# Packing Integral Tori in Del Pezzo Surfaces

Karim Boustany  Karim Boustany, Department of Mathematics, University of Notre
Dame<EMAIL_ADDRESS>
###### Abstract.

We extend a packing result of R. Hind and E. Kerman [10] for integral
Lagrangian tori in ${}^{2}\times^{2}$ to the Del Pezzo surfaces
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ for $n=1,\dots,5$. An
_integral_ torus is one whose relative area homomorphism is integer-valued,
and we seek a _maximal integral packing_. By definition, this is a disjoint
collection $\\{L_{i}\\}$ of integral Lagrangian tori with the following
property: any other integral Lagrangian torus not in this collection must
intersect at least one of the $L_{i}$. We show that one can always find such a
packing consisting of only the Clifford torus. The structure of our argument
closely follows the one in [10].

The work presented here was partially supported by NSF grant DMS-1547292.

##  1\. Introduction

The study of Lagrangian submanifolds and their properties is ubiquitous in
symplectic topology. This is partly due to the fact that many questions in the
field can be formulated as questions involving Lagrangian submanifolds. For
instance, if $(M,\omega)$ is a symplectic manifold and
$\varphi\colon(M,\omega)\to(M,\omega)$ is a symplectomorphism, then its graph
$\Gamma_{\varphi}$ is a Lagrangian submanifold of $(M\times
M,(-\omega)\oplus\omega)$. Consequently, studying the fixed points of
$\varphi$ is equivalent to studying the set $\Gamma_{\varphi}\cap\Delta$ of
intersection points in $M\times M$.

Turning therefore to the question of Lagrangian intersections, we must mention
the progress made by Lagrangian Floer theory, first developed by Floer under
some relative asphericity conditions and then generalized by Oh to the case of
monotone Lagrangians in closed symplectic manifolds. This latter work was done
in a series of papers starting with [20]. The theory has been extended
considerably since then, but we will not delve into the history of these
developments. We do mention however that one of the main motivations for the
development of Lagrangian Floer theory was the _Arnold-Givental conjecture_ ,
a version of which can be stated as follows. If $(M,\omega)$ is a closed
symplectic manifold and $L\subset M$ is the fixed point set of an anti-
symplectic involution on $M$ (so in particular, a Lagrangian submanifold),
then

$\\#(L\cap\varphi(L))\geq\dim{H_{\ast}(L)},$

where $\varphi\colon M\to M$ is any Hamiltonian diffeomorphism and $L$ and
$\varphi(L)$ intersect transversally. By the work of Oh mentioned above, this
conjecture is true when $L$ is a monotone Lagrangian torus. (The definition of
monotonicity will be recalled in Section 1.) Observe that this result gives no
information about intersections of arbitrary Lagrangians with $L$, but only
Lagrangians that are images of $L$ under Hamiltonian diffeomorphisms. It is
therefore a statement about the _nondisplaceability of $L$_.

One can ask more general questions about Lagrangian intersections. For
instance, let $(M,\omega)$ be any symplectic manifold and $\mathcal{L}$ be a
collection of Lagrangians in $M$ satisfying some set of geometric or
topological restrictions. Then the following two questions naturally present
themselves:

  1. (1)

Do any two Lagrangians in $\mathcal{L}$ intersect?

  2. (2)

Is there some fixed Lagrangian in $\mathcal{L}$ that intersects every other
Lagrangian in $\mathcal{L}$?

Evidently, question (2) is more accessible than question (1). The following
result of Gromov, see $2.3.B_{4}^{\prime\prime}$ in [9], is along these lines
and will have important ramifications in the present paper.

######  Theorem 1.1 (Gromov).

Let $\Lambda$ be a closed manifold and $L$ be a closed and exact Lagrangian
submanifold of $(T^{\ast}\Lambda,d\lambda)$, where $\lambda$ is the canonical
Liouville $1$-form on $T^{\ast}\Lambda$. Then $L$ must intersect the zero
section $\Lambda$.

For $\Lambda=\mathbb{T}^{2}$, this theorem, when combined with the work of
Dimitroglou Rizell, Goodman, and Ivrii on the Nearby Lagrangian Conjecture for
$T^{\ast}\mathbb{T}^{2}$, see Theorem B of [8], implies the following result
in the spirit of question (1).

######  Proposition 1.2.

Any two exact Lagrangian tori in $T^{\ast}\mathbb{T}^{2}$ must intersect.

###### Proof.

Let $L$ be an exact Lagrangian torus in $T^{\ast}\mathbb{T}^{2}$. We use
Theorem B of [8] to obtain a Hamiltonian diffeomorphism $\varphi\colon
T^{\ast}\mathbb{T}^{2}\to T^{\ast}\mathbb{T}^{2}$ taking $L$ to the zero
section. If $L^{\prime}$ is another Lagrangian torus in
$T^{\ast}\mathbb{T}^{2}$ that is exact and disjoint from $L$, then
$\smash{\varphi(L^{\prime})}$ would be an exact Lagrangian torus which is
disjoint from the zero section, contradicting Theorem 1.1 above. ∎

######  Remark 1.3.

The previous proposition is true whenever $\Lambda$ is a smooth manifold for
which the Nearby Lagrangian Conjecture is verified, see [8].

The objective of this paper is to study question (2) for Lagrangian tori in a
special class of symplectic $4$-manifolds, the so-called _symplectic Del Pezzo
surfaces_. These are the monotone symplectic manifold
$(^{2}\times^{2},\sigma\oplus\sigma)$, where $\sigma(^{2})=2$, the monotone
symplectic manifold $(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$,
where $\omega_{\mspace{1.0mu}\mathrm{FS}}$ is the Fubini-Study form with
$\omega_{\mspace{1.0mu}\mathrm{FS}}(\mathbb{CP}^{1})=3$, and eight other
monotone symplectic manifolds obtained by blowing up along Darboux balls of
capacity 1 in $(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$. We now
describe recent progress on answering question (2) for this class of
manifolds, together with a description of our main results.

###  1.1. Main Results

Let $(M,\omega)$ be an integral symplectic $4$-manifold. (A symplectic
manifold is _integral_ if its symplectic form takes on integer values when
integrated over any closed $2$-dimensional submanifold.) A Lagrangian torus
$L\subset M$ is said to be _integral_ if the relative area homomorphism
$\omega\colon\pi_{2}(M,L)\to\mathbb{R}$ is integer-valued. A disjoint
collection $\\{L_{i}\\}$ of such tori is said to be a _maximal integral
packing of $(M,\omega)$_ if any integral torus in $M$ not belonging to this
collection intersects at least one member of the collection. The integrality
condition is required only so that the above intersection condition is not
trivially false, since one can always embed a Lagrangian torus in any
sufficiently small Darboux chart in $(M,\omega)$ missing the collection
$\\{L_{i}\\}$. The following result was recently obtained by Richard Hind and
Ely Kerman in [10].

######  Theorem 1.4 (Hind, Kerman).

The Clifford torus is a maximal integral packing of
$(^{2}\times^{2},\sigma\oplus\sigma)$.

This answers question (2) affirmatively for integral Lagrangian tori in
$(^{2}\times^{2},\sigma\oplus\sigma)$. The present paper takes inspiration
from the above result and extends it to the symplectic Del Pezzo surfaces
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ for $n=1,\dots,5$.
These are obtained by symplectically blowing up along $n$ pairwise disjoint
Darboux balls of capacity $1$ in
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$. The resulting
symplectic manifolds are independent of the symplectic isotopy classes of the
above Darboux balls. In particular, if
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ is obtained by
symplectically blowing up along $n$ Darboux balls in $\mathbb{CP}^{2}$ that
lie in the complement of the Clifford torus $L_{1,1}$, then this lifts to a
Lagrangian torus in $\mathbb{D}_{n}$ which we also call the Clifford torus and
denote by $L_{1,1}$. This is always possible for $n\leq 3$, and more recently
was shown to be true for $n=4$ and $n=5$, see Theorem 6.1 of [14]. Our main
result is then the following.

######  Theorem 1.5.

The Clifford torus is a maximal integral packing of
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ for $n=1,\dots,5$.

In other words, every integral Lagrangian torus in
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ for $n=1,\dots,5$
must intersect $L_{1,1}$. In view of the description of blowing up and down
given in Section 2 below, the case $n=1$ turns out to imply the following
slightly weaker result about maximal integral packings in certain open
submanifolds of $\mathbb{CP}^{2}$.

######  Corollary 1.6.

If $B$ is any closed Darboux ball of capacity $1$ in
$\mathbb{CP}^{2}\smallsetminus L_{1,1}$, then the Clifford torus is a maximal
integral packing of $(\mathbb{CP}^{2}\smallsetminus
B,\omega_{\mspace{1.0mu}\mathrm{FS}})$.

###  1.2. Commentary

The conclusion of Theorem 1.5 is not known to hold for $n>5$ since, to our
knowledge, there is no known embedding of $n>5$ disjoint closed Darboux balls
of capacity $1$ in $\mathbb{CP}^{2}\smallsetminus L_{1,1}$, although we
emphasize that this is the _only_ obstruction to the above theorem holding in
those cases, since our methods reduce all considerations to the case $n=1$.

Actually, the results in this paper imply a more general statement than that
of Theorem 1.5. Namely, our work actually shows that the Clifford torus is a
maximal packing of _any_ symplectic manifold obtained from $\mathbb{CP}^{2}$
by symplectically blowing up along finitely many disjoint Darboux balls of
integral capacity in the complement of $L_{1,1}$, so long as _at least one of
them_ is of capacity $1$. This a priori sounds like a much larger class of
manifolds then the Del Pezzo surfaces, but relies on the possibility of
symplectically embedding closed balls of integral capacity greater than $1$ in
$\mathbb{CP}^{2}\smallsetminus L_{1,1}$. In fact, in joint work in progress
with Richard Hind, it is shown that such a ball must have capacity strictly
less than $2$, meaning our current result really does exhaust all
possibilities.

There are two limitations to the techniques used to prove Theorem 1.5:

  1. (1)

They require the ambient symplectic manifold to be a ruled surface over 2,
since they make heavy use of the existence of foliations by closed
pseudoholomorphic spheres, as well as some of the special features of
pseudoholomorphic curves in dimension $4$.

  2. (2)

They require the tori in a maximal integral packing to be _monotone_ , since
this restricts the kinds of degenerations of pseudoholomorphic curves that can
occur when stretching the neck, and allows one to compute the indices of
pseudoholomorphic planes given information about the relative area classes of
their compactifications.

There are two interesting situations where one of the two requirements above
are not met and for which the integral packing problem appears to be
accessible. The first is
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$, where the first
requirement fails and for which the following conjecture seems reasonable.

######  Conjecture 1.7.

$L_{1,1}$ is a maximal integral packing of
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$.

In the cases where this problem has been solved thus far and where the
symplectic manifold admits a moment image, namely
$(^{2}\times^{2},\sigma\oplus\sigma)$ and
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ for $n=1,2,3$, there
is a maximal packing consisting of precisely the integer points in the moment
image, which in all cases is the Clifford torus projecting to the point
$(1,1)$. It would be appealing to think that this holds for all (precompact)
toric symplectic manifolds, but curiously a second result from [10], see
Theorem 1.2 in that reference, shows it to be false for the polydisk $P(a,b)$
when $\min(a,b)>2$.

Another situation in which the packing problem appears accessible through the
methods of this paper are the toric manifolds $(Y_{m},\omega_{m})$ for $m\geq
2$ obtained from $\mathbb{CP}^{2}$ with lines of area $m+2$ by symplectically
blowing up along the standard Darboux ball of capacity $m$ centered at the
point $[1:0:0]$. In this case, the monotonicity of the ambient symplectic
manifold is lost but the moment image contains $m$ integer points on the line
$x+y=m+1$, whose preimages are the product tori $L_{k,l}$ with $k+l=m+1$. We
can then make the following conjecture.

######  Conjecture 1.8.

The collection $\\{L_{k,l}:k+l=m+1\\}$ is a maximal integral packing of
$(Y_{m},\omega_{m})$ for each $m\geq 2$.

If true, this would produce the first examples of maximal integral packings
consisting of more than one torus, and would also produce examples of maximal
integral packings consisting of $m$ tori for each $m\geq 2$. We hope to
address the packing problem in this and in the aforementioned case of
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$ in future work.

###  1.3. Organization of the Paper

We begin in Section 2 by reviewing some standard background material from
symplectic topology. We briefly discuss symplectic area and monotonicity,
recall some basic facts from the theory of closed pseudoholomorphic curves in
dimension $4$, describe the symplectic blowup operation in the form in which
it will be used in this paper, and conclude by describing a standard inflation
procedure that can be performed on pseudoholomorphic spheres inside symplectic
$4$-manifolds. Although much of the material is standard, this section also
serves the purpose of fixing our notations and conventions for the rest of the
paper. The reader is therefore encouraged to at least skim through it.

In Section 3, we give a rather brief exposition of the theory of punctured
pseudoholomorphic spheres, closely following the one given in [8]. This begins
with a discussion of symplectic manifolds with cylindrical ends, and proceeds
in increasing generality to culminate in the definition of a pseudoholomorphic
building tailored to our specific context. This context is the one that arises
when producing buildings as limits of closed spheres by applying the splitting
procedure and compactness results from symplectic field theory, see [3] and
[5]. We conclude by recording some index formulae to be used in later
sections.

In Section 4, we specialize to the situation of interest to us, and define the
symplectic Del Pezzo surfaces. In particular, we give a detailed description
of a specific toric model of the Del Pezzo surface
$(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$, which we denote by
$(X,\omega_{X})$. It will be our main object of interest because, as we show
in Proposition 5.1, the proof of Theorem 1.5 reduces to studying Lagrangian
tori in $(X,\omega_{X})$. This is followed by an analysis of closed
pseudoholomorphic spheres in $(X,\omega_{X})$. These spheres will be used to
construct sequences whose limits after the splitting procedure will be
pseudoholomorphic buildings. Next, we state and prove a displacement result
for Lagrangian tori in $(X,\omega_{X})$, which will greatly simplify
subsequent considerations. This being done, we introduce one of the central
techniques used in this paper, that of relative pseudoholomorphic foliations.
These will be foliations of the complement of a collection of monotone
Lagrangian tori in $(X,\omega_{X})$ by pseudoholomorphic leaves obtained as
limits of closed spheres via the splitting procedure. We carefully describe
the properties of such foliations. We then discuss punctured pseudoholomorphic
curves that are transverse to such foliations, and which we call _essential_.
Finally, we study homologicaly essential tori in certain subsets of
$(X,\omega_{X})$ and show how the above relative foliations can shed light on
the packing problem considered there.

Section 5 can be thought of as the actual starting point for the proof of
Theorem 1.5. We assume it to be false and proceed by contradiction. The first
step is Proposition 5.1, which says that under such an assumption, we can find
disjoint monotone tori $L_{\mathfrak{r}}$ and $L_{\mathfrak{s}}$ in
$(X,\omega_{X})$ satisfying some special properties. Once this is done, we
construct a relative foliation of $X\smallsetminus(L_{\mathfrak{r}}\cup
L_{\mathfrak{s}})$ and begin the analysis of punctured spheres in the
complement. Using these considerations, we are able to obtain a full
classification of building types arising from the splitting procedure. We then
show that under certain conditions, there exists buildings of one of the above
types with some quantifiable properties. The section concludes with the
statement of Proposition 5.8, a large black box result that encapsulates all
of the facts needed in order to complete the proof of Theorem 1.5.

Section 6 concerns itself with the actual proof of Theorem 1.5. Starting with
the facts established at the end of the previous section, we perform a series
of blowups, inflations, and blowdowns, keeping track of the intersection
numbers and symplectic areas of various objects along each step. Some
concluding arguments then yield the desired contradiction. This is the only
place in the paper where the technique of inflation is used.

Section 7 describes some deformation procedures that can be performed on the
top level curves of certain pseudoholomorphic buildings, and then analyzes
various intersection patterns that arise from these deformations. The entirety
of this section is devoted to proving Proposition 5.8. Finally, we construct
in Appendix A a $3$-punctured symplectic sphere in
$(T^{\ast}\mathbb{T}^{2},d\lambda)$, which is needed in the proof of
Proposition 5.1.

###  1.4. Acknowledgments

I would like to thank Richard Hind for his many invaluable suggestions and
patient guidance. I would also like to thank the participants in an AMS
special session on quantitative symplectic topology, which took place in April
of 2023 at the University of Cincinnati. Their interest in this project led to
many helpful comments and suggestions for further research.

##  2\. Background from Symplectic Topology

We assume known the basic concepts of symplectic topology and closed
pseudoholomorphic curves, as found for example in [19] and [18]. However, in
order to make the presentation more readable, we will recall some elementary
concepts and fix some notation.

###  2.1. Notation and Conventions

Let $M$ be a smooth $4$-manifold without boundary and $L\subset M$ be an
embedded submanifold of codimension $2$. We will always consider singular
homology and cohomology with integer coefficients, and so do not specify
coefficients in our notation. We also identify the de Rham and real singular
cohomologies of $M$. If $L$ is compact and endowed with a fixed Riemannian
metric, its cotangent bundle $T^{\ast}L$ also inherits a Riemannian metric,
and for $\epsilon>0$ we define

$\displaystyle D_{\epsilon}^{\ast}L=\\{(q,p)\in
T^{\ast}L:\|p\|\leq\epsilon\\},\qquad B_{\epsilon}^{\ast}L=\\{(q,p)\in
T^{\ast}L:\|p\|<\epsilon\\}$ $\displaystyle S_{\epsilon}^{\ast}L=\\{(q,p)\in
T^{\ast}L:\|p\|=\epsilon\\}.$

In particular, we can consider the _unit cotangent bundle_
$S^{\ast}L=S^{\ast}_{1}L$.

When $M$ is compact, it is well-known that any class in $H_{2}(M)$ is
realizable by a compact oriented submanifold. In view of this, we use the same
symbol to denote a submanifold and the homology class it represents. If $M$ is
also oriented, then the _intersection number_ of homology classes $A,B\in
H_{2}(M)$ is defined to be

$A\cdot B=\langle\mathrm{PD}(A)\smile\mathrm{PD}(B),[M]\rangle,$

where $\mathrm{PD}(A)\in H^{2}(M)$ denotes the Poincaré dual of $A$. The
number $A\cdot B$ is equal to the signed count of intersection points between
any two closed, oriented, embedded submanifolds of $M$ that represent the
classes $A$ and $B$ and intersect transversally. This last condition can
always be achieved generically using standard arguments from differential
topology. If $L$ is also compact and oriented, we can sometimes consider a
_relative intersection number_ $A\cdot B$, where $A,B\in H_{2}(M,L)$. The
convention we adopt is to always interpret this as a signed count of
intersection points for transversally intersecting smooth representatives, and
only when these intersection points occur away from the boundaries (if any) of
both representatives. In this paper, all relative intersection numbers will be
well defined.

When $M$ is symplectic and $L$ is a Lagrangian torus, the following local
picture will be useful. We first fix a diffeomorphism
$\psi\colon\mathbb{T}^{2}\to L$, where we view $\mathbb{T}^{2}$ as the
quotient $\mathbb{R}^{2}/\mathbb{Z}^{2}$ with coordinates $(q_{1},q_{2})$ for
$0\leq q_{1},q_{2}\leq 1$. We call $\psi$ a _parameterization of $L$_, and use
it to transport the flat metric to $L$. The Weinstein neighborhood theorem,
see [19], allows us to extend this parametrization to a symplectic embedding,
also denoted by $\psi$, of an open neighborhood of the zero section in
$T^{\ast}\mathbb{T}^{2}$ into $M$. Its image is an open neighborhood of $L$
denoted by $\mathcal{U}(L)$ and called a _Weinstein neighborhood of $L$_. It
has coordinates induced from the above symplectomorphism, which we denote by
$(q_{1},q_{2},p_{1},p_{2})$. Note that $\psi$ induces an identification of
$H_{1}(L)$ with $\mathbb{Z}^{2}$, and we denote this copy of $\mathbb{Z}^{2}$
by $H_{1}(L;\psi)$.

###  2.2. Area and Monotonicity

If $(M,\omega)$ is a symplectic $4$-manifold, there is an obvious _area
homomorphism_ $\omega\colon H_{2}(M)\to\mathbb{R}$, defined by

$\omega(A)=\int_{S}\omega,$

where $S\subset M$ is any smooth representative of $A$. We can restrict the
domain of $\omega$ to the _spherical part_ of $H_{2}(M)$, which is the image
of $\pi_{2}(M)$ in $H_{2}(M)$ under the Hurewicz homomorphism. We thus obtain
a map

$\omega\colon\pi_{2}(M)\to\mathbb{R},\qquad[f]\mapsto\omega([f])=\int_{{}^{2}}f^{\ast}\omega.$

Since $TM$ has a well defined first Chern class $c_{1}(M)\in H^{2}(M)$, we
also have a _Chern homomorphism_ $c_{1}\colon H_{2}(M)\to\mathbb{Z}$ given by
$c_{1}(A)=\langle c_{1}(M),A\rangle$, which can again be restricted to a
homomorphism on $\pi_{2}(M)$. We then call $(M,\omega)$ _monotone_ if there is
a real constant $\kappa>0$, called the _monotonicity constant_ , such that the
Chern and area homomorphisms restricted to the spherical part of $H_{2}(M)$
satisfy $c_{1}=\kappa\mspace{1.0mu}\omega$. This is sometimes referred to as
_spherical monotonicity_ in the literature, but for the manifolds we consider
in this paper the distinction will be irrelevant.

If $L\subset M$ is a Lagrangian torus, then we also have an obvious _relative
area homomorphism_ $\omega\colon H_{2}(M,L)\to\mathbb{R}$ and a restricted
homomorphism on $\pi_{2}(M,L)$. We also have a _Maslov homomorphism_
$\mu\colon\pi_{2}(M,L)\to\mathbb{Z}$, whose precise definition can be found
for instance in [20]. We then say that $L$ is _monotone_ if there is a real
constant $\lambda>0$, also called the _monotonicity constant_ , such that
$\mu=\lambda\mspace{1.0mu}\omega$. In Remark 2.3 of [20] it is shown that the
presence of such a torus forces $(M,\omega)$ to also be monotone as a
symplectic manifold, with monotonicity constant $\kappa=\lambda/2$. In
particular, all monotone tori in $M$ have the same monotonicity constant
$\lambda=2\kappa$. We record the following easy lemma for use in some later
arguments.

######  Lemma 2.1.

Let $(M,\omega)$ be monotone with monotonicity constant $\lambda/2$ and
suppose that there are smooth disks
$v_{1},v_{2}\colon(\mathbb{D}^{2},^{1})\to(M,L)$ that satisfy
$\mu([v_{i}])=\lambda\mspace{1.0mu}\omega([v_{i}])$ for $i=1,2$ and whose
boundaries form an integral basis for $H_{1}(L)$. Then $L$ is monotone with
monotonicity constant $\lambda$.

###  2.3. Pseudoholomorphic Spheres

The basic reference for this material is [18]. Let $(M,\omega)$ be a
symplectic $4$-manifold endowed with a tame almost complex structure $J$ and
let $u\colon^{2}\to M$ be a closed $J$-holomorphic sphere. Then the class
$[u]=u_{\ast}[\Sigma]\in H_{2}(M)$ is called the _class of $u$_ and satisfies
$\omega([u])\geq 0$ with equality if and only if $u$ is constant. Recall also
that the _index of $u$_ is given by

$\operatorname{ind}(u)=-2+2c_{1}([u]).$

If $(M,\omega)$ is monotone, then $c_{1}([u])>0$ whenever $u$ is nonconstant,
and consequently $\operatorname{ind}(u)\geq 0$. Frequently, we will want to
choose $J$ to make a certain symplectically embedded sphere $S\subset M$ the
image of a $J$-holomorphic sphere. In this case, by a slight abuse of
language, we will say that $J$ is chosen to be _integrable near $S$_.

We will make use of the usual machinery of pseudoholomorphic curves in
dimension $4$, such as automatic regularity, the adjunction inequality, and
positivity of intersections: any two closed, connected $J$-holomorphic curves
with nonidentical images have finitely many intersection points, and these
always contribute positively to the intersection number. In particular, we
have $[u]\cdot[v]=0$ if and only $u$ and $v$ have disjoint images, and
$[u]\cdot[v]=1$ if and only if $u$ and $v$ have a single transverse
intersection. This is a local phenomenon, and we will be able to apply it
equally well to intersections between punctured pseudoholomorphic curves in
the following section.

###  2.4. The Blowup Operation

Our primary sources for this material are [19] and [23]. The complex blowup at
a point is a classical operation in complex and algebraic geometry, which can
be generalized to the almost complex category as follows. Let $(M,J)$ be an
almost complex $4$-manifold, fix a point $p\in M$, and assume that $J$ is
integrable in an open neighborhood of $p$. Then the _almost complex blowup of
$M$ at $p$_, denoted by $M^{\prime}$, is the disjoint union of
$M\smallsetminus\\{p\\}$ and the projectivization of $T_{p}M$, which is the
set of all complex $1$-dimensional subspaces in $T_{p}M$. It can be given the
structure of a smooth manifold as well as an almost complex structure
$J^{\prime}$. Reversing this operation yields a _blowdown map_ $\beta\colon
M^{\prime}\to M$, which is a pseudoholomorphic diffeomorphism from
$M^{\prime}\smallsetminus\\{\beta^{-1}(p)\\}$ to $M\smallsetminus\\{p\\}$.
Blowing up increases the rank of the second homology group by $1$.
Furthermore, the preimage $E=\beta^{-1}(p)$, which is called the _exceptional
divisor_ , is an embedded $J^{\prime}$-holomorphic sphere in $M^{\prime}$ with
$E\cdot E=-1$.

If $u^{\prime}$ is any $J^{\prime}$-holomorphic sphere in $M^{\prime}$, then
obviously $u=\beta\circ u^{\prime}$ is a $J$-holomorphic sphere in $M$.
Conversely, a $J$-holomorphic sphere in $M$ will always lift to a
$J^{\prime}$-holomorphic sphere in $M^{\prime}$ by removal of singularities.
It is not too hard to see that if $u^{\prime}$ is immersed and transverse to
$E$, then $u$ is also immersed. We then have

$c_{1}([u])=c_{1}([u^{\prime}\mspace{1.0mu}])+[u^{\prime}\mspace{1.0mu}]\cdot
E.$

Additionally, the sphere $u$ is embedded and passes through $p$ if and only if
$u^{\prime}$ is embedded with $[u^{\prime}\mspace{1.0mu}]\cdot E=1$, in which
case we have

$[u]\cdot[u]=[u^{\prime}\mspace{1.0mu}]\cdot[u^{\prime}\mspace{1.0mu}]+1.$

These formulas also hold if $u$ and $u^{\prime}$ are only smooth maps, a fact
which we shall occasionally use. Note however that a smooth map in $M$ _need
not_ always lift to a smooth map in $M^{\prime}$.

There is a symplectic analogue of the almost complex blowup, where the role of
a point is played by a symplectically embedded ball. The following
proposition, which is essentially Proposition 9.3.3 from [18], describes it in
terms of almost complex blowups. See also Theorem 3.13 in [23]. Recall that
the _open ball of capacity $\delta$_ is

$B(\delta)=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:\pi|z_{1}|^{2}+\pi|z_{2}|^{2}<\delta\\}.$

Here is the result.

######  Proposition 2.2.

Let $(M,\omega)$ be a symplectic $4$-manifold and $B(\delta+\epsilon)\subset
M$ be a Darboux ball centered at $p\in M$ for $\delta,\epsilon>0$. Let $J$ be
a tame almost complex structure on $M$, which is integrable on
$B(\delta+\epsilon)$ and $(M^{\prime},J^{\prime})$ be the almost complex
blowup of $M$ at $p$ with blowdown map $\beta\colon M^{\prime}\to M$. Then
there exists a symplectic form $\omega^{\prime}$ on $M^{\prime}$ that tames
$J^{\prime}$, agrees with $\beta^{\ast}\omega$ on
$M^{\prime}\smallsetminus\beta^{-1}(B(\delta+\epsilon))$, and satisfies

$\omega^{\prime}(A)=\omega(\beta_{\ast}A)-\delta(A\cdot E),$

where $A\in H_{2}(M^{\prime})$. If $J$ is compatible with $\omega$, then
$J^{\prime}$ is compatible with $\smash{\omega^{\prime}}$.

The symplectic manifold $(M^{\prime},\omega^{\prime})$ in the previous
proposition will be called the _symplectic blowup of $M$ along a Darboux ball
of capacity $\delta$_. Up to symplectomorphism, it turns out to be independent
of all choices except for the parameter $\delta$ and the symplectic isotopy
class of the embedding of $B(\delta)$. However, it follows from the results of
[16] that for the manifolds we consider in this paper, there is no dependency
on the symplectic isotopy class.

If $u$ is any $J$-holomorphic curve in $M$ that lifts to a
$J^{\prime}$-holomorphic curve $u^{\prime}$ in $M^{\prime}$, the formula in
Proposition 2.2 reads

$\omega^{\prime}([u^{\prime}\mspace{1.0mu}])=\omega([u])-\delta([u^{\prime}\mspace{1.0mu}]\cdot
E).$

We note that the formula in Proposition 2.2 will also hold if $L\subset M$ is
a Lagrangian torus and $A\in H_{2}(M^{\prime},L)$, so long as $E$ is disjoint
from $L$. We will use this fact numerous times in later sections.

We now give a necessary and sufficient condition for the symplectic blowup of
a monotone symplectic manifold to be monotone.

######  Proposition 2.3.

Let $(M,\omega)$ be a monotone symplectic $4$-manifold with monotonicity
constant $\kappa>0$ and $(M^{\prime},\omega^{\prime})$ be the symplectic
blowup of $M$ along a Darboux ball of capacity $\delta>0$. Then
$(M^{\prime},\omega^{\prime})$ is monotone if and only if $\delta=1/\kappa$,
in which case the monotonicity constant of $(M^{\prime},\omega^{\prime})$ is
$\kappa$.

###### Proof.

Let $S^{\prime}$ be an embedded representative of an element of
$\pi_{2}(M^{\prime})$, and $S$ be its image under the blowdown map $\beta$.
Then $S$ represents an element of $\pi_{2}(M)$, and so
$c_{1}(S)=\kappa\mspace{1.0mu}\omega(S)$ by monotonicity. Then

$\displaystyle c_{1}(S^{\prime})$ $\displaystyle=c_{1}(S)-S^{\prime}\cdot E$
$\displaystyle=\kappa\mspace{1.0mu}\omega(S)-S^{\prime}\cdot E$
$\displaystyle=\kappa(\omega^{\prime}(S^{\prime})+\delta(S^{\prime}\cdot
E))-S^{\prime}\cdot E$
$\displaystyle=\kappa\mspace{1.0mu}\omega^{\prime}(S^{\prime})+(\kappa\delta-1)(S^{\prime}\cdot
E).$

At this point, it is clear that $\delta=1/\kappa$ is a sufficient condition
for $(M^{\prime},\omega^{\prime})$ to be monotone. To see that this condition
is also necessary, set $S^{\prime}=E$. ∎

We will also need the _symplectic blowdown_ operation. It is defined in a
completely analogous fashion to the symplectic blowup, by characterizing it in
terms of a certain _almost complex blowdown_ operation. All we will need to
know for our purposes is that this can be performed on any symplectic
$4$-manifold $(M^{\prime},\omega^{\prime})$ in the presence of a
symplectically embedded sphere $E$ with $E\cdot E=-1$. We can arrange for any
fixed collection of symplectic submanifolds that are disjoint from $E$ to
remain symplectic in the resulting blowdown $(M,\omega)$. There is also a
blowdown map $\beta\colon M^{\prime}\to M$, which is a diffeomorphism of
$M^{\prime}\smallsetminus E$ onto $M\smallsetminus\\{\beta(E)\\}$, as well as
an obvious analogue of the formula in Proposition 2.2, which we shall use
frequently. For more details on these matters, we refer the reader to Section
3.2 and in particular Theorem 3.14 of [23].

###  2.5. Inflation

We will need to make use of the _inflation_ procedure in some later arguments.
The following proposition, which is Theorem 4.9 in [1], contains all the facts
that will be relevant to us.

######  Proposition 2.4.

Let $(M,\omega)$ be a symplectic $4$-manifold together with a compatible
almost complex structure $J$, and let $S\subset M$ be an embedded
$J$-holomorphic sphere with $S\cdot S\geq-1$. Then there is a family
$\\{\omega_{t}\\}_{t\geq 0}$ of symplectic forms on $M$, with $t<\omega(S)$
when $S\cdot S=-1$, which all tame $J$ and satisfy
$[\omega_{t}]=[\omega]+t\mathrm{PD}(S)$ in $H^{2}_{\mathrm{dR}}(M)$.

We call an _inflation of capacity $t$ along $S$_ the process of replacing
$(M,\omega)$ with $(M,\omega_{t})$ in the previous proposition. For any class
$A\in H_{2}(M)$, we then have

$\omega_{t}(A)=\omega(A)+t(S\cdot A).$

In other words, inflation modifies the symplectic area of objects according to
their intersections with $A$. Furthermore, if $L\subset M$ is a Lagrangian
torus that is disjoint from $S$, then the local nature of the inflation
procedure guarantees that the above formula continues to hold when $A\in
H_{2}(M,L)$.

######  Remark 2.5.

The statement we have given in Proposition 2.4 is an amalgam of _positive
inflation_ , as found in [17], and _negative inflation_ , as found in [4].
However, our statement assumes a compatible almost complex structure $J$, and
produces a family of symplectic forms that only _tame_ $J$. This is weaker
than the results found in the above two references, but as pointed out in [1],
there are certain issues with the inflation procedure when $J$ is only assumed
to be tame that are as of yet not fully resolved. The version we are using
avoids these issues and is sufficient for our purposes.

##  3\. Punctured Pseudoholomorphic Spheres

In this section, we recall some basic definitions and facts concerning
punctured pseudoholomorphic spheres and buildings. We then discuss the so-
called _splitting procedure_ , used to produce buildings as limits of closed
pseudoholomorphic curves in an appropriate geometric setting. We also record
some standard and useful index formulas. The primary references for much of
this material are [3] and [5]. Some additional references that are closer to
the spirit of this paper are [6], [8], [11], and [12], the last three of which
we draw on heavily.

###  3.1. Cylindrical Ends

Let $(V,\alpha)$ be a contact manifold with Reeb vector field $R$, and
consider the symplectic manifold

$(\mathbb{R}\times V,d(e^{t}\alpha)).$

This is called the _symplectization of $(V,\alpha)$_. A compatible almost
complex structure $J$ is then called _cylindrical_ if it is preserved by the
flow of $\partial_{t}$, if $J(\partial_{t})=R$, and if
$J(\ker\alpha)=\ker\alpha$. All of these constructions carry over to when
$\mathbb{R}$ is replaced by an arbitrary interval.

Now consider a _noncompact_ symplectic manifold $(M,\omega)$ containing a
compact regular domain $W$ with contact-type boundary given by a disjoint
union

$(V_{+},\alpha_{+})\sqcup(V_{-},\alpha_{-}).$

We require that at least one of $V_{+}$ or $V_{-}$ be nonempty, and allow one
or both of them to be disconnected. We then say that $(M,\omega)$ has
_cylindrical ends_ if the complement
$(M\smallsetminus\operatorname{Int}(W),\omega)$ is symplectomorphic to the
disjoint union of the symplectizations

$E_{+}=([0,+\infty)\times V_{+},d(e^{t}\alpha_{+})),\qquad
E_{-}=((-\infty,0]\times V_{-},d(e^{t}\alpha_{-})).$

The connected components of $E_{+}$ are called the _positive ends of
$(M,\omega)$ over $(V_{+},\alpha_{+})$_, and those of $E_{-}$ are called the
_negative ends of $(M,\omega)$ over $(V_{-},\alpha_{-})$_. (In the literature,
these are more commonly called _concave_ and _convex_ ends, respectively.)
There is a _collapse map_ $M\to W$ defined by simply collapsing the
cylindrical ends of $M$ to the boundary of $W$. The symplectization
$(\mathbb{R}\times V,d(e^{t}\alpha))$ of any contact manifold $(V,\alpha)$
itself has exactly one positive and one negative cylindrical end, both over
$(V,\alpha)$. Here one can take $W=[-\epsilon,\epsilon]\times V$ for any
$\epsilon>0$.

Call an almost complex structure $J$ _adapted to $(M,\omega)$_ if its
restriction to $W$ is tame and its restriction to each component of $E_{+}\cup
E_{-}$ is cylindrical. It can be shown using standard arguments that these
form a nonempty and contractible set. We can require $J$ to match any given
tame almost complex structure on a proper compact subset of $W$, or some given
cylindrical almost complex structure far enough along any component of
$E_{+}\cup E_{-}$.

###  3.2. Punctured Pseudoholomorphic Spheres

Throughout, let $(M,\omega)$ be a symplectic manifold with cylindrical ends
over $(V_{\pm},\alpha_{\pm})$ and $J$ be an almost complex structure adapted
to $(M,\omega)$. An _$n$ -punctured $J$-holomorphic sphere in $(M,\omega)$_
consists of the following data:

  1. (1)

A proper $J$-holomorphic map $u\colon^{2}\smallsetminus Z\to M$, where 2 has a
given complex structure and $Z=\\{z_{1},\dots,z_{n}\\}$ is a finite set of
_punctures_. It is partitioned into $Z_{+}\sqcup Z_{-}$, where the punctures
in $Z_{+}$ are called _positive_ and those in $Z_{-}$ are called _negative_.

  2. (2)

A (possibly multiply covered) periodic Reeb orbit $\gamma_{i}$ in
$(V_{\pm},\alpha_{\pm})$ assigned to each puncture $z_{i}\in Z_{\pm}$,
parameterized by the circle ${}^{1}=\mathbb{R}/\mathbb{Z}$ as
$t\mapsto\gamma_{i}(P_{i}t)$, where $P_{i}$ is the period of $\gamma_{i}$
obtained by integrating $\alpha$ over it.

  3. (3)

The map $u$ is required to be _asymptotic to $\gamma_{i}$ at $z_{i}$_, meaning
that the restriction of $u$ to a punctured disk centered at $z_{i}\in Z_{\pm}$
has a _cylindrical representation_
$(a_{i},\tilde{u}_{i})\colon\mathbb{D}^{2}\smallsetminus\\{0\\}\to\mathbb{R}\times
V_{\pm}$ satisfying

$\lim_{r\to 0}a_{i}(re^{2\pi it})=\pm\infty,\qquad\lim_{r\to
0}\tilde{u}_{i}(re^{2\pi it})=\gamma_{i}(\pm P_{i}t).$

These limits are required to be uniform with respect to $t$.

More generally, a $J$-holomorphic map $u$ satisfying (1) and (2) also
satisfies (3) if and only if it has _finite energy_ , see [3]. This will be
the case for all punctured spheres considered in this paper. A $1$-punctured
$J$-holomorphic sphere is called a _plane_ , and a $2$-punctured
$J$-holomorphic sphere is called a _cylinder_. Furthermore, the above
definition pertains specifically to a _parameterized sphere_. Under a suitable
definition of equivalence, we can (and will) think of the image of $u$ above
as an _unparameterized sphere_. See [3] for a precise definition of this
equivalence.

The domain ${}^{2}\smallsetminus Z$ of $u$ can be compactified to a
topological surface $\Sigma$ with nonempty boundary by simply replacing each
puncture in $Z$ with a circle. The asymptotic properties of $u$ imply that it
extends continuously over $\partial\Sigma$ by mapping a boundary circle to the
corresponding asymptotic Reeb orbit. Recalling the regular domain $W\subset
M$, we can compose the extension of $u$ with the collapse map $M\to W$,
obtaining a continuous map
$\overline{u}\colon(\Sigma,\partial\Sigma)\to(M,V_{+}\cup V_{-})$, which is
called the _compactification of $u$_. Its image defines a relative class
$[\overline{u}]\in H_{2}(M,V_{+}\cup V_{-})$, called the _class of $u$_.

Assume now that all periodic Reeb orbits of $\alpha_{\pm}$ come in _Morse-
Bott_ families, see [2], and denote by $\smash{\Gamma_{i}^{\pm}}$ the
connected Morse-Bott manifold containing $\smash{\gamma_{i}^{\pm}}$, where the
sign indicates whether the orbit corresponds to a positive or negative
puncture. Any symplectic trivialization of the contact structures on
$\smash{(V_{\pm},\alpha_{\pm})}$ over these orbits extends to a symplectic
trivialization of $TM|_{\smash{\gamma_{i}^{\pm}}}\cong T(\mathbb{R}\times
V_{\pm})|_{\smash{\gamma_{i}^{\pm}}}$, since the remaining directions are
spanned by the Reeb and global coordinate vector fields. The bundle
$u^{\ast}TM$ will then inherit a symplectic trivialization over
$\partial\Sigma$, denoted by $\tau$, and we can define the following two
quantities:

  1. $\bullet$

Each orbit $\smash{\gamma_{i}^{\pm}\in\Gamma_{i}^{\pm}}$ gives rise to an
_asymptotic operator_ defined on an appropriate space of sections, see [22],
and relative to $\tau$, it corresponds to a path in $\operatorname{Sp}(2)$.
This in turn has a _Robbin-Salamon index_
$\smash{\mu_{\mathrm{RS}}(\gamma_{i}^{\pm};\tau)}$ in the sense of [21]. It is
a half-integer that turns out to depend only on $\tau$ and
$\smash{\Gamma_{i}^{\pm}}$.

  2. $\bullet$

The bundle $u^{\ast}TM$ has a _relative first Chern class_ $c_{1}(u;\tau)$. It
can be defined as an algebraic count of the zeros of a generic section
$\sigma$ of the line bundle $\Lambda^{2}_{\mathbb{C}}(u^{\ast}TM)$, where we
require $\sigma$ to be nonvanishing and constant with respect to $\tau$ near
all cylindrical ends of $(M,\omega)$.

Setting $s_{\pm}=|Z_{\pm}|$ and following [2], the _index of $u$_ (as an
_unparameterized_ curve with _unconstrained_ ends) is then given by

$\displaystyle\operatorname{ind}(u)=-2+s_{+}+s_{-}+c_{1}(u;\tau)$
$\displaystyle+\sum_{i=1}^{s_{+}}\Big{(}\mu_{\mathrm{RS}}(\gamma_{i}^{+};\tau)+\frac{1}{2}\dim\Gamma_{i}^{+}\Big{)}$
$\displaystyle-\sum_{i=1}^{s_{-}}\Big{(}\mu_{\mathrm{RS}}(\gamma_{i}^{-};\tau)-\frac{1}{2}\dim\Gamma_{i}^{-}\Big{)}.$

It follows immediately from this formula that the total contribution of the
last three terms must be independent of $\tau$.

We need the following generalization of punctured spheres. A _punctured nodal
$J$-holomorphic sphere in $(M,\omega)$_ consists of the following data:

  1. (1)

An unordered collection $\mathbf{u}=\\{u_{1},\dots,u_{k}\\}$ of punctured
$J$-holomorphic spheres in $(M,\omega)$. We write $S_{i}=^{2}\smallsetminus
Z_{i}$ for the punctured domain of the map $u_{i}$.

  2. (2)

A finite subset $\Delta\subset S_{1}\cup\cdots\cup S_{k}$ with a fixed-point
free involution. In particular, the set $\Delta$ consists of an even number of
points, called _nodes_.

  3. (3)

If $p\in S_{i}$ and $q\in S_{j}$ are nodes that are matched under the above
involution, then $u_{i}(p)=u_{j}(q)$. It is also required that the surface
obtained by taking a connected sum of the $S_{i}$ along all matching nodes be
homeomorphic to a punctured sphere.

When $\Delta=\varnothing$, then $\mathbf{u}$ is a punctured $J$-holomorphic
sphere as before. In this case, we will say $\mathbf{u}$ is _smooth_ to
emphasize that there are no nodes. As for smooth punctured spheres, we
identify $\mathbf{u}$ with the union of the images of the $u_{i}$, thinking of
$\mathbf{u}$ as an _unparameterized nodal sphere_. This can be made precise
using a suitable notion of equivalence, see [3]. We can also form the
compactification of each of the $u_{i}$ as before and take their union, which
we also call the _compactification of $\mathbf{u}$_. It gives rise to a
homology class $A\in H_{2}(M,V_{+}\cup V_{-})$, which is the sum of the
classes of all the $u_{i}$. Finally, we define the _index of $\mathbf{u}$_ to
be

$\operatorname{ind}(\mathbf{u})=\sum_{i=1}^{k}\operatorname{ind}(u_{i})+2(k-1).$

This is compatible with the definition of the index of a nodal sphere in the
closed case, as is easily checked.

###  3.3. The Cotangent Bundle

The global coordinates $(q_{1},q_{2})$ on $\smash{\mathbb{T}^{2}}$ induce
coordinates $(q_{1},q_{2},p_{1},p_{2})$ on the cotangent bundle
$\smash{T^{\ast}\mathbb{T}^{2}=\mathbb{T}^{2}\times\mathbb{R}^{2}}$. This has
a canonical symplectic structure induced by the $2$-form $d\lambda$, where

$\lambda=p_{1}d{q_{1}}+p_{2}d{q_{2}}$

is the _Liouville $1$-form_. Under the natural identification of
$S^{\ast}\mathbb{T}^{2}$ with the $3$-torus
$\mathbb{T}^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$, the restriction
$\alpha=\lambda|_{S^{\ast}\mathbb{T}^{2}}$ then takes the form

$\alpha=\cos(2\pi\theta)d{q_{1}}+\sin(2\pi\theta)d{q_{2}}.$

This is a contact form on $S^{\ast}\mathbb{T}^{2}$, which becomes a contact-
type hypersurface in $(T^{\ast}\mathbb{T}^{2},d\lambda)$. Its Reeb vector
field is given by

$R=\cos(2\pi\theta)\partial_{q_{1}}+\sin(2\pi\theta)\partial_{q_{2}}.$

Under the identification of $T^{\ast}\mathbb{T}^{2}$ with $T\mathbb{T}^{2}$
induced by the flat metric, the flow of $R$ corresponds to the geodesic flow,
and the projection $\smash{S^{\ast}\mathbb{T}^{2}\to\mathbb{T}^{2}}$ then
induces a correspondence between periodic Reeb orbits of fixed multiplicity on
$S^{\ast}\mathbb{T}^{2}$ and closed oriented geodesics on $\mathbb{T}^{2}$,
which in turn correspond to $\theta\in^{1}$ such that
$\tan(2\pi\theta)\in\mathbb{Q}\cup\\{\infty\\}$. Thus to each nonzero class
$\eta\in H_{1}(\mathbb{T}^{2})$, there corresponds a $1$-dimensional family of
periodic Reeb orbits of fixed multiplicity, with families of simple Reeb
orbits corresponding to primitive homology classes. We denote by
$\Gamma_{\eta}$ the manifold of Reeb orbits corresponding to the class $\eta$.
This is a Morse-Bott manifold in the sense of [2].

Consider the symplectomorphism $(\mathbb{R}\times
S^{\ast}\mathbb{T}^{2},d(e^{t}\alpha))\to(T^{\ast}\mathbb{T}^{2}\smallsetminus\mathbb{T}^{2},d\lambda)$
given by

$(t,q_{1},q_{2},p_{1},p_{2})\mapsto(q_{1},q_{2},e^{t}p_{1},e^{t}p_{2}).$

Restricting it to $[0,+\infty)\times S^{\ast}\mathbb{T}^{2}$ shows that
$(T^{\ast}\mathbb{T}^{2},d\lambda)$ has a single positive cylindrical end over
$S^{\ast}\mathbb{T}^{2}$, and restricting it to $(-\infty,0]\times
S^{\ast}\mathbb{T}^{2}$ shows that
$(D_{1}^{\ast}\mathbb{T}^{2}\smallsetminus\mathbb{T}^{2},d\lambda)$ has a
single negative end over $S^{\ast}\mathbb{T}^{2}$. We then define an almost
complex structure $J_{\mathrm{std}}$ on $T^{\ast}\mathbb{T}^{2}$ by

$J_{\mathrm{std}}(\partial_{q_{i}})=-\rho(\|(p_{1},p_{2})\|)\partial_{p_{i}},\qquad
i=1,2,$

where $\rho\colon[0,+\infty)\to(0,+\infty)$ is a smooth nondecreasing function
that satisfies $\rho(t)=1$ on $[0,1]$ and $\rho(t)=t$ on $[2,+\infty)$. We
also define an almost complex structure $J_{\mathrm{cyl}}$ on
$\mathbb{R}\times S^{\ast}\mathbb{T}^{2}$ by

$J_{\mathrm{cyl}}(\partial_{q_{i}})=-\|(p_{1},p_{2})\|\partial_{p_{i}},\qquad
i=1,2,$

using the identification coming from the above symplectomorphism. It is shown
in Lemma 4.1 of [8] that $J_{\mathrm{cyl}}$ is a cylindrical almost complex
structure on $(\mathbb{R}\times S^{\ast}\mathbb{T}^{2},d(e^{t}\alpha))$ and
that $J_{\mathrm{std}}$ is adapted to $(T^{\ast}\mathbb{T}^{2},d\lambda)$.

###  3.4. Pseudoholomorphic Buildings

We now narrow our focus to the setting in which buildings will arise in this
paper. Let $(M,\omega)$ be a closed symplectic $4$-manifold and
$L=L_{1}\cup\cdots\cup L_{k}$ be a union of pairwise disjoint Lagrangian tori
in $M$. We will work inside the so-called _split symplectic manifold_

$(M\smallsetminus L,\omega)\sqcup(T^{\ast}L,d\lambda),$

where $\lambda$ is the Liouville $1$-form on $T^{\ast}L$.

For $i=1,\dots,k$, fix a parametrization $\psi_{i}\colon\mathbb{T}^{2}\to
L_{i}$. We can assume the induced Weinstein neighborhoods $\mathcal{U}(L_{i})$
are pairwise disjoint, and by rescaling the flat metric on $\mathbb{T}^{2}$,
we can also assume that $\mathcal{U}(L_{i})$ contains
$\psi_{i}(D^{\ast}_{4}\mathbb{T}^{2})$. We declare $\psi_{i}$ to be an
isometry, so that $\psi_{i}(D^{\ast}_{4}\mathbb{T}^{2})=D^{\ast}_{4}L_{i}$.
The entire discussion from the previous subsection can then be carried over to
the current setting. In particular, since

$(D_{1}^{\ast}L_{i}\smallsetminus
L_{i},\omega)\cong(D_{1}^{\ast}\mathbb{T}^{2}\smallsetminus\mathbb{T}^{2},d\lambda),$

we see that $(M\smallsetminus L,\omega)$ has a negative cylindrical end and
that $(T^{\ast}L,d\lambda)$ has a positive cylindrical over $S^{\ast}L_{i}$
for each $i$. The almost complex structures $J_{\mathrm{std}}$ and
$J_{\mathrm{cyl}}$ can also be transported to almost complex structures on
$T^{\ast}L$ and $\mathbb{R}\times S^{\ast}L$, respectively. We will then say a
tame almost complex structure $J$ on $M\smallsetminus L$ is _adapted to the
parameterizations $\psi_{i}$_ if, for each $i$, it takes the form

$J(\partial_{q_{i}})=-\|(p_{1},p_{2})\|\partial_{p_{i}}$

inside $\mathcal{U}(L_{i})\smallsetminus L_{i}$. Since this set contains
$D_{1}^{\ast}L_{i}\smallsetminus L_{i}$ by assumption, it follows that $J$ is
adapted to $(M\smallsetminus L,\omega)$.

We define a _pseudoholomorphic building in $(M\smallsetminus
L,\omega)\sqcup(T^{\ast}L,d\lambda)$_ as a finite collection $\mathbf{F}$ of
punctured nodal pseudoholomorphic spheres, called the _components_ of the
building, which are divided among various _levels_ according to the following
scheme:

  1. $\bullet$

Top level: A finite number of $J$-holomorphic components contained in
$(M\smallsetminus L,\omega)$. There is always at least one such component.

  2. $\bullet$

Middle levels: A finite number of $J_{\mathrm{cyl}}$-holomorphic components
contained in consecutively ordered copies of $(\mathbb{R}\times
S^{\ast}L,d(e^{t}\alpha))$, of which there may be none. If there are however,
then each must contain at least one component that is not a trivial cylinder
over a periodic Reeb orbit.

  3. $\bullet$

Bottom level: A finite number of $J_{\mathrm{std}}$-holomorphic components
contained in $(T^{\ast}L,d\lambda)$. There may be no such components.

The smooth curves that constitute each component of $\mathbf{F}$ are
collectively referred to as the _curves of $\mathbf{F}$_. We require the
following conditions:

  1. (1)

If $u$ is a curve of $\mathbf{F}$ contained in the $i$th level and asymptotic
to a parameterized Reeb orbit in $(S^{\ast}L,\alpha)$ at some positive
puncture, there must exist a curve in the $(i+1)$th level asymptotic to that
same orbit at a negative puncture. Likewise, if $u$ is asymptotic to a
parameterized Reeb orbit in $(S^{\ast}L,\alpha)$ at some negative puncture,
there must exist a curve in the $(i-1)$th level asymptotic to that same orbit
at a positive puncture. All punctures must furthermore match bijectively in
pairs.

  2. (2)

Let $\Sigma$ be the (nodal) surface formed by taking the connected sum of all
component domains along all pairs of asymptotically matching punctures and by
identifying nodes under the involution in each domain. Then $\Sigma$ must be
homeomorphic to a (nodal) sphere.

We will say that $\mathbf{F}$ is a _broken nodal sphere_ if it consists of at
least two nonempty levels, and an _unbroken nodal sphere_ otherwise. In either
case, we replace the qualification _nodal_ by _smooth_ if each component of
$\mathbf{F}$ is smooth (that is, if its domain has no nodes). In this case,
the surface $\Sigma$ in (2) above will be homeomorphic to a genuine sphere. We
also identify $\mathbf{F}$ with its image under a suitable notion of
equivalence for buildings, see [3]. In this way, we think of $\mathbf{F}$ as
an _unparameterized building_.

If $u$ is a top level curve of $\mathbf{F}$, then its compactification
$\overline{u}$ is a map into $M$ with boundary on $S^{\ast}L$ corresponding to
its asymptotic Reeb orbits. We extend it by cylinders to a map whose boundary
components are the corresponding geodesics on $L$, obtaining a homology class
$[\overline{u}]\in H_{2}(M,L)$, also called the _class of $u$_. We then define
the _area of $u$_ to be $\omega(\overline{u})$. (Observe that when $u$ has no
punctures, this is just the usual symplectic area of a closed $J$-holomorphic
sphere.) It can then be shown that $\omega(\overline{u})\geq 0$, with equality
if and only if $u$ is constant.

On the other hand, the compactifications of curves of $\mathbf{F}$ in the
middle or bottom levels can be projected into $L$. We then obtain chains whose
boundaries are geodesics corresponding to boundaries of compactified top level
curves. Identifying these boundaries gives rise to a continuous map
$\overline{\mathbf{F}}\colon\Sigma\to M$, called the _compactification of
$\mathbf{F}$_. Its image defines a class $A\in H_{2}(M)$, which we call the
_class of $\mathbf{F}$_. The _area of $\mathbf{F}$_ is then defined to be
$\omega(A)$. It is clearly the sum of the areas of all top level curves, and
we have $\omega(A)\geq 0$ with equality if and only if $\mathbf{F}$ is
constant, see [5].

The following proposition records some additional facts about buildings.

######  Proposition 3.1.

Let $\mathbf{F}$ be a building in the class $A\in H_{2}(M)$.

  1. (a)

If $\mathbf{F}$ is broken, the it has at least two top level planes.

  2. (b)

If there is no decomposition $A=A_{1}+A_{2}$ with
$\omega(A_{1}),\omega(A_{2})>0$, then $\mathbf{F}$ is smooth.

  3. (c)

If the tori $L_{i}$ are all integral, then every top level curve of
$\mathbf{F}$ has integral area.

###### Proof.

To prove (a), first note that any nonconstant middle or bottom level curve of
$\mathbf{F}$ has at least one puncture, since otherwise by exactness, it would
have zero area. Furthermore, if it had exactly one puncture, then it would
project to a disk in $S^{\ast}L$ bounding a Reeb orbit, which is impossible
since all orbits are noncontractible for a flat torus. The curve must
therefore have at least two punctures, and middle level curves must have at
least one positive puncture by the maximum principle for subharmonic
functions. From these facts it is easy to deduce that $\mathbf{F}$ has at
least two planes in the top level.

The proof of (b) is clear if $\mathbf{F}$ is constant, so assume that
$\mathbf{F}$ is nonconstant and nodal, and let $\Sigma$ be the domain of its
compactification $\overline{\mathbf{F}}$. Then $\Sigma$ can be decomposed as
$\Sigma_{1}\cup\Sigma_{2}$, where $\Sigma_{1}\cap\Sigma_{2}$ consists of a
single point, such that the restriction of $\overline{\mathbf{F}}$ to each
$\Sigma_{i}$ is the compactification of a nonconstant pseudoholomorphic
building $\mathbf{F}_{i}$ whose homology class $A_{i}\in H_{2}(M)$ satisfies
$\omega(A_{i})>0$ by (a). But this would lead to a decomposition
$A=A_{1}+A_{2}$ with $\omega(A_{1}),\omega(A_{2})>0$, contradicting our
assumption.

Finally, to prove (c), note that $(M,\omega)$ must then be integral. This
implies that every spherical homology class in $H_{2}(M)$ has integral area,
and the result for any top level curve without punctures follows. On the other
hand, if $u$ was a top level curve with at least one puncture, then its
compactification $\overline{u}$ represents a relative class in $H_{2}(M,L)$
and has boundary components on some of the tori $L_{1},\dots,L_{k}$. We can
then choose disks in $M$ that bound these components, and these have integral
area by assumption. Gluing these disks to $\overline{u}$ yields a spherical
class in $H_{2}(M)$, which also has integral area. By additivity, it then
follows that $u$ must also have integral area. ∎

###  3.5. The Splitting Procedure

The primary method used for producing pseudoholomorphic buildings in this
paper is the _splitting procedure_ , also known as _stretching the neck_. This
can be carried out in a fairly general setting, see [3] and [5]. We will give
a description of this procedure that is tailored to our specific context,
following [8].

With the same setup and notation as the previous subsection, consider a family
$\\{J_{t}\\}_{t\geq 0}$ of tame almost complex structures on $(M,\omega)$,
which for each $i$ are defined according to the following prescription:

  1. $\bullet$

Outside all of the open sets $B_{4}^{\ast}L_{i}$, we set $J_{t}\equiv J$ for
each $t\geq 0$.

  2. $\bullet$

Inside each of the open sets $B_{2}^{\ast}L_{i}$, we set $J_{t}\equiv
J_{\mathrm{std}}$ for each $t\geq 0$.

  3. $\bullet$

Inside each $B_{4}^{\ast}L_{i}\smallsetminus B_{2}^{\ast}L_{i}$, we identify
$J_{t}$ for each $t\geq 0$ with the restriction of $J_{\mathrm{cyl}}$ to the
set $\smash{[\log{2},\log{4}+t)\times S^{\ast}L_{i}}$ under the
identifications

$\displaystyle B_{4}^{\ast}L_{i}\smallsetminus B_{2}^{\ast}L_{i}$
$\displaystyle\approx[\log{2},\log{4})\times S^{\ast}L_{i}$
$\displaystyle\approx[\log{2},\log{4}+t)\times S^{\ast}L_{i}.$

Note that only the first of these is a symplectomorphism.

We say the family $\\{J_{t}\\}_{t\geq 0}$ _stretches to $J$_, and in this
context we refer to the latter as a _stretched_ almost complex structure on
$M\smallsetminus L$. We can then extract a sequence
$(J_{t_{i}})_{i=1}^{\infty}$ such that $t_{i}\to+\infty$ as $i\to+\infty$. In
this setting, we have the following _compactness theorem_.

######  Theorem 3.2.

Let $\smash{(u_{i})_{i=1}^{\infty}}$ be a sequence of closed
$J_{t_{i}}$-holomorphic spheres in $(M,\omega)$ in a fixed class $A\in
H_{2}(M)$. Then there exists a subsequence that converges to a
pseudoholomorphic building in $(M\smallsetminus
L,\omega)\sqcup(T^{\ast}L,d\lambda)$ whose homology class is also $A$.

The building obtained in the conclusion of this theorem is called a _limit
building_. The exact definition of convergence is not relevant to us, since
all we care about is the _existence_ of such buildings. See [3] or [5] for a
more comprehensive discussion as well as proofs of the above theorem.

######  Remark 3.3.

It is possible to generalize the discussion in this and the previous
subsection by allowing $(M,\omega)$ to be noncompact with cylindrical ends.
This requires a generalization of the notions of pseudoholomorphic building
and convergence, but the only time we will need this more general theory is in
Lemma 5.2 below. There we will be applying the splitting procedure given the
contact-type hypersurfaces we have been considering so far as well as the
boundary of a symplectically embedded ball. We will study the limit of a
sequence of top level planes of minimal area, all belonging to the same
relative homology class, and thus having uniformly bounded energy. This
condition is sufficient to ensure compactness, and so no complicated limit
configurations will arise. Because of this, we will not give an exposition of
this more general setting, instead referring the reader to [3].

###  3.6. Index Calculations

With the setup and notation of the previous two subsections, let
$\mathbf{u}_{1},\dots,\mathbf{u}_{l}$ be the components of the building
$\mathbf{F}$. If $m$ is the total number of Reeb orbits at which these
components match asymptotically, then following [12], we define the _index of
$\mathbf{F}$_ to be

$\operatorname{ind}(\mathbf{F})=\sum_{i=1}^{l}\operatorname{ind}(\mathbf{u}_{i})-m.$

This is designed so that if $(u_{i})_{i=1}^{\infty}$ is a sequence of closed
pseudoholomorphic spheres in $A\in H_{2}(M)$ converging to $\mathbf{F}$ as in
Theorem 3.2, then

$\operatorname{ind}(\mathbf{F})=\operatorname{ind}(u_{i})=-2+2c_{1}(A).$

Computing the index of $\mathbf{F}$ therefore reduces to computing the index
of its components, and thus the index of its curves in various levels. Here is
one result in this direction.

######  Proposition 3.4.

Let $u$ be a punctured pseudoholomorphic sphere having $s_{+}$ positive
punctures and $s_{-}$ negative punctures in $(\mathbb{R}\times
S^{\ast}L,d(e^{t}\alpha))$ or $(T^{\ast}L,d\lambda)$, where $s_{-}=0$ in the
latter case. Then

$\operatorname{ind}(u)=-2+2s_{+}+s_{-}.$

This index is nondecreasing under multiple covers, and is positive if $u$ is a
middle or bottom level curve of a limit building.

###### Proof.

A proof of the formula can be found in [8] or [11]. The fact that it is
nondecreasing under multiple covers follows from an argument using the
Riemann-Hurwitz formula as in the proof of Lemma 3.2 of [8]. The last
statement follows from the fact that, in this case, any such curve always has
at least two punctures by (a) of Proposition 3.1. ∎

The next proposition, which follows from the discussion in Section 3.1 of [8],
provides a convenient formula for the index of a top level curve using an
extension of the Maslov index for surfaces with boundary on $L$. We refer to
Appendix C.3 of [18] for an exact definition.

######  Proposition 3.5.

Let $u$ be a punctured $J$-holomorphic sphere in $M\smallsetminus L$ with
$s_{-}$ negative punctures. Then

$\operatorname{ind}(u)=-2+s_{-}+\mu(\overline{u}).$

This index is nondecreasing under multiple covers.

##  4\. Initial Setup

In this section, we begin by setting up our primary objects of study. Next, we
discuss the elements of a _relative foliation theory_ , which will be crucial
to our arguments in later sections. We then study some special properties of
the Clifford torus in the context of such relative foliations.

###  4.1. Del Pezzo Surfaces

Consider the symplectic manifold
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$, where the Fubini-Study
form $\omega_{\mspace{1.0mu}\mathrm{FS}}$ is scaled so that
$\omega_{\mspace{1.0mu}\mathrm{FS}}(\mathbb{CP}^{1})=3$. In this way
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$ becomes a monotone
symplectic manifold with monotonicity constant $1$. Recall that the standard
torus $\mathbb{T}^{2}$ is viewed as the quotient
$\mathbb{R}^{2}/\mathbb{Z}^{2}$ with global coordinates $(q_{1},q_{2})$ for
$0\leq q_{i}\leq 1$. Following [19], the moment map
$\mu\colon\mathbb{CP}^{2}\to\mathbb{R}^{2}$ for the standard Hamiltonian toric
action of $\mathbb{T}^{2}$ on
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$ is given in homogeneous
coordinates by

$\mu([z_{0}:z_{1}:z_{2}])=3\left(\frac{|z_{1}|^{2}}{\|z\|^{2}},\frac{|z_{2}|^{2}}{\|z\|^{2}}\right).$

The image of $\mu$ is the right triangle $\\{(x,y)\in\mathbb{R}^{2}:0\leq
x+y\leq 3\\}$, and we define the subsets

$S_{\infty}=\\{x+y=3\\},\qquad T_{0}=\\{y=0\\},\qquad T_{\infty}=\\{x=0\\}$

and denote their preimages under $\mu$ by the same symbols. These are in fact
symplectically embedded spheres in $\mathbb{CP}^{2}$, the first being the
_line at infinity_. The _Clifford torus_ is defined as

$L_{1,1}=\mu^{-1}(1,1)=\\{[z_{0}:z_{1}:z_{2}]\in\mathbb{CP}^{2}:|z_{0}|=|z_{1}|=|z_{2}|\\}.$

It is a monotone Lagrangian torus with monotonicity constant $2$. The
following diagram illustrates the moment image with all of these objects:

$T_{0}$$T_{\infty}$$S_{\infty}$$L_{1,1}$$T_{0}$

By Proposition 2.3, the only way to obtain monotone symplectic manifolds from
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$ by symplectically
blowing up is to do so along Darboux balls of capacity $1$. It is a known fact
that one can embed up to $8$ such disjoint closed balls in $\mathbb{CP}^{2}$
before encountering a volume obstruction. We then obtain for $n=1,\dots,8$ a
monotone symplectic manifold
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ with monotonicity
constant $1$. These together with the symplectic manifolds
$(^{2}\times^{2},\sigma\oplus\sigma)$, where $\sigma(^{2})=2$, and
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$, are known as the
_symplectic Del Pezzo surfaces_. As we shall show in Proposition 4.4, once
Theorem 1.5 is proved for the case $n=1$ then it will follow immediately for
$n=2,\dots,5$. Because of this, we will now examine a particularly explicit
model of $(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$ and record
some of its features.

Consider the symplectic embedding $B(3)\hookrightarrow\mathbb{CP}^{2}$
centered at the point $[1:0:0]$, given by

$(z_{1},z_{2})\mapsto\left[\sqrt{(3/\pi)-|z_{1}|^{2}-|z_{2}|^{2}}:z_{1}:z_{2}\right].$

This restricts to a symplectic embedding of $B(1+\epsilon)$, with $\epsilon>0$
chosen small enough so that $B(1+\epsilon)$ misses $L_{1,1}$. We then apply
Proposition 2.2 to obtain the desired model of
$(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$. For reference, we
will henceforth denote this model by $(X,\omega_{X})$. One of its salient
features is an induced torus action with moment image obtained from that of
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$ by excising the
triangle $\\{0\leq x+y<1\\}$. The proper transforms of $S_{\infty}$, $T_{0}$,
and $T_{\infty}$ in $X$ are still symplectically embedded spheres and are
denoted by the same symbols. Additionally, the exceptional divisor in
$(X,\omega_{X})$, denoted by $S_{0}$, corresponds to the edge $\\{x+y=1\\}$ in
the moment image. The following diagram sums up these observations:

$S_{\infty}$$T_{0}$$T_{\infty}$Blowup$S_{0}$$S_{\infty}$$T_{0}$$T_{\infty}$$L_{1,1}$$L_{1,1}$$T_{0}$

The configuration of spheres $S_{0}\cup S_{\infty}\cup T_{0}\cup T_{\infty}$
will be a recurring object in this paper. We will refer to it as the _axes of
$(X,\omega_{X})$_ and denote it by $Y$.

The second homology group $H_{2}(X)$ is free abelian of rank $2$ and contains
three distinguished classes. There is the class of the spheres $T_{0}$ and
$T_{\infty}$, as well as the classes $S_{0}$ and $S_{\infty}$. We will use
$(S_{0},T_{0})$ as an ordered basis for $H_{2}(X)$, identifying it with
$\mathbb{Z}^{2}$. Thus $S_{\infty}=S_{0}+T_{0}$ is the class $(1,1)$. The
corresponding symplectic areas are

$\omega_{X}(S_{0})=1,\qquad\omega_{X}(T_{0})=2,\qquad\omega_{X}(S_{\infty})=3.$

By monotonicity, these are also the values of the corresponding Chern numbers.
We also record some intersection numbers for future reference:

$\displaystyle S_{0}\cdot S_{0}=-1,\qquad T_{0}\cdot T_{0}=0,\qquad
S_{\infty}\cdot S_{\infty}=1,$ $\displaystyle S_{0}\cdot S_{\infty}=0,\qquad
T_{0}\cdot S_{0}=1,\qquad T_{0}\cdot S_{\infty}=1.$

In later considerations, we will work with tame almost complex structures
making these spheres pseudoholomorphic, and positivity of intersections will
then allow us to deduce many important conclusions.

###  4.2. Existence of Pseudoholomorphic Spheres

For many arguments in this paper, we will need to understand which
(unparameterized) closed pseudoholomorphic spheres exist in $(X,\omega_{X})$.
The following proposition contains all the facts we will need.

######  Proposition 4.1.

Let $J$ be a tame almost complex structure on $(X,\omega_{X})$.

  1. (a)

There is a unique embedded $J$-holomorphic sphere in the class $(1,0)$.

  2. (b)

For any point $p\in X$, there is a unique embedded $J$-holomorphic sphere in
the class $(0,1)$ passing through $p$.

  3. (c)

For $d\geq 1$ and a codimension $2$ set of tuples $(p_{1},\dots,p_{2d})\in
X^{d}$, there is a unique embedded $J$-holomorphic sphere in the class $(1,d)$
passing through the points $p_{1},\dots,p_{2d}$.

###### Proof.

Let $A$ denote any of the above homology classes. We first observe that
because $A$ is primitive, every _smooth_ $J$-holomorphic sphere in the class
$A$ must be simple and hence somewhere injective. It then follows from the
adjunction inequality that any such sphere must be embedded. Furthermore, it
follows from automatic regularity, with point constraints when necessary, that
$J$ is regular for all spheres in the class $A$. The dimension of the
corresponding constrained or unconstrained moduli space is always $0$.
Furthermore, the self-intersection number $A\cdot A$ is strictly less than the
number of point constraints (if any), implying that there is at most one
sphere in the class $A$ satisfying these constraints.

Next, we prove existence of these curves in the class $A$ for the integrable
complex structure $J_{0}$ on $X$. In this case the complex manifold
$(X,J_{0})$ is known to be biholomorphic to the projectivization
$\mathbb{P}(\mathcal{O}(-1)\oplus\mathbb{C})$, where $\mathcal{O}(-1)$ is the
line bundle of degree $-1$ over $\mathbb{CP}^{1}$. The unique
$J_{0}$-holomorphic sphere in the class $(1,0)$ is then precisely the zero
section, and the sphere in the class $(0,1)$ through a fixed point is
precisely the compactified fiber of the above bundle passing through that
point. The sphere in the class $(1,d)$ is a meromorphic section passing
through $d-1$ zeros on $S_{0}$ and $d$ poles on $S_{\infty}$, together with an
additional normalizing point constraint chosen away from these two subsets.
These points have to be chosen to lie on distinct fibers, meaning we have to
avoid a codimension $2$ subset of $X^{2d}$.

Now, we show that spheres in the classes $(1,0)$ and $(0,1)$ exist for any
tame $J$ by showing that no bubbling can occur, so that all relevant
parametric moduli spaces, with path constraints in the case of (b), are
compact. When $A=(1,0)$, this follows since any decomposition $A=A_{1}+A_{2}$
with each $A_{i}$ representable by a $J$-holomorphic curve implies that
$\omega_{X}(A_{i})\geq 0$, and since $\omega_{X}(A)=1$, this implies that one
of the $A_{i}$ is the trivial class by integrality. When $A=(0,1)$, a similar
decomposition $A=A_{1}+A_{2}$ is trivial unless $\omega_{X}(A_{i})=1$ for
$i=1,2$. Writing $A_{i}=(a_{i},b_{i})$, this implies that $a_{i}+2b_{i}=1$,
showing that this homology class is primitive. We also have $a_{1}+a_{2}=0$
and $b_{1}+b_{2}=1$. If $a_{1}=0$ or $a_{2}=0$, then the above decomposition
would be trivial, so without loss of generality, we can assume that $a_{1}\geq
1$, and this forces $b_{1}\leq 0$. Then the adjunction inequality for spheres
in the class $A_{2}=(-a_{1},1-b_{1})$ eventually simplifies to

$0\leq(a_{1}+1)(2b_{1}-a_{1}),$

a contradiction since the right hand side is negative by assumption.

Finally, we show that no bubbling can occur when $A=(1,d)$ with $d\geq 1$,
provided the point constraints are chosen away from a subset of codimension
$2$ in $X^{2d}$. Using the fact that there exists $J$-holomorphic spheres in
the classes $(1,0)$ and $(0,1)$ together with positivity of intersections, one
sees that the only possible decomposition of the class $(1,d)$ into classes
representable by $J$-holomorphic spheres is

$(1,d)=(1,d_{1})+(0,d_{2})+\cdots+(0,d_{k}),$

where $d_{1}+\cdots+d_{k}=d$ and $d_{i}\geq 0$ for each $i$. This then forces
$k\leq d+1$. Letting $S$ denote the unique $J$-holomorphic sphere in the class
$(1,0)$, we pick our point constraints so that at most $d-1$ of them lie on
$S$, and such that no two of them lie on the same $J$-holomorphic sphere in
the class $(0,1)$. We thus have to avoid a codimension $2$ subset of $X^{2d}$,
and it is easy to check that any decomposition of the class $(1,d)$ as above
would contradict this choice of constraints. Since we can also find disjoint
paths of permissible point constraints, this shows that the corresponding
parametric moduli space is always compact, and hence that the desired spheres
exist for any $J$. ∎

######  Remark 4.2.

Implicit in the above proof is the fact that any two symplectic spheres in the
class $A$ can be joined by a symplectic isotopy. In certain circumstances, we
will also be able to find symplectic isotopies of certain _nodal
configurations_ of symplectic spheres, such as the axes $Y=S_{0}\cup
S_{\infty}\cup T_{0}\cup T_{\infty}$. By Corollary 3.7 in [8], these
symplectic isotopies can then be realized as Hamiltonian ones. By definition,
we require that any two spheres in a nodal configuration intersect
transversally, positively, and no more than once.

###  4.3. A Displacement Result

We now prove a displacement result, which, in addition to being needed in
later sections, will be used to prove two important facts. The first of these
will allow us to reduce the proof of Theorem 1.5 to the case $n=1$, and the
second is a striking fact about integral tori in
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$. Here is the
displacement result.

######  Proposition 4.3.

If $L$ is a Lagrangian torus in $(X,\omega_{X})$, then there is a Hamiltonian
diffeomorphism $\varphi\colon X\to X$ such that $\varphi(L)$ is disjoint from
$S_{0}$. If $L$ is also disjoint from $L_{1,1}$, the corresponding Hamiltonian
isotopy can be taken to be stationary on $L_{1,1}$.

###### Proof.

We first find an unbroken smooth symplectic sphere in the class $(1,0)$ that
is disjoint from $L$. To this end, fix any parameterization
$\psi\colon\mathbb{T}^{2}\to L$, let $J$ be an almost complex structure on
$X\smallsetminus L$ that is adapted to $\psi$ and regular for somewhere
injective curves, and let $\\{J_{t}\\}_{t\geq 0}$ be a family of tame almost
complex structures on $(X,\omega_{X})$ that stretches to $J$. We can extract a
sequence $(J_{t_{i}})_{i=1}^{\infty}$ and consider the unique sequence of
embedded $J_{t_{i}}$-holomorphic spheres in the class $(1,0)$, which exist by
(a) of Proposition 4.1. Applying Theorem 3.2, we obtain a limit building
$\mathbf{F}$ in the class $(1,0)$, which is a smooth (and possibly broken)
sphere by (b) of Proposition 3.1. Letting $u_{1},\dots,u_{k}$ be the top level
curves of $\mathbf{F}$ and $v_{1},\dots,v_{l}$ be all remaining curves, we
have

$0=-2+2c_{1}((1,0))=\operatorname{ind}(\mathbf{F})=\sum_{i=1}^{k}\operatorname{ind}(u_{i})+\sum_{i=1}^{l}\operatorname{ind}(v_{i})-m,$

where $m$ is the total number of Reeb orbits at which the above curves match
asymptotically. Now we know from Proposition 3.4 that

$\operatorname{ind}(v_{i})=-2+2s_{+}^{i}+s_{-}^{i}=-\chi(S_{i})+s_{+}^{i},$

where $S_{i}$ is the punctured domain of $v_{i}$. We also have
$s_{+}^{1}+\cdots+s_{+}^{k}=m$, so

$0=\sum_{i=1}^{k}\operatorname{ind}(u_{i})+\sum_{i=1}^{l}\operatorname{ind}(v_{i})-m=\sum_{i=1}^{k}\operatorname{ind}(u_{i})-\sum_{i=1}^{l}\chi(S_{i}).$

Since each domain $S_{i}$ has at least two punctures, we know that
$\chi(S_{i})\leq 0$ for each $i$. But now observe that
$\operatorname{ind}(u_{i})\geq 0$ for each $i$, which follows immediately from
the fact that $J$ is regular when $u_{i}$ is somewhere injective, and then by
Proposition 3.5 if $u_{i}$ is multiply covered. The above equality then forces
$\operatorname{ind}(u_{i})=0$ for each $i$. But by (a) of Proposition 3.1, at
least one of the $u_{i}$ must be a plane, and then
$\operatorname{ind}(u_{i})=\mu(\overline{u})-1$ by Proposition 3.5 again.
Since $L$ is orientable, the Maslov index $\mu(\overline{u})$ is even, which
forces $\operatorname{ind}(u_{i})>0$, a contradiction. It follows that
$\mathbf{F}$ is an unbroken smooth sphere in the class $(1,0)$ that is
disjoint from $L$.

Now extend $J$ in $\mathcal{U}(L)$ away from $\mathbf{F}$ to a smooth tame
almost complex structure defined on all of $X$, and consider a path of tame
almost complex structures $\\{J_{t}\\}_{t\in[0,1]}$ with $J_{0}$ integrable
near $S_{0}$ and $J_{1}=J$. It follows from (a) of Proposition 4.1 and
automatic regularity that the parametric moduli space for spheres in the class
$(1,0)$ is a compact, $1$-dimensional cobordism that gives rise to a
symplectic isotopy taking $\mathbf{F}$ to $S_{0}$. This can then be upgraded
to a Hamiltonian isotopy as mentioned in Remark 4.2, and the corresponding
time-one map is the desired Hamiltonian diffeomorphism $\varphi$. This proves
the first statement in the theorem.

Finally, assume that $L$ is disjoint from $L_{1,1}$ and fix a parametrization
$\psi_{1,1}$ of $L_{1,1}$. (An explicit parameterization is given in
Subsection 5.1 below). Let $J$ be tame, regular for somewhere injective
curves, and adapted to both $\psi$ and $\psi_{1,1}$. The above arguments then
go through with no modification to show that there is a unique unbroken smooth
$J$-holomorphic sphere $\mathbf{F}$ in the class $(1,0)$ that is disjoint from
both $L$ and $L_{1,1}$. Furthermore, there is such a sphere disjoint from
$L_{1,1}$ for _all_ tame almost complex structures adapted to $\psi_{1,1}$,
and not just regular ones. This is because $L_{1,1}$ is integral, which by (c)
of Proposition 3.1 implies that _any_ limit building in the class $(1,0)$ is
unbroken. As before, we take a path $\\{J_{t}\\}_{t\in[0,1]}$ of tame almost
complex structures on $X\smallsetminus L_{1,1}$, with $J_{0}$ integrable near
$S_{0}$ and $J_{1}=J$. The corresponding parametric moduli space is a compact,
$1$-dimensional cobordism giving rise to a symplectic isotopy in
$X\smallsetminus L_{1,1}$ taking $\mathbf{F}$ to $S_{0}$. The corresponding
Hamiltonian isotopy can then be extended to all of $X$ and taken to be
stationary on $L_{1,1}$, completing the proof of the second statement and
consequently of the proposition. ∎

We now present two important applications of Proposition 4.3. The first
reduces the proof of Theorem 1.5 to the case $n=1$.

######  Proposition 4.4.

If $L_{1,1}$ is a maximal integral packing of
$(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$, then it is also a
maximal integral packing of
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ for $n=2,\dots,5$.

###### Proof.

After $n\leq 5$ blowups, there are $n$ pairwise disjoint exceptional divisors
$E_{1},\dots,E_{n}$ in
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$, each of area $1$ and
with self-intersection number $-1$. These are symplectically embedded spheres,
and consequently, one can find a tame almost complex structure integrable near
all of them. The same arguments used in the proof of Proposition 4.1 then show
that for any tame $J$, there is a unique $J$-holomorphic sphere in the class
of each $E_{i}$. If $L\subset\mathbb{D}_{n}$ were an integral torus that is
disjoint from $L_{1,1}$, a straightforward adaptation of the proof of
Proposition 4.3 produces a Hamiltonian isotopy in
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ that is stationary
near $L_{1,1}$ and displaces $L$ from the configuration $E_{1}\cup\cdots\cup
E_{n}$. Blowing down each of these except $E_{1}$ then yields an integral
torus in $(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$ that is
disjoint from $L_{1,1}$, a contradiction since this latter torus is a maximal
integral packing of $(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$.
∎

The second application of Proposition 4.3 is the following nice result about
integral Lagrangian tori in
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$, which aside from
being interesting in its own right, will greatly simplify some later
considerations. Also, since the result is no harder to prove for
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$, we include that case
as well.

######  Theorem 4.5.

All integral Lagrangian tori in
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$ and
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ for $n=1,\dots,8$ are
monotone.

###### Proof.

First off, the proof of Proposition 4.4 shows that any integral torus in
$(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$ can be displaced from
the configuration $E_{1}\cup\cdots\cup E_{n}$ by a Hamiltonian isotopy.
Blowing down these exceptional divisors then yields an integral torus in
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$, and it is therefore
enough to prove the theorem in this case. Hence let $L\subset\mathbb{CP}^{2}$
be an integral torus, and use Theorem C of [8] to displace $L$ from
$S_{\infty}$ by a Hamiltonian isotopy, thereby placing it inside the open ball
$B(3)$. We can thus assume that $L$ is an integral torus in
$(\mathbb{R}^{4},\omega_{0})$, and in this setting a homological argument
allows us to recast the Maslov and area homomorphisms as maps from $H_{1}(L)$.
By Lemma 2.1, it is then enough to find an integral basis $\\{e_{1},e_{2}\\}$
of $H_{1}(L)$ such that $\mu(e_{i})=2\omega_{0}(e_{i})$ for $i=1,2$.

We now appeal to the results in Section 1.1 of [6] to obtain a smooth map
$v_{1}\colon(\mathbb{D}^{2},^{1})\to(\mathbb{CP}^{2},L)$ such that
$\omega_{\mspace{1.0mu}\mathrm{FS}}([v_{1}])=1$ and $\mu([v_{1}])=2$. More
precisely, the proofs of those results produce three maps, each satisfying
these requirements, and which are pseudoholomorphic away from $L$ and obtained
from the splitting procedure as a limit of spheres in the class $S_{\infty}$.
By positivity of intersections, only one of these can then intersect
$S_{\infty}$, and so we may assume without loss of generality that
$v_{1}(\mathbb{D}^{2})\subset B(3)$. Letting $e_{1}$ be the class of
$v_{1}|_{{}^{1}}$, we get $\mu(e_{1})=2$ and $\omega_{0}(e_{1})=1$. Now choose
any integral class $e_{2}$ such that $(e_{1},e_{2})$ is a basis for
$H_{1}(L)$. Since $L$ is orientable, we know that $\mu(e_{2})$ is even, and by
adding to it integer multiples of $e_{1}$ if necessary, we may assume that
$\mu(e_{2})=2$. We will then be done if we can show that
$\omega_{0}(e_{2})=1$. If this were not the case, then either
$\omega_{0}(e_{2})>1$ or $\omega_{0}(e_{2})<1$, in which case we redefine
$e_{2}$ to be the class $e_{2}+2(e_{1}-e_{2})$. We are then left with an
integral basis $(e_{1},e_{2})$ of $H_{1}(L)$ such that

$\omega_{0}(e_{1})=1,\qquad\omega_{0}(e_{2})\geq
2,\qquad\mu(e_{1})=2,\qquad\mu(e_{2})=2.$

This puts us in the scenario of Theorem 2 in [12], which says that in the
presence of such a basis, the inclusion $L\subset B(R)$ implies that $R>3$,
contradicting the present situation. ∎

###  4.4. Relative Foliations

Let $L=L_{1}\cup\cdots\cup L_{k}$ be a disjoint union of integral Lagrangian
tori in $(X,\omega_{X})$, itself disjoint from $S_{0}$. By Proposition 4.3,
this is not a very restrictive situation. (The proof adapts in a
straightforward way to deal with several tori.) We then fix a parametrization
$\psi_{i}\colon\mathbb{T}^{2}\to L_{i}$ for each $i$, and always assume that
the induced Weinstein neighborhoods are pairwise disjoint and miss $S_{0}$.
Finally, we choose an almost complex structure $J$ on $X\smallsetminus L$ that
is adapted to all the $\psi_{i}$ and that is integrable near $S_{0}$. We will
use this collection of data to construct a foliation of $X\smallsetminus L$ by
$J$-holomorphic spheres and planes that is in some sense _compatible with
$L$_. Its existence and properties underlie all subsequent arguments in this
paper. The construction is carefully discussed in [8] and [11], and we use
these as our primary references for this subsection. We will present all the
machinery that we will need in the form of a single all-encompassing theorem.

######  Theorem 4.6.

There is a foliation
$\mathcal{F}=\mathcal{F}(L_{1},\dots,L_{k},\psi_{1},\dots,\psi_{k},J)$ of
$X\smallsetminus L$ and a unique foliation class $\beta_{i}\in H_{1}(L_{i})$
for each $i$, which satisfy the following properties:

  1. (1)

The foliation $\mathcal{F}$ is comprised of both unbroken leaves, which are
$J$-holomorphic spheres in the class $(0,1)$, as well as broken leaves, which
are pairs of $J$-holomorphic planes asymptotic to Reeb orbits in the Morse-
Bott families $\Gamma_{\beta_{i}}$ and $\Gamma_{-\beta_{i}}$ for some $i$.

  2. (2)

The leaves of $\mathcal{F}$ are all simply covered and embedded, and the
planes comprising a broken leaf are asymptotic to simply covered orbits. In
particular, the foliation classes $\beta_{i}$ are all primitive.

  3. (3)

Every leaf of $\mathcal{F}$ has a single transverse intersection with $S_{0}$,
which for a broken leaf occurs with the plane asymptotic to an orbit in
$\Gamma_{\beta_{i}}$.

Moreover, the foliation classes $\beta_{i}$ are independent of the choice of
$J$.

###### Proof.

Take a family $\\{J_{t}\\}_{t\geq 0}$ of tame almost complex structures on
$(X,\omega_{X})$ that stretches to $J$, and consider buildings in the class
$(0,1)$ obtained as limits of $J_{t_{i}}$-holomorphic spheres in the class
$(0,1)$ passing through any fixed point in $X$, where $t_{i}\to+\infty$ as
$i\to+\infty$. These sequences exist by (b) of Proposition 4.1, and the
corresponding limit building $\mathbf{F}$ is smooth by (b) of Proposition 3.1,
and has area $2$. It is therefore either an unbroken smooth sphere in
$X\smallsetminus L$ in the class $(0,1)$, or it is broken, in which case (a)
of Proposition 3.1 implies that it must have at least two planes in its top
level. But since the $L_{i}$ are all integral, these planes will have integral
area by (c) of Proposition 3.1, and thus $\mathbf{F}$ has exactly two planes
in its top level, each of area $1$. There can be no additional top level
curves, and curves in the middle and bottom levels are all cylinders. As
explained in Section 5 of [11], a diagonal subsequence argument together with
positivity of intersections shows that the collection of top level curves of
all such buildings give a partition of $X\smallsetminus L$. That it is
actually a smooth foliation follows from automatic regularity for punctured
curves, see [22], together with an extended intersection number argument, as
in the proof of Proposition 5.1 in [11]. To avoid a digression into Siefring
intersection theory, we refer the reader to that source for the full argument.

Since $J$ is integrable near $S_{0}$, it follows from positivity of
intersections that exactly one of the top level planes of a broken sphere as
above intersects $S_{0}$. We denote this plane by $u$ and the remaining one by
$v$. Suppose these are asymptotic to Reeb orbits on $S^{\ast}L_{i}$ for some
fixed $i$. Then $u$ is asymptotic to an orbit contained in
$\Gamma_{\beta_{i}}$ for some $\beta_{i}\in H_{1}(L_{i})$, and for topological
reasons, the plane $v$ must be asymptotic to an orbit in
$\Gamma_{-\beta_{i}}$. Furthermore, both $u$ and $v$ are simply covered since
they have area $1$, and it follows that they must be embedded, being somewhere
injective components of a limit of closed embedded spheres. To see that their
asymptotic orbits are also simply covered, fix an unbroken sphere from our
partition in the class $(0,1)$ intersecting $S_{0}$ at a single point, and
follow the proof of Proposition 4.12 in [15] to construct a symplectically
embedded sphere $S$ in the class $(1,1)$ that is disjoint from $S_{0}$.
Observe that $S$ is also disjoint from $v$, and blowing down $S_{0}$ yields a
plane $v$ in
$(\mathbb{CP}^{2},\mspace{2.0mu}\omega_{\mspace{1.0mu}\mathrm{FS}})$. By
Theorem 1.4 of [16], there is a symplectomorphism of
$(\mathbb{CP}^{2},\mspace{2.0mu}\omega_{\mspace{1.0mu}\mathrm{FS}})$ with
itself taking $S$ to $S_{\infty}$. Keeping the same notation, the plane $v$ is
now contained in $\mathbb{CP}^{2}\smallsetminus S_{\infty}$, and hence a
subset of $(\mathbb{R}^{4},\omega_{0})$. Viewing $L_{i}$ as a torus in
$\mathbb{R}^{4}$, it now follows that the compactification of $v$ represents a
primitive homology class in $H_{2}(\mathbb{R}^{4},L_{i})$. But then it also
follows that its boundary $-\beta_{i}$ represents a primitive class in
$H_{1}(L_{i})$, since the connecting homomorphism
$H_{2}(\mathbb{R}^{4},L_{i})\to H_{1}(L_{i})$ is an isomorphism. Therefore
$-\beta_{i}$ and $\beta_{i}$ are primitive classes in $H_{1}(L_{i})$, implying
that the asymptotic orbits of $u$ and $v$ are simply covered.

Now consider another limit building and suppose its asymptotic orbits belonged
to $\Gamma_{\gamma_{i}}\cup\Gamma_{-\gamma_{i}}$ for $\gamma_{i}\neq\beta_{i}$
in $H_{1}(L_{i})$. As shown in Lemma 5.8 of [8], positivity of intersections
implies that no component of this building can intersect a component of our
earlier building in a discrete and nonempty set. But $\beta_{i}$ and
$\gamma_{i}$ are both primitive classes, hence not collinear, and the
description of cylinders in lower levels from Lemma 4.2 and Corollary 4.3 of
[8] would then _force_ some of these curves to have a nonempty and discrete
intersection, a contradiction. Hence $\gamma_{i}=\pm\beta_{i}$, and in fact
$\gamma_{i}=\beta_{i}$, since otherwise, after possibly reparameterizing the
plane asymptotic to an orbit in $\Gamma_{\gamma_{i}}$ and using an explicit
cylinder in $T^{\ast}L_{i}$ as described in Lemma 4.2 of [8], we can construct
a sphere in some class $(a,b)$ of area $2$ and with two intersections with
$S_{0}$. These conditions would force $a=-2/3$, which is impossible. This
shows that the foliation classes $\beta_{i}$ are well defined.

Finally, we show that the foliation classes $\beta_{i}$ are independent of
$J$. Choose a path $\\{J_{t}\\}_{t\in[0,1]}$ of tame almost complex structures
that are all adapted to the $\psi_{i}$ and integrable near $S_{0}$, with
$J_{0}=J$. The planes corresponding to broken leaves of the foliation obtained
from $J$ divide into moduli spaces $\mathcal{M}_{\beta_{i}}(J)$ and
$\mathcal{M}_{-\beta_{i}}(J)$ of planes with asymptotics in
$\Gamma_{\beta_{i}}$ and $\Gamma_{-\beta_{i}}$, respectively. The
corresponding parametric moduli spaces with asymptotics in these same families
are all smooth by automatic regularity, and compact since area $1$ planes
cannot degenerate further by the integrality of the $L_{i}$. We therefore
obtain smooth moduli spaces $\mathcal{M}_{\beta_{i}}(J_{1})$ and
$\mathcal{M}_{-\beta_{i}}(J_{1})$ of embedded, simply covered
$J_{1}$-holomorphic planes with simply covered asymptotic orbits, and the
planes in $\mathcal{M}_{\beta_{i}}(J_{1})$ have a single transverse
intersection with $S_{0}$. If we then construct a foliation of
$X\smallsetminus L$ by $J_{1}$-holomorphic planes and spheres as in the first
paragraph, positivity of intersections forces the resulting top level planes
of broken spheres to coincide with the union of the two moduli spaces above.
Hence the foliation $\mathcal{F}$ exists and has the desired properties for
$J_{1}$, concluding the proof of the theorem. ∎

The foliation $\mathcal{F}$ constructed in the previous theorem induces a
smooth projection map $\pi\colon X\to S_{0}$, which is defined as follows:

  1. $\bullet$

If $p\in X\smallsetminus L$, then $p$ lies on a unique (broken or unbroken)
leaf of $\mathcal{F}$, and this leaf has a unique component that intersects
$S_{0}$. Then $\pi(p)$ is defined to be this point of intersection.

  2. $\bullet$

If $p\in L_{i}$ for some $i$, then $p$ lies on a unique geodesic in the class
$\beta_{i}$, since these foliate the torus. This geodesic corresponds to a
unique simply covered orbit on $S^{\ast}L_{i}$, and the same argument used to
prove Lemma 5.2 in [11] shows that there is a _unique_ broken leaf of
$\mathcal{F}$ with a plane component asymptotic to the above orbit. Then
$\pi(p)$ is defined to be the unique point of intersection of this leaf with
$S_{0}$.

As shown in Section 5 of [8], the images $\pi(L_{i})\subset S_{0}$ are
pairwise disjoint embedded circles. Furthermore, it follows from Proposition
5.16 in [8] that the fiber $\pi^{-1}(p)$ for $p\in\pi(L_{i})$ is a smooth
sphere obtained by gluing compactifications of planes in
$\mathcal{M}_{\beta_{i}}(J)$ and $\mathcal{M}_{-\beta_{i}}(J)$ with matching
orbits, after perturbing these planes in an arbitrarily small neighborhood of
$L_{i}$. The intersection $L_{i}\cap\pi^{-1}(p)$ is the (unoriented) geodesic
on $L_{i}$ covered by the above orbits.

###  4.5. Essential Curves

We carry over the setup and notation from the previous subsection. If $u$ is a
punctured $J$-holomorphic sphere in $X\smallsetminus L$, it will be convenient
to say $u$ has a _puncture on $L_{i}$_ if it has a puncture at which it is
asymptotic to a Reeb orbit on $S^{\ast}L_{i}$. We will say such a puncture is
of _foliation type_ if this orbit covers a closed geodesic on $L_{i}$ that is
an integer multiple of the foliation class $\beta_{i}$. We then say the
puncture is _positive_ or _negative_ according to whether this integer is
positive or negative, respectively. (This should hopefully not cause confusion
with the standard usage of these terms, since our definition applies only to
curves in $X\smallsetminus L$.) The following lemma gives a geometric
description of the behavior of punctures of foliation type. It follows
directly from the standard exponential convergence theorem for punctured
curves, which is Theorem 1.4 in [13].

######  Lemma 4.7.

Let $u$ be a punctured $J$-holomorphic sphere in $X\smallsetminus L$ and
$\\{C_{j}\\}_{j=1}^{\infty}$ be a sequence of circles contained in a
coordinate neighborhood of a puncture of $u$ on $L_{i}$. Assume each $C_{j}$
winds once around the puncture and converges to it in the Hausdorff topology.

  1. (a)

If the puncture is of foliation type, then the sets $\pi(u(C_{j}))$ converge
to a single point on $\pi(L_{i})$. Additionally, each such set either
coincides with the point, which implies that $u$ covers a plane in a broken
leaf of $\mathcal{F}$, or it winds nontrivially around the point.

  2. (b)

If the puncture is not of foliation type, then the sets $\pi(u(C_{j}))$
converge to the entire circle $\pi(L_{i})$.

There is a special class of curves in $X\smallsetminus L$ for which the
classification of punctures according to the above scheme is particularly
transparent. A (closed or punctured) $J$-holomorphic sphere in
$X\smallsetminus L$ is said to be _essential with respect to $\mathcal{F}$_,
or simply _essential_ , if its composition with the projection $\pi\colon
X\smallsetminus L\to S_{0}$ is injective. These curves are in a sense well-
detected by the foliation. They are partly distinguished by the following
proposition.

######  Proposition 4.8.

Let $u$ be an essential $J$-holomorphic sphere in $X\smallsetminus L$. If $u$
has punctures on $L_{i}$, then they are either all of foliation type, in which
case they are either all positive or all negative, or none of them are, in
which case $u$ has a unique puncture on $L_{i}$.

###### Proof.

By Lemma 4.7, if $u$ simultaneously had punctures of and not of foliation type
on $L_{i}$, then the projection of a small circle around a puncture of
foliation type will intersect any circle sufficiently close to $\pi(L_{i})$,
including the projection of a small circle around a puncture not of foliation
type. This contradicts the essentiality of $u$ since $\pi\circ u$ is certainly
not injective. On the other hand, if $u$ had two punctures not of foliation
type on $L_{i}$, then the image of $\pi\circ u$ would have to intersect the
regions in $S_{0}$ both inside and outside of $\pi(L_{i})$, since $u$ is
essential. By connectedness, this would imply that some interior point of $u$
must intersect a fiber of $\pi$ over a point in $\pi(L_{i})$. Considering the
image of a small circle around this point as in Lemma 4.7, the argument in the
first part of this paragraph again yields a contradiction, and hence $u$
cannot have more than one puncture on $L_{i}$ that is not of foliation type.
That punctures of foliation type are either all positive or all negative is
proved in the same way as Lemma 6.2 in [11], which we refer the reader to for
additional details. ∎

The following scenario provides a rich source of essential curves. Consider a
family $\\{J_{t}\\}_{t\geq 0}$ of tame almost complex structures on
$(X,\omega_{X})$ that stretches to $J$, and extract a sequence
$(J_{t_{i}})_{i=1}^{\infty}$ as usual. We then fix $d\geq 1$ and use (c) of
Proposition 4.1 to obtain a sequence $(u_{i})_{i=1}^{\infty}$ of
$J_{t_{i}}$-holomorphic spheres in the class $(1,d)$ passing through $2d$
generic point constraints. By Theorem 3.2, this produces a limit building
$\mathbf{F}$ in the class $(1,d)$. We then have the following lemma, which
among other things shows that $\mathbf{F}$ always has an essential top level
curve.

######  Lemma 4.9.

If $u$ is a top level curve of $\mathbf{F}$, then $u$ is either essential or
else the image of $\pi\circ u$ is a single point. Moreover, essential top
level curves of $\mathbf{F}$ have disjoint open images under $\pi$ whose union
contains $S_{0}\smallsetminus\pi(L)$.

###### Proof.

If the image of $\pi\circ u$ is a point, then $u$ covers part of a broken leaf
or an unbroken leaf of $\mathcal{F}$. On the other hand, if $\pi\circ u$ is
nonconstant, then since $(1,d)\cdot(0,1)=1$, we see that $u$ intersects any
unbroken leaf of $\mathcal{F}$ once or not at all. This implies that any
double points of $\pi\circ u$ lie on $\pi(L)$. But by positivity of
intersections, the nonconstant map $\pi\circ u$ is open and hence its double
points form an open set, which is impossible in view of the above. It follows
that $u$ must be essential, and all such curves have disjoint images under
$\pi$ by applying this argument to the union of two curves. Intersection
considerations also imply that all unbroken leaves of $\mathcal{F}$ must
intersect at least one essential curve, implying the last statement. ∎

######  Remark 4.10.

At this point, we must mention that the considerations starting from
Subsection 4.2 until now were stated in the context of the toric model
$(X,\omega_{X})$ for concreteness, but they evidently apply _mutatis mutandis_
to any model of $(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$ which
contains $L_{1,1}$, and we will now take this to be understood.

###  4.6. Homologicaly Essential Tori

This subsection will shed some light on why the considerations thus far should
have any bearing on the packing problem we are considering. It begins with a
lemma whose motivation might appear unclear but whose importance will become
evident shortly. In line with Remark 4.10, we state it in the context of an
arbitrary model of $(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$
containing $L_{1,1}$. We let $\mathcal{S}_{0}$ denote the corresponding
exceptional divisor, and consider a _monotone_ torus $L$ in $\mathbb{D}_{1}$.
Assume there is a symplectically embedded sphere
$\mathcal{S}_{\infty}\subset\mathbb{D}_{1}$ in the class of the line at
infinity that is disjoint from $L$. (It turns out such a sphere can always be
found, but we will not need this fact nor elaborate on it.)

Fix a parameterization $\psi\colon\mathbb{T}^{2}\to L$, and choose a tame
almost complex structure $J$ on $\mathbb{D}_{1}\smallsetminus L$ that is
adapted to $\psi$ and integrable near
$\mathcal{S}_{0}\cup\mathcal{S}_{\infty}$. Using Theorem 4.6, we obtain a
relative foliation $\mathcal{F}=\mathcal{F}(L,\psi,J)$ of
$\mathbb{D}_{1}\smallsetminus L$ by $J$-holomorphic spheres and planes. The
broken leaves of $\mathcal{F}$ consist of pairs of embedded, simply covered,
$J$-holomorphic planes, and there is a unique primitive foliation class
$\beta\in H_{1}(L)$ such that the totality of these planes divide into moduli
spaces $\mathcal{M}_{\beta}(J)$ and $\mathcal{M}_{-\beta}(J)$. The space
$\mathcal{M}_{\beta}(J)$ consists of exactly those planes that have a single
transverse intersection with $\mathcal{S}_{0}$. Here is the aforementioned
lemma.

######  Lemma 4.11.

Assume that the planes in $\mathcal{M}_{-\beta}(J)$ are exactly those that
intersect $\mathcal{S}_{\infty}$ in a single point, and let $\mathcal{T}_{0}$
and $\mathcal{T}_{\infty}$ be two unbroken leaves of $\mathcal{F}$ whose
images under $\pi$ lie on opposite sides of $\pi(L)$. Then $L$ is homologicaly
nontrivial in
$\mathbb{D}_{1}\smallsetminus(\mathcal{S}_{0}\cup\mathcal{S}_{\infty}\cup\mathcal{T}_{0}\cup\mathcal{T}_{\infty})$.

###### Proof.

Set
$\mathcal{Y}=\mathcal{S}_{0}\cup\mathcal{S}_{\infty}\cup\mathcal{T}_{0}\cup\mathcal{T}_{\infty}$.
We construct a relative homology class in $H_{2}(\mathbb{D}_{1},\mathcal{Y})$
whose intersection number with $L$ is nonzero, thus showing that $L$ is
homologicaly nontrivial in $\mathbb{D}_{1}\smallsetminus\mathcal{Y}$. To this
end, choose an embedded path $\gamma\colon[0,1]\to\mathcal{S}_{0}$ that starts
at $\pi(\mathcal{T}_{0})=\mathcal{T}_{0}\cap\mathcal{S}_{0}$ and that ends at
$\pi(\mathcal{T}_{\infty})=\mathcal{T}_{\infty}\cap\mathcal{S}_{0}$. Since
$\pi(L)$ separates these points, we can assume that $\gamma$ intersects it
transversally at some unique time $t_{0}$. For each $t\in[0,1]$, we then
choose an embedded path $\sigma_{t}$ in $\pi^{-1}(\gamma(t))$ that starts at
$\mathcal{S}_{0}$ and that ends at $\mathcal{S}_{\infty}$. The union of these
defines a homology class $\Sigma\in H_{2}(\mathbb{D}_{1},\mathcal{Y})$. Now by
construction, the set $K=L\cap\pi^{-1}(\gamma(t_{0}))$ is a circle bounding
(perturbed) compactified planes from $\mathcal{M}_{\beta}(J)$ and
$\mathcal{M}_{-\beta}(J)$, which glue together to form a smooth sphere
intersecting $\mathcal{S}_{0}$ and $\mathcal{S}_{\infty}$. Since $K$ separates
these intersection points, it must intersect the path $\sigma_{t_{0}}$, and
this is the only intersection between $L$ and $\Sigma$. Therefore $L$ must be
homologicaly nontrivial in $\mathbb{D}_{1}\smallsetminus\mathcal{Y}$. ∎

In particular, we can apply the above lemma in the toric model
$(X,\omega_{X})$ when the set $\mathcal{Y}$ is the axes $Y=S_{0}\cup
S_{\infty}\cup T_{0}\cup T_{\infty}$. The importance of producing homologicaly
essential tori in $X\smallsetminus Y$ stems from the following crucial fact.

######  Proposition 4.12.

Any two monotone tori in $X\smallsetminus Y$ that are homologicaly nontrivial
must intersect.

###### Proof.

Let $L\subset X\smallsetminus Y$ be a monotone torus that is homologicaly
essential in $X\smallsetminus Y$. We can symplectically identify this latter
set with a symplectic manifold of the form $(\mathbb{T}^{2}\times
U,d\lambda)\subset(T^{\ast}\mathbb{T}^{2},d\lambda)$, where
$U\subset\mathbb{R}^{2}$ is a convex subset containing the origin, and by
Proposition 1.2, it is enough to show that $L$ is exact when considered in
$\mathbb{T}^{2}\times U$. By (1) of Theorem B in [7], there is a Hamiltonian
isotopy in $(T^{\ast}\mathbb{T}^{2},d\lambda)$ from $L$ to the graph of a
closed $1$-form on $\mathbb{T}^{2}$, and, crucially, this isotopy can be taken
to be supported in $\mathbb{T}^{2}\times U$. We can therefore assume without
loss of generality that $L$ is a section of
$T^{\ast}\mathbb{T}^{2}\to\mathbb{T}^{2}$. Then taking any closed curve
$\gamma\subset L$ together with a smooth bounding disk $D$ in $X$, we can
construct a cylinder $C$ in $\mathbb{T}^{2}\times U$ by projecting $\gamma$
onto the zero section and taking the trace of this projection. It is
straightforward to see that $C$ has Maslov index $0$ in
$T^{\ast}\mathbb{T}^{2}$. By additivity of Maslov indices, we then see that
$\mu(D\cup C)=\mu(D)$, and therefore the monotonicity of $L$ implies that
$d\lambda(C)=0$. Since the zero section is obviously exact in
$T^{\ast}\mathbb{T}^{2}$, it now follows from Stoke’s theorem that
$\smash{\int_{\gamma}\lambda=0}$, showing that $L$ is exact in
$\smash{T^{\ast}\mathbb{T}^{2}}$, as desired. ∎

##  5\. Towards a Contradiction

In this section, we begin working towards the proof of Theorem 1.5 in earnest.
The strategy is to assume it fails and derive a contradiction. Thus our
standing assumption for the rest of the paper will be that there is an
integral torus $L$ in $(\mathbb{D}_{n},\omega_{\mspace{1.0mu}\mathbb{D}_{n}})$
for $n\leq 5$ that is disjoint from $L_{1,1}$. By Proposition 4.4, we can take
$n=1$ and by Theorem 4.5, we know that $L$ is monotone. We will first show
that this standing assumption leads to a slightly weaker statement about tori
in the toric model $(X,\omega_{X})$ that is easier to contradict. We then use
the relative foliations developed in the previous section together with an
analysis of limit buildings to establish the existence of a distinguished
building in $(X,\omega_{X})$ with some special properties. Finally, we state a
black box result, which we will use to obtain the desired contradiction in
Section 6. The proof of this black box result will in turn be taken up in
Section 7.

###  5.1. A Simplification

This subsection is dedicated to proving the following proposition, in view of
the standing assumption mentioned in the introduction to this section. Recall
that $Y=S_{0}\cup S_{\infty}\cup T_{0}\cup T_{\infty}$ in $(X,\omega_{X})$.

######  Proposition 5.1.

There are disjoint monotone Lagrangian tori $L_{\mathfrak{r}}$ and
$L_{\mathfrak{s}}$ in $X\smallsetminus Y$ with the following properties:

  1. (a)

$L_{\mathfrak{r}}$ is homologicaly trivial in $X\smallsetminus Y$.

  2. (b)

$L_{\mathfrak{s}}$ is homologicaly nontrivial in $X\smallsetminus Y$.

  3. (c)

$T_{0}$ and $T_{\infty}$ are distinct classes in $H_{2}(X\smallsetminus
L_{\mathfrak{r}})$ and in $H_{2}(X\smallsetminus L_{\mathfrak{s}})$.

There are several steps involved in proving the above proposition. The first
is to displace $L$ from the exceptional divisor inside
$\mathbb{D}_{1}\smallsetminus L_{1,1}$ by using Proposition 4.3 to displace it
by a Hamiltonian isotopy that is stationary on $L_{1,1}$. In view of the
definition of symplectic blowup, the resulting situation is equivalent to the
existence of a monotone Lagrangian torus $L$ in
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$ that is disjoint from
$L_{1,1}$ together with a closed Darboux ball $B$ of capacity $1$ in
$\mathbb{CP}^{2}\smallsetminus(L\cup L_{1,1})$. To proceed further, we must
now take a closer look at the Clifford torus in $\mathbb{CP}^{2}$.

Define a parameterization $\psi_{1,1}\colon\mathbb{T}^{2}\to L_{1,1}$ by

$\psi_{1,1}(q_{1},q_{2})=[1:e^{-2\pi iq_{1}}:e^{-2\pi iq_{2}}].$

This extends to a symplectic embedding, denoted by the same symbol, of the
open neighborhood

$\\{(q_{1},q_{2},p_{1},p_{2})\in
T^{\ast}\mathbb{T}^{2}:|p_{1}|<1/2,|p_{2}|<1/2\\},$

which is given explicitly by

$\psi_{1,1}(q_{1},q_{2},p_{1},p_{2})=\Bigg{[}1:\sqrt{\frac{p_{1}+1}{1-p_{1}-p_{2}}}e^{-2\pi
iq_{1}}:\sqrt{\frac{p_{2}+1}{1-p_{1}-p_{2}}}e^{-2\pi iq_{2}}\Bigg{]}.$

Its image is a Weinstein neighborhood $\mathcal{U}(L_{1,1})$, which in
practice will usually be made quite smaller. Observe that in these coordinates
the moment map takes the form

$(q_{1},q_{2},p_{1},p_{2})\mapsto(p_{1}+1,p_{2}+1).$

Consider the smooth embedded disk in $\mathbb{CP}^{2}$ with boundary on
$L_{1,1}$ given by $z\mapsto[z:1:1]$. This has Maslov index $2$ and a unique
transverse and positive intersection with $S_{\infty}$. A calculation using
polar coordinates on $\mathbb{D}^{2}\smallsetminus\\{0\\}$ and coordinates
$\smash{(q_{1},q_{2},p_{1},p_{2})}$ on $\mathcal{U}(L_{1,1})\smallsetminus
L_{1,1}$ shows that $J_{\mathrm{cyl}}$ preserves the span of its differential.
We may then choose a tame almost complex structure $J_{1,1}$ on
$\mathbb{CP}^{2}\smallsetminus L_{1,1}$ that is adapted to $\psi_{1,1}$,
integrable near $S_{\infty}$, and makes the interior of the above disk, which
we denote by $u$, into a $J_{1,1}$-holomorphic plane with a puncture on
$L_{1,1}$. It is asymptotic at its unique puncture to the Reeb orbit
$\smash{(\theta,\theta,1/\sqrt{2},1/\sqrt{2})}\in S^{\ast}\mathbb{T}^{2}$.

The same considerations apply to the smooth embedded disks given by

$z\mapsto\left[1:\frac{1}{\sqrt{2-|z|^{2}}}:\frac{z}{\sqrt{2-|z|^{2}}}\right],\qquad
z\mapsto\left[1:\frac{z}{\sqrt{2-|z|^{2}}}:\frac{1}{\sqrt{2-|z|^{2}}}\right].$

The interiors of these disks are then $J_{1,1}$-holomorphic planes with
punctures on $L_{1,1}$, which we denote by $v$ and $w$, respectively. The
plane $v$ has a unique transverse and positive intersection with $T_{0}$, and
is asymptotic at its unique puncture to the Reeb orbit $(-\theta,0,-1,0)$.
Similarly, the plane $w$ has a unique transverse and positive intersection
with $T_{\infty}$, and is asymptotic at its unique puncture to the Reeb orbit
$(0,-\theta,0,-1)$. By monotonicity, all three of the above planes have index
and area $1$. These facts together with automatic regularity imply that the
corresponding moduli spaces are all compact and smooth $1$-manifolds. This
plays a key role in the following lemma.

######  Lemma 5.2.

There is a Hamiltonian diffeomorphism
$\varphi\colon\mathbb{CP}^{2}\to\mathbb{CP}^{2}$ that fixes $L_{1,1}$ and such
that $\varphi(L)$ and $\varphi(B)$ are disjoint from the planes $u$, $v$, and
$w$.

###### Proof.

The idea of the proof is similar to that of Proposition 4.3 in light of Remark
3.3. Fix a parameterization $\psi$ of $L$ in addition to $\psi_{1,1}$ above.
Then let $J$ be a tame almost complex structure on
$\mathbb{CP}^{2}\smallsetminus(L\cup L_{1,1}\cup B)$ adapted to $\psi$ and
$\psi_{1,1}$ and of appropriate form for applying the splitting procedure
along $\partial{B}$. Let $\\{J_{t}\\}_{t\geq 0}$ be a family of tame almost
complex structures on $\mathbb{CP}^{2}\smallsetminus L_{1,1}$ that stretches
to $J$ such that $J_{t}$ is adapted to $\psi_{1,1}$ for each $t$. Now deform
$J_{1,1}$ to $J_{t}$ for each $t$ through almost complex structures that are
adapted to $\psi_{1,1}$. By compactness, the moduli spaces of $u$, $v$, and
$w$ will persist for each $t$. Applying the more general splitting procedure
for symplectic manifolds with cylindrical ends mentioned in Remark 3.3, we
obtain moduli spaces of $J$-holomorphic limit planes in
$\mathbb{CP}^{2}\smallsetminus(L\cup L_{1,1}\cup B)$. Then extending $J$
smoothly past $L$ and $B$ and away from the above moduli spaces, we can deform
it back to $J_{1,1}$ through almost complex structures that are all adapted to
$\psi_{1,1}$. Finally, we can find a Hamiltonian isotopy in $\mathbb{CP}^{2}$
that fixes $L_{1,1}$ and displaces $L$ and $B$ from the planes $u$, $v$, and
$w$. Its time-one map is the desired Hamiltonian diffeomorphism. ∎

In view of the previous lemma, there is no loss of generality in assuming that
we are in the situation depicted schematically in the following figure:

$S_{\infty}$$T_{0}$$T_{\infty}$$u$$v$$w$$B$$L$$L_{1,1}$

This is evidently meant to be suggestive, as $B$ and $L$, while being disjoint
from $u$, $v$, and $w$, could still intersect any fiber lying above one of the
three lines in the above moment image.

######  Lemma 5.3.

There exists a symplectically embedded sphere in the class of $S_{\infty}$
that is disjoint from $L$, $L_{1,1}$ and $B$. Furthermore, this sphere can be
arranged to have a unique transverse and positive intersection with exactly
one of the planes $u$, $v$, or $w$.

###### Proof.

We will construct the desired sphere by examining the situation more closely
in $\mathcal{U}(L_{1,1})$. For concreteness, our proof will show how to
construct a sphere having a single intersection with $u$, but the arguments
adapt in the obvious way to the other two cases. Now by definition of
$\psi_{1,1}$, the projection of $\mathcal{U}(L_{1,1})$ onto the
$(p_{1},p_{2})$-plane is a small square in the moment image of
$(\mathbb{CP}^{2},\omega_{\mspace{1.0mu}\mathrm{FS}})$, where the origin
corresponds to $L_{1,1}$. We can then consider the product torus $L_{a,b}$ for
$(a,b)$ close enough to $(1,1)$, and translate the parametrization
$\psi_{1,1}$ in the obvious way to obtain a parameterization $\psi_{a,b}$ of
$L_{a,b}$. We assume the corresponding Weinstein neighborhood
$\mathcal{U}(L_{a,b})$ is made small enough to still be contained in
$\mathcal{U}(L_{1,1})$.

We then choose an almost complex structure $J_{a,b}$ on
$\mathbb{CP}^{2}\smallsetminus L_{a,b}$ that coincides with $J$ outside of
$\mathcal{U}(L_{1,1})$ and that is adapted to $\psi_{a,b}$, and translate the
coordinate representations of the planes $v$ and $w$ to obtain symplectic
cylinders that are $J_{a,b}$-holomorphic in $\mathcal{U}(L_{a,b})$. We also
translate the entire plane $u$ by a toric translation using the vector
$(a,b)$, obtaining a symplectic plane that is $J_{a,b}$-holomorphic in
$\mathcal{U}(L_{a,b})$. This process is illustrated in the following figure
for a specific translation:

$u$$v$$w$$\mathcal{U}(L_{1,1})$$\mathcal{U}(L_{a,b})$$T_{0}$$T_{0}$

We then construct symplectic planes in $\mathbb{CP}^{2}\smallsetminus L_{a,b}$
that are $J_{a,b}$-holomorphic in $\mathcal{U}(L_{a,b})$ and asymptotic to
Reeb orbits covering geodesics in the classes $(1,1)$, $(0,-1)$ and $(-1,0)$
in $H_{1}(L_{a,b},\psi_{a,b})$. This is done via the following toric
deformation:

$T_{0}$$T_{0}$

We have also circled a distinguished intersection point in the above figure.

We now consider the symplectically embedded surface with three cylindrical
ends in $T^{\ast}\mathbb{T}^{2}$ whose existence is guaranteed by Proposition
A.1. This can be transported to $T^{\ast}L_{a,b}$ via the embedding
$\psi_{a,b}$, and using the fact that all Reeb orbits in question match, we
can glue this surface together with the above perturbed objects in order to
construct a symplectically embedded sphere $\mathcal{S}_{\infty}$ in
$\mathbb{CP}^{2}\smallsetminus(L\cup L_{1,1}\cup B)$, which from the
intersection properties of the above planes with $S_{\infty}\cup T_{0}\cup
T_{\infty}$ must belong to the class of $S_{\infty}$. By construction, we see
that $\mathcal{S}_{\infty}$ has a single positive and transverse intersection
with $u$, as desired. ∎

At this point, we can actually employ the machinery of relative foliations. To
start, fix an almost complex structure $J$ on
$\mathbb{CP}^{2}\smallsetminus(L\cup L_{1,1})$ that is adapted to $\psi_{1,1}$
and some fixed parameterization of $L$, and that is integrable near the planes
$u$, $v$, and $w$. Then perturb $J$ near $B$ so that it it is of a suitable
form to perform a symplectic blowup along the interior of $B$. Once this is
done, we are back in $(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$,
and $J$ lifts to an almost complex structure on $\mathbb{D}_{1}$ that is
integrable near the exceptional divisor $\mathcal{S}_{0}$ and makes the planes
$u$, $v$, and $w$ into $J$-holomorphic planes with punctures on $L_{1,1}$.
There is a corresponding relative foliation $\mathcal{F}$, and we choose two
unbroken leaves $\mathcal{T}_{0}$ and $\mathcal{T}_{\infty}$ with the property
that the points $\pi(\mathcal{T}_{0})$ and $\pi(\mathcal{T}_{\infty})$ lie on
opposite sides of each of the circles $\pi(L)$ and $\pi(L_{1,1})$ in
$\mathcal{S}_{0}$.

######  Lemma 5.4.

There exists a symplectically embedded sphere $\mathcal{S}_{\infty}$ in the
class $(1,1)$ that is disjoint from $L$ and $L_{1,1}$, such that $L_{1,1}$ is
homologicaly nontrivial in
$\mathbb{D}_{1}\smallsetminus(\mathcal{S}_{0}\cup\mathcal{S}_{\infty}\cup\mathcal{T}_{0}\cup\mathcal{T}_{\infty})$.

###### Proof.

We first show that one of the planes $u$, $v$, and $w$ must be part of a
broken leaf of $\mathcal{F}$. To this end, observe that these planes
compactify to smooth disks with boundary on $L_{1,1}$, and the homology
classes of the corresponding boundaries in $H_{1}(L_{1,1};\psi_{1,1})$ sum to
$0$. Adding to this a chain in $L_{1,1}$ with matching boundary, we obtain a
chain in $\mathbb{D}_{1}$ that is smoothly embedded and $J$-holomorphic
outside of any arbitrarily small neighborhood of $L_{1,1}$. Since this chain
has area $3$ and does not intersect $\mathcal{S}_{0}$, it must represent the
class $(1,1)$. It is then a simple matter to see that the conclusions of Lemma
4.9 hold for the planes $u$, $v$, and $w$. In particular, they cannot all be
essential, as this would contradict the last part of the aforementioned lemma.
It thus follows that one of them must be part of a broken leaf of
$\mathcal{F}$. Without loss of generality, we can assume this plane to be $u$.
We then use Lemma 5.3 to produce a symplectically embedded sphere
$\mathcal{S}_{\infty}$ in $\mathbb{CP}^{2}$ in the class of $S_{\infty}$ that
is disjoint from $L$, $L_{1,1}$ and $B$, and that has a unique transverse and
positive intersection with $u$. We can also modify $J$ so that it is
integrable near $\mathcal{S}_{\infty}$ Blowing up along $B$ returns us to
$(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$, and it is now
straightforward to see that the hypotheses of Lemma 4.11 are satisfied. ∎

We can now finally prove Proposition 5.1. First, we claim that the unbroken
leaves $\mathcal{T}_{0}$ and $\mathcal{T}_{\infty}$ represent different
classes in $H_{2}(\mathbb{D}_{1}\smallsetminus L)$ and in
$H_{2}(\mathbb{D}_{1}\smallsetminus L_{1,1})$. If this failed for say $L$,
there would be a homotopy in $X\smallsetminus L$ from $\mathcal{T}_{0}$ to
$\mathcal{T}_{\infty}$, and this would project to a connected subset of
$\mathcal{S}_{0}\smallsetminus\pi(L)$ containing $\pi(\mathcal{T}_{0})$ and
$\pi(\mathcal{T}_{\infty})$, contradicting the assumption that $\pi(L)$
separates these two points. Now by Theorem 1.4 of [16], there is a
symplectomorphism from the current model of
$(\mathbb{D}_{1},\omega_{\mspace{1.0mu}\mathbb{D}_{1}})$ to the toric model
$(X,\omega_{X})$ taking $\mathcal{S}_{0}$ to $S_{0}$ and
$\mathcal{S}_{\infty}$ to $S_{\infty}$. Composing this with a Hamiltonian
diffeomorphism if necessary, we can assume that it also takes
$\mathcal{T}_{0}$ to $T_{0}$ and $\mathcal{T}_{\infty}$ to $T_{\infty}$. Let
$L_{\mathfrak{r}}$ and $L_{\mathfrak{s}}$ be the images of $L$ and $L_{1,1}$,
respectively, under the above symplectomorphism. These are now disjoint,
monotone tori in $X\smallsetminus Y$, and (c) of Proposition 5.1 is
automatically satisfied, as is (b) in view of Lemma 5.4. To see that (a)
holds, simply observe that if $L_{\mathfrak{r}}$ were homologicaly nontrivial
in $X\smallsetminus Y$, then Proposition 4.12 would imply that it must
intersect $L_{\mathfrak{s}}$, contradicting our standing assumption. This
finally completes the proof of Proposition 5.1.

###  5.2. Building Types

In view of the previous subsection, to prove Theorem 1.5 it is enough to
assume the conclusion of Proposition 5.1 and derive a contradiction. So taking
$L_{\mathfrak{r}}$ and $L_{\mathfrak{s}}$ as above, we fix parameterizations
$\psi_{\mathfrak{r}}$ and $\psi_{\mathfrak{s}}$, and shrink the corresponding
Weinstein neighborhoods so that they are disjoint from each other and from
$Y$. We then choose an almost complex structure $J$ on
$X\smallsetminus(L_{\mathfrak{r}}\cup L_{\mathfrak{s}})$ that is integrable
near $Y$ and adapted to the parameterizations $\psi_{\mathfrak{r}}$ and
$\psi_{\mathfrak{s}}$. From Theorem 4.6 we obtain a relative foliation

$\mathcal{F}=\mathcal{F}(L_{\mathfrak{r}},L_{\mathfrak{s}},\psi_{\mathfrak{r}},\psi_{\mathfrak{s}},J),$

together with foliation classes $\beta_{\mathfrak{r}}$ and
$\beta_{\mathfrak{s}}$, and a projection map $\pi\colon X\to S_{0}$ defined as
before. The sets $C_{\mathfrak{r}}=\pi(L_{\mathfrak{r}})$ and
$C_{\mathfrak{s}}=\pi(L_{\mathfrak{s}})$ are disjoint embedded circles in
$S_{0}$, By (c) of Proposition 5.1, both of these must separate the points
$\pi(T_{0})$ and $\pi(T_{\infty})$. Consequently, we can find disjoint closed
disks $D_{\mathfrak{r}}$ and $D_{\mathfrak{s}}$ in $S_{0}$ with boundaries
$C_{\mathfrak{r}}$ and $C_{\mathfrak{s}}$, respectively, such that
$\pi(T_{0})\in D_{\mathfrak{r}}$ and $\pi(T_{\infty})\in D_{\mathfrak{s}}$.
The closure of $S_{0}\smallsetminus(D_{\mathfrak{r}}\cup D_{\mathfrak{s}})$ is
a closed annulus, which we will denote by $A$.

Observe that the leaves of $\mathcal{F}$ split up into three categories. There
are unbroken leaves in the class $(0,1)$, and two categories of broken leaves.
The first consists of pairs of planes that are asymptotic to Reeb orbits in
$\Gamma_{\beta_{\mathfrak{s}}}\cup\Gamma_{-\beta_{\mathfrak{s}}}$. We know
that the totality of these planes divide into two families, which going
forward we shall denote by $\mathfrak{s}_{0}$ and $\mathfrak{s}_{\infty}$. The
family $\mathfrak{s}_{0}$ consists of those planes that intersect $S_{0}$, and
the family $\mathfrak{s}_{\infty}$ consists of those planes that intersect
$S_{\infty}$. This follows from the fact that $L_{\mathfrak{s}}$ is
homologicaly nontrivial in $X\smallsetminus Y$, since otherwise the union of
the planes in $\mathfrak{s}_{\infty}$ would compactify to a solid torus in
$X\smallsetminus Y$ whose boundary is $L_{\mathfrak{s}}$, see Proposition 5.16
of [8]. The second category of broken leaves consists of pairs of planes
asymptotic to Reeb orbits in
$\Gamma_{\beta_{\mathfrak{r}}}\cup\Gamma_{-\beta_{\mathfrak{r}}}$. These again
divide into two families $\mathfrak{r}_{0}$ and $\mathfrak{r}_{\infty}$. The
family $\mathfrak{r}_{0}$ consists of planes intersecting $S_{0}$, but the
family $\mathfrak{r}_{\infty}$ now consists of planes intersecting _neither
$S_{0}$ nor $S_{\infty}$_. This follows from Lemma 4.11, since
$L_{\mathfrak{r}}$ is homologicaly trivial in $X\smallsetminus Y$ but
$C_{\mathfrak{r}}$ still separates $\pi(T_{0})$ and $\pi(T_{\infty})$.
Consequently, the planes in $\mathfrak{r}_{0}$ intersect _both $S_{0}$ and
$S_{\infty}$_. This fact will have important ramifications.

Now consider a family $\\{J_{t}\\}_{t\geq 0}$ of tame almost complex
structures on $(X,\omega_{X})$ that stretches to $J$, and extract a sequence
$(J_{t_{i}})_{i=1}^{\infty}$ as usual. We then fix $d\geq 1$ and use (c) of
Proposition 4.1 to obtain a sequence $(u_{i})_{i=1}^{\infty}$ of
$J_{t_{i}}$-holomorphic spheres in the class $(1,d)$ passing through $2d$
generic point constraints. By Theorem 3.2, this produces a limit building
$\mathbf{F}$ in the class $(1,d)$. Using the foliation $\mathcal{F}$, we can
obtain a complete classification scheme for such buildings, and this is
content of the following proposition.

######  Proposition 5.5.

The building $\mathbf{F}$ is exactly one of five types:

######  Type 0.

$\mathbf{F}$ is a (possibly nodal) $J$-holomorphic sphere in
$X\smallsetminus(L_{\mathfrak{r}}\cup L_{\mathfrak{s}})$ in the class $(1,d)$,
with one essential sphere in the class $(1,c)$ for $1\leq c\leq d$. Any
remaining top level curves cover broken or unbroken leaves of $\mathcal{F}$.
Any middle and bottom level curves cover cylinders asymptotic to orbits
covering geodesics in multiples of $\beta_{\mathfrak{r}}$ and
$\beta_{\mathfrak{s}}$.

######  Type 1.

$\mathbf{F}$ has a unique essential curve $\underline{u}$ whose punctures on
both $L_{\mathfrak{r}}$ and $L_{\mathfrak{s}}$ are all of foliation type. The
image of $\pi\circ\underline{u}$ is $S_{0}$ minus finitely many points on
$C_{\mathfrak{r}}\cup C_{\mathfrak{s}}$. Any remaining top level curves are
either spheres covering unbroken leaves, or planes covering one of the
components in a broken leaf of $\mathcal{F}$. Any middle and bottom level
curves cover cylinders asymptotic to orbits covering geodesics in multiples of
$\beta_{\mathfrak{r}}$ and $\beta_{\mathfrak{s}}$.

######  Type 2a.

$\mathbf{F}$ has two essential curves. The first curve $u_{\mathfrak{r}}$ is a
plane with a single puncture on $L_{\mathfrak{r}}$, not of foliation type. The
closure of the image of $\pi\circ u_{\mathfrak{r}}$ is $D_{\mathfrak{r}}$. The
second curve $\underline{u}$ has a single puncture on $L_{\mathfrak{r}}$, not
of foliation type, and punctures on $L_{\mathfrak{s}}$, all of foliation type.
The closure of the image of $\pi\circ\underline{u}$ is $A\cup
D_{\mathfrak{s}}$. Any remaining top level curves cover broken or unbroken
leaves of $\mathcal{F}$. Any middle and bottom level curves in
$\mathbb{R}\times S^{\ast}L_{\mathfrak{s}}$ and $T^{\ast}L_{\mathfrak{s}}$
cover cylinders asymptotic to orbits covering geodesics in multiples of
$\beta_{\mathfrak{s}}$.

######  Type 2b.

$\mathbf{F}$ has two essential curves. The first curve $\underline{u}$ has a
single puncture on $L_{\mathfrak{s}}$, not of foliation type, and punctures on
$L_{\mathfrak{r}}$, all of foliation type. The closure of the image of
$\pi\circ\underline{u}$ is $D_{\mathfrak{r}}\cup A$. The second curve
$u_{\mathfrak{s}}$ is a plane with a single puncture on $L_{\mathfrak{s}}$,
not of foliation type. The closure of the image of $\pi\circ u_{\mathfrak{s}}$
is $D_{\mathfrak{s}}$. Any remaining top level curves cover broken or unbroken
leaves of $\mathcal{F}$. Any middle and bottom level curves in
$\mathbb{R}\times S^{\ast}L_{\mathfrak{r}}$ and $T^{\ast}L_{\mathfrak{r}}$
cover cylinders asymptotic to orbits covering geodesics in multiples of
$\beta_{\mathfrak{r}}$.

######  Type 3.

$\mathbf{F}$ has three essential curves. The first curve $u_{\mathfrak{r}}$ is
a plane with a single puncture on $L_{\mathfrak{r}}$, not of foliation type,
and the closure of the image of $\pi\circ u_{\mathfrak{r}}$ is
$D_{\mathfrak{r}}$. The second curve $\underline{u}$ is a cylinder with a
single puncture on each of $L_{\mathfrak{r}}$ and $L_{\mathfrak{s}}$, neither
of foliation type, and the closure of the image of $\pi\circ\underline{u}$ is
$A$. The third curve $u_{\mathfrak{s}}$ is a plane with a single puncture on
$L_{\mathfrak{s}}$, not of foliation type, and the closure of the image of
$\pi\circ u_{\mathfrak{s}}$ is $D_{\mathfrak{s}}$. Any remaining top level
curves cover broken or unbroken leaves of $\mathcal{F}$.

###### Proof.

By Lemma 4.9, we know that $\mathbf{F}$ has at least one essential top level
curve, which we denote by $u$. As a preliminary step, we observe that if $u$
has punctures and none of them are of foliation type, then it falls into
exactly one of three categories:

  1. $\bullet$

A $J$-holomorphic plane with a single puncture on $L_{\mathfrak{r}}$. The
closure of the image of $\pi\circ u$ is $D_{\mathfrak{r}}$ or $A\cup
D_{\mathfrak{s}}$.

  2. $\bullet$

A $J$-holomorphic plane with a single puncture on $L_{\mathfrak{s}}$. The
closure of the image of $\pi\circ u$ is $D_{\mathfrak{r}}\cup A$ or
$D_{\mathfrak{s}}$.

  3. $\bullet$

A $J$-holomorphic cylinder with a single puncture on each of
$L_{\mathfrak{r}}$ and $L_{\mathfrak{s}}$. The closure of the image of
$\pi\circ u$ is $A$.

Indeed, by Proposition 4.8, we see that $u$ can have at most one puncture on
either of $L_{\mathfrak{r}}$ or $L_{\mathfrak{s}}$, and this shows that it is
either a plane or a cylinder. Observing that the image of $u$ must project via
$\pi$ onto a connected subset of $S_{0}$ with boundary components equal to one
or both of $C_{\mathfrak{r}}$ or $C_{\mathfrak{s}}$, the three possibilities
for $u$ are easily deduced.

Now suppose that $u$ intersects the fiber $T_{0}$, itself an unbroken leaf of
$\mathcal{F}$ since we are assuming it to be $J$-holomorphic. We can then
conduct a case by case analysis based on the nature of the punctures of $u$.

Case 1: _$u$ has no punctures._ This implies that $\pi\circ u$ must be a
bijection onto $S_{0}$. It follows that $u$ is a $J$-holomorphic sphere in the
class $(1,c)$ for some $0\leq c\leq d$. Then again by Lemma 4.9, all remaining
top level curves of $\mathbf{F}$ must cover leaves of $\mathcal{F}$, and all
middle and bottom level curves cover cylinders asymptotic to orbits covering
geodesics in multiples of $\beta_{\mathfrak{r}}$ and $\beta_{\mathfrak{s}}$.
Furthermore, the top level curves that cover leaves fit together to form a
possibly disconnected curve in the class $(0,d-c)$, and if $c=d$, then
$\mathbf{F}$ consists only of the curve $u$. In any case, the building is of
type 0.

Case 2: _$u$ has punctures and they are all of foliation type._ In this case,
we will show that $\mathbf{F}$ must be of type 1. First, by Lemma 4.7, we see
that the image of $\pi\circ u$ includes points in each component of
$S_{0}\smallsetminus C_{\mathfrak{r}}\cup C_{\mathfrak{s}}$. By Lemma 4.9, it
follows that $\pi\circ u$ is actually a bijection onto $S_{0}$ minus a finite
set of points on $C_{\mathfrak{r}}\cup C_{\mathfrak{s}}$. Any remaining top
level curves must then cover unbroken leaves of $\mathcal{F}$ or be planes
covering one of the components of a broken leaf. This shows that $\mathbf{F}$
is of type 1 with $\underline{u}=u$.

Case 3: _$u$ has at least one puncture that is not of foliation type._ Since
$u$ intersects $T_{0}$, the closure of the image of $\pi\circ u$ is either
$D_{\mathfrak{r}}$ or $D_{\mathfrak{r}}\cup A$. In either case, we see by
Proposition 4.8 that $u$ does not intersect $T_{\infty}$ and has exactly one
puncture that is not of foliation type. If we suppose furthermore that the
closure of the above image is $D_{\mathfrak{r}}$, then $u$ is a plane, for it
cannot have any punctures on $L_{\mathfrak{s}}$. By Lemma 4.9, there is
another essential curve $v$ of $\mathbf{F}$ that intersects $T_{\infty}$, and
the images of $\pi\circ u$ and $\pi\circ v$ must be disjoint. This implies
that the closure of the image of the latter is either $D_{\mathfrak{s}}$ or
$A\cup D_{\mathfrak{s}}$. If it is $D_{\mathfrak{s}}$, then $v$ cannot have
any punctures on $L_{\mathfrak{r}}$ and must therefore be a plane with a
single puncture on $L_{\mathfrak{s}}$, not of foliation type. Lemma 4.9 then
guarantees the existence of yet a third essential curve $\underline{u}$, which
must therefore be a cylinder with a puncture on either torus, both not of
foliation type. The building $\mathbf{F}$ is thus of type 3 with
$u_{\mathfrak{r}}=u$ and $u_{\mathfrak{s}}=v$. If the closure of the image of
$\pi\circ v$ was instead $A\cup D_{\mathfrak{s}}$, then all its punctures on
$L_{\mathfrak{s}}$ must be of foliation type, and it must have a single
puncture on $L_{\mathfrak{r}}$, not of foliation type. The building
$\mathbf{F}$ is now of type 2a, with $u_{\mathfrak{r}}=u$ and
$\underline{u}=v$. Finally, if initially the closure of the image of $\pi\circ