Modification of Convex Ends to Cylindrical
and Symplectic Foliations

Yoshihiko MITSUMATSU111 Supported by Grant-in-Aid for Scientific Research (B)
22340015, 17H02845, 21H00985. Date: February 5, 2022. © 2022 Y. Mitsumatsu.
2020 Mathematics Subject Classification. 57R30, 57R17, 32S55.

###### Abstract

We show that the natural symplectic structure on the Milnor fiber of an
isolated singularity in complex three variables whose link fibers over the
circle can be modified into one which is cylindrical at the end. As a
consequence we see that the foliation of codimension one on $S^{5}$ which is
adapted to the Milnor open book of $S^{5}$ associated with such a singularity
admits a leafwise symplectic structure. The modification also enables us to
construct certain closed symplectic 4-manifolds.

###### Contents

  1. 0 Introduction
  2. 1 Milnor open book and associated symplectic structure on pages
    1. 1.1 Milnor fibration and open book decomposition
    2. 1.2 Exact symplectic structures on Milnor pages and open book decompositions
    3. 1.3 Simple elliptic and cusp singularities
  3. 2 Adapted foliations of open book decompositions
    1. 2.1 Mori’s criterions on open books
    2. 2.2 Leafwise symplectic structure around the binding
  4. 3 Leafwise symplectic structures on adapted foliations
  5. 4 A certain closed symplectic 4-manifold
    1. 4.1 Gluing two Milnor fibers
    2. 4.2 Circular Example
  6. 5 Other applications
    1. 5.1 $\boldsymbol{b}$-Symplectic structures
    2. 5.2 Leafwise symplectic foliations

##  0 Introduction

In the previous work [Mi2] the author constructed leafwise symplectic
structures on foliations adapted to the Milnor open books for the simple
elliptic singularities $\tilde{E}_{6},\tilde{E}_{7}$, and $\tilde{E}_{8}$. The
foliation adapted to $\tilde{E}_{6}$ is so called Lawson’s foliation (see [L])
which is known to be the first foliation of codimension one constructed on the
five sphere. In this paper we refine the construction mainly from the point of
view of open book decomposition and give a sufficient condition not only for
Milnor open book but also for general open books supporting contact structures
so that the adapted foliation exists and admits a leafwise symplectic
structure. As a consequence, for isolated hypersurface singularities of three
complex variables, Lawson type (adapted) foliations of the Milnor open books
of cusp singularities also admit leafwise symplectic structures, as well as
simple elliptic ones on which we have already worked.

The notion of leafwise symplectic foliation or foliation with leafwise
symplectic structure is essentially equivalent to that of regular Poisson
structure and is defined as follows.

###### Definition 0.1

(Leafwise symplectic structure. See [Mi2]) For a smoothly foliated manifold
$(M,{\mathcal{F}})$, a leafwise symplectic structure is a smooth 2-form
$\omega$ on $M$ which restricts to a symplectic form of each leaf of
$\mathcal{F}$.

Often it is regarded as a smooth section of 2nd exterior power of the
cotangent bundle to the foliation, i.e., an element of
$\Gamma^{\infty}(\bigwedge^{2}T^{*}{\mathcal{F}})$ whose restriction to each
leaf is symplectic on the leaf.

###### Remark 0.2

The one in the sense of the first definition restricts to one of the second
definition, the one in the sense of second definition always admits a lift to
one in the sense of the first definition. Therefore in this article we do not
distinct the two and shift from one to the other without mention.

The first definition has an importance if we would like to impose on the
leafwise symplectic form $\omega$ to be closed as a 2-form on the ambient
manifold $M$. For a foliation of codimension one with a globally closed
leafwise symplectic form, Donaldson’s theory of approximately holomorphic
geometry is applicable and we find very fine structures (see Martínez-Torres
[Mt]). On the spheres a compact leaf is an apparent obstruction.

The second definition is completely equivalent to the notion of regular
Poisson structures, i.e., Poisson structures all of whose symplectic leave has
the same dimension. From this point of view our work is paraphrased as
constructions of regular Poisson structures.

In Section 1 we review the basic materials concerning singularities, including
the Milnor fibrations and open books of isolated singularities, cusp and
simple elliptic singularities. The associated foliations (often called Lawson
type foliations) with open books and their relations with the contact
structures are reviewed in Section 2.

The key of the construction of leafwise symplectic structure on a Lawson type
foliation is the modification of symplectic structure of a Milnor fiber into
one with cylindrical end. The natural symplectic structure of Milnor fiber
which is inherited from the Euclidean space has a convex end. In order to
construct a leafwise symplectic structure on an associated foliation, the
fibers should spiral around a compact symplectic leaf, and accordingly the
fibers is required to admit a symplectically periodic end. For Milnor fibers
of simple elliptic singularities and cusp singularities we show that the
modification is possible in a rather simple way (Section 3, Theorem 3.1).

The method of modifying symplectic structures in Section 3 yields some further
applications. We give some constructions of symplectic structures in Section 4
or of $b$-symplectic structures in Section 5 on certain closed 4-manifolds. A
closed symplectic 4-manifold is obtained by gluing two copies of the Milnor
fibers of certain cusp singularities (Construction 4.2), while $b$-symplectic
structures are obtained on the doubles of the Milnor fiber of simple elliptic
or cusp singularities. This construction seems to have some relation with
recent study of homological mirror symmetry due to Hacking and Keating ([K],
[HK]). A closer look at these closed symplectic 4-manifolds is passed to a
forthcoming paper [K2M2].

Acknowledgement The author is grateful to Atsuhide Mori for his insights and
discussions as well as to Naohiko Kasuya for detailed discussions. He also is
grateful to l’Unité de Mathématique de l’Ecole Normale Supérieure de Lyon
because the main part of this work was done during the author’s sabbatical
stay in Lyon in 2012.

##  1 Milnor open book and associated symplectic structure on pages

###  1.1 Milnor fibration and open book decomposition

First we recall the construction of exact symplectic open book decomposition
from a Milnor fibration of an isolated singularity. Let $f(z_{0},...,z_{n})$
be a polynomial or holomorphic function in $n+1$ complex variables and assume
that the origin is an isolated critical point $\\{f=0\\}$.

Let $S^{2n+1}_{\rho_{0}}$ denote the sphere in ${\mathbb{C}}^{n+1}$ of radius
$\rho_{0}>0$ with center at the origin. The most classical Milnor fibration is
defined as $S^{2n+1}_{\rho_{0}}\setminus f^{-1}(0)\to S^{1}$ by taking
$\mathrm{arg}f(z)=f(z)/|f(z)|$ for $z\in S^{2n+1}_{r}\setminus f^{-1}(0)$.
Then for a small enough $\varepsilon_{0}>0$, the restriction of this fibration
to a compact manifold $M=S^{2n+1}_{\rho_{0}}\cap\\{z=(z_{0},\dots
z_{n})\,;\,|f(z)|\geq\varepsilon_{0}\\}$ with boundary is also called the
Milnor fibration of the singularity. $U=S^{2n+1}_{r}\setminus M$ is a tubular
neighborhood of the link $N=f^{-1}(0)\cap S^{2n+1}_{r}$ in the sphere
$S^{2n+1}_{r}$. As the normal bundle of the link $N$ is trivialized by the
value of $f$, $U$ is diffeomorphic to the product
$N\times\mathrm{int}D^{2}_{\varepsilon_{0}}$ where
$D^{2}_{\varepsilon_{0}}=\\{w\in{\mathbb{C}}\,;\,|w|\leq\varepsilon_{0}\\}$.
The common boundary $W=\partial\overline{U}=\partial M$ is diffeomorphic to
$S^{1}\times N$. The decomposition $S^{2n+1}_{r}=\overline{U}\cup M$ together
with the fibration $\varphi=f/|f|:M\to S^{1}$ is the Milnor open book
decomposition of the sphere $S^{2n+1}$ associated with the isolated
singularity. The binding is nothing but the link $N$.

If we choose $\rho_{0}>0$ small enough carefully, the real-hypersurface
$\\{\mathrm{arg}f=\theta\\}$ for each $\theta\in S^{1}$ is transverse to
$S^{2n+1}_{\rho^{\prime}}\setminus f^{-1}(0)$ for
$0<\rho^{\prime}\leq\rho_{0}$ so that the gradient flow of
$\rho^{2}=\sum_{j=0}^{n}|z_{j}|^{2}$ on the real-hypersurfce
$\\{\mathrm{arg}f=\theta\\}$ gives rise to a diffeomprphism between
$F_{w}=\\{f(z)=w\\}\cap D^{2(n+1)}_{\rho_{0}}$ and
$\varphi^{-1}(\theta)\subset M\subset S^{2n+1}_{\rho_{0}}$ for
$w=\varepsilon_{0}e^{i\theta}$. Therefore the fibration $\varphi:M\to S^{1}$
can be identified with $\displaystyle\cup_{|w|=\varepsilon_{0}}F_{w}\to
S^{1}$, which is also called a Milnor fibration. In this article, we mainly
use this formulation of Milnor fibration and thus it is also denoted by
$\varphi:M\to S^{1}$. When the absolute value $|w|$ is well-understood or a
Milnor fiber $F_{w}$ is regarded as a page of the open book, $F_{w}$ is also
denoted by $F_{\theta}$ where $\theta=\mathrm{arg}\,w$.

###  1.2 Exact symplectic structures on Milnor pages and open book
decompositions

The canonical symplectic structure
$\displaystyle\omega_{0}=\sum_{j=0}^{n}dx_{j}\wedge dy_{j}$ on
${\mathbb{C}}^{n+1}$ with its canonical primitive (or a Liouville form)
$\displaystyle\lambda_{0}=\frac{1}{2}\sum_{j=0}^{n}(x_{j}dy_{j}-y_{j}dx_{j})$
on ${\mathbb{C}}^{n+1}$ restricts to each Milnor fiber $F_{w}$ to be an exact
symplectic structure. We slightly modify this structure by isotopying the
embedding $F_{w}\hookrightarrow{\mathbb{C}}^{n+1}$’s.

Let us identify the compact region
$CM=\\{|f|\leq\varepsilon_{0}\\}\cap(D^{2(n+1)}_{\rho_{0}}\setminus\mathrm{int}D^{2(n+1)}_{{\rho_{0}}/2})\subset{\mathbb{C}}^{n+1}$
with the product $[{\rho_{0}}/2,{\rho_{0}}]\times
D^{2}_{\varepsilon_{0}}\times N$ as follows. The first coordinate $\rho$ is
the distance from the origin, the second is the value of $f$ in
$D^{2}_{\varepsilon_{0}}\subset{\mathbb{C}}$. Fixing the projection to $N$ is
not very important but it is better to fix it.

For example, first , on $\rho={\rho_{0}}$, namely on ${\overline{U}}\subset
S^{2n+1}_{{\rho_{0}}}$, by the exponential map in $S^{2n+1}$ from $N$ we can
define a projection to $N$. Then, for each
${\rho}^{\prime}\in[{\rho_{0}}/2,{\rho_{0}}]$ the $CM\cap\\{\
\rho={\rho}^{\prime}\\}$, applying the same construction on
$S^{2n+1}_{{\rho}^{\prime}}$, we obtain a smooth projection $CM\to N$. For the
second step we can also rely on the gradient flow of the function $\rho$ on
each $f^{-1}(w)$ ($w\in D^{2}_{\varepsilon_{0}}$) to extend the projection to
$N$. (Also in the place of the gradient flow, the Liouville vector field on
each $F_{w}=f^{-1}(w)$ is available.) This gives a smooth identification of
$CM$ with $[{\rho_{0}}/2,{\rho_{0}}]\times D^{2}_{\varepsilon_{0}}\times N$
and we regard $(\rho,w,n)\in[{\rho_{0}}/2,{\rho_{0}}]\times
D^{2}_{\varepsilon_{0}}\times N$ as the product coordinate of $CM$.

Next take a smooth function $\psi(\rho)$ on $[{\rho_{0}}/2,{\rho_{0}}]$
satisfying

$\displaystyle\bullet$ $\displaystyle\qquad\psi(\rho)\equiv
1,\qquad\rho\in[\frac{{\rho_{0}}}{2},\frac{2{\rho_{0}}}{3}],\qquad\qquad\qquad$
$\displaystyle\bullet$ $\displaystyle\quad 0\leq\psi(\rho)\leq
1,\quad\rho\in[\frac{2{\rho_{0}}}{3},\frac{5{\rho_{0}}}{6}],$
$\displaystyle\bullet$ $\displaystyle\qquad\psi(\rho)\equiv
0,\qquad\rho\in[\frac{5{\rho_{0}}}{6},{\rho_{0}}].$

With these preparations we show the following.

###### Propoition 1.1

For sufficiently small $\varepsilon_{0}$ and for any $w\in{\mathbb{C}}$ with
$|w|=\varepsilon_{0}$, the identical embedding of $F_{w}$ admits an isotopy
through symplectic embeddings with respect to $\omega_{0}$ in such a way that
on the part $\rho\leq{\rho_{0}}/2$ the isotopy is trivial, through the isotopy
$\rho$ is preserved, and the part
$F_{w}\cup\\{\rho\geq\frac{5{\rho_{0}}}{6}\\}$ is embedded onto the
corresponding part of $f^{-1}(0)$. Moreover these family of isotopies are
chosen to be smooth on the parameter $\theta\in S^{1}$ and whole through the
isotopies the projection to $N$ is stationary.

Proof. In the product coordinate, the hypersurface
$F_{w}\cap\\{{\rho_{0}}/2\leq\rho\leq{\rho_{0}}\\}$ is considered to be a
graph of the constant function $w$ from $[{\rho_{0}}/2,{\rho_{0}}]\times N$ to
$D^{2}_{\varepsilon_{0}}$.

As the modification of the identical embedding of $F_{w}$, take the function
$\psi(\rho)w$ instead of the constant function $w$ and consider its graph. Let
$s\in[0,1]$ be the parameter for the isotopy and consider the graphs of the
smooth map

$(\rho,n,w,s)\mapsto((1-s)+s\psi(\rho))w$

from $[{\rho_{0}}/2,{\rho_{0}}]\times N\times
D^{2}_{\varepsilon_{0}}\times[0,1]$ to $D^{2}_{\varepsilon_{0}}$, where
$(w,s)\in D^{2}_{\varepsilon_{0}}\times[0,1]$ are regarded as parameters so
that the graphs are considered to be drawn in $[{\rho_{0}}/2,{\rho_{0}}]\times
N\times D^{2}_{\varepsilon_{0}}$. For any $(w,s)$, on
$[{\rho_{0}}/2,\frac{2{\rho_{0}}}{3}]\cup[\frac{5{\rho_{0}}}{6},{\rho_{0}}]$
the graph is symplectic in ${\mathbb{C}}^{n+1}$. Also note that if $s=0$ or
$w=0$, the graphs are symplectic. From the compactness we can see easily that
there exists a positive $\varepsilon>0$ such that whenever
$|w|\leq\varepsilon$ regardless to $s\in[0,1]$ the graphs is symplectic. To
complete the proof take $w=\varepsilon_{0}e^{i\theta}$ for $\theta\in S^{1}$.
$\square$

Let $\Phi:M=\cup_{\theta\in S^{1}}F_{w}\cap
D^{2(n+1)}_{\rho_{0}}\to{\mathbb{C}}^{n+1}$ be the map which is identity on
$M\cap D^{2(n+1)}_{r/2}$ and is the above one with
$w=\varepsilon_{0}e^{i\theta}$ and $s=1$. On each Milnor page, we take the
restriction of $\Phi^{*}\omega_{0}$ and $\Phi^{*}\lambda_{0}$ which define a
new exact symplectic structure on each Milnor page.

Along the image of $\Phi$ take the symplectic orthonormal plane field to
$\Phi(F_{w})$. It defines a symplectic parallel transport from the Milnor page
$F_{w}\cap D^{2(n+1)}_{\rho_{0}}$, namely $\theta=0$, through the pages
$f^{-1}(\varepsilon e^{i\theta})\cap D^{2(n+1)}_{\rho_{0}}$ ($0\leq\theta\leq
2\pi$) to $F_{w}\cap D^{2(n+1)}_{\rho_{0}}$, $\theta=2\pi$, which gives rise
to a symplectic monodromy. It is clearly stationary on
$\frac{5r}{6}\leq{\rho}\leq{\rho_{0}}$ so that the monodromy is identity
around the boundary.

Each page has the identical symplectically convex end, which also coincides
with the end of $f^{-1}(0)\cap D^{2(n+1)}_{\rho_{0}}$. The boundary
$N=\partial$, which is also called the binding of the open book or also the
link of the singularity, is naturally equipped with a standard contact
structure $\xi_{N}=\mathrm{ker}\,\lambda_{0}|_{N}$.

###### Definition 1.2

(Exact symplectic open book decomposition) An open book decomposition of a
$(2n+1)$-dimensional manifold ${M}$ is an exact symplectic open book
decomposition if each pages admits exact symplectic structure with contact
type boundary (or equivalently a globally convex symplectic structure (see
[EG])) and the monodromy preserves the symplectic structure and is identity in
some neighborhood of the boundary.

Then there exists a contact structure $\xi$ on ${M}$, which is unique up to
isotopy, such that the the binding is a contact submanifold which naturally
coincides with one given as the boundary of the pages and $\xi$ is almost
tangent to the pages. It is also said that the open book is supporting the
contact structure $\xi$. For more details on these notions, see e.g., [BCS].

###### Corollary 1.3

With a Milnor open book of an isolated singularity, an exact symplectic open
book decomposition is associated.

In our construction of symplectic foliations, we need a less strict class of
open books.

###### Definition 1.4

(Convex symplectic open book decomposition) An open book decomposition of a
$(2n+1)$-dimensional manifold ${M}$ is a convex symplectic open book
decomposition if each pages admits a symplectic structure with contact type
boundary and the monodromy preserves the symplectic structure and is identity
in some neighborhood of the boundary. Compared with the exact symplectic open
books the condition is loosen from the exactness to not necessarily global
convexity of the symplectic structures on pages.

The binding $N$ of the open book which is regarded as the common boundary of
all pages naturally admits a contact structure as the boundary of convex
symplectic structure. The symplectic structure on a neighborhood of the
boundary of a page is of the form of symplectization of the contact structure
on $N$.

###  1.3 Simple elliptic and cusp singularities

The singularities with which we are mainly concerned are the following
isolated hypersurface singularities at the origin $0=(0,0,0)$ in three complex
variables $(x,y,z)\in{\mathbb{C}}^{3}$.

$\displaystyle\widetilde{E}_{6}$ $\displaystyle\colon$ $\displaystyle
x^{3}+y^{3}+z^{3}+kxyz,\;k^{3}\neq-27$ $\displaystyle\widetilde{E}_{7}$
$\displaystyle\colon$ $\displaystyle x^{2}+y^{4}+z^{4}+kxyz,\;k^{4}\neq 64,$
$\displaystyle\widetilde{E}_{8}$ $\displaystyle\colon$ $\displaystyle
x^{2}+y^{3}+z^{6}+kxyz,\;k^{6}\neq 432.$

These singularities are called simple elliptic singularities. Usually an
simple elliptic singularity is defined as a normal surface singularity whose
minimal resolution locus $E$ is a smooth elliptic curve. The self-intersection
$E\cdot E$ is equal to $-3$, $-2$, and to $-1$ respectively for
$\widetilde{E}_{6}$, $\widetilde{E}_{7}$, and for $\widetilde{E}_{8}$. The
negative of $E\cdot E$ is called the index of the simple elliptic singularity.
It is known that the simple elliptic singularities which are realized as
hypersurface singularities are analytically equivalent to one of the above
three (Saito [S]). The restriction on $k$ is to ensure that the singularity is
isolated, and $k$ can be zero. Even though the different $k$’s define
analytically different singularities, they are, from smooth topological point
of view, diffeomorphic to each other as the space of $k$ is connected.
Therefore in this article we do not pay particular attention to the value of
$k$.

The following polynomial also defines an isolated singularity at the origin,
which is called a cusp singularity:

$\displaystyle T_{pqr}\colon x^{p}+y^{q}+z^{r}+kxyz,\;k\neq
0,\;\dfrac{1}{p}+\dfrac{1}{q}+\dfrac{1}{r}<1.$

In these cases, $k$ should just avoid $0$, then for different $k$’s they are
linearly equivalent. For $k=0$ above polynomials define singularities of a
different class. A cusp singularity is also usually defined as a normal
surface singularity whose minimal resolution locus is a rational curve with a
node or a certain cycle of smooth rational curves. See, e.g., [H]. It is also
know that a cusp singularity is analytically realizable as an isolated
hypersurface singularity if and only if it is analytically equivalent to one
of the above $T_{pqr}$-singularities for $1/p+1/q+1/r=1$ (Karras [K]).

The above three simple elliptic singularities can ‘almost’ be considered as
$T_{pqr}$-singularities for $1/p+1/q+1/r=1$ with an attention to the value of
$k$. The link of these singularities are known to be $T^{2}$-bundles over the
circle with monodromy

$A_{p,q,r}=\begin{pmatrix}r-1&-1\\\ 1&0\end{pmatrix}\begin{pmatrix}q-1&-1\\\
1&0\end{pmatrix}\begin{pmatrix}p-1&-1\\\ 1&0\end{pmatrix}.$

See [La], [N], and [Ka]. For simple elliptic singularities, i.e.,
$(p,q,r)=(3,3,3)$, $(2,4,4)$, and $(2,3,6)$, $A_{p,q,r}$’s are conjugate to
$\begin{pmatrix}1&0\\\ 3&1\end{pmatrix}$, $\begin{pmatrix}1&0\\\
2&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}$, respectively,
and they are unipotent, so that the links are nil 3-manifolds and also are
considered to be the circle bundles over $T^{2}$ with euler numbers $-3$,
$-2$, and $-1$ respectively (see [Mi2]). For cusp singularities, i.e.,
$1/p+1/q+1/r<1$, as $\mathrm{tr}\,A_{p,q,r}=2+pqr(1-(1/p+1/q+1/r))$, the
monodromy is hyperbolic and the link is a solv 3-manifold.

##  2 Adapted foliations of open book decompositions

In this section, we review and discuss two criterions due to A. Mori ([Mo])
for open books supporting contact structures to be nicely foliated. The first
one is on the binding so that the open book admits an adapted foliation and
the second is on the contact structure of the binding so that the ambient
contact structure converges to the foliation. Surprisingly these conditions
give rise to leafwise symplectic foliations adapted to Milnor fibrations which
we can associate starting from isolated singularities in complex three
variables.

###  2.1 Mori’s criterions on open books

With an open book decomposition on a 3-manifold it is easy to construct an
associated foliation of codimension one, which is called a spinnable foliation
(cf. [KM3]).

In general, foliating smoothly the exterior of the binding $N$ (which is
denoted by $M$ in our context) is always easy. It suffices to make the pages
winding around the boundary. However, foliating the tubular neighborhood of
the binding is not trivial. Thurston’s $h$-principle [Th] tells us that for an
open book of an odd dimensional closed manifold there always exists a
foliation which has the boundary of the tubular neighborhood of the binding as
a compact leaf, while from the $h$-principle we do not have enough control on
the resultant foliation.

###### Theorem 2.1

(Mori’s criterion-1, [Mo], [L]) Let $M^{m}$ be a closed smooth manifold which
is given an open book decomposition. If the the binding $N^{m-2}$ of the open
book admits a Riemannian foliation ${\mathcal{G}}$ of codimension one, then
its pull-back to $U\cong N\times{\mathrm{int}}D^{2}$ can be turbulized around
the boundary $W=S^{1}\times N$ so that $\overline{U}\cong N\times D^{2}$ is
smoothly foliated and the boundary $W$ is a compact leaf. As a conclusion
there exists a foliation ${\mathcal{F}}$ adapted to the open book
decomposition.

This is a slight generalization of Lawson’s construction ([L]) of foliation
adapted to an open book, which just assumes that the binding fibers over the
circle. Then we can foliate the tubular neighborhood of the binding by so
called turbulization so as to have the boundary as a compact leaf. It is easy
to choose the holonomy around this compact leaf so that after pasting the
pages we obtain a smooth foliation. From the point of view of Mori’s
criterion, Lawson’s construction adopts the bundle foliation which the
fibration over the circle determines as the Riemannian foliation on the biding
to start with. Such a foliation which is associated with such an open book is
called a foliation of Lawson type or an adapted foliation. In this article, if
we start from Mori’s criterion-1with a Riemannian foliation, we call the
resultant foliation a generalized Lawson type foliation.

As is well known, if the binding admits a Riemannian foliation of codimension
one, a slight perturbation of the defining closed 1-form among the non-
singular closed 1-forms so as to represent a rational first cohomology class,
we obtain a Riemannian foliation determined by a fibration over the circle
(Tischler [Ti]). Therefore as the restriction of the topology of binding,
Mori’s criterion is equivalent to Lawson’s one, however, classes of foliations
on the binding and of resultant foliations are wider.

Now we explain the turbulization. First let us just assume the fibering of the
binding $N$ over the circle $S^{1}=\\{x\in{\mathbb{R}}/2\pi{\mathbb{Z}}\\}$.

Also fix a normal coordinate $w\in D^{2}_{\varepsilon_{0}}\subset{\mathbb{C}}$
of the fiber as the normal bundle to $N\subset U$ in the tubular neighborhood
${\overline{U}}$ and put $r=|w|$. Let $\psi(r)$ be a smooth function on
$[0,\varepsilon_{0}]$ satisfying

$\displaystyle\bullet$ $\displaystyle\qquad\,\psi(r)\equiv 1,\qquad
r\in[0,\varepsilon_{0}/2],\qquad\qquad\qquad$ $\displaystyle\bullet$
$\displaystyle\qquad\psi^{\prime}(r)<0,\qquad
r\in(\varepsilon_{0}/2,\varepsilon_{0}),$ $\displaystyle\bullet$
$\displaystyle\qquad\psi(\varepsilon_{0})=0,$ $\displaystyle\bullet$
$\displaystyle\qquad\psi(r)\mathrm{\,\,is\,\,flat\,\,to\,\,}0\mathrm{\,\,at\,\,}\varepsilon_{0}.$

Then take

$\alpha_{U}=\psi dx+(1-\psi)dr$

as the defining $1$-form of the desired foliation on ${\overline{U}}$.

###### Remark 2.2

Consider the solid 3-torus $S^{1}\times D^{2}_{\varepsilon_{0}}$ with the
coordinate $(x,w)$ and set $r=|w|$ as above. Then the $1$-form $\alpha$ as
above defines a so called Reeb component, i.e., a typical foliation on the
solid torus. Two copies of Reeb components are glued together to form a Reeb
foliation on $S^{3}$. The foliation on ${\overline{U}}$ constructed above can
be regarded as the pull back of the Reeb component (or its open neighborhood)
by the projection $p:{\overline{U}}\to S^{1}$ ($p(n,w)=(x(p),w)$) with the
same monodromy and the fiber as those of $N\to S^{1}$.

For a generalized Lawson type foliation, first take a defining closed 1-form
$\alpha_{N}$ of the Riemanian foliation on $N$ and take a Tischler fibration
to $S^{1}$. Then the place of $dx$ use $\alpha_{N}$ and take

$\alpha_{U}=\psi\alpha_{N}+(1-\psi)dr.$

Then it suffices to obtain a smooth foliation which is extendable even to the
other side of the boundary as a trivial foliation defined by $dr$.

The reason why the foliation with which we start should be Riemannian is as
follows. If we trivially extend the defining $1$-form $\alpha_{N}$ of the
foliation on $N$ to ${\overline{U}}=D^{2}\times N$ it also defines a foliation
on $\partial U$. Due to Rosenberg and Thurston [RT], for the smooth
turbulization which can be extended to the other side as a trivial foliation,
it should be Riemannian.

The fibration of the binding over the circle works not only for foliating the
tubular neighborhood of binding but also plays an important roll for existence
of the leafwise symplectic structures.

###### Theorem 2.3

(Mori’s criterion-2, [Mo]) Let $\xi$ be a contact structure on a closed
manifold $M^{5}$ In the above criterion, i.e., the existence of a Riemannian
foliation on the binding $N^{3}$, we also assume that there exists a contact
1-form $\alpha$ defining $\xi$ which is adapted to the supporting open book,
the contact form $\alpha|_{N}$ of the restricted contact structure $\xi|_{N}$
has the Reeb vector field is tangent to ${\mathcal{G}}$. Then there exists a
one-parameter family $\xi_{t}$ of contact structures, which is an isotopic
deformation of $\xi$, converges to the Lawson type foliation ${\mathcal{F}}$
as hyperplane field.

###### Remark 2.4

This criterion applies to the Milnor open books of all simple elliptic and
cusp singularities of hypersurfaces. However, the exact 2-forms $d\alpha$’s do
not converge to the leafwise symplectic structures which are constructed in
this article. The compact leaf is an apparent obstruction.

For the simple elliptic singularities, the natural contact structure obtained
as the convex boundary of the Milnor page or the singular surface admits a
Reeb vector field which is tangent to the circle fiber if the link is
considered as a circle bundle over $T^{2}$. The circle fiber is contained in
the $T^{2}$ fiber of the fibration to the circle. See [Mi2].

In the cusp case, a natural contact structure on the link which is a solv
manifold admits a Reeb vector field tangent to one of the two eigen directions
of the monodromy in each $T^{2}$ fiber. See [Ka] for this.

Mori [Mo] constructed a family of open books of $S^{4}\times S^{1}$ supporting
contact structures whose bindings are nil 3-manifolds, where any of $-\ell$
for $\ell\geq 4$ are realized as the euler number of the link as circle bundle
over $T^{2}$.

###  2.2 Leafwise symplectic structure around the binding

In this subsection we explain the existence of leafwise symplectic structures
on the tubular neighborhood of the binding.

Surprisingly we will see the fibering of binding over the circle works further
until the existence of the leafwise symplectic structures if we start from
isolated singularities of complex three variables.

In this subsection we assume

  * (o)

the manifold $M^{2n+1}$ is closed and oriented,

  * (i)

$M$ admits an open book satisfying with a fixed fibration $p_{N}:N^{2n-1}\to
S^{1}=\\{x\in{\mathbb{R}}/2\pi{\mathbb{Z}}\\}$ of the binding over the circle
with the fiber $\Sigma^{2n-2}$ and the monodromy
$\varphi_{\Sigma}\curvearrowright\Sigma$,

  * (ii)

the fiber $\Sigma$ is equipped with a symplectic structure $\omega_{\Sigma}$
and the monodromy $\varphi_{\Sigma}$ preserves $\omega_{\Sigma}$,

and let the fibration $p_{N}:N\to S^{1}$ trivially extend to the closed
tubular neighborhood as $p:{\overline{U}}=N^{2n-1}\times
D^{2}_{{\varepsilon_{0}}}\to S^{1}\times D^{2}_{{\varepsilon_{0}}}$.

###### Remark 2.5

To obtain a leafwise symplectic structure on a Lawson type foliation, we need
that the compact leaf $\partial U=S^{1}\times N$ admits a symplectic
structure. For this, the conditions (i) and (ii) are sufficient, because we
can take $\omega_{\partial U}=dx\wedge d\theta+\tilde{\omega}_{\Sigma}$ where
$\theta=\mathrm{arg}\,w$ ($w\in\partial
D^{2}_{{\varepsilon_{0}}}\subset{\mathbb{C}}$), where
$\tilde{\omega}_{\Sigma}$ is an extension of ${\omega}_{\Sigma}$ on a fiber to
the total space of the fibration as a closed 2-form, which we have thanks to
the condition (ii).

For $n=2$, (i) implies (ii) because any orientation preserving mapping class
of a closed oriented surface contains an area preserving representative.
Moreover, (i) is also necessary for the symplectic structure on $\partial U$
due to Friedl-Vidussi [FV].

Moreover, if we start from an isolated singularity of three complex variables
($n=2$), simple elliptic and cusp singularities are exactly those whose Milnor
open book satisfy the condition (ii) (Neumann [N]). For a simple elliptic
singularity $\partial U$ is diffeomorphic to a so called Kodaira-Thurston nil-
manifold which is also known as a Kodaira’s primary surface.

###### Theorem 2.6

(Friedl-Vidussi, [FV]) A closed oriented 4-manifold $W^{4}=S^{1}\times N^{3}$
admits a symplectic structure if and only if $N$ fibers over the circle.

In the case the symplectic structure on $W$ is not necessarily in the above
form which we adopt in this article.

###### Theorem 2.7

(Neumann, [N]) The link of an isolated singularity of three complex variables
is Seifert fibered or a graph manifold. If it fibers over the circle, it
should be a nil 3-manifold or a solv 3-manifold, i.e., a torus bundle over the
circle, and the singularity is a simple elliptic one or a cusp.

###### Construction 2.8

(Leafwise symplectic structures on ${\overline{U}}$, [Mi2]) We adopt the
notations in the above remark. It is easy to take a leafwise symplectic
structure $\omega_{R}$ on the Reeb component so that on the toral leaf
$(\theta,\,x)\in S^{1}\times S^{1}$ it restricts to $d\theta\wedge dx$. As the
Lawson type foliation is the pull back of the Reeb component by the projection
$p$, which is a trivial extension of $p:N\to S^{1}$ to the
$D^{2}_{{\varepsilon_{0}}}$ factor, we also have an extension of
$\tilde{\omega}_{\Sigma}$ which is denoted the same. Then, for any non-zero
constants $a$ and $b$,

$\omega_{{\overline{U}}}=a(d\theta\wedge dx)+b(\tilde{\omega}_{\Sigma})$

is a leafwise symplectic form which restricts to the boundary as desired
above.

##  3 Leafwise symplectic structures on adapted foliations

We state the main results in this article and give their proofs .

###### Theorem 3.1

Let $M$ be a closed oriented manifold of dimension $5$ equipped with a convex
symplectic open book decomposition. Assume the following conditions.

  * (1)

The binding $N$ admits a fibration $p_{N}:N\to S^{1}$ over the circle.
Cnsequently, there is a closed 2-form $\tilde{\omega}_{\Sigma}$ on $N$ it is
restricted to an area form on each fiber of $p_{N}$.

  * (2)

The contact structure $\xi_{N}$ which the convex open book defines on $N$
admits a Reeb vector field which is tangent to the fibers of $p_{N}$. (Mori’s
criterion-2)

  * (3)

The cohomology class $[\tilde{\omega}_{\Sigma}]$ is in the image of the
restriction map $H^{2}(F;{\mathbb{R}})\to H^{2}(N;{\mathbb{R}})$ in the real
cohomology of the Milnor fiber $F$ to its boundary $N\cong\partial F$. This is
equivalent to that the fundamental class $[\Sigma]$ of the fiber is non-
trivial in $H_{2}(F;{\mathbb{R}})$.

Then the open book admits a leafwise symplectic foliation of Lawson type.

###### Remark 3.2

We can formulate the theorem in a slightly generalized way, by Mori’s
criterion-1 with a Riemannian foliation in the place of the foliation by the
fibration $p_{N}$ in the conditions (1) and (2). In that case, the arguments
in Subsection 2.2 is fixed exactly as in Remark 2.2. For the arguments below
we need to further assume that
$\tilde{\omega}_{\Sigma}\wedge\alpha_{{\mathcal{G}}}$ is a volume form of $N$
and to replace $dx$ with $\alpha_{{\mathcal{G}}}$, where
$\alpha_{{\mathcal{G}}}$ is a closed 1-form on $N$ which defines the
Riemannian foliation ${\mathcal{G}}$.

If we try to generalize these constructions to the Milnor open books of higher
dimensional isolated singularities, for four or more complex variables,
unfortunately the Milnor open book never satisfies not only (3) but also (1).
That is because the Milnor fiber is 2-connected and the link is 1-connected.

###### Corollary 3.3

Any isolated hypersurface singularity of three complex variables which is
simple elliptic or cusp has the Milnor open book of $S^{5}$ which admits
leafwise symplectic foliation of Lawson type.

Proof of Corollay 3.3. For simple elliptic and cusp singularities of three
complex variables, the condition (1) and the first half of (3) hold as
explained in 1.3. The condition (2) is, as mentioned in Remark 2.4 observed in
[Mi2] for the simple elliptic case and in [Ka] for the cusp case. Therefore we
need to check the condition (3). In [Mi2] it is proved by a Meyer-Vietoris
argument in the simple elliptic case, while including that case it is easily
shown from the 1-connectedness of the Milnor fiber $F$ is in this dimension
([M]), a part of the cohomology long exact sequence for the pair
$(F,N=\partial F)$ and the Poincare duality of $(F,N)$ as follows.

$H^{2}(F;{\mathbb{R}})\to H^{2}(N;{\mathbb{R}})\to
H^{3}(F,N;{\mathbb{R}})\cong H_{1}(F;{\mathbb{R}})=0.\vspace*{-5pt}$

3.3$\square$
Proof of Theorem 3.1 goes along the three steps. The first two are
preparatory, and in the third step we construct symplectic forms on the pages
which are naturally glued smoothly to the leafwise symplectic structure on the
closed tubular neighborhood ${\overline{U}}$. The construction is similar to
that in [Mi2].
Step 1. We put a better coordinates on the ends of pages by a standard
procedure in symplectic geometry. Let $(F_{\theta},\omega_{\theta}$
($\theta\in S^{1}$) be the exact symplectic pages. The tubular neighborhoods
of the boundaries of the pages are identified by the monodromy as symplectic
manifolds so that we can assume they all are of the form
$(1-\varepsilon_{1},1]\times N,d(\varrho_{1}\alpha_{1}))$ of a part of
symplectization of a contact structure $\xi_{N}=\mathrm{ker}\,\alpha_{1}$ on
the binding $N=\partial F_{\theta}$, where $\varepsilon_{1}>0$ is some
constant and $\varrho_{1}$ is the coordinate of $(1-\varepsilon_{1},1]$. Once
extend the end to $\varrho_{1}<\infty$ with the symplectic form
$d(\varrho_{1}\alpha_{1})$ and take a contact form $\alpha_{N}$ for $\xi_{N}$
which satisfies the condition (4). Then, if necessary by taking a constant
multiplication, $\alpha_{N}$ can be regarded as the graph of the function
$\varrho_{1}=\alpha_{N}/\alpha_{1}$ on $N$ with values in $[1,c]$ for some
$c\geq 1$. Putting $\varrho\alpha_{N}=\varrho_{1}\alpha_{1}$ and taking
$\varrho=(\alpha_{1}/\alpha_{N})\varrho_{1}$ as the new coordinate function in
the place of $\varrho_{1}$, the symplectic form is $d(\varrho\alpha_{N})$.
Remark that in the case of Milnor open book $\varrho_{1}=\rho^{2}$.
Step 2. From the conditions (1) and (2) the binding $N$ is equipped with a
closed 2-form $\tilde{\omega}_{\Sigma}$ which restricts to a symplectic form
on each fiber $\Sigma_{x}$. The preimage
$r_{\theta}^{-1}(\tilde{\omega}_{\Sigma})\subset H^{2}(F;{\mathbb{R}})$ is
non-vacant affine space by the condition (3) where
$r_{\theta}:H^{2}(F;{\mathbb{R}})\to H^{2}(N;{\mathbb{R}})$ is the restriction
map, even though the monodromy of the open book might act, it is not difficult
to find a family of closed 2-forms $\kappa_{\theta}$ on each page $F_{\theta}$
satisfying the following conditions:

  * (a)

On the end $\varrho\geq 1$ $\kappa_{\theta}=\tilde{\omega}_{\Sigma}$, where by
abuse of notation $\tilde{\omega}_{\Sigma}$ also denotes the pull back of
$\tilde{\omega}_{\Sigma}$ by the projection $[1,\infty)\times N\to N$.

  * (b)

On $\bigcup_{{\theta\in S^{1}}}\\!F_{\theta}\subset M$,
$\kappa\\!=\\!\\{\kappa_{\theta}\\}_{\theta\in S^{1}}$ is a smooth family of
pagewise closed 2-forms.

Step 3. First remark that there is a constant $b>0$ so that
$||\omega_{\theta}^{2}||-b||\omega_{\theta}||\cdot||\kappa_{\theta}||>0$ at
any point of $F_{\theta}$. Consequently,
$\omega_{1}=\omega_{\theta}+b\kappa_{\theta}$ is symplectic. For this we have
to check $\omega_{1}^{2}>0$ on $\bigcup_{{\theta\in
S^{1}}}\\!F_{\theta}\setminus\\{\varrho>1\\}$ and on $\\{\varrho\geq 1\\}$. On
$\bigcup_{{\theta\in S^{1}}}\\!F_{\theta}\setminus\\{\varrho>1\\}$ the
condition is open for $b$ and of course $b=0$ satisfies it. Therefore there
exists $b_{0}>0$ such that $b$ with $0<b<b_{0}$ satisfies this open condition.
On $\\{\varrho\geq 1\\}$, for a product type Riemannian metric on
$[1,\infty)\times N$ where we assume $||d\varrho||\equiv 1$, once we can
verify $||d\varrho\wedge\alpha_{N}\wedge
d\alpha_{N}||>b\cdot||d\varrho\wedge\alpha_{N}\wedge\kappa_{\theta}||$ on
$\\{\varrho=1\\}$, the same inequality holds on $\\{\varrho\geq 1\\}$. Remark
that on $\\{\varrho\geq 1\\}$ $\kappa_{\theta}={\tilde{\omega}}_{\Sigma}$.
This inequality implies $\omega_{1}^{2}>0$ on $\\{\varrho\geq 1\\}$ because we
have $(d\alpha_{N})^{2}=d\alpha_{N}\wedge{\tilde{\omega}}_{\Sigma}=0$. We
prepare four positive constants

$\displaystyle{\mathbf{a}}$ $\displaystyle=$ $\displaystyle\min_{n\in
N}||\alpha_{N}\wedge d\alpha_{N}||_{n},\qquad{\mathbf{C}}=\max_{n\in
N}||\alpha_{N}\wedge{\tilde{\omega}}_{\Sigma}||_{n},$
$\displaystyle{\mathbf{A}}$ $\displaystyle=$ $\displaystyle\max_{n\in
N}||\alpha_{N}\wedge d\alpha_{N}||_{n},\qquad{\mathbf{m}}=\min_{n\in
N}||dx\wedge\tilde{\omega}_{\Sigma}||_{n}.$

We take $b>0$ satisfying $b<{\mathbf{a}}/{\mathbf{C}}$ and $b\leq b_{0}$. This
is not only for that $\omega_{1}$ is symplectic but also for the following
arguments.

Now we deform the symplectic structures on the pages $F_{\theta}$, only on
their ends which are identical independently of $\theta$, we omit $\theta$
from the description. On the end $[1,\infty)\times N$ the symplectic form is
$\omega=d(\varrho\alpha_{N})$. As in [Mi2] for some smooth functions
$K(\varrho)$ and $L(\varrho)$, consider the following 2-form

$\omega_{E}=d(K(\rho)\wedge\alpha_{N})+b{\tilde{\omega}}_{\Sigma}+L(\varrho)d\varrho\wedge
dx$

on $F\cap\\{\varrho\geq 1\\}$. Let us take $K(\rho)$ which satisfies the
following conditions.
(K-1) $K(\rho)=\rho$ $1\leq\varrho\leq 3$,
(K-2) $0\leq K^{\prime}(\rho)\leq 1$ $3\leq\varrho<4$,
(K-3) $-1\leq K^{\prime}(\rho)\leq 0$ $4<\varrho\leq 8$,
(K-4) $K(\rho)=0$ $8\leq\varrho$.
Then, we take $L(\varrho)$ which satisfies the following conditions.
(L-1) $L(\rho)=0$ $1\leq\varrho\leq 2$,
(L-2) $0<L^{\prime}(\rho)\leq$ $\,\,\,\,2\leq\varrho<3$,
(L-3) $L(\rho)\equiv a$ $3\leq\varrho$,
where $a$ is any constant satisfying
$a>\dfrac{2{\mathbf{A}}+b{\mathbf{C}}}{b{\mathbf{m}}}$.

On the end $[8,\infty)\times N$ the 2-form $\omega_{E}$ is a cylindrical
symplectic form $ad\varrho\wedge dx+b{\tilde{\omega}}_{\Sigma}$ as desired,
because in making the ends of pages winding around the boundary $\partial
U=S^{1}\times N\ni(\theta,n)$ of the tubular neighborhood of the binding $N$,
we take $\theta=\varrho$ (mod $2\pi$).

We have to check that $\omega_{E}$ is symplectic on $[1,8]\times N$. The most
important property of this construction is that at any point we have

$d(K(\rho)\wedge\alpha_{N})\wedge(d\varrho\wedge dx)=0\quad\cdots\quad(*).$

This follows from the condition (2) (Mori’s criterion-2) which implies
$d\alpha_{N}$ is divisible by $dx$ as both of them annihilate the Reeb vector
field of $\xi_{N}$, and just because
$d(K(\rho)\alpha_{N})=K^{\prime}(\varrho)d\varrho\wedge\alpha_{N}+K(\rho)d\alpha_{N}$.
Therefore we have

$\displaystyle\omega_{E}^{2}$ $\displaystyle=$
$\displaystyle(d(K(\rho)\alpha_{N}))^{2}+2b(d(K(\rho)\alpha_{N}))\wedge{\tilde{\omega}}_{\Sigma}+2bL(\varrho)d\varrho\wedge
dx\wedge{\tilde{\omega}}_{\Sigma}$ $\displaystyle=$ $\displaystyle
2d\varrho\wedge\left(K^{\prime}(\varrho)K(\varrho)\alpha_{N}\wedge
d\alpha_{N}+bK^{\prime}(\varrho)\alpha_{N}\wedge{\tilde{\omega}}_{\Sigma}+bL(\varrho)dx\wedge{\tilde{\omega}}_{\Sigma}\right)$

and we also remark that $|K^{\prime}(\varrho)K(\varrho)|<4$ holds.

  * •

On $[1,2]\times N$, $\omega_{P}=\omega_{1}$ is symplectic.

  * •

On $[2,3]\times N$, still
$d(K(\rho)\wedge\alpha_{N})+b{\tilde{\omega}}_{\Sigma}=\omega_{1}$ holds and
thus the sum of the first two terms is $\omega_{1}^{2}>0$ and the third term
is also positive.

  * •

On $[3,8]\times N$, the first two terms might be negative, while the third
term is already big enough to conclude $\omega_{E}^{2}>0$.

On each page $F_{\theta}$, $\omega_{\theta}+b\kappa_{\theta}$ extends to the
end to be $\omega_{E}$ as a symplectic form ${\omega_{P}}_{\theta}$ and we
obtain a leafwise symplectic structure
${\omega_{P}}=\\{{\omega_{P}}_{\theta}\\}$ on $\bigcup_{\theta\in
S^{1}}F_{\theta}$. Remark here that the subscript $\,{*}_{\theta}\,$ indicates
the ‘page number’. This completes Step 3.

As is already remarked above, the end $[8,\infty)\times N$ with the
cylindrical (i.e., $\varrho$-translation invariant) symplectic form
$ad\varrho\wedge dx+b{\tilde{\omega}}_{\Sigma}$ is exactly the covering of
$\partial U=S^{1}\times N$ with $ad\theta\wedge dx+b{\tilde{\omega}}_{\Sigma}$
by $\theta=\varrho$ (mod $2\pi$), and naturally after the tubulization, by
putting $\omega_{U}$ on ${\overline{U}}$ and $\omega_{P}$ on
$M\setminus{\overline{U}}$ we naturally obtain a smooth leafwise symplectic
structure after the tubulization (see [Mi2] for more detail). 3.1$\square$

###### Remark 3.4

For higher dimensional generalization, we need not only to assume (ii) in
Subsection 2.2 but to put a stronger condition than (2) in order to achieve a
good modification of the symplectic structures on the pages. We do not know
good examples nor geometric way to state the extra condition which we need.

If we take $\omega_{E}$ in the same way, e.g., in the case $n=3$, in computing
$\omega_{E}^{3}$, we would have terms $bKL\,d\varrho\wedge dx\wedge
d\alpha_{N}\wedge{\tilde{\omega}}_{\Sigma}$ or
$bKK^{\prime}\,d\varrho\wedge\alpha_{N}\wedge
d\alpha_{N}\wedge{\tilde{\omega}}_{\Sigma}$ of which we do not know the
positivity and thus they should be compared with $bLd\varrho\wedge
dx\wedge{{\tilde{\omega}}_{\Sigma}}^{2}$. In the case of general dimensions
there might appear terms
$b^{i}K^{n-i-1}L\,d\varrho\wedge
dx\wedge(d\alpha_{N})^{n-i-1}\wedge{{\tilde{\omega}}_{\Sigma}}^{i}$
or
$b^{i}K^{n-i-1}K^{\prime}\,d\varrho\wedge\alpha_{N}\wedge(d\alpha_{N})^{n-i-1}\wedge{{\tilde{\omega}}_{\Sigma}}^{i}$
for $i=1,2,\cdots,n-2$ and should be compared with $b^{n-1}L\wedge
d\varrho\wedge dx\wedge{{\tilde{\omega}}_{\Sigma}}^{n-i}$. As we have to take
$b$ small enough, we rather have to assume that the above terms vanish. Mori’s
criterion-2 kills the term of the above form of $i=0$ and in the case of $d=2$
we do not have other terms. So we would like to put the following condition

$(d\alpha_{N})^{n-i-1}\wedge{{\tilde{\omega}}_{\Sigma}}^{i}=0\quad\mathrm{on}\,\,N,\qquad
i=1,2,\cdots,n-2.$

Then all the arguments work exactly in the same way as in the case $d=2$ and
we obtain a leafwise symplectic structure on the associated foliation with the
open book.

##  4 A certain closed symplectic 4-manifold

In this section, we present two constructions of closed symplectic manifolds
as application of the construction in the previous section.

###  4.1 Gluing two Milnor fibers

The first application of the end-cylindrical symplectic structure is a
construction of a particular closed symplectic 4-manifold, which has similar
properties as K3-surfaces, starting from a particular cusp singularity,
$T_{2,3,7}$ or $T_{4,4,4}$.

###### Propoition 4.1

For the cusp singularity $T_{2,3,7}$ or $T_{4,4,4}$, the monodromy of the link
$N$ if the singularity is conjugate to its inverse as an element in
$\mathit{SL}(2;{\mathbb{Z}})$.

Proof of 4.1. The monodromy of the link of such a $T_{p,q,r}$-singularity is
geievn as $A_{p,q,r}=\begin{pmatrix}r-1&-1\\\
1&0\end{pmatrix}\begin{pmatrix}q-1&-1\\\
1&0\end{pmatrix}\begin{pmatrix}p-1&-1\\\ 1&0\end{pmatrix}$ as mentioned in
Subsection . Therefore we have $A_{2,3,7}=\begin{pmatrix}5&-11\\\
1&-2\end{pmatrix}$ and $A_{4,4,4}=\begin{pmatrix}21&-8\\\ 8&-3\end{pmatrix}$
and they are conjugate to $\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}$ and
$\begin{pmatrix}13&8\\\ 8&5\end{pmatrix}$. As they are unimodular and
symmetric, conjugate to their own inverses $\begin{pmatrix}1&-1\\\
-1&2\end{pmatrix}$ and $\begin{pmatrix}5&-8\\\ -8&13\end{pmatrix}$
respectively. 4.1$\square$

###### Construction 4.2

Let us take $T_{2,3,7}$ and one of its Milnor fiber, e.g., $F_{0}$ with the
end-cylindrical symplectic structure $\omega_{P\,0}$ constructed in the proof
of Theorem 3.1 and consider the restriction $\omega(1)=\omega_{P\,0}|_{F(1)}$
to the truncated Milnor fiber $F(1)=F_{0}\cap\\{\varrho\leq 10\\}$. Let
$(F(2),\omega(2))$ be the identical copy of $(F(1),\omega(1))$. We glue them
along their boundaries. The boundary $N$ is the suspension of $T^{2}$ by the
monodromy matrix $A_{2,3,7}\in\mathit{SL}(2;{\mathbb{Z}})$, we can assume that
the closed 2-form $\tilde{\omega}_{T^{2}}$ (here the fiber $\Sigma$ is
$T^{2}$) is the ‘pul-back’ by the suspension of the standard area form of
$T^{2}$. The fact that the monodromy is conjugate to its inverse in
$\mathit{SL}(2;{\mathbb{Z}})$ implies that even if we invert the orientation
of the base circle, still it is isomorphic to the original as oriented
$T^{2}$-bundle. We identify the boundary $N(2)=\partial F(2)$ with the
$N(1)=\partial F(1)$ by a map $\Phi:N(2)\to N(1)$ in this way, namely, the
$T^{2}$-fiber orientation is preserved, but the orientation of the base circle
is reversed as $x(1)\circ\Phi=-x(2)\in S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}$.
Here $x(i)$ denotes the $x$-coordinate which we have been fixed in previous
sections on $N(i)=N$. Similar notations are used here for other coordinates.
Therefore we have $\Phi^{*}dx(1)=-dx(2)$ while
$\Phi^{*}\tilde{\omega}_{T^{2}}=\tilde{\omega}_{T^{2}}$, and . $\Phi:N(2)\to
N(1)$ is orientation reversing. Now in order to glue $F(1)$ and $F(2)$, around
the boundaries take the identification of $\varrho$-coordinate as
$\varrho(2)=20-\varrho(1)$. In this way not only without changing the
orientation as 4-manifolds but also without changing the symplectic structures
around the boundaries, we can glue two copies $F(1)$ and $F(2)$ to obtain a
closed symplectic manifold $W_{2,3,7}$, because we have
$\Phi^{*}\tilde{\omega}_{T^{2}}=\tilde{\omega}_{T^{2}}$ and $d\varrho(1)\wedge
dx(1)=(-d\varrho(2))\wedge(-dx(2))=d\varrho(2)\wedge dx(2)$.

Remark that this construction is not taking the ‘double’. Of course also to
$T_{4,4,4}$ the same construction applies to obtain a closed symplectic
4-manifold $W_{4,4,4}$.

In both cases of $T_{2,3,7}$ and $T_{4,4,4}$ the Milnor fiber is simply
connected and has Euler characteristic $12$. In fact it is not difficult to
compute the Milnor number $\mu(T_{p,q,r})=p+q+r-1$ (see [Ga], [K2M2]).
Therefore the resultant 4-manifolds $W_{2,3,7}$ and $W_{4,4,4}$ are simply
connected and have the Euler characteristic 24. Also their Milnor lattice (the
intersection form) is also computed (also see [Ga], [K2M2]), and from these
informations we see the integral cohomology ring of $W_{2,3,7}$ and the
rational cohomology ring of $W_{4,4,4}$ coincide with those of a K3-surface
respectively. In particular, we see $W_{2,3,7}$ is homeomorphic to a
K3-surface. These cohomology computations and the arguments in 4-dimensional
differential topology are informed to the author by M. Ue. Therefore it is
strongly expected that they are diffeomorphic to a K3-surface. This strongly
motivates the forthcoming paper [K2M2] which in fact achieved it.

###  4.2 Circular Example

In this subsection, we slightly generalize the method in Step 3 of the proof
of Theorem 3.1, namely a modification of symplectic structure on the end, and
construct symplectic structures on $W=S^{1}\times N$. Here we present a
construction for a Riemannian foliation on $N$ (see Remark 3.2). We start with
the following objects.
$\bullet$ $N$ : a $(2n-1)$-dimensional closed oriented manifold,
$\bullet$ $p_{N}:N\to S^{1}=\\{x\in{\mathbb{R}}/2\pi{\mathbb{Z}}\\}$ : a
smooth fibration,
$\bullet$ $\tilde{\omega}_{\Sigma}$ : a closed 2-form on $N$ which restricts
to a symplectic form on each
fiber of $p_{N}$,
$\bullet$ $\xi_{N}$ : a positive contact structure on $N$,
$\bullet$ $\alpha_{N}$ : a contact 1-form defining $\xi_{N}$,
$\bullet$ ${\mathcal{G}}$ : a Riemannian foliation of codimension one on $N$,
$\bullet$ $\alpha_{{\mathcal{G}}}$ : a non-singular 1-form which defines
${\mathcal{G}}$.
We assume Mori’s criterion 2 for the Reeb vector field of $\xi_{N}$ and
${\mathcal{G}}$. We also assume the following condition.
(*) : $(\tilde{\omega}_{\Sigma})^{n-1}\wedge\alpha_{{\mathcal{G}}}$ is a
volume form on $N$.
In the case that the Riemannian foliation ${\mathcal{G}}$ is chosen to be the
foliation given by the fibration $p_{N}$, this condition is automatically
satisfied (see Remark 3.2). For the case $n\geq 3$ we further assume not only
the condition (ii) in Subsection 2.2 for a Tischler fibration $N\to S^{1}$
associated with the Riemannian foliation ${\mathcal{G}}$ but the codition in
Remark 3.4 with $dx$.

In the case of $n=2$, for all the nil 3-manifolds $N=\mathit{Nil}^{3}(-l)$
with ($l\in{\mathbb{N}}$) and all the solv manifolds $N=\mathit{Solv}_{A}$
with hyperbolic monodoromy $A\in\mathit{SL}(2;{\mathbb{Z}})$, namely,
$\mathrm{tr}\,A\geq 3$, any of them admits a contact structure $\xi_{N}$ ] and
a contact 1-form $\alpha_{N}$ whose Reeb vector field is tangent to the fibers
of the fibration $p_{N}:N\to S^{1}$. In these cases the condition is satisfied
by taking the foliation by fibration as the Riemannian foliation
${\mathcal{G}}$.

Following the construction of symplectic form $\omega_{E}$ on the end of
Milnor fiber in the proof of 3.1, let us consider the closed 2-form

$\omega=d(K(\theta)\alpha_{N})+\tilde{\omega}_{\Sigma}+L(\theta)d\theta\wedge\alpha_{{\mathcal{G}}}$

determined by smooth functions $K(\theta)$ and $L(\theta)$ on $S^{1}$ and
consider when

$w^{n}=\left(K^{\prime}(\theta)d\theta\wedge\alpha_{N}+K(\theta)d\alpha_{N}+\tilde{\omega}_{\Sigma}+L(\theta)d\theta\wedge\alpha_{{\mathcal{G}}}\right)^{n}$

gives an everywhere positive volume form. In the cohmology we have

$[\omega]=[\tilde{\omega}_{\Sigma}]+\left(\int_{S^{1}}L(\theta)\,d\theta\right)[d\theta\wedge\alpha_{{\mathcal{G}}}]\in
H^{2}(W;{\mathbb{R}}).$

Because of Mori’s criterion-2 the term
$d\theta\wedge\alpha_{{\mathcal{G}}}\wedge(d\alpha_{N})^{n-1}$ vanishes. Then
as we have an extra condition that we assume for the case $n\geq 3$, $w^{n}$
has only three terms as follows.

$\displaystyle w^{n}$
$\displaystyle\\!\\!=\,\,K^{\prime}(\theta)K(\theta)^{n-1}d\theta\wedge\alpha_{N}\wedge(d\alpha_{N})^{n-1}+\,L(\theta)d\theta\wedge\alpha_{{\mathcal{G}}}\wedge{\tilde{\omega}_{\Sigma}}^{n-1}$
$\displaystyle\,\,+\,K^{\prime}(\theta)d\theta\wedge\alpha_{N}\wedge{\tilde{\omega}_{\Sigma}}^{n-1}.$

We have a wide variety for the choices of $K(\theta)$ and $L(\theta)$ which
make $\omega$ symplectic. For any choice of $K(\theta)$, let us consider the
ratio of the sun of the first and the third terms against
$d\theta\wedge\alpha_{{\mathcal{G}}}\wedge{\tilde{\omega}_{\Sigma}}^{n-1}$ as
follows

$\lambda(\theta)=-\min_{n\in
N}\frac{K^{\prime}(\theta)\alpha_{N}\wedge\left(K(\theta)^{n-1}(d\alpha_{N})^{n-1}+{\tilde{\omega}_{\Sigma}}^{n-1}\right)_{(\theta,n)}}{{\alpha_{{\mathcal{G}}}\wedge{\tilde{\omega}_{\Sigma}}^{n-1}}_{(\theta,n)}}$

where the ratio is taken as the that of $2n-1$ forms on $N$, abbreviating
$d\theta$ in common. Then apparently $\omega$ is symplectic if and only if
$L(\theta)$ satisfies $L(\theta)>\lambda(\theta)$ for any $\theta\in S^{1}$.

Remark that the choice of $K(\theta)$ and that of $L(\theta)$ can be made
quite independently. For example, such a choice of $K(\theta)$ and $L(\theta)$
is possible that $K(\theta)\equiv 0$ on some intervals while $L(\theta)\equiv
0$ on some other intervals.

###### Theorem 4.3

For a closed 3-dimensional solv or nil manifold $N$, $W=S^{1}\times N$ admits
a symplectic form which looks like a co-symplectic form (see 5.1) on some
intervals in $S^{1}$ and like a slight deformation of a symplectization of the
canonical contact structure on $N$ on some other intervals.

##  5 Other applications

###  5.1 $\boldsymbol{b}$-Symplectic structures

As another application of the construction of end-cylindrical symplectic
structure, we remark in this section that certain $b$-symplectic manifolds are
easily constructed.

A b-symplectic structure is a Poisson structure (Poisson bi-vector field)
$\Pi\in C^{\infty}(W;\bigwedge^{2}TW)$ on a $2n$-dimensional manifold $W$ such
that the graph of $\wedge^{n}\Pi$ is transverse to the zero section in
$C^{\infty}(W;\bigwedge^{n}TW)$. Therefore the singular locus $Z=\\{w\in
W\,|\,\wedge^{n}\Pi(w)=0\\}$ is a smooth hypersurface of $W$. This notion was
introduced by Melrose in [Me] and recently is drawing more attensions from
Poisson geometry. For more details, see e.g., [GMP].

For a $b$-symplectic manifold, a Darboux theorem holds around the singular
locus $Z$ in the following a way. For a point in $Z$ there exists a coordinate
neighborhood $(x_{1},y_{1},\ldots,x_{n},y_{n})$ so that the Poisson bi-vector
$\Pi$ is expressed as

$\Pi=x_{1}\frac{\partial}{\partial x_{1}}\wedge\frac{\partial}{\partial
y_{1}}+\sum_{i=2}^{n}\frac{\partial}{\partial
x_{i}}\wedge\frac{\partial}{\partial y_{i}}$

where $Z=\\{x_{1}=0\\}$. This is equivalent to say that the corresponding
symplectic form $\omega$ has the expression

$\omega=\frac{1}{x_{1}}dx_{1}\wedge dy_{1}+\sum_{i=2}^{n}dx_{i}\wedge dy_{i}.$

If we allow the degeneration (or the divergence) along the codimension one
singular locus $Z$ to be the $\ell$-th order $\ell\in{\mathbb{N}}$ in
transverse direction, it is called a $b^{\ell}$-symplectic structure and has a
local expression replacing the coefficient $x_{1}^{\pm 1}$ above with
$x_{1}^{\pm\ell}$ ($\pm$ correspond to Poisson/symplectic formulations).

Let $W$ be the double of the Milnor page of one of simple elliptic or cusp
singularities. Namely, take two copies of a compact Milnor page $F_{\theta}$
with boundary and paste them along the boundaries with identity to obtain
$W=-F_{\theta}\cup_{\partial}F_{\theta}$.

###### Theorem 5.1

1) $W$ admits a $b$-symplectic structure whose singular locus is $\partial
F_{\theta}=N$.
2) Exactly the same holds for $b^{\ell}$-symplectic structures for odd
$\ell\in{\mathbb{N}}$.
3) For any even $\ell\in{\mathbb{N}}$, $W_{2,3,7}$ and $W_{4,4,4}$ in
Subsection 4.1 admit $b^{\ell}$-symplectic structures whose singular loci are
$\partial F(i)=N$.

Now our purpose in this section is a consequence of the following rather
general remark .

###### Propoition 5.2

Let $(W_{\pm},\Omega_{\pm})$ be a pair of $2n$-dimensional symplectic
manifolds with boundary which have the identical collar neighborhoods of the
boundaries as follows.

  1. (1)

$\partial W_{+}$ and $\partial W_{-}$ admit collar neighborhoods both of which
are symplectomorphic to $([0,2)\times N,\Omega)$.

  2. (2)

There exist a non-singular closed $1$-form $\alpha_{N}$ and a closed $2$-form
$\Omega_{N}$ on $N$ satisfying

$\Omega=dt\wedge{P_{N}}^{*}\alpha_{N}+{P_{N}}^{*}\Omega_{N}\vspace{-5pt}$

where ${P_{N}}$ denotes the projection $[0,2)\times N\to N$ and $t$ denotes
the coordinate on $[0,2)$.

Then, the manifold $W=W_{-}\cup_{\partial W_{-}=\partial W_{+}}W_{+}$ obtained
by identifying the boundaries admits a $b$-symplectic structure whose singular
locus is $\partial W_{-}=\partial W_{+}$.

###### Remark 5.3

1) The condition around the boundary is also said that the boundary admits a
cosymplectic structure. More precisely, if a compact boundary $N$ (or a
compact closed hypersurface) of a symplectic manifold $(W,\Omega)$ admits a
closed one form $\alpha_{N}$ with $\alpha_{N}\wedge{\Omega_{N}}^{n-1}$ a
volume form of $N$, then $N$ is called of coymplectic type, and the pair
$(\alpha_{N},\Omega_{N})$ is called a cosymplectic structure. From the normal
form theorem due to Gotay [Go], $N$ has a collar neighborhood where $\Omega$
is described exactly as in the above Proposition. For our application we can
directly arrange the situation in a stronger form as in the proposition.
2) For $\ell\geq 2$, the claim on $b^{\ell}$-symplectic structures is almost
direct. Even for $\ell=0$ the claim is nothing but Construction 4.2.
3) The assumption on such a closed 1-form $\alpha_{N}$ is nothing but Mori’s
criterion-1.

Proof of Proposition 5.2. Let $t_{\pm}\in[0,2)$ denote the each coordinate for
$W_{\pm}$ corresponding to $t$. For the resultant manifold $W$, let
$\tau\in(-2,2)$ be the corresponding coordinate and we construct $W$ by gluing
$W_{\pm}$ by defining $\tau=-t_{-}$ on $(-2,0]$ and $\tau=t_{+}$ on $(-2,0]$.
The two symplectic forms $\Omega_{\pm}=\Omega\,\,\mathrm{on}\,\,W_{\pm}$ do
not coincide at $\tau=0$ because they are indicated as $\Omega_{\pm}=\pm
dt\wedge{P_{N}}^{*}\alpha_{N}+{P_{N}}^{*}\Omega_{N}$. Remark that the
orientations of $W_{\pm}$ are opposite to each other in $W$. We do not need to
orient $W$ but if we define it as that of $W_{+}$, the original orientation of
$W_{-}$ is negative.

In order to obtain a $b$-symplectic structure whose singular locus is
$N\times\\{0\\}$, Take a smooth positive function $p(\tau)$ on $(-2,2)$
satisfying
(i) $p(\tau)=1/t$ $\tau\in(-1/2,0)\cup(0,1/2)$,
(ii) $p(\tau)\equiv-1$ $\tau\in(-2,-3/2)$,
(iii) $p(\tau)\equiv 1$ $\tau\in(3/2,2)$,
(iv) $p^{\prime}(\tau)<-0$ $\tau\in(-3/2,-1/2)\cup(1/2,3/2)$.
Then replace $\Omega$ with
$\Omega_{b}=p(t)dt\wedge{P_{N}}^{*}(\alpha_{N})+{P_{N}}^{*}\Omega_{N}$ on
$(-2,2)$ and

the collar neighborhoods and put them on $W_{i}$’s. They naturally give rise
to a $b$-symplectic structure on $W$. 5.2$\square$

###  5.2 Leafwise symplectic foliations

Taking the double is useful in constructing regular leafwise symplectic
foliations of codimension one. Because it concerns only the cylindrical end of
the symplectic manifold, let us start only with the cylindrical end of the
setting of Proposition 5.2 and follow some notations there.

###### Construction 5.4

Let $W_{a}=(a,\infty)\times N$ ($a$ has no particular meaning) be a
$2n$-dimensional symplectic manifold with a symplectic form

$\Omega=dt\wedge{P_{N}}^{*}\alpha_{N}+{P_{N}}^{*}\Omega_{N}$

as in Proposition 5.2 (2). We construct a symplectic foliation of codimension
one on $M^{\prime}=(-2,2)\times N\times S^{1}$. Take a smooth function
$\varphi(\tau)$ on $(-2,2)$ satisfying

$\displaystyle\mathrm{(i)}\,\,\varphi(\tau)\equiv
0\,\,\,\,(\tau\leq-1),\qquad\quad\quad\mathrm{(ii)}\,\,\varphi(\tau)\equiv\pi\,\,\,\,(\tau\geq
1)\qquad\quad$
$\displaystyle\mathrm{(iii)}\,\,\varphi(0)=\frac{\pi}{2}\,,\qquad\qquad\qquad\qquad\mathrm{(iv)}\,\,\varphi^{\prime}(\tau)\geq
0\,\,\,\,(-1<\tau<1),$
$\displaystyle\mathrm{(v)}\,\,\varphi(\tau)\equiv\frac{\pi}{4}\,\,\,\,(-\frac{2}{3}<\tau<-\frac{1}{3}),\quad\mathrm{(vi)}\,\,\varphi(\tau)\equiv\frac{3\pi}{4}\,\,\,\,(\frac{1}{3}<\tau<\frac{2}{3}),$

and consider the hyperplane field
$\tau{\mathcal{F}}^{\prime}=\mathrm{ker}\,\,\alpha^{\prime}$ defined by the
1-form $\alpha^{\prime}=\cos\varphi(\tau)d\theta-\sin\varphi(\tau)dt$ where
$\tau$ and $\theta$ are the coordinates of $(-2,2)$ and $S^{1}$ respectively
and extend them to. Apparently the hyperplane field is integrable and defines
a foliation ${\mathcal{F}}^{\prime}$ of codimension one which is tangent to
‘$\mathrm{arg}=\varphi$’ in $\tau\theta$-plane. $M^{\prime}$ is covered by
three open sets
$M_{-}=\\{-2<\tau<-1/3\\}$,
$M_{0}\,=\\{-2/3<\tau<2/3\\}$, and
$M_{+}=\\{1/3<\tau<2\\}$.
We define the leafwise symplectic form $\Omega^{\prime}$ on
$(M^{\prime},{\mathcal{F}}^{\prime})$ by restricting the 2-forms
$\Omega_{-}=\,\,\,\,d\tau\wedge{P_{N}}^{*}\alpha_{N}+{P_{N}}^{*}\Omega_{N}$ on
$M_{-}$,
$\Omega_{0}\,=\,\,\,\,\,d\theta\wedge{P_{N}}^{*}\alpha_{N}+{P_{N}}^{*}\Omega_{N}$
on $M_{-}$, and
$\Omega_{+}=-d\tau\wedge{P_{N}}^{*}\alpha_{N}+{P_{N}}^{*}\Omega_{N}$ on
$M_{+}$
to the leaves. Thanks to the conditions (v) and (vi), on $M_{-}\cap M_{0}$ and
on $M_{0}\cap M_{+}$ the two restrictions coincide to each other. Remark that
the restriction of $(M^{\prime},{\mathcal{F}}^{\prime},\Omega^{\prime})$ to
$M^{\prime}\cap\\{\tau<0\\}$ and $M^{\prime}\cap\\{\tau>0\\}$ are both
isomorphic to $(W_{a}\times S^{1},\,\,{\mathcal{F}}=\\{W_{a}\times\\{\theta\in
S^{1}\\}\\},\,\,\Omega=dt\wedge{P_{N}}^{*}\alpha_{N}+{P_{N}}^{*}\Omega_{N})$
as leafwise symplectic foliations.

As we have seen, we can modify the symplectic structures of the Milnor fibers
of the simple elliptic or cusp singularities of hypersurface type so as to
have such ends.

###### Theorem 5.5

Let $F_{\theta}$ be the Milnor fiber of one of $T_{p,q,r}$ singularities for
$1/p+1/q+1/r<1$ or $\widetilde{E}_{k}$ for $k=6,7,8$, and
$W=-F_{\theta}\cup_{\partial F_{\theta}}\\!F_{\theta}$ be its double. Then,
$W\times S^{1}$ admits a leafwise symplectic foliation of codimension one.

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Yoshihiko MITSUMATSU

Department of Mathematics,
Chuo University
1-13-27 Kasuga Bunkyo-ku,
Tokyo, 112-8551,
Japan

<EMAIL_ADDRESS>