# Hamiltonian loops on the symplectic blow up along a submanifold

Andrés Pedroza  Facultad de Ciencias
Universidad de Colima
Bernal Díaz del Castillo No. 340
Colima, Col., Mexico 28045<EMAIL_ADDRESS>
###### Abstract.

We prove that the fundamental group of the group of Hamiltonian
diffeomorphisms of the symplectic manifold that is obtain by blowing up a
submanifold contains an element of infinite order. We prove this using
Weinstein’s morphism and by constructing explicitly such loop of Hamiltonian
diffeomorphisms.

###### Key words and phrases:

Symplectic blow up, Hamiltonian diffeomorphism group, Weinstein’s morphism.

###### 1991 Mathematics Subject Classification:

Primary: 57S05 Secondary: 53D35.

The author was supported by CONACYT-CB-2017-2018-A1-S-8830 grant

##  1\. Introduction

In [12], it was shown that one-point blow up
$(\widetilde{M},\widetilde{\omega}_{\rho})$ of a closed 4-dimensional
symplectic manifold $(M,\omega)$ has a loop $\\{\widetilde{\psi}_{t}\\}$ of
Hamiltonian diffeomorphisms such that the class $[\widetilde{\psi}_{t}]$ has
infinite order in
$\pi_{1}(\textup{Ham}(\widetilde{M},\widetilde{\omega}_{\rho})).$ Actually the
argument presented in [12] works for any dimension as long as it is greater
than two. The restriction to the 4-dimensional case had a specific purpose; it
was related to the study of the abelian group
$\pi_{1}(\textup{Ham}(\widetilde{M},\widetilde{\omega}_{\rho})).$ Namely, for
any positive integer $k$ that there exists closed symplectic 4-manifold such
that the rank of $\pi_{1}(\textup{Ham})$ is at least $k$. This phenomena about
the rank of $\pi_{1}(\textup{Ham}(\widetilde{M},\widetilde{\omega}_{\rho}))$
being positive was first discovered by D. McDuff in [9]; the difference
between the results of [12] and those of [9] is that the element of infinite
order in $\pi_{1}(\textup{Ham})$ is written explicitly in [12]. The goal of
this article consists on extending this result. Instead of blowing up a zero-
dimensional submanifold, we blow up a symplectic submanifold
$N\subset(M,\omega)$ of codimension $2k>2$. In this way we also obtain a loop
of Hamiltonian diffeomorphisms on $(M,\omega)$, that induces an element of
infinite order in
$\pi_{1}(\textup{Ham}(\widetilde{M},\widetilde{\omega}_{N,\rho}))$ where
$(\widetilde{M},\widetilde{\omega}_{N,\rho})$ stands for the symplectic blow
up along the submanifold $N$.

######  Theorem 1.1.

Let $(M,\omega)$ be a rational closed symplectic manifold and
$N\subset(M,\omega)$ a closed symplectic submanifold of codimension $2k>2$.
Let $(\widetilde{M},\widetilde{\omega}_{N,\rho})$ be the symplectic blow up of
$(M,\omega)$ along $N$. Then for some small values of $\rho$ the rank of
$\pi_{1}(\textup{Ham}(\widetilde{M},\widetilde{\omega}_{N,\rho}))$ is
positive.

In $(\widetilde{M},\widetilde{\omega}_{N,\rho})$, the symplectic blow up of
$(M,\omega)$ along $N$, the parameter $\rho$ in the description of
$(\widetilde{M},\widetilde{\omega}_{N,\rho})$ is related to the
$\omega^{n}$-volume of a tubular neighborhood of $N\subset(M,\omega)$.
Therefore, in the case when $N$ is a single point, it corresponds to the
weight of the one-point blow up.

Although, Theorem 1.1 is a statement about the topology of the group
$\textup{Ham}(\widetilde{M},\widetilde{\omega}_{N,\rho})$, that is as much as
we know about its topology. Nonetheless, the study of the homotopy type of
$\textup{Ham}(M,\omega)$ has succeeded in some cases, all of which are
4-dimensional. For instance the homotopy type of
$(\mathbb{C}P^{2},\omega_{FS})$ and
$(\mathbb{C}P^{1}\times\mathbb{C}P^{1},\omega_{FS}\oplus\omega_{FS})$ was
computed by M. Gromov [5]; in [1], M. Abreu and D. McDuff settled the problem
for the one-point blow up of $(\mathbb{C}P^{2},\omega_{\textup{FS}})$. For
other examples of 4-dimensional symplectic manifolds see the works of S. Anjos
and S. Eden [2], S. Anjos and M. Pinsonnault [3], J. Evans [4] and F. Lalonde
and M. Pinsonnault [6]. As for computations of the fundamental group of
$\textup{Ham}(M,\omega)$, there are the works of J. Li and T.-J. Li [7] and J.
Li, T.-J. Li and W. Wu [8].

The techniques that are used to prove Theorem 1.1 are soft techniques.
Furthermore, we give an explicit description of the loop
$\\{\widetilde{\psi}_{N,t}\\}$ of Hamiltonian diffeomorphisms on
$(\widetilde{M},\widetilde{\omega}_{N,\rho})$ that induces the element of
infinite order in its fundamental group. In fact, the loop
$\\{\widetilde{\psi}_{N,t}\\}$ is induced from a Hamiltonian loop
$\\{\psi_{t}\\}$ on $(M,\omega)$ that is supported in a tubular neighborhood
of $N$. Finally, in order determine that an element on the fundamental group
of $\textup{Ham}(M,\omega)$ has infinite order, we relay on Weinstein’s
morphism,

$\mathcal{A}:\pi_{1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}(M,\omega).$

In this direction we have the following relation between Hamiltonian
diffeomorphism on $(M,\omega)$ and on
$(\widetilde{M},\widetilde{\omega}_{N,\rho})$ for a particular class of
Hamiltonian loops. In the formula of the above result,
${\Phi(\mathcal{U}_{\rho}(\nu_{0}))}$ stands for a particular tubular
neighborhood of the symplectic submanifold $N\subset(M,\omega)$.

######  Theorem 1.2.

Let $(M,\omega)$ be a closed symplectic manifold and $N\subset(M,\omega)$ a
symplectic submanifold of codimension $2k>2$. If $\\{\psi_{t}\\}$ is a loop of
Hamiltonian diffeomorphisms on $(M,\omega)$ that is $N$-liftable to a loop
$\\{\widetilde{\psi}_{N,t}\\}$ on
$(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$, then

$\displaystyle\mathcal{A}([\widetilde{\psi}_{N,t}])=\left[\mathcal{A}([\psi_{t}])+\frac{1}{\textup{Vol}(\widetilde{M}_{N},(\widetilde{\omega}_{N,\rho})^{n})}\int_{0}^{1}\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H_{t}\,\omega^{n}dt\right]$

in $\mathbb{R}/\mathcal{P}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$,
where $H_{t}$ if the normalized Hamiltonian function of the loop
$\\{\psi_{t}\\}$.

Loosely speaking, the reason why the results of [12] extend to the case of a
submanifold of positive dimension is because many of the arguments that are
involved in the definition of symplectic blow up are in a sense
$U(k)$-equivariant. Furthermore, the diffeomorphisms $\\{\psi_{t}\\}$ and
$\\{\widetilde{\psi}_{N,t}\\}$ that appear in the above result are induced by
matrices that lie in the center of $U(k)$. This condition turns to be crucial,
since in many instances the commutativity of the matrices is fundamental.

The paper is organized as follows. In Section 2 we review the definition of
the symplectic one-point blow up as well as some facts of [11] and [12]. The
definition of the symplectic blow up a long a submanifold is review in Section
3. In Section 4 we give conditions that guarantee that a Hamiltonian
diffeomorphisms can be lifted to the symplectic blow up. Finally, Sections 5
and 6 deal with normalization of Hamiltonian functions and the proofs of the
main results respectively.

##  2\. The case of a one-point blow up

###  2.1. Symplectic structure on the one-point blow up.

In this section we review how the symplectic structure on the blow up of the
origin in $(\mathbb{C}^{k},\omega_{0})$ is defined. Following [11, Sec. 2], we
also review the conditions that a Hamiltonian diffeomorphism on
$(\mathbb{C}^{k},\omega_{0})$ needs to satisfy in order to induced a
Hamiltonian diffeomorphism on the one-point blow up. Later, we will impose
similar conditions on a Hamiltonian diffeomorphism in order to induced a
Hamiltonian on the symplectic manifold that is obtain by blowing up a compact
symplectic submanifold.

As a manifold, the one-point blow up of $\mathbb{C}^{k}$ at the origin is the
total space of the tautological line bundle,

$\widetilde{\mathbb{C}}^{k}:=\\{(z,\ell)\in\mathbb{C}^{k}\times\mathbb{C}P^{k-1}\mid
z\in\ell\\}.$

The projection maps are denoted by
$\textup{pr}\colon\widetilde{\mathbb{C}}^{k}\to\mathbb{C}P^{k-1}$ and
$\pi\colon\widetilde{\mathbb{C}}^{k}\to\mathbb{C}^{k}$. Here, $\pi$ and
$E:=\pi^{-1}(0)$ are called the blow up map is the exceptional divisor
respectively. Let $\omega_{\textup{FS}}$ be the Fubini-Study symplectic form
on $\mathbb{C}P^{k-1}$ normalized so that
$\langle\omega_{\textup{FS}},\mathbb{C}P^{1}\rangle=\pi$. For each $\rho>0$
define on $\widetilde{\mathbb{C}}^{k}$ the symplectic form

$\omega_{\rho}:=\pi^{*}\omega_{0}+\rho^{2}\textup{pr}^{*}\omega_{\textup{FS}}.$

Notice that the $\omega_{\rho}$-area of a line in the exceptional divisor is
$\pi\rho^{2}$. Denote by $B_{r}\subset\mathbb{C}^{k}$ the open ball of radius
$r$ centered at the origin and its preimage under $\pi$ by

$L_{r}:=\pi^{-1}(B_{r}).$

According to [10, Prop. 7.1.13] for each $\epsilon\in(0,1)$ there exists a
symplectic form $\widetilde{\omega}_{\rho,\epsilon}$ on
$\widetilde{\mathbb{C}}^{k}$ such that

  * •

$\widetilde{\omega}_{\rho,\epsilon}=\omega_{\rho}$ on $L_{\epsilon}$ and

  * •

$\widetilde{\omega}_{\rho,\epsilon}=\pi^{*}(\omega_{0})$ on
$\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon}.$

Later in this note it will be important to know specific details on how the
symplectic form $\widetilde{\omega}_{\rho,\epsilon}$ is defined. To that end,
one considers a diffeomorphism
$F_{\rho,\epsilon}:\mathbb{C}^{k}\setminus\\{0\\}\to\mathbb{C}^{k}\setminus\overline{B}_{\rho}$
that consists of stretching the punctured space, defined as

(1) $\displaystyle F_{\rho,\epsilon}(z):=f_{\rho,\epsilon}(|z|)\frac{z}{|z|}$

where $f_{\rho,\epsilon}:\mathbb{R}\to\mathbb{R}$ is a smooth function such
that $f_{\rho,\epsilon}(x)=\sqrt{\lambda^{2}+x^{2}}$ on $(0,\epsilon)$ and is
equal to the identity on $[\rho+\epsilon,\infty)$. Then the above symplectic
form $\widetilde{\omega}_{\rho,\epsilon}$ is defined as
$\widetilde{\omega}_{\rho,\epsilon}:=(\pi^{*}\circ
F_{\rho,\epsilon}^{*})(\omega_{0})$ on $\widetilde{\mathbb{C}}^{k}\setminus
E$.

In this case $(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})$
is defined as the symplectic blow up of the origin in
$(\mathbb{C}^{k},\omega_{0})$ of weight $\rho$. It is important to note that
the blow up map
$\pi\colon(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to(\mathbb{C}^{k},\omega_{0})$
is a symplectic diffeomorphism on the complement of
$\overline{L}_{\rho+\epsilon}\subset\widetilde{\mathbb{C}}^{k}.$

The complete details of symplectic one-point blow up, including the exact
definitions of $\widetilde{\omega}_{\rho,\epsilon}$ and $f_{\rho,\epsilon}$
appear in [10, Ch. 7].

###  2.2. Induced Hamiltonian on the one-point blow up.

Next we review some results of [11] that deal with the problem of lifting a
Hamiltonian diffeomorphism on $(\mathbb{C}^{k},\omega_{0})$ to a Hamiltonian
on $(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})$.

Let $\phi$ and $\widetilde{\phi}$ be Hamiltonian diffeomorphisms on
$(\mathbb{C}^{k},\omega_{0})$ and
$(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})$
respectively. We say that the diffeomorphism $\widetilde{\phi}$ lifts the
diffeomorphism $\phi$ if the following diagram commutes

$\textstyle{(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\widetilde{\phi}}$$\textstyle{(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{({\mathbb{C}}^{k},{\omega}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{({\mathbb{C}}^{k},{\omega}_{0}).}$

This means that $\pi\circ\widetilde{\phi}=\phi\circ\pi$.

Consider $\\{\psi_{t}\\}_{0\leq t\leq 1}$ a path of Hamiltonian
diffeomorphisms on $(\mathbb{C}^{k},\omega_{0})$ with Hamiltonian function
$H_{t}$ such that $\psi_{0}=1_{\mathbb{C}^{k}}$. The goal is to define a path
of Hamiltonian diffeomorphisms $\\{\widetilde{\psi}_{t}\\}$ on
$(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})$ that lifts
the path $\\{\psi_{t}\\}$; that is
$\pi\circ\widetilde{\psi}_{t}=\psi_{t}\circ\pi$ for each $t\in[0,1]$. At this
point it is important to note that the Hamiltonian path induced by the
function
$H_{t}\circ\pi:(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to\mathbb{R}$
fails to lift the path $\\{\psi_{t}\\}$. Away from the exceptional divisor is
true that the path induced by $H_{t}\circ\pi$ is a lift of the original path;
the problem occurs near the exceptional divisor. Note for instance that
$H_{t}\circ\pi$ is independent of the parameter $\rho$ and such parameter is
relevant in the symplectic structure on the one-point blow up. As we will see,
the Hamiltonian function that induces the desired Hamiltonian path will depend
on the parameter $\rho$.

The required condition for a path of Hamiltonian diffeomorphisms to lift to
the one-point blow up is that for each $t\in[0,1]$ there exists a matrix
$U_{t}\in U(k)$ such that $\psi_{t}=U_{t}$ on
${B}_{\rho+\epsilon}\subset\mathbb{C}^{k}$.

######  Lemma 2.3.

Let $\psi_{t}:(B_{r},\omega_{0})\to(B_{r},\omega_{0})$ be a Hamiltonian path
given by unitary matrices starting at the identity matrix with Hamiltonian
function $H_{t}$. Then

$H_{t}(z)=H_{t}(\lambda z)$

for $z\in B_{r}$ and $\lambda\in S^{1}$.

Proof. Denote by $X_{t}$ the time-dependent vector field of the path
$\\{\psi_{t}\\}$. For $\lambda\in S^{1}$, let $\phi_{\lambda}:B_{r}\to B_{r}$
be matrix multiplication by $\lambda I$. Since $\phi_{\lambda}$ is in the
center of $U(k)$ it follows that

$\displaystyle
X_{t}\circ\phi_{\lambda}=\left.\frac{d}{ds}\right|_{s=t}\psi_{s}\circ\psi_{t}^{-1}\circ\phi_{\lambda}=\left.\frac{d}{ds}\right|_{s=t}\phi_{\lambda}\circ\psi_{s}\circ\psi_{t}^{-1}=(\phi_{\lambda})_{*}X_{t}.$

Therefore

$d(H_{t}\circ\phi_{\lambda})=\omega_{0}(X_{t},(\phi_{\lambda})_{*}(\cdot))=(\phi_{\lambda})^{*}\omega_{0}(X_{t},\cdot)=dH_{t}.$

Since both functions $H_{t}$ and $H_{t}\circ\phi_{\lambda}$ agree at the
origin, it follows that $H_{t}(z)=H_{t}(\lambda z)$. $\Box$

Let $\\{\psi_{t}\\}$ be a path of Hamiltonian diffeomorphisms on
$(\mathbb{C}^{k},\omega_{0})$ such that for each $t$ the restriction of
$\psi_{t}$ to ${B}_{\rho+\epsilon}\subset\mathbb{C}^{k}$ is given by a unitary
matrix. If $H_{t}:(\mathbb{C}^{k},\omega_{0})\to\mathbb{R}$ is the Hamiltonian
function of the path, define the function
$\widetilde{H}_{t}:(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to\mathbb{R}$
by

(4)
$\displaystyle\widetilde{H}_{t}(z,\ell):=\left\\{\begin{array}[]{ll}H_{t}\circ
F_{\rho,\epsilon}\circ\pi(z,\ell)&\textup{if
}(z,\ell)\in\widetilde{\mathbb{C}}^{k}\setminus E\\\
H_{t}\left(\frac{\rho}{|w|}w\right)&\textup{if $z=0$ and
$[w]=\ell\in\mathbb{C}P^{k-1}$}.\\\ \end{array}\right.$

It follows from Lemma 2.3 and the definition of $F_{\rho,\epsilon}$ that
$\widetilde{H}_{t}$ is well defined. Note that $\widetilde{H}_{t}$ depends on
the diffeomorphism $F_{\rho,\epsilon}$ and hence on the parameter $\rho$. The
path of Hamiltonian diffeomorphisms on
$(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})$ induced by
$\widetilde{H}_{t}$ is the path that lifts the initial path on
$({\mathbb{C}}^{k},{\omega}_{0})$. The proof of this fact as well as the proof
of the next proposition appears in [11, Sec. 3].

######  Proposition 2.4.

The time-dependent vector field $\widetilde{X}_{t}$ and path of
diffeomorphisms $\\{\widetilde{\psi}_{t}\\}$ induced by the function
$\widetilde{H}_{t}:(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to\mathbb{R}$,
defined in (4), are such that

  1. (1)

$\pi_{*}(\widetilde{X}_{t})=X_{t}$ for each $t\in[0,1]$ and

  2. (2)

$\widetilde{\psi}_{t}$ is a lift of $\psi_{t}$ for each $t\in[0,1]$.

In particular, $\\{\widetilde{\psi}_{t}\\}$ is a Hamiltonian path of
$(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})$ that lifts
the Hamiltonian path $\\{\psi_{t}\\}$.

Thus if $\psi$ is a Hamiltonian diffeomorphism on
$(\mathbb{C}^{k},\omega_{0})$ such that it can be joined to the identity by a
path of Hamiltonian diffeomorphisms such that each one of them is equal to a
unitary matrix on the ball ${B}_{\rho+\epsilon}$, then $\psi$ admits a
Hamiltonian lift to the one-point blow up
$(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})$.

This definition can be extended to the one-point blow up of weight $\rho$ of
an arbitrary symplectic manifold $(M,\omega).$ For the purpose of this note it
is not relevant to review such definition. For further details see [11].

##  3\. Review of the blow up along a submanifold

Following [10, Ch. 7], we review the construction of the symplectic blow up
along a compact symplectic submanifold. The complete details of this
construction are relevant for the Hamiltonian diffeomorphism that we will
defined on the blown up manifold.

###  3.1. The blow up along a compact submanifold.

Let $N\subset(M,\omega)$ be a compact symplectic submanifold of codimension
$2k>2$. Then $\pi_{N}:TN^{\omega}\to N$ corresponds to its normal bundle which
is a symplectic vector bundle. Since the symplectic linear group contracts to
the unitary group, $\pi_{N}$ is in fact a complex vector bundle. For
simplicity, write $\nu$ for $TN^{\omega}$. Thus there exists a Hamiltonian
linear action of $U(k)$ on
$(\mathbb{R}^{2k},\omega_{0})=(\mathbb{C}^{k},\omega_{0})$ and a principal
$U(k)$-bundle $P\to N$ such that the associated fibration coincides with the
normal bundle, $\nu\simeq P\times_{U(k)}\mathbb{C}^{k}$. Let
$\mu:(\mathbb{C}^{k},\omega_{0})\to\mathfrak{u}(k)^{*}$ be the corresponding
moment map and fix a connection 1-form $A\in\Omega^{1}(P,\mathfrak{u}(k))$ on
$P\to N$. On $P\times\mathbb{C}^{k}$ define the 1-form

$\langle\mu,A\rangle_{(p,z)}(u+v):=\langle\mu(z),A_{p}(u)\rangle$

where $u\in T_{p}P$ and $v\in T_{z}\mathbb{C}^{k}$. Here
$\langle\cdot,\cdot\rangle$ stands for the pairing between
$\mathfrak{u}(k)^{*}$ and $\mathfrak{u}(k)$. Notice that the 1-form
$\langle\mu,A\rangle$ is independent of the vectors that lie in
$T\mathbb{C}^{k}$. For $\xi\in\mathfrak{u}(k)$, let $X_{\xi}$ be the vector on
$\mathbb{C}^{k}$ induced by the action of $U(k)$. Then on
$P\times\mathbb{C}^{k}$ define the 2-form

$\displaystyle\omega_{A}(u_{1}+v_{1},u_{2}+v_{2})$ $\displaystyle:=$
$\displaystyle\omega_{0}(v_{1}+X_{A(u_{1})},v_{2}+X_{A(u_{2})})-$
$\displaystyle d\langle\mu,A\rangle(u_{1}+v_{1},u_{2}+v_{2}).$

So defined, the 2-form $\omega_{A}$ is closed. Moreover, according to [10,
Thm. 6.6.3] $\omega_{A}$ is $U(k)$-invariant and horizontal. Henceforth, it
induces a closed 2-form on $\nu\simeq P\times_{U(k)}\mathbb{C}^{k}$. In order
to avoid cumbersome notation, we will make no distinction between a basic form
and the induced form on the orbit space. Note that on each fibre of
$\pi_{N}:\nu\to N$ the form $\omega_{A}$ restricts to $\omega_{0}$. Finally,
define the 2-form

(5) $\omega_{\nu,A}:=\omega_{A}+\pi_{N}^{*}(\omega)$

on $\nu.$ Notice that $\omega_{\nu,A}$ also restricts to $\omega_{0}$ on each
fibre. Furthermore, if $\nu_{0}$ stands for the zero-section of $\nu$ then
$\omega_{\nu,A}$ restricted to $\nu_{0}$ is equal to $\pi_{N}^{*}(\omega)$.
These observations imply the following lemma. First, for $\epsilon>0$ denote
by $\mathcal{U}_{\epsilon}(\nu_{0}):=P\times_{U(k)}B_{\epsilon}$ the
$\epsilon$-disk subbundle of $\nu\to N$.

######  Lemma 3.5.

Let $N\subset(M,\omega)$ be a compact symplectic submanifold of codimension
$2k$. Then there exist $\epsilon_{0}>0$ such that the 2-form

$\omega_{\nu,A}=\omega_{A}+\pi_{N}^{*}(\omega)$

is symplectic on the disk subbundle $\mathcal{U}_{\epsilon_{0}}(\nu_{0})$.
Moreover $\omega_{\nu,A}$ restricts to $\omega_{0}$ on each fibre and to
$\pi_{N}^{*}(\omega)$ on the zero-section.

It follows by the symplectic neighborhood theorem there exists a symplectic
diffeomorphism

(6)
$\displaystyle\Phi:(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})\to(\mathcal{U}(N),\omega)$

where $\mathcal{U}(N)\subset(M,\omega)$ is some neighborhood of $N$.

The outline of the construction of the symplectic blow up of $(M,\omega)$
along the symplectic submanifold $N$ is as follows: The neighborhood
$(\mathcal{U}(N),\omega)$ of $N\subset(M,\omega)$ is identified with the open
set $(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})$ via the symplectic
diffeomorphism $\Phi$ and thus the zero-section $\nu_{0}$ in $\nu$ is
identified with the submanifold $N$ is blown up. Hence we give a description
of the blow up of the zero-section in $\nu$. By the blow up of $\nu_{0}$ we
mean the total space of the associated bundle that is obtained by
$\widetilde{\mathbb{C}}^{k}$ and $P\to N$ which is denoted by
$\widetilde{\nu}$. Finally, $\widetilde{\nu}$ carries a symplectic form
$\widetilde{\omega}_{\widetilde{\nu},A,\rho}$ and such model is glued back to
$(M\setminus N,\omega)$.

The complex-linear action of $U(k)$ on $\mathbb{C}^{k}$ has a canonical lift
to $\widetilde{\mathbb{C}}^{k}$ defined as

$U.(z,\ell)\mapsto(U(z),U(\ell))\hskip 42.67912pt\textup{ for }U\in
U(k)\textup{ and }(z,\ell)\in\widetilde{\mathbb{C}}^{k}.$

This $U(k)$-space together with the $U(k)$-principal bundle $P\to N$ induce
the associated fibration $\widetilde{\pi}_{N}:\widetilde{\nu}\to N$ where
$\widetilde{\nu}:=P\times_{U(k)}\widetilde{\mathbb{C}}^{k}$. Since the blow up
map $\pi:\widetilde{\mathbb{C}}^{k}\to\mathbb{C}^{k}$ is $U(k)$-equivariant,
it induces the map

$\widetilde{\pi}:P\times_{U(k)}\widetilde{\mathbb{C}}^{k}\to
P\times_{U(k)}{\mathbb{C}}^{k}$

defined as $\widetilde{\pi}[p,(z,\ell)]:=[p,\pi(z,\ell)]=[p,z].$ Notice that
$\widetilde{\pi}:\widetilde{\nu}\to\nu$ corresponds to the blow up of the
zero-section $\nu_{0}$ in $\nu$. As in the case of the one-point blow up,
$\widetilde{\pi}$ is a diffeomorphism on the complement of
$\widetilde{\pi}^{-1}(\nu_{0})\subset\widetilde{\nu}$.

For any $\epsilon>0$, the ball $B_{\epsilon}\subset\mathbb{C}^{k}$ is
$U(k)$-invariant and hence $L_{\epsilon}\subset\widetilde{\mathbb{C}}^{k}$ is
also $U(k)$-invariant. Write
${\mathcal{U}}_{\epsilon}(\widetilde{\pi}^{-1}(\nu_{0}))$ for the subbundle
$P\times_{U(k)}L_{\epsilon}$. Notice that $\widetilde{\pi}$ maps
${\mathcal{U}}_{\epsilon}(\widetilde{\pi}^{-1}(\nu_{0}))$ onto
$\mathcal{U}_{\epsilon}(\nu_{0})$. Furthermore, $\widetilde{\pi}$ maps
${\mathcal{U}}_{\epsilon}(\widetilde{\pi}^{-1}(\nu_{0}))\setminus\widetilde{\pi}^{-1}(\nu_{0})$
diffeomorphically onto $\mathcal{U}_{\epsilon}(\nu_{0})\setminus\nu_{0}$.
Summarizing, we have the following commutative diagram of maps

$\textstyle{{\mathcal{U}}_{\epsilon}(\widetilde{\pi}^{-1}(\nu_{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}}$$\scriptstyle{\widetilde{\pi}_{N}}$$\textstyle{\mathcal{U}_{\epsilon}(\nu_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{N}}$$\scriptstyle{\Phi}$$\textstyle{\nu}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{U}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M.}$

We are now ready to define the manifold that results by blowing up
$(M,\omega)$ along the submanifold $N$. If $\epsilon_{0}>0$ is as before, the
blow up of $M$ along the compact symplectic submanifold $N$ is defined as

(7) $\displaystyle\widetilde{M}_{N}:=M\setminus N\ \
\cup_{\Phi\circ\widetilde{\pi}}\ \
{\mathcal{U}}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))$

where
${\mathcal{U}}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))\setminus\widetilde{\pi}^{-1}(\nu_{0})$
is identified with $\mathcal{U}(N)\setminus N$ via the diffeomorphism
$\Phi\circ\widetilde{\pi}$.

As in the case of the one-point blow up, there is a projection map
$\widetilde{\pi}_{(M,N)}:\widetilde{M}_{N}\to M$, also called the blow up map,
defined as

(10)
$\displaystyle\widetilde{\pi}_{(M,N)}(p):=\left\\{\begin{array}[]{ll}p&\textup{if
}p\in M\setminus N\\\ \Phi\circ\widetilde{\pi}(p)&\textup{if
}p=[p^{\prime},(z,\ell)]\in{\mathcal{U}}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0})).\end{array}\right.$

Note that $\widetilde{\pi}_{(M,N)}$ depends on the disk subbundle
${\mathcal{U}}_{\epsilon_{0}}(\nu_{0})$. We omit such dependency from the
notation of the map. The preimage of the blown up submanifold
$E_{N}:={\widetilde{\pi}}_{(M,N)}^{-1}(N)$ is called the exceptional divisor
and ${\widetilde{\pi}}_{(M,N)}$ restricted to the exceptional divisor,
$E_{N}\to N$, is a fibration with fiber $\mathbb{C}P^{k-1}$ that corresponds
to the projectivization of the normal bundle $\nu\to M$.

It remains to define a symplectic structure on $\widetilde{M}_{N}$. In the
definition of $\widetilde{M}_{N}$, the parameter $\epsilon_{0}>0$ of the disk
subbundle was irrelevant. Its importance will become apparent in the
description of the symplectic structure on it.

###  3.2. The symplectic structure on $\widetilde{M}_{N}$.

Let
$\widetilde{\mu}_{\rho}:(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to\mathfrak{u}(k)^{*}$
be the moment map of the induced action of $U(k)$ on
$(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho})$ described above.
Using the same connection 1-form $A$ on the principal bundle $P\to N$, we have
the analog of the form $\omega_{A}$ defined before, but now on
$P\times\widetilde{\mathbb{C}}^{k}$. That is, on
$P\times\widetilde{\mathbb{C}}^{k}$ we have the following 2-form

$\displaystyle\widetilde{\omega}_{A,\rho}(u_{1}+v_{1},u_{2}+v_{2})$
$\displaystyle:=$
$\displaystyle\widetilde{\omega}_{\rho,\epsilon}(v_{1}+X_{A(u_{1})},v_{2}+X_{A(u_{2})})-$
$\displaystyle
d\langle\widetilde{\mu}_{\rho},A\rangle(u_{1}+v_{1},u_{2}+v_{2})$

that is closed and descends to the orbit space
$\widetilde{\nu}:=P\times_{U(k)}\widetilde{\mathbb{C}}^{k}$. Moreover it
restricts to $\widetilde{\omega}_{\rho,\epsilon}$ on each fibre of
$\widetilde{\pi}_{N}:\widetilde{\nu}\to N$. As before, on $\widetilde{\nu}$ we
define the 2-form

(11)
$\widetilde{\omega}_{\widetilde{\nu},A,\rho}:=\widetilde{\omega}_{A,\rho}+\widetilde{\pi}_{N}^{*}(\omega).$

Recall that
$\pi:(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to(\mathbb{C}^{k},\omega_{0})$
is a symplectic diffeomorphism on the complement of
$\overline{L}_{\rho+\epsilon}$.

######  Lemma 3.6.

Let $\rho$ and $\epsilon\in(0,1)$ such that $\rho+\epsilon<\epsilon_{0}$. Then
under the map $\widetilde{\pi}:\widetilde{\nu}\to\nu$ we have that

$\widetilde{\pi}^{*}(\omega_{\nu,A})=\widetilde{\omega}_{\widetilde{\nu},A,\rho}$

on
$P\times_{U(k)}(\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon})$.

###### Proof.

In order to prove the claim, consider the forms on $P\times{\mathbb{C}}^{k}$
instead of $P\times_{U(k)}{\mathbb{C}}^{k}$. Similarly for the forms on
$P\times_{U(k)}\widetilde{\mathbb{C}}^{k}$. Hence we must show that

$(1\times\pi)^{*}(\omega_{\nu,A})=\widetilde{\omega}_{\widetilde{\nu},A,\rho}.$

First note that since $\pi_{N}$ and $\widetilde{\pi}_{N}$ are fibrations over
$N$ and $\widetilde{\pi}$ is a bundle map, it follows that
$\widetilde{\pi}_{N}^{*}(\omega)=\widetilde{\pi}^{*}\circ\pi_{N}^{*}(\omega)$
on all $\widetilde{\nu}$.

$\textstyle{P\times\widetilde{\mathbb{C}}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times\pi}$$\textstyle{\widetilde{\nu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}_{N}}$$\scriptstyle{\widetilde{\pi}}$$\textstyle{P\times{\mathbb{C}}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\nu\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{N}}$$\textstyle{N}$

Therefore
$(1\times\pi)^{*}(\pi_{N}^{*}(\omega))=\widetilde{\pi}_{N}^{*}(\omega)$. It
remains to show that
$(1\times\pi)^{*}(\omega_{A})=\widetilde{\omega}_{A,\rho}.$ In this step the
parameter $\rho$ will be relevant. Since
$\pi:(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to(\mathbb{C}^{k},\omega_{0})$
is a symplectic diffeomorphism on the complement of
$\overline{L}_{\rho+\epsilon}$ and is $U(k)$-equivariant, the moment maps
satisfy $\widetilde{\mu}_{\rho}=\mu\circ\pi$ on
$P\times(\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon})$.
This in turn imply that
$(1\times\pi)^{*}\langle\mu,A\rangle=\langle\widetilde{\mu}_{\rho},A\rangle$
also on
$P\times(\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon})$.

Once again, using the fact that the blow up map is a symplectic diffeomorphism
on the complement of $\overline{L}_{\rho+\epsilon}$ and the fact that
$\pi_{*}(\widetilde{X}_{\xi})=X_{\xi}$ it follows that

$\displaystyle\omega_{0}(\pi_{*}(v_{1})+X_{A(u_{1})},\pi_{*}(v_{2})+X_{A(u_{2})})$
$\displaystyle=$
$\displaystyle\omega_{0}(\pi_{*}(v_{1})+\pi_{*}(\widetilde{X}_{A(u_{1})}),$
$\displaystyle\hskip
17.07182pt\pi_{*}(v_{2})+\pi_{*}(\widetilde{X}_{A(u_{2})}))$ $\displaystyle=$
$\displaystyle\pi^{*}(\omega_{0})(v_{1}+\widetilde{X}_{A(u_{1})},v_{2}+\widetilde{X}_{A(u_{2})})$
$\displaystyle=$
$\displaystyle\omega_{\rho,\epsilon}(v_{1}+\widetilde{X}_{A(u_{1})},v_{2}+\widetilde{X}_{A(u_{2})})$

where $u_{j}\in TP$ and $v_{j}\in T\widetilde{\mathbb{C}}^{k}$.

All these facts show that
$(1\times\pi)^{*}(\omega_{\nu,A})=\widetilde{\omega}_{\widetilde{\nu},A,\rho}$
on $P\times(\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon})$.
Henceforth,
$\widetilde{\pi}^{*}(\omega_{\nu,A})=\widetilde{\omega}_{\widetilde{\nu},A,\rho}$
on
$P\times_{U(k)}(\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon})$.
∎

For the rest of the article we fix the parameters $\rho>0$ and
$\epsilon\in(0,1)$ such that $\rho+\epsilon<\epsilon_{0}$. Remember that
$\Phi:(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})\to(\mathcal{U}(N),\omega)$
is a symplectic diffeomorphism and by Lemma 3.6 that
$\widetilde{\pi}^{*}(\omega_{\nu,A})=\widetilde{\omega}_{\widetilde{\nu},A,\rho}$
on
$P\times_{U(k)}(\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon})$.
Therefore on
$P\times_{U(k)}(\widetilde{\mathbb{C}}^{k}\setminus\overline{L}_{\rho+\epsilon})$
we have that
$\widetilde{\pi}^{*}\circ\Phi^{*}(\omega)=\widetilde{\omega}_{\widetilde{\nu},A,\rho}$.
Hence it follows that $\widetilde{M}_{N}$ carries a symplectic form
$\widetilde{\omega}_{N,\rho}$ defined as

(14)
$\displaystyle\widetilde{\omega}_{N,\rho}:=\left\\{\begin{array}[]{ll}\omega&\textup{on
}M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})\\\
\widetilde{\omega}_{\widetilde{\nu},A,\rho}&\textup{on
}\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))\end{array}\right.$

where $\epsilon^{\prime}$ is such that
$\rho+\epsilon<\epsilon^{\prime}<\epsilon_{0}$. Note that
$\widetilde{\pi}_{(M,N)}$ is a symplectic diffeomorphism on
$M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})$.

##  4\. Induced Hamiltonians on the blown up manifold

In this section we give conditions on a Hamiltonian diffeomorphism on
$(M,\omega)$ in order to ensure that it lifts to a Hamiltonian diffeomorphism
on the blown up manifold $(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$. As
stated in the previous section, a diffeomorphism $\widetilde{\phi}$ on
$\widetilde{M}_{N}$ is said to be a lift of the diffeomorphism $\phi$ on $M$
if the following diagram commutes

$\textstyle{\widetilde{M}_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}_{(M,N)}}$$\scriptstyle{\widetilde{\phi}}$$\textstyle{\widetilde{M}_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}_{(M,N)}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{M.}$

By the nature of the blow up construction, these conditions will only take
place in a neighborhood of the symplectic submanifold $N\subset M$. To that
end, fix a tubular neighborhood $\mathcal{U}(N)$ of $N$ and a symplectic
diffeomorphism
$\Phi:(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})\to(\mathcal{U}(N),\omega)$
as before.

######  Definition 1.

Let $N\subset(M,\omega)$ be a compact symplectic submanifold of codimension
$2k$. A symplectic diffeomorphism $\psi:(M,\omega)\to(M,\omega)$ is called
$N$-liftable if

  * a)

$\psi(\mathcal{U}(N))\subset\mathcal{U}(N)$ and

  * b)

the map
$\Phi^{-1}\circ\psi\circ\Phi:(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})\to(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})$
takes the form

$[p,z]\mapsto[p,U(z)]$

where $U$ lies in the center of $U(k)$.

Remark.

  * •

The above definition depends on the tubular neighborhood $\mathcal{U}(N)$ and
the diffeomorphism $\Phi$. In order to avoid cumbersome notation we omit such
dependencies from the definition.

  * •

The reason that the matrix $U$ must lie in the center of $U(k)$ is to ensure
that $\widetilde{\mu}_{\rho}(U(z))=\widetilde{\mu}_{\rho}(z)$, where
$\widetilde{\mu}_{\rho}:(\widetilde{\mathbb{C}}^{k},\widetilde{\omega}_{\rho,\epsilon})\to\mathfrak{u}(k)^{*}$
is the moment map of the induced action used in the definition of the
symplectic form on described in Sec. 3.2. It will also be useful when proving
that the induced diffeomorphisms on the blown up manifold by a Hamiltonian is
also Hamiltonian.

  * •

If $\phi$ is a gauge transformation of $P\to N$, then the map
$[p,z]\mapsto[\phi(p),U(z)]$ is a diffeomorphism of
$\mathcal{U}_{\epsilon_{0}}(\nu_{0})$. However, if $\phi\neq 1_{P}$, such
diffeomorphism on $(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})$ will
not be symplectic.

If $\psi:(M,\omega)\to(M,\omega)$ is $N$-liftable, denote by $\psi_{\nu}$ the
diffeomorphism $\Phi^{-1}\circ\psi\circ\Phi$ defined on
$(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})$. Hence $\psi_{\nu}$
induces the diffeomorphism
$\widetilde{\psi}_{\nu}:\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))\to\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))$
given by

$\widetilde{\psi}_{\nu}[p,(z,\ell)]:=[p,(U(z),U(\ell))].$

Since $\psi$ is $N$-liftable it maps the complement of
$\;\mathcal{U}(N)\subset M$ to itself. Therefore $\widetilde{\psi}_{\nu}$
together with $\psi$ give rise to a well-defined diffeomorphism on the blown
up manifold $\widetilde{M}_{N}$ that we denote by $\widetilde{\psi}_{N}$ . So
defined $\widetilde{\psi}_{N}$ is a lift of $\psi$, that is
$\widetilde{\pi}_{(M,N)}\circ\widetilde{\psi}_{N}=\psi\circ\widetilde{\pi}_{(M,N)}$.
It remains to show that it is a symplectic diffeomorphism.

######  Proposition 4.7.

If $\psi:(M,\omega)\to(M,\omega)$ is a symplectic diffeomorphism that is
$N$-liftable, then $\widetilde{\psi}_{N}$ is a symplectic diffeomorphism of
$(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}).$

###### Proof.

By definition of the symplectic form $\widetilde{\omega}_{N,\rho}$ in Eq. (14)
and of the diffeomorphism $\widetilde{\psi}_{N}$, if follows that
$\widetilde{\psi}_{N}$ is a symplectic diffeomorphism on
$M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})$. It only
remains to show that $\widetilde{\psi}_{N}$ is a symplectic diffeomorphism on
$\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))$.

Since $\psi$ is $N$-liftable, on
$\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))$ the diffeomorphism
$\widetilde{\psi}_{N}$ is given by $\widetilde{\psi}_{\nu}$. That is

$\widetilde{\psi}_{\nu}[p,(z,\ell)]=[p,(U(z),U(\ell))]$

where $U$ lies in the center of $U(k)$. In order to show that
$\widetilde{\psi}_{\nu}^{*}(\widetilde{\omega}_{\widetilde{\nu},A,\rho})=\widetilde{\omega}_{\widetilde{\nu},A,\rho}$,
we regard the symplectic form on $P\times L_{\epsilon_{0}}$ instead of
$P\times_{U(k)}L_{\epsilon_{0}}$. In particular,
$\widetilde{\omega}_{\widetilde{\nu},A,\rho}$ is a basic form. On $P\times
L_{\epsilon_{0}}$ the diffeomorphism $\widetilde{\psi}_{\nu}$ is given by
$1_{P}\times U$, where $U$ lies in the center of $U(k)$. Thus $(1\times
U)^{*}\widetilde{\omega}_{A,\rho}=\widetilde{\omega}_{A,\rho}$ and
$\widetilde{\psi}_{N}$ is a symplectic diffeomorphism of
$(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}).$

∎

Next we prove that the lift of a Hamiltonian diffeomorphism, under some
constraints, is again a Hamiltonian diffeomorphism on the blown up manifold.
To that end, consider $\psi$ a Hamiltonian diffeomorphism on $(M,\omega)$ such
that there exists a path $\\{\psi_{t}\\}_{0\leq t\leq 1}$ that start at the
identity, ends at $\psi$ and for each $t$ the diffeomorphism $\psi_{t}$ is
$N$-liftable. Hence, $\\{\psi_{\nu,t}\\}$ is a Hamiltonian path on
$(\mathcal{U}_{\epsilon_{0}}(\nu_{0}),\omega_{\nu,A})$. Moreover for each
$t\in[0,1]$ there exist a matrix $U_{t}$ in the center of $U(k)$ such that

$\psi_{\nu,t}[p,z]=[p,U_{t}(z)].$

Since each matrix $U_{t}$ is of the form $\lambda_{t}\cdot 1_{k\times k}$ for
$\lambda_{t}\in\mathbb{C}$ and $|\lambda_{t}|=1$, it follows that the induced
Hamiltonian function of the path of unitary matrices is autonomous. Moreover,
it is given by $H(z):=c\sum_{j}(x_{j}^{2}+y_{j}^{2})=c|z|^{2}$ for some
$c\in\mathbb{R}$. Recall from Section 2.2 that such Hamiltonian function $H$
induces the Hamiltonian $\widetilde{H}:L_{\epsilon_{0}}\to\mathbb{R}$ defined
in Eq. (4). Further it also satisfies the relation
$\widetilde{H}(U(z),U(\ell))=\widetilde{H}(z,\ell)$. Henceforth, we define the
function
$\widetilde{H}_{t,N}:(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})\to\mathbb{R}$
by

(17)
$\displaystyle\widetilde{H}_{N,t}(q):=\left\\{\begin{array}[]{ll}H_{t}(\widetilde{\pi}_{(M,N)}(q))&\textup{if
}q\in M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})\\\
\widetilde{H}(z,\ell)&\textup{if
}q=[p,(z,\ell)]\in\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0})).\end{array}\right.$

Next we show that $\widetilde{H}_{N,t}$ is the Hamiltonian function of the
lifted path of symplectic diffeomorphisms.

######  Proposition 4.8.

Let $\psi_{t}:(M,\omega)\to(M,\omega)$ be a path of Hamiltonian
diffeomorphisms such that each $\psi_{t}$ is $N$-liftable and $\psi_{0}=1$.
Then
$\widetilde{\psi}_{N,t}:(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})\to(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$
is a Hamiltonian path induced by $\widetilde{H}_{N,t}$.

###### Proof.

Since the Hamiltonian path $\\{\psi_{t}\\}$ is $N$-liftable, by Prop. 4.7 the
path $\\{\widetilde{\psi}_{N,t}\\}$ consists of symplectic diffeomorphisms on
$(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$ and satisfies
$\psi_{t}\circ\widetilde{\pi}_{(M,N)}=\widetilde{\pi}_{(M,N)}\circ\widetilde{\psi}_{N,t}$
for each $t$. Thus if $\widetilde{X}_{t}$ is the time-dependent vector field
induced by $\\{\widetilde{\psi}_{N,t}\\}$, then
$(\widetilde{\pi}_{(M,N)})_{*}\widetilde{X}_{t}=X_{t}$. Thus by the definition
of the function $\widetilde{H}_{N,t}$ on (17) and the fact that
$\widetilde{\pi}_{(M,N)}$ is a symplectic diffeomorphism on
$M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})$ it
follows that

$\widetilde{\omega}_{N}(\widetilde{X}_{t},\cdot)=d\widetilde{H}_{t,N}$

holds on
$\widetilde{M}_{N}\setminus\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))$.

Now on $\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))\simeq
P\times_{U(k)}L_{\epsilon_{0}}$ the path $\\{\widetilde{\psi}_{N,t}\\}$ is
given by $[p,(z,\ell)]\mapsto[p,(U_{t}(z),U_{t}(\ell))]$. Thus the time-
dependent vector field is actually independent of time and takes the form
$0+\widetilde{X}$ where $\widetilde{X}$ is a vector field on
$L_{\epsilon_{0}}$. Also, on
$\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))$ the symplectic
form $\widetilde{\omega}_{N}$ is equal to
$\widetilde{\omega}_{\widetilde{\nu},A,\rho}=\widetilde{\omega}_{A,\rho}+\widetilde{\pi}_{N}^{*}(\omega)$.
Therefore,

$\displaystyle\widetilde{\omega}_{\widetilde{\nu},A,\rho}(\widetilde{X}_{t},\cdot)$
$\displaystyle=$
$\displaystyle\widetilde{\omega}_{\widetilde{\nu},A,\rho}(0+\widetilde{X},\cdot)$
$\displaystyle=$
$\displaystyle\widetilde{\omega}_{A,\rho}(0+\widetilde{X},\cdot)+\widetilde{\pi}_{N}^{*}(\omega)(0+\widetilde{X},\cdot)$
$\displaystyle=$
$\displaystyle\widetilde{\omega}_{\rho,\epsilon}(\widetilde{X}+0,\cdot)-d\langle\widetilde{\mu}_{\rho},A\rangle(0+\widetilde{X},\cdot)+\widetilde{\pi}_{N}^{*}(\omega)(0+\widetilde{X},\cdot)$
$\displaystyle=$
$\displaystyle\widetilde{\omega}_{\rho,\epsilon}(\widetilde{X}+0,\cdot)$

By [11, Prop. 3.7], on $L_{\epsilon_{0}}$ we have that
$\widetilde{\omega}_{\rho,\epsilon}(\widetilde{X},\cdot)=d\widetilde{H}$ where
the function $\widetilde{H}$ was defined in Eq. (4). Thus the proposition
follows. ∎

Notice that if $\\{\psi_{t}\\}$ is a loop of Hamiltonian diffeomorphisms on
$(M,\omega)$ that is $N$-liftable, then the lift
$\\{\widetilde{\psi}_{N,t}\\}$ is also loop of Hamiltonian diffeomorphisms. We
are interested in understanding the the behavior of the loop
$\\{\widetilde{\psi}_{N,t}\\}$ in the fundamental group of
$\textup{Ham}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}).$

##  5\. Weinstein’s morphism

The period group $\mathcal{P}(M,\omega)$ of $(M,\omega)$ is defined as the
image of the pairing $\langle\omega,\cdot\rangle\colon
H_{2}(M;\mathbb{Z})\to\mathbb{R}$. Then Weinstein’s morphism [13],

$\mathcal{A}:\pi_{1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}(M,\omega)$

is defined via the action functional on a loop $\psi=[\\{\psi_{t}\\}]$ as

$\mathcal{A}(\psi)=\left[-\int_{D}u^{*}(\omega)+\int_{0}^{1}H_{t}(\psi_{t}(x_{0}))dt\right].$

Here $D$ is the unit closed disk and $u:D\to M$ is a smooth function such that
$u(\partial D)$ agrees with the loop $\\{\psi_{t}(x_{0})\\}$, $x_{0}\in M$ a
base point and $H_{t}$ is the 1-periodic Hamiltonian function induced by the
Hamiltonian loop $\\{\psi_{t}\\}$ subject to the normalized condition

$\int_{M}H_{t}\,\omega^{n}=0$

for every $t\in[0,1]$.

The aim is to relate Weinstein’s morphism on $\pi_{1}(\textup{Ham}(M,\omega))$
and $\pi_{1}(\textup{Ham}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}))$ on
a particular class of loops. To that end, recall that we must consider a
Hamiltonian diffeomorphisms on $(M,\omega)$ that are $N$-liftable. As seen
above, such Hamiltonians are induced by Hamiltonians functions on
$(\mathbb{C}^{n},\omega)$ that on a neighborhood of the origin take the form
$c\sum_{j}(x_{j}^{2}+y_{j}^{2})=c|z|^{2}$ for some $c\in\mathbb{R}$.

######  Lemma 5.9.

Let $H:M\to\mathbb{R}$ be a smooth function that is induced by a Hamiltonian
diffeomorphism that is $N$-liftable, $\rho$ as before and
$\widetilde{H}_{N}:\widetilde{M}_{N}\to\mathbb{R}$ the induced function
defined in (17). Then

$\int_{\widetilde{M}_{N}}\widetilde{H}_{N}\,\widetilde{\omega}_{N,\rho}^{n}=\int_{M}H\,\omega^{n}-\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H\,\omega^{n}.$

###### Proof.

First note that $\widetilde{M}_{N}$ is the union of the disjoint sets
$M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})$ and
$\mathcal{U}_{\epsilon^{\prime}}(\widetilde{\pi}^{-1}(\nu_{0}))$. Recall that
$\epsilon^{\prime}$ is such that
$\rho+\epsilon<\epsilon^{\prime}<\epsilon_{0}.$ Then from the definition of
$\widetilde{H}_{N}$ in (17) and of $\widetilde{H}$ in (4) it follows that

$\displaystyle\int_{\widetilde{M}_{N}}\widetilde{H}_{N}\,\widetilde{\omega}_{N,\rho}^{n}$
$\displaystyle=$
$\displaystyle\int_{M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})}\widetilde{H}_{N}\,\widetilde{\omega}_{N,\rho}^{n}+\int_{\mathcal{U}_{\epsilon^{\prime}}(\widetilde{\pi}^{-1}(\nu_{0}))}\widetilde{H}_{N}\,\widetilde{\omega}_{N,\rho}^{n}$
$\displaystyle=$
$\displaystyle\int_{M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})}H\circ\widetilde{\pi}_{(M,N)}\,\widetilde{\omega}_{N,\rho}^{n}+\int_{\mathcal{U}_{\epsilon^{\prime}}(\widetilde{\pi}^{-1}(\nu_{0}))}\widetilde{H}\,\widetilde{\omega}_{N,\rho}^{n}$
$\displaystyle=$
$\displaystyle\int_{M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})}H\,\omega^{n}+\int_{\mathcal{U}_{\epsilon^{\prime}}(\widetilde{\pi}^{-1}(\nu_{0}))}\widetilde{H}\,\widetilde{\omega}_{\widetilde{\nu},A,\rho}^{n}$
$\displaystyle=$
$\displaystyle\int_{M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})}H\,\omega^{n}+\int_{(P\times_{U(k)}L_{\epsilon^{\prime}})\setminus\nu_{0}}\widetilde{H}\,\widetilde{\omega}_{\widetilde{\nu},A,\rho}^{n}$

Recall that the diffeomorphism
$F_{\rho,\epsilon}:\mathbb{C}^{k}\setminus\\{0\\}\to\mathbb{C}^{k}\setminus\overline{B}_{\rho}$
defined in (1) is $U(k)$-equivariant. Hence it induces a diffeomorphism ${\bf
F}_{\rho,\epsilon}:P\times_{U(k)}(\mathbb{C}^{k}\setminus\\{0\\})\to
P\times_{U(k)}(\mathbb{C}^{k}\setminus\overline{B}_{\rho})$ and

$\displaystyle{\bf F}_{\rho,\epsilon}^{*}(\omega_{\nu,A})$ $\displaystyle=$
$\displaystyle{\bf F}_{\rho,\epsilon}^{*}(\omega_{A})+{\bf
F}_{\rho,\epsilon}^{*}(\pi_{N}^{*}(\omega))$ $\displaystyle=$
$\displaystyle\widetilde{\omega}_{A,\rho}+\widetilde{\pi}_{N}^{*}(\omega)$
$\displaystyle=$ $\displaystyle\widetilde{\omega}_{\widetilde{\nu},A,\rho}.$

It then follows that

$\displaystyle\int_{(P\times_{U(k)}L_{\epsilon^{\prime}})\setminus\nu_{0}}\widetilde{H}\,\widetilde{\omega}_{\widetilde{\nu},A,\rho}^{n}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{U}_{\epsilon^{\prime}}(\widetilde{\pi}^{-1}(\nu_{0}))}\widetilde{H}\,{\bf
F}_{\rho,\epsilon}^{*}\omega_{\nu,A}^{n}$ $\displaystyle=$
$\displaystyle\int_{P\times_{U(k)}(B_{\epsilon^{\prime}}\setminus\overline{B_{\rho}})}H\,\omega_{\nu,A}^{n}.$

Hence

$\displaystyle\int_{\widetilde{M}_{N}}\widetilde{H}_{N}\,\widetilde{\omega}_{N,\rho}^{n}$
$\displaystyle=$
$\displaystyle\int_{M\setminus\Phi(\overline{\mathcal{U}_{\epsilon^{\prime}}(\nu_{0})})}H\,\omega^{n}+\int_{P\times_{U(k)}(B_{\epsilon^{\prime}}\setminus\overline{B_{\rho}})}H\,\omega_{\nu,A}^{n}$
$\displaystyle=$
$\displaystyle\int_{M}H\,\omega^{n}-\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H\,\omega^{n}.$

∎

##  6\. Proof of Theorems 1.1 and 1.2

Now we prove the main results of this article that appeared at the
Introduction.

Theorem 1.2. Let $(M,\omega)$ be a closed symplectic manifold and
$N\subset(M,\omega)$ a symplectic submanifold of codimension $2k>2$. If
$\\{\psi_{t}\\}$ is a loop of Hamiltonian diffeomorphisms on $(M,\omega)$ that
is $N$-liftable to a loop $\\{\widetilde{\psi}_{N,t}\\}$ on
$(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$, then

$\displaystyle\mathcal{A}([\widetilde{\psi}_{N,t}])=\left[\mathcal{A}([\psi_{t}])+\frac{1}{\textup{Vol}(\widetilde{M}_{N},(\widetilde{\omega}_{N,\rho})^{n})}\int_{0}^{1}\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H_{t}\,\omega^{n}dt\right]$

in $\mathbb{R}/\mathcal{P}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$,
where $H_{t}$ if the normalized Hamiltonian function of the loop
$\\{\psi_{t}\\}$.

###### Proof.

Let $\\{\psi_{t}\\}$ be a loop of Hamiltonian diffeomorphisms on $(M,\omega)$
that is $N$-liftable with normalized Hamiltonian function $H_{t}$. If
$\\{\widetilde{\psi}_{N,t}\\}$ is the induced loop on
$(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$, denote by
$\widetilde{H}_{N,t}$ its Hamiltonian function defined by Eq. (17).

It follows from Lemma 5.9 that for each $t$,

(18)
$\displaystyle\int_{\widetilde{M}_{N}}\widetilde{H}_{N,t}\,\widetilde{\omega}_{N,\rho}^{n}=-\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H_{t}\,\omega^{n}.$

Denote such number by $c(\rho,t)\in\mathbb{R}$. Hence the normalized
Hamiltonian function of the loop $\\{\widetilde{\psi}_{N,t}\\}$ is

$\widetilde{\bf
H}_{N,t}:=\widetilde{H}_{N,t}-\frac{1}{\textup{Vol}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}^{n})}c(\rho,t).$

Thus, taking a disk $D$ and a point $x_{0}$ in
$\widetilde{M}_{N}\setminus\mathcal{U}_{\epsilon_{0}}(\widetilde{\pi}^{-1}(\nu_{0}))$,

$\displaystyle\mathcal{A}([\widetilde{\psi}_{N,t}])$ $\displaystyle=$
$\displaystyle\left[-\int_{D}u^{*}(\widetilde{\omega}_{N,\rho})+\int_{0}^{1}\widetilde{\bf
H}_{N,t}(\widetilde{\psi}_{N,t}(x_{0}))dt\right]$ $\displaystyle=$
$\displaystyle\left[-\int_{D}u^{*}(\omega)+\int_{0}^{1}\widetilde{H}_{N,t}(\widetilde{\psi}_{N,t}(x_{0}))dt\right.$
$\displaystyle\left.-\frac{1}{\textup{Vol}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}^{n})}\int_{0}^{1}c(\rho,t)dt\right].$

Hence from Eq. (18) we get

$\displaystyle\mathcal{A}([\widetilde{\psi}_{N,t}])=\left[\mathcal{A}([\psi_{t}])+\frac{1}{\textup{Vol}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}^{n})}\int_{0}^{1}\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H_{t}\,\omega^{n}dt\right].$

∎

In order to give meaning to the expression that was proven above, now we
relate the periods groups of $(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$
and $(M,\omega)$. Recall that the blow up map
$\widetilde{\pi}_{(M,N)}:\widetilde{M}_{N}\to M$ is a diffeomorphism on the
complement of $\widetilde{\pi}_{(M,N)}^{-1}(N)$. Moreover,
$\widetilde{\pi}_{(M,N)}$ induces a symplectic diffeomorphism on the
complement of a neighborhood of $\widetilde{\pi}_{(M,N)}^{-1}(N)$. Therefore,
the maps

$\langle\widetilde{\omega}_{N,\rho},\cdot\rangle:H_{2}(\widetilde{M}_{N}\setminus\widetilde{\pi}_{(M,N)}^{-1}(N);\mathbb{Z})\to\mathbb{R}\textup{
and }\langle\omega,\cdot\rangle:H_{2}(M_{N};\mathbb{Z})\to\mathbb{R}$

have the same image. It remains to determine the value of
$\langle\widetilde{\omega}_{N,\rho},\cdot\rangle$ on the kernel of
$\widetilde{\pi}_{(M,N)*}:H_{2}(\widetilde{M}_{N};\mathbb{Z})\to
H_{2}(M;\mathbb{Z})$. From the commutative diagram

$\textstyle{H_{1}(\nu\setminus\nu_{0};\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{H_{2}(\widetilde{M}_{N};\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}_{(M,N)*}}$$\textstyle{H_{2}(M\setminus
N;\mathbb{Z})\oplus
H_{2}(\widetilde{\nu};\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\oplus\widetilde{\pi}_{*}}$$\textstyle{H_{1}(\nu\setminus\nu_{0};\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}(M;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}(M\setminus
N;\mathbb{Z})\oplus
H_{2}(\nu;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

it suffices to look at the map
$\widetilde{\pi}_{*}:H_{2}(\widetilde{\nu};\mathbb{Z})\to
H_{2}(\nu;\mathbb{Z})$.

Recall that the closed 2-form $\widetilde{\omega}_{\widetilde{\nu},A,\rho}$ on
the total space of the fibration $\widetilde{\pi}_{N}:\widetilde{\nu}\to N$
restricts to $\widetilde{\omega}_{\rho,\epsilon}$ on each fibre.

The relation between the homology of $\widetilde{M}_{N}$ and $M$ is as follows

$H_{2}(\widetilde{M}_{N},\mathbb{Z})\simeq
H_{2}(M,\mathbb{Z})\oplus\textup{ker}\\{{\widetilde{\pi}}_{N,*}:H_{2}(\widetilde{\nu},\mathbb{Z})\to
H_{2}(N,\mathbb{Z})\\}.$

Moreover, if $\alpha\in H_{2}(\widetilde{M}_{N};\mathbb{Z})$ takes the form
$\widetilde{\pi}_{(M,N)*}(\alpha)+0$ in the above decomposition, then

$\langle\widetilde{\omega}_{N,\rho},\alpha\rangle=\langle\omega,\widetilde{\pi}_{(M,N)*}(\alpha)\rangle.$

Since $\widetilde{\pi}_{(M,N)*}:H_{2}(\widetilde{M}_{N};\mathbb{Z})\to
H_{2}(M;\mathbb{Z})$ is surjective, we have that

$\mathcal{P}(M,\omega)\subset\mathcal{P}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}).$

Finally,
$\textup{ker}\\{{\widetilde{\pi}}_{N,*}:H_{2}(\widetilde{\nu};\mathbb{Z})\to
H_{2}(N;\mathbb{Z})\\}$ is generated by a single class that its restriction to
the fiber is the generator of $H_{2}(\mathbb{C}P^{k-1};\mathbb{Z})$. Hence we
have the following result.

######  Lemma 6.10.

Let $(M,\omega)$ and $(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$ as
above. Assume that $N$ is compact, then

$\displaystyle\mathcal{P}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})=\mathcal{P}(M,\omega)+\mathbb{Z}\langle\pi\rho^{2}\rangle.$

Recall from Lemma 3.5 that on a neighborhood $P\times_{U(k)}B_{\epsilon_{0}}$
of the zero-section of $\nu=P\times_{U(k)}\mathbb{C}^{k}$ the symplectic form
is given by

$\displaystyle\omega_{\nu,A}(u_{1}+v_{1},u_{2}+v_{2})$ $\displaystyle=$
$\displaystyle\omega_{0}(v_{1}+X_{A(u_{1})},v_{2}+X_{A(u_{2})})-$
$\displaystyle d\langle\mu,A\rangle(u_{1}+v_{1},u_{2}+v_{2})+$
$\displaystyle\pi_{N}^{*}(\omega)(u_{1}+v_{1},u_{2}+v_{2})$

for $u_{j}\in TP$ and $v_{j}\in T\mathbb{C}^{k}$. Hence on the normal bundle
$\nu$ we have that

(20) $\displaystyle\omega_{\nu,A}^{n}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\omega_{0}^{j}\;\pi_{N}^{*}(\omega)^{n-j}+d\alpha$

for some $(2n-1)$-form $\alpha$. (Keep in mind that in this equation the
2-form $\omega_{0}$ must be evaluated as in Eq. (6)).

######  Lemma 6.11.

Let $H:\nu\to\mathbb{R}$ be a smooth function. Then,

$\displaystyle\int_{\nu}H\;\omega_{\nu,A}^{n}$ $\displaystyle=$
$\displaystyle\int_{N}(\pi_{N})_{*}(H\;\omega_{0}^{k})\;\omega^{n-k}.$

###### Proof.

First we integrate along the fiber with respect to $\pi_{N}:\nu\to N$ and then
use the commutativity of the diagram

$\textstyle{\nu\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{N}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{pt\\}.}$

Therefore by the projection formula we get that

$\displaystyle(\pi_{N})_{*}(H\;\omega_{\nu,A}^{n})$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}(\pi_{N})_{*}\left(H\;\omega_{0}^{j}\;\pi_{N}^{*}(\omega)^{n-j}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{n}(\pi_{N})_{*}(H\;\omega_{0}^{j})\;\omega^{n-j}.$

Note that in this expression, the symplectic form $\omega$ is restricted to
the submanifold $N$.

Next we integrate on the submanifold $N$,

$\displaystyle\int_{\nu}H\;\omega_{\nu,A}^{n}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\int_{N}(\pi_{N})_{*}(H\;\omega_{0}^{j})\;\omega^{n-j}$
$\displaystyle=$
$\displaystyle\sum_{j=k}^{n}\int_{N}(\pi_{N})_{*}(H\;\omega_{0}^{j})\;\omega^{n-j}$
$\displaystyle=$
$\displaystyle\int_{N}(\pi_{N})_{*}(H\;\omega_{0}^{k})\;\omega^{n-k}.$

The last equality follows from the fact that the form $\omega_{0}$ comes from
the fiber $\mathbb{C}^{k}$ of $\pi_{N}$. ∎

Notice that from this result the evaluation of $\omega_{0}$ as in Eq. (6)
becomes irrelevant.

In particular, restrict the above integral to ${\mathcal{U}_{\rho}(\nu_{0})}$
and take $H\equiv 1$. Then are able to compute the volume of
${\mathcal{U}_{\rho}(\nu_{0})}$,

$\displaystyle\textup{Vol}({\mathcal{U}_{\rho}(\nu_{0})},\omega_{\nu,A}^{n})$
$\displaystyle=$
$\displaystyle\int_{{\mathcal{U}_{\rho}(\nu_{0})}}\omega_{\nu,A}^{n}$
$\displaystyle=$
$\displaystyle\int_{N}(\pi_{N}|_{\mathcal{U}_{\rho}(\nu_{0})})_{*}(\omega_{0}^{k})\;\omega^{n-k}$
$\displaystyle=$
$\displaystyle\textup{Vol}(B_{\rho},\omega_{0}^{k})\cdot\textup{Vol}(N,\omega^{n-k})$
$\displaystyle=$
$\displaystyle\frac{\pi^{k}\rho^{2k}}{k!}\cdot\textup{Vol}(N,\omega^{n-k}).$

###  6.1. Description of the loop.

In order to prove Thm. 1.1, we need to defined a loop of Hamiltonian
diffeomorphisms on $(M,\omega)$ supported on $P\times_{U(k)}B_{\epsilon_{0}}$,
a neighborhood of $N$. The definition of such loop follows the same line of
thought as the Hamiltonian loop constructed in [12, Sec. 2].

Let $\alpha:\mathbb{R}\to\mathbb{R}$ such that $\alpha(0)=0$ and
$\alpha(1)\neq 0$. Then consider the path of diagonal matrices
$A_{t}:=\exp{(2\pi i\alpha(t))}\cdot 1_{k\times k}$. Hence $A_{t}$ lies in the
center of $U(k)$ and it induces on $(\mathbb{C}^{k},\omega_{0})$ a path
$\\{\psi^{\alpha}_{t}\\}_{0\leq t\leq 1}$ of Hamiltonian diffeomorphisms
starting at the identity and with Hamiltonian function

$H^{\alpha}_{t}(z_{1},\ldots,z_{k}):=\pi\;\alpha^{\prime}(t)\;\sum_{j=1}^{k}|z_{j}|^{2}.$

As before we have the positive parameters $\rho,\epsilon_{0}$ and
$\epsilon\in(0,1)$ such that $\rho+\epsilon<\epsilon_{0}$, that are used in
the definition of $(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho})$. Now fix a
$U(k)$-invariant smooth function $g:\mathbb{C}^{k}\to\mathbb{R}$ supported in
$B_{\epsilon_{0}}$ such that $g\equiv 1$ on $B_{\rho}$. Consider the
Hamiltonian function

$H^{\alpha,g}_{t}:=g\cdot H^{\alpha}_{t}$

and let $\\{\psi^{\alpha,g}_{t}\\}_{0\leq t\leq 1}$ be the induced Hamiltonian
path. Since $\omega_{0}$ and $H^{\alpha,g}_{t}$ are $U(k)$-equivariant, it
follows that each $\psi^{\alpha,g}_{t}$ is also $U(k)$-equivariant.
Furthermore, on $B_{\rho}$ we have that
$\psi^{\alpha,g}_{t}\equiv\psi^{\alpha}_{t}$ and $H^{\alpha,g}_{t}\equiv
H^{\alpha}_{t}$. Note that as in [12, Lemma 2.4],

$\displaystyle\int_{0}^{1}\int_{B_{\rho}}\;H^{\alpha,g}_{t}\;\omega_{0}^{k}\;dt$
$\displaystyle=$
$\displaystyle(\alpha(1)-\alpha(0))\;\pi\int_{B_{\rho}}\sum_{j=1}^{k}x_{j}^{2}+y_{j}^{2}\frac{1}{k!}dx_{1}\cdots
dy_{k}$ $\displaystyle=$
$\displaystyle\alpha(1)\cdot\frac{\pi}{k!}\cdot\frac{\pi^{k}\rho^{2k+2}}{k+1}.$

Since each $\psi^{\alpha,g}_{t}$ is $U(k)$-equivariant, it induces a path of
diffeomorphisms $1\times\psi^{\alpha,g}_{t}$ on
$\nu=P\times_{U(k)}\mathbb{C}^{k}.$ Furthermore, each diffeomorphism is
supported on $\mathcal{U}_{\epsilon}(\nu_{0})=P\times_{U(k)}B_{\epsilon_{0}}$.

######  Lemma 6.12.

The path $1\times\psi^{\alpha,g}_{t}$ restricted to
$\mathcal{U}_{\epsilon_{0}}(\nu_{0})$ is a Hamiltonian path with Hamiltonian
function $1\times H^{\alpha,g}_{t}.$

###### Proof.

Recall that the 2-form $\omega_{\nu,A}$ is a basic form on $P\times
B_{\epsilon_{0}}$, in particular is $U(k)$-invariant. Therefore,
$1\times\psi^{\alpha,g}_{t}$ preserves the form $\omega_{\nu,A}$ and is a
symplectic diffeomorphism on $\mathcal{U}_{\epsilon_{0}}(\nu_{0})$.

It remains to prove that is Hamiltonian. Note that by definition on
$(B_{\epsilon_{0}},\omega_{0})$ we have that $\\{\psi^{\alpha,g}_{t}\\}$ is
Hamiltonian with Hamiltonian function $H^{\alpha,g}_{t}$. Let
$X^{\alpha,g}_{t}$ be the associated Hamiltonian vector field,
$\omega_{0}(X^{\alpha,g}_{t},\cdot)=dH^{\alpha,g}_{t}$. Recall from Eq. (6)
that

$\displaystyle\omega_{\nu,A}(u_{1}+v_{1},u_{2}+v_{2})$ $\displaystyle=$
$\displaystyle\omega_{0}(v_{1}+X_{A(u_{1})},v_{2}+X_{A(u_{2})})-$
$\displaystyle d\langle\mu,A\rangle(u_{1}+v_{1},u_{2}+v_{2})+$
$\displaystyle\pi_{N}^{*}(\omega)(u_{1}+v_{1},u_{2}+v_{2})$

for $u_{j}\in TP$ and $v_{j}\in TB_{\epsilon_{0}}$. If $X_{\xi}$ is the vector
field on $B_{\epsilon_{0}}$ induced by $\xi\in\mathfrak{u}(k)$ we have that
$\omega_{0}(X_{\xi_{1}},X_{\xi_{2}})=\langle\mu,[{\xi_{1}},{\xi_{2}}]\rangle$
for any $\xi_{1},\xi_{2}\in\mathfrak{u}(k)$. In particular, the vector field
$X^{\alpha,g}_{t}$ is of the form $X_{\xi}$ for ${\xi}$ in the center of
$\mathfrak{u}(k)$ and therefore $\omega_{0}(X^{\alpha,g}_{t},X_{\xi})=0$ for
any ${\xi}$. Then for $u\in TP$ and $v\in TB_{\epsilon_{0}}$

$\displaystyle\omega_{\nu,A}(X^{\alpha,g}_{t},u+v)$ $\displaystyle=$
$\displaystyle\omega_{0}(X^{\alpha,g}_{t},v+X_{A(u)})$ $\displaystyle=$
$\displaystyle\omega_{0}(X^{\alpha,g}_{t},v)$ $\displaystyle=$ $\displaystyle
dH^{\alpha,g}_{t}(v)$ $\displaystyle=$ $\displaystyle d(1\times
H^{\alpha,g}_{t})(u,v)$

where $\langle\mu,A\rangle$ vanishes since $v_{1}=0$. ∎

Now we will consider another path of Hamiltonian diffeomorphisms in order to
create a loop by concatenating it with the path
$\\{\psi^{\alpha,g}_{t}\\}_{0\leq t\leq 1}$. Hence, consider
$\beta:\mathbb{R}\to\mathbb{R}$ such that $\beta(0)=0$ and
$\alpha(t)-\beta(t)\equiv 1$ in a neighborhood of 1. As above denote by
$\\{\psi^{\beta,g}_{t}\\}_{0\leq t\leq 1}$ the induced the path of Hamiltonian
diffeomorphisms with Hamiltonian function $H^{\beta,g}_{t}$ supported on
$B_{\epsilon_{0}}$. Then the desired loop of Hamiltonian diffeomorphisms
$\\{\psi^{\mathbb{C}^{k}}_{t}\\}_{0\leq t\leq 2}$ is defined as

(23)
$\displaystyle\psi^{\mathbb{C}^{k}}_{t}:=\left\\{\begin{array}[]{ll}\psi^{\alpha,g}_{t}&t\in[0,1]\\\
\psi^{\beta,g}_{2-t}&t\in[1,2]\end{array}\right.$

with Hamiltonian function

(26) $\displaystyle
H^{\mathbb{C}^{k}}_{t}:=\left\\{\begin{array}[]{ll}H^{\alpha,g}_{t}&t\in[0,1]\\\
H^{\beta,g}_{2-t}&t\in[1,2].\end{array}\right.$

Furthermore, we have that

$\displaystyle\int_{0}^{2}\int_{B_{\rho}}\;H^{\mathbb{C}^{k}}_{t}\;\omega_{0}^{k}\;dt=(\alpha(1)-\beta(1))\;\frac{\pi^{k+1}\rho^{2k+2}}{(k+1)!}=\frac{(\pi\rho^{2})^{k+1}}{(k+1)!}.$

We have all the prerequisites to proceed with the proof of the main theorem.

Theorem 1.1. Let $(M,\omega)$ be a rational closed symplectic manifold and
$N\subset(M,\omega)$ a closed symplectic submanifold of codimension $2k>2$.
Let $(\widetilde{M},\widetilde{\omega}_{N,\rho})$ be the symplectic blow up of
$(M,\omega)$ along $N$. Then for some small values of $\rho$ the rank of
$\pi_{1}(\textup{Ham}(\widetilde{M},\widetilde{\omega}_{N,\rho}))$ is
positive.

###### Proof.

Consider the loop of Hamiltonian diffeomorphisms
$\\{\psi^{\mathbb{C}^{k}}_{t}\\}_{0\leq t\leq 2}$ defined above. Since each
$\psi^{\mathbb{C}^{k}}_{t}$ is $U(k)$-equivariant, it induces a loop of
diffeomorphisms $\\{\psi^{\nu}_{t}\\}$ on $\nu=P\times_{U(k)}\mathbb{C}^{k}.$
Furthermore, each diffeomorphism is supported on
$P\times_{U(k)}B_{\epsilon_{0}}$.

From Lemma 6.12, we have that $\\{1\times\psi^{\nu}_{t}\\}$ is a loop of
Hamiltonian diffeomorphisms on $P\times_{U(k)}B_{\epsilon_{0}}$ with
Hamiltonian function $H^{\nu}_{t}=1\times H^{\mathbb{C}^{k}}_{t}.$ Since
$P\times_{U(k)}B_{\epsilon_{0}}$ is symplectomorphic to a neighborhood
$\mathcal{U}(N)$, we consider $\\{\psi^{\nu}_{t}\\}$ as a loop of Hamiltonian
diffeomorphisms on $(M,\omega)$. Denote such loop by $\\{\psi^{M}_{t}\\}$ with
Hamiltonian function $H^{M}_{t}$. Notice that such loop is contractible in
$\textup{Ham}(M,\omega)$ and the Hamiltonian function is not normalized. Thus
consider the function

(27) $\displaystyle
c_{t}:=\int_{M}H^{M}_{t}\;\omega^{n}=\int_{\mathcal{U}(N)}H^{M}_{t}\;\omega^{n}.$

Then the normalized Hamiltonian function of $\\{\psi^{M}_{t}\\}$ is defined as

$\displaystyle
H^{M,\textup{norm}}_{t}:=H^{M}_{t}-\frac{c_{t}}{\textup{Vol}(M,\omega^{n})}.$

From its definition, the loop $\\{\psi^{M}_{t}\\}_{0\leq t\leq 2}$ is
$N$-liftable. Denote by $\\{\widetilde{\psi}_{N,t}\\}_{0\leq t\leq 2}$ the
induced loop of Hamiltonian diffeomorphisms. Hence by Thm 1.2 we have that

$\displaystyle\mathcal{A}([\widetilde{\psi}_{N,t}])=\left[\frac{1}{\textup{Vol}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}^{n})}\int_{0}^{2}\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H^{M,\textup{norm}}_{t}\,\omega^{n}dt\right]$

in $\mathcal{P}(M,\omega)+\mathbb{Z}\langle\pi\rho^{2}\rangle.$ From the
definition of the symplectic blow up we know that
$\textup{Vol}(\widetilde{M}_{N},\widetilde{\omega}_{N,\rho}^{n})$ is equal to
$\textup{Vol}(M,\omega^{n})-\textup{Vol}(\mathcal{U}_{\rho}(\nu_{0}),\omega_{\nu,A}^{n})$.

Next we compute the integrals. To that end, by the definition of the
Hamiltonian function on the neighborhood $\Phi(\mathcal{U}_{\rho}(\nu_{0}))$
of $N$ we have that

$\displaystyle\int_{0}^{2}\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H^{M,\textup{norm}}_{t}\,\omega^{n}\,dt$
$\displaystyle=$
$\displaystyle\int_{0}^{2}\int_{\Phi(\mathcal{U}_{\rho}(\nu_{0}))}H^{M}_{t}-\frac{c_{t}}{\textup{Vol}(M,\omega^{n})}\,\omega^{n}\,dt$
$\displaystyle=$
$\displaystyle\int_{0}^{2}\int_{\mathcal{U}_{\rho}(\nu_{0})}1\times
H^{\mathbb{C}^{k}}_{t}\,\omega_{\nu,A}^{n}\,dt$
$\displaystyle-\frac{\textup{Vol}(\mathcal{U}_{\rho}(\nu_{0}),\omega_{\nu,A}^{n})}{\textup{Vol}(M,\omega^{n})}\int_{0}^{2}c_{t}\,dt.$

By replacing the normal bundle $\nu$ by the disk bundle
$\mathcal{U}_{\rho}(\nu_{0})$ on the first sumand, we get by Lemma 6.11 that

$\displaystyle\int_{\mathcal{U}_{\rho}(\nu_{0})}1\times
H^{\mathbb{C}^{k}}_{t}\omega_{\nu,A}^{n}$ $\displaystyle=$
$\displaystyle\int_{N}(\pi_{N}|_{\mathcal{U}_{\rho}(\nu_{0})})_{*}(1\times
H^{\mathbb{C}^{k}}_{t}\;\omega_{0}^{k})\;\omega^{n-k}.$

Since

$\displaystyle(\pi_{N}|_{\mathcal{U}_{\rho}(\nu_{0})})_{*}(1\times
H^{\mathbb{C}^{k}}_{t}\;\omega_{0}^{k})$ $\displaystyle=$
$\displaystyle\int_{B_{\rho}}\;H^{\mathbb{C}^{k}}_{t}\;\omega_{0}^{k},$

it follows that

$\displaystyle\int_{0}^{2}\int_{\mathcal{U}_{\rho}(\nu_{0})}1\times
H^{\mathbb{C}^{k}}_{t}\omega_{\nu,A}^{n}\,dt$ $\displaystyle=$
$\displaystyle\frac{(\pi\rho^{2})^{k+1}}{(k+1)!}\textup{Vol}(N,\omega^{n-k})$

since $\alpha(1)-\beta(1)=1.$

Notice that $c_{t}$ can be rewritten as

$\displaystyle{c_{t}}$ $\displaystyle=$
$\displaystyle\int_{M}H^{M}_{t}\;\omega^{n}$ $\displaystyle=$
$\displaystyle\int_{\mathcal{U}(N)}H^{M}_{t}\;\omega^{n}$ $\displaystyle=$
$\displaystyle\int_{\nu}1\times H^{\mathbb{C}^{k}}_{t}\;\omega_{\nu,A}^{n}$
$\displaystyle=$ $\displaystyle\int_{N}(\pi_{N})_{*}(1\times
H^{\mathbb{C}^{k}}_{t}\;\omega_{0}^{k})\;\omega^{n-k}.$

Before the fiber was $(B_{\rho},\omega_{0})$, in this case the fiber is
$(\mathbb{C}^{k},\omega_{0})$ and $H^{\mathbb{C}^{k}}_{t}$ has compact
support. But on $(\mathbb{C}^{k},\omega_{0})$, we know that the Calabi
morphism vanishes. Therefore we have that

$\int_{0}^{2}{c_{t}}\;dt=\int_{0}^{2}(\pi_{N})_{*}(1\times
H^{\mathbb{C}^{k}}_{t}\;\omega_{0}^{k})\;dt=0$

and

$\displaystyle\mathcal{A}([\widetilde{\psi}_{N,t}])=\left[\frac{1}{\textup{Vol}(M,\omega^{n})-\frac{\pi^{k}\rho^{2k}}{k!}\cdot\textup{Vol}(N,\omega^{n-k})}\cdot\frac{(\pi\rho^{2})^{k+1}}{(k+1)!}\textup{Vol}(N,\omega^{n-k})\right]$

in $\mathcal{P}(M,\omega)+\mathbb{Z}\langle\pi\rho^{2}\rangle.$ By hypothesis,
$\textup{Vol}(M,\omega^{n}),\textup{Vol}(N,\omega^{n-k})$ and
$\mathcal{P}(M,\omega)$ are in $\mathbb{Q}$. Therefore, it follows from the
above expression that if $\rho$ is such that $\pi\rho^{2}$ is a transcendental
number then $\mathcal{A}([\widetilde{\psi}_{N,t}]^{m})\neq 0$ in
$\mathcal{P}(M,\omega)+\mathbb{Z}\langle\pi\rho^{2}\rangle$ for all non zero
$m\in\mathbb{Z}$. Henceforth,
$\pi_{1}(\textup{Ham}(\widetilde{M},\widetilde{\omega}_{N,\rho}))$ has
positive rank. ∎

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