\)-orthogonal to \(H\) in \(\mathbb{CP }^{2}\# k\overline{\mathbb{CP }}^{2}\).

  • Push-off system for \(S\), i.e. sets of \(k\) exceptional classes that are \(\mathbb{Z }\)-orthogonal to \(S=-H+2E_1-E_2\) in \(\mathbb{CP }^{2}\# (k+1)\overline{\mathbb{CP }}^{2}\).

For ease of notation, when \(k\ge 4\), we will replace \(S\) with the Cremona equivalent \(S^{\prime } = E_1-E_2-E_3-E_4\). In the following \(\mathcal O _{k} = \{E_i\}_{i=1}^k\) will be a \(\mathbb{Z }_2\)-orthogonal system for \(H\) in \(\mathbb{CP }^{2}\#k\overline{\mathbb{CP }}^{2}\), while \(\mathcal P _{k} = \{H-E_1-E_i\}_{i=2}^{4}\cup \{E_j\}_{j=5}^{k+1}\) will be a push-off system for \(S^{\prime }\) in \(\mathbb{CP }^{2}\#(k+1)\overline{\mathbb{CP }}^{2}\).

  • For \(k \le 3\) : \(H\) has \(\mathcal O _{k}\) and \(S\) has \(\{H-E_1-E_2\} \cup \{E_{i}\}_{i=3}^{k+1}\).

  • For \(4 \le k \le 5\) : \(H\) has \(\mathcal O _{k}\) and \(S^{\prime }\) has \(\mathcal P _{k}\).

  • For \(k=6\) : \(H\) has \(\mathcal O _{6}\) and its Cremona transform along \(2H-\sum _{i=1}^6 E_i\). While \(S^{\prime }\) has \(\mathcal P _{6}\) and its Cremona transform along \(H-E_5-E_6-E_7\).

  • For \(k=7\) : There are \(8\) systems: \(H\) has \(\mathcal O _{7}\) and its \(7\) Cremona transforms along

    $$\begin{aligned} 2H-(E_1+\cdots +E_7)+E_i \quad \text{ for} \quad 1\le i\le 7. \end{aligned}$$

    While \(S^{\prime }\) has \(\mathcal P _{7}\), its \(4\) Cremona transforms along

    $$\begin{aligned} H-(E_5+E_6+E_7+E_8)+E_j \quad \text{ for} \quad j=5, 6, 7, 8 \end{aligned}$$

    and its \(3\) Cremona transforms along

    $$\begin{aligned} 2H-E_1-(E_5+E_6+E_7+E_8)+E_j \quad \text{ for} \quad j=2, 3,4. \end{aligned}$$
  • For \(k = 8\) : There are \(29\) systems: \(H\) has \(\mathcal O _{8}\) and its \(28\) Cremona transforms along

    $$\begin{aligned} 2H-(E_1+\cdots +E_7+E_8)+E_i+E_j \quad \text{ for} \quad 1\le i\ne j\le 8. \end{aligned}$$

    While \(S^{\prime }\) has \(\mathcal P _{8}\), its \(10\) Cremona transforms along

    $$\begin{aligned} H-(E_5+E_6+E_7+E_8+E_9)+E_i+E_j \quad \text{ for} \quad 5\le i\ne j\le 9, \end{aligned}$$

    its \(15\) Cremona transforms along

    $$\begin{aligned} 2H\!-\!E_1\!-\!(E_5\!+\!E_6\!+\!E_7\!+\!E_8\!+\!E_9)\!+\!E_j\!+\!E_p \quad \text{ for} \quad 2\le j\le 4,\, 5\le p\le 9, \end{aligned}$$

    and its \(3\) Cremona transforms along

    $$\begin{aligned} 3H\!-\!2E_1\!-\!(E_2\!+\!E_3\!+\!E_4)-(E_5\!+\!E_6\!+\!E_7\!+\!E_8\!+\!E_9)+E_q \quad \text{ for} \quad 2\le q\le 4. \end{aligned}$$

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Borman, M.S., Li, TJ. & Wu, W. Spherical Lagrangians via ball packings and symplectic cutting. Sel. Math. New Ser. 20, 261–283 (2014). https://doi.org/10.1007/s00029-013-0120-z

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Keywords

  • Symplectic manifolds
  • Symplectic ball packing
  • Lagrangian knots
  • Symplectic cutting
  • Rational manifolds

Mathematics Subject Classification

  • 53Dxx
  • 53D35
  • 53D12