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.

Appendices

Appendix A: Lagrangian spheres in rational manifolds

1.1 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 $M$.$\square$

1.2 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

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