# Liouville polarization and the rigidity of their Lagrangian skeleta in
dimension $4$

E. Opshtein and F. Schlenk

###### Abstract.

In [3], Biran introduced polarizations of closed symplectic manifolds and
showed that their Lagrangian skeleta exhibit remarkable rigidity properties.
He in particular found that their complements contain only small balls. In
this paper, we introduce so-called Liouville polarizations of certain open
4-dimensional symplectic manifolds. This leads to several symplectic embedding
results, that in turn lead to new Lagrangian non-removable intersections and a
novel phenomenon of Legendrian barriers.

We show for instance that given any connected symplectic 4-manifold
$(M,\omega)$ and a 4-ball of smaller volume, their exists an explicit finite
union of Lagrangian discs in the 4-ball such that their complement
symplectically embeds into $(M,\omega)$, extending a result by Sackel-Song-
Varolgunes-Zhu and Brendel.

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Symplectic embeddings
    2. 1.2 Rigidity properties of Lagrangian skeleta
    3. 1.3 Legendrian barriers
    4. 1.4 A general embedding result.
    5. 1.5 Related results and references
  2. 2 Polarizations of symplectic manifolds
    1. 2.1 Neighborhoods of symplectic curves in dimension $4$
    2. 2.2 Polarizations (closed case)
    3. 2.3 Liouville polarizations (open case)
  3. 3 Some explicit polarizations and skeleta
    1. 3.1 Polarizations in dimension $2$
    2. 3.2 Bidiscs
    3. 3.3 Surgery on Liouville polarizations
  4. 4 Symplectic embeddings
    1. 4.1 The main embedding result (Theorem 6)
      1. 4.1.1 The monotone case
      2. 4.1.2 Reduction to the monotone setting
      3. 4.1.3 Proof of Theorem 6
    2. 4.2 Proof of Theorem 2
    3. 4.3 Proof of Theorem 3
  5. 5 Lagrangian rigidity
  6. 6 Legendrian barriers

##  1\. Introduction and main results

A polarization of a closed symplectic manifold $(M,\omega)$ is a decomposition
of the symplectic class $[\omega]$ along positive real multiples of the
Poincaré-dual classes of codimension 2 symplectic submanifolds. Closed
symplectic manifolds always admit polarizations, and these have proven useful
to study the flexible side of symplectic embedding problems [6, 25]. In this
work, we introduce polarizations of open symplectic manifolds and show that
this concept offers new opportunities, at least in dimension $4$. Referring to
$\S$ 2.3 for the definition, we focus in this introduction on the
applications.

###  1.1. Symplectic embeddings

As usual, $D(a)$, $B^{4}(a)$, $C^{4}(a)$, and $Z^{4}(a)$ denote the open round
disc of area $a$, the open 4-ball of capacity $\pi r^{2}=a$, the open “cube”
$D(a)\times D(a)$, and the open cylinder
$D(a)\times\mathbb{C}\subset\mathbb{C}\times\mathbb{C}$, all endowed with the
standard symplectic form
$\omega_{{\operatorname{st}}}=\sum_{j=1}^{2}dx_{j}\wedge dy_{j}$ on
$\mathbb{R}^{4}$ with primitive
$\alpha_{{\operatorname{st}}}\,=\,\frac{1}{2}\sum_{j=1}^{2}x_{j}dy_{j}-y_{j}dx_{j}$.

A cornerstone of modern symplectic geometry is Gromov’s non-squeezing theorem
from [17], stating in dimension four that $B^{4}(a)$ symplectically embeds
into $Z^{4}(1)$ only for $a\leq 1$. In [29], Sackel, Song, Varolgunes, and Zhu
investigated a refinement of this result, asking how big a set (in the sense
of the Hausdorff dimension, for instance) one has to remove from $B^{4}(a)$
such that the complement symplectically embeds into $Z^{4}(1)$. They first
proved that it does not suffice to remove a set of lower Minkowski dimension
$<2$, and then showed optimality of this result by proving that for $a\leq 2$
it suffices to remove a Lagrangian coordinate disc from $B^{4}(a)$. Later,
Brendel in the appendix to [29] showed that for $a<3$ it suffices to remove
three Lagrangian pinwheels (certain simple Lagrangian CW-complexes) and a
symplectic torus. In their Question 3, these authors then asked:

######  Question 1.1.

What is the largest $a$ such that the complement of a 2-dimensional subset of
$B^{4}(a)$ symplectically embeds into $Z^{4}(1)$ ?

We denote by $\Gamma_{\\!\frac{1}{k}}$ the union of the $k$ half-lines
emanating from the origin in $\mathbb{C}$ and cutting the plane into isometric
sectors, as partly shown in (i), (ii), and (iii) of Figure 1.1 on page 1.1 for
$k=2,3,4$, and write

$\Delta_{k}:=\Gamma_{\\!\frac{1}{k}}\times\Gamma_{\\!\frac{1}{k}}\subset\mathbb{C}^{2}.$

Then $\Delta_{k}$ is the orbit of the Lagrangian quadrant $\mathbb{R}_{\geq
0}^{2}\subset\mathbb{C}^{2}$ under the subgroup $G$ of $\operatorname{U}(2)$
generated by $(e^{2\pi i/k},1)$ and $(1,e^{2\pi i/k})$. When $k$ is even, then
$\Delta_{k}$ is also the $G$-orbit of the standard Lagrangian plane
$\mathbb{R}^{2}\subset\mathbb{C}^{2}$.

A symplectic embedding
$\varphi\colon(M,d\alpha)\to(M^{\prime},d\alpha^{\prime})$ between exact
symplectic manifolds is $(\alpha,\alpha^{\prime})$-exact if
$\varphi^{*}\alpha^{\prime}=\alpha+df$ for a smooth function $f$ on $M$. Since
$B^{4}(1)\subset C^{4}(1)$, our first symplectic embedding result give a
positive answer to Question 1.1 for every $a$.

######  Theorem 1.

There exists an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding

$C^{4}(1)\backslash\Delta_{k}\,\to\,Z^{4}\bigl{(}\tfrac{2}{k}\bigr{)}.$

######  Remarks 1.2.

(i) There exists a symplectic embedding of $D(\frac{1}{k})\times
D(\frac{1}{k})$ into $C^{4}(1)\setminus\Delta_{k}$, so
$C^{4}(1)\setminus\Delta_{k}$ does not symplectically embed into $Z^{4}(A)$
for $A<\frac{1}{k}$ by the nonsqueezing theorem. Thus Theorem 1 is sharp up to
a possible factor of $2$. We do not know the exact size of the thinnest
cylinder into which $C^{4}(1)\backslash\Delta_{k}$ or
$B^{4}(1)\backslash\Delta_{k}$ symplectically embed, except for the optimal
embedding $B^{4}(1)\setminus\Delta_{2}\to Z^{4}(\frac{1}{2})$ from [29]. For a
different such embedding see Remark 4.7.

(ii) A different positive answer to Question 1.1 was independently given by
Haim-Kislev, Hind, and Ostrover in [18], and it is interesting to compare the
two constructions, see $\S$ 1.5.

Our construction also applies to unbounded domains. For instance, consider the
union of Lagrangian planes $\Gamma\times\Gamma$ in $\mathbb{R}^{4}$, where
$\Gamma\subset\mathbb{R}^{2}$ is the square grid

$\Gamma=\bigcup_{(n,m)\in\mathbb{Z}^{2}}\\{n\\}\times\mathbb{R}\cup\mathbb{R}\times\\{m\\}.$

######  Theorem 2.

There exists an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding

$\mathbb{R}^{4}\setminus(\Gamma\times\Gamma)\,\to\,Z^{4}(1).$

We can construct similar embeddings when the source or the target have
topology. If we take the 4-ball as domain, we can fill “anything” after
removing sufficiently many Lagrangian planes:

######  Theorem 3.

Let $(M,\omega)$ be a connected symplectic 4-manifold of finite volume. Let
$B^{4}(a)$ be the ball of the same volume. Then for every $\varepsilon>0$
there exist an even number $k\in\mathbb{N}$ such that
$B^{4}(a-\varepsilon)\setminus\Delta_{k}$ symplectically embeds into
$(M,\omega)$.

In fact a much more general statement holds. Any affine part of a closed
symplectic 4-manifold (defined as the complement of a polarizing divisor, see
Definition 2.2) verifies the same kind of flexibility: After removing a
suitable Lagrangian CW-complex, it can be embedded into any other affine part
of a closed symplectic 4-manifold of larger volume. We shall give the proof of
this result in a separate paper, [27], because in addition to the embedding
method developped here the proof relies on a relative version of Giroux’s
result in [16] and requires specific techniques. In contrast to the results of
this paper, the removed CW-complex cannot be explicitly described, however.

###  1.2. Rigidity properties of Lagrangian skeleta

Following Biran [2], we notice that if the complement of a Lagrangian CW-
complex $\Delta$ in a domain $U$ symplectically embeds into a cylinder, then
any domain $V\subset U$ that does not symplectically embed into that cylinder
must intersect $\Delta$. The results of the previous paragraph therefore have
implications on non-removable intersections of $\Delta$ with various subsets,
and even with Lagrangian submanifolds. We shall remove Lagrangian CW-complexes
more general than the $\Delta_{k}$, namely products of arbitrary connected
graphs.

######  Definition 1.3.

A grid $\Gamma$ in $D(A)$ is the part in $D(A)$ of a connected graph
$\Gamma\cup\partial D(A)$ in $\overline{D(A)}$ that has no 1-valent vertex,
contains the boundary of $D(A)$, and has smooth edges.

Figure 1.1. Examples of grids.

Note that the complement of a grid consists of topological discs. Figure 1.1
shows examples of grids. Note that for two grids $\Gamma$ and
$\Gamma^{\prime}$, the Lagrangian CW-complex $\Gamma\times\Gamma^{\prime}$ is
smooth if and only if both $\Gamma$ and $\Gamma^{\prime}$ have no vertex, as
in (i), (iv), and (vii).

Cieliebak and Mohnke in [9] defined for every closed Lagrangian submanifold
$L\subset\mathbb{R}^{4}$ its minimal symplectic area by

(1.1)
${\mathcal{A}}_{\min}(L):=\inf\left\\{\int_{\sigma}\omega_{{\operatorname{st}}}\mid[\sigma]\in\pi_{2}(\mathbb{R}^{4},L),\,\int_{\sigma}\omega_{{\operatorname{st}}}>0\right\\}\;\in\;[0,\infty].$

######  Theorem 4.

Let $\Gamma_{\\!\leq a}\subset D(A)$ and $\Gamma_{\\!\leq b}\subset D(B)$ be
any two grids whose complements are a union of topological discs of area
$\leq\\!a$ and $\leq\\!b$, respectively. Then a closed Lagrangian submanifold
$L$ of $D(A)\times D(B)$ with

${\mathcal{A}}_{\min}(L)\geq a+b$

cannot be mapped to $\bigl{(}D(A)\times
D(B)\bigr{)}\setminus\bigl{(}\Gamma_{\\!\leq a}\times\Gamma_{\\!\leq
b}\bigr{)}$ by a Hamiltonian diffeomorphism of $\mathbb{R}^{4}$.

######  Example.

For every Markov triple ${\mathfrak{m}}$ there exists a monotone Lagrangian
torus $\widehat{L}({\mathfrak{m}})$ in the complex projective plane endowed
with the Fubini–Study form integrating to $3$ over complex lines. Each such
torus gives rise to a Lagrangian torus $L({\mathfrak{m}})\subset B^{4}(3)$,
and $L({\mathfrak{m}})$ is Hamiltonian isotopic to
$L({\mathfrak{m}}^{\prime})$ in $B^{4}(3)$ only if
${\mathfrak{m}}={\mathfrak{m}}^{\prime}$, see [31]. For these Lagrangians,
$A_{\min}(L({\mathfrak{m}}))=1$. If $\phi_{H}$ is a Hamiltonian diffeomorphism
of $\mathbb{R}^{4}$ such that $\phi_{H}(L({\mathfrak{m}}))\subset
B^{4}(A)\setminus\Delta_{k}$, then $A\geq\frac{k}{2}$. $\diamond$

###  1.3. Legendrian barriers

Lagrangian CW-complexes $\Delta$ with symplectically small complement were
called Lagrangian barriers in [3] (cf. $\S$ 1.5). A further consequence of the
phenomenon of Lagrangian barriers, that apparently went unnoticed, is a
similar Legendrian barrier phenomenon in contact dynamics. A grid is radial if
it consists of straight rays emanating from the origin.

######  Theorem 5.

Let $\Gamma_{\\!\leq\delta_{1}}$ and $\Gamma_{\\!\leq\delta_{2}}$ be two
radial grids that divide $D(1)$ into sectors of area $\leq\delta_{1}$ and area
$\leq\delta_{2}$, respectively. Let $U\subset C^{4}(1)$ be a starshaped domain
with smooth boundary $S$, endowed with the usual contact form
$\lambda_{S}=\alpha_{{\operatorname{st}}}|_{S}$, and consider the connected
Legendrian graph

$\Lambda_{\delta}\,=\,\bigl{(}\Gamma_{\\!\leq\delta_{1}}\\!\times\Gamma_{\\!\leq\delta_{2}}\bigr{)}\cap
S.$

Then any Legendrian knot $\Lambda$ in $S$ has a Reeb chord from $\Lambda$ to
$\Lambda\cup\Lambda_{\delta}$ of length $\leq\delta_{1}+\delta_{2}$ for the
contact form $\lambda_{S}$.

######  Remarks 1.4.

(i) While the proof of Theorem 5 relies on tools from hard symplectic
geometry, the case of the round sphere can be shown by elementary geometric
arguments, see Section 6.

(ii) The smooth arcs of $\Lambda_{\delta}$ are indeed Legendrian curves on
$S$, because
$\iota_{r\partial_{r}}\omega_{{\operatorname{st}}}=\alpha_{{\operatorname{st}}}$
and $\partial_{r}$ is tangent to
$\Gamma_{\\!\leq\delta_{1}}\\!\times\Gamma_{\\!\leq\delta_{2}}$. The theorem
says that for any Legendrian knot $\Lambda$, every embedded Reeb cylinder of
length $\geq\delta_{1}+\delta_{2}$ over $\Lambda$ must intersect
$\Lambda_{\delta}$, hence the name Legendrian barrier for $\Lambda_{\delta}$.
It follows in particular that for $a>\frac{2}{k}$ there exists no exact
contact embedding

$\bigl{(}D(a)\times\mathbb{R}/\mathbb{Z},\alpha_{{\operatorname{st}}}+dt\bigr{)}\,\hookrightarrow\,\bigl{(}S\backslash\Delta_{k},\lambda_{S}\bigr{)}.$

It is then natural to ask whether there is a purely contact topological
statement of this kind. For instance, using the Reeb flows one constructs
exact contact embeddings

$\bigl{(}D(a)\times\mathbb{R}/\mathbb{Z},\alpha_{{\operatorname{st}}}+dt\bigr{)}\,\hookrightarrow\,\bigl{(}S^{3}(1),\lambda_{S}\bigr{)}$

for all $a\leq 1$. Is it true that for $a\in(\frac{2}{k},1]$ the image of
these embeddings cannot be displaced from $\Delta_{k}\cap S^{3}(1)$ by a (not
necessarily exact) contact isotopy?

(iii) In fact, the proof will show that there exist both forwards and
backwards Reeb chord from $\Lambda$ to $\Lambda\cup\Lambda_{\delta}$ of length
$\leq\delta_{1}+\delta_{2}$.

(iv) (Improved return time) Theorem 5 states an alternative: There must be
short Reeb chords from $\Lambda$ to $\Lambda$ or $\Lambda_{\delta}$. This is a
relative version of the solution of the Arnold chord conjecture by
Cieliebak–Mohnke in [22, 9]. They proved that for $\Lambda\subset S=\partial
U$ as in the theorem, there must be a Reeb chord from $\Lambda$ to $\Lambda$
of length at most

$\min\left\\{e(U),\tfrac{1}{2}\right\\}$

where $e(U)$ denotes the displacement energy of $U$. Theorem 5 improves this
bound on the return time for all those $\Lambda$ that lie in the complement of
the Reeb cylinder of $\Lambda_{\delta}$ of length $\delta_{1}+\delta_{2}$,
whenever $\delta_{1}+\delta_{2}<\min\left\\{e(U),\tfrac{1}{2}\right\\}$.

###  1.4. A general embedding result.

The upper bounds on the cylindrical capacity of the complement of our
Lagrangian CW-complexes all rely on a single embedding result. To state it, we
need to introduce the subclass of regular grids.

######  Definition.

A grid $\Gamma$ on $D(A)$ (as defined in 1.3) is regular if every vertex of
$\Gamma\cup\partial D(A)$ has a Darboux chart in which $\Gamma$ consists of
radial rays that cut the disc into equal sectors. When the vertex belongs to
the boundary, the Darboux chart must be understood with image
$D(\varepsilon)\cap\\{\operatorname{im}z\geq 0\\}\subset\mathbb{H}$.

In Figure 1.1, all grids are regular except for (iv) and (v) – and possibly
(ix).

######  Theorem 6.

Let $\Gamma_{\\!\leq a}\subset D(A)$ and $\Gamma_{\\!\leq b}\subset D(B)$ be
any two regular grids whose complements are a union of topological discs of
area $\leq a$ and $\leq b$, respectively. Then there exists an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding

$\bigr{(}D(A)\times D(B)\bigr{)}\setminus\bigl{(}\Gamma_{\\!\leq
a}\\!\times\Gamma_{\\!\leq b}\bigr{)}\,\hookrightarrow\,Z^{4}(a+b).$

Theorem 1 is a special case of this theorem, and Theorem 2 is a generalization
to a non-compact setting. The exactness in this statement is essential to
derive Theorems 4 and 5 on Lagrangian rigidity and Legendrian barriers.

###  1.5. Related results and references

Biran’s Lagrangian barriers. As we have already alluded to, the present paper
is in many ways a sequel of Biran’s work [3], that is useful to have in mind
while reading our paper. We therefore briefly recall it, integrating Giroux’s
results from [15, 16] that allow to generalize Biran’s results from the Kähler
to the symplectic setting. Let $(M,\omega)$ be a closed symplectic manifold
with integral symplectic class (meaning that $[\omega]\in
H^{2}(M,\mathbb{Z})$). For every sufficiently large $k$, Donaldson [10]
produced a symplectic hypersurface $\Sigma$ Poincaré-dual to $k[\omega]$,
called a polarization of degree $k$ of $(M,\omega)$. The complement
$M\setminus\Sigma$ was proven to be Weinstein by Giroux [16]. Hence there
exists an isotropic CW-complex (called the skeleton) $\Delta\subset
M\backslash\Sigma$ to which $M\backslash\Sigma$ retracts. Furthermore, any
compact subset of $M\backslash\Delta$ embeds into a ruled symplectic manifold,
namely a symplectic sphere bundle over $\Sigma$ whose fibers have area
$\frac{1}{k}$, see [2, 3]. A fundamental example to have in mind along this
discussion is the following:

######  Example 1.5.

([3, $\S$ 3.1.3]) In the complex projective space
$\operatorname{\mathbb{C}P}^{n}$ with Fubini–Study form integrating to 1 over
complex lines, the hypersurface

(1.2) $\Sigma_{2k}\,=\,\left\\{\sum_{j=0}^{n}z_{j}^{2k}=0\right\\}$

is a symplectic polarization of degree $2k$, and one can arrange the skeleton
of $\operatorname{\mathbb{C}P}^{n}\setminus\Sigma_{2k}$ to be the union
$\Delta_{2k}(\operatorname{\mathbb{C}P}^{n})$ of $k^{n}$ equally distributed
images of $\operatorname{\mathbb{R}P}^{n}$ by projectivized unitary maps. The
complement of $2k$ fibers of the symplectic disc bundle over $\Sigma_{2k}$ is
symplectomorphic to $B^{2n}(1)\setminus\Delta_{2k}^{n}$, where

$\Delta_{2k}^{n}:=(\Gamma_{2k})^{n}$

is the union of $k^{n}$ unitary images of $\mathbb{R}^{n}$. For $n=2$ this is
our set $\Delta_{2k}^{2}=\Delta_{2k}$.

Ruled symplectic manifolds are known to have many symplectic invariants
bounded by the area of their sphere fibres, which allowed Biran to get
interesting non-removable intersections with the skeleton. For instance, for a
polarization of degree $k$ with skeleton $\Delta$, every symplectic ball of
capacity $\geq\frac{1}{k}$ in $M$ must intersect $\Delta$. He therefore called
these skeleta Lagrangian barriers. In particular, such balls cannot be
displaced from $\Delta$ by a Hamiltonian isotopy. Combined with the work [9]
by Cieliebak and Mohnke, Biran’s decomposition result also implies non-
displaceability of small Lagrangian submanifolds from the skeleton. Namely,
applying the neck-stretching in [9] to the set-up in [2, 3], one finds that
the general Cieliebak–Mohnke Lagrangian width

${\mathcal{A}}_{\min}(L,M):=\inf\left\\{\int_{\sigma}\omega\mid[\sigma]\in\pi_{2}(M,L),\,\int_{\sigma}\omega>0\right\\}$

can be estimated as follows.

######  Theorem 7 (Biran–Cieliebak–Mohnke).

Let $\Delta$ be the skeleton of a polarization of degree $k$ of $(M,\omega)$.
Then for any closed Lagrangian submanifold $L\subset M\backslash\Delta$ it
holds that

${\mathcal{A}}_{\min}(L,M)<\frac{1}{k}.$

In other words, a closed Lagrangian submanifold with
${\mathcal{A}}_{\min}(L,M)\geq\frac{1}{k}$ cannot be displaced from the
skeleton $\Delta$ by a Hamiltonian isotopy.

For instance, for $x\geq\frac{1}{k}$ the product torus
$\mathbb{T}^{n}(x)\subset B^{2n}(1)$ whose factors are circles enclosing area
$x$ cannot be displaced from $\Delta_{2k}^{n}$ inside $B^{2n}(1)$. Again,
Theorem 7 has a Legendrian corollary:

######  Theorem 8.

Let $U\subset B^{2n}(1)$ be a starshaped domain with smooth boundary $S$,
endowed with the usual contact form
$\lambda_{S}=\alpha_{{\operatorname{st}}}|_{S}$, and consider the Legendrian
CW-complex $\Lambda_{2k}=\Delta_{2k}^{n}\cap S$. Then for every Legendrian
submanifold $\Lambda\subset S$ there exists a Reeb chord from $\Lambda$ to
$\Lambda\cup\Lambda_{2k}$ of length $T\leq\frac{1}{2k}$.

Biran’s decomposition theorem thus implies the same kind of applications that
we have drawn from our embedding results, in any dimension and with better
constants. We should therefore explain what this paper adds beyond the
embedding results from Section 1.1. Here is a first answer. Biran’s
intersection result and Theorem 7 state that some symplectic capacities of
$M\backslash\Delta_{k}$ are smaller than $\frac{1}{k}$, namely Gromov’s ball
embedding capacity

$c_{B}(M)\,:=\,\sup\left\\{a\mid\mbox{$B^{2n}(a)$ symplectically embeds into
$M$}\right\\},$

which is the smallest normalized symplectic capacity, and the Cieliebak–Mohnke
capacity

$c_{\textnormal{Lag}}(M)\,:=\,\sup\left\\{{\mathcal{A}}_{\min}(L,M)\mid
L\subset M\right\\}$

where the supremum runs over all closed Lagrangian submanifolds. Since the
cylindrical capacity

$c^{Z}(U)\,:=\,\inf\left\\{A\mid\mbox{$U$ symplectically embeds into
$Z^{2n}(A)$}\right\\}$

is the largest normalized capacity, Theorem 1 states instead that all
symplectic capacities of $B^{4}(1)\backslash\Delta_{k}$ are small, and our
work [27] generalizes this property to the affine part of any closed
symplectic manifold. There are many different capacities, whose estimates have
different implications. For instance, applied to the Hofer–Zehnder capacity,
Theorem 1 says that the flow of any compactly supported Hamiltonian function
on $C^{4}(1)$ that vanishes on a neighbourhood of $\Delta_{k}$ and whose
maximum is at least $\frac{2}{k}$ has a non-trivial closed orbit of period
$\leq 1$.

Another, more important, gain is that our approach allows to study the
rigidity properties of certain singular polarizations. The role of singular
polarizations for symplectic embeddings has been observed earlier, and they
were used in several works [6, 7, 25, 26]. But the study of their skeleta was
left aside in these works. The question of rigidity of Lagrangian skeleta
associated to singular polarizations is more subtle than for smooth
polarizations. Indeed, the complement of the skeleton of a singular
polarization is not ruled anymore, at if its degree is $k$ the Gromov width
may not be $\frac{1}{k}$ but may be arbitrarily large [26, Theorem 1]. The
present work opens a way to identify singular polarizations for which the
complement of the skeleton has small width. It may also be worth noting that
some very natural Lagrangian CW-complexes are the skeleta of singular
polarizations and not of smooth ones. For instance, the Clifford torus in
$\operatorname{\mathbb{C}P}^{2}$ is the skeleton of a polarization by three
lines, but cannot be the skeleton of a smooth polarization, see $\S$ 2.2. And
even if the skeleton of a smooth polarization is computable (like
$\Delta_{2k}$ for $\Sigma_{2k}$), it may be much easier to obtain it from a
singular polarization, see e.g. $\S$ 3.2.

Around Theorems 4 and 5. Traditionally, Lagrangian intersection results were
proven for smooth closed monotone Lagrangians. For non-smooth Lagrangians, the
Floer machinery does not work directly (see however [13]), and for non-
monotone Lagrangians it is more difficult [14]. Our Lagrangian skeleta
$\Gamma_{\\!\leq a}\times\Gamma_{\\!\leq b}$ may be smooth or not, are not
closed, and they may be very far from monotone.

Theorem 4 says that even rather small Lagrangians must intersect
$\Gamma_{\\!\leq a}\times\Gamma_{\\!\leq b}$. Such Lagrangian rigidity at a
small scale has been observed recently in [19, 28]. In these papers, the role
of our $\Delta$ is played by Lagrangian submanifolds of $S^{2}\times S^{2}$
that are products of one circle in the first factor with a collection of
circles in the second factor which, in contrast to our grids, decompose
$S^{2}$ into components that may have arbitrary topology. Another difference
is that these Lagrangian submanifolds are secretly monotone (when lifted to a
symmetric product), whereas our skeleta do not need to be monotone in any
sense: Already each disc $D(A)$ or $D(B)$ may contain several discs bounded by
$\Gamma_{\\!\leq a}$ or $\Gamma_{\\!\leq b}$ whose areas are completely
independent. Theorem 4 therefore suggests that rigidity at a small scale is
not intrinsically related to monotonicity.

Singular Lagrangian were first studied in [12, 13], where it is shown that the
product of the 1-skeleton of a fine enough triangulation of the 2-sphere is
super-heavy and hence must intersect any heavy Lagrangian. Recall that heavy
sets are non-displaceable. Our intersection condition on the minimal action is
thus very often much weaker.

Theorem 6 versus the main result of [18]. A different positive answer to
Question 1.1 was given by Haim-Kislev, Hind, and Ostrover afterwards in [18].
They showed that for every $a$ one can remove from the ball $B^{4}(a)$ a
finite number (depending on $a$) of parallel symplectic planes such that the
complement symplectically embeds into $Z^{4}(1)$. Their embedding has very
different properties from ours: The removed set is symplectic instead of
Lagrangian, and the construction is rigid in the sense that one cannot alter
the position of the planes, while our grids are rather arbitrary in view of
Theorem 6. The key difference is that in contrast to theirs, our embeddings
are exact, a property that we crucially need for deriving Theorems 4 and 5.

Organisation of the paper. The paper is organized as follows. In Section 2 we
recall the results from [24, 25] on symplectic polarisations in dimension
four, introduce Liouville polarizations and explain their main relations with
symplectic embeddings. In Section 3 we compute some explicit pairs
polarizations/skeleta. Combining these tools, we prove our embedding results
in Section 4, deduce the Lagrangian rigidities in Section 5 and the Legendrian
ones in Section 6.

Notation and conventions. Since the paper is long, we list here some of our
terminology and conventions, to which the reader may return when necessary.

  * $\bullet$

A symplectic embedding
$\varphi\colon(M,d\alpha)\to(M^{\prime},d\alpha^{\prime})$ between exact
symplectic manifolds is $(\alpha,\alpha^{\prime})$-exact if
$\varphi^{*}\alpha^{\prime}=\alpha+df$ for a smooth function $f$ on $M$.
Equivalently, $\int_{\gamma}\alpha=\int_{\varphi(\gamma)}\alpha^{\prime}$ for
all closed curves $\gamma$ in $M$. It is enough to check this equality of
actions on a set of closed curves that generate $H_{1}(M;\mathbb{R})$. If
$H^{1}(M;\mathbb{R})=0$, then $(\alpha,\alpha^{\prime}$)-exactness is
automatic; and otherwise, the notion of $(\alpha,\alpha^{\prime}$)-exactness
may depend on the choice of primitives $\alpha,\alpha^{\prime}$. We sometimes
write $\lhook\joinrel\xrightarrow{\;\\!\alpha_{{\operatorname{st}}}\,}$ to
abbreviate “there exists an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding”.

  * $\bullet$

A Liouville form on a symplectic manifold $(M,\omega)$ is just a primitive
$\lambda$ of the symplectic form: $d\lambda=\omega$. The associated Liouville
vector field $X_{\lambda}$ is defined by
$\iota_{X_{\lambda}}d\lambda=\lambda$. Its flow, the Liouville flow, is
conformally symplectic:
$\left(\phi_{X_{\lambda}}^{t}\right)^{*}\omega=e^{t}\,\omega$.

  * $\bullet$

By a symplectic curve in a symplectic 4-manifold we mean a 2-dimensional
embedded symplectic submanifold, which is closed if the ambient symplectic
manifold is, or properly embedded if it is open. A normal crossing between
symplectic curves $\Sigma,\Sigma^{\prime}$ is an intersection point
$p\in\Sigma\cap\Sigma^{\prime}$ such that $T_{p}\Sigma$ and
$T_{p}\Sigma^{\prime}$ are $\omega$-orthogonal.

  * $\bullet$

A symplectic multi-curve with normal crossings $\Sigma$ is a union of
symplectic curves $\Sigma_{i}$ (called the components of $\Sigma$) whose
pairwise intersections are all normal crossings. The singularities of a multi-
curve is the set of intersection points between components and is denoted
$\operatorname{Sing}(\Sigma)$. The complement of the singular locus is called
the regular part of $\Sigma$.

  * $\bullet$

Usual objects of differential geometry are generalized to multi-curves by just
taking a collection of such objects on each component. For instance, a
differential $k$-form $\alpha$ on $\Sigma=\cup\Sigma_{i}$ is a collection of
differential $k$-forms on the $\Sigma_{i}$. The morphisms between multi-curves
$\Sigma,\Sigma^{\prime}$ are smooth maps from $\Sigma$ to $\Sigma^{\prime}$
(in the sense that they are smooth when restricted to each components), and
send the regular part and singular locus of $\Sigma$ to those of
$\Sigma^{\prime}$. A symplectic embedding between multi-curves is an injective
such morphism that pulls back $\omega^{\prime}|_{\Sigma^{\prime}}$ to
$\omega|_{\Sigma}$.

  * $\bullet$

A weighted symplectic multi-curve ${\bf\Sigma}:=\\{(\Sigma_{i},\mu_{i})\\}$
with normal crossings is a collection of symplectic curves $\Sigma_{i}$
weighted by real numbers $\mu_{i}$, whose total space $\Sigma:=\cup\Sigma_{i}$
is a symplectic multi-curve with normal crossings.

  * $\bullet$

A symplectic morphism from ${\bf\Sigma}$ to ${\bf\Sigma}^{\prime}$ is a
symplectic embedding of $\Sigma$ into $\Sigma^{\prime}$ that sends each
component to a component of the same weight and sends the regular part and
singular locus of $\Sigma$ to those of $\Sigma^{\prime}$. We write
${\bf\Sigma}\to{\bf\Sigma}^{\prime}$.

  * $\bullet$

Symplectic multi-curves will also be called symplectic divisors, or simply
divisors.

  * $\bullet$

Given a subset $S$ of a manifold $M$, we write $\operatorname{Op}(S,M)$ or
just $\operatorname{Op}(S)$ instead of “some open neighbourhood of $S$ in
$M$”.

Acknowledgment. EO thanks Kai Cieliebak for an interesting discussion during a
visit to Strasbourg Unversity. FS is grateful to Viktor Ginzburg, Basak Gürel,
Marco Mazzucchelli, and Alfonso Sorrentino for organizing the workshop
Symplectic Dynamics at Rome in May 2023, where some of the results of this
work were presented, and to Umut Varolgunes for many interesting discussions
and a great week in June 2023 at Istanbul. FS also cordially thanks Yasha
Eliashberg and the Department of Mathematics at Stanford University for their
wonderful hospitality during the spring term 2024.

##  2\. Polarizations of symplectic manifolds

In this section we review those results from [3, 24, 25, 26] that we need in
this paper, in a form useful for us. For simplicity, we stick to dimension
$4$, although most of the statements can be adapted to higher dimensions.

###  2.1. Neighborhoods of symplectic curves in dimension $4$

Let $\pi\colon{\mathcal{L}}\to(\Sigma,\tau)$ be a complex line bundle with
first Chern class $c_{1}$ over a symplectic curve of area
${\mathcal{A}}=\int_{\Sigma}\tau$. The multiplication by $e^{2\pi i\theta}$
defines a vector field $\frac{\partial}{\partial\theta}$ which, together with
the complex structure on the fiber, provides a closed $1$-form $d\theta$ on
the punctured fibers $F\setminus\\{0\\}$. We now in addition endow
${\mathcal{L}}$ with a Hermitian metric. This provides a radial coordinate
$r$, and throughout the paper we write $R:=r^{2}$. Chern–Weil theory
guarantees the existence of a connection $1$-form with curvature
$\frac{c_{1}}{{\mathcal{A}}}\,\tau$. This is a $1$-form $\Theta$ on
${\mathcal{L}}\setminus\Sigma$ that satisfies

$\Theta|_{F\setminus\\{0\\}}=d\theta\,\mbox{ for each fiber $F$ }\quad\mbox{
and }\quad d\Theta=-\frac{c_{1}}{{\mathcal{A}}}\,\pi^{*}\tau\,\mbox{ on
}{\mathcal{L}}.$

The $2$-form

(2.1)
$\omega_{0}\,:=\,\pi^{*}\tau+d(R\Theta)\,=\,\left(1-\frac{c_{1}R}{{\mathcal{A}}}\right)\pi^{*}\tau+dR\wedge\Theta$

is defined on all of ${\mathcal{L}}$, is closed, and is symplectic on
$\bigl{\\{}R<\frac{{\mathcal{A}}}{c_{1}}\bigr{\\}}$. Furthermore, $\omega_{0}$
is exact on ${\mathcal{L}}\setminus\Sigma$ (where we identify ${\bf
0}_{\mathcal{L}}$ with $\Sigma$), with Liouville form

(2.2) $\lambda_{0}:=\left(R-\frac{{\mathcal{A}}}{c_{1}}\right)\Theta.$

The associated Liouville vector field is

(2.3) $\left(R-\frac{{\mathcal{A}}}{c_{1}}\right)\frac{\partial}{\partial R}.$

The 1-form $d\theta$ is defined on the punctured fibers $F\setminus\\{0\\}$,
and in a local trivialization of ${\mathcal{L}}$ over $U\subset\Sigma$ it is
defined on ${\mathcal{L}}|_{U}\setminus U$, but not on all of
${\mathcal{L}}\setminus\Sigma$. A transition function between two local
trivializations is of the form $(z,w)\mapsto(z,g(z)\;\\!w)$, where $g$ is a
smooth $\mathbb{C}\setminus\\{0\\}$-valued function, and $d\theta$ is pulled
back by this map to $d\theta+\operatorname{im}\frac{dg}{g}$. We can therefore
define the angular class $\\{d\theta\\}$ as the set of $1$-forms on
${\mathcal{L}}\setminus\Sigma$ that locally are equal to $d\theta$ up to a
smooth 1-form on ${\mathcal{L}}$. The form $\Theta$ is a representative of
$\\{d\theta\\}$. Recall that we have fixed a Hermitian metric on
${\mathcal{L}}$.

######  Definition 2.1.

The _symplectic disc bundle_ over $(\Sigma,\tau)$ with Chern class $c_{1}$ is

$\textnormal{\small
SDB}(\Sigma,\tau,c_{1})\,:=\,\left(\left\\{R<\frac{{\mathcal{A}}}{c_{1}}\right\\},\omega_{0},\lambda_{0}\right)\,\subset\,{\mathcal{L}}.$

It comes with a distinguished class of $1$-forms on $\textnormal{\small
SDB}(\Sigma,\tau,c_{1})\setminus\Sigma$ modulo smooth forms on
$\textnormal{\small SDB}(\Sigma,\tau,c_{1})$, called the angular class. Any
connection $1$-form provides a representative of the angular class. For
$\varepsilon\leq\frac{{\mathcal{A}}}{c_{1}}$, we denote

$\textnormal{\small
SDB}_{\varepsilon}(\Sigma,\tau,c_{1})\,:=\,\big{(}\\{R<\varepsilon\\},\omega_{0},\lambda_{0}\big{)}\,\subset\,{\mathcal{L}}.$

By the symplectic neighbourhood theorem, for any symplectic curve
$\Sigma\subset(M^{4},\omega)$ and for $\varepsilon>0$ sufficiently small, an
open neighbourhood of $\Sigma$ is symplectomorphic to

$\textnormal{\small SDB}_{\varepsilon}(\Sigma,M)\,:=\,\textnormal{\small
SDB}_{\varepsilon}\bigl{(}\Sigma,\omega|_{\Sigma},c_{1}(T\Sigma^{\omega})\bigr{)}$

by a symplectomorphism that lifts any given symplectomorphism between the zero
section of $\textnormal{\small SDB}_{\varepsilon}(\Sigma,M)$ and
$\Sigma\subset M$.

###  2.2. Polarizations (closed case)

In [3], Biran introduced the notion of polarization for closed symplectic
manifolds with rational symplectic class. This definition was later extended
in [25] as follows.

######  Definition 2.2.

A _polarization_ ${\bf\Sigma}:=\\{(\Sigma_{i},\mu_{i})\\}$ of a closed
$4$-dimensional symplectic manifold $(M,\omega)$ is a finite collection of
closed 2-dimensional symplectic submanifolds $\Sigma_{i}$ that intersect
$\omega$-orthogonally, weighted by real coefficients $\mu_{i}>0$, such that

(2.4)
$[{\bf\Sigma}]\,:=\,\sum_{i}\mu_{i}\;\\![\Sigma_{i}]\,=\,\operatorname{PD}([\omega])\,\in\,H_{2}(M;\mathbb{R}).$

In other terms, a polarization is a weighted symplectic divisor with normal
crossings Poincaré-dual to $[\omega]$ in the sense of (2.4). We write
$\Sigma:=\bigcup_{i}\Sigma_{i}$. We say that the polarization is _smooth_ if
it consists of a single component. In this case, $\mu$ is determined by
$\Sigma$ and $[\omega]$, and we say that $\Sigma$ has degree $d=1/\mu$.

Let $(M,\omega,{\bf\Sigma})$ be a polarized closed symplectic manifold. The
symplectic form $\omega$ is exact on the complement of $\Sigma$, hence
$\omega$ has primitives on $M\setminus\Sigma$. Throughout the paper, we
consider only Liouville forms that satisfy a regularity assumption near
$\Sigma$, that we discuss now.

When ${\bf\Sigma}$ is smooth, a neighbourhood can be modeled on

$\textnormal{\small SDB}_{\varepsilon}(\Sigma,M)\,:=\,\textnormal{\small
SDB}_{\varepsilon}\big{(}\Sigma,\omega|_{\Sigma},c_{1}(T\Sigma^{\omega})\big{)}$

for $\varepsilon$ small enough, and is therefore endowed with a radial
coordinate $R$ and a connection $1$-form $\Theta$, as in Section 2.1. When
${\bf\Sigma}$ consists of several pairwise $\omega$-orthogonal components
$\Sigma_{i}$, we follow [25, Section 3.1] and make a neighbourhood of $\Sigma$
a symplectic plumbing of the $\varepsilon$-disc bundles over the $\Sigma_{i}$,
which we still denote $\textnormal{\small SDB}_{\varepsilon}(\Sigma,M)$: a
deleted neighbourhood of each $\Sigma_{i}$ can be endowed with local
coordinates $(R_{i},\theta_{i})$, $d\theta_{i}$ belonging to the angular class
around $\Sigma_{i}$, such that on a neighbourhood of an intersection point of
$\Sigma_{i}$ and $\Sigma_{j}$ we have $\Sigma_{i}=\\{R_{j}=0\\}$,
$\Sigma_{j}=\\{R_{i}=0\\}$ and $\omega=d(R_{i}d\theta_{i}+R_{j}d\theta_{j})$.

######  Definition 2.3.

A $1$-form $\lambda$ on $M\setminus\Sigma$ is tame

  * $\bullet$

at a regular point $p\in\Sigma_{i}\setminus\bigcup_{j\neq i}\Sigma_{j}$, if
there exists a real number $a_{i}$ and a bounded $1$-form $\lambda^{\prime}$
on $\textnormal{Op}\,(p)\setminus\Sigma_{i}$ such that
$\lambda=a_{i}\,d\theta_{i}+\lambda^{\prime}$ on
$\textnormal{Op}\,(p)\setminus\Sigma_{i}$ ($\lambda^{\prime}$ is then smooth
on $\textnormal{Op}\,(p)\backslash\Sigma_{i}$). In other terms, $\lambda$
locally represents a multiple of a connection form modulo bounded forms.

  * $\bullet$

at an intersection point $p\in\Sigma_{i}\cap\Sigma_{j}$, if there exist real
numbers $a_{i},a_{j}$ such that
$\lambda=(R_{i}+a_{i})\,d\theta_{i}+(R_{j}+a_{j})\,d\theta_{j}$ on
$\textnormal{Op}\,(p)\setminus(\Sigma_{i}\cup\Sigma_{j})$ in the plumbing
coordinates.

  * $\bullet$

along $\Sigma$, if it is tame at each point of $\Sigma$.

######  Remarks 2.4.

(i) The class of tame forms around $\Sigma$ depends on the choice of the
Hermitian metrics on the ${\mathcal{L}}_{i}$, and on the identification of our
neighbourhood of $\Sigma$ in $M$ with the plumbing of the $\varepsilon$-disc
bundles over $\Sigma_{i}$. Given a polarization, we fix these choices, whether
implicitely or after a construction. The notion of tameness will always refer
to these choices.

(ii) Liouville forms in the complement of a normal crossing divisor
$\Sigma\subset M$ have been used in many works, for instance in [20, 5]. In
these works, the main interest was the symplectic homology of
$M\setminus\Sigma$, whence it sufficed to understand that the Liouville flow
is pointing towards $\Sigma$. Variants of the notion of tameness, that is
crucial here, were introduced in [26]. $\diamond$

It is not hard to check that the number $a_{i}$ depends only on the component
$\Sigma_{i}$ and not on the point $p$ on this component (see [26, Lemma 4.1]).
We call $a_{i}$ the residue of $\lambda$ at $\Sigma_{i}$ and denote it
$\textnormal{Res}\,(\lambda,\Sigma_{i})$. By [25, Lemma 3.2] plumbings of
symplectic disc bundles have tame Liouville forms with arbitrary residues.
When, moreover, the homological polarizing condition (2.4) holds for
$\mu_{i}=-\textnormal{Res}\,(\lambda,\Sigma_{i})$, then these tame Liouville
forms extend to Liouville forms on $M\setminus\Sigma$, see the proof of [26,
Lemma 4.1 (iii)]. For further reference we state:

######  Proposition 2.5.

Let $(M,\omega,{\bf\Sigma})$ be a polarized closed symplectic manifold with
${{\bf\Sigma}=\\{(\Sigma_{i},\mu_{i})\\}}$. Then

  * (i)

There exists a tame Liouville form $\lambda$ on $M\setminus\Sigma$ with
residues $\textnormal{Res}\,(\lambda,\Sigma_{i})=-{\mu_{i}}$.

  * (ii)

Let $U$ be an open subset of $M$. Two Liouville forms on $U$ tame along
$\Sigma$ with the same residues differ on $U\backslash\Sigma$ by a bounded
smooth closed 1-form $\vartheta$.

  * (iii)

Let $\lambda$ be a tame Liouville form on $M\backslash\Sigma$. For every small
enough $\varepsilon>0$ the vector field $X_{\lambda}$ is pointing inwards
along the boundary of $\operatorname{SDB}_{\varepsilon}(\Sigma,M)$, and for
every point $p\in\operatorname{SDB}_{\varepsilon}(\Sigma,M)\setminus\Sigma$
the flow line of $X_{\lambda}$ starting at $p$ hits $\Sigma$ in finite
positive time. The Liouville field $X_{\lambda}$ is therefore backward
complete, meaning that any trajectory is defined on $\mathbb{R}_{\leq 0}$.

Proof. (i) has been discussed above and (ii) holds by definition of tameness.
We prove (iii). Let $\lambda$ be a Liouville form tame along $\Sigma$. Near a
regular point of $\Sigma_{i}$, $\lambda$ is then equal to
$-\mu_{i}\>\\!\Theta_{i}$ modulo a bounded one-form on
$M\backslash\Sigma_{i}$, and we have seen that $\Theta_{i}$ is $\omega$-dual
to $\frac{\partial}{\partial R_{i}}$, so

$X_{\lambda}\,=\,-\mu_{i}\,\frac{\partial}{\partial R_{i}}+Z$

for a bounded vector field $Z$. Since $\frac{\partial}{\partial R_{i}}$ has
norm of order $\frac{1}{r_{i}}$ and since $\mu_{i}>0$, $X_{\lambda}\cdot
R_{i}<0$ near $\Sigma_{i}=\\{R_{i}=0\\}$. A similar argument applies near a
singular point. $\square$

######  Definition 2.6 (Biran decomposition).

Given a polarized closed symplectic manifold $(M,\omega,{\bf\Sigma})$ and a
tame Liouville form $\lambda$ on $M\setminus\Sigma$, we denote

$\displaystyle{\mathcal{B}}({\bf\Sigma},\lambda,M)$ $\displaystyle:=$
$\displaystyle\Sigma\cup\\{p\in M\mid\mbox{$\exists$ $t(p)>0$,\;
}\lim_{t\nearrow t(p)}\phi^{t}_{X_{\lambda}}(p)\in\Sigma\\},$
$\displaystyle\textnormal{Skel}({\bf\Sigma},\lambda,M)$ $\displaystyle:=$
$\displaystyle M\setminus{\mathcal{B}}(M,{\bf\Sigma},\lambda).$

More generally, for any subset $X\subset\Sigma$, we define

${\mathcal{B}}(X,\lambda,M):=X\cup\\{p\in M\mid\mbox{$\exists$ $t(p)>0$,\;
}\lim_{t\nearrow t(p)}\phi^{t}_{X_{\lambda}}(p)\in X\\}.$

There is a natural continuous projection map
$\pi\colon{\mathcal{B}}(X,\lambda,M)\to X$ that associates to $x$ the point on
$X$ to which its flow line arrives.

By (iii) of Proposition 2.5, the set ${\mathcal{B}}({\bf\Sigma},\lambda,M)$ is
open. It is the basin of attraction of $\Sigma$ under the Liouville flow of
$\lambda$. The closed set $\textnormal{Skel}({\bf\Sigma},\lambda,M)$ is the
maximal compact subset of $M\setminus\Sigma$ that is invariant under the
Liouville flow. It is called the skeleton of $(M,\omega)$ with respect to the
polarization ${\bf\Sigma}$ and the Liouville form $\lambda$. Finally, point
(iii) also guarantees that
$\overline{{\mathcal{B}}(X,\lambda,M)}\cap\Sigma=\overline{X}$.

The following examples should be useful for readers unfamiliar with these
notions.

######  Example 2.7.

If $(\Sigma,\mu)$ is a smooth polarization of $(M,\omega)$, then the basin
${\mathcal{B}}({\bf\Sigma},\lambda,M)$ is symplectomorphic to the symplectic
disc bundle $\textnormal{\small SDB}(\Sigma,M)=\textnormal{\small
SDB}_{\mu}\bigl{(}\Sigma,\omega|_{\Sigma},c_{1}(T\Sigma^{\omega})\bigr{)}$. It
therefore depends only very mildly on $M$ or $\lambda$ from a symplectic
perspective.

######  Examples 2.8.

As before, endow the complex projective plane $\operatorname{\mathbb{C}P}^{2}$
with the Fubini–Study symplectic form $\omega_{\operatorname{FS}}$ that
integrates to $1$ over every projective line. Every smooth algebraic curve in
$\operatorname{\mathbb{C}P}^{2}$ is symplectic and (since
$H_{2}(\operatorname{\mathbb{C}P}^{2})$ has rank $1$) is a polarization when
weighted by the inverse of its algebraic degree. For instance:

  * $\bullet$

When $\Sigma$ is a line, then
$(\operatorname{\mathbb{C}P}^{2}\setminus\Sigma,\omega_{\operatorname{FS}})$
is symplectomorphic to the ball $(B^{4}(1),\omega_{{\operatorname{st}}})$.
Using the usual toric description of
$(\operatorname{\mathbb{C}P}^{2},\omega_{{\operatorname{FS}}})$, one checks
that the Liouville form
$\lambda_{0}=\alpha_{{\operatorname{st}}}=R_{1}\,d\theta_{1}+R_{2}\,d\theta_{2}$
defined on $B^{4}(1)\cong\operatorname{\mathbb{C}P}^{2}\backslash\Sigma$ is
tame along $\Sigma$. The skeleton of $\lambda_{0}$ is a single point, namely
the origin of $B^{4}(1)$.

  * $\bullet$

When
$\Sigma=\Sigma_{k}:=\\{z_{0}^{k}+z_{1}^{k}+z_{2}^{k}=0\\}\subset\operatorname{\mathbb{C}P}^{2}$,
then $\Sigma$ is the vanishing locus of a holomorphic section $s_{k}$ of the
Hermitian line bundle ${\mathcal{O}}(k)\to\operatorname{\mathbb{C}P}^{2}$ of
curvature $-k\;\\!\omega_{{\operatorname{FS}}}$. The form
$-\frac{1}{k}d^{c}\log\|s_{k}\|$, defined on
$\operatorname{\mathbb{C}P}^{2}\backslash\Sigma$, provides a tame Liouville
form with residue $-\frac{1}{k}$. For $k$ even, its skeleton is the set
$\Delta_{k}(\operatorname{\mathbb{C}P}^{2})$ described in Example 1.5, and for
all $k$ the intersection of the skeleton with
$B^{4}(1)=\operatorname{\mathbb{C}P}^{2}\setminus\operatorname{\mathbb{C}P}^{1}$
is $\Delta_{k}$.

For these examples in $\operatorname{\mathbb{C}P}^{2}$, notice the discrepancy
between the “algebraic degree” of a curve and its degree as a polarizartion
(the “symplectic degree”). These two degrees coincide when the Fubini–Study
form is normalized so that the area of a line is $1$, but they differ for
different normalizations. For instance, if $\operatorname{\mathbb{C}P}^{2}$ is
obtained as the compactification of $B^{4}(2)$, so that the symplectic form
becomes $2\,\omega_{\operatorname{FS}}$, a curve of algebraic degree $2$ is a
polarization of degree $1$.

######  Examples 2.9.

  * $\bullet$

$\left(\operatorname{\mathbb{C}P}^{1}\times\operatorname{\mathbb{C}P}^{1},a\;\\!\omega_{\operatorname{FS}}\oplus
b\;\\!\omega_{\operatorname{FS}}\right)$ is polarized by

${\bf\Sigma_{1}}\,:=\,\Bigl{\\{}\left(\operatorname{\mathbb{C}P}^{1}\times\\{0\\},b\right),\left(\\{0\\}\times\operatorname{\mathbb{C}P}^{1},a\right)\Bigr{\\}}.$

For $\lambda=(R_{1}-a)\,d\theta_{1}+(R_{2}-b)\,d\theta_{2}$ the skeleton is
the single point $\\{(\infty,\infty)\\}$.

  * $\bullet$

$\left(\operatorname{\mathbb{C}P}^{1}\times\operatorname{\mathbb{C}P}^{1},a\;\\!\omega_{\operatorname{FS}}\oplus
b\;\\!\omega_{\operatorname{FS}}\right)$ is polarized by ${\bf\Sigma_{2}}:=$

$\Bigl{\\{}\left(\operatorname{\mathbb{C}P}^{1}\times\\{0\\},\tfrac{b}{2}\right),\left(\\{0\\}\times\operatorname{\mathbb{C}P}^{1},\tfrac{a}{2}\right),\left(\operatorname{\mathbb{C}P}^{1}\times\\{\infty\\},\tfrac{b}{2}\right),\left(\\{\infty\\}\times\operatorname{\mathbb{C}P}^{1},\tfrac{a}{2}\right)\Bigr{\\}}.$

For a suitable choice of $\lambda$ the skeleton is the Clifford torus, i.e.
the product of the two equators.

  * $\bullet$

$\left(\operatorname{\mathbb{C}P}^{2},\omega_{\operatorname{FS}}\right)$ is
polarized by

${\bf\Sigma}\,:=\,\Bigl{\\{}\left(\\{z_{0}=0\\},\tfrac{1}{3}\right),\left(\\{z_{1}=0\\},\tfrac{1}{3}\right),\left(\\{z_{2}=0\\},\tfrac{1}{3}\right)\Bigr{\\}}.$

For a suitable choice of $\lambda$ the skeleton is the Clifford torus.

We end this section by keeping a promise made in the introduction:

######  Proposition 2.10.

There exists no smooth polarization of $\operatorname{\mathbb{C}P}^{2}$ whose
skeleton is the Clifford torus.

Proof. Assume that the Clifford torus $L$ is the skeleton of a smooth
polarization of $\operatorname{\mathbb{C}P}^{2}$ of degree $k$. By [2], the
Gromov width of the complement is
$c_{{\operatorname{G}}}(\operatorname{\mathbb{C}P}^{2}\setminus
L)\leq\frac{1}{k}$. On the other hand,
$\operatorname{\mathbb{C}P}^{2}\setminus L$ is symplectomorphic to the
standard symplectic disc bundle
$\operatorname{SDB}_{\frac{1}{k}}(\Sigma,\tau,k)$ over a closed surface of
area $k$ with disc fibres of area $\frac{1}{k}$. By [24, Proposition 1.3] this
disc bundle contains a symplectic ellipsoid $E(k,\frac{1}{k})$ and hence a
ball $B^{4}(\frac{1}{k})$. It follows that
$c_{{\operatorname{G}}}(\operatorname{\mathbb{C}P}^{2}\setminus
L)=\frac{1}{k}$. This is in contradiction to the fact that
$c_{{\operatorname{G}}}(\operatorname{\mathbb{C}P}^{2}\setminus
L)=\frac{2}{3}$, proven in [4, p. 2887]. $\square$

###  2.3. Liouville polarizations (open case)

Let $(\Omega,\omega)$ be a connected symplectic manifold (without boundary)
and with an exact symplectic form $\omega$. Let $\Sigma\subset\Omega$ be a
symplectic divisor with normal crossings. This now means that each of the
finitely many components $\Sigma_{i}$ is a properly embedded symplectic
surface and that the singular locus $\operatorname{Sing}(\Sigma)$ is compact.

Tameness of a 1-form $\lambda$ on $\Omega\setminus\Sigma$ along $\Sigma$ is
defined as in the closed case (Definition 2.3), in terms of Hermitian line
bundles ${\mathcal{L}}_{i}\to\Sigma_{i}$ modeled on the symplectic normal
bundles, and an identification of a neighbourhood of $\Sigma$ in $\Omega$ with
the plumbing of disc bundles over the $\Sigma_{i}$.

######  Definition 2.11.

A _Liouville polarization_ $({\bf\Sigma},\lambda)$ of $(\Omega,\omega)$
consists of ${\bf\Sigma}=\\{(\Sigma_{i},\mu_{i})\\}$, where
$\Sigma=\cup_{i}\Sigma_{i}\subset\Omega$ is a symplectic divisor as above and
$\mu_{i}>0$, together with a Liouville form $\lambda$ on
$\Omega\setminus\Sigma$ that is tame along $\Sigma$, with residues $-\mu_{i}$
at $\Sigma_{i}$, and such that the Liouville flow $\phi_{X_{\lambda}}^{t}$ on
$\Omega\setminus\Sigma$ is backward complete and “forward complete up to
hitting $\Sigma$”: For every point $p\in\Omega\setminus\Sigma$, the flow line
starting at $p$ is defined for $t\in(-\infty,t^{+}(p))$, where either
$t^{+}(p)=+\infty$ or $\lim_{t\nearrow
t^{+}(p)}\phi_{X_{\lambda}}^{t}(p)\in\Sigma$.

The points $p$ with $t^{+}(p)<\infty$ form the basin of attraction
${\mathcal{B}}({\bf\Sigma},\lambda,\Omega)$. The complement of the basin of
attraction of $\Sigma$ under the Liouville flow is the _skeleton_ of
$(\Omega,\omega,{\bf\Sigma},\lambda)$. This is the set of points whose
Liouville flow line is defined for all times. If ${\bf\Sigma}$ has only one
component $\Sigma$, we call the polarization _smooth_ , and in analogy with
the closed case say that it has _degree $d=1/\mu$_.

######  Remarks 2.12.

(i) In the context of closed manifolds, we have called polarization the sole
weighted divisor ${\bf\Sigma}$, which could be easily separated from the
Liouville form. In the context of open manifolds, the situation is more
intricate because the homological condition
$[{\bf\Sigma}]=\operatorname{PD}([\omega])$ does not make sense anymore. This
condition is replaced by the residue and integrability conditions on $\lambda$
and its Liouville flow, making impossible to separate ${\bf\Sigma}$ from
$\lambda$ in the above definition.

(ii) Whether or not an exact symplectic manifold $(\Omega,\omega)$ has a
Liouville polarization does not seem to be easy to decide, in general. We
think that if $\Omega$ is the interior of a compact manifold with smooth
boudary, this only holds when $(\Omega,\omega)$ is the interior of a very
special kind of Liouville domain $(\overline{\Omega},d\beta)$. Note that for
Liouville domains, the Liouville vector field $X_{\beta}$ is transverse to the
boundary, pointing outwards. This is in sharp contrast to our Liouville vector
fields $X_{\lambda}$, that in view of the integrability condition on
$X_{\lambda}$ are “tangent at infinity”.

(iii) The reader may wonder why we insist on assuming $\Omega$ open, and
propose a definition that may look a bit artificial. Indeed, if $\Omega$ were
assumed compact with smooth boundary, we could as well define our Liouville
polarization in terms of a Liouville form on $\Omega\backslash\Sigma$ whose
Liouville vector field is tangent to $\partial\Omega$. This makes a perfectly
sound definition in this case, that would however discard important examples
that we want to consider: open manifolds whose boundaries have corners, like
the bidisc and, more generally, symplectic disc bundles over open surfaces as
in Example 2.14 below, and $\mathbb{R}^{4}$, which is not compact.

######  Definition 2.13.

A Liouville polarization $({\bf\Sigma},\lambda)$ of $(\Omega,\omega)$ is
extendable if there exists an exact symplectic manifold
$(\widehat{\Omega},\widehat{\omega})$ with $\Omega\subset\widehat{\Omega}$ and
$\widehat{\omega}|_{\Omega}=\omega$, a symplectic divisor
$\widehat{\Sigma}\subset\widehat{\Omega}$ with $\Sigma\subset\widehat{\Sigma}$
and such that the closure $\overline{\Sigma}$ of $\Sigma$ in
$\widehat{\Sigma}$ is a compact surface with boundary and $\widehat{\Sigma}$
is a collar neighbourhood of $\overline{\Sigma}$ and a tame Liouville form
$\widehat{\lambda}$ on $\widehat{\Omega}\backslash\widehat{\Sigma}$ with
$\widehat{\lambda}|_{\widehat{\Omega}\setminus\widehat{\Sigma}}=\lambda|_{\Omega\setminus\Sigma}$
such that the boundary $\partial\Omega$ of $\Omega$ in $\widehat{\Omega}$ is a
smooth manifold to which $\widehat{\Sigma}$ is transverse (cf. Figure 2.1).
For an extendable polarization $({\bf\Sigma},\lambda)$, we denote
$\overline{\Sigma}$ the closure of $\Sigma$ in $\widehat{\Sigma}$. $\diamond$

The completeness assumption on $X_{\lambda}$ implies that
$X_{\widehat{\lambda}}$ is tangent to the boundary of $\Omega$ in
$\widehat{\Omega}$.

Figure 2.1. An extension $(\widehat{\Omega},\widehat{\Sigma})$ of
$(\Omega,\Sigma)$.

In [27] we give a large class of examples of extendable Liouville
polarizations. For the moment, we look at the following model example.

######  Example 2.14.

Let $(X,\tau)$ be a compact symplectic surface with smooth non-empty boundary,
and let $\mu>0$. Consider an extension $(\Sigma,\tau)$ of $(X,\tau)$ to a
closed symplectic surface, whose area ${\mathcal{A}}_{\tau}(\Sigma)$ is an
integral multiple $c_{1}$ of $\mu$. Let $\pi\colon\bigl{(}\textnormal{\small
SDB}(\Sigma,\tau,c_{1}),\lambda_{0}\bigr{)}\to\Sigma$ be the symplectic disc
bundle defined in Section 2.1, together with its Liouville form of residue
$-\mu=-\frac{{\mathcal{A}}_{\tau}(\Sigma)}{c_{1}}$. Then $X_{\lambda_{0}}$
vanishes on the boundary of $\textnormal{\small SDB}(\Sigma,\tau,c_{1})$ by
(2.3) and is tangent to the fibers. Hence
$\bigl{(}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\mu),\lambda_{0}\bigr{)}$
is a Liouville polarization of

$\textnormal{\small
SDB}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu):=\pi^{-1}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X})\;\subset\;\textnormal{\small
SDB}(\Sigma,\tau,c_{1}).$

The basin of attraction is the whole manifold, and the skeleton is empty.
Taking an open collar neighbourhood of $X\subset\Sigma$ we see that this
Liouville polarization is extendable.

From the differentiable view point, $\textnormal{\small
SDB}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu)$
is just
$\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X}\times\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip
2.0mu}{\mathbb{D}}$ with a twisted symplectic form, with the particularity of
having a Liouville vector field tangent to the fibers and vanishing at their
boundaries. $\diamond$

The above construction does not rule out the possibility that different
extensions $(\Sigma,\tau,c_{1})$ of $(X,\tau)$ lead to different symplectic
manifolds $\textnormal{\small
SDB}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu)$.
This is not so. While this fact is not crucial for us, its proof is a good
warm-up for the main result of this section (Theorem 2.20), and it leads to
two lemmas needed later on.

######  Lemma 2.15.

Let $(X,\tau)$ be a compact symplectic surface with boundary, and let
$(\widehat{X},\widehat{\tau})$ be an open collar neighbourhood of $X$. Let $W$
be a neighbourhood of $\widehat{X}$ in
$\operatorname{SDB}(\widehat{X},\hat{\tau},\mu)$, and let $\vartheta$ be a
bounded closed 1-form on $W\setminus\widehat{X}$. Let $\lambda$ be a Liouville
1-form for $\operatorname{SDB}(\widehat{X},\hat{\tau},\mu)$ as constructed in
Example 2.14.

(i) There exist neighbourhoods $V,V^{\prime}$ of $X$ in $W$ and a
$C^{1}$-smooth symplectomorphism $f\colon(V,\omega)\to(V^{\prime},\omega)$
such that $f|_{X}=\operatorname{id}$ and $f^{*}(\lambda+\vartheta)=\lambda$ on
$V\setminus\widehat{X}$.

(ii) If, moreover, $\vartheta$ vanishes near a union $\partial^{\prime}X$ of
components of the boundary $\partial X$, then we can choose $V,V^{\prime}$ and
$f$ such that $f=\operatorname{id}$ on a neighbourhood of
$\pi^{-1}(\partial^{\prime}X)\cap V$.

Figure 2.2. The symplectomorphism $f\colon V\to V^{\prime}$ with
$f^{*}(\lambda+\vartheta)=\lambda$.

Proof. This is proven in [24, Lemma A.1] in the case that $\widehat{X}$ is a
closed surface. The same proof yields Lemma 2.15. $\square$

######  Proposition 2.16.

The symplectic manifold $\textnormal{\small
SDB}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu)$
does not depend on the choice of the extension $(\Sigma,\tau,c_{1})$.

Proof. For $i=0,1$, let $\bigl{(}\textnormal{\small
SDB}_{i}(X,\tau,\mu),\lambda_{i}\bigr{)}$ be defined as above by the inclusion
$\iota_{i}\colon(X,\tau)\to(\Sigma_{i},\tau_{i})$. We need to show that
$\textnormal{\small
SDB}_{0}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu)$
is symplectomorphic to $\textnormal{\small
SDB}_{1}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu)$.
Let $\widehat{X}_{i}\subset\Sigma_{i}$ be open collar neighbourhoods of
$X\subset\Sigma_{i}$. We can assume that there exists a symplectomorphism
$\varphi\colon(\widehat{X}_{0},\tau_{0})\to(\widehat{X}_{1},\tau_{1})$ that is
the identity on $X$. By the symplectic neighbourhood theorem, there exist open
neighbourhoods $U_{i}$ of $\widehat{X}_{i}$ in
$\operatorname{SDB}(\widehat{X}_{i},\tau_{i},\mu)$ and a symplectic
diffeomorphism $\psi\colon U_{0}\to U_{1}$ that extends $\varphi$.

Figure 2.3. The extension $\psi\colon U_{0}\to U_{1}$ of
$\varphi\colon\widehat{X}_{0}\to\widehat{X}_{1}$.

By Proposition 2.5 (ii) there exists a bounded smooth closed one-form
$\vartheta$ on $\operatorname{im}\psi\backslash\widehat{X}_{1}$ such that
$\psi_{*}\lambda_{0}-\lambda_{1}=\vartheta$. Let $\widetilde{X}_{1}$ be a
closed collar neighbourhood of $X$ that lies in $\widehat{X}_{1}$. Applying
Lemma 2.15 (i) with
$(X,\widehat{X},W):=(\widetilde{X}_{1},\widehat{X}_{1},\operatorname{im}\psi)$,
we obtain open neighbourhoods $V,V^{\prime}$ of $X$ in $\operatorname{im}\psi$
and a $C^{1}$-smooth symplectomorphism $f\colon V\to V^{\prime}$ that equals
the identity on $\widehat{X}_{1}$ and takes $\lambda_{1}+\vartheta$ to
$\lambda_{1}$. Choose $\varepsilon>0$ so small that
$\psi\bigl{(}\textnormal{\small
SDB}_{0,\varepsilon}(X,\tau,\mu)\bigr{)}\subset\textnormal{Dom}\,f$. Then the
map

$\varphi:=f\circ\psi\colon\textnormal{\small
SDB}_{0,\varepsilon}(X,\tau,\mu)\,\hookrightarrow\,\textnormal{\small
SDB}(\Sigma_{1},\tau_{1},c_{1}^{1})$

is well-defined, covers $\iota_{1}\circ\iota_{0}^{-1}$, and verifies
$\varphi_{*}\lambda_{0}=\lambda_{1}$. Define

$\begin{array}[]{rcccl}\Phi&:&\textnormal{\small
SDB}_{0}(X,\tau,\mu)&\longrightarrow&\textnormal{\small
SDB}_{1}(X,\tau,\mu)\\\
&&x&\longmapsto&\phi_{X_{\lambda_{1}}}^{-t}\circ\varphi\circ\phi_{X_{\lambda_{0}}}^{t}(x),\end{array}$

where for $x\in X$ we take $t=0$ and for $x\notin X$ we take any $t\geq 0$
such that $\phi_{X_{\lambda_{0}}}^{t}(x)\in\text{Dom}(\varphi)\setminus X$.
Then $\Phi(x)$ is well-defined because $\varphi_{*}\lambda_{0}=\lambda_{1}$,
and a $C^{1}$-smooth symplectomorphism (see e.g. [24, Section 2.1] if needed).
By restriction, we obtain a $C^{1}$-smooth symplectomorphism
$\textnormal{\small
SDB}_{0}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu)\to\textnormal{\small
SDB}_{1}(\overset{\raisebox{-0.86108pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-3.0mu}{X},\tau,\mu)$.
The claim in the $C^{\infty}$-category now follows from the Smoothing Lemma
2.21 relegated to the end of this section. $\square$

Almost the same proof yields the following statement.

######  Lemma 2.17.

Let $((\Sigma,\mu),\lambda)$ be a smooth and extendable Liouville polarization
of $(\Omega,\omega)$. Then there exists $\varepsilon>0$ and a $C^{1}$-smooth
symplectic embedding

$\Phi\colon\operatorname{SDB}_{\varepsilon}(\Sigma,\omega|_{\Sigma},\mu)\,\to\,{\mathcal{B}}(\Sigma,\lambda,\Omega)$

that is onto a neighbourhood of $\Sigma$ in
${\mathcal{B}}(\Sigma,\lambda,\Omega)$ and is such that
$\Phi|_{\Sigma}=\operatorname{id}_{\Sigma}$ and $\Phi^{*}\lambda=\lambda_{0}$.

Proof. The previous proof shows the existence of a symplectic embedding

$\Phi\colon\textnormal{\small
SDB}_{\varepsilon}(\Sigma,\omega|_{\Sigma},\mu)\,\to\,{\mathcal{B}}(\widehat{\Sigma},\hat{\lambda},\widehat{\Omega}),$

where $(\widehat{\Sigma},\hat{\lambda},\widehat{\Omega})$ is an extension of
the Liouville polarization $(\Sigma,\lambda)$, with $\Phi|_{\Sigma}=\mbox{id}$
and $\Phi^{*}\widehat{\lambda}=\lambda_{0}$. The map $\Phi$ is constructed by
a dynamical conjugacy procedure that guarantees that
$\operatorname{im}\Phi\subset{\mathcal{B}}(\Sigma,\widehat{\lambda},\widehat{\Omega})$.
Since the Liouville flow associated to $\widehat{\lambda}$ is tangent to
$\partial\Omega$ in a neighbourhood of $\widehat{\Sigma}\cap\partial\Omega$,
this basin of attraction lies in $\Omega$ and coincides with
${\mathcal{B}}(\Sigma,\lambda,\Omega)$. $\square$

Putting Lemmas 2.15 and 2.17 together, we obtain the following lemma.

######  Lemma 2.18.

Let $({\bf\Sigma},\lambda)$ be an extendable polarization of an exact
symplectic manifold $\Omega$. Let
$(\widehat{\Omega},\widehat{\Sigma},\widehat{\lambda})$ be an extension and
let $\vartheta$ be a bounded closed $1$-form on a deleted neighbourhood of
$\widehat{\Sigma}$ in $\widehat{\Omega}\setminus\widehat{\Sigma}$ that
vanishes near $\operatorname{Sing}(\Sigma)$. Then there exists a
$C^{1}$-smooth symplectomorphism $\Phi$ between neighbourhoods of
$\overline{\Sigma}$ in $\widehat{\Omega}$ such that:

  * $\bullet$

$\Phi$ is the identity on $\overline{\Sigma}$,

  * $\bullet$

$\Phi^{*}(\widehat{\lambda}+\vartheta)=\widehat{\lambda}$.

Proof. For each $i$ choose small open discs $\coprod_{j}D_{i}^{j}$ around the
points in $\Sigma_{i}\cap\operatorname{Sing}(\Sigma)$ whose closures are
disjoint from the support of $\vartheta$, and take the compact surface
$X_{i}:=\widetilde{\Sigma}_{i}\setminus\coprod_{j}D_{i}^{j}$, where
$\widetilde{\Sigma}_{i}$ is a closed collar neighbourhood of
$\overline{\Sigma}_{i}$ in $\widehat{\Sigma}_{i}$. Then $X:=\coprod_{i}X_{i}$
is a disjoint union, and so is the basin
${\mathcal{B}}(X,\widehat{\lambda})=\sqcup_{i}{\mathcal{B}}(X_{i},\widehat{\lambda})$.

Figure 2.4.
${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}X}}\subset\widehat{\Sigma}$
and $\operatorname{supp}\vartheta$.

By Lemma 2.17 we can identify a neighbourhood of $X_{i}$ in
${\mathcal{B}}(X_{i},\widehat{\lambda})$ with
$\operatorname{SDB}_{\varepsilon_{i}}(X_{i},\widehat{\lambda})$. By Lemma 2.15
(ii) there exists a $C^{1}$-smooth symplectomorphism $\Phi_{i}$ between
neighbourhoods of $X_{i}$ in $\widehat{\Omega}$ such that
$\Phi_{i}^{*}(\widehat{\lambda}+\vartheta)=\widehat{\lambda}$ and
$\Phi_{i}=\mbox{id}$ on $X_{i}$ and near the boundaries of the discs
$D_{i}^{j}$. Extend $\Phi_{i}$ by the identity to an open neighbourhood of all
of $\widetilde{\Sigma}_{i}$. Then the map $\Phi$ obtained by gluing together
the $\Phi_{i}$ is as required. $\square$

Our main interest in the definition of Liouville polarizations is the next
result, that allows to use the techniques set up to study the symplectic
embedding problems in [24, 25] in the context of exact symplectic manifolds.

######  Definition 2.19.

A symplectic morphism $\phi\colon{\bf\Sigma}\to{\bf\Sigma}^{\prime}$ between
weighted symplectic divisors ${\bf\Sigma}$, ${\bf\Sigma}^{\prime}$ is an
injective continuous map $\phi\colon\Sigma\to\Sigma^{\prime}$ with the
following two properties:

  * $\bullet$

For every component $(\Sigma_{i},\mu_{i})$ of ${\bf\Sigma}$ there exists a
component $(\Sigma_{i^{\prime}}^{\prime},\mu_{i^{\prime}})$ of
${\bf\Sigma}^{\prime}$ with $\mu_{i}=\mu_{i^{\prime}}$ such that
$\phi|_{\Sigma_{i}}\colon\Sigma_{i}\to\Sigma_{i^{\prime}}^{\prime}$ is a
symplectic embedding.

  * $\bullet$

$\phi|_{\Sigma_{i}}\colon\Sigma_{i}\to\Sigma_{i^{\prime}}^{\prime}$ takes
smooth points to smooth points, i.e.,

$\phi\bigl{(}\Sigma_{i}\setminus\bigcup_{j\neq
i}\Sigma_{j}\bigr{)}\,\subset\,\Sigma_{i^{\prime}}^{\prime}\setminus\bigcup_{j^{\prime}\neq
i^{\prime}}\Sigma_{j^{\prime}}^{\prime}.$

Recall that an embedding $\varphi$ is $(\alpha,\alpha^{\prime})$-exact if
$\varphi^{*}\alpha^{\prime}=\alpha+df$ for a smooth function $f$.

######  Theorem 2.20.

Let $(\Omega,\omega=d\alpha)$ and
$(\Omega^{\prime},\omega^{\prime}=d\alpha^{\prime})$ be exact symplectic
manifolds with Liouville polarizations $({\bf\Sigma},\lambda)$ and
$({\bf\Sigma}^{\prime},\lambda^{\prime})$, respectively. Assume that
$({\bf\Sigma},\lambda)$ is extendable, with extension
$(\widehat{\bf\Sigma},\widehat{\lambda})$. Assume further that there exists a
symplectic morphism $\phi\colon\widehat{\bf\Sigma}\to{\bf\Sigma}^{\prime}$
that is $(\alpha|_{\Sigma},\alpha^{\prime}|_{\Sigma^{\prime}}$)-exact. Then
there exists an $(\alpha,\alpha^{\prime})$-exact symplectic embedding

$\Phi\colon{\mathcal{B}}({\bf\Sigma},\lambda)=\Omega\setminus\textnormal{Skel}(\Omega,{\bf\Sigma},\lambda)\,\to\,{\mathcal{B}}({\bf\Sigma}^{\prime},\lambda^{\prime})=\Omega^{\prime}\setminus\textnormal{Skel}(\Omega^{\prime},{\bf\Sigma}^{\prime},\lambda^{\prime}).$

Proof. Let $(\widehat{\Omega},\widehat{\Sigma},\widehat{\lambda})$ be the
extension of $(\Omega,\Sigma,\lambda)$. Let $\widetilde{\Sigma}$ be a closed
collar neighbourhood of $\overline{\Sigma}$ in $\widehat{\Sigma}$. By the
symplectic neighbourhood theorem, there exists an open neighbourhood $V$ of
$\widetilde{\Sigma}$ in $\widehat{\Omega}$ and a symplectic extension
$\psi\colon
V\hookrightarrow{\mathcal{B}}({\bf\Sigma}^{\prime},\lambda^{\prime})\subset\Omega^{\prime}$
of $\phi$ that preserves the symplectic plumbing structure (i.e., the Darboux
coordinates $(R_{1},\theta_{1},R_{2},\theta_{2})$ near the intersections).
Since $\psi^{*}\lambda^{\prime}$ and $\widehat{\lambda}$ are tame and have the
same residues,

$\psi^{*}\lambda^{\prime}-\widehat{\lambda}=\vartheta$

where $\vartheta$ is a smooth bounded closed $1$-form on
$V\backslash\widehat{\Sigma}$. Since $\psi$ preserves the plumbing structure,
we also have $\vartheta=0$ on $\textnormal{Op}\,(\operatorname{Sing}\Sigma)$.

By Lemma 2.18 there exists a $C^{1}$-smooth symplectomorphism $\Phi_{1}$
between open neighbourhoods of $\overline{\Sigma}$ in
${\mathcal{B}}(\widetilde{\bf\Sigma},\hat{\lambda})$ such that
$\Phi_{1}^{*}\widehat{\lambda}=\widehat{\lambda}+\vartheta$ and
$\Phi_{1}|_{\widetilde{\Sigma}}=\mbox{id}$. Then the map
$\Phi:=\psi\circ\Phi_{1}^{-1}$ provides a symplectic extension of $\phi$ such
that $\Phi^{*}\lambda^{\prime}=\widehat{\lambda}$. As in the proof of
Proposition 2.16 define

$\begin{array}[]{rcccl}\Psi_{1}&:&{\mathcal{B}}({\bf\Sigma},\lambda)&\longrightarrow&{\mathcal{B}}({\bf\Sigma}^{\prime},\lambda^{\prime})\\\
&&x&\longmapsto&\phi_{X_{\lambda^{\prime}}}^{-t}\circ\Phi\circ\phi_{X_{\lambda}}^{t}(x),\end{array}$

where for $x\in\Sigma$ we take $t=0$ and for $x\notin\Sigma$ we take any
$t\geq 0$ such that
$\phi_{X_{\lambda}}^{t}(x)\in\text{Dom}(\Phi)\setminus\Sigma$. Then $\Psi_{1}$
is a $C^{1}$-smooth symplectic embedding. It is
$(\alpha,\alpha^{\prime})$-exact. Indeed, since
${\mathcal{B}}({\bf\Sigma},\lambda)$ retracts to $\Sigma$, its first homology
is generated by cycles in $\Sigma$, on which $\Psi_{1}=\phi$. Since $\phi$ is
$(\alpha|_{\Sigma},\alpha^{\prime}|_{\Sigma^{\prime}})$-exact by assumption,
$\Psi_{1}$ is $(\alpha,\alpha^{\prime})$-exact. The claim in the smooth
category now follows from Lemma 2.21. $\square$

######  Lemma 2.21.

Let $\Psi_{1}\colon(M,\omega)\to(M^{\prime},\omega^{\prime})$ be a
$C^{1}$-smooth symplectomorphism of symplectic manifolds without boundary.
Then there exists a $C^{\infty}$-smooth symplectomorphism
$\Psi\colon(M,\omega)\to(M^{\prime},\omega^{\prime})$. Furthermore, if
$\Psi_{1}$ is $(\alpha,\alpha^{\prime})$-exact, then $\Psi$ can be taken
$(\alpha,\alpha^{\prime})$-exact.

Proof. It is shown in [32, pp. 831–836] that there exists a
$C^{\infty}$-smooth symplectomorphism
$\Psi\colon(M,\omega)\to(M^{\prime},\omega^{\prime})$ which is $C^{1}$-close
to $\Psi_{1}$. The proof shows that $\Psi$ is $(\alpha,\alpha^{\prime})$-exact
whenever $\Psi_{1}$ is $(\alpha,\alpha^{\prime})$-exact: Take Darboux charts
$\varphi_{i}\colon B^{4}(a_{i})\to U_{i}\subset M$ such that
$\varphi_{i}\bigl{(}B^{4}(a_{i}/2)\bigr{)}=:U_{i}^{\prime}$ form a locally
finite covering of $M$. Smoothing a suitable generating function, one replaces
$\Psi_{1}$ by $\Psi_{1}^{1}$ that is $C^{\infty}$-smooth on $U_{1}^{\prime}$
and agrees with $\Psi_{1}$ outside $U_{1}$. In fact, convex interpolation of
the two generating functions yields a smooth symplectic isotopy $\Psi_{1}^{t}$
from $\Psi_{1}$ to $\Psi_{1}^{1}$ such that $\Psi^{-1}_{1}\circ\Psi_{1}^{t}$
is supported in $U_{1}$. Since $U_{1}=\varphi_{1}(B^{4}(a_{1}))$ is simply-
connected, this isotopy is Hamiltonian, and hence
$\Psi^{-1}_{1}\circ\Psi_{1}^{1}$ is $(\alpha,\alpha)$-exact, see e.g. [21,
Proposition 9.3.1]. Since $\Psi_{1}$ is $(\alpha,\alpha^{\prime})$-exact, we
find that $\Psi^{1}_{1}$ is also $(\alpha,\alpha^{\prime})$-exact. In the same
way, smoothen $\Psi_{1}^{1}$ to $\Psi_{1}^{2}$ on $U_{2}^{\prime}$, $\dots$,
$\Psi_{1}^{k}$ to $\Psi_{1}^{k+1}$ on $U_{k+1}^{\prime}$, $\dots$. The limit
map $\Psi$ is then $(\alpha,\alpha^{\prime})$-exact. $\square$

##  3\. Some explicit polarizations and skeleta

We start our computation of explicit polarizations with the simplest case, in
dimension $2$. This case is then used in $\S$ 3.2 to construct Liouville
polarizations on bidiscs. In $\S$ 3.3 we show how certain singular Liouville
polarizations can be surgered to smooth Liouville polarizations.

###  3.1. Polarizations in dimension $2$

Let $(S,\omega)$ be a 2-dimensional compact symplectic surface with non-empty
boundary $\partial S$. We define a Liouville polarization of $S$ to be a
finite set of points $p_{i}$ in $\operatorname{Int}S$ with weights $\mu_{i}>0$
together with a tame Liouville form $\lambda$ defined on
$S\setminus\bigcup_{i}p_{i}$ such that $X_{\lambda}$ is tangent to $\partial
S$. Tameness now means that for each $i$ there are symplectic polar
coordinates $(R,\theta)$ near $p_{i}$ such that
$\lambda=-\mu_{i}d\theta+\lambda^{\prime}$ on
$\operatorname{Op}(p_{i})\setminus\\{p_{i}\\}$, where $\lambda^{\prime}$ is a
bounded 1-form on $\operatorname{Op}(p_{i})\setminus\\{p_{i}\\}$. One easily
checks that $\sum_{i}\mu_{i}={\mathcal{A}}_{\omega}(S)$. Conversely, any
finite set of positively weighted points $\\{(p_{i},\mu_{i})\\}$ on
$(S,\omega)$ that satisfies $\sum_{i}\mu_{i}={\mathcal{A}}_{\omega}(S)$ admits
a Liouville polarization, as is easy to see. Because of this very direct link
between the residues and the area in dimension $2$, we switch notation from
$\mu_{i}$ to $a_{i}$ throughout this section. Its main purpose is to explain
that in dimension $2$, the skeleton can be prescribed, which seems much harder
in higher dimensions. We recall the notion of regular grid in a slightly more
general setting.

######  Definition 3.1.

A regular grid $\Gamma$ on a compact surface $S$ with boundary is a finite
connected graph $\Gamma\subset S$ with smooth edges, whose edges cover
$\partial S$ and such that each vertex has a Darboux chart on which $\Gamma$
is a union of radial rays that cut the disc into equal sectors. When the
vertex belongs to the interior of $S$, the Darboux chart takes values in
$D(\varepsilon)\subset\mathbb{C}$, while it takes values in
$D(\varepsilon)\cap\\{\operatorname{im}z\geq 0\\}\subset\mathbb{H}$ if the
vertex belongs to $\partial S$.

######  Proposition 3.2.

Let $(S,\omega)$ be a compact surface with boundary, and let $\Gamma$ be a
regular grid that decomposes $S$ into $m$ discs of area $a_{i}$, in each of
which we chose a point $p_{i}$. Then there exists a Liouville polarization
$\bigl{(}\\{(p_{i},a_{i})\\},\lambda\bigr{)}$ of $S$ whose skeleton is
$\Gamma$. This Liouville form can be required to coincide with
$(R-a_{i})d\theta$ in Darboux coordinates near each $p_{i}$.

In the case of a closed disc $\overline{D}(A)$ (which will be the example we
will use later on) one may attempt to prove the proposition as follows: Just
define a first tame Liouville form with residues $-a_{i}$ at $p_{i}$ by
$\lambda:=\alpha_{{\operatorname{st}}}-\sum a_{i}\,d\theta_{i}$, where
$\theta_{i}$ is the pull-back by the translation of the vector $-p_{i}$ of the
standard angular coordinate in $\mathbb{C}\setminus\\{0\\}$. Then the periods
of $\lambda$ vanish on each loop of $\Gamma$, so we can correct $\lambda$ by
adding the differential of a smooth function in order to make $\lambda$ vanish
on $\Gamma$. Then the Liouville flow of $\lambda$ fixes $\Gamma$ pointwise,
and so $\bigl{(}\\{(p_{i},a_{i})\\},\lambda\bigr{)}$ provides a Liouville
polarization of $\overline{D}(A)$ whose skeleton contains $\Gamma$.
Unfortunately, this proof does not constrain the skeleton enough: it could be
strictly bigger than $\Gamma$. We therefore proceed in a different manner.
Recall that $X_{\lambda}$ is tangent to $\ker\lambda$, since
$\lambda(X_{\lambda})=d\lambda(X_{\lambda},X_{\lambda})=0$. Our strategy is
therefore to first construct a (singular) foliation ${\mathcal{F}}$ to which
we wish $X_{\lambda}$ to be tangent to. Once this “vanishing foliation” is
constructed, the following lemma will readily provide the Liouville form
itself.

######  Lemma 3.3.

Let $\gamma_{\theta}\colon[0,1]\to\mathbb{C}$, $\theta\in[0,\alpha]$, be a
family of smooth rays emanating from the origin of $\mathbb{C}$, and set
$U:=\left\\{\gamma_{\theta}(t)\mid\theta\in[0,\alpha],\,t\in[0,1]\right\\}$.
Fix $a\in\mathbb{R}$. We assume that

  * $\bullet$

$(t,\theta)\mapsto\gamma_{\theta}(t)$ is smooth and a diffeomorphism except at
$t=0$. Thus the set of curves $\\{\operatorname{im}\gamma_{\theta}\\}$
provides a foliation ${\mathcal{F}}$ of $U\setminus\\{0\\}$.

  * $\bullet$

There exists $\varepsilon>0$ such that $\gamma_{\theta}(t)=t\>\\!e^{i\theta}$
for $t\leq\varepsilon$.

Then there exists a unique smooth Liouville form $\lambda_{a}$ on
$(U\setminus\\{0\\},\omega_{{\operatorname{st}}})$ such that

  * $\bullet$

$\ker\lambda_{a}=T{\mathcal{F}}$,

  * $\bullet$

$\lambda_{a}(\gamma_{\theta}(t))=(R-a)d\theta$ for $t$ sufficiently small.

Proof. This follows at once from Stokes’ theorem. $\square$

Proof of Proposition 3.2: We write the proof for the disc $\overline{D}(A)$.
The adaptation to a general surface $(S,\omega)$ will be clear. We first set
some notation. We write $D$ instead of $\overline{D}(A)$. For each point
$p_{i}$ let ${\mathcal{D}}_{i}$ be the closure in $\mathbb{R}^{2}$ of the
connected component of $p_{i}$ in $D\setminus\Gamma$ (recall that $\Gamma$
contains $\partial D$ by assumption). Write $\operatorname{Sing}(\Gamma)$ for
the set of non-smooth points of $\Gamma$, and

${\mathcal{S}}_{i}:=\operatorname{Sing}(\Gamma)\cap{\mathcal{D}}_{i}$

for the set of singularities of $\Gamma$ in the boundary of
${\mathcal{D}}_{i}$. For each $p_{i}$ and a sufficiently small $\varepsilon$
the translation $(D(p_{i}),p_{i})\to(D(\varepsilon),0)$ is a Darboux chart
that induces coordinates $(R,\theta)$ on $D(p_{i})$. Similarly, by our
assumption on $\Gamma$ we can consider for each
$q\in\operatorname{Sing}(\Gamma)\setminus\partial D$ symplectic coordinates
$\phi_{q}=(R,\theta)\colon D(q)\to D(\varepsilon)$ on a disk around $q$ such
that

$\Gamma\cap
D(q)\,=\,\Bigl{\\{}\theta=\frac{k}{m_{q}},\,k\in[0,m_{q}-1]\Bigr{\\}},$

where $m_{q}$ is the number of branches of $\Gamma$ at $q$. The sectors in
$D(q)$ delimited by the rays $\\{\theta=\frac{k}{m_{q}}\\}$ are denoted by

$S_{i}(q):={\mathcal{D}}_{i}\cap D(q).$

Also, when $q\in\operatorname{Sing}(\Gamma)\cap\partial D$, we have symplectic
coordinates $(R,\theta)$ on a neighbourhood $D^{+}(q)$ of $q$ in $D$, with
values in $\\{R<\varepsilon,\theta\in[0,\pi]\\}$ such that

$\Gamma\cap
D^{+}(q)\,=\,\Bigl{\\{}\theta=\frac{k}{2m_{q}},\,k\in[0,m_{q}-1]\Bigr{\\}},$

where $m_{q}$ is the number of branches of $\Gamma$ at $q$. The sectors
delimited by the rays $\theta=\frac{k}{2m_{q}}$ are again denoted by

$S_{i}(q):={\mathcal{D}}_{i}\cap D^{+}(q).$

The radial ray that cuts the sector $S_{i}(q)$ into two equal sectors is
called a local separatrix at $q$ and is denoted $s_{i}(q)$.

Finally, we fix a symplectic diffeomorphism
$f_{i}\colon{\mathcal{D}}_{i}\setminus{\mathcal{S}}_{i}\to\overline{D}(a_{i})\setminus{\mathcal{P}}_{i}$
where ${\mathcal{P}}_{i}$ is a finite set of points, and set
$\lambda_{i}^{f}:=f_{i}^{*}\alpha_{{\operatorname{st}}}$. Then
$\lambda_{i}^{f}$ is positive on
$\partial{\mathcal{D}}_{i}\setminus{\mathcal{S}}_{i}$. The notations are
illustrated on the right part of Figure 3.1.

Step 1: Constructing the vanishing foliation near the $p_{i}$ and
$\operatorname{Sing}(\Gamma)$ (see Figure 3.1)

In $D(p_{i})$, we fix ${\mathcal{F}}$ to be the radial foliation
$\\{\theta=\operatorname{const}\\}$.

For a point $q\in\operatorname{Sing}(\Gamma)\setminus\partial D$, let
$\rho_{q}\colon D(\varepsilon)\to\mathbb{C}$ be the “ramified covering”

$\rho_{q}(R,\theta)\,:=\,\left(\frac{2R}{m_{q}},\frac{m_{q}}{2}\theta\right).$

Note that $\rho_{q}$ is smooth and symplectic on
$D(\varepsilon)\setminus\\{0\\}$, but only continuous at $0$. The sectors
$S_{i}(q)$ around $q$ are the preimages under $\rho_{q}\circ\phi_{q}$ of
$\\{y\geq 0\\}$ or $\\{y\leq 0\\}$, and the separatrices $s_{i}(q)$ are sent
by $\rho_{q}\circ\phi_{q}$ to $\\{x=0\\}$. We define the foliation in $D(q)$
to be the pull-back by $\rho_{q}\circ\phi_{q}$ of the vertical foliation:

${\mathcal{F}}|_{D(q)}\,:=\,(\rho_{q}\circ\phi_{q})^{*}\\{x=c\\}.$

This local foliation has remarkable properties:

  1. (i)

The local separatrices emanating from $q$ are leaves of ${\mathcal{F}}$; they
are the pull-back by $\rho_{q}\circ\phi_{q}$ of the curves $\\{x=0\\}$.

  2. (ii)

Since $\rho_{q}$ sends concentric circles around the origin to concentric
circles, and since $\rho_{q}\circ\phi_{q}$ is a diffeomorphism between
$S_{i}(q)\setminus\\{q\\}$ and $\\{y\geq 0\\}\setminus\\{0\\}$ or $\\{y\leq
0\\}\setminus\\{0\\}$, the leaves $\\{x=c\\}$ with $c\neq 0$ are pulled back
by $\rho_{q}\circ\phi_{q}$ to smooth curves that intersect $\Gamma$
orthogonally (in the chart $\phi_{q}$).

  3. (iii)

As a result, the foliation ${\mathcal{F}}$ can be extended to a smooth
foliation on $\textnormal{Op}\,(\Gamma)\backslash\operatorname{Sing}(\Gamma)$
that is transverse to the regular part of $\Gamma$.

Finally, for a point $q\in\operatorname{Sing}(\Gamma)\cap\partial D$, the
model is the same. We leave it to the reader to complete the picture, cf.
Figure 3.2.

Figure 3.1. Notation, and the foliation on $D(\varepsilon)$ for $m_{q_{1}}=3$.

Step 2: Interpolating the separatrices (see Figure 3.2)

Fix $i\in\\{1,\dots,m\\}$. (In Figure 3.2, $i=1$.) Fix a reference point
$q_{0}\in{\mathcal{S}}_{i}$, denote $q_{1},\dots,q_{\ell}$ the points of
${\mathcal{S}}_{i}$ enumerated by going around $\partial{\mathcal{D}}_{i}$ in
the anti-clockwise sense from $q_{0}$, and let
$\partial{\mathcal{D}}_{i}(q_{j}\to q_{j+1})$ be the smooth arc in $\Gamma$
joining $q_{j}$ to $q_{j+1}$ on $\partial{\mathcal{D}}_{i}$. Now interpolate
between the local separatrix $s_{i}(q_{0})$ and the ray $\\{\theta=0\\}$ in
$D(p_{i})$ by a smoothly embedded curve. Set

$\beta_{j}\,:=\,\frac{1}{a_{i}}\int_{\partial{\mathcal{D}}_{i}(q_{j}\to
q_{j+1})}\lambda_{i}^{f},\qquad j=0,\dots,|{\mathcal{S}}_{i}|-1,$

and

$\theta_{j}\,:=\,\sum_{k=0}^{j-1}\beta_{k}.$

Interpolate now inductively between the local separatrix $s_{i}(q_{j})$ and
the ray $\\{\theta=\theta_{j}\\}$ by disjoint smoothly embedded curves (for
different $j$) in ${\mathcal{D}}_{i}$ such that the area enclosed by the three
arcs $\partial{\mathcal{D}}_{i}(q_{j}\to q_{j+1})$, $s(p_{i}\to q_{j})$, and
$s(p_{i}\to q_{j+1})$ is $\beta_{j}\>\\!a_{i}$. Here, $s(p_{i}\to q_{j})$,
$j=1,\dots,\ell$, denote the global separatrices thus constructed. These
interpolations can be found because the total area enclosed by
$\partial{\mathcal{D}}_{i}$ is $a_{i}$. Interpolations $s(p_{1}\to q_{1})$ and
$s(p_{1}\to q_{2})$ are drawn in pink in Figure 3.2. We declare the
$s(p_{i}\to q_{j})$ to be leaves of ${\mathcal{F}}$ in ${\mathcal{D}}_{i}$.

Step 3: Interpolating between the local foliations in ${\mathcal{D}}_{i}$

The foliation ${\mathcal{F}}_{i}$ on ${\mathcal{D}}_{i}$ is then taken to be
any (singular) foliation in ${\mathcal{D}}_{i}$ that verifies the following
properties:

  1. (1)

The points $p_{i}$ and $q_{j}\in{\mathcal{S}}_{i}$ are the only singularities
of ${\mathcal{F}}|_{{\mathcal{D}}_{i}}$, so
${\mathcal{F}}|_{{\mathcal{D}}_{i}\setminus\\{p_{i}\\}}$ is smooth.

  2. (2)

The curves $s(p_{i}\to q_{j})$ are leaves.

  3. (3)

On the $D(p_{i})$, $D(q_{j})$, $D^{+}(q_{k})$, and
$\textnormal{Op}\,(\Gamma)\backslash\operatorname{Sing}(\Gamma)$ the leaves of
${\mathcal{F}}$ are the ones described in Step 1.

  4. (4)

All leaves join $p_{i}$ to $\partial{\mathcal{D}}_{i}$ and are transverse to
$\partial{\mathcal{D}}_{i}\setminus{\mathcal{S}}_{i}$.

  5. (5)

For $c\in(\beta_{j},\beta_{j+1})$ the leaf extending the ray
$\\{\theta=c\\}\subset D(p_{i})$ joins $p_{i}$ to the unique point
$q\in\partial{\mathcal{D}}_{i}(q_{j}\to q_{j+1})$ such that

$\int_{\partial{\mathcal{D}}_{i}(q_{j}\to
q_{j+1})}\lambda_{i}^{f}\,=\,c-\beta_{j}.$

Figure 3.2 shows in black seven curves that belong to leaves of
${\mathcal{F}}_{1}$ from $p_{1}$ to $\partial{\mathcal{D}}_{1}(q_{1}\to
q_{2})$. By (3) and (4) we can successively construct foliations
${\mathcal{F}}_{1},{\mathcal{F}}_{2},\dots,{\mathcal{F}}_{m}$ such that all
non-singular leaves are $C^{\infty}$-smooth.

Figure 3.2. The foliation ${\mathcal{F}}$ on the $D(p_{i})$, $D(q_{j})$, and
$D^{+}(q_{k})$ (blue and green), part of the foliation ${\mathcal{F}}_{1}$,
and ${\mathcal{D}}(p_{1},p_{3})$ (yellow).

Step 4: Joining the foliations ${\mathcal{F}}_{i}$

Since ${\mathcal{F}}_{i}$ is a smooth foliation on
${\mathcal{D}}_{i}\setminus\bigl{(}\\{p_{i}\\}\cup{\mathcal{S}}_{i}\bigr{)}$
and by the smooth fitting of the leaves across the
$\partial{\mathcal{D}}_{i}(q_{j}\to q_{j+1})$, the foliation ${\mathcal{F}}$
on
$D\setminus\bigl{(}\cup\\{p_{i}\\}\bigcup\operatorname{Sing}(\Gamma)\bigr{)}$
defined by

${\mathcal{F}}|_{{\mathcal{D}}_{i}\setminus\left(\\{p_{i}\\}\cup{\mathcal{S}}_{i}\right)}\,:=\,{\mathcal{F}}_{i}$

is $C^{\infty}$-smooth, and has a unique extension to a singular foliation on
the whole disc $D$ whose singularities are
$\cup\\{p_{i}\\}\bigcup\operatorname{Sing}(\Gamma)$. The different
separatrices $s(p_{i}\to q)$, $q\in{\mathcal{S}}_{i}$, are smooth leaves. The
leaves of ${\mathcal{F}}$ that pass through a point in
$\Gamma\setminus\operatorname{Sing}(\Gamma)$ that separates between
${\mathcal{D}}_{i}$ and ${\mathcal{D}}_{j}$ join $p_{i}$ to $p_{j}$. Finally,
the leaves through a sequence of regular points on
${\mathcal{D}}_{i}\cap{\mathcal{D}}_{j}$ that converge to a singular point
$q\in\operatorname{Sing}(\Gamma)$ break into the two separatrices $s(p_{i}\to
q)$ and $s(p_{j}\to q)$.

We denote by ${\mathcal{D}}(p_{i},p_{j})$ the closure of the union of leaves
joining $p_{i}$ to $p_{j}$. Figure 3.2 shows ${\mathcal{D}}(p_{1},p_{3})$ in
yellow.

Step 5: Fixing the Liouville form itself

By applying Lemma 3.3 to the curves of the foliation ${\mathcal{F}}$ that
emanate either from $(p_{i},a_{i})$ or from $(p_{j},a_{j})$, we obtain on the
interior
$\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}(p_{i},p_{j})$
two Liouville forms $\lambda_{i},\lambda_{j}$. We claim that they coincide. To
see this, we first check that these forms vanish on $\Gamma$. Indeed, let
$q\in\Gamma\cap\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}(p_{i},p_{j})$.
Since $\lambda_{i}$ vanishes along ${\mathcal{F}}$, we already have
$\lambda_{i}(q)v=0$ for $v\in T_{q}\gamma$, where $\gamma$ is the leaf through
$q$. On the other hand, parametrize the leaves through
$q\in\Gamma\cap\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}(p_{i},p_{j})$
by curves $\gamma(\theta,t)$, where $\theta$ represents the angle of the leaf
at $p_{i}$ and the parameter $t$ verifies $\gamma_{\theta}(1)\in\Gamma$, and
put
$\frac{\partial}{\partial\theta}(q):=\frac{\partial\gamma}{\partial\theta}(\theta(q),1)\in
T_{q}\Gamma$. Since $\lambda_{i}$ vanishes along ${\mathcal{F}}$, and by
Stokes’ theorem and the second property of $\lambda_{i}$ from Lemma 3.3,

(3.1)
$\lambda_{i}(q)\,\frac{\partial}{\partial\theta}(q)\,=\,\frac{d{\mathcal{A}}_{i}}{d\theta}(\theta(q))-a_{i},$

where ${\mathcal{A}}_{i}(\theta)$ is the area of the disc bounded by $\Gamma$,
$\gamma(\theta,[0,1])$, and $\gamma(\theta(q),[0,1])$. By the choice of our
foliation (point $5$ in Step 3), the right hand side of (3.1) vanishes, so
$\lambda_{i}(q)$ vanishes. Now, since $\lambda_{i}$ and $\lambda_{j}$ coincide
on ${\mathcal{F}}$ and on $\Gamma$, Stokes’ theorem shows that they coincide
on
$\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}(p_{i},p_{j})$.
We therefore define

$\lambda|_{\overset{\raisebox{-0.90417pt}[0.0pt][-0.90417pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}(p_{i},p_{j})}\,:=\,\lambda_{i}=\lambda_{j}.$

Since $\lambda_{i}$ is defined and smooth on
$\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}_{i}\setminus\\{p_{i}\\}$,
and not only on
$\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}(p_{i},p_{j})$,
we can glue the $\lambda_{i}$ to the smooth Liouville form $\lambda$ on
$D\setminus\operatorname{Sing}(\Gamma)$. Notice that for $\gamma(\theta,t)$ as
defined above, we have by the same argument

$\lambda(\gamma(\theta,t))\,\frac{\partial}{\partial\theta}<0\qquad\forall\,t<1.$

Hence on
$\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip-2.0mu}{{\mathcal{D}}}_{i}\setminus\\{p_{i}\\}$
we have
$d\lambda(X_{\lambda},\frac{\partial}{\partial\theta})=\lambda(\frac{\partial}{\partial\theta})<0$,
which shows that the coordinate $t$ is (negative) gradient-like for the
Liouville flow $X_{\lambda}$ of $\lambda$, and so
${\mathcal{B}}(p_{i},\lambda)\supset{\mathcal{D}}_{i}$. On the other hand,
$\lambda$ vanishes on $\Gamma$, so $\Gamma$ is invariant under the Liouville
flow of $\lambda$. Summarizing, we have therefore proved that

  * $\bullet$

$\lambda$ is a smooth Liouville form on
$D\setminus\bigl{(}\operatorname{Sing}\Gamma\cup\\{p_{1},\dots,p_{m}\\}\bigr{)}$,

  * $\bullet$

$\lambda$ is tame at each $p_{i}$ with residue $-a_{i}$, in fact
$\lambda=(R-a_{i})d\theta$ near $p_{i}$,

  * $\bullet$

$X_{\lambda}$ vanishes on $\Gamma\backslash\operatorname{Sing}\Gamma$,

  * $\bullet$

every trajectory starting in $D\setminus\Gamma$ converges to some $p_{i}$.

Step 6: Smoothening $\lambda$ at $\operatorname{Sing}(\Gamma)$

We only treat the case of $q\in\operatorname{Sing}(\Gamma)\setminus\partial
D$, the other case being similar. We recall that a neighbourhood $D(q)$ of
$q$, that is identified with $D(\varepsilon)$, covers a neighbourhood of
$0\in\mathbb{C}$ via the map
$\rho_{q}(R,\theta)=(\frac{2R}{m_{q}},\frac{m_{q}}{2}\theta)$, and that in
these coordinates, ${\mathcal{F}}$ is the foliation
$\rho_{q}^{*}\\{x=\text{const}\\}$. This foliation is thus also tangent to the
kernel of the local Liouville form
$\lambda_{q}:=\rho_{q}^{*}(-\tfrac{1}{\pi}\;\\!y\>\\!dx)$. Away from $q$, this
form $\lambda_{q}$ also vanishes on $\rho_{q}^{*}\\{y=0\\}=\Gamma\cap D(q)$.
Away from $q$, both forms $\lambda$ and $\lambda_{q}$ are thus smooth and
vanish along ${\mathcal{F}}$ and on $\Gamma$. Stokes’ theorem therefore
guarantees that they coincide on $D(q)\backslash\\{q\\}$. None of the two
forms smoothly extends to $q$, however, since at $(x,y)\neq(0,0)$,

$\displaystyle\lambda_{q}$ $\displaystyle=$
$\displaystyle\rho_{q}^{*}(-\tfrac{1}{\pi}\;\\!y\>\\!dx)$ $\displaystyle=$
$\displaystyle\rho_{q}^{*}\big{(}\tfrac{1}{2\pi}(x\>\\!dy-y\>\\!dx)-\tfrac{1}{2\pi}(x\>\\!dy+y\>\\!dx)\big{)}$
$\displaystyle=$
$\displaystyle\rho_{q}^{*}\big{(}R\;\\!d\theta-\tfrac{1}{2\pi}d(xy)\big{)}$
$\displaystyle=$
$\displaystyle\rho_{q}^{*}\big{(}R\;\\!d\theta-\tfrac{1}{2\pi}d(R\sin(2\pi\theta)\cos(2\pi\theta))\big{)}$
$\displaystyle=$ $\displaystyle
R\;\\!d\theta-\tfrac{1}{4\pi}\rho_{q}^{*}d(R\sin(4\pi\theta))$
$\displaystyle=$ $\displaystyle R\;\\!d\theta-\tfrac{1}{2\pi
m_{q}}d(R\sin(2\pi m_{q}\theta))$

and the function $R\sin(2\pi m_{q}\theta)$ is not smooth at the origin for
$m_{q}\geq 2$. We thus alter $\lambda_{q}$ in $D(q)$ so as to make it smooth
at $q$.

The function $r^{m_{q}}\sin(2\pi m_{q}\theta)$ is smooth on $\mathbb{R}^{2}$,
since it is the imaginary part of $z\mapsto z^{m_{q}}$. Let
$\chi\colon[0,2\varepsilon]\to\mathbb{R}_{\geq 0}$ be a function that
coincides with $r^{m_{q}}$ near $0$ and with $R$ on
$[\varepsilon,2\varepsilon$], is positive and smooth except at $0$, and meets
$\chi(R)<R$ on $(0,\varepsilon)$. Define the smooth 1-form $\lambda^{\prime}$
on $D$ by $\lambda^{\prime}=\lambda$ on
$D\setminus\cup_{q\in\operatorname{Sing}\Gamma}D(q)$ and

$\lambda^{\prime}\,=\,R\;\\!d\theta-\frac{1}{2\pi
m_{q}}d\bigl{(}\chi(R)\sin(2\pi m_{q}\theta)\bigr{)}\;\text{ on }\,D(q).$

Then $\lambda^{\prime}$ is a smooth Liouville form on
$D\setminus\\{p_{1},\dots,p_{m}\\}$, tame at $p_{i}$ with residue $-a_{i}$. We
claim that the skeleton of $\lambda^{\prime}$ is $\Gamma$, as required. To see
this, notice that $X_{\lambda^{\prime}}=X_{\lambda}$ on
$D\setminus\cup_{q\in\operatorname{Sing}\Gamma}D(q)$, and

$X_{\lambda^{\prime}}=(R-\chi(R)\cos(2\pi
m_{q}\theta))\frac{\partial}{\partial R}+\frac{1}{2\pi
m_{q}}\chi^{\prime}(R)\sin(2\pi m_{q}\theta)\frac{\partial}{\partial\theta}$

on $D(q)$. We first look at the $\frac{\partial}{\partial\theta}$-component of
$X_{\lambda^{\prime}}$. Since

$\Gamma\,=\,\\{\sin(\pi m_{q}\theta)=0\\}\,\subset\,\\{\sin(2\pi
m_{q}\theta)=0\\},$

$X_{\lambda^{\prime}}$ is radial on $\Gamma\cap D(q)$ (hence its flow
preserves $\Gamma$), and the trajectories of the points in
$D(q)\setminus\Gamma$ flow away from $\Gamma$. From the
$\frac{\partial}{\partial R}$-component, and from $\Gamma\supset\\{\cos(2\pi
m_{q}\theta)=1\\}$ and $\chi(R)<R$ on $(0,\varepsilon)$, we infer that the
trajectories of the points in $D(q)\setminus\Gamma$ all leave $D(q)$, and
hence reach the set $\\{\lambda=\lambda^{\prime}\\}\setminus\Gamma$ (since
$\Gamma$ is invariant). It follows that
$\textnormal{Skel}(\lambda^{\prime})=\Gamma$. $\square$

The following easy result will be needed in the next subsection.

######  Lemma 3.4.

Let $(S,\omega)$ be a compact symplectic surface with or without boundary. Let
$\bigl{(}{\mathcal{P}}:=\\{(p_{i},a_{i}),\;i=1,\dots,n\\},\lambda\bigr{)}$
with $p_{i}\in S\setminus\partial S$ be a polarization of $S$ (where $\lambda$
is a tame Liouville form with residues $a_{i}$ at $p_{i}$ and with flow
tangent to $\partial S$), and set
$\Gamma:=\textnormal{Skel}(S,{\mathcal{P}},\lambda)$.

For $k\leq n$ and any collection $\\{(a_{i}^{1},a_{i}^{2}),i=1,\dots,k\\}$ of
pairs of positive real numbers with $a_{i}=a_{i}^{1}+a_{i}^{2}$ there exist:

  * $\bullet$

a collection of open embedded discs $D_{i}$ around $p_{i}$, in any prescribed
neighbourhood of $p_{i}$

  * $\bullet$

for $i\leq k$: points $p_{i}^{1},p_{i}^{2}\in D_{i}$

  * $\bullet$

and a tame Liouville form $\lambda^{\prime}$ with residues $a_{i}^{1}$ at
$p_{i}^{1}$ and $a_{i}^{2}$ at $p_{i}^{2}$ for $i\leq k$ and $a_{i}$ at
$p_{i}$ for $i\in\\{k+1,\dots,n\\}$

such that

${\mathcal{B}}(D_{i},\lambda^{\prime})\,:=\,\left\\{x\in
S\mid\exists\,t>0\mbox{ such that }\phi_{\lambda^{\prime}}^{t}(x)\in
D_{i}\right\\}\,=\,{\mathcal{B}}(p_{i},\lambda).$

Proof. Since $\lambda$ is tame at $p_{i}$, its dual vector field points
towards $p_{i}$ on a neighbourhood of $p_{i}$, so there exist arbitrarily
small disjoint open discs $D_{i}\subset S\setminus(\partial S\cup\Gamma)$
around $p_{i}$ with

${\mathcal{B}}(D_{i},\lambda)={\mathcal{B}}(p_{i},\lambda).$

For $i\leq k$ take any pair of points $p_{i}^{j}\in D_{i}$, $j=1,2$, and a
Hamiltonian diffeomorphisms $\phi_{i}^{j}$ with compact support in $D_{i}$
such that $\phi_{i}^{j}(p_{i}^{j})=p_{i}$. Define

$\lambda^{\prime}(x)\,:=\,\left\\{\begin{array}[]{ll}\lambda(x)&\text{ if
}x\in S\setminus\bigcup_{i=1}^{k}D_{i},\\\\[1.99997pt]
\frac{a_{i}^{1}}{a_{i}}\,\phi_{i}^{1*}\lambda(x)+\frac{a_{i}^{2}}{a_{i}}\,\phi_{i}^{2*}\lambda(x)&\text{
if }x\in D_{i}\mbox{ for some $i\in\\{1,\dots,k\\}$}.\end{array}\right.$

Then $\lambda^{\prime}$ is a Liouville form tame at $p_{i}^{j}$ with residue
$a_{i}^{j}$. Since it coincides with $\lambda$ on $\Gamma\cup\partial S$, its
flow preserves both $\Gamma$ and $\partial S$, so
$\left({\mathcal{P}}^{\prime}:=\left\\{(p_{i}^{j},a_{i}^{j})\right\\},\lambda^{\prime}\right)$
is a Liouville polarization of $S$. Since $\lambda^{\prime}$ coincides with
$\lambda$ outside the discs $D_{i}$,

${\mathcal{B}}(D_{i},\lambda^{\prime})={\mathcal{B}}(D_{i},\lambda)={\mathcal{B}}(p_{i},\lambda).$

The lemma follows. $\square$

###  3.2. Bidiscs

As before we denote by $\overline{D}(A)$ the closed disc of area $A$, and by
$D(A)$ its interior.

######  Lemma 3.5.

Let $\Gamma_{1}\subset D(A)$ and $\Gamma_{2}\subset D(B)$ be two regular grids
containing $\partial D(A),\partial D(B)$ whose complements are a union of
discs of area $a_{i},b_{j}$ on which we choose points $p_{i},q_{j}$,
respectively. Then

${\bf\Sigma}:=\left\\{\bigl{(}{p_{i}}\times
D(B),a_{i}\bigr{)},\bigl{(}D(A)\times{q_{j}},b_{j}\bigr{)}\right\\}$

is an extendable Liouville polarization of $D(A)\times D(B)$, and there is a
tame Liouville form $\lambda$ whose skeleton is $\Gamma_{1}\times\Gamma_{2}$.

Proof. By Proposition 3.2 there exist Liouville forms
$\lambda_{1},\lambda_{2}$ on $D(A),D(B)$ that are tame along the
$p_{i},q_{j}$, with residues $a_{i},b_{j}$ and skeleton
$\Gamma_{1},\Gamma_{2}$, respectively. Then the restriction of the form
$\lambda:=\pi_{1}^{*}\lambda_{1}\times\pi_{2}^{*}\lambda_{2}$ to ${\bf\Sigma}$
is a Liouville form tame along ${\bf\Sigma}$ with the correct residues. Since
the associated Liouville flow is simply the product of the flows in each
factor, it clearly preserves

$\partial\bigl{(}D(A)\times D(B)\bigr{)}\,=\,\bigl{(}\partial
D(A)\bigr{)}\times D(B)\cup D(A)\times\bigl{(}\partial D(B)\bigr{)}.$

Moreover, for $x\notin\Gamma_{1}\times\Gamma_{2}$, we have $\pi_{1}(x)\in
D(A)\setminus\Gamma_{1}$ or $\pi_{2}(x)\in D(B)\setminus\Gamma_{2}$, so the
trajectory of $\pi_{1}(x)$ or $\pi_{2}(x)$ under the Liouville flow of
$\lambda_{1}$ or $\lambda_{2}$ is forward attracted by one of the $p_{i}$ or
$q_{j}$, which shows that the complement of $\Gamma_{1}\times\Gamma_{2}$ in
$D(A)\times D(B)$ is forward attracted by ${\bf\Sigma}$, whence
$\textnormal{Skel}({\bf\Sigma},\lambda)\subset\Gamma_{1}\times\Gamma_{2}$.
Finally, for a point in $\Gamma_{1}\times\Gamma_{2}$, both components remain
in $\Gamma_{1},\Gamma_{2}$, so $\Gamma_{1}\times\Gamma_{2}$ is invariant under
the Liouville flow and
$\textnormal{Skel}({\bf\Sigma},\lambda)\supset\Gamma_{1}\times\Gamma_{2}$. The
extendability is obvious: take $\widehat{\Omega}:=D(A+\varepsilon)\times
D(B+\varepsilon)$ and $\widehat{\Sigma}:=\left\\{p_{i}\times
D(B+\varepsilon),D(A+\varepsilon)\times q_{j}\right\\}$. $\square$

Combining Lemma 3.4 and Lemma 3.5 we obtain:

######  Lemma 3.6.

Let $\bigl{(}\left\\{(p_{i},a_{i})\right\\}_{i=1,\dots,m},\lambda_{A}\bigr{)}$
and $\bigl{(}\left\\{(q_{j},b_{j})\right\\}_{j=1,\dots,n},\lambda_{B}\bigr{)}$
be Liouville polarizations of $D(A)$ and $D(B)$ (so $\sum a_{i}=A$ and $\sum
b_{j}=B$) with skeleton $\Gamma_{A}$ and $\Gamma_{B}$, respectively. Let
$m^{\prime}\leq m$, $n^{\prime}\leq n$, and assume that for $i\leq m^{\prime}$
and $j\leq n^{\prime}$ we are given decompositions

$a_{i}=a_{i}^{1}+a_{i}^{2}\quad\text{and}\quad
b_{j}=b_{j}^{1}+b_{j}^{2},\quad\mbox{ with }\;a_{i}^{\ell},b_{j}^{\ell}>0.$

Then there exist:

  * $\bullet$

open discs $D_{i}^{A}$, $D_{j}^{B}$ for $i\leq m$, $j\leq n$ around $p_{i}$,
$q_{j}$, which lie in any prescribed neighbourhood of $p_{i}$, $q_{j}$

  * $\bullet$

for $i\leq m^{\prime}$, $j\leq n^{\prime}$: points $p_{i}^{1},p_{i}^{2}\in
D_{i}^{A}$ and $q_{j}^{1},q_{j}^{2}\in D_{j}^{B}$

  * $\bullet$

and a tame Liouville form $\lambda^{\prime}$ on the complement of

$\displaystyle\qquad\quad{\bf\Sigma}^{\prime}$ $\displaystyle\\!\\!:=\\!\\!$
$\displaystyle\bigl{\\{}\bigl{(}p_{i}^{\ell}\times
D(B),a_{i}^{\ell}\bigr{)}_{i\leq m^{\prime}}^{\ell=1,2},\,\bigl{(}D(A)\times
q_{j}^{\ell},b_{j}^{\ell}\bigr{)}_{j\leq n^{\prime}}^{\ell=1,2},$
$\displaystyle\phantom{\bigl{\\{}}\bigl{(}p_{i}\times
D(B),a_{i}\bigr{)}_{i>m^{\prime}},\;\bigl{(}D(A)\times
q_{j},b_{j}\bigr{)}_{j>n^{\prime}}\bigr{\\}}$

that makes $({\bf\Sigma}^{\prime},\lambda^{\prime})$ a Liouville polarization
of $D(A)\times D(B)$

such that

${\mathcal{B}}\Bigl{(}\cup_{i}D_{i}^{A}\times D(B)\bigcup\cup_{j}D(A)\times
D_{j}^{B},\lambda^{\prime}\Bigr{)}\,=\,\bigl{(}D(A)\times
D(B)\bigr{)}\setminus(\Gamma_{A}\times\Gamma_{B}).$

Proof. Take small open discs $D_{i}^{A}$ around $p_{i}$, for $i\leq
m^{\prime}$ take points $p_{i}^{1},p_{i}^{2}\in D_{i}^{A}$, and let
$\lambda_{A}^{\prime}$ be the tame Liouville form on

$D(A)\setminus\left(\\{p_{i}^{\ell}\\}_{i\leq
m^{\prime}}^{\ell=1,2}\cup\\{p_{i}\\}_{i>m^{\prime}}\right)$

provided by Lemma 3.4, with residues $a_{i}^{\ell}$ at $p_{i}^{\ell}$ for
$i\leq m^{\prime}$ and $a_{i}$ at $p_{i}$ for $i>m^{\prime}$, and such that

${\mathcal{B}}\bigl{(}\cup_{i}D_{i}^{A},\lambda_{A}^{\prime}\bigr{)}\,=\,D(A)\backslash\Gamma_{A}.$

Construct similarly $\lambda_{B}$ on

$D(B)\setminus\left(\\{q_{j}^{\ell}\\}_{i\leq
n^{\prime}}^{\ell=1,2}\cup\\{q_{j}\\}_{j>n^{\prime}}\right).$

As we have seen in the proof of Lemma 3.5, the 1-form
$\lambda^{\prime}:=\lambda_{A}^{\prime}\oplus\lambda_{B}^{\prime}$ is tame
along ${\bf\Sigma}^{\prime}$, and

${\mathcal{B}}\Bigl{(}\cup_{i}D_{i}^{A}\times D(B)\bigcup\cup_{j}D(A)\times
D_{j}^{B},\lambda^{\prime}\Bigr{)}\,=\,\bigl{(}D(A)\times
D(B)\bigr{)}\setminus(\Gamma_{A}\times\Gamma_{B}).$

The lemma is proved. $\square$

###  3.3. Surgery on Liouville polarizations

The aim of this section is Proposition 3.8 that allows to glue certain
components of a Liouville polarization, to obtain a “smoother” Liouville
polarization. We will need the following rather obvious statement.

######  Lemma 3.7.

Let $\Sigma\subset B^{4}(1)$ be a smooth symplectic curve that near the
boundary of $B^{4}(1)$ agrees with a collection of complex lines
(1-dimensional complex vector spaces). Let $\mu\in\mathbb{R}$. Then there
exists a Liouville form $\lambda$ on $B^{4}(1)\backslash\Sigma$, tame along
$\Sigma$, with residue $\mu$. (We do not claim that
$(B^{4}(1),\Sigma,\lambda)$ is a Liouville polarization.)

Proof. Since $\Sigma$ coincides with a union of complex lines near $\partial
B^{4}(1)$, compactifying $B^{4}(1)$ into $\operatorname{\mathbb{C}P}^{2}(1)$
provides a smooth symplectic curve $\Sigma^{\prime}$ in
$\operatorname{\mathbb{C}P}^{2}$. This is a polarization, so by Proposition
2.5 (i) there exists a Liouville form $\lambda^{\prime}$ on
$\operatorname{\mathbb{C}P}^{2}\backslash\Sigma^{\prime}$ that is tame along
$\Sigma^{\prime}$. Restricting $\lambda^{\prime}$ to
$B^{4}(1)\subset\operatorname{\mathbb{C}P}^{2}(1)$ provides a Liouville form
$\lambda$ on $B^{4}(1)\backslash\Sigma$ tame along $\Sigma$. It has a well-
defined residue which is a constant $\nu$ because $\Sigma$ is assumed to be
smooth. Then the form

$\lambda_{{\operatorname{st}}}+\frac{\mu}{\nu}(\lambda-\lambda_{{\operatorname{st}}})$

is a Liouville form on $B^{4}(1)\backslash\Sigma$, tame along $\Sigma$ because
it coincides with a multiple of $\lambda$ up to a smooth form, with residue
$\mu$. $\square$

######  Proposition 3.8.

Let $(M,\omega,{\bf\Sigma},\lambda)$ be a polarized symplectic manifold,
either closed or Liouville polarized. Assume that the weights
$\mu_{i},\mu_{j}$ of two components $\Sigma_{i},\Sigma_{j}$ of ${\bf\Sigma}$
coincide. Then for any neighbourhood $U$ of $\Sigma_{i}\cap\Sigma_{j}$ there
exist:

  * $\bullet$

A smooth symplectic curve $\Sigma_{ij}$ that coincides with
$\Sigma_{i}\cup\Sigma_{j}$ in the complement of $U$. We then define

${\bf\Sigma}^{\prime}:=\\{(\Sigma_{k},\mu_{k}^{\prime}=\mu_{k})_{k\neq
i,j},(\Sigma_{ij},\mu_{ij}^{\prime}=\mu_{i}=\mu_{j})\\}.$

  * $\bullet$

A Liouville form $\lambda^{\prime}$ on $M\backslash\Sigma^{\prime}$ tame along
$\Sigma^{\prime}$ with residues $\mu_{k}^{\prime}$ that coincides with
$\lambda$ on the complement of $U$ and is such that
$(M,\omega,{\bf\Sigma}^{\prime},\lambda^{\prime})$ is again a (Liouville)
polarized symplectic manifold.

Proof. We first consider a model in $\mathbb{R}^{4}$. Let
$\Sigma_{1}=\\{z_{2}=0\\}$ and $\Sigma_{2}=\\{z_{1}=0\\}$ be the complex
coordinate lines in $\mathbb{C}^{2}$. For $\varepsilon>0$ let
$\chi\colon[0,2\varepsilon]\to[0,1]$ be a smooth function that is $1$ near $0$
and vanishes on $[\varepsilon,2\varepsilon]$. For $\delta>0$ define the smooth
cylinder

$Z_{\delta}:=\\{z_{1}z_{2}=\delta\>\\!\chi(R)\\}.$

Note that $Z_{\delta}=\Sigma_{1}\cup\Sigma_{2}$ outside $B^{4}(\varepsilon)$,
see Figure 3.3. As is well known, $Z_{\delta}$ is symplectic for
$\delta=\delta(\varepsilon)$ small enough.

Figure 3.3. From $\Sigma_{1}\cup\Sigma_{2}$ to $Z_{\delta}$, schematically.

Consider the Liouville form

$\lambda_{0}\,=\,\lambda_{{\operatorname{st}}}-\mu(d\theta_{1}+d\theta_{2})$

on $\mathbb{R}^{4}\setminus(\Sigma_{1}\cup\Sigma_{2})$, tame along
$\Sigma_{1}\cup\Sigma_{2}$ with residue $-\mu$. Since
$Z_{\delta}=\Sigma_{1}\cup\Sigma_{2}$ on $B^{4}(2\varepsilon)\setminus
B^{4}(\varepsilon)$, Lemma 3.7 provides a tame Liouville form
$\lambda_{\delta}$ on $B^{4}(2\varepsilon)\setminus Z_{\delta}$ with residue
$-\mu$. Consider the shell

$V\,=\,\\{R\in(\varepsilon,2\varepsilon)\\}\,=\,B^{4}(2\varepsilon)\setminus\overline{B}{}^{4}(\varepsilon).$

Then $\lambda_{0}$ and $\lambda_{\delta}$ are Liouville forms on $V\setminus
Z_{\delta}$, tame along $Z_{\delta}$, with equal residue $-\mu$. By
Proposition 2.5 (ii) there exist a smooth bounded closed $1$-form $\vartheta$
on $V\backslash Z_{\delta}$ such that

$\lambda_{\delta}=\lambda_{0}+\vartheta.$

Since $H_{1}(V\backslash Z_{\delta})$ is generated by two small loops around
$Z_{\delta}$ on which $\vartheta$ integrates to $0$, $\vartheta$ is the
derivative of a smooth function $f\colon V\backslash Z_{\delta}\to\mathbb{R}$.
Since $\vartheta$ is bounded, $f$ extends to a Lipschitz function on $V$. Let
now $\rho\colon[\varepsilon,2\varepsilon]\to[0,1]$ be a smooth function that
equals $1$ near $\varepsilon$ and vanishes near $2\varepsilon$, and define the
smooth Liouville form $\lambda^{\prime}$ on $B^{4}(2\varepsilon)\setminus
Z_{\delta}$ by

$\left\\{\begin{array}[]{lcl}\lambda^{\prime}|_{\\{R\leq\varepsilon\\}}&:=&\lambda_{\delta}\\\\[3.00003pt]
\lambda^{\prime}|_{\\{\varepsilon\leq R\leq
2\varepsilon\\}}&:=&\lambda_{0}+d(\rho(R)f).\end{array}\right.$

Since $\rho(R)f$ is a Lipschitz function, smooth on $V\setminus Z_{\delta}$,
$\lambda^{\prime}$ is tame on $V\setminus Z_{\delta}$ with residue $-\mu$, and
the same holds on $\\{R\leq\varepsilon\\}$. Moreover,
$\lambda^{\prime}=\lambda_{0}$ near the boundary of $B^{4}(2\varepsilon)$.
This finishes the construction of the interpolating model
$\bigl{(}Z_{\delta}\cap B^{4}(2\varepsilon),\lambda^{\prime}\bigr{)}$.

Now take two components $\Sigma_{i},\Sigma_{j}$ of $({\bf\Sigma},\lambda)$
with equal weights $\mu_{i}=\mu_{j}$ as in the proposition. Since $\Sigma$ has
normal crossings and $\lambda$ is tame along $\Sigma$ by assumption, there
exists for each $p\in\Sigma_{i}\cap\Sigma_{j}$ a Darboux chart $\phi_{p}\colon
B^{4}(3\varepsilon_{p})\to(M,\omega)$ centered at $p$ such that

$\begin{array}[]{l}\phi_{p}^{-1}(\Sigma)=\phi_{p}^{-1}(\Sigma_{i}\cup\Sigma_{j})=\\{z_{1}z_{2}=0\\},\\\\[3.99994pt]
\phi_{p}^{*}\lambda=\lambda_{{\operatorname{st}}}-(\mu_{i}d\theta_{1}+\mu_{j}d\theta_{2})=\lambda_{{\operatorname{st}}}-\mu_{i}(d\theta_{1}+d\theta_{2}).\end{array}$

Now choose the $\varepsilon_{p}$ so small that the Darboux balls
$\phi_{p}(B^{4}(3\varepsilon_{p}))$ are disjoint, and apply to each ball
$B^{4}(2\varepsilon_{p})$ the above model interpolation. When transplanted to
$M$, this yields a Liouville form $\lambda^{\prime}$ on
$M\setminus\Sigma^{\prime}$, tame along $\Sigma^{\prime}$, with correct
residues. Also notice that in the open case the Liouville flow of
$\lambda^{\prime}$ is backward complete, because $\lambda^{\prime}=\lambda$
outside $\bigcup_{p}\phi_{p}(B^{4}(2\varepsilon))$. $\square$

##  4\. Symplectic embeddings

In this section we prove the symplectic embedding results Theorems 1, 2, 3,
and 6. The proofs are all based on the same principle, stated in Theorem 2.20,
which reduces the proofs to finding convenient polarizations. We first prove
Theorem 6. While Theorem 1 is a direct corollary, Theorem 2 deals with an
unbounded domain, whence the proof needs some adjustments. We finally prove
Theorem 3, using explicit smooth polarizations of the ball.

###  4.1. The main embedding result (Theorem 6)

We first prove a special “monotone” case of Theorem 6, and then reduce the
general case to this monotone case.

####  4.1.1. The monotone case

######  Proposition 4.1.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two regular grids in $D(A)$ and $D(B)$,
respectively, that cut $D(A)$ into $m$ topological discs of equal area $a$ and
$D(B)$ into $n$ topological discs of equal area $b$. Then there exists an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding

$\bigl{(}D(A)\times
D(B)\bigr{)}\setminus(\Gamma_{1}\times\Gamma_{2})\,\to\,Z^{4}(a+b).$

Proof. Let ${\mathcal{P}}=\\{p_{1},\dots,p_{m}\\}$,
${\mathcal{Q}}=\\{q_{1},\dots,q_{n}\\}$,

${\bf\Sigma}:=\left\\{\bigl{(}{\mathcal{P}}\times
D(B),a\bigr{)},\bigl{(}D(A)\times{\mathcal{Q}},b\bigr{)}\right\\},$

and let $\lambda$ be the Liouville form on $\bigl{(}D(A)\times
D(B)\bigr{)}\backslash\Sigma$ provided by Lemma 3.5 so that
$({\bf\Sigma},\lambda)$ is an extendable Liouville polarization of $D(A)\times
D(B)$ with skeleton $\Gamma_{1}\times\Gamma_{2}$.

We now construct a Liouville polarization of $\Omega^{\prime}:=D(a+b)\times
D(Kb)$ for $K$ large enough, to be fixed later in the proof. We refer to
Figure 4.1 for an illustration of the construction. Take a regular grid (a
smooth line) in $D(a+b)$ that divides $D(a+b)$ into topological discs of area
$a$ and $b$, and take a point $p_{1}^{\prime}$ in the disc of area $a$ and
$p_{2}^{\prime}$ in the disc of area $b$. For a large integer $K$ (to be
chosen later), take a regular grid in $D(Kb)$ that divides $D(Kb)$ into $K$
topological discs of area $b$, and choose a point $q_{j}^{\prime}$ in each
disc. Lemma 3.5 yields a Liouville polarization

$\left(\Bigl{\\{}\bigl{(}D(a+b)\times\\{q_{j}^{\prime}\\},b\bigr{)},\bigl{(}\\{p_{1}^{\prime}\\}\times
D(Kb),a\bigr{)},\bigl{(}\\{p_{2}^{\prime}\\}\times
D(Kb),b\bigr{)}\Bigr{\\}},\lambda^{\prime}\right)$

of $\Omega^{\prime}=D(a+b)\times D(Kb)$, consisting of $K+2$ discs.

Define now $\Sigma_{1}^{\prime}:=\bigl{\\{}\\{p_{1}^{\prime}\\}\times
D(Kb)\bigr{\\}}$ and let $\Sigma_{2}^{\prime}$ be the symplectic curve
obtained by resolving all the intersection points
$(p_{2}^{\prime},q_{j}^{\prime})$ of the $K+1$ remaining discs. We can assume
that the smoothening takes place in balls around these points that are
disjoint and do not intersect $\partial\Omega^{\prime}$. Since
$\lambda^{\prime}$ has residue $b$ along each component used to define
$\Sigma_{2}^{\prime}$, Proposition 3.8 shows that it can be modified in this
union of balls to a Liouville form $\lambda^{\prime\prime}$, tame along
$\Sigma^{\prime}_{1}\cup\Sigma_{2}^{\prime}$, with residue $a$ along
$\Sigma^{\prime}_{1}$ and $b$ along $\Sigma^{\prime}_{2}$. Since
$\lambda^{\prime\prime}$ coincides with $\lambda^{\prime}$ near
$\partial\Omega^{\prime}$, its Liouville flow is also tangent to
$\partial\Omega^{\prime}$. In other terms,

$\bigl{(}{\bf\Sigma}^{\prime}:=\\{(\Sigma_{1}^{\prime},a),(\Sigma^{\prime}_{2},b)\\},\lambda^{\prime\prime}\bigr{)}$

is a Liouville polarization of $\Omega^{\prime}$. Now $\Sigma_{1}^{\prime}$
has area $Kb$, $\Sigma_{2}^{\prime}$ has area $K(a+2b)$, and
$\\#(\Sigma_{1}^{\prime}\cap\Sigma_{2}^{\prime})=K$. For $K$ large enough we
therefore find $m$ disjoint topological discs $D_{1}^{i}$ of area strictly
larger than $B$ in $\Sigma_{1}^{\prime}$ and $n$ disjoint topological discs
$D_{2}^{j}$ of area strictly larger than $A$ in $\Sigma_{2}^{\prime}$ such
that each $D_{1}^{i}$ intersects each $D_{2}^{j}$ exactly once. Hence there
exists a symplectic morphism
$\phi\colon\widehat{\bf\Sigma}\to{\bf\Sigma}^{\prime}$. Since the first
cohomology of a disc vanishes, $\phi$ is
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact. Theorem
2.20 now guarantees an exact symplectic embedding of $\bigl{(}D(A)\times
D(B)\bigr{)}\setminus\left(\Gamma_{1}\times\Gamma_{2}\right)$ into
$D(a+b)\times D(Kb)\subset Z^{4}(a+b)$. $\square$

Figure 4.1. $\Sigma_{1}^{\prime}$ and $\Sigma_{2}^{\prime}$ in
$\Omega^{\prime}=D(a+b)\times D(Kb)$ for $K=4$.

####  4.1.2. Reduction to the monotone setting

######  Proposition 4.2.

Let $\Gamma_{1}\subset D(A)$ and $\Gamma_{2}\subset D(B)$ be regular grids
whose complements are a union of $m$ topological discs of area $a_{i}\leq a$
and of $n$ topological discs of area $b_{j}\leq b$, respectively.

Then there exists a regular grid $\Gamma^{\prime}_{1}\subset D(ma)$ that cuts
$D(ma)$ into $m$ topological discs of area $a$ and a regular grid
$\Gamma_{2}^{\prime}\subset D(nb)$ that cuts $D(nb)$ into $n$ topological
discs of area $b$, and an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding

$\bigl{(}D(A)\times
D(B)\bigr{)}\setminus\bigl{(}\Gamma_{1}\times\Gamma_{2}\bigr{)}\,\to\,\bigl{(}D(ma)\times
D(nb)\bigr{)}\setminus\bigl{(}\Gamma_{1}^{\prime}\times\Gamma_{2}^{\prime}\bigr{)}.$

Proof. If $a_{i}=a$ and $b_{j}=b$ for all $i,j$, then there is nothing to
prove. So after switching the factors and re-indexing we can assume that
$a_{i}<a$ for $1\leq i\leq m^{\prime}\leq m$ and $a_{i}=a$ for $i\geq
m^{\prime}+1$. Define $n^{\prime}\leq n$ for the $b_{i}$ in the same way,
where however $n^{\prime}=0$ if $b_{i}=b$ for all $i$.

Write ${\mathcal{P}}:=\\{p_{1},\dots,p_{m}\\}\subset D(A)$,
${\mathcal{Q}}:=\\{q_{1},\dots,q_{n}\\}\subset D(B)$, and let

$\bigl{(}{\bf\Sigma}:=\bigl{\\{}(p_{i}\times D(B),a_{i}),(D(A)\times
q_{j},b_{j})\bigr{\\}},\lambda\bigr{)}$

be the extendable Liouville polarization of $D(A)\times D(B)$ with skeleton
$\Gamma_{1}\times\Gamma_{2}$ provided by Lemma 3.5. We recall that in this
case, an extension is obtained by taking somewhat larger discs, so
$\overline{\Sigma}$ is just the closure of $\Sigma$ in $\mathbb{C}^{2}$, i.e.
the closed discs. Similarly, let
${\mathcal{P}}^{\prime}:=\\{p_{1}^{\prime},\dots,p_{m}^{\prime}\\}\subset
D(ma)$,
${\mathcal{Q}}^{\prime}:=\\{q_{1}^{\prime},\dots,q_{n}^{\prime}\\}\subset
D(nb)$, and let

$\bigl{(}{\bf\Sigma}^{\prime}:=\bigl{\\{}({\mathcal{P}}^{\prime}\times
D(nb),a),(D(ma)\times{\mathcal{Q}}^{\prime},b)\bigr{\\}},\lambda^{\prime}\bigr{)}$

be the Liouville polarization of $D(ma)\times D(nb)$ with skeleton
$\Gamma^{\prime}_{\\!1}\times\Gamma^{\prime}_{\\!2}$. Since we are free to
choose $\Gamma^{\prime}_{\\!1}$ and $\Gamma^{\prime}_{\\!2}$, and since
$D(A)\subset D(ma)$ and $D(B)\subset D(nb)$, we may as well assume that
$p_{i}^{\prime}=p_{i}$ and $q_{j}^{\prime}=q_{j}$. This is not an important
assumption but will simplify the proof at some point. We will nevertheless
keep the primes in order to distinguish the source and the target. Lemma 3.6
provides disjoint open discs $D_{i}^{A}$ in $D(ma)\setminus\Gamma_{1}$,
disjoint open discs $D_{j}^{B}$ in $D(nb)\setminus\Gamma_{2}$, points
$p_{i}^{\prime 1},p_{i}^{\prime 2}\in D_{i}^{A}$ for $i\leq m^{\prime}$,
$q_{j}^{\prime 1},q_{j}^{\prime 2}\in D_{j}^{B}$ for $j\leq n^{\prime}$, and a
Liouville polarization $({\bf\Sigma}^{\prime\prime},\lambda^{\prime\prime})$
of $D(ma)\times D(nb)$ with

$\displaystyle{\bf\Sigma}^{\prime\prime}$ $\displaystyle\,:=\,\Bigl{\\{}$
$\displaystyle\bigl{(}p_{i}^{\prime 1}\times D(nb),a_{i}\bigr{)}_{i\leq
m^{\prime}},\bigl{(}p_{i}^{\prime 2}\times D(nb),a-a_{i}\bigr{)}_{i\leq
m^{\prime}},$ $\displaystyle\bigl{(}D(ma)\times q_{j}^{\prime
1},b_{j}\bigr{)}_{j\leq n^{\prime}},\bigl{(}D(ma)\times q_{j}^{\prime
2},b-b_{j}\bigr{)}_{j\leq n^{\prime}},$
$\displaystyle\bigl{(}p_{i}^{\prime}\times D(nb),a\bigr{)}_{i\geq
m^{\prime}+1},\bigl{(}D(ma)\times q_{j}^{\prime},b\bigr{)}_{j\geq
n^{\prime}+1}\Bigr{\\}}$

and such that

$\displaystyle{\mathcal{B}}\left(\cup_{i}D_{i}^{A}\times
D(nb)\bigcup\cup_{j}D(ma)\times D_{j}^{B},\lambda^{\prime\prime}\right)$
$\displaystyle\subset$ $\displaystyle\bigl{(}D(ma)\times
D(nb)\bigr{)}\setminus(\Gamma_{1}^{\prime}\times\Gamma_{2}^{\prime}).$

As is clear from the proof of Lemma 3.6 we can chose $p_{i}^{\prime
1}=p_{i}^{\prime}$ and $q_{i}^{\prime 1}=q_{i}^{\prime}$. We can find
$\varepsilon>0$ and two area preserving embeddings

$\begin{array}[]{l}\sigma\colon\overline{D(B)}\to\overline{D(nb)}\\\\[1.99997pt]
\tau\colon D(A+\varepsilon)\to D(ma)\end{array}$

such that $\sigma(q_{j})=q_{j}^{\prime}=q_{j}$ for all $j\leq n$ and the image
$\operatorname{im}\sigma$ avoids all the $q_{j}^{\prime 2}$, and such that
$\tau(p_{i})=p_{i}^{\prime}$ for all $i\leq m$ and $\operatorname{im}\tau$
avoids all the $p_{i}^{\prime 2}$. This is obvious for $\tau$ because $A<ma$
by assumption. It is also true for $\sigma$ because if $B<nb$ we can even
extend $\sigma$ to $\hat{\sigma}\colon D(B+\varepsilon)\to D(nb)$ with the
required properties, and if $B=nb$ then $b_{j}=b$ for all $j$, so there is no
$q_{j}^{\prime 2}$ to avoid, and $\sigma$ can be taken to be the identity. The
two maps $\sigma$ and $\tau$ induce an embedding
$\phi\colon\Sigma\to\Sigma^{\prime\prime}$ defined by

$\begin{array}[]{l}\phi|_{p_{i}\times
D(B)}=p_{i}^{\prime}\times\sigma\\\\[1.99997pt] \phi|_{D(A)\times
q_{j}}=\tau\times q_{j}^{\prime}.\end{array}$

There are now two cases.

Case $B<nb$: As already noticed, in this case $\sigma$ can be extended to an
area preserving embedding $\hat{\sigma}\colon D(B+\varepsilon)\to D(nb)$, so
$\phi$ can be extended to a smooth area preserving embedding
$\widehat{\Sigma}\to\Sigma^{\prime\prime}$ which is a symplectic morphism
between our polarizations. It is
$(\alpha_{\operatorname{st}},\alpha_{\operatorname{st}})$-exact because all
components of $\Sigma$ are discs. Theorem 2.20 now implies that

(4.2) $\bigl{(}D(A)\times
D(B)\bigr{)}\setminus(\Gamma_{1}\times\Gamma_{2})={\mathcal{B}}({\bf\Sigma},\lambda)\,\lhook\joinrel\xrightarrow{\;\\!\alpha_{{\operatorname{st}}}\,}\,{\mathcal{B}}({\bf\Sigma}^{\prime\prime},\lambda^{\prime\prime}).$

Moreover, since

${\bf\Sigma}^{\prime\prime}\,\subset\,\cup_{i}D_{i}^{A}\times D(nb)\bigcup
D(ma)\times\cup_{j}D_{j}^{B},$

we obtain together with (4.1.2) the inclusion

$\displaystyle{\mathcal{B}}({\bf\Sigma}^{\prime\prime},\lambda^{\prime\prime})$
$\displaystyle\subset$
$\displaystyle{\mathcal{B}}\Bigl{(}\cup_{i}D_{i}^{A}\times D(nb)\bigcup
D(ma)\times\cup_{j}D_{j}^{B}\Bigr{)}$ $\displaystyle\subset$
$\displaystyle\bigl{(}D(ma)\times
D(nb)\bigr{)}\setminus(\Gamma_{1}^{\prime}\times\Gamma_{2}^{\prime}).$

Composing (4.1.2) with (4.1.2) we obtain the asserted exact symplectic
embedding.

Case $B=nb$: In this case, $\sigma=\mbox{id}$ does not extend to a larger disc
with image still in $D(nb)$, so formally the above argument does not apply. We
solve this difficulty by revisiting the proof of Theorem 2.20 in this
particular case: Although $\phi$ does not extend to an extended polarization
$\widehat{\Sigma}$ as before, the product map $\psi:=\tau\times\mbox{id}$
extends $\phi$ to a small neighbourhood of $\Sigma$ in $D(A)\times D(b)$,
still with image in $D(ma)\times D(nb)$. The forms $\lambda$ on $D(A)\times
D(B)$ and $\lambda^{\prime\prime}$ on $D(ma)\times D(nb)$ are both split:

$\lambda=\lambda_{1}\oplus\lambda_{2},\qquad\lambda^{\prime\prime}=\lambda_{1}^{\prime\prime}\oplus\lambda_{2}^{\prime\prime},$

where $\lambda_{1},\lambda_{2}$ are defined on $D(A)\backslash\\{p_{i}\\}$,
$D(B)\backslash\\{q_{j}\\}$ and
$\lambda_{1}^{\prime\prime},\lambda_{2}^{\prime\prime}$ are defined on
$D(ma)\backslash\\{p_{i}^{\prime},p_{i}^{\prime 2}\\}$,
$D(mb)\backslash\\{q_{j}^{\prime}\\}$. Since $B=mb$ and $q_{j}=q_{j}^{\prime}$
we can take $\lambda_{2}=\lambda_{2}^{\prime\prime}$, and since $\lambda_{1}$
and $\lambda_{1}^{\prime\prime}$ have the same residues at $p_{i}$ and
$p_{i}^{\prime}=p_{i}$, we can make them coincide on a small neighbourhood of
the $p_{i}$. Since $\tau$ can clearly be taken to be the identity on a
neighbourhood of the points $p_{i}$, the map $\psi=\tau\times\mbox{id}$
(defined on a neighbourhood of $\Sigma$) then pulls back
$\lambda^{\prime\prime}$ to $\lambda$. Thus the basic conjugacy procedure
described in the proof of Theorem 2.20 to embed
${\mathcal{B}}({\bf\Sigma},\lambda)$ into
${\mathcal{B}}({\bf\Sigma}^{\prime\prime},\lambda^{\prime\prime})$ applies,
and now the argument in the previous case applies. $\square$

####  4.1.3. Proof of Theorem 6

Let $\Gamma_{1}\subset D(A)$ and $\Gamma_{2}\subset D(B)$ be two regular grids
that cut the discs into topological discs of area $\leq a$ and $\leq b$,
respectively. By Proposition 4.2,

$\bigr{(}D(A)\times
D(B)\bigr{)}\setminus(\Gamma_{1}\times\Gamma_{2})\,\lhook\joinrel\xrightarrow{\;\\!\alpha_{{\operatorname{st}}}\,}\,\bigl{(}D(ma)\times
D(nb)\bigr{)}\setminus(\Gamma_{a}\times\Gamma_{b})$

where $\Gamma_{a}$ and $\Gamma_{b}$ cut $D(ma)$ and $D(nb)$ into topological
discs of area $a$ and $b$, respectively. By Proposition 4.1, we also have

$\bigl{(}D(ma)\times
D(nb)\bigr{)}\setminus(\Gamma_{a}\times\Gamma_{b})\,\lhook\joinrel\xrightarrow{\;\\!\alpha_{{\operatorname{st}}}\,}\,Z^{4}(a+b).$

Composing these two maps we obtain the searched exact symplectic embedding

$f\colon\bigl{(}D(A)\times
D(B)\bigr{)}\setminus(\Gamma_{1}\times\Gamma_{2})\lhook\joinrel\xrightarrow{\;\\!\alpha_{{\operatorname{st}}}\,}Z^{4}(a+b).$

$\square$

###  4.2. Proof of Theorem 2

Let $\Gamma:=\Gamma_{1}\times\Gamma_{1}\subset\mathbb{R}^{4}$, where
$\Gamma_{1}\subset\mathbb{R}^{2}$ is the regular grid

$\Gamma_{1}:=\bigcup_{(n,m)\in\mathbb{Z}^{2}}\\{n\\}\times\mathbb{R}\cup\mathbb{R}\times\\{m\\}.$

Let also

$\Sigma:=\bigcup_{(n,m)\in\mathbb{Z}^{2}}\bigl{\\{}(n+\tfrac{1}{2},m+\tfrac{1}{2})\bigr{\\}}\times\mathbb{C}\cup\mathbb{C}\times\bigl{\\{}(n+\tfrac{1}{2},m+\tfrac{1}{2})\bigr{\\}}.$

######  Lemma 4.3.

There exists a Liouville form $\lambda$ on $\mathbb{R}^{4}\backslash\Sigma$
with the following properties:

  * •

$\lambda$ has residue $1$ along $\Sigma$.

  * •

For each $p\in\mathbb{C}^{2}\backslash\Sigma$, the Liouville trajectory
$\Phi^{t}_{X_{\lambda}}(p)$ is defined for $t\in\;]-\infty,t^{+}(p)[$, where
$t^{+}(p)\in\;]0,+\infty]$. If $t^{+}(p)<+\infty$, then $\lim_{t\to
t^{+}(p)}\Phi^{t}_{X_{\lambda}}(p)$ exists and belong to $\Sigma$.

  * •

$\Gamma$ is the skeleton of $(\Sigma,\lambda)$:

$\Gamma=\\{p\;|\;t^{+}(p)=+\infty\\}=\mathbb{R}^{4}\backslash{\mathcal{B}}(\Sigma,\lambda).$

  * •

$X_{\lambda}$ is tangent to the hyperplanes $\\{x_{1}\in\mathbb{Z}\\}$,
$\\{x_{2}\in\mathbb{Z}\\}$, $\\{y_{1}\in\mathbb{Z}\\}$,
$\\{y_{2}\in\mathbb{Z}\\}$.

Proof. Again, the form $\lambda$ is a product:

$\lambda_{(z,w)}(u,v)=\lambda_{z}(u)+\lambda_{w}(v),$

where $\lambda_{z}=\lambda_{w}$ is a Liouville form on
$\mathbb{R}^{2}\backslash\bigl{(}\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2})\bigr{)}$,
with $\Gamma_{1}$ as skeleton. We construct this Liouville form exactly as in
Section 3.1, where compactness plays no role. Alternatively, we can do the
construction in the proof of Proposition 3.2 on $[-2,2]^{2}$ in a
$\mathbb{Z}^{2}$-periodic way. We then obtain a smooth Liouville form on
$[0,1]^{2}$ that extends to a smooth $\mathbb{Z}^{2}$-periodic Liouville form
on
$\mathbb{R}^{2}\backslash\bigl{(}\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2})\bigr{)}$
with skeleton $\Gamma_{1}$. $\square$

From Lemma 4.3 we obtain the following two facts:

  * •

For $N\in\mathbb{N}$ define

$\Sigma_{N}:=\Sigma\;\\!\cap\,]-N,N[^{4}\quad\mbox{and}\quad\Gamma_{N}:=\Gamma\times\;\\!]-N,N[^{4}.$

Then $\bigl{(}(\Sigma_{N},1),\lambda\bigr{)}$ is an extendable Liouville
polarization of $]-N,N[^{4}$ whose skeleton is $\Gamma_{N}$. Notice that
$\Sigma_{N}\subset\Sigma_{N+1}$.

  * •

Define

$Z(2):=\>]-1,1[\;\\!\times\;\\!]0,1[\;\\!\times\mathbb{R}^{2}\quad\mbox{and}\quad\Sigma_{Z}:=\Sigma\cap
Z(2).$

Then $\bigl{(}(\Sigma_{Z},1),\lambda\bigr{)}$ is a Liouville polarization of
$Z(2)$.

Now $\Sigma_{Z}$ is made of two planes
$\\{(\pm\frac{1}{2},\frac{1}{2})\\}\times\mathbb{R}^{2}$ and infinitely many
discs

$D_{n,m}:=\>]-1,1[\;\\!\times\;\\!]0,1[\;\\!\times\bigl{\\{}(n+\tfrac{1}{2},m+\tfrac{1}{2})\bigr{\\}},\quad
n,m\in\mathbb{Z}.$

As in the proof of Proposition 4.1, we keep
$\Sigma_{1}^{\prime}:=\bigl{\\{}(-\frac{1}{2},\frac{1}{2})\bigr{\\}}\times\mathbb{R}^{2}$
and glue all other components together to form a symplectic curve
$\Sigma_{2}^{\prime}$, which is diffeomorphic to a plane with infinitely many
punctures arranged on a lattice. Each puncture corresponds to a boundary
component of action $2$ (see Figure 4.2). Since $\lambda$ has residue $1$
along each component of $\Sigma_{Z}$, Proposition 3.8 (applied with $U$ the
union of small balls around the singular points of $\Sigma_{2}^{\prime}$)
yields a Liouville form $\lambda^{\prime}$ on
$Z(2)\backslash(\Sigma_{1}^{\prime}\cup\Sigma^{\prime}_{2})$ tame along
$\Sigma^{\prime}:=\Sigma_{1}^{\prime}\cup\Sigma_{2}^{\prime}$ with residue $1$
and equal to $\lambda$ near $\partial Z(2)$. In other terms,

$\bigl{(}\\{(\Sigma^{\prime}_{1},1),(\Sigma^{\prime}_{2},1)\\},\lambda^{\prime}\bigr{)}$

is a Liouville polarization of $Z(2)$.

######  Lemma 4.4.

For each $N$ there exists a symplectic morphism
$\phi_{N}\colon\widehat{\Sigma}_{N}\to\Sigma^{\prime}$ such that

$\phi_{N+1}|_{\widehat{\Sigma}_{N}}=\phi_{N}.$

Proof. For describing the construction of $\phi_{N}$ it is convenient to
redescribe the components $\Sigma^{\prime}_{1}$ and $\Sigma_{2}^{\prime}$ of
$\Sigma^{\prime}$:

  * •

$\Sigma_{1}^{\prime}=\\{(-\frac{1}{2},\frac{1}{2})\\}\times\mathbb{R}^{2}$ is
symplectic with standard area form on the $\mathbb{R}^{2}$-factor. Its
intersections with $\Sigma_{2}^{\prime}$ are the points

$I_{n,m}:=(-\tfrac{1}{2},\tfrac{1}{2},n+\tfrac{1}{2},m+\tfrac{1}{2}),\quad(n,m)\in\mathbb{Z}^{2}.$

  * •

$\Sigma_{2}^{\prime}$ is obtained by gluing
$(\frac{1}{2},\frac{1}{2})\times\mathbb{R}^{2}$ with the topological discs
$D_{n,m}$, $(n,m)\in\mathbb{Z}^{2}$. This can be done by replacing each round
disc $(\frac{1}{2},\frac{1}{2})\times
D\bigl{(}(n+\frac{1}{2},m+\frac{1}{2}),\varepsilon\bigr{)}$ by its gluing with
the disc $D_{n,m}$. These gluings can be parameterized by
$\mathbb{Z}^{2}$-translations of a punctured disc
$D(0,\varepsilon)\backslash\\{(0,\frac{\varepsilon}{2}\\}$ with an area form
$\omega_{\varepsilon}$ whose total area is $2+\varepsilon$, and that is
standard near the boundary $S^{1}(\varepsilon)$. Each point $I_{n,m}$ is again
parameterized by $(n+\frac{1}{2},m+\frac{1}{2})$.

We also define
$\Sigma_{1}^{\prime}(N):=\\{(-\frac{1}{2},\frac{1}{2})\\}\times[-N,N]^{2}$ and
$\Sigma_{2}^{\prime}(N)\subset\Sigma_{2}^{\prime}$ the part of
$\Sigma_{2}^{\prime}$ parameterized by
$[-N,N]^{2}\backslash(\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2}+\frac{\varepsilon}{2}))$,
and notice that they correspond exactly to the intersections of these curves
with $Z(2)\cap[-N,N]^{4}$ (see Figure 4.2).

Figure 4.2. The parametrization of the curve $\Sigma_{2}^{\prime}$. Each
depicted disc $D(\varepsilon)$ has area $\varepsilon$ on the picture, but
carries the symplectic form $\omega_{\varepsilon}$ of area $2+\varepsilon$.
The black points are the punctures, and the red crosses are the intersection
points with $\Sigma_{1}^{\prime}$.

Now divide $\Sigma_{N}$ into its $4N^{2}$ horizontal components

$\Sigma_{N}^{H}(n,m):=\;]-N,N[^{2}\times\bigl{\\{}(n+\tfrac{1}{2},m+\tfrac{1}{2})\bigr{\\}},\quad(n,m)\in[-N,N-1]^{2}$

and its $4N^{2}$ vertical components

$\Sigma_{N}^{V}(n,m):=\bigl{\\{}(n+\tfrac{1}{2},m+\tfrac{1}{2})\bigr{\\}}\times\,]-N,N[^{2},\quad(n,m)\in[-N,N-1]^{2}.$

Each such component is a disc of area $4N^{2}$, the horizontal and vertical
components do not intersect pairwise, but each horizontal component intersects
each vertical component exactly once, thus each component passes through
$4N^{2}$ singular points of $\Sigma_{N}$.

We construct $\phi_{N}$ by sending a slightly larger disc around the closure
of each $\Sigma_{N}^{V}(n,m)$ into $\Sigma_{1}^{\prime}(2N^{2}+1)$, and by
sending a slightly larger disc around the closure of each
$\Sigma_{N}^{H}(n,m)$ into $\Sigma^{\prime}_{2}(2N^{2}+1)$. The areas of these
surfaces are

$\begin{array}[]{l}{\mathcal{A}}(\Sigma_{1}^{\prime}(2N^{2}+1))=(2(2N^{2}+1))^{2}>16N^{4}={\mathcal{A}}(\Sigma^{V}_{N})\\\\[3.00003pt]
{\mathcal{A}}(\Sigma_{2}^{\prime}(2N^{2}+1))=(2(2N^{2}+1))^{2}+2(2N^{2}+1)^{2}>16N^{4}={\mathcal{A}}(\Sigma_{N}^{H}),\end{array}$

where $\Sigma_{N}^{H}:=\coprod\Sigma_{N}^{H}(n,m)$ and
$\Sigma_{N}^{V}:=\coprod\Sigma_{N}^{V}(n,m)$, and

$\\#\bigl{(}\Sigma_{1}^{\prime}(2N^{2}+1)\cap\Sigma_{2}^{\prime}(2N^{2}+1)\bigr{)}=(2(2N^{2}+1))^{2}>16N^{4}=\\#(\Sigma_{N}^{V}\cap\Sigma_{N}^{H}).$

We can therefore define $\phi_{N}$ by its restrictions to $\Sigma_{1}^{V}$ and
$\Sigma_{1}^{H}$ as illustrated in Figures 4.3 and 4.4, where $\phi_{N}$ is
explained in blue and $\phi_{N+1}$ in yellow for $N=1$. These are area
preserving embeddings that take the points at the red crosses to points at the
red crosses. This defines symplectic morphisms
$\phi_{N}\colon\widehat{\Sigma}_{N}\to\Sigma^{\prime}$, and we can clearly
choose $\phi_{N+1}$ such that $\phi_{N+1}|_{\widehat{\Sigma}_{N}}=\phi_{N}$.
$\square$

Figure 4.3. The restriction of $\phi_{1}$ to $\Sigma_{1}^{V}$ and of
$\phi_{2}$ to $\Sigma_{2}^{V}$. Each band has height slightly less than $1$,
so that we can construct a larger band of height less than $1$ in the next
step $N+1$. Figure 4.4. The restriction of $\phi_{1}$ to $\Sigma_{1}^{H}$ and
of $\phi_{2}$ to $\Sigma_{2}^{H}$. These are almost the same embeddings as in
the previous figure, except that they avoid the punctures. Note that the area
$2+\varepsilon$ of the discs is not even used since for $\varepsilon>0$ small
enough the area of $\Sigma_{2}^{\prime}(2N^{2}+1)$ without the discs is
already larger than the area of $\Sigma_{N}^{H}$.

Proof of Theorem 2. We can now proceed as in the proof of Theorem 2.20. First
we extend each $\phi_{N}$ from Lemma 4.4 to a symplectic embedding

$\psi_{N}\colon V_{N}\to\operatorname{Op}\bigl{(}\Sigma^{\prime},Z(2)\bigr{)}$

of an open neighbourhood $V_{N}$ of $\widehat{\Sigma}_{N}$. Set
$U_{N}:=V_{N}\cap[-N,N]^{4}$. Since
$\phi_{N+1}|_{\widehat{\Sigma}_{N}}=\phi_{N}$, Moser’s method shows that the
maps $\psi_{N}$ can be chosen such that

$U_{N+1}\supset U_{N}\quad\text{ and
}\quad\psi_{N+1}|_{U_{N}}=\psi_{N}|_{U_{N}}.$

We can therefore define the $C^{\infty}$-smooth symplectic embedding

$\psi\colon\bigl{(}U,\Sigma\bigr{)}\to\bigl{(}Z(2),\Sigma^{\prime}\bigr{)}$

on $U:=\bigcup U_{N}$ by $\psi|_{U_{N}}:=\psi_{N}|_{U_{N}}$.

In the construction of the maps $\phi_{N}$ of Lemma 4.4 we can assume that
their images avoid a whole neighbourhood of the punctures. After choosing $U$
smaller, if necessary, the closure of $\psi(U)$ is then disjoint from the