\section*{Abstract}
\end{centering}

We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of a prequantization bundle over a closed monotone toric manifold. Namely we show that any contactomorphism of this prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern number of the toric manifold. The proof relies on the theory of generating functions coupled with equivariant cohomology, whereby we adapt Givental's approach to the Arnold conjecture for rational symplectic toric manifolds to the context of prequantization bundles.

\vspace{1cm}



\section{Introduction and result}{\label{sec1}}

\subsection{The main result}{\label{sec1.1}}


A major driving force in symplectic topology is the celebrated Arnold conjecture \cite{arnold1965propriete}:
\begin{center}
\textit{The number of fixed points of a Hamiltonian symplectomorphism of a closed symplectic manifold is at least the number of critical points of a smooth function.}
\end{center}
While in general diffeomorphisms and even symplectomorphisms have far fewer fixed points, it has now been proved in full generality in a homological version using Floer homology (see for instance \cite{Flo89b}, \cite{Hof-Sal1995}, \cite{Ono95}, \cite{liu98}, \cite{Ken99}): on any closed symplectic manifold $(M,\omega)$, \textit{non-degenerate} Hamiltonian symplectomorphisms have at least $\dim H^*(M,\mathbb{Q})$ fixed points. Results were also obtained about general, not necessarily non-degenerate Hamiltonian symplectomorphisms, by Oh \cite{Oh90}, Schwarz \cite{Schwarz98}, and Givental \cite{Giv1995}. The present paper is closer in spirit to the latter results.

The analogue of the Arnold conjecture in contact topology was introduced by S.\ Sandon \cite{sandon2013morse}. Recall that a cooriented \textbf{contact manifold} \footnote{In this paper we will simply say \textit{contact manifold}.} is a pair $(V,\xi)$, where $V$ is an odd-dimensional manifold, and $\xi$ is a maximally non-integrable cooriented hyperplane field, called a \textbf{contact structure}. A \textbf{contactomorphism} of $(V,\xi)$ is a diffeomorphism preserving $\xi$ and its coorientation. Since contact manifolds are odd-dimensional, in general contactomorphisms have no fixed points. Instead, Sandon \cite{Sandon2009contact} introduced the notion of \textit{translated points}. To define it, fix a \textbf{contact form} $\alpha$, which is a $1$-form such that $\xi = \ker \alpha$ and $\mathrm{d} \alpha_{|\xi}$ is non-degenerate. The \textbf{Reeb vector field $R_{\alpha}$ of $\alpha$} is defined by 
$$
\alpha(R_{\alpha}) = 1 \quad \text{ and } \quad \iota_{R_{\alpha}} \mathrm{d} \alpha = 0,
$$
and its flow is denoted $\{\phi_{\alpha}^t\}_{t \in \mathbb{R}}$. Given a contactomorphism $\phi$, a point $x \in V$ is called an \textbf{$\alpha$-translated point} of $\phi$ if $x$ and $\phi(x)$ belong to the same Reeb orbit and if moreover $\phi$ preserves the contact form $\alpha$ at $x$:
$$
\exists \ s \in \mathbb{R} \quad \text{such that} \quad \phi(x) = \phi_{\alpha}^s(x) \quad \text{and} \quad (\phi^* \alpha)_x = \alpha_x.
$$
We let $\text{Cont}(V,\xi)$ be the group of contactomorphisms of $(V,\xi)$ and $\text{Cont}_0(V,\xi)$ its identity component.

\begin{conjecture}[\cite{sandon2013morse}, Conjecture 1.2]
Let $(V,\xi)$ be a closed contact manifold, and $\phi \in \normalfont{\text{Cont}_0(V,\xi)}$. For any choice of contact form $\alpha$, the number of $\alpha$-translated points of $\phi$ is at least the number of critical points of a function.
\end{conjecture}
A symplectic manifold $(M,\omega)$ is called \textbf{monotone} if the cohomology class of its symplectic form is \textit{positively proportional to the first Chern class $c$ of $M$}. We denote $N_M$ the \textbf{minimal Chern number of $M$}, that is the positive generator of $\langle c, H_2(M,\mathbb{Z}) \rangle \subset \mathbb{Z}$. A \textbf{toric} manifold $(M^{2d}, \omega, \mathbb{T})$ is a symplectic manifold endowed with an effective Hamiltonian action of a torus $\mathbb{T}$ of dimension $d$. A \textbf{prequantization space} over a symplectic manifold $(M,\omega)$ is a contact manifold $(V, \xi := \ker \alpha)$ along with a principal $S^1$-bundle $\pi : (V,\alpha) \to (M,\omega)$ such that $\pi^* \omega = \mathrm{d} \alpha$, and the Reeb vector field $R_{\alpha}$ induces the free $S^1$-action on $V$, where $S^1 = \mathbb{R} / \hbar \mathbb{Z}$, $\hbar > 0$ being the minimal period of a closed Reeb orbit. Our main result is the following.

\begin{comment}
The main result of this paper is concerned with the following contact manifold: 
consider the product $S^{2n+1} \times S^{2n+1} \subset \mathbb{C}^{n+1} \times \mathbb{C}^{n+1}$ of unit spheres. Let $S^1$ act on $\mathbb{C}^{n+1} \times \mathbb{C}^{n+1}$ by $e^{i\theta}(z,w) = (e^{i\theta}z,e^{-i\theta}w)$. Then the restriction of the standard $1$-form $\alpha_{\text{std}} := \frac{1}{2} \underset{j=1}{\overset{2n+2} \sum} (x_j \mathrm{d} y_j - y_j \mathrm{d} x_j)$ to $S^{2n+1} \times S^{2n+1}$ descends to a contact form $\overline{\alpha}_{\text{std}}$ on the quotient $S^{2n+1} \times S^{2n+1} / S^1$. 
\end{comment}

\begin{theoremsubsec}{\label{theo1.1.1}}
Let $(M,\omega)$ be a closed monotone toric manifold different from $\mathbb{C} P^n$. There exists a natural prequantization space $(V,\xi := \ker \alpha)$ over $(M,\omega)$ such that any $\phi \in \normalfont{\text{Cont}_0}(V,\xi)$ has at least $N_M$ $\alpha$-translated points.
\end{theoremsubsec}

\noindent Sandon's conjecture was previously established for numerous prequantization spaces over $\mathbb{C} P^n$: for the standard contact sphere $S^{2n+1}$ and real projective space $\mathbb{R} P^{2n+1}$ in \cite{sandon2013morse}, and later for all lens spaces in \cite{GKPS17}. Note that $\mathbb{C} P^n$ is monotone, however we will see that in this setting our arguments do not apply. The existence of translated points was also obtained in other contexts, for instance in \cite{Alb2013}, \cite{Shel17}, and \cite{MN18}.

Our approach to Theorem \ref{theo1.1.1} is based on the theory of \textit{generating functions} and \textit{equivariant cohomology}, as developed by A.\ Givental \cite{Giv1995}. In the next section, we will give an overview of the constructions and of the proof of Theorem \ref{theo1.1.1}. 

\subsection{Overview of the paper and proof of the theorem}{\label{sec1.2}}

Generating functions were extensively used in the eighties and nineties by numerous authors (see for instance \cite{Sik85}, \cite{Giv1990}, \cite{Giv1995}, \cite{theret1995utilisation}, \cite{theret1998rotation}). They provide a powerful tool when the manifold can be obtained somehow from a symplectic vector space. In \cite{Giv1995}, Givental used this approach along with equivariant cohomology to establish the Arnold conjecture for rational toric manifolds. A symplectic manifold $(M,\omega)$ is called \textbf{rational} if the cohomology class $[\omega]$ of its symplectic form lies in the image of the natural homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$. Note that a symplectic manifold is the base of a prequantization space if and only if it is rational, and that a monotone symplectic manifold is in particular rational.

We explain here the main lines of our construction. The reader shall notice that although Theorem \ref{theo1.1.1} is proved over monotone toric manifolds, the construction itself carries over for a natural prequantization space which is defined over \textit{any} closed rational toric manifold $(M,\omega)$. We begin by viewing $M$ as a symplectic reduction of the standard symplectic vector space $\mathbb{C}^n$ by the Hamiltonian action of a torus $\mathbb{K} \subset \mathbb{T}^n :=(S^1)^n$. Based on a procedure of Borman-Zapolsky \cite{borman2015quasimorphisms}, the prequantization space $(V, \xi := \ker \alpha)$ over $(M,\omega)$ is then obtained by contact reduction of a contact sphere in $\mathbb{C}^n$ by the action of a codimension $1$ subtorus $\mathbb{K}_0 \subset \mathbb{K}$. Given a contact isotopy $\{ \phi^t \}_{t \in [0,1]}$ of $V$, we then lift it up to an $\mathbb{R}_{>0}$ and $\mathbb{K}_0$-equivariant Hamiltonian isotopy $\{ \Phi^t \}_{t \in [0,1]}$ of $\mathbb{C}^n$, thus reducing the problem of finding translated points on $V$ to a problem on a symplectic vector space. This procedure is carried out in sections \ref{sec3.1} and \ref{sec3.2}.

In sections \ref{sec3.3} - \ref{sec3.6}, we mainly follow \cite{Giv1995}. We first decompose the isotopy $\{ \Phi^t \}_{t \in [0,1]}$ into $2N_1$ Hamiltonian symplectomorphisms $\Phi^1 = \Phi_{2N_1} \circ ... \circ \Phi_1$ which are close to $Id_{\mathbb{C}^n}$. Taking the torus action into account, we come to the definition of a family of \textbf{generating functions}
$$ 
\mathcal{F}_N : \mathbb{C}^{2nN} \times \Lambda_N \to \mathbb{R}, \quad \mathcal{F}_{\lambda}^{(N)} := \mathcal{F}_N(.,\lambda), \quad N \geq N_1,
$$
parametrized by compact subsets $\Lambda_N \subset \text{Lie}(\mathbb{K})$ that exhaust the Lie algebra $\text{Lie}(\mathbb{K})$ of $\mathbb{K}$ when $N \to \infty$. Each $\mathcal{F}_{\lambda}^{(N)}$ is $\mathbb{K}_0$-invariant and homogeneous of degree $2$ with respect to $\mathbb{R}_{>0}$. The key observation is that \textit{certain critical points of $\mathcal{F}_{\lambda}^{(N)}$ correspond to translated points of the time $1$ map $\phi^1$ on $V$}. Let $S_N \subset \mathbb{C}^{2nN}$ denote the unit sphere. For any real number $\nu \in \mathbb{R}$, we consider the following sets:
\begin{equation}{\label{eq1}}
F_N^-(\nu) := \lbrace (x,\lambda) \in S_N \times \Lambda_N : \mathcal{F}_N(x,\lambda) \leq 0 \text{ and } p(\lambda) = \nu \rbrace, \quad \partial F_N^-(\nu) := F_N^-(\nu) \cap (S_N \times \partial \Lambda_N),
\end{equation}
where $p = [\omega]$ denotes the cohomology class of the symplectic form $\omega$, viewed as an linear function on $\text{Lie}(\mathbb{K})$ (see Remark \ref{rem3.1.1}). By construction, the torus $\mathbb{K}_0$ has $\mathfrak{k}_0:=\ker p \subset \text{Lie}(\mathbb{K})$ as its Lie algebra. When $\nu$ is generic, we construct natural homomorphisms of $\mathbb{K}_0$-equivariant relative cohomology groups over the field $\mathbb{C}$: 
$$
H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2nN'}(F_{N'}^-(\nu), \partial F_{N'}^-(\nu)), \quad N \leq N'.
$$
In the limit $N \to \infty$, we obtain a cohomology group 
$$
\mathcal{H}_{\mathbb{K}_0}^*(F^-(\nu)) := \underset{N \to \infty}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu))
$$
naturally associated with the Hamiltonian isotopy $\{\Phi^t\}_{t \in [0,1]}$. This group comes along with certain structures. First, one can define similar sets as in equation $(\ref{eq1})$ without restricting the generating functions to the sphere $S_N$ (so they are subsets of the product $\mathbb{C}^{2nN} \times \Lambda_N$). The latter contract onto $\Lambda_N \cap p^{-1}(\nu)$ and $\partial \Lambda_N \cap p^{-1}(\nu)$ respectively, and by the same limit process as above, one obtains a cohomology group which is \textit{independent of $\nu$ and the isotopy $\{\Phi^t\}_{t \in [0,1]}$}. Let us denote it $\mathcal{R}_0$. One then easily sees that there is a natural homomorphism called the \textit{augmentation map}
$$
\mathcal{R}_0 \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu)).
$$
Moreover, the groups $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ are endowed with the \textbf{Novikov action} of $H_2(M,\mathbb{Z})$:
$$
m.\Hcal_{\mathbb{K}_0}^*(F^-(\nu)) = \Hcal_{\mathbb{K}_0}^{* + 2c(m)}(F^-(\nu + p(m))), \quad m \in H_2(M,\mathbb{Z}),
$$
where $c$ is the first Chern class of $M$ (see equation $(\ref{eq2})$ below). Let us denote $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ the kernel of the augmentation map, and describe the restriction of the Novikov action to it. Let $\mathfrak{k}$ denote the Lie algebra of the torus $\mathbb{K}$, $\mathfrak{k}_{\mathbb{Z}} := \ker (\exp : \mathfrak{k} \to \mathbb{K})$ the kernel of the exponential map, and $\iota : \mathfrak{k} \hookrightarrow \mathbb{R}^n$ the inclusion of $\mathfrak{k}$ into $\mathbb{R}^n \simeq \text{Lie}((S^1)^n)$. There are natural isomorphisms
\begin{equation}{\label{eq2}}
H_2(M,\mathbb{Z}) \simeq \mathfrak{k}_{\mathbb{Z}} \quad \text{and} \quad H^2(M,\mathbb{Z}) \simeq \mathfrak{k}_{\mathbb{Z}}^*,
\end{equation}
under which the first Chern class of $M$ writes
$$c(m) = \underset{j=1}{\overset{n} \sum} m_j, \quad \text{where} \quad \iota(m) = (m_1,...,m_n).$$
We denote $H_{\mathbb{T}^n}^*(\pt)$ and $H_{\mathbb{K}_0}^*(\pt)$ the singular cohomology of the classifying spaces $B \mathbb{T}^n$ and $B \mathbb{K}_0$ respectively. The cohomology groups $\mathcal{R}_0$, $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$, and $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ are naturally endowed with the structure of $H_{\mathbb{K}_0}^*(\pt)$-modules, and therefore with that of $H_{\mathbb{T}^n}^*(\pt)$-modules under the natural homomorphism
$$
H_{\mathbb{T}^n}^*(\pt) \to H_{\mathbb{K}_0}^*(\pt).
$$
Note that the cohomology $H_{\mathbb{T}^n}^*(\pt)$ is naturally isomorphic to the ring $\mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$, while $H_{\mathbb{K}_0}^*(\pt)$ is a quotient of this ring, namely the ring of regular functions on $\mathfrak{k}_0 \otimes \mathbb{C}$. 

\textit{From now on, we assume that $M \neq \mathbb{C} P^n$}. This implies in particular that $H_{\mathbb{K}_0}^*(\pt) \neq \{0\}$ (if $\mathfrak{k}_0 = \{0\}$, then $\mathbb{K}_0 = \{\pt\}$ and $\mathbb{K} = S^1$, in which case $M = \mathbb{C} P^n$). Note that by compactness of $M$, \textit{the above homomorphism sends any monomial $u_i$ to a positive degree element in $H_{\mathbb{K}_0}^*(\pt)$}. 

A key novelty in this paper is the use of a Gysin-type long exact sequence for equivariant cohomology (section \ref{sec4}), relating our cohomology groups $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ to the ones constructed by Givental in \cite{Giv1995}. Using this sequence, we will see that $\mathcal{R}_0$ can be viewed as the ring of regular functions on the intersection $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$, where $(\mathbb{C}^{\times})^n$ is the complex torus. The kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ of the augmentation map is then a submodule of this ring, and one can interpret the Novikov action on $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ as follows. Let 
$$
\mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}]
$$ 
be the ring of regular functions on $(\mathbb{C}^{\times})^n$. Then the Novikov action on $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ is given by the multiplication by $u_1^{m_1}...u_n^{m_n}$:
\begin{equation}{\label{eq3}}
m. \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) = u_1^{m_1}...u_n^{m_n} \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \simeq \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(F^-(\nu + p(m)), \quad m \in H_2(M,\mathbb{Z}),
\end{equation}
where $u_1^m...u_n^{m_n}$ is identified with its image under the natural ring homomorphism from the regular functions on $(\mathbb{C}^{\times})^n$ to those on $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$.

Consider now two generic numbers $\nu_0 \leq \nu_1$. There is a natural homomorphism
$$
\Hcal_{\mathbb{K}_0}^*(F^-(\nu_1)) \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0)).
$$
Recall the contact analogue of the action spectrum: given a contactomorphism $\phi$ of a contact manifold $(V,\xi)$, the \textbf{translated spectrum} (or simply \textbf{spectrum}) of $\phi$ is the set
\begin{equation}{\label{eq4}}
\Spec(\phi) := \lbrace s \in \mathbb{R} : \exists \ x \in V, \text{ $x$ is a translated point of $\phi$ with } \phi(x) = \phi_{\alpha}^s(x) \rbrace.
\end{equation}
Note that on a prequantization space, the spectrum of any contactomorphism is periodic of period $\hbar$. In the sequel we assume that $\hbar = 1$, which is equivalent to $p$ being a primitive integral vector in $\mathfrak{k}_{\mathbb{Z}}^*$. Let us denote $\phi := \phi^1$ the time $1$ map of the original contact isotopy of $V$. The cohomology groups $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ and the translated points of $\phi$ are related by the two following analogues of \cite[Propositions $6.2$, $6.3$]{Giv1995}, which are proved in section \ref{sec3.6}.

\begin{propsubsec}{\label{prop1.2.1}}
Suppose that $[\nu_0,\nu_1] \cap \Spec(\phi) = \emptyset$. Then the homomorphism above is an isomorphism{}
$$\Hcal_{\mathbb{K}_0}^*(F^-(\nu_1)) \simeq \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0)).$$
\end{propsubsec}

\begin{propsubsec}{\label{prop1.2.2}}
Suppose that the segment $[\nu_0,\nu_1]$ contains only one value $\nu \in \Spec(\phi)$, which corresponds to a finite number of translated points. Let $v \in H_{\mathbb{K}_0}^*(\pt)$ be an element of positive degree, and $q \in \mathcal{R}_0$. Suppose that $q \in \mathcal{J}_{\mathbb{K}_0}^*(\nu_0)$. Then $v q \in \mathcal{J}_{\mathbb{K}_0}^*(\nu_1)$.
\end{propsubsec}
\noindent In other words, the image of the augmentation map $\mathcal{R}_0 \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu_1))$ cannot "differ too much" from that of $\mathcal{R}_0 \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0))$. Similar statements (in finite dimensional settings) are used for instance in \cite[Proposition $5.2$ (c)]{theret1998rotation}, \cite[section $8$]{Giv1990}, and \cite[Proposition $4.8$]{GKPS17}.
\begin{comment}
We now assume that $M = (\mathbb{C} P^n \times \mathbb{C} P^n, \overline{\omega}_{\text{std}} \oplus \overline{\omega}_{\text{std}})$ In thi{}s setting, $M$ is the symplectic reduction of $\mathbb{C}^{2n+2}$ by the torus $\mathbb{K} = S^1 \times S^1$, acting diagonally on each factor of $\mathbb{C}^{n+1} \times \mathbb{C}^{n+1}$. We fix the cohomology class of the symplectic form to be $p = (1,1) \in \mathbb{R}^{2*} \simeq \mathfrak{k}^*$. The Chern class $c$ applied to an element $m = (m_1,m_2) \in \mathbb{R}^2$ gives $c(m) = (n+1)(m_1+m_2)$, and therefore $c(m) = (n+1)p(m)$. 
\end{comment}

\textit{We now assume that $(M,\omega)$ is monotone}. Since $p$ is \textbf{primitive}, we have 
$$
p = \frac{c}{N_M}.
$$
We will see in section \ref{sec5} that, in this situation, the kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ admits in some sense \textit{elements of minimal degree}:
\begin{propsubsec}{\label{prop1.2.3}}
There exists $q \in \mathcal{R}_0$, such that $q \notin \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, but $u_i q \in \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ for all $i=1,...,n$.
\end{propsubsec} 

\begin{remarksubsec}
\normalfont Note that in the above proposition the monotonicity assumption cannot be lifted, since in general, it can happen that $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) = \mathcal{R}_0$. This is the reason why the proof of Theorem \ref{theo1.1.1} does not work in this case. Note also that if $M$ was the complex projective space $\mathbb{C} P^n$, the ring $\mathcal{R}_0$ would be trivial, since we would have $\mathfrak{k}_0 = \{0\}$, and in particular $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) = \{0\}$, so the above statement would not hold either.
\end{remarksubsec}

The above discussion allows us to prove our main result, Theorem \ref{theo1.1.1}.

\begin{proof}{(Theorem \ref{theo1.1.1})}
We adapt the proof of {\cite[Theorem 1.1]{Giv1995}} to the contact setting. Consider the prequantization space $(V, \xi := \ker \alpha)$ constructed by the procedure described above. Let $\{ \phi^t \}_{t \in [0,1]}$ be a contact isotopy of $V$. For any generic $\nu \in \mathbb{R}$, we have a cohomology group $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$, and the kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ of the augmentation map satisfies Proposition \ref{prop1.2.3}. Suppose that $\nu \notin \Spec(\phi)$. Since $\Spec(\phi)$ is periodic of period $1$, it is enough to count the number of elements of $\Spec(\phi)$ between $\nu$ and $\nu + 1$. To every such element corresponds at least one $\alpha$-translated point of $\phi$ on $V$. Therefore, to one $\alpha$-translated point of $\phi$ there correspond $l$ elements of the spectrum between $\nu$ and $\nu + l$, where $l \in \mathbb{Z}_{>0}$. Assume that $\phi$ has a finite number of $\alpha$-translated points (otherwise the statement is trivial).

Let $m \in \mathfrak{k}_{\mathbb{Z}} \setminus \{ 0 \}$ such that $\iota(m) =  (m_1,...,m_n) \in \mathbb{R}_{\geq 0}^n$. Suppose that the number $\#$ of elements in $\Spec(\phi)$ between $\nu$ and $\nu + p(m)$ is strictly less than $c(m) = \underset{j=1}{\overset{n} \sum} m_j$. Let $q \in \mathcal{R}_0$ be such that $q \notin \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, but $u_i q \in \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ for any $i = 1,...,n$. Since $\# < c(m)$, Propositions \ref{prop1.2.1} and \ref{prop1.2.2} imply that $u_1^{m_1}...u_n^{m_n}q \in \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(\nu+p(m))$. This is precisely the Novikov action $(\ref{eq3})$ of $m$, which is an isomorphism
$$\mathcal{J}_{\mathbb{K}_0}^*(\nu) \overset{u_1^{m_1}...u_n^{m_n}}{\overset{\sim} \longrightarrow} \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(\nu+p(m)).$$
In particular, $u_1^{m_1}...u_n^{m_n}q \notin \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(\nu+p(m))$, which is a contradiction. This means that the number $\#$ of elements of $\Spec(\phi)$ between $\nu$ and $\nu + p(m)$ is not less that $c(m)$, and thus the number of $\alpha$-translated points of $\phi$ is not less than $N_M$.
\end{proof}

 
\noindent \textbf{{\large Discussion.}} \normalfont In \cite{Giv1995}, Givental constructed the generating functions as a finite dimensional approximation of action functionals defined on the free loop space of $\mathbb{C}^n$ (see \cite[section $2$]{Giv1995}). Following such an analogy, one could think of the limit $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ as a kind of $\mathbb{R}$-filtered Floer cohomology group associated with the contactomorphism $\phi$. There exist several Floer-type constructions that pertain to translated points, see for instance \cite{Alb2013}, \cite{MU2017}. Moreover, there are at least two ongoing projects, by Albers-Shelukhin-Zapolsky and Leclercq-Sandon, concerned with comparable constructions and applications. Note that, even in the symplectic setting of \cite{Giv1995}, building a precise relation between Floer homology and the limit of equivariant cohomology groups seems highly complicated, for Floer's construction is of much different nature than Givental's. Still, we observe a similar behaviour, at least in the study of translated points. In addition, our limit has the very convenient property of being defined for any contactomorphism in $\text{Cont}_0(V,\xi)$, and our result holds in this level of generality. In contrast, all the other constructions use a non-degeneracy condition analoguous to that of the symplectic setting. Another remark is that we currently don't know if the limit $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ is non-trivial over a non-monotone base. If it is trivial, the above analogy raises the question of the existence of Floer homology groups for prequantization spaces over non-monotone symplectic manifolds.\\


\noindent \textbf{{\large Acknowlegements.}} I would like to warmly thank my advisor Frol Zapolsky for his support throughout this project. His professionalism and patience were crutial in every step of this work. I would also like to thank Leonid Polterovich and Michael Entov for listening to a preliminary version of this work and their interest, and Alexander Givental for his generous help regarding the paper \cite{Giv1995}. I would also like to thank the participants of the workshop Floer homology and contact topology, held at the University of Haifa in September $2017$, for their questions and attention during my talk about this project. Finally, I would like to thank the organizers and participants of the seminar Symplectix at the Institut Henri Poincaré in Paris, for their invitation and interest during my talk in January $2019$, more particularly Vincent Humilière for his suggestion regarding the presentation of the main result of this paper, and Alexandru Oancea and Claude Viterbo for their questions during and after my talk. I was supported by grant number $1825/14$ from the Israel Science Foundation.


\begin{comment}
Any time dependent function $h : V \times [0,1] \to \mathbb{R}$, called a \textbf{contact Hamiltonian}, gives rise to a unique time-dependent vector field satisfying
$$
\alpha(X_h^t) = h_t \quad \text{and} \quad \mathrm{d} \alpha(X_h^t,.) = - \mathrm{d} h_t + \mathrm{d} h_t(R_{\alpha}) \alpha.
$$
The vector field $\{ X_h^t \}_{t \in [0,1]}$ preserves $\xi$ and integrates into a contact isotopy denoted $\{ \phi_h^t \}_{t \in [0,1]}$. This establishes a bijection, depending on the contact form $\alpha$, between smooth time-dependent functions $h : V \times [0,1] \to \mathbb{R}$ and the identity componant $\text{Cont}_0(V,\xi)$ of the group of contactomorphisms.

Since contact manifolds are odd-dimensional, generic contactomorphisms do not have any fixed points. The contact analogue of the Arnold conjecture was introduced by S.\ Sandon \cite{Sandon2009contact}, via the notion of \textit{translated points}: given a contactomorphism $\phi \in \text{Cont}(V,\xi)$ and a contact form $\alpha$ such that $\xi = \ker \alpha$, we denote by $g : V \to \mathbb{R}$ the function satisfying $\phi^* \alpha = e^g \alpha$. A point $x \in V$ is called an \textbf{$\alpha$-translated point of $\phi$} if $x$ and $\phi(x)$ belong to the same Reeb orbit and if moreover $\phi$ preserves the contact form $\alpha$ at $x$, that is $g(x) = 0$:
$$
\exists s \in \mathbb{R} \quad \text{such that} \quad \phi(x) = \phi_{\alpha}^s(x) \quad \text{and} \quad (\phi^* \alpha)_x = \alpha_x.
$$
The point $x$ is called a \textbf{discriminant point} if it is a translated point and if moreover it is fixed by $\phi$, that is $\phi(x) = x$. Note that $x$ is an $\alpha$-translated point of $\phi$ if and only if there exists $s \in \mathbb{R}$ such that $x$ is a discriminant point of $\phi_{\alpha}^{-s} \circ \phi$.

\begin{conjecture}[\cite{sandon2013morse} conjecture $1.2$]
Let $(V,\xi)$ be a compact contact manifold, and $\phi \in \normalfont{\text{Cont}_0(V,\xi)}$. For any choice of contact form $\alpha$, the number of $\alpha$-translated points of $\phi$ is at least the number of critical points of a function.
\end{conjecture}
\noindent In \cite{sandon2013morse}, Sandon proved her conjecture for the standard contact sphere $(S^{2n-1}, \alpha_{\text{std}})$, where $\alpha_{\text{std}}$ is the standard $1$-form on $\mathbb{C}^{2n}$, and the real projective space $(\mathbb{R} P^{2n-1}, \overline{\alpha_{\text{std}}})$, where $\overline{\alpha_{\text{std}}}$ is the contact form induced by the antipodal map 
$$
S^{2n-1} \to \mathbb{R} P^{2n-1} := S^{2n-1} / \mathbb{Z}_2.
$$
It was later established for all lens spaces in \cite{GKPS17} via the extention of Givental's nonlinear Maslov index \cite{Giv1990} to this setting. The existence of translated points was also obtained in other contexts, for instance in \cite{Alb2013}, \cite{Shel17}, and \cite{MN18}.

Given a contactomorphism $\phi \in \text{Cont}(V,\alpha)$, we call \textbf{translated spectrum of $\phi$} (or simply \textbf{spectrum of $\phi$}) the set
\begin{equation}{\label{eq1}}
\Spec(\phi) := \lbrace s \in \mathbb{R}: \exists x \in V, \text{ $x$ is an $\alpha$-translated point of $\phi$ with $\phi(x) = \phi_{\alpha}^s(x)$} \rbrace.
\end{equation}
It is a \textit{nowhere dense closed} subset of $\mathbb{R}$.

\subsection{Givental's construction for prequantization spaces}{\label{sec1.2}}

A \textbf{prequantization space} over a symplectic manifold $(M,\omega)$ is a contact manifold $(V,\xi := \ker \alpha)$ along with a principal $S^1$-bundle $\pi : (V,\alpha) \to (M,\omega)$ such that $\pi^* \omega = \mathrm{d} \alpha$, and the Reeb vector field $R_{\alpha}$ induces the free $S^1$-action on $V$, where $S^1 = \mathbb{R} / \hbar \mathbb{Z}$, $\hbar > 0$ being the minimal period of a closed Reeb orbit.  Note that on a prequantization space, the spectrum of any contactomorphism is periodic of period $\hbar$. In the sequel we assume that $\hbar = 1$.

In this paper, we extend the construction of Givental \cite{Giv1995} to a prequantization space over a general rational toric manifold. As an application, we prove a version of Sandon's conjecture for a prequantization space over a product of monotone projective spaces. A symplectic manifold $(M,\omega)$ is called \textbf{monotone} if the cohomology class $[\omega]$ is proportional to the first Chern class $c$ of $M$. 

\begin{theoremsubsec}{\label{theo1.2.1}}
Let $(M,\omega) = (\mathbb{C} P^n \times \mathbb{C} P^n, \omega)$ be a product of complex projective spaces, endowed with a rational monotone symplectic form. There exists a prequantization space $(V,\alpha)$ over $(M,\omega)$, such that for any contactomorphism $\phi \in \normalfont{\text{Cont}}_0(V,\xi)$, $\phi$ has at least $n+1$ $\alpha$-translated points.
\end{theoremsubsec}
\end{comment}
\begin{comment}
by lifting a contact Hamiltonian $h$ on $S^{2n-1}$ (resp. $\mathbb{R} P^{2n-1}$) to a $\mathbb{R}_{>0}$-invariant (resp. $\mathbb{Z}_2$ and $\mathbb{R}_{>0}$-invariant) Hamiltonian $H$ on $(\mathbb{C}^n, \omega_{\text{std}})$, and investigating the Morse theory of a $1$-parameter family of homogeneous generating functions (resp. homogeneous and $\mathbb{Z}_2$-invariant generating functions) associated with the Hamiltonian symplectomorphism $\Phi_H$ generated by H. In (GKPS) Granja-Karshon-Pabiniak-Sandon contructed a Maslov index for Lens spaces, adapting the construction of Givental \cite{Giv1990}, which led them to a confirmation of the conjecture in this setting.
\end{comment}


\begin{comment} 
a number of ideas ranging from Conley and Zehnder's proof of the conjecture for tori \cite{Con-Zeh1983}, B.Fortune and A.Weinstein's approach in the case of $\mathbb{C} P^n$ \cite{For-Wei1985}, and M.Chaperon and Yu.Chekanov's idea of discrete loops as finite approximations of periodic Hamiltonian trajectories \cite{chaperon1984idee}, \cite{chekanov1987legandrova}. Given a Hamiltonian $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ which is homogeneous of degree $2$ with respect to the standard $\mathbb{R}_{>0}$-action on $\mathbb{C}^n$ (or simply homogeneous), and invariant by the action of a subtorus $\mathbb{K} \subset (S^1)^n$ of $(S^1)^n$ Givental defines the following \textbf{action functional}
\begin{equation}
\mathcal{A}_H : \mathcal{L{}} \mathbb{C}^n \times \mathfrak{k} \to \mathbb{R}, \quad \mathcal{A}(x,\lambda) := \int_0^1 x^*\omega_{\text{std}} \mathrm{d} t - \int_0^1 H(x(t),t) \mathrm{d} t - \int_0^1 \langle P_{\mathbb{K}}(x(t)), \lambda \rangle \mathrm{d} t,
\end{equation}
where $\mathcal{L} \mathbb{C}^n$ is the free loop space on $\mathbb{C}^n$, $\mathfrak{k}$ denotes the Lie algebra of $\mathbb{K}$, $\omega_{\text{std}}$ is the standard symplectic structure on $\mathbb{C}^n$, and $P_{\mathbb{K}} : \mathbb{C}^n \to \mathfrak{k}$ is the momentum map of the $\mathbb{K}$-action on $\mathbb{C}^n$ (see section \ref{sec1}). In this setting, the restricted action functionals $\mathcal{A}_{\lambda} := \mathcal{A}_{H | \mathcal{L} \mathbb{C}^n \times \{ \lambda \}}$ are homogeneous and $\mathbb{K}$-invariant, and critical loops of $\mathcal{A}_{\lambda}$ correspond to fixed points of the Hamiltonian symplectomorphism $\exp(\lambda) \circ \Phi_H$, where $\exp(\lambda) := e^{2i \pi \lambda}$, and $\Phi_H$ is the Hamiltonian symplectomorphism generated by $H$. 

Let $(M,\omega)$ be a closed rational toric manifold. By Delzant \cite{delzant1988hamiltoniens}, there exists a regular value\\

$p$ of $P_{\mathbb{K}}$,  Given a toric manifold $(M,\omega)$, Givental views it as the symplectic reduction $M := P_{\mathbb{K}}^{-1}(p) / \mathbb{K}$, endowed with the symplectic form induced by $\mathrm{d} \alpha_{\text{std}| P^{-1}(p)}$, where $p$ is a regular value of $P_{\mathbb{K}}$. Given a Hamiltonian $h : M \times [0,1] \to \mathbb{R}$, he obtains, by a lifting procedure, a homogeneous $\mathbb{K}$-invariant Hamiltonian on $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$. He observes that the fixed points of the Hamiltonian symplectomorphism $\phi_h$ generated by $h$ correspond to fixed points of Hamiltonian symplectomorphisms $\exp(\lambda) \circ \Phi_H$, where $\Phi_H$ is generated by $H$. This defines a relation between fixed points on $M$ and critical points of the action functional $\mathcal{A}_H$ (see \cite{Giv1995} section $2$). 

The Floer theoretic approach to the study of action functionals requires some assumptions on the Hamiltonian $H$. Indeed, the construction of a Floer complex imposes that the fixed points of the Hamiltonian symplectomorphism $\Phi_H$ be isolated. However, in the generic setting investigated by Givental, no such assumption is made, for the action functional is replaced by finite dimensional approximations called \textit{generating functions}, as developped in \cite{Giv1990} by the same author, and loops by discrete trajectories. More precisely, given a homogeneous and $\mathbb{K}$-invariant Hamiltonian $H$ on $\mathbb{C}^n$ which lifts a Hamiltonian $h$ on $M$, Givental constructs a sequence $\lbrace \mathcal{F}_N \rbrace_N$ of families
\begin{equation}
\mathcal{F}_N : \mathbb{C}^{2nN} \times \Lambda_N \to \mathbb{R}, \quad \mathcal{F}_N(z,\lambda) := \mathcal{F}_{\lambda}^{(N)}(z),
\end{equation}
of homogeneous and $\mathbb{K}$-invariant generating functions $\mathcal{F}_{\lambda}^{(N)}$ parametrized by compact subspaces $\Lambda_N \subset \mathfrak{k}$ exhausting the Lie algebra $\mathfrak{k}$, each $N$ corresponding to a decomposition of the Hamiltonian symplectomorphism $\Phi_H$ into $N$ "small" parts. He then investigates the changes in the $\mathbb{K}$-equivariant cohomology of suitable sublevel sets of $\mathcal{F}_N$. In the limit $N \rightarrow \infty$, he comes to the definition of a \textit{semi-infinite cohomology group} $\mathcal{H}_{\mathbb{K}}^*(F^-(\nu))$ associated with $H$ and levels $p^{-1}(\nu)$ of the regular value $p$ of $P_{\mathbb{K}}$. A key point in this approach is the notion of front. The $\textbf{front}$ of a Hamiltonian $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ (or equivalently of the action functional $\mathcal{A}_H$) can be defined as the set
\begin{equation}
L_H := \lbrace  \lambda \in \mathfrak{k} : \exp(\lambda) \circ \Phi_H  \text{ admits a non-trivial fixed point} \rbrace.
\end{equation}
Notice that, since $\exp(\lambda + m) = \exp(\lambda)$ for any $m \in \mathfrak{k}_{\mathbb{Z}} := \ker(\exp : \mathfrak{k} \to \mathbb{K})$, the front is $\mathfrak{k}_{\mathbb{Z}}$-periodic. It appears that, whenever $H$ is the lift of a Hamiltonian $h : M \times [0,1]$, tangency events of levels $p^{-1}(\nu)$ with $L_H$ correspond to fixed points of the Hamiltonian symplectomorphism $\phi_h : M \to M$ generated by $h$ (see \cite{Giv1995} section 3). The result of such a tangency event is a change in the cohomology $\mathcal{H}_{\mathbb{K}}^*(F^-(\nu))$. In particular, he shows in the proof of his main theorem that, for any $\nu \in \mathbb{R}$ such that $p^{-1}(\nu)$ is tranversal to $L_H$, and for any $m \in \mathfrak{k}_{\mathbb{Z}} \setminus \lbrace 0 \rbrace$ such that $m_j \geq 0$ there are at least $\underset{j=1}{\overset{n}\sum} m_j$ tangency events between the levels $p^{-1}(\nu)$ and $p^{-1}(\nu + p(m))$.
\end{comment}




\begin{comment}
\subsection{Approach and proof of the theorem}{\label{sec1.3}}
We briefly introduce here the constructions and the concepts used in this paper. We delibarately avoid making precise definitions, for the purpose of this section is to give a proof of \ref{theo1.2.1}. 

We view $M$ as a symplectic reduction of $\mathbb{C}^n$ by the Hamiltonian action of a torus $\mathbb{K} \subset \mathbb{T}^n :=(S^1)^n$. Based on a procedure of Borman-Zapolsky \cite{borman2015quasimorphisms}, we associated to $(M,\omega)$ a prequantization space $(V,\alpha)$, obtained by contact reduction of a contact sphere in $\mathbb{C}^n$ by the action of a codimension $1$ subtorus $\mathbb{K}_0 \subset \mathbb{K}$. 

Given a contact Hamiltonian $h : V \times [0,1] \to \mathbb{R}$, this procedure allows us to lift $h$ to a $\mathbb{K}_0$-invariant Hamiltonian $\widetilde{H} : \mathbb{C}^n \times [0,1] \to \mathbb{R}$. We then associate to $\widetilde{H}$ a family $\{ \mathcal{F}_N \}_{N \in \mathbb{N}}$ of functions, the critical points of which bear information about the translated points of $\phi_h$. We investigate the $\mathbb{K}_0$-equivariant cohomology of suitable sublevel sets of the functions $\mathcal{F}_N$, and by a limit process in $N \to \infty$, we come to the definition of a cohomology group $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$, naturally associated with $\widetilde{H}$ and a \textit{generic} real number $\nu \in \mathbb{R}$. The latter comes along with a homomorphism called \textit{augmentation map}
$$
\Hcal_{\mathbb{K}_0}^*(\nu) \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu)),
$$
and is endowed with the action of the \textbf{Novikov ring} $H_2(M,\mathbb{Z})$:
$$
m.\Hcal_{\mathbb{K}_0}^*(F^-(\nu)) = \Hcal_{\mathbb{K}_0}^{* + 2c(m)}(F^-(\nu + p(m))), \quad m \in H_2(M,\mathbb{Z}),
$$
where $c$ and $p$ denote respectively the first Chern class of $M$ and the cohomology class of the symplectic form $\omega$. Let us denote $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ the kernel of the augmentation map, and describe the restriction of the Novikov action to it. Let $\mathfrak{k}$ denote the Lie algebra of the torus $\mathbb{K}$, $\mathfrak{k}_{Z} := \ker (\exp : \mathfrak{k} \to \mathbb{K})$ the kernel of the exponential map, and $\iota : \mathfrak{k} \hookrightarrow \mathbb{R}^n$ the inclusion of $\mathfrak{k}$ into $\mathbb{R}^n \simeq \text{Lie}((S^1)^n)$. There are natural isomorphisms
\begin{equation}{\label{eq2}}
H_2(M,\mathbb{Z}) \simeq \mathfrak{k}_{Z} \quad \text{ and } \quad H^2(M,\mathbb{Z}) \simeq \mathfrak{k}^*_{Z},
\end{equation}
under which $c$ writes
$$c(m) = \underset{j=1}{\overset{n} \sum} m_j, \quad \iota(m) = (m_1,...,m_n).$$
The cohomology groups $\Hcal_{\mathbb{K}_0}^*(\nu)$, $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$, and $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ are endowed with that of $H_{\mathbb{K}_0}^*(\pt)$-modules, and therefore with the structure of $H_{\mathbb{T}^n}^*(\pt)$-modules under the natural homomorphism

$$H_{\mathbb{T}^n}^*(\pt) \to H_{\mathbb{K}_0}^*(\pt).$$
The $\mathbb{T}^n$-equivariant cohomology of a point is naturally isomorphic to the ring $\mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$. Note that by compactness of $M$, the above homomorphism sends any monomial $u_i$ to a positive degree element in $H_{\mathbb{K}_0}^*(\pt)$. We will see that $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ is in fact a quotient of the ring $\mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}]$ of regular functions on the complexified torus $(\mathbb{C}^{\times})^n$. In this setting, the Novikov action is given by the multiplication by $u_1^{m_1}...u_n^{m_n}$:
\begin{equation}{\label{eq3}}
m. \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) = u_1^{m_1}...u_n^{m_n} \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \simeq \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(F^-(\nu + p(m)).
\end{equation}

Let $\nu_0 < \nu_1$ be generic. There is a natural homomorphism
$$
\Hcal_{\mathbb{K}_0}^*(F^-(\nu_1)) \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0)).
$$
The cohomology groups and the translated points of $\phi_h$ are related by the two following analogues of propositions $6.2$ and $6.3$ of \cite{Giv1995}.

\begin{propsubsec}{\label{prop1.3.1}}
Suppose that $[\nu_0,\nu_1] \notin \Spec(\phi_h)$. Then the homomorphism above is an isomorphism
$$\Hcal_{\mathbb{K}_0}^*(F^-(\nu_1)) \simeq \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0)).$$
\end{propsubsec}

\begin{propsubsec}{\label{prop1.3.2}}
Suppose that the segment $[\nu_0,\nu_1]$ contains only one value $\nu \in \Spec(\phi_h)$, and that all the associated translated points are isolated. Let $v \in H_{\mathbb{K}_0}^*(\pt)$ be an element of positive degree, and $q \in \Hcal_{\mathbb{K}_0}^*(\pt)$. Suppose that $q \in \mathcal{J}_{\mathbb{K}_0}^*(\nu_0)$. Then $v q \in \mathcal{J}_{\mathbb{K}_0}^*(\nu_1)$.
\end{propsubsec}

We now assume the $M$ is a \textit{monotone} product of projective spaces $M = \mathbb{C} P^n \times \mathbb{C} P^n$. In this setting, $M$ is the symplectic reduction of $\mathbb{C}^{2n+2}$ by the torus $\mathbb{K} = S^1 \times S^1$, acting diagonally on each factor $\mathbb{C}^{n+1} \times \mathbb{C}^{n+1}$. We fix the class of the symplectic form to be $p = (1,1) \in \mathbb{R}^{2*}$. The Chern class $c$ applied to an element $m = (m_1,m_2) \in \mathbb{R}^2$ gives $c(m) = (n+1)(m_1+m_2)$, and therefore $c(m) = (n+1)p(m)$. In this situation, the kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ admits in some sense \textit{elements of minimal degree}:
\begin{propsubsec}{\label{prop1.3.3}}
There exists $q \in \Hcal_{\mathbb{K}_0}^*(\nu)$, such that $q \notin \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, but for any $i=1,...,n$ and any monomial $u_i \in \mathbb{C}[u_1,...,u_n]$, $u_i q \in \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$.
\end{propsubsec} 

\begin{proof}{(theorem \ref{theo1.2.1})}
We adapt the proof of \cite{Giv1995} theorem $1.1$ to the contact setting. Consider the prequantization space $(V,\alpha)$ constructed by the procedure described above. Let $h : V \times [0,1] \to \mathbb{R}$ be a contact Hamiltonian such that $\phi = \phi_h$. For any Hamiltonian lift $\widetilde{H}$ of $h$ and any generic $\nu$, we have a cohomology group $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$, and the kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ of the augmentation map satisfies proposition \ref{prop1.3.3}. Let $\nu \notin \Spec(\phi_h)$. Since $\Spec(\phi_h)$ is periodic of period $1$, it is enough to count the number of elements of $\Spec(\phi_h)$ between $\nu$ and $\nu + 1$. To every such element corresponds at least one $\alpha$-translated point of $\phi_h$ on $V$. Therefore, to one $\alpha$-translated point of $\phi_h$ there are $l$ elements of the spectrum between $\nu$ and $\nu + l$, where $l \in \mathbb{Z}_{>0}$. Let us assume that all the $\alpha$-translated points of $\phi$ are isolated (otherwise the statement is trivial).

Let $m \in \mathfrak{k}_{Z} \setminus \{ 0 \}$ such that $\iota(m) =  (m_1,...,m_n) \in \mathbb{R}_{\geq 0}^n$. Suppose that the number $\#$ of elements in $\Spec(\phi_h)$ between $\nu$ and $\nu + p(m)$ is strictly less than $c(m) = \underset{j=1}{\overset{n} \sum} m_j$. Let $q \in \Hcal_{\mathbb{K}_0}^*(\nu)$ such that $q \notin \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, but $u_i q \in \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ for any $i = 1,...,n$. Since $\# < c(m)$, propositions \ref{prop1.3.1} and \ref{prop1.3.2} imply that $u_1^{m_1}...u_n^{m_n}q \in \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(\nu+p(m))$. This is precisely the Novikov action $\ref{eq3}$ of $m$, which is an isomorphism
$$\mathcal{J}_{\mathbb{K}_0}^*(\nu) \overset{u_1^{m_1}...u_n^{m_n}}{\overset{\sim} \longrightarrow} \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(\nu+p(m)).$$
In particular, $u_1^{m_1}...u_n^{m_n}q \notin \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(\nu+p(m))$, which is a contradiction. This means that the number $\#$ of elements of $\Spec(\phi_h)$ between $\nu$ and $\nu + p(m)$ is not less that $c(m)$, that is the number of $\alpha$-translated points of $\phi_h$ is not less than $\frac{c(m)}{p(m)} = n+1$.
\end{proof}



The paper is organized as follows. In section $2$ we recall several technical results from \cite{Giv1995}, and state some generalities about equivariant cohomology. Section $3$ is devoted to the constructions of the objects introduced above. In sections $3.1$ to $3.5$ we adapt Givental's construction to our setting. In section $3.6$ we come to the definition of the cohomology group $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$, interpret the Novikov action, and prove propositions \ref{prop1.3.1} and \ref{prop1.3.2}. In section $4$ we relate our construction to that of Givental \cite{Giv1995}. In section $5$ we compute the kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ in the projective setting, and prove proposition \ref{prop1.3.3}. 
\end{comment}
\begin{comment}
In this paper, we extend the construction of Givental \cite{Giv1995} to prequantization spaces over toric manifolds. Let $(M,\omega)$ be a rational toric manifold. Based on a procedure of Borman-Zapolsky \cite{borman2015quasimorphisms}, we construct a prequantization space $(V, \alpha)$ over $(M,\omega)$. Given a contactomorphism $\phi_h \in \text{Cont}_0(V,\xi)$, $\xi = \ker \alpha$, and a \textit{generic} real number $\nu \in \mathbb{R}$, we construct a cohomology group  
\end{comment}


\begin{comment}
as a contact reduction $V := P_{\mathbb{K}}^{-1}(p) / \mathbb{K}_0$, where $\mathbb{K}_0 \subset \mathbb{K}$ is a subtorus of codimension $1$. Given a contact Hamiltonian $h$ on $V$, we lift it up to a homogeneous and $\mathbb{K}_0$-invariant Hamiltonian $H$ of $\mathbb{C}^n$. Following \cite{Giv1995}, we then associate to $H$ a sequence of generating families $\mathcal{F}_N : \mathbb{C}^{2nN} \times \Lambda_N \to \mathbb{R}$, with the difference that, in our case, even if $\Lambda_N \subset \mathfrak{k}$ still exhauts the Lie algebra of $\mathbb{K}$, the functions $\mathcal{F}_{\lambda}^{(N)}$ are $\mathbb{K}_0$-invariant, and the investigation of tangency events between level sets $p^{-1}(\nu)$ and the front $L_H$ now correspond to translated points of the Hamiltonian contactomorphism generated by $h$. 
\end{comment}
\begin{comment}
a cohomology group $HF_{\leq \nu}^*(\widetilde{\phi}_h)$ such that, for generic $\nu \leq \nu'$, there is a natural homomorphism 
\begin{equation}
HF_{\leq \nu'}^*(\widetilde{\phi}_h) \to HF_{\leq \nu}^*(\widetilde{\phi}_h).
\end{equation}
We show that the limit $\underset{\nu \to \infty}{\varprojlim} HF_{\leq \nu}^*(\widetilde{\phi}_h)$ is independent of $\widetilde{\phi}_h$, and thus gives rise to a cohomology group 
\begin{equation}
HF^*(V) := \underset{\nu \to \infty}{\varprojlim} HF_{\leq \nu}^*(\widetilde{\phi}_h),
\end{equation}
naturally associated with $V$. This defines naturally a so-called \textbf{action filtration}
\begin{equation}
\iota_h^{\nu} :  HF^*(V) \to HF_{\leq \nu}^*(\widetilde{\phi}_h).
\end{equation}
\end{comment}
\begin{comment}
where $HF_{\leq \nu}^*(\widetilde{\phi}_h)$ depend on a real numbers $\nu$, and an element $\widetilde{\phi}_h \in \Cont(V)$ in the universal cover of the identity component of the group $\text{Cont}(V,\xi)$ of contactomorphisms of $(V, \xi)$ which are contact isotopic to the identity. These groups correspond to a contact analogue of the semi-infinite cohomology groups $\mathcal{H}_{\mathbb{K}}^*(F^-(\nu))$ from \cite{Giv1995}. 
\end{comment}
\begin{comment}
We compute the group $HF^*(V)$ and show that \textit{it is non-trivial whenever $M \neq \mathbb{C} P^n$}. In this case, following the works of Viterbo \cite{viterbo1992symplectic}, Schwarz \cite{schwarz2000action}, and Oh \cite{oh2005construction}, we define a function called \textbf{spectral invariant}: 
\begin{equation}
l : HF^*(V) \times \Cont(V,\xi) \to \mathbb{R} \cup \lbrace -\infty \rbrace, \quad  l(A,\widetilde{\phi}_h) := \sup \lbrace \nu \text{ generic} : \iota_h^{\nu}(A) \neq 0 \rbrace,
\end{equation}
satisfying the following properties:
\begin{enumerate}
\item $l$ is \textit{finite} on non-zero classes in $HF^*(V)$: $l(A,\widetilde{\phi}_h) = - \infty \iff A = 0$;

\item it is \textit{monotone}: $h \leq h' \implies \forall A \in HF^*(V) \setminus \{ 0 \}, \  l(A,h) \leq l(A,h')$;

\item it is \textit{continuous} with respect to the $C^0$-norm in $C^{\infty}(V \times [0,1])$: $\forall A \in HF^*(V)$, we have $$|l(A,\widetilde{\phi}_h) - l(A,\widetilde{\phi}_h')| \leq \frac{|h-h'|}{\underset{V \times [0,1]}{\min} e^{g_h}},$$ where $g_h : V \times [0,1] \to \mathbb{R}$ is the function satisfying $\phi_h^{t*} \alpha = e{g_h^t} \alpha;$

\item it is \textit{spectral}: $\forall A \in HF^*(V) \setminus \{ 0 \}, l(A,\widetilde{\phi}_h) \in \Spec(\phi_h)$;

\item it is \textit{computable} on the Reeb flow: $\exists B \in HF^*(V) \text{ such that } l(B, \widetilde{\phi}_{\alpha}^t) = t$. 
\end{enumerate}

The first application of this spectral invariant concerns the \textit{dynamics} of the group $\Cont(V,\xi)$, and constitutes the main result of this paper. Notice that non-triviality of $HF^*(V)$ and property $4$ imply in a straightforward manner \textit{the existence of at least one $\alpha$-translated point for any $\phi \in \Cont(V,\xi)$}. We have the following improvement.

\begin{theoremsec}{\label{theo1}}
Let $\phi \in \normalfont{\text{Cont}_0(V,\xi)}$. For any choice of contact form $\beta$ such that $\xi = \ker \beta$, $\phi$ has a $\beta$-translated point.
\begin{proof}
Let $\beta$ be a contact form on $V$, and $f : V \to \mathbb{R}_{>0}$ be the function satisfying $\alpha = f \beta$. Let $h: V \times [0,1] \to \mathbb{R}$ be a contact Hamiltonian such that $\phi = \phi_h$. For any $s \in \mathbb{R}$, we denote by $\Psi^s$ the time $1$ map of the following contact isotopy
\begin{equation}
t \mapsto \phi_{\beta}^{-st} \circ \phi_h^t, \quad t \in [0,1], 
\end{equation}
and consider its class $\widetilde{\Psi^s} \in \Cont(V,\xi)$. We use the fact that having a discriminant point is independent of the choice of a contact form. Note that $x$ is a discriminant point of $\Psi^s$ if and only if $x$ is a $\beta$-translated point of $\phi_h$. Consider the class $B \in HF^*(V)$ from property $5$. Notice first that if there exists $s \in \mathbb{R}$ such that $l(B, \widetilde{\Psi^s}) \in \hbar \mathbb{Z}$, then $\Psi^s$ has a discriminant point. Indeed, by property $4$, this condition means that $\Psi^s$ has an $\alpha$-translated point $x \in V$ such that
$$\Psi^s(x) = \phi_{\alpha}^{l(B,\widetilde{\Psi^s})}(x).$$
But by periodicity of $\Spec(\Psi^s)$, we have $\phi_{\alpha}^{l(B,\widetilde{\Psi^s})}(x) = \phi_{\alpha}^{0 \hbar}(x) = x$. We are thus reduced to find $s$ such that $l(B, \widetilde{\Psi^s}) \in \hbar \mathbb{Z}$. For the contact form $\alpha$, the contact Hamiltonian generating $\Psi^s$ is of the form $g_s -sf$, where $g_s : [0,1] \times V \to \mathbb{R}$. Let us bound the functions $g_s$ and $f$ in the following way: $$\underline{C} \leq g_s \leq \overline{C}, \quad \epsilon \leq f \leq E, \quad \underline{C}, \overline{C}, \epsilon, E \in \mathbb{R} \setminus \{ 0 \}.$$ By properties $2$ and $5$, we have
$$\underline{C} - s E \leq l(B, \widetilde{\Psi^s}) \leq \overline{C} - sE.$$
Finally, by property $3$, we can move $s$ continuously so that $l(B, \widetilde{\Psi^s})$ eventually hits the set $\hbar \mathbb{Z}$.
\end{proof}
\end{theoremsec}

\noindent A discussion is required. Note first that lens spaces are prequantization spaces over $\mathbb{C} P^n$. We treat prequantization spaces over \textit{any toric manifold different from $\mathbb{C} P^n$}. Let $h : M \times [0,1] \to \mathbb{R}$ be a Hamiltonian on $M$. It lifts naturally to an $S^1$-invariant contact Hamiltonian $\widetilde{h} : V \times [0,1] \to \mathbb{R}$ such that $\phi_{\widetilde{h}}$ preserves the contact form $\alpha$, and any $\alpha$-translated point of $\phi_{\widetilde{h}}$ projects to a fixed point of $\phi_h$. Conversely, a fixed point of $\phi_h$ lifts to a translated point of $\phi_{\widetilde{h}}$. By \cite{Giv1995}, Hamiltonian symplectomorphism of $M$ might have far more than one fixed point. This means that our contribution rather concerns contactomorphisms which do not come from Hamiltonian symplectomorphisms of $M$, and one might wonder if the lower bound from \cite{Giv1995} holds in general, for any $\phi \in \text{Cont}_0(V,\xi)$.\\


Our second result is rather \textit{geometrical}, and involves the notion of orderability, introduced in \cite{eliashberg2000partially}. Denoting by $\text{id} \in \Cont(V,\xi)$ the neutral element, we will write $\text{id} \preceq \widetilde{\phi}$ if there is a nonnegative Hamiltonian $h$ such that $\widetilde{\phi} = \widetilde{\phi}_h$ in $\Cont(V,\xi)$. The nonnegativity of $h$ is equivalent to $X_h^t$ being nowhere negatively transverse to $\xi$, and therefore it is independent of $\alpha$. This induces a reflexive and transitive relation on $\Cont(V,\xi)$, where
\begin{equation}
\widetilde{\phi} \preceq \widetilde{\psi} \text{ if and only if } \text{id} \preceq \widetilde{\phi}^{-1} \widetilde{\psi},
\end{equation}
which is also bi-invariant \cite{eliashberg2000partially}. The contact manifold $(V,\xi)$ is called \textbf{orderable} if $\preceq$ is also anti-symmetric. In \cite{eliashberg2000partially}, Eliashberg-Polterovich proved that this is equivalent to the condition that there are no positive contractible loops in $\text{Cont}_0(V,\xi)$: for any $h > 0$, $\widetilde{\phi}_h \neq \text{id} \in \Cont(V,\xi)$.

\begin{theoremsec}{\label{theo2}}
Suppose that $HF^*(V) \neq 0$. Then $(V,\xi)$ is orderable.

\begin{proof}
Suppose that there exists a contact Hamiltonian $h > 0$ such that $\widetilde{\phi}_h = \text{id}$. By properties $2$ and $5$, we would have
$$0 = l(B, \widetilde{\phi}_{\alpha}^0) = l(B, \text{id}) = l(B, \widetilde{\phi}_h) > 0.$$
\end{proof}
\end{theoremsec}
Orderability of prequantization spaces was confirmed in \cite{eliashberg2000partially} for the real projective space $\mathbb{R} P^{2n-1}$, and extended in (GKPS) to lens spaces other than the sphere $S^{2n-1}$, which was shown to be non-orderable in (EKP06). We prove this property for prequantization spaces over \textit{all toric manifolds different from $\mathbb{C} P^n$}. 

\begin{remarksec}
\noindent \normalfont Notice that since the contact sphere $S^{2n-1}$ is non-orderable, Theorem \ref{theo2} implies that the cohomology group $HF^*(S^{2n-1})$ vanishes.
\end{remarksec}
\end{comment}




\section{Preliminaries}{\label{sec2}}

\subsection{Fronts and deformation of sublevel sets}{\label{sec2.1}}

The main symplectic ingredient of our construction is a family of generating functions associated with the lift of a contactomorphism (section \ref{sec3.3}). We will study suitable sublevel sets of these functions, and more particularly how they behave under certain deformations. We describe here the general setting and relevant results. This section is added only for the sake of completeness; we strictly follow \cite[section $3$]{Giv1995}, except for some notations that are adapted to this paper. Let $F : X \times \Lambda \to \mathbb{R}$ be a family of functions $F_{\lambda}$ on a compact manifold $X$, where $\Lambda$ is a parametrizing manifold. Suppose that $F \in C^{1,1}$, that is $F$ is differentiable with Lipschitz derivatives (so that gradient flow deformations apply), and $F$ is smooth at all points $(x,\lambda)$ such that $x$ is a critical point of $F_{\lambda}$ with critical value zero. We consider restrictions of $F$ to submanifolds $\Gamma \subset \Lambda$ with boundary $\partial \Gamma$, and look at the sublevel sets
\begin{equation}{\label{eq5}}
F_{\Gamma}^- := \lbrace (x,\lambda) \in X \times \Lambda \ | \ \lambda \in \Gamma, F(x,\lambda) \leq 0 \rbrace, \quad \quad 
F_{\Gamma}^+ := \lbrace (x,\lambda) \in X \times \Lambda \ | \ \lambda \in \Gamma, F(x,\lambda) \geq 0 \rbrace.
 \end{equation}
Assume that $0$ is a regular value of $F$. The \textbf{front of $F$} is defined as
$$
L := \lbrace \lambda \in \Lambda : 0 \text{ is a singular value of } F_{\lambda} \rbrace.
$$
It is the singular locus of the projection
$$F^{-1}(0) \to \Lambda,$$
and since $F$ is smooth at all critical points with critical value zero, it is of \textit{zero-measure}. To understand its structure, it is convenient to view it as a singular hypersurface in $\Lambda$, provided at every point with tangent hyperplanes (there can be more than one hyperplane at each point). The construction goes as follows: let $N$ be a smooth neighborhood in $X \times \Gamma$ of the set of critical points of $F$ with critical value $0$. The zero set $F_{|N}^{-1}(0) \subset X \times \Lambda$ gives rise naturally to a Legendrian submanifold

$$\mathcal{L} := \lbrace (x,\lambda,T_{(x,\lambda)}F_{|N}^{-1}(0)) \ |\  (x,\lambda) \in X \times \Lambda \rbrace \subset PT^*(X \times \Lambda)$$
of the space $PT^*(X \times \Lambda)$ of contact elements. Let $\mathcal{P}$ denote the space of vertical contact elements (also called the \textit{mixed space} \cite{AN2001}): a hyperplane $H_{(x,\lambda)}$ is in $\mathcal{P}$ if and only if $T_x X \times \{ \lambda \} \subset H_{(x,\lambda)}$. Then the intersection $ \hat{\mathcal{L}} := \mathcal{L} \cap \mathcal{P}$ projects to a singular Legendrian submanifold $\hat{L} \subset PT^*(\Lambda)$ in the space of contact elements $PT^*(\Lambda)$. The front $L$ is then defined as the projection of $\hat{L}$ to the base $\Lambda$, whereas a tangent hyperplane at a point $\lambda \in L$ is an element of $\hat{L} \cap PT^*_{\lambda} (\Lambda)$ \footnote{The term \textit{front} comes from the study of wave fronts in geometrical optics. Given a Legendrian submanifold $L$ of a Legendrian fibration $E \to B$, the front of $L$ is defined as the projection of $L$ to the base $B$ (see \cite[chapter $5$]{AN2001}). If $L$ projects injectively onto its front, the latter is also called the \textit{generating hypersurface} of $L$. In our case $E = PT^*(\Lambda)$, and the (singular) Legendrian submanifold $\hat{L}$ is the projection of the (singular) intersection $\mathcal{L} \cap \mathcal{P} \subset PT^*(X \times \Lambda)$ to $PT^*(\Lambda)$. The zero set $F^{-1}(0) \subset X \times \Lambda$ is called a \textit{family of generating hypersurfaces} for $\hat{L}$. Note that if $\mathcal{L}$ was transversal to $\mathcal{P}$, $\hat{L}$ would be an immersed Legendrian submanifold in $PT^*(\Lambda)$. In fact, one can show that any immersed Legendrian submanifold of a space of contact elements can be obtained via such a procedure.}.

\begin{propsubsec}[\cite{Giv1995} Proposition $3.1$]{\label{prop2.1.1}}
A submanifold $\Gamma \subset \Lambda$ is transversal to $L$ if and only if the hypersurface $F^{-1}(0)$ is transversal to $X \times \Gamma$.
\begin{proof}
$X \times \Gamma$ is tangent to $F^{-1}(0)$ at a point $(x,\lambda) \in X \times \Lambda$ if and only if $T_{(x,\lambda)} (X \times \Gamma) \subset T_{(x,\lambda)} F^{-1}(0)$, if and only if $T_{(x,\lambda)} (X \times \Gamma) \subset \hat{\mathcal{L}}$, if and only if $T_{\lambda} \Gamma \subset \hat{L} \cap PT^*(\Lambda)$.
\end{proof}
\end{propsubsec}
\begin{corsubsec}[\cite{Giv1995} Proposition $3.2$]{\label{cor2.1.1}}
Let $\Gamma_t := \rho^{-1}(t)$ be non-singular levels of some smooth map $\rho : \Gamma \to \mathbb{R}^m$. Then almost all $\Gamma_t$ are transversal to $L$. 
\end{corsubsec}
The two following propositions can be proved using standard gradient flow deformations.
\begin{propsubsec}{\label{prop2.1.2}}
Let $F : X \times \Lambda \times [0,1] \to \mathbb{R}$ be a $C^{1,1}$ family of functions $F_s : X \times \Lambda \to \mathbb{R}$ satisfying similar properties as above. Suppose that for any $s \in [0,1]$, $\Gamma \subset \Lambda$ and $\partial \Gamma$ are transversal to $L_s$. Then there exists a Lipschitz isotopy $I : X \times \Lambda \times [0,1] \to X \times \Lambda$ such that
$$I_s(F_{\Gamma,0}^{\pm}, \partial F_{\partial \Gamma,0}^{\pm}) = (F_{\Gamma,s}^{\pm}, \partial F_{\partial \Gamma,s}^{\pm}),$$
where we have denoted by $F_{\Gamma,s}^{\pm}, F_{\partial \Gamma,s}^{\pm}$ the sublevel sets defined for $F_s$ as in equation (\ref{eq5}). If moreover each $F_s$ is invariant under the action of a compact Lie group $G$ on $X$, the isotopy can be made $G$-equivariant.
\end{propsubsec}
\begin{propsubsec}{\label{prop2.1.3}}
Let $\lbrace \Gamma_s \rbrace_{s \in [0,1]}$ be a $C^{1,1}$ family of submanifolds $\Gamma_s \subset \Lambda$, such that for any $s \in [0,1]$, $\Gamma_s$ and $\partial \Gamma_s$ are transversal to $L$. Then there exists a Lipschitz isotopy $I : X \times \Lambda \times [0,1] \to X \times \Lambda$ such that
$$I_s(F_{\Gamma_0}^{\pm}, F_{\partial \Gamma_0}^{\pm}) = (F_{\Gamma_s}^{\pm}, F_{\partial \Gamma_s}^{\pm}).$$
If moreover $F$ is invariant under the action of a compact Lie group $G$ on $X$, the isotopy can be made $G$-equivariant.
\end{propsubsec}

\begin{remarksubsec}
\normalfont In the sequel $\Gamma$ will be a \textit{stratified} manifold. In this setting, one must improve the notion of transversality: $\Gamma$ is transversal to $L$ if each of its strata is. Then the results above remain valid (we refer to \cite{goresky1988stratified} for a detailed treatment of stratified Morse theory).
\end{remarksubsec}
\subsection{Equivariant cohomology and conical spaces}{\label{sec2.2}}

In the sequel we will study the equivariant homotopy type of conical sublevel sets of generating families. This section is meant to recall several facts from equivariant cohomology, fix some notations, and describe a simple identification in equivariant cohomology which holds for conical spaces. The latter will be convenient for defining homomorphisms of equivariant cohomology groups (see section \ref{sec3.6}). We refer to \cite{BGS13} for a complete treatment of equivariant cohomology theory. Let $X$ be a topological space provided with the action of a compact Lie group $G$. Equivariant cohomology $H^*_G(X;\mathbb{C})$ with complex coefficients is defined as the singular cohomology $H^*(X_G;\mathbb{C})$ of the quotient
$$X_G := (X \times EG) / G,$$
where $EG \to BG$ is the universal principal $G$-bundle over the classifying space $BG$ of $G$. The canonical projection
$$(X \times EG)/BG \to EG/G$$
provides $H^*_G(X;\mathbb{C})$ with the structure of a module over $H^*(BG;\mathbb{C})$, which plays the role of the coefficient ring $H^*_G(\pt;\mathbb{C})$ in equivariant cohomology theory. If $G$ is the $n$-dimensional torus $\mathbb{T}^n = (S^1)^n$, the coefficient ring $H^*_{\mathbb{T}^n}(\pt;\mathbb{C})$ is naturally isomorphic to a polynomial algebra in $n$ variables $u = (u_1,...,u_n)$ of degree $2$. More precisely, there is a natural algebra isomorphism, called the \textit{Chern-Weil isomorphism}
$$\chi : H_{\mathbb{T}^n}^*(\pt;\mathbb{C}) \simeq \mathbb{C}[\mathbb{R}^{n*}],$$
between the cohomology of the classifying space $B \mathbb{T}^n$ and the polynomial algebra on $\mathbb{R}^{n*}$ over $\mathbb{C}$.

Throughout this paper we will only deal with singular cohomology with complex coefficients, and will use the notation $H^*(Y) := H^*(Y;\mathbb{C})$. If $(u_1,...,u_n)$ denotes the standard basis of $\mathbb{R}^{n*}$, the Chern-Weil isomorphism writes
$$
H^*_{\mathbb{T}^n}(\pt) \simeq \mathbb{C}[u_1,...,u_n].
$$

Let now $A \subset X$ be a $G$-invariant subspace of $X$, and $pr : X \times EG \to X_G$ denote the canonical projection. We introduce the following notations:
\begin{itemize}
\item $C_G^*(X) := C^*(X_G)$ and $C_G^*(A) := C^*(A_G)$;
\item $C_G^*(X, A) := \ker C_G^*(X) \to C_G^*(A)$, where the map is induced by the inclusion $A \subset X$;
\item $C_{G,c}^*(X) := \lbrace \sigma \in C_G^*(X) : \exists K \subset X \text{ compact such that } pr^* \sigma \text{ vanishes on } (X \setminus K) \times EG \rbrace$, the complex of equivariant cocycles with compact support;
\item $C_{G,c}^*(A) := \lbrace \sigma \in C_G^*(A) : \exists K \subset A \text{ compact such that } pr^*_{| A \times EG} \sigma \text{ vanishes on } (A \setminus K) \times EG \rbrace$;
\item $C_{G,c}^*(X,A) := \ker C^*_{G,c}(X) \to C^*_{G,c}(A)$;
\item if $X \to M$ is a fiber bundle of $G$-spaces, $C_{G,cv}^*(X)$, $C_{G,cv}^*(A)$, $C_{G,cv}^*(X,A)$, the complexes of equivariant (relative) cocycles with vertical compact support, that is we replace the term compact above by compact in the fibers direction. We will denote $H_{\mathbb{K}_0,cv}^*$ the corresponding equivariant cohomology groups.  
\end{itemize}
Suppose that $X$ and $A$ are closed conical subspaces of $\mathbb{R}^m$, that is $tX \subset X$ and $tA \subset A$ for any $t > 0$, and that $G$ is a torus. Then the natural retraction of $\mathbb{R}^m$ onto the closure $\overline{B}$ of its unit ball $B$ induces a $G$-equivariant retraction of $X$ (resp. $A$) onto $X \cap \overline{B}$ (resp. $A \cap \overline{B}$), and of $X \setminus (X \cap B)$ (resp. $A \setminus (A \cap B)$) onto $X \cap S$ (resp. $A \cap S$), where $S = \partial \overline{B}$.
\begin{comment}
Thus the commutative diagram
\begin{equation}
\begin{tikzcd}
0 \ar[r] & C_G^*(X, X \setminus X \cap B) \ar[r] \ar[d] & C_G^*(X) \ar[r] \ar[d] & C_G^*(X \setminus X \cap B) \ar[r] \ar[d] & 0 \\
0 \ar[r] & C_G^*(A, A \setminus A \cap B) \ar[r] & C_G^*(A) \ar[r] & C_G^*(A \setminus A \cap B) \ar[r] & 0 
\end{tikzcd}
\end{equation}
naturally identifies with the diagram
\begin{equation}
\begin{tikzcd}
0 \ar[r] & C_G^*(X \cap \overline{B},  X \cap S) \ar[r] \ar[d] & C_G^*(X \cap \overline{B}) \ar[r] \ar[d] & C_G^*(X \cap S) \ar[r] \ar[d] & 0 \\
0 \ar[r] & C_G^*(A \cap \overline{B}, A \cap S) \ar[r] & C_G^*(A \cap \overline{B}) \ar[r] & C_G^*(A \cap S) \ar[r] & 0 
\end{tikzcd}
\end{equation}
the vertical maps being induced by inclusions, and the horizontal maps being standard short exact sequences. 
\end{comment}
Moreover, the inclusion $C_G^*(X, X \setminus X \cap B) \hookrightarrow C_{G,c}^*(X)$ is a $G$-equivariant homotopy equivalence, since any compact subset $K \subset X$ is included in an intersection $X \cap \widetilde{B}$, where $\widetilde{B}$ is a ball centered at $0$, and $X \cap \widetilde{B}$ is $G$-equivariantly homotopic to $X \cap B$ ($X$ is conical and $G$ commutes with $\mathbb{R}_{>0}$). The same argument applies when replacing $X$ with $A$. Putting all these simple facts together yields a natural isomorphism of cohomology groups
$$
H^*(C_G^*(X \cap S, A \cap S)) \simeq H^*(C_G^*(X,A) / C_{G,c}^*(X,A)).
$$


\begin{comment}
\subsection{The Hamilton-Jacobi equation}

We recall here the Hamilton-Jacobi identity. This 
We start by considering a closed manifold $N$, and we endow $T^*N$ with its canonical symplectic form $\omega = \mathrm{d} \lambda$, where $\lambda = p \mathrm{d} q$ is the Liouville form. Consider the action functional of a Hamiltonian $H : T^*N \times [0,1] \to \mathbb{R}$, defined on the space $\mathcal{P}$ of paths in $T^*N$ starting at the zero section:
\begin{equation}
\mathcal{A}_H : \mathcal{P} \to \mathbb{R}, \quad \mathcal{A}_H(\gamma) = \int_0^1 \lambda(\dot{\gamma}(t)) \mathrm{d} t - \int_{0}^1 H_t(\gamma(t)) \mathrm{d} t.
\end{equation} 
For a function $F : N \to \mathbb{R}$, we denote the graph of its differential by $\Gamma_F$, that is
\begin{equation}
\Gamma_F := \lbrace \mathrm{d}_qF | q \in N \rbrace.
\end{equation}
Let $\pi : T^*N \to N$ denote the canonical projection. For $q \in N$, we define $\gamma_q \in \mathcal{P}$ via
\begin{equation}
\gamma_q(t) = \Phi_H^t(q).
\end{equation}

\begin{lemmasec}{\label{lemma31}}
Suppose that $L := \Phi_H(0_N)$ is graphical in $T^*N$, that is the restriction $\pi_{|L} : L \to N$ is a diffeomorphism. Then the function
\begin{equation}
F(q) = \mathcal{A}_H(\gamma_{\Phi_H^{-1}\pi_{|L}^{-1}(q)})
\end{equation}
generates $L$, that is $L = \Gamma_F$.

\begin{proof}
We define $F$ as the following composition:
\begin{equation}
N \xrightarrow{\pi_{|L}^{-1}} L \xrightarrow{\phi_H^{-1}} N \xrightarrow{\gamma} \mathcal{P} \xrightarrow{\mathcal{A}_H} \mathbb{R},
\end{equation}
and therefore its differential is the composition of the differentials of these maps.

The differential of $\mathcal{A}_H$ is given by 
\begin{equation}
\mathrm{d}_{\gamma} \mathcal{A}(\xi) = \langle \lambda_{\gamma(1)}, \xi(1) \rangle - \int_0^1 \omega(\xi(t),X_H(\gamma(t))-\dot{\gamma}(t)) \mathrm{d} t.
\end{equation}

The differential of the third map
\begin{equation}
N \to \mathcal{P}, \quad q \mapsto \gamma_q
\end{equation}
is the linearized map of the Hamiltonian flow of $H$, that is it maps a tangent vector $u \in T_q N$ to the vector field
\begin{equation}
t \mapsto (\Phi_H^t)_*(u) \in T_{\gamma_q(t)} T^*N.
\end{equation}
Since $\gamma_{\phi_H^{-1}(\pi_{|L}^{-1}(q))}$ is a Hamiltonian trajectory, the differential of $\mathcal{A}_H$ at it reduces to the evaluation of $\lambda$ at its endpoint, which is $\pi_{|L}^{-1}(q)$. Thus for the differential of $F$ we have the following: if $u \in T_q N$, then
\begin{equation}
\mathrm{d}_q F(u) = \langle \lambda_{\pi_{|L}^{-1}(q)}, \Phi_{H*}\Phi_{H*}^{-1}\pi_{|L *}^{-1}(u) \rangle = \langle \lambda_{\pi_{|L}^{-1}(q)}, \pi_{|L *}^{-1}(u) \rangle.
\end{equation}
the definition of $\lambda$ implies that this is equal to the value of the functional $\pi_{|L}^{-1}(q) \in T^*_q N$ evaluated on the vector $\pi_* \pi_{|L *}^{-1}(u) = u$, that is
\begin{equation}
\mathrm{d}_q F(u) = \pi_{|L}^{-1}(q)(u),
\end{equation} 
in other words
\begin{equation}
\mathrm{d}_q F = \pi_{|L}^{-1}(q),
\end{equation}
which means that $\mathrm{d}_q F$ is precisely the point of $L$ lying above $q$, that is $F$ generates $L$.
\end{proof}
\end{lemmasec}

Now, for $t \in [0,1]$, we can apply the above construction to the Lagrangian $L_t := \Phi_H^t(0_N)$, obtaining the generating functions $F_t : N \to \mathbb{R}$, provided $L_t$ is graphical over $N$. Let us denote 
\begin{equation}
\pi_t := \pi_{|L_t} : L_t \to N,
\end{equation}
which is a diffeomorphism by assumption.

\begin{propsec}[Hamilton-Jacobi equation]{\label{prop32}}
We have
\begin{equation}
\frac{\partial F_t}{\partial t}(q) = - H_t(\pi_t^{-1}(q)).
\end{equation}
\begin{proof}
Consider $T^*[0,1]$ with coordinates $(t,\tau)$ and Liouville form $\tau \mathrm{d} t$. Define
\begin{equation}
Q = \lbrace  (z;t,-H_t(z)) | t \in [0,1], z \in L_t \rbrace \subset T^*N \times T^*[0,1].
\end{equation}
A direct computation shows that this is a Lagrangian submanifold. We claim that it is exact. First, for $t \in [0,1]$, put $Q_t := Q \cap T^*N \times T_t^*[0,1]$. Note that the projection of $Q_t$ to $T^*N$ (that is forgetting about the $\tau$ coordinate) is exactly $L_t$. Clearly $\tau \mathrm{d} t$ vanishes when restricted to $Q_t$, and we see that the restriction of $\lambda \oplus \tau \mathrm{d} t$ to $Q_t$ equals, via the identification $Q_t \simeq L_t$, the restriction $\lambda_{| L_t}$. In particular for $t=0$, we have $\lambda_{|L_0} = \lambda_N = 0$.

To show that $Q$ is exact it suffices to show that the integral of $\lambda \oplus \tau \mathrm{d} t$ over any loop in $Q$ vanishes. Since $Q$ is Lagrangian, $\lambda \oplus \tau \mathrm{d} t$ is closed when restricted to $Q$, and therefore the integral over any loop only depends on its free homotopy class. Any loop in $Q$ can be deformed into a loop contained entirely in $Q_0$, and so the integral of $\lambda \oplus \tau \mathrm{d} t$ vanishes, because, as we've just seen, the restriction of $\lambda \oplus \tau \mathrm{d} t$ to $Q_0$ is zero.

Since by assumption all the manifolds $L_t$ are graphs over $N$, it follows that $Q$ is a graph over $N \times [0,1]$. Since it is also exact, it follows that we can find a function $G : N \times [0,1] \to \mathbb{R}$ such that
\begin{equation}
(\lambda \oplus \tau \mathrm{d} t)_{|Q} = \mathrm{d} (\Pi^*G)_{|Q},
\end{equation}
where $\Pi : T^*N \times T^*[0,1] \to N \times [0,1]$ is the projection. The primitive of an exact $1$-form is determined up to a constant, and we normalize the choice of $G$ as follows. We have 
\begin{equation}
0 = (\lambda \oplus \tau \mathrm{d} t)_{|Q_0} = \mathrm{d} (\Pi^*G)_{|Q_0},
\end{equation}
meaning that $G$ is constant on the slice $N \times \lbrace 0 \rbrace$, and we normalize $G$ by setting this constant to zero. We claim that with this normalization we have
\begin{equation}
G(q,t) = F_t(q).
\end{equation}
Indeed, fix $q \in N$ and consider the path $s \mapsto (\Phi_H^s(q); s, -H_s(\Phi_H^s(q))) \in Q$, for $s \in [0,t]$. A direct computation shows that the integral of $\lambda \oplus \tau \mathrm{d} t$ over it for $s \in [0,t]$ equals the action of the path $s \mapsto \Phi_H^s(q) \in T^*N$ (for $s \in [0,t]$), which, as we have seen, equals $F_t(\pi(\Phi_H^t(q)))$. On the other hand, using the fact that $(\lambda \oplus \tau \mathrm{d} t)_{|Q} = \mathrm{d} (\Pi^* G)_{|Q}$ and the above normalization, we see that the integral also equals $G(\pi(\Phi_H^t(q)),t)$, meaning 
\begin{equation}
G(\pi(\Phi_H^t(q)),t) = F_t(\pi(\Phi_H^t(q))).
\end{equation}
Substituing $q = (\Phi_H^{t})^{-1} \pi_t^{-1}(x)$ for $x \in N$, we obtain the desired result $G(x,t) = F_t(x)$.

The tangent space to $Q$ at a point $(z; t, -H_t(z)) = (\Phi_H^t(x); t, -H_t(\Phi_H^t(x))) \in Q$, $x \in N$ is 
\begin{equation}
T_{(z;t,-H_t(z))}Q = \lbrace (\Phi^t_{H*}(Z) + T X_H^t(\Phi_H^t(x)); T, - \mathrm{d}_{x}(H_t \circ \Phi_H^t)(Z) - T \frac{\partial H_t}{\partial t}(\Phi_H^t(x))) | T \in \mathbb{R}, Z \in T_x N \rbrace.
\end{equation}
Setting $Z = - \Phi_H^{t*} \pi_t^* \pi_* X_H^t(z) \in T_x N$, and $T=1$, we get on one hand
\begin{equation}
\begin{array}{lcl}
(\lambda \oplus \tau \mathrm{d} t)_{(z;t,-H_t(z))}(\Phi_{H*}^tZ + X_H^t(z); 1, - \mathrm{d}_{z} H_t(\Phi_{H*}^tZ) - \frac{\partial H_t}{\partial t}(z)) = & \\ \\
\underbrace{\lambda_z(\Phi_{H*}^tZ + X_H^t(z))}_{=0 \text{ since $\lambda_{(q,p)} = p \circ \pi_*$}} - H_t(z) =  -H_t(z),
\end{array}
\end{equation}
and on the other hand
\begin{equation}
\begin{array}{lcl}
\mathrm{d} (\Pi^* G)(\Phi_{H*}^tZ + X_H^t(z); 1, - \mathrm{d}_{z} H_t(\Phi_{H*}^tZ) - \frac{\partial H_t}{\partial t}(z)) = \\ \\
\mathrm{d}_{(\pi(z),t)} G (\pi_*(\Phi_{H*}^tZ + X_H^t(z)), 1) = \\ \\
\mathrm{d}_{(\pi(z),t)} G (0,1) = \frac{\partial G}{\partial t} (\pi(z),t).
\end{array}
\end{equation}
We therefore obtain:
\begin{equation}
\frac{\partial F_t}{\partial t}(\pi(z)) = \frac{\partial G}{\partial t}(\pi(z),t) = - H_t(z),
\end{equation}
as claimed.
\end{proof}
\end{propsec}
\end{comment}

\section{The constructions}{\label{sec3}}

\subsection{The prequantization space}{\label{sec3.1}}

In this section, we describe a natural construction of a prequantization space over a rational toric symplectic manifold $(M^{2d},\omega, \mathbb{T})$, mainly following \cite{borman2015quasimorphisms} (see Remark \ref{rem3.1.2}). The action of $\mathbb{T}$ is induced by a momentum map $M \to \mathfrak{t}^*$, where $\mathfrak{t}^*$ is the dual to the Lie algebra $\mathfrak{t}$ of $\mathbb{T}$. The image $\Delta$ of the momentum map is called the moment polytope. If $\Delta$ has $n$ facets, it is given by
$$
\Delta = \lbrace x \in \mathfrak{t}^* \ |\  \langle x,v_j \rangle + a_j \geq 0 \text{ for } j=1,...,n \rbrace,
$$
where the conormals $v_j$ are primitive vectors in the integer lattice $\mathfrak{t}_{\mathbb{Z}} := \ker(\exp : \mathfrak{t} \to \mathbb{T})$, and $a := (a_1,...,a_n) \in (\mathbb{R}_{> 0})^n$. The polytope $\Delta$ is compact and smooth, that is each $k$-codimensional face of $\Delta$ is the intersection of exactly $k$ facets and the $k$ associated conormals $\lbrace v_{l_1}, ..., v_{l_k} \rbrace$ can be extended to an integer basis for the lattice $\mathfrak{t}_{\mathbb{Z}}$.

\subsubsection{Delzant's construction of symplectic toric manifolds}{\label{sec3.1.1}}

Let us first recall Delzant's construction of toric manifolds \cite{delzant1988hamiltoniens}. The standard Hamiltonian action of the torus $\mathbb{T}^n := \mathbb{R}^n / \mathbb{Z}^n$ on $(\mathbb{C}^n, \omega_{\text{std}} := \underset{j=1}{\overset{n} \sum} \mathrm{d} x_j \wedge \mathrm{d} y_j)$ by rotation in each coordinate is induced by the momentum map
\begin{equation}{\label{eq6}}
P : \mathbb{C}^n \to \mathbb{R}^{n*}, \quad \text{where} \quad \langle \lambda, P \rangle (z) = \pi \underset{j=1}{\overset{n} \sum} \lambda_j |z_j|^2, \quad \text{and} \quad \lambda = (\lambda_1,...,\lambda_n) \in \mathbb{R}^n. 
\end{equation}
Consider the following surjective linear map:
$$\beta_{\Delta} : \mathbb{R}^n \to \mathfrak{t}, \quad \epsilon_j \mapsto v_j, \quad \text{for any} \quad j=1,...,n,$$
where $(\epsilon_1,...,\epsilon_n)$ is the standard basis of $\mathbb{R}^n$, and $v_j \in \mathfrak{t}_{\mathbb{Z}}$ are the conormals. Since $\Delta$ is compact and smooth, the map $\beta_{\Delta}$ satisfies $\beta_{\Delta} (\mathbb{Z}^n) = \mathfrak{t}_{\mathbb{Z}}$, and therefore it induces a homomorphism $[ \beta_{\Delta} ] : \mathbb{T}^n \to \mathbb{T}$. We define the connected subtorus 
$$\mathbb{K} \subset \mathbb{T}^n$$
as the kernel of $[\beta_{\Delta}]$. It has Lie algebra
$$\mathfrak{k} := \ker(\beta_{\Delta} : \mathbb{R}^n \to \mathfrak{t}),$$
and if $\iota : \mathfrak{k} \hookrightarrow \mathbb{R}^n$ denotes the inclusion, the momentum map for the action of $\mathbb{K}$ on $\mathbb{C}^n$ is given by
$$
P_{\mathbb{K}} := \iota^* \circ P : \mathbb{C}^n \to \mathfrak{k}^*.
$$
The torus $\mathbb{K}$ acts freely on the regular level set
$$
P_{\mathbb{K}}^{-1}(p), \quad \text{where} \quad p := \iota^*(a) \in \mathfrak{k}^* \setminus \lbrace 0 \rbrace,
$$
and if $X_{\lambda}(z) = 2 i \pi (\lambda_1 z_1,...,\lambda_n z_n) \in \mathbb{C}^n = T_z \mathbb{C}^n$ denotes the Hamiltonian vector field for the function $\langle \lambda, P \rangle : \mathbb{C}^n \to \mathbb{R}$, and $\alpha_{\text{std}} = \frac{1}{2} \underset{j=1}{\overset{n}{\sum}}(x_j \mathrm{d} y_j - y_j \mathrm{d} x_j)$ is the standard $1$-form on $\mathbb{C}^n$ so that $\mathrm{d} \alpha_{\text{std}} = \omega_{\text{std}}$, we have
\begin{equation}{\label{eq7}}
\alpha_{\text{std}}(X_{\lambda}) = \langle \lambda, P \rangle \quad \text{and} \quad \iota_{X_{\lambda}} \mathrm{d} \alpha_{\text{std}} = \iota_{X_{\lambda}} \omega_{\text{std}} = - \mathrm{d} \langle \lambda, P \rangle.
\end{equation}
In particular, $(\mathcal{L}_{X_{\lambda}} \omega_{\text{std}})|_{P_{\mathbb{K}}^{-1}(p)} = 0$. Therefore, symplectic reduction gives rise to a symplectic manifold $(M_{\Delta}, \omega_{\Delta})$, where
$$
M_{\Delta} := P_{\mathbb{K}}^{-1}(p) / \mathbb{K}, \quad \text{and the symplectic form $\omega_{\Delta}$ is induced by} \quad \omega_{\text{std}}|_{P_{\mathbb{K}}^{-1}(p)}.
$$
Finally, Delzant's theorem \cite{delzant1988hamiltoniens} shows that $(M_{\Delta},\omega_{\Delta}, \mathbb{T}^n / \mathbb{K})$ and $(M,\omega, \mathbb{T})$ are equivariantly symplectomorphic as toric manifolds.

\begin{remarksubsec}{\label{rem3.1.1}}
\normalfont This construction provides natural isomorphisms
$$
H_2(M;\mathbb{R}) \simeq \mathfrak{k} \quad \text{and} \quad H^2(M;\mathbb{R}) \simeq \mathfrak{k}^*,
$$
which restrict to the isomorphisms from equation $(\ref{eq2})$, and under which $p \in \mathfrak{k}^*$ identifies with the cohomology class $[\omega] \in H^2(M;\mathbb{R})$. Note that $M$ being a toric manifold, the natural homomorphisms $H_2(M;\mathbb{Z}) \to H_2(M;\mathbb{R})$ and $H^2(M;\mathbb{Z}) \to H^2(M;\mathbb{R})$ are injective.
\end{remarksubsec}



\subsubsection{The contact sphere}{\label{seq3.1.2}}

The generating families of our paper differ from those defined in \cite{Giv1995} for they are related to a contactomorphism of a prequantization space over $M$, rather than to a symplectomorphism of $M$. Yet, we will see now that the sublevel set $P_{\mathbb{K}}^{-1}(p)$ lies in a contact sphere in $\mathbb{C}^n$ (in fact it lies in many). Since the latter is the symplectization of the former, this will allow us to lift the contactomorphism of the prequantization space to a symplectomorphism of $\mathbb{C}^n$, (see section \ref{sec3.2.2}), and thus we will be able to define the generating families as in the aforementioned paper. The existence of such a sphere is ensured by compactness of the toric manifold $M$. There is no canonical way of choosing it if $M$ is non-monotone, however the prequantization space won't depend on such a choice (in fact it does not depend on the sphere at all, see Remark \ref{rem3.1.2}). we refer to \cite{Aud2012} for more details on toric manifolds.

The image of the momentum map $P$ from equation $(\ref{eq6})$ is the first orthant $\mathbb{R}^{n*}_{\geq 0}$, and the polytope $\Delta$ can be identified with 
$$(\iota^*)^{-1}(p) \cap \mathbb{R}^{n*}_{\geq 0}.$$
Moreover, compactness of $M_{\Delta}$ is equivalent to that of $\Delta$, which is ensured by the condition 
$$\ker \iota^* \cap \mathbb{R}^{n*}_{\geq 0} = \lbrace 0 \rbrace.$$
Since $\ker \iota^*$ is equal to the annihilator of $\iota(\mathfrak{k})$ in $\mathbb{R}^{n*}$, this implies in particular that $\iota(\mathfrak{k}) \cap \mathbb{R}^n_{\geq 0} \neq \lbrace 0 \rbrace$. Moreover, $p$ lies in the image of $\mathbb{R}_{\geq 0}^{n*}$, and therefore it is positive on $\iota(\mathfrak{k}) \cap \mathbb{R}^n_{\geq 0}$. Choose any vector $b \in \mathfrak{k} \setminus \{0\}$ such that $\iota(b) = (b_1,...,b_n) \in \mathbb{R}^n_{\geq 0}$. Then one can define the contact sphere as
\begin{equation}{\label{eq8}}
S_p := \lbrace z \in \mathbb{C}^n : \underset{j=1}{\overset{n} \sum} b_j \pi |z_j|^2 = p(b) \rbrace, \quad \text{with contact form} \quad \alpha_p := \alpha_{\text{std} | S_p}.
\end{equation}

\begin{comment}
\begin{remarksubsec} \normalfont The positivity of $p$ on $\iota(\mathfrak{k}) \cap \mathbb{R}^n_{>0}$ will be important in the sequel, when we compare sublevel sets of generating families.
\end{remarksubsec}
\end{comment}


\begin{comment}
Here we prove the existence of a certain vector in $\mathfrak{k}_{\mathbb{Z}}$ which, on one hand defines a sphere from which we will construct the contact prequantization, and on the other hand is such that the generating families which we will construct in section \ref{sec2} decrease along this direction. The statement here is that such a vector exists in $\mathfrak{k}_{\mathbb{Z}}$, which will be critical in the sequel. Recalling the description (\ref{eq1}) of $\Delta$, we denote by $V_j$ the hyperplane with normal $v_j$, for $j=1,...,n$.


\begin{propsubsec}{\label{prop1}}
There exists $b \in \mathfrak{k}_{\mathbb{Z}}$ which, seen as a vector $b := (b_1,...,b_n) \in \mathbb{R}^n$, satisfies $b_j > 0$, and such that $p(b) > 0$.

\begin{proof}
By compactness, a translation of $V_j$ in the direction of $v_j$ eventually becomes another supporting affine hyperplane $\widetilde{V_j}$ of $\Delta$ (put figure), the supported face of which is of dimension, say, $r \leq d-1$. Let $p_1, ..., p_{r+1}$ be the vertices of this face. Since $\Delta$ is smooth, we can write $v_j$ as $r+1$ linear equations:
\begin{equation}
\begin{array}{lcl}
v_j &= - \underbrace{\sum_{i=1}^d b_i^{1,j} v_i^1}_{L_j^1}\\
& \vdots\\
&= - \underbrace{\sum_{i=1}^d b_i^{{r+1},j} v_i^{r+1}}_{L_j^{r+1}},
\end{array}
\end{equation}
where $\lbrace v_1^l,...,v_d^l \rbrace$ is the integer basis for $\mathfrak{t}_{\mathbb{Z}}$ formed by the conormals to the $d$ facets containing $p_l$, and for any $l$, the coefficients $b_j^l$ are non-negative integers, and at least one of them is positive. We get
\begin{equation}
B_j := (r+1)v_j + \sum_{l=1}^{r+1} L_j^l = 0.
\end{equation}
By construction, we can write 
\begin{equation}
\underset{j=1}{\overset{n}{\sum}} B_j = \underset{j=1}{\overset{n}{\sum}} b_j v_j = 0,
\end{equation}
where $b:= (b_1,...,b_n) \in (\mathbb{Z}_{>0})^n$. Thus, we have 
\begin{equation}
b \in \mathfrak{k}_{\mathbb{Z}} = \ker (\beta_{\Delta}), \quad \text{and} \quad p(b) = \underset{j=1}{\overset{n}{\sum}}a_j b_j > 0,
\end{equation}
as we wanted.
\end{proof}
\end{propsubsec}
As a direct consequence, the regular level set $P_{\mathbb{K}}^{-1}(p)$ is a submanifold of the $\mathbb{K}$-invariant sphere
\begin{equation}
S_p := \lbrace z \in \mathbb{C}^n : \sum_{j=1}^n b_j \pi |z_i|^2 = p(b) > 0 \rbrace.
\end{equation}
\end{comment}


\subsubsection{The prequantization space}{\label{sec3.1.3}}

We now construct our prequantization space. From now on we assume that the toric manifold $(M,\mathbb{T},\omega)$ is rational. Notice that since $\beta_{\Delta}(\mathbb{Z}^n) = \mathfrak{t}_{\mathbb{Z}}$, the inclusion $\iota$ satisfies
$$\iota(\mathfrak{k}_{\mathbb{Z}}) \subset \mathbb{Z}^n.$$
Thus, the rationality condition is equivalent to
$$
p \in \mathfrak{k}_{\mathbb{Z}}^*.
$$
Let $\mathfrak{k}_0$ denote the kernel of $p : \mathfrak{k} \to \mathbb{R}$. Then $\mathfrak{k}_{0,\mathbb{Z}} := \ker (p : \mathfrak{k}_{\mathbb{Z}} \to \mathbb{Z}) \subset \mathfrak{k}_0$ is a sublattice of $\mathfrak{k}_{\mathbb{Z}}$. This means that $\mathbb{K}_0 := \exp (\mathfrak{k}_0) \subset \mathbb{K}$ is a subtorus of codimension $1$ of $\mathbb{K}$ which, in particular, acts freely on the regular level set $P_{\mathbb{K}}^{-1}(p)$. Let $j : \mathfrak{k}_0 \hookrightarrow \mathfrak{k}$ denote the inclusion of Lie algebras. Then $P_{\mathbb{K}}^{-1}(p)$ is the zero-level of the momentum map
$$
\mu : S_p \to \mathfrak{k}_0^*, \quad \mu(z) := j^* \circ P_{\mathbb{K}}(z)
$$
associated with the action of $\mathbb{K}_0$ on the contact sphere $(S_p, \alpha_p)$. Therefore, contact reduction yields a contact manifold
$$
(V := P_{\mathbb{K}}^{-1}(p) / \mathbb{K}_0, \xi := \ker \alpha), \quad \rho^* \alpha = \alpha_p,
$$
where $\rho : P_{\mathbb{K}}^{-1}(p) \to V$ denotes the canonical projection (see \cite{geiges2008introduction}  
for more on contact reductions). For the circle $S^1 := \mathbb{K} / \mathbb{K}_0$, the projection
$$
\pi : (V, \alpha) \to (M_{\Delta},\omega_{\Delta})
$$
defines a principal $S^1$-bundle, and satisfies $\pi^* \omega_{\Delta} = \mathrm{d} \alpha$, since $\omega_{\text{std}} = \mathrm{d} \alpha_{\text{std}}$. Finally, one can choose an equivariant symplectomorphism $(M_{\Delta},\omega_{\Delta},\mathbb{T}^n / \mathbb{K}) \simeq (M,\omega,\mathbb{T})$, and obtain a prequantization space over the toric manifold $(M,\omega,\mathbb{T})$.

\begin{remarksubsec}{\label{rem3.1.2}}
\normalfont The construction of $V$ is independent of the sphere $S_p$. Indeed, equation (\ref{eq7}) shows that the infinitesimal action of $\mathbb{K}_0$ is tangent to $\ker \alpha_{\text{std}}$ along $P_{\mathbb{K}}^{-1}(p)$, which implies that the contact form $\alpha$ is well defined. We view $V$ as a contact reduction of $S_p$ only for the lifting procedure of the following section.
\end{remarksubsec}



\subsection{Lifting contact isotopies}{\label{sec3.2}}

In this section we explain the procedure for lifting a contact isotopy of $V$ to a Hamiltonian isotopy of $\mathbb{C}^n$. Recall that for any contact manifold $(V,\xi)$ and any choice of contact form $\alpha$, any \textbf{contact Hamiltonian} $h : V \times [0,1] \to \mathbb{R}$ gives rise to a unique time-dependent vector field satisfying
$$
\alpha(X_h^t) = h_t \quad \text{and} \quad \mathrm{d} \alpha(X_h^t,.) = - \mathrm{d} h_t + \mathrm{d} h_t(R_{\alpha}) \alpha, \quad h_t := h(.,t).
$$
The vector field $\{ X_h^t \}_{t \in [0,1]}$ preserves $\xi$ and integrates into a contact isotopy denoted $\{ \phi_h^t \}_{t \in [0,1]}$. This establishes a bijection, depending on the contact form $\alpha$, between smooth time-dependent functions $h : V \times [0,1] \to \mathbb{R}$ and the identity component $\text{Cont}_0(V,\xi)$ of the group of contactomorphisms.

\subsubsection{Lift to the contact sphere}{\label{sec3.2.1}}

Following \cite[Definition $1.6$]{borman2015quasimorphisms}, we say that a closed submanifold $Y \subset V$ transverse to $\xi$ is \textit{strictly coisotropic with respect to $\alpha$} if it is coisotropic, that is the subbundle $TY \cap \xi$ of the symplectic vector bundle $(\xi_{|Y},\mathrm{d} \alpha)$ is coisotropic:
$$
\lbrace X \in \xi_y | \iota_X \mathrm{d} \alpha = 0 \text{ on } T_yY \cap \xi_y \rbrace \subset T_yY \cap \xi_y, \quad \text{for all} \quad y \in Y,
$$
and additionally $R_{\alpha} \in T_yY$ for all $y \in Y$, that is the Reeb vector field is tangent to $Y$. 

Consider the setting
$$
(S_p, \alpha_p) \supset (P_{\mathbb{K}}^{-1}(p),\alpha_{p | P_{\mathbb{K}}^{-1}(p)}) \overset{\rho}{\to} (V,\alpha).
$$
Then $P_{\mathbb{K}}^{-1}(p)$ is strictly coisotropic with respect to $\alpha_p$.
Let $h : V \times [0,1] \to \mathbb{R}$ be a time-dependent contact Hamiltonian of $V$, and let $\phi_h := \phi_h^1$ denote the time-$1$ map of the contact isotopy $\lbrace \phi_h^t \rbrace_{t \in [0,1]}$ generated by $h$. We first lift $h_t := h(.,t)$ to a $\mathbb{K}_0$-invariant function 
$$
\tilde{h}_t : P_{\mathbb{K}}^{-1}(p) \to \mathbb{R}, \quad \tilde{h}_t := \rho^* h_t,
$$
and then extend $\tilde{h}$ to a $\mathbb{K}_0$-invariant contact Hamiltonian $\overline{h} : S_p \times [0,1] \to \mathbb{R}$, so that $\overline{h}_{t | P_{\mathbb{K}}^{-1}(p)} = \tilde{h}_t$. By \cite[Lemma $3.1$ and $3.2$]{borman2015quasimorphisms}, the contact isotopy $\lbrace \phi_{\overline{h}}^t \rbrace_{t \in [0,1]}$ generated by $\overline{h}$ and the Reeb flow $\lbrace \phi_{\alpha_p}^t \rbrace_{t \in \mathbb{R}}$ of $\alpha_p$ preserve $P_{\mathbb{K}}^{-1}(p)$, and project to the contact isotopy $\lbrace \phi_h^t \rbrace_{t \in [0,1]}$ and the Reeb flow $\lbrace \phi_{\alpha}^t \rbrace_{t \in \mathbb{R}}$ of $\alpha$ respectively. More precisely, the following equalities hold:
$$
\begin{array}{lcl}
\rho \circ \phi_{{\alpha_p} | P_{\mathbb{K}}^{-1}(p)}^t = \phi_{\alpha}^t \circ \rho : P_{\mathbb{K}}^{-1}(p) \to V \quad \text{for all $t \in \mathbb{R}$};\\
\rho \circ \phi_{\overline{h} | P_{\mathbb{K}}^{-1}(p)}^t = \phi_h^t \circ \rho : P_{\mathbb{K}}^{-1}(p) \to V \quad \text{for all $t \in [0,1]$}.\\
\end{array}
$$
	
Let $q \in V$ be an $\alpha$-translated point of $\phi_h$, that is
$$
\phi_h(q) = \phi_{\alpha}^s(q) \quad \text{for some $s \in \mathbb{R}$, and} \quad (\phi_h^* \alpha)_q = \alpha_q.
$$
Then $q$ is a \textbf{discriminant point} of $\phi_{\alpha}^{-s} \circ \phi_h$, that is an $\alpha$-translated point which is also a fixed point of $\phi_{\alpha}^{-s} \circ \phi_h$.
For any $z \in P_{\mathbb{K}}^{-1}(p)$ such that $\rho(z) = q$, the equations above show moreover that
$$
\rho (\phi_{\alpha_p}^{-s} \circ \phi_{\overline{h}}(z)) = \rho(z).
$$
In other words, $\phi_{\alpha_p}^{-s} \circ \phi_{\overline{h}}(z)$ and $z$ lie in the same $\mathbb{K}_0$-orbit in $P_{\mathbb{K}}^{-1}(p)$. Since the Reeb orbits of $\alpha$ generate the circle $\mathbb{K} / \mathbb{K}_0$, there exists $\lambda \in \mathfrak{k}$ such that $p(\lambda) = -s$ and $\exp(\lambda) \circ \phi_{\overline{h}}(z) = z$. Moreover,
$$
(\phi_{\overline{h}}^* \alpha_p)_z = (\phi_{\overline{h}}^* \rho^* \alpha)_z = ((\rho \circ \phi_{\overline{h}})^* \alpha)_z = ((\phi_h \circ \rho)^* \alpha)_z = (\rho^* \phi_h^* \alpha)_z = (\rho^* \alpha)_z = (\alpha_p)_z,
$$
and $\mathbb{K}$ acts by $\alpha_p$-preserving transformations (see equation (\ref{eq7})). In other words, \textit{$z$ is a discriminant point of $\exp(\lambda) \circ \phi_{\overline{h}}$}.\\

\begin{comment}
\begin{remarksubsec}{\label{remark231}}
\normalfont The Reeb flow $\lbrace \phi_{\alpha_p}^t \rbrace_{t \in \mathbb{R}}$ is given by
\begin{equation}
\phi_{\alpha_p}^t(z) = (e^{2i \pi \frac{b_1}{p(b)}t}z_1,...,e^{2i \pi \frac{b_n}{p(b)}t}z_n), \text{ for any $z \in S_p$},
\end{equation}
and we have
\begin{equation}
p(\frac{t}{p(b)}(b_1,...,b_n)) = t.
\end{equation}
Let $q \in V$ be a translated point $\phi_h$, and let $s \in \mathbb{R}$ be such that
\begin{equation}
\phi_h(q) = \phi_{\alpha_p}^s(q).
\end{equation}
Then for any $z \in P_{\mathbb{K}}^{-1}(p)$ such that $\rho(z)=q$, there exists $\lambda \in \mathfrak{k}$ such that $z$ is a discriminant point of 
\begin{equation}
\exp(\lambda) \circ \phi_{\overline{h}}. 
\end{equation}
Moreover, we have
\begin{equation}
p(\lambda) = -s.
\end{equation}
\end{remarksubsec}
\end{comment}

\subsubsection{Lift to a symplectic vector space}{\label{sec3.2.2}}

A convenient way to pass from the contact to the symplectic setting consists of associating to a contact manifold its symplectization. We briefly recall this procedure, and apply it to lift contactomorphisms of the contact sphere $(S_p, \alpha_p)$ to symplectomorphisms of $\mathbb{C}^n$.

Let $(N,\xi = \ker \alpha)$ be a contact manifold. its \textbf{symplectization} is the symplectic manifold
$$
SN := N \times \mathbb{R} \quad \text{with symplectic form} \quad \mathrm{d} (e^r \alpha),
$$
where $r$ is a coordinate on $\mathbb{R}$. Let $\phi$ be a contactomorphism of $N$. Then $\phi$ lifts up to a symplectomorphism 
$$
\begin{array}{lcl}
\Phi : SN \to SN \\
\quad (x,r) \to (\phi(x),r - g(x)),
\end{array}
$$
where $g : N \to \mathbb{R}$ is the function satisfying $\phi^* \alpha = e^g \alpha$. In particular, a discriminant point $q$ of $\phi$ corresponds to a $\mathbb{R}$-line $\lbrace (q,r) \in SN \ |\ r \in \mathbb{R} \rbrace$ of fixed points of $\Phi$. If $\phi := \phi_h$ is the time-$1$ map of a contact isotopy $\lbrace \phi_h^t \rbrace_{t \in [0,1]}$ generated by a contact Hamiltonian $h : N \times [0,1] \to \mathbb{R}$, $\Phi$ is the time-$1$ map $\Phi = \Phi_H := \Phi_H^1$ of the Hamiltonian isotopy $\lbrace \Phi_H^t \rbrace_{t \in [0,1]}$ generated by the Hamiltonian
$$
H_t(x,r) := e^r h_t(x), \quad t \in [0,1].
$$

Suppose now that $N \subset \mathbb{C}^n$ is a star-shaped hypersurface, that is the image of a map 
$$
\begin{array}{lcl}
S^{2n-1} \to \mathbb{C}^n\\
\quad z \to f(z)z := (f_1(z)z_1,...,f_n(z)z_n),
\end{array}
$$
where $f:=(f_1,...,f_n)$, and each $f_j$ is a smooth positive function on $S^{2n-1}$. One can show that the standard Liouville form $\alpha_{\text{std}}$ on $\mathbb{C}^n$ restricts to a contact form on $N$, and the symplectization $(SN, \mathrm{d}(e^r \alpha_{\text{std}|N}))$ of $(N, \alpha_{\text{std}|N})$ is symplectomorphic to $(\mathbb{C}^n \setminus \lbrace 0 \rbrace, \omega_{\text{std}|\mathbb{C}^n \setminus \lbrace 0 \rbrace})$, via the symplectomorphism
$$
\begin{array}{lcl}
\Psi : N \times \mathbb{R} \to \mathbb{C}^n \setminus \lbrace 0 \rbrace. \\
\quad \quad (x,r) \to e^{\frac{r}{2}} x
\end{array}
$$
We then have
$$
\begin{array}{lcl}
\Psi \Phi \Psi^{-1} : \mathbb{C}^n \setminus \lbrace 0 \rbrace \to \mathbb{C}^n \setminus \lbrace 0 \rbrace,\\
\quad \quad \quad \quad \quad \ \  z \to \frac{|z|}{|pr(z)|e^{ \frac{1}{2} g(pr(z))}} \phi(pr(z))
\end{array}
$$
where 
$$
\begin{array}{lcl}
pr : \mathbb{C}^n \setminus \lbrace 0 \rbrace \to N \\
\quad \quad \quad \ z \to f(\frac{z}{|z|})\frac{z}{|z|}
\end{array}
$$
is the radial projection, and $|z| := \sqrt{\underset{j=1}{\overset{n} \sum} |z_j|^2}$. For the time-$1$ map $\phi_h$ of a contact isotopy $\{ \phi_h^t \}_{t \in [0,1]}$, the Hamiltonian of $\mathbb{C}^n \setminus \lbrace 0 \rbrace$ generating the Hamiltonian isotopy $\lbrace \Psi \Phi_H^t \Psi^{-1} \rbrace_{t \in [0,1]}$ is of the form
$$
\widetilde{H}_t(z) := \frac{|z|^2}{|pr(z)|^2} h_t(pr(z)). 
$$
It is homogeneous of degree $2$ with respect to the action of $\mathbb{R}_{>0}$ on $\mathbb{C}^n \setminus \lbrace 0 \rbrace$, that is
$$
\widetilde{H}_t(r z) = r^2 \widetilde{H}_t(z), \quad \text{for any} \quad r>0.
$$
The Hamiltonian isotopy $\lbrace \Phi_{\widetilde{H}}^t \rbrace_{t \in [0,1]} = \lbrace \Psi \Phi^t_H \Psi^{-1} \rbrace_{t \in [0,1]}$ is therefore $\mathbb{R}_{>0}$-equivariant, and we can extend $\widetilde{H}_t$ and $\Phi^t_{\widetilde{H}}$  continuously to $\mathbb{C}^n$ by 
\begin{align*}
\begin{array}{lcl}
\widetilde{H}_t(0)=0 \\
\Phi_{\widetilde{H}}^t(0)=0.
\end{array}
\end{align*}

Back to our setting, the contact sphere $(S_p,\alpha_p)$ is a star-shaped hypersurface of $\mathbb{C}^n$, with
$$
f_i(z) := \sqrt{\frac{p(b)}{\pi b_i}}, \quad \text{for all $i=1,...,2n$}.
$$
The action of $\mathbb{K}$ on $S_p$ lifts to the linear action of $\mathbb{K}$ on $\mathbb{C}^n$, and $pr$ is $\mathbb{K}$-equivariant. Consider a contact Hamiltonian $h : V \times [0,1] \to \mathbb{R}$, and its $\mathbb{K}_0$-invariant lift $\overline{h} : S_p \times [0,1] \to \mathbb{R}$. The Hamiltonian $\widetilde{H}$ extending $\overline{h}$ to $\mathbb{C}^n$ is $\mathbb{K}_0$-invariant, and therefore the Hamiltonian isotopy $\lbrace \Phi_{\widetilde{H}}^t \rbrace_{t \in [0,1]}$ is $\mathbb{K}_0$-equivariant. We call $\widetilde{H}$ \textbf{a Hamiltonian lift of $h$}.

Let $q \in V$ be an $\alpha$-translated point of $\phi_h$. Any $z \in P_{\mathbb{K}}^{-1}(p)$ such that $\rho(z) = q$ is a discriminant point of $\exp(\lambda) \circ \phi_{\overline{h}}$, for some $\lambda \in \mathfrak{k}$. On $\mathbb{C}^n$, the latter becomes a ray of fixed points of $\exp(\lambda) \circ \Phi_{\widetilde{H}}$:
$$
\exp(\lambda) \circ \Phi_{\widetilde{H}}(rz)=rz, \quad \text{for any} \quad r > 0.
$$

\begin{remarksubsec}{\label{rem3.2.1}}
\normalfont Notice that $\lambda$ is not unique, since $\exp(\lambda + \lambda_0) = \exp(\lambda)$, for any $\lambda_0 \in \mathfrak{k}_{\mathbb{Z}}$.
\end{remarksubsec}

\noindent Conversely, let $z \in \mathbb{C}^n$ be a fixed point of $\exp(\lambda) \circ \Phi_{\widetilde{H}}$ such that $r z \in P^{-1}(p)$, for some $r > 0$. Then $r z \in S_p$ is a discriminant point of $\exp(\lambda) \circ \phi_{\overline{h}}$, and therefore $\rho(r z)$ is a discriminant point of 
$$
\phi_{\alpha}^{p(\lambda)} \circ \phi_h.
$$
In other words, \textit{$\rho(pr(z))$ is an $\alpha$-translated point of $\phi_h$}. 




\subsection{The generating families}{\label{sec3.3}}

In this section, we introduce the generating families from which we will derive our cohomology groups. For the sake of completeness, we first recall the general construction of a generating function from a Hamiltonian symplectomorphism of $\mathbb{C}^n$, following \cite{Giv1995}. We then apply it to the lifts of the previous section. Finally, we add the torus action to this construction and come to the notion of generating family. In constrast with the aforementioned paper, our generating functions are $\mathbb{K}_0$-invariant, and critical points are related to translated points on $V$, rather than fixed points on $M$.
\begin{comment}
Generating functions have been extensively studied over the past decades in symplectic topology, and have proved to be a highly powerful tool to study the geometry of exact Lagrangian submanifolds of cotangent bundles. Numerous authors have contributed to the developement of this approach, especially in the context where the Lagrangian submanifold is associated with a Hamiltonian symplectomorphism. In this setting, the critical points of the generating function are in one-to-one correspondence with the fixed points of the Hamiltonian symplectomorphism (see \cite{chaperon1984idee}, \cite{viterbo1992symplectic}, \cite{theret1995utilisation}, \cite{theret1998rotation}, \cite{chekanov1987legandrova} for more on this approach), and therefore investigating the topology of the generating function, for instance by means of Morse-theoretic arguments, can provide information about the dynamics of the Hamiltonian isotopy. The following procedure, introduced in \cite{Giv1990}, associates a generating function to a Hamiltonian symplectomorphism of $\mathbb{C}^n$. In practice, constructing a genuine generating function requires that the Hamiltonian isotopy be complactly supported. Our setting is quite different: our Hamiltonian symplectomorphisms are $\mathbb{R}_{>0}$-equivariant (thus non complactly supported), and therefore the classical procedure doesn't work. Yet, the one-to-one correspondence between critical points and fixed points of the Hamiltonian symplectomorphisms still holds, whence the slight language abuse.
\end{comment}

\subsubsection{General construction}{\label{sec3.3.1}}

Let $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ be a time-dependent Hamiltonian, and $\lbrace \phi_H^t \rbrace_{t \in [0,1]}$ be the Hamiltonian isotopy generated by $H$. Dividing the interval $[0,1]$ into an even number of parts, say $2N$, we decompose the time-$1$ map $\phi_H := \phi_H^1$ as follows:
$$
\phi_H = \phi_{2N} \circ ... \circ \phi_1,
$$
where $\phi_j := \phi_H^{\frac{j}{2N}} \circ (\phi_H^{\frac{j-1}{2N}})^{-1}$. If $N$ is big enough so that, for any $z$, $-1$ is not an eigen value of $\mathrm{d}_z \phi_j$, the graph 
$$
Gr_{\phi_j} := \lbrace (z,\phi_j(z)) : z \in \mathbb{C}^n \rbrace \subset \overline{\mathbb{C}^n} \times \mathbb{C}^n,
$$
projects diffeomorphically onto the diagonal
$$
\Delta := \lbrace (z,z) : z \in \mathbb{C}^n \rbrace \subset \overline{\mathbb{C}^n} \times \mathbb{C}^n,
$$
where $\overline{\mathbb{C}^n}$ denotes the symplectic vector space $(\mathbb{C}^n, - \omega_{\text{std}})$. The linear symplectomorphism
$$
\begin{array}{lcl}
\Psi : \overline{\mathbb{C}^n} \times \mathbb{C}^n \to (T^* \mathbb{C}^n, - \mathrm{d} (p\mathrm{d} q)) \\
\quad \quad (z,w) \mapsto (\frac{z+w}{2}, i(z-w))
\end{array}
$$
sends the diagonal $\Delta$ to the zero-section $0_{\mathbb{C}^n} \subset T^* \mathbb{C}^n$, and therefore $\Psi(Gr_{\phi_j})$ is the graph of a closed $1$-form. This form is exact (either because $H^1(\mathbb{C}^n,\mathbb{R}) = \lbrace 0 \rbrace$ or $\phi_j$ is Hamiltonian): 
$$
\Psi(Gr_{\phi_j}) = Gr_{\mathrm{d} \mathcal{H}_j}.
$$
The function $\mathcal{H}_i$ is a \textit{generating function} for the Lagrangian submanifold $\Psi(Gr_{\phi_j})$. In particular, the critical points of $\mathcal{H}_j$ are, through $\Psi$, in bijection with the points of the intersection $Gr_{\phi_j} \cap \Delta$, that is with the fixed points of $\phi_j$.

Consider now the Lagrangian product
$$
Gr_{\phi_1} \times ... \times Gr_{\phi_{2N}} \subset (\overline{\mathbb{C}^n} \times \mathbb{C}^n)^{2N}.
$$
Applying the above to each component and the identification $(T^* \mathbb{C}^n)^{2N} = T^* \mathbb{C}^{2nN}$, we can write
$$
\underset{j=1}{\overset{2N} \prod} \Psi(Gr_{\phi_j}) := \Psi(Gr_{\phi_1}) \times ... \times \Psi(Gr_{\phi_{2N}}) = Gr_{\mathrm{d} \mathcal{H}},
$$
were 
$$
\begin{array}{lcl}
\mathcal{H} : \mathbb{C}^{2nN} \to \mathbb{R}\\
\ \  (x_1,...,x_{2N}) \mapsto \underset{j=1}{\overset{2N} \sum} \mathcal{H}_j(x_j).
\end{array}
$$
The critical points of $\mathcal{H}$ are in bijection with the fixed points of the product
$$
\underset{j=1}{\overset{2N} \prod} \phi_j := \phi_1 \times ... \times \phi_{2N} : (\mathbb{C}^n)^{2N} \to (\mathbb{C}^n)^{2N}.
$$
Yet, they do not correspond to fixed points of $\phi_H$, which are rather in bijection with the solutions of the equation
$$
(z_2,...,z_{2N},z_1) = (\phi_1(z_1),...,\phi_N(z_{2N})).
$$
The graph $Gr_q \subset (\overline{\mathbb{C}^n})^{2N} \times (\mathbb{C}^n)^{2N} \simeq (\overline{\mathbb{C}^n} \times \mathbb{C}^n)^{2N}$ of the "twisted" cyclic shift 
$$
\begin{array}{lcl}
q : (\mathbb{C}^n)^{2N} \to (\mathbb{C}^n)^{2N} \\
(z_1,...,z_{2N}) \mapsto (z_2,...,z_{2N},-z_1)
\end{array}
$$
corresponds, through $\underset{j=1}{\overset{2N} \prod} \Psi$, to a Lagrangian subvector space of $T^* \mathbb{C}^{2nN}$, which has a generating quadratic form. Since we have decomposed $\phi_H$ into an even number of parts and have a factor $-1$ in the map $q$ above, the graph $Gr_q$ has trivial intersections with both the multi-diagonal and multi-antidiagonal
$$
(\pm \Delta)^{2N} := \lbrace (z_1,\pm z_1,...,z_{2N}, \pm z_{2N}) : z_j \in \mathbb{C}^n \rbrace.
$$
The latter are sent, through $\underset{j=1}{\overset{2N} \prod} \Psi$, to the zero-section $0_{\mathbb{C}^{2nN}}$ and the fiber $\lbrace 0 \rbrace \times \mathbb{C}^{2nN} \subset T^* \mathbb{C}^{2nN}$ respectively. Therefore, in $T^* \mathbb{C}^{2nN}$, we can write
$$
\underset{j=1}{\overset{2N} \prod} \Psi (Gr_q) = Gr_{\mathrm{d} \mathcal{Q}},
$$
where $\mathcal{Q} : \mathbb{C}^{2nN} \to \mathbb{R}$ is a \textit{non-degenerate} quadratic form. The intersection points of $Gr_q$ with $\underset{j=1}{\overset{2N} \prod} Gr_{\phi_j}$ are in bijection with the fixed points of the Hamiltonian symplectomorphism
$$
-Id_{\mathbb{C}^n} \circ \phi_H.
$$
Consider the function
$$
\mathcal{F}^{(N)} : \mathbb{C}^{2nN} \to \mathbb{R}, \quad \mathcal{F}^{(N)} := \mathcal{Q} - \mathcal{H}.
$$
The critical points of $\mathcal{F}^{(N)}$ are in bijection with the fixed points of $- Id_{\mathbb{C}^n} \circ \phi_H$. The function $\mathcal{F}^{(N)}$ is called the \textbf{generating function} associated with the decomposition $\phi_H = \phi_{2N} \circ ... \circ \phi_1$.\\

\begin{comment}
\begin{remarksubsec}
\normalfont The function $\mathcal{F}^{(N)}$ depends on the maps $\phi_j$ of the decomposition (\ref{eq4}) only, rather than on the whole isotopy $\lbrace \phi_h^t \rbrace_{t \in [0,1]}$. Moerover, we have not used the fact that each $\phi_j$ is Hamiltonian, since $H^1(\mathbb{C}^n,\mathbb{R}) = \lbrace 0 \rbrace$ (in general, a Lagrangian section in a cotangent bundle admits a generating closed $1$-form, which is not necessarily exact). In particular, any even composition of small symplectomorphisms of $\mathbb{C}^n$ admits a generating function as above.
\end{remarksubsec}
\end{comment}

\subsubsection{Generating functions for the Hamiltonian lifts}{\label{sec3.3.2}}


Let $h : V \times [0,1] \to \mathbb{R}$ be a contact Hamiltonian, and $\widetilde{H} : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ a Hamiltonian lift of $h$. We apply the construction of the previous section to the Hamiltonian symplectomorphism
$$
\exp(\lambda) \circ \Phi_{\widetilde{H}}, \quad \lambda \in \mathfrak{k}
$$
from section \ref{sec3.2.2}. Consider a decomposition
$$
\Phi_{\widetilde{H}} = \Phi_{2N_1} \circ ... \circ \Phi_1
$$
of the Hamiltonian symplectomorphism $\Phi_{\widetilde{H}}$ into $2N_1$ small parts $\Phi_j := \Phi_{\widetilde{H}}^{\frac{j}{2N_1}} \circ \Phi_{\widetilde{H}}^{\frac{j-1}{2N_1}}$, and similarly, a decomposition
$$
\exp(\lambda) = \underbrace{\exp(\frac{\lambda}{2N_2}) \circ ... \circ \exp(\frac{\lambda}{2N_2})}_{2N_2-\text{times}}
$$
of the Hamiltonian symplectomorphism $\exp(\lambda)$. We denote
$$
\mathcal{F}_{\lambda}^{(N)} := \mathcal{Q} - \mathcal{H}_{\lambda}: \mathbb{C}^{2nN} \to \mathbb{R}, \quad N = N_1+N_2, 
$$
the generating function associated with the decomposition 
$$
\exp(\lambda) \circ \Phi_{\widetilde{H}} = \underbrace{\exp(\frac{\lambda}{2N_2}) \circ ... \circ \exp(\frac{\lambda}{2N_2})}_{2N_2-\text{times}} \circ \Phi_{2N_1} \circ ... \circ \Phi_1.
$$
The function $\mathcal{H}_{\lambda}$ is of the form
$$
\mathcal{H}_{\lambda}(x_1,...,x_{2N}) := \underset{j=1}{\overset{2N_1} \sum} \mathcal{H}_j(x_j) + \underset{j=2N_1+1}{\overset{2N_2} \sum} \mathcal{T}_{\lambda}(x_j), \quad x_j \in \mathbb{C}^n,
$$
where $\mathcal{H}_j$ and $\mathcal{T}_{\lambda}$ are the generating functions of $\Phi_j$ and $\exp(\frac{\lambda}{2N_2})$ respectively. Moreover, a direct computation shows that
$$
\mathcal{T}_{\lambda}(x_j) = \underset{k=1}{\overset{n} \sum} \tan (\frac{\pi \lambda_k}{2N_2})|q_j^k|^2, \quad \text{where} \quad x_j = (q_j^1,...,q_j^n) \in \mathbb{C}^n \quad \text{and} \quad \lambda = (\lambda_1,...,\lambda_n) \in \mathfrak{k} \subset \mathbb{R}^n.
$$

Let us list several properties of the function $\mathcal{F}_{\lambda}^{(N)}$:
\begin{enumerate}
\item the Hamiltonian lift $\widetilde{H}$ is smooth on $\mathbb{C}^n \setminus \lbrace 0 \rbrace$, and is $C^2$ at $0$ only if it is quadratic. However, it is homogeneous of degree $2$, and therefore it is $C^1$ on $\mathbb{C}^n$ with Lipschitz derivative near $0$. Therefore, for any $\lambda \in \mathfrak{k}$, $\mathcal{F}_{\lambda}^{(N)}$ is $C^1$ on $\mathbb{C}^{2nN}$ with Lipschitz derivatives, and smooth on $(\mathbb{C}^n \setminus \lbrace 0 \rbrace)^{2N}$;

\item for any $\lambda \in \mathfrak{k}$, $\mathcal{F}_{\lambda}^{(N)}$ is homogeneous of degree $2$ and $\mathbb{K}_0$-invariant. In particular, the critical points of $\mathcal{F}_{\lambda}^{(N)}$ appear as \textit{rays of $\mathbb{K}_0$-orbits in $\mathbb{C}^{2nN}$}, and have \textit{critical value $0$};

\item the function $\mathcal{F}_{\lambda}^{(N)}$ is well-defined as long as $\iota(\lambda) \in (]-N_2, N_2[)^n \subset \mathbb{R}^n$;

\item{\label{p1}} the family $\lambda \mapsto \mathcal{F}_{\lambda}^{(N)}$ decreases in positive directions: for any $s \in \mathfrak{k}$ such that $\iota(\lambda + s) \in (]-N_2, N_2[)^n$ and $\iota(s) \in (\mathbb{R}_{>0})^n$, we have
$$
\mathcal{F}_{\lambda + s}^{(N)} \leq  \mathcal{F}_{\lambda}^{(N)};
$$

\item for any $\lambda \in \mathfrak{k}$, the critical points of $\mathcal{F}_{\lambda}^{(N)}$ are in bijection with the fixed points of the Hamiltonian symplectomorphism
$$
-Id_{\mathbb{C}^n} \circ \exp(\lambda) \circ \Phi_{\widetilde{H}}.
$$
\end{enumerate}
The symplectomorphism $- Id_{\mathbb{C}^n}$ is Hamiltonian and $\mathbb{K}$-equivariant. It preserves the sphere $S_p$ and the contact form $\alpha_p$, as well as the level set $P_{\mathbb{K}}^{-1}(p)$. Therefore, it projects to an $\alpha$-preserving contactomorphism $g \in \text{Cont}_0(V,\xi)$. In particular, any estimate for the number of translated points of all the compositions of $g$ with contactomorphisms in $\text{Cont}_0(V,\xi)$ will give rise to the same estimate for the contactomorphisms in $\text{Cont}_0(V,\xi)$ themselves. In other words, the twist by $-Id_{\mathbb{C}^n}$ won't affect the estimation on the number of translated points.


\subsubsection{Adding the torus action}{\label{sec3.3.3}}

We are looking for all the possible translated points of the time $1$-map $\phi_h$ on $V$. By the discussion of section \ref{sec3.2.2}, we must look for fixed points of the composition
$$
\exp(\lambda) \circ \Phi_{\widetilde{H}},
$$
for all values of $\lambda$. To that aim, we shall consider $\mathfrak{k}$ as the space of Lagrange multipliers. For any subset $\Lambda_N \subset \mathfrak{k}$ such that $\iota(\Lambda_N) \subset (]-N_2,N_2[)^n$, we can consider the following function
$$
\mathcal{F}_N : \mathbb{C}^{2nN} \times \Lambda_N \to \mathbb{R}, \quad \mathcal{F}_N(x,\lambda) := \mathcal{F}_{\lambda}^{(N)}(x).
$$
Let $S_N$ denote the unit sphere $S^{4nN-1} \subset \mathbb{C}^{2nN}$, and $F_N$ be the restriction of $\mathcal{F}_N$ to $S_N \times \Lambda_N$. We call $F_N$ (resp. $\mathcal{F}_N$) the \textbf{generating family} (resp. \textbf{homogeneous generating family}) associated with the decomposition $\Phi_{\widetilde{H}} = \Phi_{2N_1} \circ ... \circ \Phi_1$. In the sequel, we fix this decomposition, and for any $N > N_1$, we take $\Lambda_N$ to be a cube in $\mathfrak{k}$ with boundary $\partial \Lambda_N$, centered at the origin, of fixed size growing linearly with $N$, and such that $\underset{N}{\cup} \Lambda_N = \mathfrak{k}$.
\begin{comment}
of the form
\begin{equation}
D_N^{\pm}(b) := p^{-1}(\pm s_N p(b)) \cap D_N,
\end{equation}
where $D_N$ denotes a disk centered at $0$ in $p^{-1}(0) \simeq p^{-1}(\pm s_N p(b))$, and $s_N > 0$, and is small enough so that $\iota(D_N^{\pm} (b)) \subset C_N$. We assume that $s_N$ and the diameter of $D_N$ grow linearly with $N$. This way, we obtain the following properties:
\begin{enumerate}
\item the volume of $\Lambda_N$ grows linearly with $N$, and $\underset{N}{\cup} \Lambda_N = \mathfrak{k}$;

\item for any $N$, the boundary $\partial \Lambda_N$ of $\Lambda_N$ is parallel to $b \in \mathfrak{k}_{\mathbb{Z}}$ from section \ref{seq312}.
\end{enumerate}
For convenience, we define $\Lambda_N$ as the full 
\end{comment}

\begin{remarksubsec} 
\normalfont In the sequel we will study the equivariant homotopy type of sublevel sets of the generating family relatively to the boundary $\partial \Lambda_N$. When studying the regularity of $\mathcal{F}_N$ and the critical point sets of the functions $\mathcal{F}_{\lambda}^{(N)}$, we will keep in mind that they are actually defined for any $\lambda \in \mathfrak{k}$ such that $\iota(\lambda) \in (]-N_2,N_2[)^n$.
\end{remarksubsec}

We are looking for critical points of the functions $\mathcal{F}_{\lambda}^{(N)}$, which lie in rays that, through $\Psi$, intersect $P_{\mathbb{K}}^{-1}(p)$. By homogeneity, any critical point of $\mathcal{F}_{\lambda}^{(N)}$ has critical value $0$, and moreover the zero-set $F_N^{-1}(0) \subset S_N \times \Lambda_N$ is $\mathbb{K}_0$-invariant. Consider the function
$$
\begin{array}{lcl}
\hat{p}_N : F_N^{-1}(0) \to \mathbb{R}.\\
\quad \quad \ (x,\lambda) \mapsto p(\lambda)
\end{array}
$$
It is $\mathbb{K}_0$-invariant, and we have the following contact analogue of \cite[Proposition $4.3$]{Giv1995}.
\begin{propsubsec}{\label{prop3.3.1}}
$0$ is a regular value of $F_N$, and critical $\mathbb{K}_0$-orbits of $\hat{p}_N$ correspond to translated points of $\phi_h$.
\begin{proof}
Notice first that by homogeneity of $\mathcal{F}_N$, we have $\mathrm{d}_x \mathcal{F}_{\lambda}^{(N)} = \mathrm{d}_x F_{\lambda}^{(N)}$ for any $x \in (F_{\lambda}^{(N)})^{-1}(0)$. Let $(x,\lambda) \in F_N^{-1}(0)$ be a critical point of $\mathcal{F}_N$. Then $x \in S_N$ is a critical point of $\mathcal{F}_{\lambda}^{(N)}$. On $\mathbb{C}^n$, it corresponds to a fixed point $z$ of the decomposition
$$
- Id_{\mathbb{C}^n} \circ \underbrace{\exp(\frac{\lambda}{2N_2}) \circ ... \circ \exp(\frac{\lambda}{2N_2})}_{2N_2 \text{ times}} \circ \Phi_{2N_1} \circ ... \circ \Phi_1.
$$
Let us denote $(z_1,...,z_{2N} = -z_1)$ the corresponding discrete trajectory in $\mathbb{C}^{2nN}$, that is $z_1=z$, and $z_j$ is obtained by applying the $(j-1)$-th symplectomorphism of the above decomposition to $z_{j-1}$. Choose coordinates on $\mathfrak{k} \simeq \mathbb{R}^k$, and write $\lambda = (\lambda_1,...,\lambda_k)$. By the Hamilton-Jacobi equation, the derivative of $\mathcal{F}_N$ in $\lambda$ is given by minus the Hamiltonian associated with the infinitesimal action of $\exp(\lambda)$ on the $N$ last coordinates of $x$. Through $\Psi$, this means that
$$
\frac{\partial \mathcal{F}_N}{\partial \lambda_j} (x,\lambda) = - P_{\mathbb{K}}^j(z_{N+1}) - ... - P_{\mathbb{K}}^j(z_{2N}),
$$ 
where $P_{\mathbb{K}} = (P_{\mathbb{K}}^1,...,P_{\mathbb{K}}^k) : \mathbb{C}^n \to \mathbb{R}^{k*}$. On the other hand, we have $z_{2N_1+i} = \exp(\frac{\lambda}{2N_2})z_{2N_1+i-1}$, for any $i = 1,...,2N_2$. Since $P_{\mathbb{K}}^j$ is $\mathbb{K}$-invariant, we obtain
$$
\frac{\partial \mathcal{F}_N}{\partial \lambda_j} (x,\lambda) = - N P_{\mathbb{K}}^j(z_{2N_1+1}).
$$
If $(x,\lambda)$ is a critical point of $\mathcal{F}_N$, then $P_{\mathbb{K}}^j(z_{2N_1+1}) = 0$, for any $j=1,...,k$. In particular, this means that $P(z_{2N_1+1}) \in \ker \iota^*$, and by compactness of the toric manifol $M$, this is possible only if $z_{2N_1+1} = 0$. By homogeneity of the symplectomorphisms in the decomposition above, this implies that $z = 0$, and thus $x=0$, which is impossible on $S_N$. Thus $0$ is a regular value of $F_N$.

By the method of Lagrange multipliers, the critical points of $\hat{p}_N$ are the points $(x,\lambda) \in F_N^{-1}(0)$ such that $x$ is a critical point of $\mathcal{F}_{\lambda}^{(N)}$, and $\frac{\partial \mathcal{F}_N}{\partial \lambda_j} (x,\lambda)$ is proportional to $p_j$, where $p = (p_1,...,p_k) \in \mathbb{R}^{k*}$. By the discussion above, this means that $x$ corresponds to a discrete trajectory $(z_1,...,z_{2N} = -z_1)$ satisfying 
$$
P_{\mathbb{K}}^j(z_{2N_1+1}) \sim p_j, \text{ for any $j=1,...,k$}.
$$
But then, $P_{\mathbb{K}}^j(z_{2N}) = P_{\mathbb{K}}^j(z) \sim p_j$. In other words, $z$ lies in a ray which intersects $P_{\mathbb{K}}^{-1}(p)$.
\begin{comment}
Then a direct calculation shows that 
\begin{equation}
\frac{\partial \mathcal{F}_N}{\partial \lambda_j} (x,\lambda) = \frac{\pi}{N} \frac{1}{\cos^2(\frac{\pi \lambda_j}{N})} \underset{i=N+1}{\overset{2N}\sum}|q_i^j|^2, \quad \lambda = (\lambda_1,...,\lambda_n),
\end{equation}
where $x = (q_1^1,..., q_1^n, ..., q_{2N}^1,..., q_{2N}^n) \in \mathbb{C}^{2nN}$. If $\mathfrak{k} \subset \mathbb{R}^n$, we have seen that there always exists a non-trivial vector $X \in T_{\lambda} \mathfrak{k} = \mathfrak{k}$  such that $\iota(X) \in \mathbb{R}^n_{> 0}$. In particular, $\mathrm{d}_{(x,\lambda)}\mathcal{F}_N(0,X)$ is a non-negative linear combination of the expressions from the above equation. This implies that the last $nN$ coordinates of $x$ are $0$. We will see in the next section that the only critical point of $\mathcal{F}_{\lambda}^{(N)}$ lying on the coordinate cross is $0$. As a consequence, $x$ must be $0$, which is impossible on $S_N$. Therefore, $0$ is a regular value of $F_N$. 

\item Let $x = (x_1,...,x_{2N}) \in \mathbb{C}^{2nN}$ be a critical point of $F_{\lambda}^{(N)}$. It corresponds to a closed trajectory in  last $N$ components are given by the successive actions of $\exp(\frac{\lambda}{N})$ on  is a discrete trajectory whose $N$ last components are
\begin{equation}
x_j = \frac{z_j(e^{\frac{2i \pi \lambda}{N}} + 1)}{2}.
\end{equation}
Therefore, we have $\frac{\partial \mathcal{F}_N}{\partial \lambda_j}(x,\lambda) = \frac{\pi}{N} \underset{i=N+1}{\overset{2N} \sum} |z_i^j|^2$, where
Via the method of Lagrange multipliers, we see that a critical point $(x,\lambda)$ of $\hat{p}_N$ is a critical point $x$ of $F_{\lambda}^{(N)}$ such that
\begin{equation}
\frac{\partial F_N}{\partial \lambda}(x,\lambda) = \frac{1}{N} P_{\mathbb{K}}(z) \sim p,
\end{equation}
that is, $z$ lies in a ray intersecting $P_{\mathbb{K}}^{-1}(p)$. 
\end{enumerate}
\end{comment}
\end{proof}
\end{propsubsec}

\begin{comment}
Let $L_N$ denote the front of $F_N$. Recall that the homogeneous generating function $\mathcal{F}_{\lambda}^{(N)}$ is smooth on $(\mathbb{C}^n \setminus \lbrace 0 \rbrace)^{4N}$. If $x$ be a critical point of $\mathcal{F}_{\lambda}^{(N)}$, it corresponds, through $\Psi$, to a solution of the equation
\begin{equation}
(z_2,...,z_{4N}, -z_1) = (\gamma_1(z_1), ..., z_{4N}, \gamma_{4N}(z_{4N}))
\end{equation}
where $\gamma_j = \Phi_j$ if $j \leq 2N$, and $\gamma_j = \exp(\frac{\lambda}{2N})$ otherwise. If $x$ lies on the coordinate cross, there exists $j$ such that $\gamma_j(z_j) = -z_j$. Since $\gamma_j$ is close to $Id_{\mathbb{C}^n}$, this can happen only if $z_j = 0$. Since $\gamma_j(0) = 0$ for all $j$, this implies that $x = 0$. Therefore, $\mathcal{F}_{\lambda}^{(N)}$ is smooth at any non-zero critical points. In particular, $L_N$
 is of zero-measure. 
\end{comment}


\subsection{From a decomposition to another}{\label{sec3.4}}

Our cohomology group is defined as a limit in $N \to \infty$ of equivariant cohomology groups associated with the generating families $F_N$. Therefore we must describe how $F_N$ changes when $N$ grows. The key ingredient is that we can deform the generating functions in a controllable way, as long as the front of the associated generating family remains unchanged during the deformation. Recall that the front of $F_N$ is given by: 
$$
L_N := \lbrace \lambda \in \Lambda_N : 0 \text{ is a singular value of } F_{\lambda}^{(N)} \rbrace.
$$
By construction of $F_N$, it will remain unchanged as long as the time $1$ map $\Phi_{\widetilde{H}}$ remains unchanged as well. We closely follow \cite{Giv1995}, and begin with an observation. 
\begin{comment} 
In this section, we investigate the change in the homotopy type of the pair $(F_N^-(\nu), \partial F_N^-(\nu))$. More precisely, we study how these sets are related for different decompositions of the associated Hamiltonian isotopy. 

Proposition \ref{prop7} shows in particular that the equivariant homotopy type of the pair $(F_N^-(\nu), \partial F_N^-(\nu))$ remains unchanged under reparametrization of the lifting isotopy $\lbrace \widetilde{\Phi}_{\overline{h}}^t \rbrace_{t \in [0,1]}$. This fact, along with the following move from \cite{Giv1995} section $4$, will reflect how this pair changes when $N$ grows. 
\end{comment}
Let $\Phi_H = \Phi_{2N'} \circ ... \circ \Phi_1$ be the decomposition of a Hamiltonian isotopy $\lbrace \Phi_H^t \rbrace_{t \in [0,1]}$, such that the first $2(N'-N)$ parts consist of a loop
$$
Id_{\mathbb{C}^n} = \Phi_{2(N'-N)} \circ ... \circ \Phi_1.
$$
We relate the generating function $\mathcal{F}^{(N')}$ associated with the whole decomposition
$$
\Phi_H = \Phi_{2N'} \circ ... \circ \Phi_1,
$$
to the generating functions $\mathcal{F}^{(N)}$ and $\mathcal{G}^{(N'-N)}$ associated with the parts
$$
\Phi_{2N'} \circ ... \circ \Phi_{2(N'-N)+1} \quad \text{and} \quad \Phi_{2(N'-N)} \circ ... \circ \Phi_1
$$
respectively. Consider the following deformation of the graph $Gr_q$ from section \ref{sec3.3.1}:
$$
\begin{array}{lcl}
Q_{\epsilon} := \lbrace (z_1,w_1,...,z_{2N'},w_{2N'}) \ |\  (z_j,w_j) \in \overline{\mathbb{C}^n} \times \mathbb{C}^n \rbrace, & \quad w_j = z_{j+1} \text{ for } j \notin \{ 2(N'-N), 2N' \} \\ &  w_{2(N'-N)} = \epsilon(z_{2(N'-N)+1} - w_{2N'}) - z_1 \\ & w_{2N'} = \epsilon (w_{2(N'-N)} - z_1) - z_{2(N'-N)+1},
\end{array}
$$
where $\epsilon \in [0,1]$. Then $Q_{\epsilon}$ is a Lagrangian subspace of $(\overline{\mathbb{C}^n} \times \mathbb{C}^n)^{2N'}$, which is transversal both to the multi-diagonal $\Delta^{2N'}$ and the multi-antidiagonal $(-\Delta)^{2N'}$. Thus, it corresponds in $T^* \mathbb{C}^{2N'}$ to the graph of a non-degenerate quadratic form $\mathcal{Q}_{\epsilon}$. Consider the generating function
$$
\mathcal{F}^{(N')}_{\epsilon} : \mathcal{Q}_{\epsilon} - \mathcal{H}.
$$
Its critical points are in bijection with the points of the intersection
$$
\underset{j=1}{\overset{2N'} \prod} \Psi(Gr_{\phi_j}) \cap Q_{\epsilon}.
$$
For $\epsilon = 0$, $Q_{\epsilon}$ is the product $Q_1 \times Q_2$, where $Q_1$ and $Q_2$ correspond to the twisted cyclic shifts in  $(\mathbb{C}^n)^{2N}$ and $(\mathbb{C}^n)^{2(N'-N)}$ respectively. Since $\mathcal{H}$ is the direct sum of the generating functions associated with the small Hamiltonian symplectomorphisms $\Phi_j$ for $j = 1,...,2N'$, we get
$$
\mathcal{F}^{(N')}_0 = \mathcal{F}^{(N)} \oplus \mathcal{G}^{(N'-N)} : \mathbb{C}^{2nN} \times \mathbb{C}^{2n(N'-N)} \to \mathbb{R}.
$$
Moreover, $\mathcal{F}^{(N')}_0$ admits a critical point if and only if $\mathcal{F}^{(N)}$ does, if and only if $\mathcal{F}^{(N')}$ does. For $\epsilon = 1$, we have $Q_{\epsilon} = Gr_q$, so that
$$
\mathcal{F}^{(N')}_1 = \mathcal{F}^{(N')}.
$$
Finally, for $\epsilon \in ]0,1[$, notice that
$$
\begin{array}{lcl}
(z_1,w_1,...,z_{2N'},w_{2N'}) \in \underset{j=1}{\overset{2N'} \prod} \Psi(Gr_{\phi_j}) \cap Q_{\epsilon}  \iff \\  (\frac{1}{\epsilon}z_1,\frac{1}{\epsilon}w_1,...,\frac{1}{\epsilon}z_{2(N'-N)},\frac{1}{\epsilon}w_{2(N'-N)},z_{2(N'-N)+1},w_{2(N'-N)+1},...,z_{2N'},w_{2N'}) \in \underset{j=1}{\overset{2N'} \prod} \Psi(Gr_{\phi_j}) \cap Q,
\end{array}
$$
that is the critical points of $\mathcal{F}^{N'}_{\epsilon}$ are in bijection with the critical points of $\mathcal{F}^{(N')}$, for any $\epsilon \in [0,1]$.


Observe now that given two decompositions 
$$
\begin{array}{lcl}
\Phi_H^1 = \Phi_{2N} \circ ... \circ \Phi_1, \quad \quad \Phi_H^1 = \Phi_{H'}^1\\ \\
\Phi_{H'}^1  = \Phi'_{2N'} \circ ... \circ \Phi'_1
\end{array}
$$
of the same Hamiltonian isotopy $\lbrace \Phi_H^t \rbrace_{t \in [0,1]} = \lbrace \Phi_{H'}^t \rbrace_{t \in [0,1]}$ of $\mathbb{C}^n$ with, say, $N' > N$, one can always use a reparametrization $H_t^s$ of $H'_t$, so that $H_t^0 = H'_t$, and $H_t^1$ generates the Hamiltonian symplectomorphism
$$
\Phi_{H^s}^1 = \Phi_{2N} \circ ... \circ \Phi_1 \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{2(N'-N)-\text{times}}.
$$
In particular, consider 
\begin{comment}
Let us deform the bigger decomposition keeping the time-$1$ map unchanged. We use a reparametrization
\begin{equation}
\begin{array}{lcl}
\widetilde{H}^s_t : \mathbb{C}^n \to \mathbb{R} \\
\quad \quad \  z \mapsto \frac{\mathrm{d}}{\mathrm{d} t}(a^{N,N'}_s(t))H'_{a^{N,N'}_s(t)}(z),
\end{array}
\end{equation}
where $a_{N,N'}$ is a smooth function satisfying
\begin{equation}
a_{N,N'}(t) = \begin{cases}
               0 \text{ if } t \in [0, \frac{N'-N}{N'}]\\
               \frac{N'}{N}t - \frac{N'-N}{N} \text{ if } t \in [\frac{N'-N+1}{N'},\frac{N'-1}{N'}]\\
               1 \text{ if } t=1               
       \end{cases},
\end{equation}
and $a^{N,N'}_s(t) := sa_{N,N'}(t)+(1-s)t$. Then, for any $s \in [0,1]$, we have $\Phi^1_{\widetilde{H}^s} = \Phi_H^1 = \Phi_{H'}^1$, and moreover $\widetilde{H}_t^0 = H'_t$. The Hamiltonian $\widetilde{H}_t^1$ yields the decomposition
\begin{equation}
\Phi_{\widetilde{H}^s}^1 = \Phi_N \circ ... \circ \Phi_1 \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{(N'-N)-\text{times}}.
\end{equation}
\end{comment}
the homogeneous generating family $\mathcal{F}_N$ associated with the decomposition 
$$
\Phi_{\widetilde{H}} = \Phi_{N_1} \circ ... \circ \Phi_1,
$$
where $\widetilde{H}$ is a Hamiltonian lift of $h$. We will denote 
$$\mathcal{G}_{\lambda}^{(K)} : \mathbb{C}^{2nK} \to \mathbb{R}$$ 
the generating function associated with the decomposition
$$
\underbrace{\exp(\frac{\lambda}{2K}) \circ ... \circ \exp(\frac{\lambda}{2K})}_{2K-\text{times}},
$$
as defined in section \ref{sec3.3.1}. Note that the front of the restriction $F_{N+K|\Lambda_N}$ is equal to the front of $F_N$, that is we have $L_{N+K} \cap \Lambda_N = L_N$.
\begin{propsubsec}{\label{prop3.4.1}}
There exists a fiberwise $\mathbb{K}_0$-invariant smooth homotopy between the restricted homogeneous family
$$
\mathcal{F}_{N+K |\Lambda_N}
$$
and the fiberwise direct sum
$$
\mathcal{F}_N \oplus_{\Lambda_N} \mathcal{G}_0^{(K)} : \mathbb{C}^{2n(N+K)} \times \Lambda_N \to \mathbb{R},
$$
in a way that the front of the corresponding generating families on $S_{N+K} \times \Lambda_N$ remains unchanged during the deformation.
\begin{proof}
For any $\lambda \in \Lambda_N$, the generating functions $\mathcal{F}_{\lambda}^{(N+K)}$ and $\mathcal{F}_{\lambda}^{(N)}$ are associated respectively with the decompositions
$$
\begin{array}{lcl}
\exp(\lambda) \circ \Phi_{\widetilde{H}} = \underbrace{\exp(\frac{\lambda}{2(N_2+K)}) \circ ... \circ \exp(\frac{\lambda}{2(N_2+K)})}_{2(N_2+K)-\text{times}} \circ \Phi_{2N_1} \circ ... \circ \Phi_1\\ \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{ and} \\ \\
\exp(\lambda) \circ \Phi_{\widetilde{H}} = \underbrace{\exp(\frac{\lambda}{2N_2}) \circ ... \circ \exp(\frac{\lambda}{2N_2})}_{2N_2-\text{times}} \circ \Phi_{2N_1} \circ ... \circ \Phi_1.
\end{array}
$$
Up to reparametrization of the $2(N_2+K)$ last factors, the first decomposition becomes
\begin{align*}
\begin{array}{lcl}
\exp(\lambda) \circ \Phi_{\widetilde{H}} & = 
\underbrace{\exp(\frac{\lambda}{2N_2}) \circ ... \circ \exp(\frac{\lambda}{2N_2})}_{2N_2-\text{times}} \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{2K-times} \circ \Phi_{2N_1} \circ ... \circ \Phi_1\\ 
& = \underbrace{\exp(\frac{\lambda}{2N_2}) \circ ... \circ \exp(\frac{\lambda}{2N_2})}_{2N_2-\text{times}} \circ \Phi_{2N_1} \circ ... \circ \Phi_1 \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{2K-times}.
\end{array}
\end{align*}
By the discussion above, the generating function associated with this last decomposition is $\mathbb{K}_0$-invariantly homotopic to $\mathcal{F}_{\lambda}^{(N)} \oplus \mathcal{G}_0^{(K)}$, and this homotopy does not change the front of the family. Thus, we obtain a smooth $\mathbb{K}_0$-invariant homotopy between $\mathcal{F}_{\lambda}^{(N+K)}$ and $\mathcal{F}_{\lambda}^{(N)} \oplus \mathcal{G}_0^{(K)}$. Since the time-$1$ map $\exp(\lambda) \circ \Phi_{\widetilde{H}}$ remains unchanged during the reparametrization, the front remains unchanged during the whole process. 
\end{proof}
\end{propsubsec}

\remarksubsec \normalfont Notice that the result above is independent of the reparametrization, since any two such reparametrizations are always homotopic. Moreover, one can show by a similar argument that the front of the generating family $F_N$ remains unchanged if one modifies the decomposition of $\Phi_{\widetilde{H}}$ \textit{into $2N_2$ parts}.

\begin{comment}
\begin{remarksubsec} 
\normalfont The correspondence is not one-to-one, but rather the following. Let $q \in V$ be a translated point of $\phi_h$ with
\begin{equation}
\phi_h(q) = \phi_{\alpha}^s(q).
\end{equation}
If $N$ is big enough so that the intersection
\begin{equation}
\Lambda_N \cap p^{-1}(-s)
\end{equation}
is not empty, there exists $(x,\lambda) \in S_N \times \Gamma_N(-s)$ such that the orbit $\mathbb{K}_0(x,\lambda)$ is made of critical points of $\hat{p}_N$. Moreover, for any $\lambda_0 \in \mathfrak{k}_{\mathbb{Z}}$, the orbit $\mathbb{K}_0(x,\lambda + \lambda_0)$ is made of critical points of $\hat{p}_{N'}$, for some $N'$ big enough so that $\Lambda_{N'} \cap p^{-1}(-s + p(\lambda_0)) \neq \emptyset$. In other words, the translated points of $\phi_h$ are in one-to-one correspondence with the $\mathfrak{k}_{\mathbb{Z}}$-lattices of $\mathbb{K}_0$-orbits of critical points of the functions \hat{p}_N$, for .
\end{remarksubsec}
\end{comment}

\subsection{Sublevel sets and transversality}{\label{sec3.5}}

In this section, we introduce the sublevel sets which will be used to define the cohomology groups. Similarly as with the generating families, we describe how they behave when $N$ grows, using the results from section \ref{sec2.1}. In particular, we define a basis of $\mathbb{C}^{2nN}$ in which the quadratic generating functions associated with the torus action are diagonal, so that we have a canonical (independent of the given element of the torus) identification between their non-positive sublevel sets and their non-positive eigenspaces. Consider the following sublevel sets
$$
\mathcal{F}_N^{\pm} := \lbrace \mathcal{F}_N \geq 0 \text{ (resp. $\leq 0$)} \rbrace, \quad F_N^{\pm} := \lbrace F_N \geq 0 \text{ (resp. $\leq 0$)} \rbrace.
$$
For any $\nu \in \mathbb{R}$, we denote $\Gamma_N(\nu)$ the intersection $\Lambda_N \cap p^{-1}(\nu)$, and define the sets
$$
\begin{array}{lcl}
\mathcal{F}_N^{\pm}(\nu) := \mathcal{F}_N^{\pm} \cap (\mathbb{C}^{2nN} \times \Gamma_N(\nu)), &
\partial \mathcal{F}_N^{\pm}(\nu) := \mathcal{F}_N^{\pm}(\nu) \cap (\mathbb{C}^{2nN} \times \partial \Gamma_N(\nu)), \\ \\
F_N^{\pm}(\nu) := F_N^{\pm} \cap (S_N \times \Gamma_N(\nu)), & 
\partial F_N^{\pm}(\nu) := F_N^{\pm}(\nu) \cap (S_N \times \partial \Gamma_N(\nu)),
\end{array}
$$
where
$$
\partial \Gamma_N(\nu) := \Gamma_N(\nu) \cap \partial \Lambda_N.
$$
Recall that the generating function $\mathcal{F}_{\lambda}^{(N)}$ is smooth on $(\mathbb{C}^n \setminus \lbrace 0 \rbrace)^{2N}$. If $x$ is a critical point of $\mathcal{F}_{\lambda}^{(N)}$, it corresponds, through the linear symplectomorphism $\Psi$ of section \ref{sec3.3.1}, to a solution of the equation
$$
(z_2,...,z_{2N}, -z_1) = (\gamma_1(z_1), ..., z_{2N}, \gamma_{2N}(z_{2N})),
$$
where $\gamma_j = \Phi_j$ if $j \leq 2N_1$, and $\gamma_j = \exp(\frac{\lambda}{2N_2})$ otherwise. If $x$ lies on the coordinate cross, there exists $j$ such that $\gamma_j(z_j) = -z_j$. Since $\gamma_j$ is close to $Id_{\mathbb{C}^n}$, this can happen only if $z_j = 0$. Moreover, $\gamma_j(0) = 0$ for all $j$, and therefore $x = 0$. Thus $\mathcal{F}_{\lambda}^{(N)}$ is smooth at any non-zero critical points. In particular, \textit{the front $L_N$ is of zero-measure}.

The relation between the front $L_N$ and the spectrum $\Spec(\phi_h)$ (equation $(\ref{eq4})$) shall be understood as follows: the front is made of all the elements $\lambda \in \Lambda_N$ such that the Hamiltonian diffeomorphism
$$
-Id_{\mathbb{C}^n} \circ \exp(\lambda) \circ \Phi_{\widetilde{H}} : \mathbb{C}^n \to \mathbb{C}^n 
$$
admits a fixed point. However, this fixed point might not correspond to a point on $V$ (if it does not lie in a ray that intersects $P_{\mathbb{K}}^{-1}(p)$). In contrast, the spectrum is made of real numbers $\nu \in \mathbb{R}$ for which an $\alpha$-translated point of $\phi_h$ appears on $V$. In other words, meeting the front corresponds to the notion of fixed points on $\mathbb{C}^n$, whereas by Propositions \ref{prop2.1.1} and \ref{prop3.3.1}, being tangent to the front corresponds to the notion of fixed points on $P_{\mathbb{K}}^{-1}(p)$, that is to that of translated points on $V$. 

Note however that $\nu \notin \Spec(\phi_h)$ does not mean that the boundary $\partial \Gamma_N(\nu)$ is transversal to $L_N$. By Corollary \ref{cor2.1.1}, the set of $\nu$ such that $\Gamma_N(\nu)$ and $\partial \Gamma_N(\nu)$ are transversal to the front $L_N$ is of full measure. We will say that $\nu \in \mathbb{R}$ \textbf{is generic} if $\Gamma_N(\nu)$ \textit{and} $\partial \Gamma_N(\nu)$ are transversal to $L_N$ for all $N$ (we use here that a countable intersection of sets of full measure is of full measure). Note that this notion depends on $\Phi_{\widetilde{H}}$ and the family of cubes $\{ \Lambda_N \}_N$.

\begin{comment}
In this section, we introduce the sublevel sets of the generating families which will be used to define the cohomology groups. We consider the following sets
\begin{equation}
\mathcal{F}_N^{\pm} := \lbrace \mathcal{F}_N \geq 0 \text{ (resp. $\leq 0$)} \rbrace, \quad F_N^{\pm} := \lbrace F_N \geq 0 \text{ (resp. $\leq 0$)} \rbrace,
\end{equation}
For any $\nu \in \mathbb{R}$, we denote by $\Gamma_N(\nu)$ the intersection $\Lambda_N \cap p^{-1}(\nu)$, and define
\begin{equation}
\begin{array}{lcl}
\mathcal{F}_N^{\pm}(\nu) := \mathcal{F}_N^{\pm} \cap (\mathbb{C}^{2nN} \times \Gamma_N(\nu)), &
\partial \mathcal{F}_N^{\pm}(\nu) := \mathcal{F}_N^{\pm}(\nu) \cap (\mathbb{C}^{2nN} \times \partial \Gamma_N(\nu)), \\ \\
F_N^{\pm}(\nu) := F_N^{\pm} \cap (S_N \times \Gamma_N(\nu)), & 
\partial F_N^{\pm}(\nu) := F_N^{\pm}(\nu) \cap (S_N \times \partial \Gamma_N(\nu)),
\end{array}
\end{equation}
where
\begin{equation}
\partial \Gamma_N(\nu) := \Gamma_N(\nu) \cap \partial \Lambda_N.
\end{equation}
Let $L_N$ denote the front of $F_N$: 
\begin{equation}
L_N := \lbrace \lambda \in \Lambda_N : 0 \text{ is a singular value of } F_{\lambda}^{(N)} \rbrace.
\end{equation}
We claim that it is of zero-measure. Indeed, recall that the homogeneous generating function $\mathcal{F}_{\lambda}^{(N)}$ is smooth on $(\mathbb{C}^n \setminus \lbrace 0 \rbrace)^{2N}$. If $x$ be a critical point of $\mathcal{F}_{\lambda}^{(N)}$, it corresponds, through $\Psi$, to a solution of the equation
\begin{equation}
(z_2,...,z_{2N}, -z_1) = (\gamma_1(z_1), ..., z_{2N}, \gamma_{2N}(z_{2N}))
\end{equation}
where $\gamma_j = \Phi_j$ if $j \leq N$, and $\gamma_j = \exp(\frac{\lambda}{N})$ otherwise. If $x$ lies on the coordinate cross, there exists $j$ such that $\gamma_j(z_j) = -z_j$. Since $\gamma_j$ is close to $Id_{\mathbb{C}^n}$, this can happen only if $z_j = 0$. Since $\gamma_j(0) = 0$ for all $j$, this implies that $x = 0$. Therefore, $\mathcal{F}_{\lambda}^{(N)}$ is smooth at any non-zero critical points.

One can make a similar definition for any Hamiltonian $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$: the \textbf{front of $H$} is defined as the set 
\begin{equation}
L_H := \lbrace \lambda \in \mathfrak{k} : - Id_{\mathbb{C}^n} \circ \exp(\lambda) \circ \Phi_{\widetilde{H}} \text{ admits a non-trivial fixed point} \rbrace.
\end{equation}
Note that for the Hamiltonian lift $\widetilde{H}$ if a contact Hamiltonian $h$, we have $L_N = L_{\widetilde{H}} \cap \Lambda_N$. 

From now on, we assume that for any $N$, $\Lambda_N$ is chosen so that $\partial \Lambda_N$ is transversal to $L_N$. ........
\end{comment}

\begin{comment}
Recall that the \textbf{spectrum} of a contactomorphism $\phi$ on a contact manifold $(V, \xi = \ker \alpha)$ is the set
\begin{equation}
\Spec(\phi) := \lbrace s \in \mathbb{R} : \text{ there exists a translated point $x \in V$ such that } \phi(x) = \phi_{R_{\alpha}}^s(x) \rbrace.
\end{equation}
\begin{defsubsec}
\normalfont We define the \textbf{spectra} of $L_H$ and $L_N$ to be respectively the sets
\begin{equation}
\begin{array}{lcl}
\Spec (L_H) := \lbrace \nu \in \mathbb{R} : p^{-1}(\nu) \text{ is tangent to } L_H \rbrace\\ \\
\Spec (L_N) := \lbrace \nu \in \mathbb{R} : \Gamma_N(\nu) \text{ is tangent to } L_N \rbrace.
\end{array}
\end{equation}
\end{defsubsec}
\noindent By the very definition of $\Gamma_N(\nu)$ and Propositions \ref{prop2} and \ref{prop3}, we have
\begin{equation}{\label{eq12}}
\Spec(L_{\widetilde{H}}) = \underset{N}{\cup} \Spec(L_N) = \Spec(\phi_h).
\end{equation}
In particular, $\Spec(L_{\widetilde{H}})$ and $\Spec(L_N)$ are nowhere dense subsets of $\mathbb{R}$. The relation between the front and its spectrum shall be understood as follows: the front is made of all the elements $\lambda$ such that the Hamiltonian diffeomorphism
\begin{equation}
-Id_{\mathbb{C}^n} \circ \exp(\lambda) \circ \Phi_{\widetilde{H}} : \mathbb{C}^n \to \mathbb{C}^n 
\end{equation}
admits a fixed point, whereas the spectrum is made of real numbers for which a translated point appears on $V$. In other words, meeting the front corresponds to the notion of fixed points on $\mathbb{C}^n$, whereas being tangent to the front corresponds to the notion of fixed point on $P_{\mathbb{K}}^{-1}(p)$, that is, to that of translated points on $V$. 
\end{comment}





\begin{comment}
\begin{propsubsec}{\label{prop7}}
Let $H : \mathbb{C}^n \times [0,1]^2 \to \mathbb{R}$ be a $C^{1,1}$ family of $\mathbb{K}_0$-invariant homogeneous of degree $2$ Hamiltonians $H^s : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ such that $L_{H^s}$ is independent of $s$.  Let $\nu \in \mathbb{R}$ be generic. If $N$ is big enough, there exists a $\mathbb{K}_0$-equivariant $C^{0,1}$ isotopy $I : S_N \times \Lambda_N \times [0,1] \to S_N \times \Lambda_N$, such that
\begin{equation}
I_s(F_{N,0}^{\pm}(\nu), \partial F_{N,0}^{\pm}(\nu)) = (F_{N,s}^{\pm}(\nu), \partial F_{N,s}^{\pm}(\nu)),
\end{equation}
where $F_{N,s} : S_N \times \Lambda_N \to \mathbb{R}$ is the generating family associated with the decomposition of the Hamiltonian symplectomorphisms $\exp(\lambda) \circ \Phi_{H^s} : \mathbb{C}^n \to \mathbb{C}^n$ into $2N$ parts.
\begin{proof}
If $N$ is big enough, then for any $s \in [0,1]$, the Hamiltonian symplectomorphisms
\begin{equation}
\Phi^s_j := \phi_{H^s}^{\frac{j}{N}} \circ (\phi_{H^s}^{\frac{j-1}{N}})^{-1}, \quad j=1,...,N
\end{equation}
are all small enough, and therefore the decomposition 
\begin{equation}
\exp(\lambda) \circ \Phi_H^s = \underbrace{\exp(\frac{\lambda}{N}) \circ ... \circ \exp(\frac{\lambda}{N})}_{N-\text{times}} \circ \Phi^s_{N} \circ ... \circ \Phi^s_1
\end{equation}
yiels a  $C^{1,1}$ homogeneous generating family $\mathcal{F}_{N,s} : \mathbb{C}^{2nN} \times \Lambda_N  \to \mathbb{R}$ such that for any $s$, $\Gamma_N(\nu)$  and $\partial \Gamma_N(\nu)$ are transversal to $L_N^s := L_{H^s} \cap \Lambda_N$. It remains to apply proposition \ref{prop4}.
\end{proof}
\end{propsubsec}
\begin{propsubsec}
Let $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ be a $\mathbb{K}_0$-invariant homogeneous Hamiltonian, and let $\nu_0 < \nu_1$ be generic. Suppose that $[\nu_0,\nu_1] \cap \Spec(L_H) = \emptyset$. For $N$ big enough, there a $C^{1,1}$ $\mathbb{K}_0$-equivariant isotopy $I : S_N \times \Lambda_N \times [0,1] \to S_N \times \Lambda_N$, such that
\begin{equation}
I_1(F_N^{\pm}(\nu_0), \partial F_N^{\pm}(\nu_0)) = (F_N^{\pm}(\nu_1), \partial F_N^{\pm}(\nu_1)).
\end{equation}
\begin{proof}
We apply proposition \ref{prop5} to the family $\lbrace \Gamma_N(\nu_s) \rbrace_{s \in [0,1]}$. For any generic $\nu \in [\nu_0,\nu_1]$, the boundary $\partial \Gamma_N(\nu)$ is transversal to $L_N$. Therefore it remains transversal in an open neighborhood $[\nu-\epsilon, \nu + \epsilon]$. Choosing a finite subcovering of $[\nu_0,\nu_1]$ by such segments implies that the boundaries $\Gamma_N(\nu_s)$ are all transversal to $L_N$.
\end{proof}
\end{propsubsec}
\end{comment}
\begin{comment}
\subsection{From a decomposition to another}

In this section, we investigate the change in the homotopy type of the pair $(F_N^-(\nu), \partial F_N^-(\nu))$. More precisely, we study how these sets are related for different decompositions of the associated Hamiltonian isotopy. 

Proposition \ref{prop7} shows in particular that the equivariant homotopy type of the pair $(F_N^-(\nu), \partial F_N^-(\nu))$ remains unchanged under reparametrization of the lifting isotopy $\lbrace \widetilde{\Phi}_{\overline{h}}^t \rbrace_{t \in [0,1]}$. This fact, along with the following move from \cite{Giv1995} section $4$, will reflect how this pair changes when $N$ grows. 

Suppose that a Hamiltonian isotopy $\lbrace \Phi_H^t \rbrace_{t \in [0,1]}$ consists of two parts and that the first one is a loop (decomposed into $2(N'-N)$ parts)
\begin{equation}
Id_{\mathbb{C}^n} = \Phi_{2(N'-N)} \circ ... \circ \Phi_1.
\end{equation}
We relate the generating function $\mathcal{F}^{(N')}$ associated with the whole decomposition
\begin{equation}
-Id_{\mathbb{C}^n} \circ \Phi_H = -Id_{\mathbb{C}^n} \circ \Phi_{2N'} \circ ... \circ \Phi_1,
\end{equation}
and the generating functions $\mathcal{F}^{(N)}$ and $\mathcal{G}^{(N'-N)}$ of the parts
\begin{equation}
-Id_{\mathbb{C}^n} \circ \Phi_{2N'} \circ ... \circ \Phi_{2(N'-N)+1}, \quad -Id_{\mathbb{C}^n} \circ \Phi_{2(N'-N)} \circ ... \circ \Phi_1,
\end{equation}
in the following way. Consider the following deformation of the graph $Gr_q$ of section \ref{sec3}:
\begin{equation}
\begin{array}{lcl}
Q_{\epsilon} := \lbrace (z_1,w_1,...,z_{2N'},w_{2N'}) : (z_j,w_j) \in \overline{\mathbb{C}^n} \times \mathbb{C}^n \rbrace, & \quad w_j = z_{j+1} \text{ for } j \notin \{ 2(N'-N), 2N' \} \\ &  w_{2(N'-N)} = \epsilon(z_{2(N'-N)+1} - w_{2N'}) - z_1 \\ & w_{2N'} = \epsilon (w_{2(N'-N)} - z_1) - z_{2(N'-N)+1},
\end{array}
\end{equation}
for $\epsilon \in [0,1]$. Then $Q_{\epsilon}$ is a Lagrangian subspace of $(\overline{\mathbb{C}^n} \times \mathbb{C}^n)^{2N'}$, which is transversal both to the multi-diagonal $\Delta^{2N'}$ and the multi-antidiagonal $(-\Delta)^{2N'}$. Thus, it corresponds in $T^* \mathbb{C}^{2N'}$ to a non-degenerate quadratic form $\mathcal{Q}_{\epsilon}$, and we can define the function
\begin{equation}
\mathcal{F}^{(N')}_{\epsilon} : \mathcal{H} - \mathcal{Q}_{\epsilon},
\end{equation}
the critical points of which correspond to intersections
\begin{equation}
\underset{j=1}{\overset{2N'} \prod} \Psi(Gr_{\phi_j}) \cap Q_{\epsilon}.
\end{equation}
For $\epsilon = 0$, $Q_{\epsilon}$ is the product $Q_1 \times Q_2$, where $Q_1$ and $Q_2$ correspond to the twisted cyclic shifts in $(\mathbb{C}^n)^{2(N'-N)}$ and $(\mathbb{C}^n)^{2N}$ respectively. Since $\mathcal{H}$ is the sum of the generating functions associated with the small Hamiltonian parts $\Phi_j$, we get
\begin{align}
\mathcal{F}^{(N')}_0 = \mathcal{F}^{(N)} \oplus \mathcal{G}^{(N'-N)} : \mathbb{C}^{2nN} \times \mathbb{C}^{2n(N'-N)} \to \mathbb{R},
\end{align} 
and moreover, $\mathcal{F}^{(N')}_0$ admits a critical point if and only if $\mathcal{F}^{(N)}$ does, if and only if $\mathcal{F}^{(N')}$ does. For $\epsilon = 1$, we have $Q_{\epsilon} = Gr_q$, so that
\begin{equation}
\mathcal{F}^{(N')}_1 = \mathcal{F}^{(N')}.
\end{equation}
Finally, for $\epsilon \in ]0,1[$, notice that
\begin{equation}
\begin{array}{lcl}
(z_1,w_1,...,z_{2N'},w_{2N'}) \in \underset{j=1}{\overset{2N'} \prod} \Psi(Gr_{\phi_j}) \cap Q_{\epsilon}  \iff \\  (\frac{1}{\epsilon}z_1,\frac{1}{\epsilon}w_1,...,\frac{1}{\epsilon}z_{2(N'-N)},\frac{1}{\epsilon}w_{2(N'-N)},z_{2(N'-N)+1},w_{2(N'-N)+1},...,z_{2N'},w_{2N'}) \in \underset{j=1}{\overset{2N'} \prod} \Psi(Gr_{\phi_j}) \cap Q,
\end{array}
\end{equation}
that is, critical points of $\mathcal{F}^{N'}_{\epsilon}$ are in one-to-one correspondence with critical points of $\mathcal{F}^{N'}$, for any $\epsilon \in [0,1]$.


The observation now, is that given two decompositions 
\begin{equation}{\label{eq6}}
\begin{array}{lcl}
\Phi_H^1 = \Phi_N \circ ... \circ \Phi_1, \quad \quad \Phi_H^1 = \Phi_{H'}^1\\ \\
\Phi_{H'}^1  = \Phi'_{N'} \circ ... \circ \Phi'_1
\end{array}
\end{equation}
of the same Hamiltonian isotopy $\lbrace \Phi_H^t \rbrace_{t \in [0,1]} = \lbrace \Phi_{H'}^t \rbrace_{t \in [0,1]}$ on $\mathbb{C}^n$ with, say, $N' > N$, one can always use a reparametrization $\widetilde{H}_t^s$ of $H'_t$, such that $\widetilde{H}_t^0 = H'_t$, and $\widetilde{H}_t^1$ generated the decomposition
\begin{equation}
\Phi_{\widetilde{H}^s}^1 = \Phi_N \circ ... \circ \Phi_1 \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{(N'-N)-\text{times}}.
\end{equation}
In particular, consider 
\end{comment}

\begin{comment}
Let us deform the bigger decomposition keeping the time-$1$ map unchanged. We use a reparametrization
\begin{equation}
\begin{array}{lcl}
\widetilde{H}^s_t : \mathbb{C}^n \to \mathbb{R} \\
\quad \quad \  z \mapsto \frac{\mathrm{d}}{\mathrm{d} t}(a^{N,N'}_s(t))H'_{a^{N,N'}_s(t)}(z),
\end{array}
\end{equation}
where $a_{N,N'}$ is a smooth function satisfying
\begin{equation}
a_{N,N'}(t) = \begin{cases}
               0 \text{ if } t \in [0, \frac{N'-N}{N'}]\\
               \frac{N'}{N}t - \frac{N'-N}{N} \text{ if } t \in [\frac{N'-N+1}{N'},\frac{N'-1}{N'}]\\
               1 \text{ if } t=1               
       \end{cases},
\end{equation}
and $a^{N,N'}_s(t) := sa_{N,N'}(t)+(1-s)t$. Then, for any $s \in [0,1]$, we have $\Phi^1_{\widetilde{H}^s} = \Phi_H^1 = \Phi_{H'}^1$, and moreover $\widetilde{H}_t^0 = H'_t$. The Hamiltonian $\widetilde{H}_t^1$ yields the decomposition
\begin{equation}
\Phi_{\widetilde{H}^s}^1 = \Phi_N \circ ... \circ \Phi_1 \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{(N'-N)-\text{times}}.
\end{equation}
\end{comment}
\begin{comment}
the homogeneous generating family $\mathcal{F}_L$ associated with a decomposition of a Hamiltonian lift
\begin{equation}
\widetilde{\Phi_{\overline{h}}} = \Phi_L \circ ... \circ \Phi_1.
\end{equation}
Let us denote by $\mathcal{G}_{\lambda}^{(K)} : \mathbb{C}^{2nK} \to \mathbb{R}$ the generating function associated with the decomposition
\begin{equation}
\underbrace{\exp(\frac{\lambda}{2K}) \circ ... \circ \exp(\frac{\lambda}{2K})}_{2K-\text{times}},
\end{equation}
as constructed in \ref{sec3}. In particular $\mathcal{G}_0^{(K)}$ is associated with the tirivial loop $\underbrace{Id \circ ... \circ Id}_{2K-\text{times}}$.
\begin{propsubsec}{\label{prop242}}
There exists a fiberwise $\mathbb{K}_0$-invariant smooth homotopy between the restricted homogeneous family
\begin{equation}
\mathcal{F}_{N+K |\Lambda_N}
\end{equation}
and the fiberwise direct sum
\begin{equation}
\mathcal{F}_N \oplus_{\Lambda_N} \mathcal{G}_0^{(K)} : \mathbb{C}^{2n(N+K)} \times \Lambda_N \to \mathbb{R},
\end{equation}
in a way that the front of the associated generating families on $S_{N+K} \times \Lambda_N$ remains unchanged during the deformation.
\begin{proof}
Recall that $\mathcal{F}_{\lambda}^{(N+K)}$ and $\mathcal{F}_{\lambda}^{(N)}$ are the generating functions associated respectively with the decompositions
\begin{equation} 
\begin{array}{lcl}
\exp(\lambda) \circ \widetilde{\Phi_{\overline{h}}} = \underbrace{\exp(\frac{\lambda}{N+K}) \circ ... \circ \exp(\frac{\lambda}{N+K})}_{(N+K)-\text{times}} \circ \Phi_{N+K} \circ ... \circ \Phi_1\\\\
\exp(\lambda) \circ \widetilde{\Phi_{\overline{h}}} = \underbrace{\exp(\frac{\lambda}{N}) \circ ... \circ \exp(\frac{\lambda}{N})}_{N-\text{times}} \circ \Phi_N \circ ... \circ \Phi_1.
\end{array}
\end{equation}
Up to reparametrization, we obtain a decomposition
\begin{align}
\begin{array}{lcl}
\exp(\lambda) \circ \widetilde{\Phi_{\overline{h}}} = &
\underbrace{\exp(\frac{\lambda}{N}) \circ ... \circ \exp(\frac{\lambda}{N})}_{N-\text{times}} \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{K-times} \circ \Phi_{N} \circ ... \circ \Phi_1 \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{K-times}\\ 
& = \underbrace{\exp(\frac{\lambda}{N}) \circ ... \circ \exp(\frac{\lambda}{N})}_{N-\text{times}} \circ \Phi_{N} \circ ... \circ \Phi_1 \circ \underbrace{Id_{\mathbb{C}^n} \circ ... \circ Id_{\mathbb{C}^n}}_{2K-times}.
\end{array}
\end{align}
This, along with the move above, yields the result.
\end{proof}
\end{propsubsec}

\remarksec \normalfont Notice that the result above is independent of the reparametrization, since any two such reparametrizations are always homotopic.\\
\end{comment}

We have the following "homogeneous" version of \cite[Proposition $5.1$]{Giv1995}.  
\begin{propsubsec}{\label{prop3.5.1}}
Let $\mathcal{F} : \mathbb{C}^M \to \mathbb{R}$ be a homogeneous of degree $2$ function, and $\hat{\mathcal{F}} : \mathbb{C}^{M+1} \to \mathbb{R}$ be its suspension
$$
\hat{\mathcal{F}}(x,z) := \mathcal{F}(x) + |z|^2.
$$
We denote $\mathcal{F}^{\pm}$ and $\hat{\mathcal{F}}^{\pm}$ the sublevel sets
$$
\mathcal{F}^{\pm} := \lbrace x \in \mathbb{C}^M :  \mathcal{F}(x) \geq 0 \ (\text{resp. $\leq 0$}) \rbrace, \quad \quad \hat{\mathcal{F}}^{\pm} := \lbrace (x,z) \in \mathbb{C}^{M+1} :  \hat{\mathcal{F}}(x) \geq 0 \ (\text{resp. $\leq 0$}) \rbrace.
$$
\begin{comment}
and by $F^{\pm}$, $\hat{F}^{\pm}$ the sets
\begin{equation}
F^{\pm} := \lbrace x \in \S^{2M-1} :  \mathcal{F}(x) \geq 0 \ (\text{resp. $\leq 0$}) \rbrace, \quad \quad \hat{F}^{\pm} := \lbrace (x,z) \in S^{2(M+1)-1} :  \hat{\mathcal{F}}(x) \geq 0 \ (\text{resp. $\leq 0$}) \rbrace,
\end{equation}
\end{comment}
Then there exist natural homotopy equivalences
$$
\hat{\mathcal{F}}^- \simeq \mathcal{F}^-, \quad \hat{\mathcal{F}}^+ \simeq \mathcal{F}^+ \times \mathbb{C}.
$$
Moreover, if $\mathcal{F}$ is invariant relatively to an $S^1$-action on $\mathbb{C}^M$, then the above homotopy equivalences can be made equivariant with respect to the diagonal $S^1$ action on $\mathbb{C}^{M+1}$. If $\mathcal{F}$ depends continuously on additional parameters, then the homotopy equivalences depend continuously on them.

\begin{proof}
We first treat the case $z \in \mathbb{R}_{\geq 0}$.
\begin{comment}
Since $\mathcal{F}$ and $\hat{\mathcal{F}}$ are homogeneous, we can restrict to the upper semi-sphere 
\begin{equation}
D := \lbrace (x,z) \in S^{2(M+1)-1} : x \in \mathbb{C}^M, z \in [0,1] \rbrace \subset \mathbb{C}^{M+1},
\end{equation} 
\end{comment}
The meridional contraction from the North pole $P = (0,1)$ of the unit sphere $S^{2M+1}$ preserves $\hat{\mathcal{F}}^- \cap (S^{2M-1} \times \lbrace 0 \rbrace)$, and therefore $\hat{\mathcal{F}}^- \cap S^{2M+1}$ retracts onto $\hat{\mathcal{F}}^- \cap (S^{2M-1} \times \lbrace 0 \rbrace)$. Extending this contraction homogeneously provides a deformation retraction from $\hat{\mathcal{F}}^-$ to $\mathcal{F}^-$ as well:
$$
\hat{\mathcal{F}}^- \simeq \mathcal{F}^-.
$$
Moerover, each meridional arc from the North pole to $\mathcal{F}^+ \cap S^{2M-1}$ lies in $\hat{\mathcal{F}}^+ \cap S^{2M+1}$, and therefore the same contraction provides a homotopy equivalence of pairs:
$$
(\hat{\mathcal{F}}^+, \mathcal{F}^+) \simeq (\mathbb{C}^M \times \mathbb{R}_{\geq 0}, \mathcal{F}^+).
$$
Now, the pairs $(\mathbb{C}^M \times \mathbb{R}_{\geq 0},\mathcal{F}^+)$ and $(\mathcal{F}^+ \times \mathbb{R}_{\geq 0},\mathcal{F}^+)$ are naturally homotopy equivalent, and therefore we have 
$$
(\hat{\mathcal{F}}^+, \mathcal{F}^+) \simeq (\mathcal{F}^+ \times \mathbb{R}_{\geq 0}, \mathcal{F}^+).
$$

For the general case $z \in \mathbb{C}$, we view $\mathbb{C}^M \times \mathbb{C}$ as the quotient $\mathbb{C}^M \times \mathbb{R}_{\geq 0} \times S^1 / \sim$, with the identification $\mathbb{C}^M \times \lbrace 0 \rbrace \times S^1 \sim \mathbb{C}^M$. Then $\mathcal{F}^{\pm}$ and $\hat{\mathcal{F}}^{\pm}$ are simply given by their restrictions to $\mathbb{C}^M \times \mathbb{R}_{\geq 0}$ multiplied by $S^1$, with the relevant identifications. We obtain  
$$
\hat{\mathcal{F}}^- \simeq \mathcal{F}^-, \quad \hat{\mathcal{F}}^+ \simeq \mathcal{F}^+ \times \mathbb{C}.
$$
Since all the homotopies from above are carried out in a canonical way, they respect group actions and parametric dependence. 
\end{proof}
\end{propsubsec}

\begin{comment}
The join $S^l * S^m$ of two spheres of respective dimensions $l$ and $m$, is a sphere $S^{l+m+1}$ of dimension $l+m+1$. Moreover, $\mathcal{G}_{\lambda,K}$ is the direct sum of $1$-dimensional $\mathbb{K}_0$-invariant Hermitian forms, where the $\mathbb{K}_0$-action on $\mathbb{C}^1$ is given by
\begin{equation}
\gamma.z_j = e^{2 i \pi \gamma_j} z_j, \quad z = (z_1,...,z_n) \in \mathbb{C}^n, \quad \gamma = (\gamma_1, ..., \gamma_n) \in \mathbb{K}_0 \subset \mathbb{T}^n.
\end{equation}
\end{comment}

Recall that $\mathcal{G}_{\lambda}^{(N)}$ denotes the generating function of the decomposition
$$
\underbrace{\exp(\frac{\lambda}{2N}) \circ ... \circ \exp(\frac{\lambda}{2N})}_{2N-\text{times}},
$$
as defined in section \ref{sec3.3.1}. It is a quadratic form. Let us denote $\mathcal{G}_{\lambda}^{(N) \pm}$ its non-negative and non-positive eigenspaces, after the following diagonalization.
\begin{lemmasubsec}
There exists an equivariant linear isomorphism
$$
\mathbb{C}^{2nN} \simeq \underset{j,k}{\overset{2nN}{\bigoplus}} \mathbb{C} v_j^k, \quad j=1,...,n, \quad k=1,...,2N,
$$
where $v_j^k$ are vectors in $\mathbb{C}^{2nN}$, such that:
\begin{enumerate}
\item for any $k$, the maximal torus $\mathbb{T}$ acts on the complex line generated by $v_j^k$ via its standard character $\chi_j : (e^{i\theta_1},...,e^{i\theta_n}) \mapsto e^{i\theta_j}$. In particular, the direct sum $\underset{k}{\oplus}\mathbb{C} v_j^k$ is $\mathbb{T}$-invariant;
\item for any $\lambda$, the quadratic form $\mathcal{G}_{\lambda}^{(N)}$ is diagonal in the basis $(v_j^k)_{j,k}$, and we have
$$
\dim(\mathcal{G}_{\lambda}^{(N)-}) = \underset{j=1}{\overset{n} \sum}2(N + \lfloor \lambda_j + \frac{1}{2} \rfloor), \quad \text{where} \quad \lambda = (\lambda_1,...,\lambda_n) \in \Lambda_N,
$$
where $\lfloor . \rfloor$ denotes the integer part.
\end{enumerate}
\begin{proof}
Recall that for any $\lambda \in \mathfrak{k} \subset \mathbb{R}^n$, the quadratic form $\mathcal{G}_{\lambda}^{(N)}$ is of the form
$$
\mathcal{G}_{\lambda}^{(N)} = \mathcal{Q} - \mathcal{H}_{\lambda}.
$$
We have $\mathcal{H}_{\lambda} = - \underset{i=1}{\overset{n} \sum} \tan(\frac{\pi \lambda_i}{2N}) (|q_1^i|^2 + ... + |q_{2N}^i|^2)$, where $q = (q_1^1,...,q_1^n,...,q_{2N}^1,...,q_{2N}^n) \in \mathbb{C}^{2nN}$, and moreover, a direct calculation shows that the non-degenerate quadratic form $\mathcal{Q}$ writes $\mathcal{Q}(q) = \langle C q, q \rangle$, where
$$
C = i(Id-A)(Id+A)^{-1}, \quad \quad A = \begin{bmatrix}
 
0&1& & & & &\\
 &.&.& & & &\\
 & &.&.& & &\\
 & & &.&.& &\\
 & & & &.&.&\\
 & & & & &.&1\\
-1 & & & & & & 0
\end{bmatrix} \in M_{2N \times 2N}(\mathbb{C}^{n}).
$$
For any $k = -N,...,N-1$, and $j=1,...,n$, we define the following vector:
$$
X_j^k = (e_j,e^{i\frac{(2k+1)\pi}{2N}} e_j,e^{i(2N-1)\frac{(2k+1)\pi}{2N}} e_j) \in (\mathbb{C}^{n})^{2N},
$$
where $e_j = (0,...,0,1,0,...,0)$ is the $j$-th standard vector in $\mathbb{C}^n$. The reader may check that the following points hold:
\begin{itemize}
\item we have $i(A-Id)X_j^k = - \tan(\frac{(2k+1)\pi}{4N})(A+I)X_j^k$, and therefore the family $\lbrace v_j^k := (A+Id)(X_j^k) \rbrace_{j,k}$ is a $\mathbb{C}$-basis of $\mathbb{C}^{2nN}$ made of eigenvectors of $C$, with eigenvalues $- \tan(\frac{(2k+1)\pi}{4N})$;
\item the maximal torus $\mathbb{T}$ acts on the complex line $\mathbb{C} v_j^k$ via the character $\chi_j$. In particular, $\mathbb{C} v_j^k$ is $\mathbb{T}$-invariant;
\end{itemize}
Put $V_j = \underset{k=1}{\overset{2N}{\oplus}} \mathbb{C} v_j^k$. Since $\mathcal{Q}$ is $\mathbb{T}$-invariant, each two lines $\mathbb{C} v_i^k$ and $\mathbb{C} v_j^l$ are orthogonal whenever $i \neq j$. It remains to diagonalize the restrictions $\mathcal{Q}_{|V_j}$. The action of $\mathbb{T}$ on the basis $(v_j^1,...,v_j^{2N})$ is given by $\chi_j$. Since it is diagonal and linear, one can find a new basis (still denoted $(v_j^1,...,v_j^{2N})$) of $V_j$, on which $\mathbb{T}$ acts via the character $\chi_j$, and such that $C_{| V_j}$ is diagonal. It remains to concatenate these bases for $j=1,...,n$. 

As for the second point, the spectrum of $\mathcal{Q}$ is given by
$$
\Spec(\mathcal{Q}) = \lbrace -\tan(\frac{\pi (2k+1)}{4N}) \ | \  -N \leq k \leq N-1 \rbrace,
$$
and therefore the spectrum of $\mathcal{G}_{\lambda}^{(N)}$ is given by
$$
\Spec(\mathcal{G}_{\lambda}^{(N)}) = \lbrace \tan(\frac{\pi (2k+1)}{4N}) - \tan(\frac{\pi \lambda_j}{2N}) \ | \  -N \leq k \leq N-1, \  j=1,...,n \rbrace.
$$
We have
$$
\tan(\frac{\pi (2k+1)}{4N}) - \tan(\frac{\pi \lambda_j}{2N}) < 0 \iff k < \lambda_j - \frac{1}{2}.
$$
There are $N + \lfloor \lambda_j + \frac{1}{2} \rfloor$ such $k$'s, and therefore the real dimension of $\mathcal{G}_{\lambda,N}^-$ is:
$$
\dim(\mathcal{G}_{\lambda}^{(N)-}) = \underset{j=1}{\overset{n}\sum}2(N + \lfloor \lambda_j + \frac{1}{2} \rfloor),
$$
as claimed.
\end{proof}
\end{lemmasubsec}
This change of basis is canonical, that is it depends only on $N$ and the non-degenerate quadratic form $\mathcal{Q}$. Consider the fiberwise direct sum $\mathcal{F}_N \oplus_{\Lambda_N} \mathcal{G}_0^{(K)}$. The non-degenerate quadratic form $\mathcal{G}_0^{(K)}$ has $2nK$ negative eigenvalues, and is diagonal in the basis of the above proposition. Applying Proposition \ref{prop3.5.1} multiple times, we get
\begin{corsubsec}{\label{cor3.5.1}}
There exists a fiberwise $\mathbb{K}_0$-equivariant homotopy equivalence
$$
\lbrace \mathcal{F}_N \oplus_{\Lambda_N} \mathcal{G}_0^{(K)} \leq 0 \rbrace \simeq \mathcal{F}_N^- \times \mathcal{G}_0^{(K)-} \simeq \mathcal{F}_N^- \times \mathbb{C}^{nK}.
$$
\end{corsubsec}
Applying Proposition \ref{prop3.4.1} along with Proposition \ref{prop2.1.2}, we get
\begin{propsubsec}{\label{prop3.5.2}}
If $\nu$ is generic, there exists a fiberwise $\mathbb{K}_0$-equivariant homotopy equivalence
$$
(\mathcal{F}^-_{N+K | \Gamma_N(\nu)}, \mathcal{F}^-_{N+K | \partial \Gamma_N(\nu)}) \simeq (\mathcal{F}^-_N(\nu) \times \mathbb{C}^{nK}, \partial \mathcal{F}^-_N(\nu) \times \mathbb{C}^{nK}).
$$
\end{propsubsec}


\subsection{The cohomology groups}{\label{sec3.6}}

In this section we study the equivariant cohomology of the sublevel sets introduced in the previous section, and come to the definition of our cohomology group by taking a limit in $N \to \infty$. In a first step we use the identification of section \ref{sec2.2} for conical spaces in order to define a natural homomorphism from which we derive our limit. The latter comes along with a natural so-called augmentation map, as well as with several algebraic structures, namely the action of coefficient rings in equivariant cohomology, as well as a Novikov action of $H_2(M,\mathbb{Z})$. We interpret the latter on the kernel of the augmentation map. Finally, using transversality arguments we prove Proposition \ref{prop1.2.1}, whereas by means of algebraic-topological considerations we prove Proposition \ref{prop1.2.2}.  
\begin{comment}
Let $X$ be a topological space provided with the action of a compact Lie group $G$. Equivariant cohomology $H^*_G(X)$ is defined to be the usual cohomology $H^*(X_G)$ of the quotient
\begin{equation}
X_G := (X \times EG) / G,
\end{equation}
where $EG \to BG$ is the universal principal $G$-bundle over the classifying space $BG$. The canonical projection
\begin{equation}
(X \times EG)/BG \to EG/G
\end{equation}
provides $H^*_G(X)$ with the structure of a module over $H^*(BG)$, which plays the role of the graded coefficient ring $H^*_G(\pt)$ in equivariant cohomology theory. If $G$ is the $n$-dimensional torus $\mathbb{T}^n = (S^1)^n$, the coefficient ring $H^*_{\mathbb{T}^n}(\pt)$ is naturally isomorphic to a polynomial algebra in $n$ variables $u = (u_1,...,u_n)$ of degree $2$. More precisely, there is a natural algebra isomorphism, called the \textit{Chern-Weil isomorphism} 
\begin{equation}
\chi : \mathbb{R}^{n*} \simeq H_{\mathbb{T}^n}^*(\pt),
\end{equation}
between polynomial functions on $\mathbb{R}^{n*}$, and the cohomology of the classifying space $B \mathbb{T}^n$. If $(e_1,...,e_n)$ denotes the standard basis on $\mathbb{R}^n$, and $(u_1,...,u_n)$ is its dual, each $u_i$ corresponds, through $\chi$, to the Euler class of the $S^1$-bundle $E S^1 \to B S^1$.

We consider singular cohomology with complex coefficients, so that $H^*(Y) := H^*(Y,\mathbb{C})$, unless otherwise is stated. Therefore we have a natural identification
\begin{equation}
H^*_{\mathbb{T}^n}(\pt) \simeq \mathbb{C}[u_1,...,u_n].
\end{equation}
\end{comment}
\begin{comment}
The following properties hold for $\mathbb{T}$-euivariant cohomology.
\begin{enumerate}
\item Let $X$ be a $\mathbb{T}$-space, and $\mathbb{K}_0 \subset \mathbb{T}$ be a subroup. The homotopy quotients $X_{\mathbb{T}}$ and $X_{\mathbb{K}_0}$ form the bundle
\begin{equation}
X_{\mathbb{K}_0} \overset{\mathbb{T} / \mathbb{K}_0}{\longrightarrow} X_{\mathbb{T}},
\end{equation}
since $E \mathbb{T}$ can be viewed as $E \mathbb{K}_0$ provided with the subgroup action. The spectral sequence of this bundle has $E_2$-term
\begin{equation}
E_2^{p,q} = H^p_{\mathbb{T}}(X) \otimes H^q(\mathbb{T} / \mathbb{K}_0)
\end{equation}
and converges to $H^*_{\mathbb{K}_0}(X)$. 
\end{enumerate}
\end{comment}
\begin{comment}
The Kunneth spectral sequence for equivariant cohomology is obtained when one considers the product $X \times Y$ of two $G$-spaces as a $G \times G$-space provided with the action of the diagonal subgroup $G \subset G \times G$. It has $E_2$-term
\begin{equation}
E_2^{*,q} = \Tor^q_{H^*(BG \times BG)}(H^*(BG) \otimes H^*(X_G \times Y_G)).
\end{equation}
In the special case when $H_G^*(Y)$ is a free $H^*_G(pt)$-module the spectral sequence reduces to one line $q = 0$, provided at least that $G$ is torus. Indeed, a resolution of $H^*(X_G \times Y_G)$ over $H^*(BG \times BG) = \mathbb{C}[u,v]$ is just the tensor product of a resolution of $H^*(X_G)$ over $\mathbb{C}[u]$ and a free $\mathbb{C}[v]$-module $H^*(Y_G)$ while the $\mathbb{C}[u,v]$-module $H^*(BG)$ is $\mathbb{C}[u,v] / (u_i = v_i)$. This results in $\Tor^q = 0$, for $q > 0$. Therefore
\begin{equation}
H^*_G(X \times Y) = H^*_G(X) \otimes H_G^*(Y),
\end{equation}
when $G$ is a torus and $H^*_G(Y)$ is free.
\end{comment}
As above, we consider the homogeneous generating family
$$
\mathcal{F}_N : \mathbb{C}^{2nN} \times \Lambda_N \to \mathbb{R},
$$
associated with a decomposition of the Hamiltonian symplectomorphism $\Phi_{\widetilde{H}}$, where $\widetilde{H}$ is a Hamiltonian lift of a contact Hamiltonian $h$ of $V$, and we fix a generic $\nu \in \mathbb{R}$. We look at the $\mathbb{K}_0$-equivariant cohomology groups
$$
H_{\mathbb{K}_0}^*(F_N^-(\nu),\partial F_N^-(\nu)).
$$

\begin{comment}
Let $A \subset X$ be topological spaces, acted on by a topological group $G$. Denote by $pr : X \times EG \to X_G$ the canonical projection. We introduce the following notations
\begin{itemize}
\item $C_G^*(X) := C^*(X_G)$ and $C_G^*(A) := C^*(A_G)$;
\item $C_G^*(X, A) := \ker C_G^*(X) \to C_G^*(A)$, where the map is induced by the inclusion $A \subset X$;
\item $C_{G,c}^*(X) := \lbrace \sigma \in C_G^*(X) : \exists K \subset X \text{ compact such that } pr^* \sigma \text{ vanishes on } (X \setminus K) \times EG \rbrace$, the complex of equivariant cocycles with compact support;
\item $C_{G,c}^*(A) := \lbrace \sigma \in C_G^*(A) : \exists K \subset A \text{ compact such that } pr^*_{| A \times EG} \sigma \text{ vanishes on } (A \setminus K) \times EG \rbrace$;
\item $C_{G,c}^*(X,A) := \ker C^*_{G,c}(X) \to C^*_{G,c}(A)$;
\item if $X \to M$ is a fiber bundle of $G$-spaces, $C_{G,cv}^*(X)$, $C_{G,cv}^*(A)$, $C_{G,cv}^*(X,A)$, the complexes of equivariant (relative) cocycles with vertical compact support, that is, we replace the term compact above by compact in the fibers direction. We will use the notation $H_{\mathbb{K}_0,cv}^*$ for the associated equivariant cohomology groups.  
\end{comment}
Using the notations from section \ref{sec2.2}, we work with the following short exact sequence
$$
0 \rightarrow C_{\mathbb{K}_0,c}^*(\mathcal{F}_N^-(\nu), \partial \mathcal{F}_N^-(\nu)) \rightarrow C_{\mathbb{K}_0}^*(\mathcal{F}_N^-(\nu), \mathcal{F}_N^-(\nu)) \rightarrow C_{\mathbb{K}_0}^*(\mathcal{F}_N^-(\nu), \mathcal{F}_N^-(\nu)) / C_{\mathbb{K}_0,c}^*(\mathcal{F}_N^-(\nu), \partial \mathcal{F}_N^-(\nu)) \rightarrow 0,
$$
and identify the cohomology of the third term with that of the complex $C_{\mathbb{K}_0}^*(F_N^-(\nu), \partial F_N^-(\nu))$, that is with $H_{\mathbb{K}_0}^*(F_N^-(\nu),\partial F_N^-(\nu))$. Consider the $\mathbb{C}^{nK}$-bundle
$$
(\mathcal{F}_N^-(\nu) \times \mathbb{C}^{nK})_{\mathbb{K}_0} \to \mathcal{F}_N^-(\nu)_{\mathbb{K}_0}.
$$
The Thom isomorphism of this bundle
$$
H_{\mathbb{K}_0}^*(\mathcal{F}_N^-(\nu), \partial \mathcal{F}_N^-(\nu)) \simeq H_{\mathbb{K}_0,cv}^{*+2nK}(\mathcal{F}_N^-(\nu) \times \mathbb{C}^{nK}, \partial \mathcal{F}_N^-(\nu) \times \mathbb{C}^{nK}),
$$
and the natural homomorphism
$$
H_{\mathbb{K}_0,cv}^{*+2nK}(\mathcal{F}_N^-(\nu) \times \mathbb{C}^{nK}, \partial \mathcal{F}_N^-(\nu) \times \mathbb{C}^{nK}) \to H_{\mathbb{K}_0}^{*+2nK}(\mathcal{F}_N^-(\nu) \times \mathbb{C}^{nK}, \partial \mathcal{F}_N^-(\nu) \times \mathbb{C}^{nK})
$$
preserve compact supports, and therefore they induce a homomorphism
$$
H_{\mathbb{K}_0}^*(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2nK}(F_N^-(\nu) \times \mathbb{C}^{nK}, \partial F_N^-(\nu) \times \mathbb{C}^{nK}).
$$
Along with Proposition \ref{prop3.5.2}, we obtain a homomorphism
$$
H_{\mathbb{K}_0}^*(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2nK}(F_{N+K | \Gamma_N(\nu)}^-, F_{N+K | \partial \Gamma_N(\nu)}^-).
$$
We now apply the equivariant version of the excision formula to the triple
$$
(\mathcal{F}_{N+K}^-(\nu), \mathcal{F}^-_{N +K | \Gamma_{N+K}(\nu) \setminus \mathring{\Gamma}_N(\nu)}, \mathcal{F}^-_{N +K | \Gamma_{N+K}(\nu) \setminus \Gamma_N(\nu)}),
$$
where $\mathring{\Gamma}_N(\nu)$ denotes the interior of $\Gamma_N(\nu)$ (the reader interested in entering the details of this argument shall note the importance of the fact that transversality is an open condition). It induces an isomorphism
$$
H_{\mathbb{K}_0}^*(\mathcal{F}_{N+K | \Gamma_N(\nu)}, \mathcal{F}_{N+K | \partial \Gamma_N(\nu)}) \simeq H_{\mathbb{K}_0}^*(\mathcal{F}_{N+K}^-(\nu), \mathcal{F}^-_{N +K | \Gamma_{N+K}(\nu) \setminus \mathring{\Gamma}_N(\nu)}),
$$
which preserves compact supports. Moreover, there is an inclusion of pairs
$$
(\mathcal{F}_{N+K}^-(\nu), \partial \mathcal{F}_{N+K}^-(\nu)) \subset (\mathcal{F}_{N+K}^-(\nu), \mathcal{F}^-_{N +K | \Gamma_{N+K}(\nu) \setminus \mathring{\Gamma}_N(\nu)}).
$$
Putting all  these maps together, we obtain a homomorphism
$$
H_{\mathbb{K}_0}^*(F_N^-(\nu), F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2nK}(F_{N+K}^-(\nu), \partial F_{N+K}^-(\nu)).
$$
We will take a limit in $N \to \infty$, and therefore it will be convenient to shift the grading by $2nN$. Thus we have built a homomorphism
$$
f_N^{N+K} : H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2n(N+K)}(F_{N+K}^-(\nu), \partial F_{N+K}^-(\nu)).
$$
Notice that all the maps involved in the construction of $f_N^{N+K}$ are natural in cohomology: they involve topological inclusions, the excision formula, the Thom isomorphism, and the deformation of Proposition $\ref{prop3.5.2}$. In particular, we have the following cocycle condition
$$
f_{N+K}^{N+K+K'}\circ f_N^{N+K} = f_N^{N+K+K'}.
$$

\begin{defsubsec}
We define the \textbf{cohomology of $\widetilde{H}$ of level $\nu$} as the limit
$$
\mathcal{H}_{\mathbb{K}_0}^*(F^-(\nu)) := \underset{N \to \infty}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu)).
$$
\end{defsubsec}

\begin{remarksubsec}
\normalfont This definition is independent of the choice of a sequence $\{\Lambda_N\}_N$. Indeed, for any other sequence of cubes $\{\Lambda'_N\}_N$, and for any $N$, there exists $N'$ such that $\Lambda_N \subset \Lambda'_{N'}$. Applying the Thom isomorphism and the excision formula, one can then build a homomorphism from one limit to the other, and similarly in the other direction which, by naturality, are inverse of one another.
\end{remarksubsec}
The above limit comes along with certain structures. First, note that there is an inclusion of pairs $(F_N^-(\nu), \partial F_N^-(\nu)) \subset (\mathcal{F}_N^-(\nu), \partial \mathcal{F}_N^-(\nu))$, and the latter is equivariantly homotopic to the pair $(\Gamma_N(\nu),\partial \Gamma_N(\nu))$. In particular, there is a natural homomorphism
$$
H_{\mathbb{K}_0}^*(\Gamma_N(\nu), \partial \Gamma_N(\nu)) \to H_{\mathbb{K}_0}^*(F_N^-(\nu), \partial F_N^-(\nu)).
$$
We denote $\mathcal{H}_{\mathbb{K}_0}^*(\nu)$ the limit 
$$
\mathcal{H}_{\mathbb{K}_0}^*(\nu) := \underset{N \to \infty}{\lim} H_{\mathbb{K}_0}^{*+2nN}(\Gamma_N(\nu), \partial \Gamma_N(\nu)),
$$
and call the induced homomorphism 
$$
\Hcal_{\mathbb{K}_0}^*(\nu) \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu))
$$
the \textbf{augmentation map}. Note that the groups $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ and $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ inherit from there finite parts the structure of $H_{\mathbb{K}_0}^*(\pt)$-modules, and that the augmentation map is a module homomorphism. We denote $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ its kernel. Recall that under the isomorphism $H^2(M,\mathbb{Z}) \simeq \mathfrak{k}_{\mathbb{Z}}^*$ from equation $(\ref{eq2})$, the first Chern class
$c$ of $M$ writes 
$$
c(m) = \underset{j=1}{\overset{n} \sum} m_j, \quad \text{for any} \quad m \in \mathfrak{k}_{\mathbb{Z}}, \quad \iota(m) = (m_1,...,m_n).
$$ 
Notice moreover that since $\nu$ is generic and the front $L_N$ of $F_N$ is $\mathfrak{k}_{\mathbb{Z}}$-invariant (Remark \ref{rem3.2.1}), all translations $\Gamma_N(\nu) + m$ and $\partial \Gamma_N(\nu) + m$ by elements $m \in \mathfrak{k}_{\mathbb{Z}}$ are transversal to $L_N$, for any $N$. Thus we can apply Proposition \ref{prop3.5.2}, replacing $0$ with $m \in \mathfrak{k}_{\mathbb{Z}}$, and along with the Thom isomorphism and the excision formula we obtain a homomorphism in equivariant cohomology
$$
H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2n(N+K)+2c(m)}(F_{N+K}^-(\nu + p(m)), \partial F_{N+K}^-(\nu + p(m))).
$$
In the limit $N \to \infty$, we obtain
$$
\Hcal_{\mathbb{K}_0}^*(F^-(\nu)) \to \Hcal_{\mathbb{K}_0}^{*+2c(m)}(F^-(\nu +p(m))).
$$
It is an isomorphism (with inverse given by applying Proposition \ref{prop3.5.2} replacing $0$ with $-m$), which reflects the \textit{Novikov action} of $H_2(M,\mathbb{Z})$. The latter induces an isomorphism between the kernels
\begin{equation}{\label{eq9}}
\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \simeq \mathcal{J}_{\mathbb{K}_0}^{*+2c(m)}(F^-(\nu + p(m))).
\end{equation}
Notice that the torus $\mathbb{K}_0$ acts trivially on the pair $(\Gamma_N(\nu), \partial \Gamma_N(\nu))$, and therefore the cohomology group $H_{\mathbb{K}_0}^*(\Gamma_N(\nu), \partial \Gamma_N(\nu))$ is a free $H_{\mathbb{K}_0}^*(\pt)$-module of rank $1$ generated by the fundamental cocycle of the sphere $\Gamma_N(\nu) / \partial \Gamma_N(\nu)$. Under this identification, the homomorphism
$$
H_{\mathbb{K}_0}^{*+2nN}(\Gamma_N(\nu), \partial \Gamma_N(\nu)) \to H_{\mathbb{K}_0}^{* + 2n(N+K) + 2c(m)}(\Gamma_{N+K}(\nu + p(m)), \partial \Gamma_{N+K}(\nu + p(m)))
$$
is simply the multiplication by the Euler class of the bundle 
$$
(\mathbb{C}^{nK + c(m)})_{\mathbb{K}_0} \to B \mathbb{K}_0.
$$
Recall the Chern-Weil isomorphism
$$
H_{\mathbb{T}^n}^*(\pt) \simeq \mathbb{C}[u_1,...,u_n].
$$
Under the surjective ring homomorphism
$$
H_{\mathbb{T}^n}^*(\pt) \to H_{\mathbb{K}_0}^*(\pt),
$$
the Euler class of the above bundle is given by $u_1^{K + m_1}...u_n^{K + m_n}$. In the limit $N \to \infty$, this implies that the isomorphism $(\ref{eq9})$ is the multiplication by $u_1^{m_1}...u_n^{m_n}$ (in Proposition \ref{prop4.2} we will see that $\Hcal_{\mathbb{K}_0}^*(\nu)$ is the ring of regular functions on $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$, which will in particular clarify why this multiplication makes sense). 

To close this section, we now prove Propositions \ref{prop1.2.1} and \ref{prop1.2.2}. They are included here only for the sake of completeness, for they are similar to the proofs of \cite[Propositions $6.2$, $6.3$]{Giv1995}.

\begin{proof}{(Proposition \ref{prop1.2.1})}
Assume that $[\nu_0, \nu_1] \cap \Spec(\phi_h) = \emptyset$. For any $\nu \in [\nu_0,\nu_1]$ and any $N$, $\Gamma_N(\nu)$ is transversal to the front $L_N$. Moreover, the set of generic $\nu$ is of full measure. Let $\nu \in [\nu_0,\nu_1]$ be in this set. Since transversality is an open condition, by Proposition \ref{prop2.1.3}, there exists $\epsilon > 0$ and a $\mathbb{K}_0$-equivariant homotopy equivalence
$$
(F_N^-(\nu + \epsilon), \partial F_N^-(\nu + \epsilon)) \simeq (F_N^-(\nu - \epsilon), \partial F_N^-(\nu - \epsilon)).
$$
In the limit $N \to \infty$ of $\mathbb{K}_0$-equivariant cohomology groups, this yields an isomorphism
$$
\Hcal_{\mathbb{K}_0}^*(F^-(\nu + \epsilon)) \simeq \Hcal_{\mathbb{K}_0}^*(F^-(\nu - \epsilon)).
$$
Choosing a finite subcovering of $[\nu_0,\nu_1]$ by such segments $[\nu - \epsilon, \nu + \epsilon]$ yields an isomorphism 
$$\Hcal_{\mathbb{K}_0}^*(F^-(\nu_1)) \simeq \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0)),$$ 
as claimed.
\end{proof}

\begin{proof}{(Proposition \ref{prop1.2.2})}
We rephrase the statement as follows: suppose that $q_1 \in \Hcal_{\mathbb{K}_0}^*(F^-(\nu_1))$ and $q_0 \in \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0))$ are the images of $q$ under the augmentation maps:
$$
\begin{tikzcd}
q \in \mathcal{R}_0 \ar[r] \ar[dr] & \Hcal_{\mathbb{K}_0}^*(F^-(\nu_1)) \ni q_1 \ar[d] \\
& \Hcal_{\mathbb{K}_0}^*(F^-(\nu_0)) \ni q_0.
\end{tikzcd}
$$
Then $q_0 = 0$ implies $v q_1 = 0$. 

Without loss of generality, we can assume that $\nu_0 = \nu - \epsilon$, $\nu_1 = \nu + \epsilon$, and that there is only one $\mathbb{K}_0$-orbit of fixed points associated with $\nu$. Using Proposition \ref{prop2.1.3}, one can show that for any $N$, the pair $(F_N^-(\nu_0), \partial F_N^-(\nu_0))$ is embedded into $(F_N^-(\nu_1), \partial F_N^-(\nu_1))$ as the complement of a neighborhood to the $\mathbb{K}_0$-orbit of critical points of $\hat{p}_N$. There exists a non-zero representative $\hat{q} \in H_{\mathbb{K}_0}^*(F_N^-(\nu_1),F_N^-(\nu_1))$ of $q_1$ (otherwise $q_1 = 0$ and the statement is trivial) which vanishes when restricted to $(F_N^-(\nu_0), \partial F_N^-(\nu_0))$. In particular, $\hat{q}$ is the image of some element $\alpha \in H_{\mathbb{K}_0}^*(F_N^-(\nu_1), F_N^-(\nu_0))$ under the long exact sequence:
$$
\begin{array}{lcl}
\hdots \longrightarrow H_{\mathbb{K}_0}^*(F_N^-(\nu_1), F_N^-(\nu_0)) \overset{f}{\longrightarrow} H_{\mathbb{K}_0}^*(F_N^-(\nu_1), \partial F_N^-(\nu_1)) \longrightarrow H_{\mathbb{K}_0}^*(F_N(\nu_0), \partial F_N^-(\nu_0)) \longrightarrow \\ \\
\quad \quad \quad  H_{\mathbb{K}_0}^{*+1}(F_N^-(\nu_1), F_N^-(\nu_0)) \longrightarrow \hdots.
\end{array}
$$
\textit{The torus $\mathbb{K}_0$ acts freely in a neighborhood of the $\mathbb{K}_0$-orbit of critical points of $\hat{p}_N$} (since $M$ is compact), hence the equivariant cohomology $H_{\mathbb{K}_0}^*(F_N^-(\nu_1),F_N^-(\nu_0))$ is simply the singular cohomology $H^*(F_N^-(\nu_1) / \mathbb{K}_0, F_N^-(\nu_0) / \mathbb{K}_0)$. Now, the coefficient ring $H_{\mathbb{K}_0}^*(\pt)$ acts by $0$ on the usual cohomology, and thus
$$
v \alpha = 0 \quad \text{and} \quad v \hat{q} = f(v \alpha) = 0.
$$
\end{proof} 



\begin{comment}
Under the identification $(\mathcal{F}_N^-(\nu), \partial \mathcal{F}_N^-(\nu)) \simeq (\Gamma_N(\nu), \partial \Gamma_N(\nu))$, the map $f_N^{N+K}$ is simply the multiplication by $u_1^{nK}...u_n^{nK} \in \mathbb{C}[u_1,...,u_n]$ (see section $2.2$). Similarly as in \cite{Giv1995} section $5$, one can show that at the limit $HF_{\leq \nu}^*(\Gamma) := \underset{N \to \infty}{\lim} H_{\mathbb{K}_0}^*(\pt)$ is given by
\begin{equation}
HF_{\leq \nu}^*(\Gamma) \simeq \mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}] / I_0 \mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}], 
\end{equation}
where $I_0$ is the ideal in $\mathbb{C}[u_1,...,u_n]$ generated by the polynomials vanishing on $\mathfrak{k}_0 \otimes \mathbb{C}$. This is nothing but the set of regular functions on the complexified torus $\mathbb{K}_0 \otimes \mathbb{C}$, seen as a module over the ring of regular functions on $\mathfrak{k}_0 \otimes \mathbb{C}$, which is naturally identified with the quotient $\mathbb{C}[u_1,...,u_n] / I_0$.
\end{comment}

\begin{comment}
\begin{defsec}
We define the \textbf{degree $\nu$ filtrated cohomology of $\widetilde{\phi}_h$} as the limit
\begin{equation}
HF_{\leq \nu}^*(\widetilde{\phi}_h) := \underset{N \to \infty}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu)).
\end{equation}
\end{defsec}
\begin{propsec}{\label{prop31}}
The cohomology groups $HF_{\leq \nu}^*(\widetilde{\phi}_h)$ are well-defined.
\begin{proof}
Note first that the limit on the right handside of the above equation is independent of the choice of a sequence $\{ \Lambda_N \}_N$. Indeed, for any other sequence $\{ \Lambda'_N \}_N$, and for any $N$, there always exist $N'$ such that $\Lambda_N \subset \Lambda'_{N'}$. Applying the Thom isomorphism and excision, one can then build a map from one limit to the other, and similarly in the other direction. Then, one conclude by naturality of the maps involved.

Let $g$ be a contact Hamiltonian such that $\widetilde{\phi}_g = \widetilde{\phi}_h \in \Cont(V,\xi)$, and $\{ \phi_s^t \}_{s,t \in [0,1]}$ be a contact isotopy between representatives
\begin{equation}
\lbrace \phi_h^t \rbrace_{t \in [0,1]}, \lbrace \phi_g^t \rbrace_{t \in [0,1]}
\end{equation}
of $\widetilde{\phi}_g$ and $\widetilde{\phi}_h$ respectively,
\begin{equation}{\label{eq11}}
\phi_s^1 = \phi_h^1 = \phi_g^1, \text{ for any } s \in [0,1].
\end{equation}
It is generated by a $C^{1,1}$ family $h : V \times [0,1]^2 \to \mathbb{R}$ of contact Hamiltonians $h^s$. Let $\widetilde{H}^s$ be a family of associated Hamiltonian lifts. Fix a sequence $\{ \Lambda_N \}_N$. For $N$ big enough, this yields a family $\mathcal{F}_N^s : \mathbb{C}^{2nN} \times \Lambda_N \times [0,1] \to \mathbb{R}$ of homogeneous generating families. Let $L_N^s$ be the associated family of fronts. Fix a sequence $\{ \Lambda_N \}_N$. If $\nu$ is generic with respect to the front $L_N^0$, $\phi_h$ does not have any translated point with Reeb shift equal to $-\nu$ (see \ref{rem321}). Since $\phi_s^1$ is independent of $s$, this implies that $\nu$ remains generic with respect to $L_N^s$, for all $s$. We can then apply proposition \ref{prop4}: there exists a $C^{1,1}$ $\mathbb{K}_0$-equivariant isotopy $I_N : S_N \times \Lambda_N \times [0,1] \to S_N \times \Lambda_N$ such that
\begin{equation}
I_{N,s}(F_{N,0}^{\pm}(\nu), \partial F_{N,0}^{\pm}(\nu)) = (F_{N,s}^{\pm}(\nu), \partial F_{N,s}^{\pm}(\nu)).
\end{equation}
This holds for any $N' \geq N$, and yields isomorphisms
\begin{equation}
H_{\mathbb{K}_0}^*(F_{N',0}^-(\nu), \partial F_{N',0}^-(\nu)) \simeq H_{\mathbb{K}_0}^*(F_{N',1}^-(\nu), \partial F_{N',1}^-(\nu)),
\end{equation}
of $\mathbb{K}_0$-equivariant cohomology groups. The latters obviously commute with the maps $f_N^{N+K}$ so that, at the limit, we obtain an isomorphism
\begin{equation}
\underset{\underset{N}{\longrightarrow}}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_{N,0}(\nu), \partial F_{N,0}(\nu)) \simeq \underset{\underset{N}{\longrightarrow}}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_{N,1}(\nu), \partial F_{N,1}(\nu)).
\end{equation}
\end{proof}
\end{propsec}

\begin{remarksec}
\normalfont The proof above shows in particular the independence of the groups $HF_{\leq \nu}^*(\widetilde{\phi}_h)$ on the decomposition of the lift $\Phi_{\widetilde{H}}$. Similarly, it shows that they are independent of the way the function $\tilde{h} : P_{\mathbb{K}} \to \mathbb{R}$ is extended to the sphere $S_p$ (see section \ref{sec5}). Indeed, if $\overline{h}_1$ and $\overline{h}_2$ are two such extentions, then there respective Hamiltonian lifts $\widetilde{H}_1,\widetilde{H}_2 : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ yield a family
\begin{equation}
H^s := s\widetilde{H}_1 + (1-s)\widetilde{H}_2 : \mathbb{C}^n \times [0,1] \to \mathbb{R}
\end{equation}
of Hamiltonians on $\mathbb{C}^n$, such that the restriction $H^s_{| P_{\mathbb{K}}^{-1}(p)}$ does not depend on $s$, since $\widetilde{H}_1$ and $\widetilde{H}_2$ coincide on $P_{\mathbb{K}}^{-1}(p)$. This means that for any decomposition of the Hamiltonian symplectomorphism $\Phi_{H^s}$, the induced front $L_N^s$ is transversal to given $\Gamma_N(\nu)$ and $\partial \Gamma_N(\nu)$ if and only if so is $L_{s'}$, for any $s'$.  to the front  of $H^s$ is independent of $s$, and we can conclude again with proposition
\end{remarksec}

The final step in our construction is a limit in $\nu \to \infty$. Recall from section \ref{sec2} that the generating functions decrease in positive directions. In particular, this means that for any two real numbers $\nu_0 < \nu_1$, we have an inclusion of pairs
\begin{equation}
(F_N^-(\nu_0), \partial F_N^-(\nu_0)) \subset (F_N^-(\nu_1), \partial F_N^-(\nu_1)),
\end{equation}
and therefore we have a natural homomorphism
\begin{equation}
H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu_1), \partial F_N^-(\nu_1)) \to H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu_0), \partial F_N^-(\nu_0)).
\end{equation}
If $\nu_0$ and $\nu_1$ are generic, taking limit in $N$ yields a homomorphism $HF_{\leq \nu_1}^*(\widetilde{\phi}_h) \to HF_{\leq \nu_0}^*(\widetilde{\phi}_h)$, which gives rise to the following definition.
\begin{defsec}
We define the \textbf{semi-infinite cohomology of $V$} as the limit
\begin{equation}
HF^*(V) := \underset{\nu \to \infty}{\varprojlim}HF_{\leq \nu}^*(\widetilde{\phi}_h)).
\end{equation}
\end{defsec}
The remaining of this section is devoted to prove that the above cohomology groups are well-defined.
\end{comment}
\begin{comment}
\newpage
More precisely, for any $\nu \notin \Spec(\Phi_H)$, $[\nu_0,\nu_1] \subset \mathbb{R}$ with $\nu_0,\nu_1 \notin \Spec(\Phi_H)$ and such that $\Gamma_N(\nu) \subset \Lambda_N$ for any $\nu \in [\nu_0,\nu_1]$, and for any $N' = N+K$, we define commutative diagrams
\begin{equation}{\label{eq9}}
\begin{tikzcd}
H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu)) \arrow{r} & H_{\mathbb{K}_0}^{*+2nN'}(F_{N'}^-(\nu), \partial F_{N'}^-(\nu))\\
H_{\mathbb{K}_0}^{*+2nN}(\Gamma_N(\nu), \partial \Gamma_N(\nu)) \arrow{r} \arrow[swap]{u} & H_{\mathbb{K}_0}^{*+2nN'}(\Gamma_{N'}(\nu), \partial \Gamma_{N'}(\nu)) \arrow{u},
\end{tikzcd}
\end{equation}
and
\begin{equation}{\label{eq10}}
\begin{tikzcd}
H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu_0), \partial F_N^-(\nu_0)) \arrow{r} & H_{\mathbb{K}_0}^{*+2nN'}(F_{N'}^-(\nu_0), \partial F_{N'}^-(\nu_0))\\
H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu_1), \partial F_N^-(\nu_1)) \arrow{r} \arrow[swap]{u} & H_{\mathbb{K}_0}^{*+2nN'}(F_{N'}^-(\nu_1), \partial F_{N'}^-(\nu_1)) \arrow{u}.
\end{tikzcd}
\end{equation}
We have natural maps 
\begin{equation}{\label{eq7}}
(F^-_N *_{\Lambda_N} G_{0,K}^-) \to (F_N^- *_{\Lambda_N} G_{0,K}^-, G_{0,K}^-) \to (F_N^- \times G_{0,K}^- \times [0,1] \Big/
\begin{array}{lcl}
\text{\footnotesize{$(x,s,0) \sim (x,s',0)$}}\\
\text{\footnotesize{$F_N^- \times G_{0,K}^- \times \{ 1 \}$}}
\end{array}
, \pt)
\equiv (T(F_N^-), \pt)
\end{equation} 
where $T(F_N^-)$ is the Thom space of the $\mathbb{K}_0$-equivariant trivial disk bundle over $F_N^-$
\begin{equation}{\label{eq17}}
F_N^- \times D \to F_N^-. 
\end{equation}
The $\mathbb{K}_0$-equivariant version of the Thom isomorphism holds for this bundle, and therefore we get a natural isomorphism of relative equivariant cohomology groups
\begin{equation}{\label{eq16}}
H_{\mathbb{K}_0}^*(F_N^-, \partial {}F_N^-) \simeq H_{\mathbb{K}_0}^{*+2nK}(T(F_N^-), T(\partial F_N^-)).
\end{equation}
Composing with equation (\ref{eq7}), and applying proposition \ref{prop8}, we obtain, for any $\nu \notin \Spec(\Phi_H)$, a homomorphism
\begin{equation}{\label{eq8}}
H_{\mathbb{K}_0}^*(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2nK}(F^-_{N+K | \Gamma_N(\nu)}, F^-_{N+K | \partial \Gamma_N(\nu)}),
\end{equation}  
The $\mathbb{K}_0$-action on $\Lambda_N$ being trivial, we can apply the excision formula in equivariant cohomology, and get an isomorphism
\begin{equation}{\label{eq13}}
H_{\mathbb{K}_0}^*(F_{N+K | \Gamma_N(\nu)}^-, F_{N+K | \partial \Gamma_N(\nu)}^-) \simeq H_{\mathbb{K}_0}^*(F_{N+K}^-(\nu), F_{N+K | \Gamma_{N+K}(\nu) \setminus \overset{\mathrm{o}}{\Gamma_N}(\nu)}^-).
\end{equation} 
The pair $(F_{N+K}^-(\nu), \partial F_{N+K}^-(\nu))$ lies in $(F_{N+K}^-(\nu), F_{N+K | \Gamma_{N+K}(\nu) \setminus \overset{\mathrm{o}}{\Gamma_N}(\nu)}^-)$, and therefore, combined with (\ref{eq8}) we get the upper horizontal arrow of diagram (\ref{eq9}) (and the horizontal arrows of diagram (\ref{eq10})):
\begin{equation}{\label{eq15}}
H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2n(N+K)}(F_{N+K}(\nu), \partial F_{N+K}(\nu)).
\end{equation}
Moerover, recall that the functions $\mathcal{F}_{\lambda}^{(N)}$ decrease in positive directions (by proposition \ref{prop1} and property \ref{p1} in section \ref{sec2}). In particular, for any segment $[\nu_0,\nu_1]$ with $\nu_0, \nu_1 \notin \Spec(\Phi_H)$ and such that $\Gamma_N(\nu) \subset \Lambda_N$ for any $\nu \in [\nu_0,\nu_1]$, we have an injection of pairs
\begin{equation}
(F_N(\nu_0),F_N(\nu_0)) \hookrightarrow (F_N(\nu_1),F_N(\nu_1)),
\end{equation}
which yields the vertical arrows of diagram (\ref{eq10}). The reader may check that each map composing the horizontal arrow commutes with the injection
\begin{equation}{\label{eq20}}
(F_N^-(\nu_0), \partial F_N^-(\nu_0)) \hookrightarrow (F_N^-(\nu_1), \partial F_N^-(\nu_1)). 
\end{equation}
All the equivariant maps involved in the horizontal arrows of diagram ($\ref{eq10}$) are whether the Thom isomorphism, the excision formula, an injection, fixed fiberwise deformations (see section $\ref{subsec243}$), or the gradient flow deformation of proposition $\ref{prop4}$, for which the choice of an invariant metric can be made dependent on $\nu$ only. This means that all commute with the injection $(\ref{eq20})$, and therefore diagram $(\ref{eq10})$ commutes.\\

COMPLETE FROM HERE...\\

We now complete diagram $(\ref{eq9})$. We introduce the pair $(CF_N^-,F_N^-)$, where $CF_N^- \subset \mathbb{C}^{2nN} \times \Lambda_N$ is the mapping cylinder over of the map $F_N^- \to \Lambda_N$, and denote by $\Sigma F_N^-$ the cone $CF_N^- / F_N^-$, which we see as a pointed space with basepoint $F_N^-$. Note that $(CF_N^-, C \partial F_N^-)$ is $\mathbb{K}_0$-equivariantly homotopic to $(\Lambda_N, \partial \Lambda_N)$. The long exact sequence of relative $\mathbb{K}_0$-equivariant cohomology groups of the pair $(CF_N^-, F_N^-)$ is of the form
\begin{equation}{\label{eq19}}
... \longrightarrow H_{\mathbb{K}_0}^* (\Sigma F_N^- , \Sigma \partial F_N^-) \longrightarrow H_{\mathbb{K}_0}^*(CF_N^-, C \partial F_N^-) \longrightarrow H_{\mathbb{K}_0}^*(F_N^-, \partial F_N^-) \longrightarrow ...
\end{equation} 
The vertical arrows of diagram (\ref{eq9}) are defined, for any $N > 0$, as the map
\begin{equation}
H^{*+2nN}_{\mathbb{K}_0}(CF_N^-(\nu), C \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^{*+2nN}(F_N^-(\nu), \partial F_N^-(\nu))
\end{equation}
in the long exact sequence above, where $\Gamma_N(\nu)$ (resp. $\partial \Gamma_N(\nu)$) is naturally identified with the mapping cone $CF_N^-(\nu)$ (resp. $C \partial F_N^-(\nu)$) of the map $F_N^-(\nu) \to \Gamma_N(\nu)$ (resp. $\partial F_N^-(\nu) \to \partial \Gamma_N(\nu)$). To see that diagram (\ref{eq9}) commutes, it sufficies to show that the natural Thom class of the trivial disk bundle $(\ref{eq17})$ is $u$. The fibers in the Thom space $T(F_N^-(\nu))$ are given by the spheres $G^-_{0,K}$, and we will see in the next proposition that $u$ is the generator, in every degree, of the equivariant cohomology of the fibers, and therefore it is the suitable Thom class of the associated trivial disk bundle. 
\begin{propsubsec}{\label{prop262}}
Let $\mathcal{G}_{\lambda,K} : S_K : \to \mathbb{R}$ be the generating Hermitian form  associated with the decomposition
\begin{equation}
\underbrace{\exp(\frac{\lambda}{2K}) \circ ... \circ \exp(\frac{\lambda}{2K})}_{2K-\text{times}},
\end{equation}
Then
\begin{equation}
H_{\mathbb{K}_0}^*(G_{\lambda,K}^-) = \mathbb{C}[u] / (u^{K- \lfloor \lambda + \frac{1}{2} \rfloor}).
\end{equation}
\begin{proof}
We begin with the case $G_{\lambda,K}^- = \emptyset$. In this case, the fact that $H_{\mathbb{K}_0}^*(G_{\lambda,K}^-) = \mathbb{C}[u]$ is trivial. If $G_{\lambda,K}^- \neq \emptyset$, we can always decompose the Hermitian form $\mathcal{G}_{\lambda,K} : \mathbb{C}^{2nK} \to \mathbb{R}$ into the direct sum of $\mathbb{K}_0$-invariant one-dimensional components, and thus restrict to the case where $\mathbb{K}_0 = S^1$, and $\mathcal{G}_{\lambda,K}$ is of the form 
\begin{equation}
a |z|^2, a \leq 0.
\end{equation}

NOT GOOD !

\end{proof}
\end{propsubsec}


\begin{comment}
It will be important in the sequel to understand how the kernel and cokernel of this homomorphism change when $N$ grows. From the long exact sequence $(149)$, we extract the short exact sequence
\begin{equation}
0 \longrightarrow \widetilde{H}^{*-1}_{\mathbb{K}_0}(F_N^-, \partial F_N^-) \longrightarrow H^*_{\mathbb{K}_0}(\Sigma F_N^-, \Sigma \partial F_N^-) \longrightarrow J_{F_N} \longrightarrow 0,
\end{equation}
where $\widetilde{H}_{\mathbb{K}_0}^*(F_N^-, \partial F_N^-)$ and $J_{F_N}$ are respectively the cokernel and kernel of the homomorphism.
\begin{equation}
H_{\mathbb{K}_0}^* (\Lambda_N , \partial \Lambda_N) \to H_{\mathbb{K}_0}^*(F_N^-, \partial F_N^-).
\end{equation}
Since $\Lambda_N / \partial \Lambda_N$ is homeomorphic to a sphere with the trivial torus action, $H_{\mathbb{K}_0}^*(\Lambda_N, \partial \Lambda_N)$ is a free $H_{\mathbb{K}_0}^*(\pt)$-module of rank $1$, generated by the fundamental cocycle of the sphere, and $J_{F_N}$ is a submodule of this module. 
\begin{propsec}[prop. 5.2, GIV]
Let $\hat{F}_{N+K} = \mathcal{F}_N \oplus_{\Lambda_N} \mathcal{G}_m^{(K)}$ be the fiberwise direct sum of the family $\mathcal{F}_N$ with the Hermitian form $\mathcal{G}_m^{(K)}$, and let $\hat{F}_{N+K}^-$ and $F_N^-$ denote the corresponding ray spaces. Then there exist natural isomorphisms
\begin{equation}
\begin{array}{lcl}
\widetilde{H}^*_{\mathbb{K}_0}(\hat{F}_{N+K}^-, \partial \hat{F}_{N+K}^-) \simeq \widetilde{H}^*_{\mathbb{K}_0}(F_N^-, \partial F_N^-) \otimes_{H_{\mathbb{K}_0}^*(\pt)} I^{K+m},\\
J_{\hat{F}_{N+K}} \simeq J_{F_N} \otimes_{H_{\mathbb{K}_0}^*(\pt)} I^{K+m},
\end{array}
\end{equation}
where $I^{K+m}$ is the principal ideal in $H_{\mathbb{K}_0}^*(\pt)$ generated by the monomial $u^{K+m} := u_1^{K+m}u_2^{K+m}...u_n^{K+m}$.
\end{propsec}




\section{The cohomology}

Based on diagrams (\ref{eq9}) and (\ref{eq10}), we can now define our filtrated cohomology theory.
\begin{defsec}
For $\nu \notin \Spec(\phi_h)$, we define
\begin{equation}
\begin{array}{lcl}
HF_{\leq \nu}^*(\widetilde{\phi}_h) := \underset{\underset{N}{\longrightarrow}}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_N(\nu), \partial F_N(\nu)),\\
HF^*(V) := \underset{\underset{\nu}{\longleftarrow}}{\lim} HF_{\leq \nu}^*(\widetilde{\phi}_h).
\end{array} 
\end{equation}
\end{defsec}

\begin{propsec}{\label{prop31}}
The cohomology groups $HF_{\leq \nu}^*(\widetilde{\phi}_h)$ are well-defined.
\begin{proof}
Let $\lbrace \phi_s^t \rbrace_{s,t \in [0,1]}$ be a contact isotopy  between two representatives 
\begin{equation}
\lbrace \phi_0^t \rbrace_{t \in [0,1]}, \lbrace \phi_1^t \rbrace_{t \in [0,1]}
\end{equation}
of the same class in $\Cont(V)$, that is
\begin{equation}{\label{eq11}}
\phi_s^1 = \phi_0^1 = \phi_1^1, \text{ for any } s \in [0,1].
\end{equation}
It is generated by a $C^{1,1}$ family $h : V \times [0,1]^2 \to \mathbb{R}$ of contact Hamiltonians $h^s$, such that $\Spec(\phi_h^s)$ does not depend on $s$, by (\ref{eq11}). Since each step in the process of lifting $h^s$ to a Hamiltonian $H^s$ can be done smoothly, and by (\ref{eq12}) and proposition \ref{prop7}, one can find $N$ such that for any $N' \geq N$ there exists a $C^{1,1}$ $\mathbb{K}_0$-equivariant isotopy $I_{N'} : S_{N'} \times \Lambda_{N'} \times [0,1] \to S_{N'} \times \Lambda_{N'}$ such that
\begin{equation}
I_{N',s}(F_{N',0}^{\pm}(\nu), \partial F_{N',0}^{\pm}(\nu)) = (F_{N',s}^{\pm}(\nu), \partial F_{N',s}^{\pm}(\nu)),
\end{equation}
where $\mathcal{F}_{N',s}$ is the generating function associated with the decomposition of the Hamiltonian lifts
\begin{equation}
\exp(\lambda) \circ \widetilde{\Phi_{\overline{h^s}}}
\end{equation}
into $2N'$ small Hamiltonian diffeomorphisms. This way we obtain, for any $N' \geq N$, isomorphisms
\begin{equation}
H_{\mathbb{K}_0}^*(F_{N',0}^-(\nu), \partial F_{N',0}^-(\nu)) \simeq H_{\mathbb{K}_0}^*(F_{N',1}^-(\nu), \partial F_{N',1}^-(\nu)),
\end{equation}
of $\mathbb{K}_0$-equivariant cohomology groups, which of course commute with the horizontal map of diagram $(\ref{eq9})$. At the limit, this yields an isomorphism
\begin{equation}
\underset{\underset{N}{\longrightarrow}}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_{N,0}(\nu), \partial F_{N,0}(\nu)) \simeq \underset{\underset{N}{\longrightarrow}}{\lim} H_{\mathbb{K}_0}^{*+2nN}(F_{N,1}(\nu), \partial F_{N,1}(\nu)).
\end{equation}
\end{proof}
\end{propsec}

\begin{remarksec}
\normalfont Note that the above proof shows in particular the independence of $HF_{\leq \nu}^*(\widetilde{\phi}_h)$ on the decomposition of the lift $\widetilde{\Phi_{\overline{h}}}$, since the spectrum of the front of any reparametrization of the associated Hamiltonian will remain unchanged. Similarily, it shows that the groups $HF_{\leq \nu}^*(\widetilde{\phi}_h)$ are independent of the way the function $\tilde{h} : P_{\mathbb{K}} \to \mathbb{R}$ is extended to the sphere $S_p$ (see section \ref{sec5}). Indeed, if $\overline{h}_1$ and $\overline{h}_2$ are two such extentions, then there respective Hamiltonian lifts $H_1,H_2 : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ yield a family
\begin{equation}
H^s := sH_1 + (1-s)H_2 : \mathbb{C}^n \times [0,1] \to \mathbb{R}
\end{equation}
of Hamiltonians on $\mathbb{C}^n$, such that the restriction $H^s_{| P_{\mathbb{K}}^{-1}(p)}$ does not depend on $s$, since $H_1$ and $H_2$ coincide on $P_{\mathbb{K}}^{-1}(p)$. It means that the spectrum $\Phi_{H^s}$ of $H^s$ is independent of $s$, and we can conclude again with proposition \ref{prop7}.
\end{remarksec}
\end{comment}
\begin{comment}
We start by considering a closed manifold $N$, and we endow $T^*N$ with its canonical symplectic form $\omega = \mathrm{d} \lambda$, where $\lambda = p \mathrm{d} q$ is the Liouville form. Consider the action functional of a Hamiltonian $H : T^*N \times [0,1] \to \mathbb{R}$, defined on the space $\mathcal{P}$ of paths in $T^*N$ starting at the zero section:
\begin{equation}
\mathcal{A}_H : \mathcal{P} \to \mathbb{R}, \quad \mathcal{A}_H(\gamma) = \int_0^1 \lambda(\dot{\gamma}(t)) \mathrm{d} t - \int_{0}^1 H_t(\gamma(t)) \mathrm{d} t.
\end{equation} 
For a function $F : N \to \mathbb{R}$, we denote the graph of its differential by $\Gamma_F$, that is
\begin{equation}
\Gamma_F := \lbrace \mathrm{d}_qF | q \in N \rbrace.
\end{equation}
Let $\pi : T^*N \to N$ denote the canonical projection. For $q \in N$, we define $\gamma_q \in \mathcal{P}$ via
\begin{equation}
\gamma_q(t) = \Phi_H^t(q).
\end{equation}

\begin{lemmasec}{\label{lemma31}}
Suppose that $L := \Phi_H(0_N)$ is graphical in $T^*N$, that is the restriction $\pi_{|L} : L \to N$ is a diffeomorphism. Then the function
\begin{equation}
F(q) = \mathcal{A}_H(\gamma_{\Phi_H^{-1}\pi_{|L}^{-1}(q)})
\end{equation}
generates $L$, that is $L = \Gamma_F$.

\begin{proof}
We define $F$ as the following composition:
\begin{equation}
N \xrightarrow{\pi_{|L}^{-1}} L \xrightarrow{\phi_H^{-1}} N \xrightarrow{\gamma} \mathcal{P} \xrightarrow{\mathcal{A}_H} \mathbb{R},
\end{equation}
and therefore its differential is the composition of the differentials of these maps.

The differential of $\mathcal{A}_H$ is given by 
\begin{equation}
\mathrm{d}_{\gamma} \mathcal{A}(\xi) = \langle \lambda_{\gamma(1)}, \xi(1) \rangle - \int_0^1 \omega(\xi(t),X_H(\gamma(t))-\dot{\gamma}(t)) \mathrm{d} t.
\end{equation}

The differential of the third map
\begin{equation}
N \to \mathcal{P}, \quad q \mapsto \gamma_q
\end{equation}
is the linearized map of the Hamiltonian flow of $H$, that is it maps a tangent vector $u \in T_q N$ to the vector field
\begin{equation}
t \mapsto (\Phi_H^t)_*(u) \in T_{\gamma_q(t)} T^*N.
\end{equation}
Since $\gamma_{\phi_H^{-1}(\pi_{|L}^{-1}(q))}$ is a Hamiltonian trajectory, the differential of $\mathcal{A}_H$ at it reduces to the evaluation of $\lambda$ at its endpoint, which is $\pi_{|L}^{-1}(q)$. Thus for the differential of $F$ we have the following: if $u \in T_q N$, then
\begin{equation}
\mathrm{d}_q F(u) = \langle \lambda_{\pi_{|L}^{-1}(q)}, \Phi_{H*}\Phi_{H*}^{-1}\pi_{|L *}^{-1}(u) \rangle = \langle \lambda_{\pi_{|L}^{-1}(q)}, \pi_{|L *}^{-1}(u) \rangle.
\end{equation}
the definition of $\lambda$ implies that this is equal to the value of the functional $\pi_{|L}^{-1}(q) \in T^*_q N$ evaluated on the vector $\pi_* \pi_{|L *}^{-1}(u) = u$, that is
\begin{equation}
\mathrm{d}_q F(u) = \pi_{|L}^{-1}(q)(u),
\end{equation} 
in other words
\begin{equation}
\mathrm{d}_q F = \pi_{|L}^{-1}(q),
\end{equation}
which means that $\mathrm{d}_q F$ is precisely the point of $L$ lying above $q$, that is $F$ generates $L$.
\end{proof}
\end{lemmasec}

Now, for $t \in [0,1]$, we can apply the above construction to the Lagrangian $L_t := \Phi_H^t(0_N)$, obtaining the generating functions $F_t : N \to \mathbb{R}$, provided $L_t$ is graphical over $N$. Let us denote 
\begin{equation}
\pi_t := \pi_{|L_t} : L_t \to N,
\end{equation}
which is a diffeomorphism by assumption.

\begin{propsec}[Hamilton-Jacobi equation]{\label{prop32}}
We have
\begin{equation}
\frac{\partial F_t}{\partial t}(q) = - H_t(\pi_t^{-1}(q)).
\end{equation}
\begin{proof}
Consider $T^*[0,1]$ with coordinates $(t,\tau)$ and Liouville form $\tau \mathrm{d} t$. Define
\begin{equation}
Q = \lbrace  (z;t,-H_t(z)) | t \in [0,1], z \in L_t \rbrace \subset T^*N \times T^*[0,1].
\end{equation}
A direct computation shows that this is a Lagrangian submanifold. We claim that it is exact. First, for $t \in [0,1]$, put $Q_t := Q \cap T^*N \times T_t^*[0,1]$. Note that the projection of $Q_t$ to $T^*N$ (that is forgetting about the $\tau$ coordinate) is exactly $L_t$. Clearly $\tau \mathrm{d} t$ vanishes when restricted to $Q_t$, and we see that the restriction of $\lambda \oplus \tau \mathrm{d} t$ to $Q_t$ equals, via the identification $Q_t \simeq L_t$, the restriction $\lambda_{| L_t}$. In particular for $t=0$, we have $\lambda_{|L_0} = \lambda_N = 0$.

To show that $Q$ is exact it suffices to show that the integral of $\lambda \oplus \tau \mathrm{d} t$ over any loop in $Q$ vanishes. Since $Q$ is Lagrangian, $\lambda \oplus \tau \mathrm{d} t$ is closed when restricted to $Q$, and therefore the integral over any loop only depends on its free homotopy class. Any loop in $Q$ can be deformed into a loop contained entirely in $Q_0$, and so the integral of $\lambda \oplus \tau \mathrm{d} t$ vanishes, because, as we've just seen, the restriction of $\lambda \oplus \tau \mathrm{d} t$ to $Q_0$ is zero.

Since by assumption all the manifolds $L_t$ are graphs over $N$, it follows that $Q$ is a graph over $N \times [0,1]$. Since it is also exact, it follows that we can find a function $G : N \times [0,1] \to \mathbb{R}$ such that
\begin{equation}
(\lambda \oplus \tau \mathrm{d} t)_{|Q} = \mathrm{d} (\Pi^*G)_{|Q},
\end{equation}
where $\Pi : T^*N \times T^*[0,1] \to N \times [0,1]$ is the projection. The primitive of an exact $1$-form is determined up to a constant, and we normalize the choice of $G$ as follows. We have 
\begin{equation}
0 = (\lambda \oplus \tau \mathrm{d} t)_{|Q_0} = \mathrm{d} (\Pi^*G)_{|Q_0},
\end{equation}
meaning that $G$ is constant on the slice $N \times \lbrace 0 \rbrace$, and we normalize $G$ by setting this constant to zero. We claim that with this normalization we have
\begin{equation}
G(q,t) = F_t(q).
\end{equation}
Indeed, fix $q \in N$ and consider the path $s \mapsto (\Phi_H^s(q); s, -H_s(\Phi_H^s(q))) \in Q$, for $s \in [0,t]$. A direct computation shows that the integral of $\lambda \oplus \tau \mathrm{d} t$ over it for $s \in [0,t]$ equals the action of the path $s \mapsto \Phi_H^s(q) \in T^*N$ (for $s \in [0,t]$), which, as we have seen, equals $F_t(\pi(\Phi_H^t(q)))$. On the other hand, using the fact that $(\lambda \oplus \tau \mathrm{d} t)_{|Q} = \mathrm{d} (\Pi^* G)_{|Q}$ and the above normalization, we see that the integral also equals $G(\pi(\Phi_H^t(q)),t)$, meaning 
\begin{equation}
G(\pi(\Phi_H^t(q)),t) = F_t(\pi(\Phi_H^t(q))).
\end{equation}
Substituing $q = (\Phi_H^{t})^{-1} \pi_t^{-1}(x)$ for $x \in N$, we obtain the desired result $G(x,t) = F_t(x)$.

The tangent space to $Q$ at a point $(z; t, -H_t(z)) = (\Phi_H^t(x); t, -H_t(\Phi_H^t(x))) \in Q$, $x \in N$ is 
\begin{equation}
T_{(z;t,-H_t(z))}Q = \lbrace (\Phi^t_{H*}(Z) + T X_H^t(\Phi_H^t(x)); T, - \mathrm{d}_{x}(H_t \circ \Phi_H^t)(Z) - T \frac{\partial H_t}{\partial t}(\Phi_H^t(x))) | T \in \mathbb{R}, Z \in T_x N \rbrace.
\end{equation}
Setting $Z = - \Phi_H^{t*} \pi_t^* \pi_* X_H^t(z) \in T_x N$, and $T=1$, we get on one hand
\begin{equation}
\begin{array}{lcl}
(\lambda \oplus \tau \mathrm{d} t)_{(z;t,-H_t(z))}(\Phi_{H*}^tZ + X_H^t(z); 1, - \mathrm{d}_{z} H_t(\Phi_{H*}^tZ) - \frac{\partial H_t}{\partial t}(z)) = & \\ \\
\underbrace{\lambda_z(\Phi_{H*}^tZ + X_H^t(z))}_{=0 \text{ since $\lambda_{(q,p)} = p \circ \pi_*$}} - H_t(z) =  -H_t(z),
\end{array}
\end{equation}
and on the other hand
\begin{equation}
\begin{array}{lcl}
\mathrm{d} (\Pi^* G)(\Phi_{H*}^tZ + X_H^t(z); 1, - \mathrm{d}_{z} H_t(\Phi_{H*}^tZ) - \frac{\partial H_t}{\partial t}(z)) = \\ \\
\mathrm{d}_{(\pi(z),t)} G (\pi_*(\Phi_{H*}^tZ + X_H^t(z)), 1) = \\ \\
\mathrm{d}_{(\pi(z),t)} G (0,1) = \frac{\partial G}{\partial t} (\pi(z),t).
\end{array}
\end{equation}
We therefore obtain:
\begin{equation}
\frac{\partial F_t}{\partial t}(\pi(z)) = \frac{\partial G}{\partial t}(\pi(z),t) = - H_t(z),
\end{equation}
as claimed.
\end{proof}
\end{propsec}
\end{comment}
\begin{comment}
Let $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ be a Hamiltonian. We have
\begin{equation}
\Phi_{0 \oplus H}^t = Id_{\overline{\mathbb{C}^n}} \times \Phi_H^t,
\end{equation}
and therefore
\begin{equation}
\Psi(Gr_{\Phi_H^t}) = \Psi(\Phi_{0 \oplus H}^t(\Delta)) = \Phi_G^t(0_{\mathbb{C}^n}),
\end{equation}
where $G := \Psi^{-1*}(0 \oplus H)$. Suppose that $H$ is positive. Then so is $G$, and if $\Phi_H^t$ is close enough to the identity, the Hamilton-Jacobi equation implies that there exists a \textit{positive} function 
\begin{equation}
F_t : \mathbb{C}^n \to \mathbb{R}
\end{equation} 
generating $\Psi(Gr_{\Phi_H^t})$. In particular, given any homogeneous positive Hamiltonian $H : \mathbb{C}^n \times [0,1] \to \mathbb{R}$, a decomposition of $\Phi_H$ into small Hamiltonian symplectomorphisms $\Phi_i$ will yield positive generating functions $\mathcal{H}_i : \mathbb{C}^n \to \mathbb{R}$ (recall that two functions generating the same graph differ by a constant, and here this constant is $0$, by homogeneity).

\begin{lemmasec}{\label{lemma32}}
Given a positive contact Hamiltonian $h : V \times [0,1] \to \mathbb{R}$, and $\nu$ generic, there is a natural homomorphism
\begin{equation}
HF^*_{\leq \nu}(\widetilde{\phi}_h) \to HF^*_{\leq \nu}(\widetilde{\phi}_0).
\end{equation}
\begin{proof}
Let us lift $h$ to a positive Hamiltonian $\widetilde{H} : \mathbb{C}^n \times [0,1] \to \mathbb{R}_{>0}$. By the discussion above, if $N$ is big enough and $\lambda \in \mathfrak{k}$, the generating function $\mathcal{F}_{\lambda}^{(N)}$ associated with a decomposition of $\exp(\lambda) \circ \Phi_{\widetilde{H}}$ into $2N$-parts is strictly smaller than the generating function $\mathcal{G}^{(N)}_{\lambda} : \mathbb{C}^n \to \mathbb{R}$ associated with a decomposition of $ \exp(\lambda)$ into $2N$-parts. In particular, we have a natural homomorphism
\begin{equation}
H^{*+2nN}_{\mathbb{K}_0}(F_N^-(\nu),\partial F_N^-(\nu)) \to  H^{*+2nN}_{\mathbb{K}_0}(G_N^-(\nu), \partial G_N^-(\nu)),
\end{equation}
which at the limit yield the result.
\end{proof}
\end{lemmasec}

\begin{propsec}
The cohomology groups $HF^*(V)$ are well-defined.

\begin{proof}
Let us begin by noticing that, on every contact manifold $(V,\xi := \ker (\alpha))$, and for any two contact Hamiltonians $h, h' : V \times [0,1] \to \mathbb{R}$ such that $h < h'$, there exists a positive contact Hamiltonian $g : V \times [0,1] \to \mathbb{R}$ such that 
\begin{equation}
\widetilde{\phi}_{h'} = \widetilde{\phi}_g \widetilde{\phi}_h = \widetilde{\phi}_{h \# g} \in \Cont(V),
\end{equation}  
where $h \# g$ is the concatenation
\begin{equation}
(h \# g)_t(x) = \begin{cases}
               2h_{2t}(x) \text{ if } t \in [0, \frac{1}{2}]\\
               2g_{2t-1}(x) \text{ if } t \in [\frac{1}{2}, 1]\\ 
               \end{cases}.
\end{equation}

Let $\widetilde{\phi}_h$ and $\widetilde{\phi}_{h'}$ be two classes in $\Cont(V)$, and $l : V \times [0,1] \to \mathbb{R}$ be the contact Hamiltonian such that
\begin{equation}
\widetilde{\phi}_{h'} = \widetilde{\phi}_l \widetilde{\phi}_h = \widetilde{\phi}_{h \# l}.
\end{equation}
We claim that we can find $c > 0$ such that there exists a homomorphism
\begin{equation}
HF_{\leq \nu}^*(\widetilde{\phi}_{h'}) \to  HF^*_{ \leq \nu + c}(\widetilde{\phi}_h).
\end{equation}
Choose a constant $c \in \mathbb{R}$ which is stricly greater than $l$. By the remark above, there exists a positive contact Hamiltonian $g : V \times [0,1] \to \mathbb{R}$ such that 
\begin{equation}
\widetilde{\phi}_c = \widetilde{\phi}_{l \# g}.
\end{equation}
On one hand, we have isomorphisms
\begin{equation}
HF_{\leq \nu}^*(\widetilde{\phi}_{h \# l \# g}) \simeq HF_{\leq \nu}^*(\widetilde{\phi}_{h \# c}) \simeq HF_{\leq \nu + c}^*(\widetilde{\phi}_h).
\end{equation}
On the other hand, the above lemma gives a homomorphism
\begin{equation}{\label{eq20}}
HF_{\leq \nu}^*(\widetilde{\phi}_{h \# l \# 0}) \to HF_{\leq \nu}^*(\widetilde{\phi}_{h \# l \# g}).
\end{equation}
Combining all, we get a homomorphism
\begin{equation}
HF^*_{\leq \nu}(\widetilde{\phi}_{h'}) \to  HF_{\leq \nu + c}^*(\widetilde{\phi}_h),
\end{equation}
Since all the maps involved are natural in cohomology, we conclude that for any three contact Hamiltonians $h,k,l$, any generic $\nu \in \mathbb{R}$, there exist $c_1, c_2 > 0$ such that the following diagram commutes:
\begin{equation}
\begin{tikzcd}
& HF_{\leq \nu + c_2}^*(\widetilde{\phi}_l)\\
HF_{\leq \nu}^*(\widetilde{\phi}_h) \arrow[r] \arrow[ur] & HF_{\leq \nu + c_1}^*(\widetilde{\phi}_k) \arrow[u].
\end{tikzcd}
\end{equation}
In particular, we have a natural isomorphism
\begin{equation}
\underset{\nu \to \infty}{\lim} HF_{\leq \nu}^*(\widetilde{\phi}_h) \simeq \underset{\nu \to \infty}{\lim} HF_{\leq \nu}^*(\widetilde{\phi}_k),
\end{equation}
as claimed.

\end{proof}
\end{propsec}
\end{comment}

\section{The Gysin sequence}{\label{sec4}}

In the previous section we have seen two algebraic actions on the kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, namely of the ring $H_{\mathbb{T}^n}^*(\pt)$, and the Novikov action of $H_2(M,\mathbb{Z})$. In order to prove Proposition \ref{prop1.2.3}, which is the last result needed to complete the proof of Theorem \ref{theo1.1.1}, we shall describe $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ in a more instrinsic way, at least when the homogeneous generating family $\mathcal{F}_N$ is associated with a decomposition of the Hamiltonian symplectomorphism $\Phi_0 = Id_{\mathbb{C}^n}$, $0$ being a Hamiltonian lift of the trivial contact Hamiltonian $0$ on $V$. In this case, up to a reparametrization of the decomposition, the function $\mathcal{F}_{\lambda}^{(N)}$ is simply the generating function $\mathcal{G}_{\lambda}^{(N)}$ associated with the decomposition
$$\underbrace{\exp(\frac{\lambda}{2N}) \circ ... \exp(\frac{\lambda}{2N})}_{2N-\text{times}},$$
as defined in section \ref{sec3.3.1}, so we shall use the notations $\mathcal{G}_N := \mathcal{F}_N$ and $G_N := F_N$. Note that for any $\lambda \in \Lambda_N$, $\mathcal{G}_{\lambda}^{(N)}$ is $\mathbb{K}$-invariant, so that one can consider either the $\mathbb{K}_0$-equivariant cohomology groups $H_{\mathbb{K}_0}^*(G_N^-(\nu), \partial G_N^-(\nu))$, or the $\mathbb{K}$-equivariant cohomology groups $H_{\mathbb{K}}^*(G_N^-(\nu),\partial G_N^-(\nu))$. In this last case, the functions $f_N^{N+K}$ from section \ref{sec3.6} can still be constructed in the exact same way as above, and one can define limits of $\mathbb{K}$-equivariant cohomology groups:
$$
\begin{array}{lcl}
\mathcal{H}_{\mathbb{K}}^*(G^-(\nu)) := \underset{N \to \infty}{\lim} H_{\mathbb{K}}^{*+2nN}(G_N^-(\nu),\partial G_N^-(\nu))\\ \\
\mathcal{H}_{\mathbb{K}}^*(\nu) := \underset{N \to \infty}{\lim} H_{\mathbb{K}}^{*+2nN}(\Gamma_N(\nu), \partial \Gamma_N(\nu)).
\end{array}
$$
The latter inherit the structure of $H_{\mathbb{K}}^*(\pt)$-modules, and therefore of $H_{\mathbb{T}^n}^*(\pt)$-modules as well, through the surjective homomorphism
$$
H_{\mathbb{T}^n}^*(\pt) \to H_{\mathbb{K}}^*(\pt).
$$
In \cite{Giv1995}, Givental showed that the augmentation map
$$
\Hcal_{\mathbb{K}}^*(\nu) \to \Hcal_{\mathbb{K}}^*(G^-(\nu))
$$
has \textit{trivial cokernel}, and he described its kernel in terms of Newton diagrams associated with the level $p^{-1}(\nu)$ (see \cite[Proposition $5.4$]{Giv1995}). We will see now that there is a Gysin-type long exact sequence relating the cohomology groups $\Hcal_{\mathbb{K}}^*(G^-(\nu))$ of Givental \cite{Giv1995} to our cohomology groups $\Hcal_{\mathbb{K}_0}^*(G^-(\nu))$. Gysin sequences were already used in symplectic topology to relate different kinds of Floer-type homologies, for instance in \cite{Per2008}, \cite{BO2013}, \cite{BK2013}. We give here another example of such a use, and by further analyzing the maps involved, we will be able to describe the kernel of the augmentation map $\Hcal_{\mathbb{K}_0}^*(\nu) \to \Hcal_{\mathbb{K}_0}^*(G^-(\nu))$. Recall that given any oriented $S^1$-bundle $\pi : E \to M$, the cohomology groups of $E$ and $M$ are related by the \textit{Gysin long exact sequence}
$$
\hdots \longrightarrow H^*(M) \overset{\cup eu}{\longrightarrow} H^{*+2}(M) \overset{\pi^*}{\longrightarrow}  H^{*+2}(V) \overset{\pi_*}{\longrightarrow} H^{*+1}(M) \longrightarrow \hdots,
$$
where $eu \in H^2(M)$ is the Euler class of the bundle $\pi$, $\cup$ denotes the cup-product, $\pi^*$ is the pull-back and $\pi_*$ the push-forward.

For the sake of clarity, we will use the notation $H_{\mathbb{T}^n}^*(\pt) \simeq \mathbb{C}[u]$, where $u = (u_1,...,u_n)$. Let $I$ (resp. $I_0$) denote the ideal of $\mathbb{C}[u]$ generated by polynomials vanishing on the complexified Lie algebra $\mathfrak{k} \otimes \mathbb{C} \subset \mathbb{C}^n$ (resp. $\mathfrak{k}_0 \otimes \mathbb{C} \subset \mathbb{C}^n$). There are natural isomorphisms
$$
H_{\mathbb{K}}^*(\pt) \simeq \mathbb{C}[u] / I \quad \text{and} \quad H_{\mathbb{K}_0}^*(\pt) \simeq \mathbb{C}[u] / I_0.
$$
In other words, $H_{\mathbb{K}}^*(\pt)$ (resp. $H_{\mathbb{K}_0}^*(\pt)$) is the ring of regular functions on $\mathfrak{k} \otimes \mathbb{C}$ (resp. $\mathfrak{k}_0 \otimes \mathbb{C}$). Recall that the ring of regular functions on the complex torus $(\mathbb{C}^{\times})^n$ is given by $\mathbb{C}[u,u^{-1}]$. Let us denote $\mathcal{R}$ and $\mathcal{R}_0$ the rings of regular functions on the intersections $(\mathfrak{k} \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$ and $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$ respectively. We have
\begin{center}
$\mathcal{R} = \mathbb{C}[u] / I \otimes \mathbb{C}[u,u^{-1}] \simeq \mathbb{C}[u,u^{-1}] / I \mathbb{C}[u,u^{-1}]$ \\ and \\ $\mathcal{R}_0 = \mathbb{C}[u] / I_0 \otimes \mathbb{C}[u,u^{-1}] \simeq \mathbb{C}[u,u^{-1}] / I_0 \mathbb{C}[u,u^{-1}]$.
\end{center}
Let $J^*(\nu)$ denote the $\mathbb{C}[u]$-submodule of $\mathbb{C}[u,u^{-1}]$ generated by monomials whose degrees lie in $\mathfrak{k}_{\mathbb{Z}}$ above the level $p^{-1}(\nu)$:
\begin{equation}{\label{eq10}}
J^*(\nu) := \langle u^{\iota(m)} \ |\  m \in \mathfrak{t}_{\mathbb{Z}}, \ p(m) \geq \nu \rangle, \quad \iota(m)=(m_1,...,m_n),
\end{equation}
and let $\mathcal{J}_{\mathbb{K}}^*(\nu)$ denote its projection to $\mathcal{R}$. Givental proved the following
\begin{propsec}[\cite{Giv1995} Proposition $5.4$]{\label{prop4.1}}
There are isomorphisms
$$
\Hcal_{\mathbb{K}}^*(\nu) \simeq \mathcal{R} \quad \text{and} \quad \Hcal_{\mathbb{K}}^*(G^-(\nu)) \simeq \mathcal{R} / \mathcal{J}_{\mathbb{K}}^*(\nu).
$$
\end{propsec}

We denote $\mathcal{J}_{\mathbb{K}_0}^*(\nu)$ the projection of $J^*(\nu)$ to $\mathcal{R}_0$. We claim that $\mathcal{J}_{\mathbb{K}_0}^*(\nu)$ is the kernel $\mathcal{J}_{\mathbb{K}_0}^*(G^-(\nu))$ of the augmentation map
$$
\Hcal_{\mathbb{K}_0}^*(\nu) \to \Hcal_{\mathbb{K}_0}^*(G^-(\nu)).
$$
Note that if $EG \to BG$ denotes the universal bundle associated with a Lie group $G$, then $E \mathbb{K}_0$ may be thought as $E \mathbb{K}$, since $\mathbb{K}_0$ acts freely on the latter and it is contractible. Therefore, one can define the homotopy quotient $(G_N^-(\nu))_{\mathbb{K}_0}$ by
$$
(G_N^-(\nu))_{\mathbb{K}_0} := (G_N^-(\nu) \times E \mathbb{K}) / \mathbb{K}_0,
$$
and construct the following principal $S^1 := \mathbb{K} / \mathbb{K}_0$-bundle:
$$
\pi_N : (G_N^-(\nu))_{\mathbb{K}_0} \to (G_N^-(\nu))_{\mathbb{K}}.
$$
In relative cohomology, one can then relate the $\mathbb{K}$-equivariant and $\mathbb{K}_0$-equivariant cohomology groups above by the Gysin sequence
$$
\begin{array}{lcl}
\hdots \longrightarrow H_{\mathbb{K}}^{*+2nN}(G_N^-(\nu), \partial G_N^-(\nu)) \overset{\cup eu}{\longrightarrow} H_{\mathbb{K}}^{*+ 2nN +2}(G_N^-(\nu), \partial G_N^-(\nu)) \overset{\pi_N^*}{\longrightarrow} \\ \\
\quad \quad \quad  H_{\mathbb{K}_0}^{*+ 2nN +2}(G_N^-(\nu), \partial G_N^-(\nu)) \overset{\pi_{N*}}{\longrightarrow} H_{\mathbb{K}}^{*+ 2nN +1}(G_N^-(\nu), \partial G_N^-(\nu)) \longrightarrow \hdots.
\end{array}
$$
Let us interpret the Euler class $eu$ in this sequence. The classifying map of the bundle $\pi_N$ is of the form
$$
l_N : (G_N^-(\nu))_{\mathbb{K}} \to B(\mathbb{K} / \mathbb{K}_0).
$$
Moreover, the group $H^*(B(\mathbb{K}/\mathbb{K}_0))$ can be identified with the polynomial algebra $\mathbb{C}[(\mathfrak{k} / \mathfrak{k}_0)^*] \simeq \mathbb{C}[p]$ (here $p$ is assigned the degree $2$, and is viewed as a $\mathbb{C}$-valued functional on $\mathfrak{k} \otimes \mathbb{C}$). In relative equivariant cohomology, one gets a map
$$
l_N^* : \mathbb{C}[p] \to H_{\mathbb{K}}^*(G_N^-(\nu), \partial G_N^-(\nu)).
$$
The Euler class in the above Gysin sequence is thus given by the pull-back $l_N^*(p)$ of the regular value $p$ by the classifying map $l_N$. The other two maps involved are the pull-back $\pi_N^*$ and the push-forward $\pi_{N*}$. The latter are canonical, and therefore \textit{they commute with the limit $N \to \infty$}. Moreover, the cup-product by the Euler class is \textit{functorial}. Since directs limits of exact sequences are exact sequences, we obtain a Gysin sequence in the limit $N \to \infty$:
$$
\hdots \longrightarrow \Hcal_{\mathbb{K}}^*(G^-(\nu)) \longrightarrow \Hcal_{\mathbb{K}}^{*+2}(G^-(\nu)) \longrightarrow \Hcal_{\mathbb{K}_0}^{*+2}(G^-(\nu)) \longrightarrow \Hcal_{\mathbb{K}}^{*+1}(G^-(\nu)) \longrightarrow \hdots.
$$
The same applies for the equivariant cohomology groups of the pair $(\Gamma_N(\nu), \partial \Gamma_N(\nu))$, for which the Gysin sequence in the limit is given by 
\begin{equation}{\label{eq11}}
\hdots \longrightarrow \Hcal_{\mathbb{K}}^*(\nu) \longrightarrow \Hcal_{\mathbb{K}}^{*+2}(\nu) \longrightarrow \Hcal_{\mathbb{K}_0}^{*+2}(\nu) \longrightarrow \Hcal_{\mathbb{K}}^{*+1}(\nu) \longrightarrow \hdots.
\end{equation}
\begin{propsec}{\label{prop4.2}}
There is a natural isomorphism
$$
\Hcal_{\mathbb{K}_0}^*(\nu) \simeq \mathcal{R}_0.
$$
\begin{proof}
By Proposition $\ref{prop4.1}$, $\Hcal_{\mathbb{K}}^*(\nu)$ is the ring $\mathcal{R}$ of regular functions on the intersection $(\mathfrak{k} \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$. The latter is an irreductible variety, and thus there are no zero divisors in $\Hcal_{\mathbb{K}}^*(\nu)$. In particular, the map induced in the limit by the cup-product with the Euler class is injective. This reduces the Gysin sequence $(\ref{eq11})$ to the following short exact sequence
$$
0 \longrightarrow \Hcal_{\mathbb{K}}^*(\nu) \longrightarrow \Hcal_{\mathbb{K}}^{*+2}(\nu) \longrightarrow \Hcal_{\mathbb{K}_0}^{*+2}(\nu) \longrightarrow 0.
$$
Now, for any $N$, recall that the torus $\mathbb{K}$ acts trivially on the pair $(\Gamma_N(\nu), \partial \Gamma_N(\nu))$. This makes $H_{\mathbb{K}}^*(\Gamma_N(\nu), \partial \Gamma_N(\nu))$ into a free $H_{\mathbb{K}}^*(\pt)$-module of rank $1$, generated by the fundamental cocycle of the sphere $\Gamma_N(\nu) / \partial \Gamma_N(\nu)$ (see section \ref{sec3.6}). The same applies for the action of $\mathbb{K}_0$, and therefore the principal $S^1$-bundle to be considered here reduces to
$$
B \mathbb{K}_0 \to B \mathbb{K}.
$$
In cohomology, this map becomes $\mathbb{C}[\mathfrak{k}^*] \to \mathbb{C}[\mathfrak{k}_0^*]$. For the map $l^* : \mathbb{C}[\mathfrak{k}^*] \to \mathbb{C}[p]$ induced in cohomology by the classifying map in this case, the Euler class $l^*(p)$ is simply the generator of the kernel $\mathfrak{k}^* \to \mathfrak{k}_0^*$, that is $p$. Thus in $\Hcal_{\mathbb{K}}^*(\nu)$, the map induced by the cup-product with the Euler class is simply the multiplication by $p$. By the very definition of $I_0$ and $I$, one gets
$$
\Hcal_{\mathbb{K}_0}^*(\nu) \simeq \mathcal{R} / p \mathcal{R} \simeq \mathcal{R}_0.
$$ 
\end{proof}
\end{propsec}

\begin{remarksec}
\normalfont Notice that if $M = \mathbb{C} P^n$, then $\mathfrak{k} = \mathbb{R}$, and $\mathfrak{k}_0 = \{0\}$. Therefore $I_0 = \mathbb{C}[u,u^{-1}]$, and $\mathcal{R}_0 = \{0\}$. Moreover, the Euler class $p$ in this case is the generator $v$ of $\mathcal{R} \simeq \mathbb{C}[v,v^{-1}]$, and thus it is invertible. Therefore the Gysin sequence in the limit $N \to \infty$ gives $\Hcal_{\mathbb{K}_0}^*(\nu) = \{0\}$.
\end{remarksec}

By the above discussion, we have the following commutative diagram
$$
\begin{tikzcd}
   &          &          &        \vdots    \ar[d]                    & \\ 
   &  0 \ar[d] & 0 \ar[d] & \Hcal_{\mathbb{K}_0}^{*+1}(G^-(\nu)) \ar[d] & \\ 
  0 \ar[r] & \mathcal{J}_{\mathbb{K}}^*(\nu) \ar[r] \ar[d,"p"] & \Hcal_{\mathbb{K}}^*(\nu) \ar[r] \ar[d,"p"] & \Hcal_{\mathbb{K}}^*(G^-(\nu)) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \mathcal{J}_{\mathbb{K}}^{*+2}(\nu) \ar[r] \ar[d,"f"] & \Hcal_{\mathbb{K}}^{*+2}(\nu) \ar[r] \ar[d,"g"] & \Hcal_{\mathbb{K}}^{*+2}(G^-(\nu)) \ar[r] \ar[d,"h"] & 0 \\
0 \ar[r] & \mathcal{J}_{\mathbb{K}_0}^{*+2}(G^-(\nu)) \ar[r] \ar[d, dotted, red] & \Hcal_{\mathbb{K}_0}^{*+2}(\nu) \ar[r] \ar[d] & \Hcal_{\mathbb{K}_0}^{*+2}(G^-(\nu)) \ar[d] \\
&  0  & 0  & \Hcal_{\mathbb{K}}^{*+1}(G^-(\nu)) \ar[d]. \\
& & & \vdots
\end{tikzcd}
$$
The vertical right and middle sequences are Gysin sequences, and the horizontal sequences are induced by the augmentation maps. Note that the left vertical sequence is not necessarily exact at the middle term. The diagram is clear from the discussion above, except for the surjectivity of the map $f$. From the two bottom rows of the diagram and the snake lemma, we have the following short exact sequence:
$$
0 \longrightarrow \ker f \longrightarrow \ker g \longrightarrow \ker h \longrightarrow \text{coker} f \longrightarrow 0.
$$
The homomorphism $\ker g \to \ker h$ is simply the surjection $\cup l^*(p) \Hcal_{\mathbb{K}}^*(\nu) \to \cup l^*(p) \Hcal_{\mathbb{K}}^*(G^-(\nu))$, where $\cup l^*(p)$ denotes the map induced in the limit $N \to \infty$ by the cup-product with the Euler class (on $\Hcal_{\mathbb{K}_0}^*(\nu)$, as we've seen above, it is the multiplication by $p$). We conclude that $f$ is onto. In particular, the kernel $\mathcal{J}_{\mathbb{K}_0}^*(G^-(\nu))$ is the image of $\mathcal{J}_{\mathbb{K}}^*(\nu) \subset \Hcal_{\mathbb{K}}^*(\nu)$ under the map $g$. In other words, it is the projection to $\mathcal{R}_0$ of the module $J^*(\nu)$:
$$
\mathcal{J}_{\mathbb{K}_0}^*(G^-(\nu)) \simeq \mathcal{J}_{\mathbb{K}_0}^*(\nu).
$$

\section{Elements of minimal degree in the monotone case}{\label{sec5}}

The proof of Theorem \ref{theo1.1.1} relies on a strong algebraic property of the kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$. In the $\mathbb{K}$-equivariant case, Givental showed that the kernel $\mathcal{J}_{\mathbb{K}}^*(F^-(\nu))$ of the augmentation map
$$
\mathcal{R} \simeq \Hcal_{\mathbb{K}}^*(\nu) \to \Hcal_{\mathbb{K}}^*(F^-(\nu))
$$
admits, in some sense, elements of minimal degree ({\cite[Corollary $1.3$]{Giv1995}}). It appears that this is not always true for the $\mathbb{K}_0$-equivariant kernel $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ when the manifold is a prequantization space over a toric manifold. For instance, if $(M,\omega)$ is $\mathbb{C} P^1 \times \mathbb{C} P^1$ endowed with the \textit{non-monotone} symplectic form whose cohomology class is, say, $p = (1,2) \in \mathbb{Z}^{2*} \simeq \mathfrak{k}_{\mathbb{Z}}^*$, the prequantization space $(V,\xi := \ker \alpha)$ and the cohomology group $\Hcal_{\mathbb{K}_0}^*(F^-(\nu))$ can be constructed as above, but the kernel $\mathcal{J}_{\mathbb{K}_0}^*(\nu)$ of the augmentation map
$$
\mathcal{R}_0 \to \Hcal_{\mathbb{K}_0}^*(G^-(\nu))
$$
in this case is the whole ring $\mathcal{R}_0$. In this section, we prove that a similar statement as \cite[Corollary $1.3$]{Giv1995} holds for $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, provided that the toric manifold $(M,\omega)$ is \textit{monotone}. To that aim, we start by bounding $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ from below and above with the projections to $\mathcal{R}_0$ of two modules of a form similar to that from equation $(\ref{eq10})$. This simplifies the analysis of $\mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ to that of much explicit modules. The proof of Proposition \ref{prop1.2.3} is then a careful analysis of the degrees of the polynomials in these submodules. 

Assume that $(M,\omega)$ is \textit{monotone}. Recall that through the isomorphism $H^2(M,\mathbb{R}) \simeq \mathfrak{k}^*$ of Remark \ref{rem3.1.1}, the regular value $p$ represents the cohomology class of $\omega$. Suppose that $p$ is \textit{integral} and \textit{primitive}, and let $N_M$ denote the minimal Chern number of $M$. Then we have 
$$
p = \frac{c}{N_M}.
$$
Let $(V,\xi := \ker \alpha)$ be the prequantization space over $(M,\omega)$ constructed in section \ref{sec3.1.3}, $h$ be a contact Hamiltonian on $V$, and $\overline{h}$ a lift of $h$ to the sphere $S_p$ from equation $(\ref{eq8})$. Given two constants $c_-,c_+ \in \mathbb{R}$ such that $c_- < \overline{h} < c_+$, we denote $H_-$, $H_+$ be the quadratic Hamiltonians on $\mathbb{C}^n$ generating respectively the Hamiltonian symplectomorphisms 
$$
\Phi_{H_-}(z) = \exp(c_-\frac{b}{p(b)})z \quad \text{and} \quad \Phi_{H_+}(z) = \exp(c_+\frac{b}{p(b)})z.
$$
If $\widetilde{H}$ is the Hamiltonian lift of $h$ obtained from extending $\overline{h}$ to $\mathbb{C}^n$, we have naturally
$$
H_- < \widetilde{H} < H_+,
$$
and for $\nu_- := \nu + c_- < \nu < \nu_+ := \nu + c_+$, this yields inclusions of pairs 
$$
(G_N^-(\nu_-), \partial G_N^-(\nu_-)) \subset (F_N^-(\nu), \partial F_N^-(\nu)) \subset (G_N^-(\nu_+), \partial G_N^-(\nu_+)),
$$
and thus homomorphisms
$$
H_{\mathbb{K}_0}^*(G_N^-(\nu_+), \partial G_N^-(\nu_+)) \to H_{\mathbb{K}_0}^*(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^*(G_N^-(\nu_-), \partial G_N^-(\nu_-)).
$$
If $\nu_-, \nu$ and $\nu_+$ are generic, we obtain homomorphisms in the limit $N \to \infty$:
$$
\Hcal_{\mathbb{K}_0}^*(G-(\nu_+)) \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu)) \to \Hcal_{\mathbb{K}_0}^*(G^-(\nu_-)),
$$
which lead to inclusions of kernels
$$
\mathcal{J}_{\mathbb{K}_0}^*(\nu_+) \subset \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \subset \mathcal{J}_{\mathbb{K}_0}^*(\nu_-).
$$
Note that as in \cite[equation $(5.5)$]{Giv1995}, for any $\nu \in \mathbb{R}$, the $\mathbb{C}[u]$-module $J^*(\nu)$ from equation $(\ref{eq10})$ lies between two modules of the form
$$
J_r := \langle u^{\iota(m)} \ |\ m \in \mathfrak{k}_{\mathbb{Z}},\ p(m) \geq r \rangle.
$$
More precisely, one can find $r_- < r_+$ such that we have inclusions
$$
J_{r_+} \subset J^*(\nu_+) \subset J^*(\nu_-) \subset J_{r_-}.
$$
The latter lead to inclusions
$$
\mathcal{J}_{r_+}^0 \subset \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \subset \mathcal{J}_{r_-}^0,
$$
where $\mathcal{J}_{r_{\pm}}^0$ denote the images of $J_{r_{\pm}}$ by the quotient map $\mathbb{C}[u,u^{-1}] \to \mathbb{C}[u,u^{-1}] / I_0 \mathbb{C}[u,u^{-1}] \simeq \mathcal{R}_0$.

\begin{proof}{(Proposition \ref{prop1.2.3})}. \textbf{Step $1$}: in Step $5$ we will use a dimensionality result from \cite{Giv1995}, which holds for the ideal $\mathbb{C}[u] \cap J_r + I$. In our case, we will deal with the ideal $\mathbb{C}[u] \cap J_r + I_0$, which we now relate to the module $\mathcal{J}_r^0$. Consider the projection 
$$
\text{pr} : \mathbb{C}[u] \subset \mathbb{C}[u,u^{-1}] \to \mathbb{C}[u,u^{-1}] / I_0\mathbb{C}[u,u^{-1}] \simeq \mathcal{R}_0.
$$ 
For $r \in \mathbb{R}$, the preimage $\text{pr}^{-1}(\mathcal{J}_r^0)$ is the intersection of $\mathbb{C}[u]$ with the preimage of $\mathcal{J}_r^0$ by the quotient map $\mathbb{C}[u,u^{-1}] \to \mathcal{R}_0$, which equals $J_r + I_0 \mathbb{C}[u,u^{-1}]$. Therefore, we have
$$
\text{pr}^{-1}(\mathcal{J}_r^0) = \mathbb{C}[u] \cap (J_r + \mathbb{C}[u,u^{-1}]I_0) \supset \mathbb{C}[u] \cap J_r + I_0.
$$

\noindent \textbf{Step $2$.} Here we show that all the polynomials in $J_r$ have minimal degree $rN_M$. For $m \in \mathfrak{k}_{\mathbb{Z}}$, we saw that $p(m) = \frac{1}{N_M} \underset{i=1}{\overset{n} \sum} m_i$. Then $J_r$ consists of polynomials whose monomials are of the form $u^{\iota(m)+m'}$, where $m \in \mathfrak{k}_{\mathbb{Z}}$ is such that $p(m) \geq r$, and $m' = (m'_1,...,m'_n) \in \mathbb{Z}^n_{\geq 0}$. In particular, we have
$$
\underset{i=1}{\overset{n} \sum} m_i + \underset{i=1}{\overset{n} \sum} m'_i
\geq \underset{i=1}{\overset{n} \sum} m_i = p(m)N_M \geq rN_M.
$$
Therefore, letting $\mathbb{C}[u,u^{-1}]^{\geq d}$ denote the submodule generated by monomials of total degree at least $d$, we obtain
$$
J_r \subset \mathbb{C}[u,u^{-1}]^{\geq rN_M}.
$$

\noindent \textbf{Step $3$.} Here we show that, for the notion of degree induced on $\mathcal{R}_0$ by that on $\mathbb{C}[u,u^{-1}]$, the elements of the module $\mathcal{J}_r^0$ have minimal degree $rN_M$. The quotient map $\mathbb{C}[u,u^{-1}] \to \mathbb{C}[u,u^{-1}] / I_0 \mathbb{C}[u,u^{-1}]$ is the restriction map from the ring of regular functions on the complex torus $(\mathbb{C}^{\times})^n$ to the ring of regular functions on the intersection $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$. If $f$ is a homogeneous regular function of degree $d$ on $(\mathbb{C}^{\times})^n$, then $\mathbb{C}^{\times}$ acts on $f$ by $\mu.f = \mu^d f$, for any $\mu \in \mathbb{C}^{\times}$. This characterizes entirely the degree of $f$. Moreover, $\mathbb{C}^{\times}$ acts on the ring of regular functions on $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$ in the same way, and the restriction is equivariant with respect to this action. This means that $f$ restricts to a regular function on $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$ which is of same degree, or equals $0$. Thus, if $\mathcal{R}_0^{\geq d}$ denotes the ring of regular functions of degree at least $d$ on $(\mathfrak{k}_0 \otimes \mathbb{C}) \cap (\mathbb{C}^{\times})^n$, we have
$$
\mathcal{J}_r^0 \subset \mathcal{R}_0^{\geq rN_M}.
$$

\noindent \textbf{Step $4$.} It is clear from the definition of $J_r$ that for any $b \in \mathfrak{k}_{\mathbb{Z}}$, we have $u^b J_r = J_{r+r_0}$, where $r_0 = p(b)$. In particular
$$
u^b \mathcal{J}_r^0 = \mathcal{J}_{r+r_0}^0.
$$
Therefore we can "move $\mathcal{J}_r^0$ above a certain minimal degree". This will serve us in Step $5$. Pick $b \in \mathfrak{k}_{\mathbb{Z}}$ such that $(r_- + r_0)N_M \geq 1$. Then
$$
u^b \mathcal{J}_{r_-}^0 = \mathcal{J}_{r_- + r_0}^0 \subset \mathcal{R}_0^{\geq (r_-+r_0)N_M} \subset \mathcal{R}_0^{\geq 1}.
$$
In particular $1 \notin u^b \mathcal{J}_{r_-}^0$, which means that $1 \in \mathbb{C}[u]$ is not mapped to $u^b \mathcal{J}_{r_-}^0$ by the projection $\text{pr} : \mathbb{C}[u] \to \mathcal{R}_0$, and thus is also not mapped to $u^b \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \subset u^b \mathcal{J}_{r_-}^0$.\\

\noindent \textbf{Step $5$.} Let 
$$
A := \lbrace u^a \in \mathbb{C}[u] \  | \ \text{pr}(u^a) \notin u^b \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \rbrace.
$$
In the previous step we saw that $1 = u^0 \in A$, so $A \neq \emptyset$. We claim that the maximal degree $\underset{i=1}{\overset{n} \sum} a_i$ of any element of $A$ is bounded from above. Since $u^b \mathcal{J}_{r_+}^0 \subset u^b \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, we see that $\text{pr}(u^a) \notin u^b \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ implies $\text{pr}(u^a) \notin u^b \mathcal{J}_{r_+}^0 = \mathcal{J}_r^0$, where $r = r_+ + r_0$. Thus
$$
u^a \notin \text{pr}^{-1}(\mathcal{J}_r^0) \supset \mathbb{C}[u] \cap J_r + I_0, \quad \text{and} \quad u^a \notin \mathbb{C}[u] \cap J_r + I_0.
$$
Therefore, $A$ lies in the complement in $\mathbb{C}[u]$ of the ideal $\mathbb{C}[u] \cap J_r + I_0$. By {\cite[Proposition $1.2$]{Giv1995}}, the zero set $Z(\mathbb{C}[u] \cap J_r + I)$ of the ideal $\mathbb{C}[u] \cap J_r + I$ has at most one point, the origin. Since $I_0 \supset I$, we have
$$
Z(\mathbb{C}[u] \cap J_r + I_0) \subset Z(\mathbb{C}[u] \cap J_r + I) \subset \{0\}.
$$
By the Nullstellensatz, this implies that for every $i$, there exists $m_i \geq 0$ such that $u_i^{m_i} \in \mathbb{C}[u] \cap J_r + I_0$, and it is easy to see that every monomial of total degree $ \geq \underset{i=1}{\overset{n} \sum} m_i$ must then also belong to the ideal. The conclusion is that $\mathbb{C}[u] \cap J_r + I_0$ contains all monomials of sufficiently high degree, and as a result the maximal degree of a monomial $u^a \in A$ is bounded from above.\\

\noindent \textbf{Conclusion.} Let $u^a \in A$ have maximal degree. Then $u_i u^a \notin A$ for all $i=1,...,n$. This means that $u^a \notin u^b \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, while $u_i u^a \in u^b \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$. Therefore $q = u^{a-b} \in \mathcal{R}_0 \setminus \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$, but $u_i q \in \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu))$ for all $i=1,...,n$, as claimed.
\end{proof}

\begin{comment}
\newpage
Let $M = \mathbb{C} P^n \times \mathbb{C} P^n$, endowed with the \textit{monotone} symplectic form whose corresponding cohomology class is $p = (1,1)$. The toric manifold $M$ is the symplectic reduction of $\mathbb{C}^{2n+2}$ by the diagonal action of the torus $\mathbb{K} = S^1 \times S^1 \subset \mathbb{T}^{n+1} \times \mathbb{T}^{n+1}$ on each component $\mathbb{C}^{n+1} \times \mathbb{C}^{n+1}$, and $\mathbb{K}_0$ is the circle $S^1$, which acts by the anti-diagonal action. The Lie algebras $\mathfrak{k}$ and $\mathfrak{k}_0$ are embedded into $\mathbb{R}^{2n+2}$ as:
$$
\begin{array}{lcl}
\mathfrak{k} = \lbrace (s,...,s,t,...,t) \in \mathbb{R}^{2n+2} \ | \ s,t \in \mathbb{R} \rbrace \simeq \mathbb{R}^2 \\ \\
\mathfrak{k}_0 = \lbrace (s,...,s,-s,...,-s) \in \mathbb{R}^{2n+2} \ | \ s \in \mathbb{R} \rbrace \simeq \mathbb{R}.
\end{array}
$$
Here, the ring $\mathbb{C}[u]$ of regular functions on $\mathbb{C}^{2n+2}$ has variable $u=(u_1,...,u_{2n+2})$. Let us write
$$
\mathcal{R} \simeq \mathbb{C}[v_1,v_2] \quad \text{and} \quad \mathcal{R}_0 \simeq \mathbb{C}[w]
$$
for the rings of regular functions on $\mathbb{K} \otimes \mathbb{C} = $ and $\mathbb{K}_0 \otimes \mathbb{C}$ respectively.
Then $\mathcal{R}$ (resp. $\mathcal{R}_0$) is obtained by sending $u_1,...,u_{n+1} \to v_1$ (resp. $w$), and $u_{n+2},...,u_{2n+2} \to v_2$ (resp. $-w$).

The $\mathbb{C}[u]$-module $J^*(\nu)$ is given by
$$
\begin{array}{lcl}
J^*(\nu) = \mathbb{C} \langle u^m | m \in \mathfrak{k}_{Z}, p(m) = \nu \rangle \\ \\
\quad \quad \quad  =  \mathbb{C} \langle u_1^{m_1}...u_{n+1}^{m_1} u_{n+2}^{m_2}...u_{2n+2}^{m_2} \ | \ m = (m_1,m_2) \in \mathfrak{k}_{Z}, \ m_1 + m_2 = \nu \rangle \\ \\
\quad \quad \quad  = \mathbb{C} \langle u_1^{m}...u_{n+1}^{m} u_{n+2}^{\nu - m}...u_{2n+2}^{\nu - m} \ | \ m \in \mathfrak{k}_{Z} \rangle.
\end{array}
$$
Its projections to $\mathcal{R}$ and $\mathcal{R}_0$ give
$$
\mathcal{J}_{\mathbb{K}}^*(\nu) = \mathbb{C} \langle v_1^{(n+1)m} v_2^{(n+1)(\nu-m)} \  | \ m \in \mathbb{Z} \rangle \quad \text{and} \quad \mathcal{J}_{\mathbb{K}_0}^*(\nu) = w^{\nu} \mathbb{C}[w].
$$


Let $(V,\alpha)$ be the prequantization space over $\mathbb{C} P^n \times \mathbb{C} P^n$ constructed in section \ref{sec3.1.3}, $h$ be a contact Hamiltonian on $V$, and $\overline{h}$ a lift of $h$ to the sphere $S_p$ from equation $(\ref{eq8})$. Given two constants $c_-,c_+ \in \mathbb{R}$ such that $c_- < \overline{h} < c_+$, we denote by $H_-$, $H_+$ be the quadratic Hamiltonians on $\mathbb{C}^n$ generating respectively the Hamiltonian symplectomorphisms 
$$
\Phi_{H_{c_-}}(z) = \exp(c_-\frac{b}{p(b)})z \quad \text{and} \quad \Phi_{H_{c_-}}(z) = \exp(c_+\frac{b}{p(b)})z.
$$
Let $\widetilde{H}$ denote the Hamiltonian lift of $h$ obtained from $\overline{h}$. We have naturally
$$
H_- < \widetilde{H} < H_+,
$$
and for $\nu_- := \nu + c_- < \nu < \nu_+ := \nu + c_+$, this yields inclusions of pairs 
$$
(G_N^-(\nu_-), \partial G_N^-(\nu_-)) \subset (F_N^-(\nu), \partial F_N^-(\nu)) \subset (G_N^-(\nu_+), \partial G_N^-(\nu_+)),
$$
and therefore to homomorphisms
$$
H_{\mathbb{K}_0}^*(G_N^-(\nu_+), \partial G_N^-(\nu_+)) \to H_{\mathbb{K}_0}^*(F_N^-(\nu), \partial F_N^-(\nu)) \to H_{\mathbb{K}_0}^*(G_N^-(\nu_-), \partial G_N^-(\nu_-)).
$$
If $\nu_-, \nu$ and $\nu_+$ are generic, we obtain homomorphisms in the limit $N \to \infty$:
$$
\Hcal_{\mathbb{K}_0}^*(G-(\nu_+)) \to \Hcal_{\mathbb{K}_0}^*(F^-(\nu)) \to \Hcal_{\mathbb{K}_0}^*(G^-(\nu_-)),
$$
and thus inclusions of kernels
$$
w^{\nu_+} \mathbb{C}[w] \simeq \mathcal{J}_{\mathbb{K}_0}^*(\nu_+) \subset \mathcal{J}_{\mathbb{K}_0}^*(F^-(\nu)) \subset \mathcal{J}_{\mathbb{K}_0}^*(\nu_-) \simeq w^{\nu_-} \mathbb{C}[w].
$$
This proves proposition \ref{prop1.2.3}.
\end{comment}




\begin{comment}
In this section, we compute the groups $HF^*(V)$. We show in particular that there are trivial if and only if $M = \mathbb{C} P^n$. Our computation is based on that of the semi-infinite cohomology defined in \cite{Giv1995} for the toric manifold $M$, and the properties of a Gysin sequence naturally appearing in this context. We have proved in the previous section that the groups $HF^*(V)$ are independent of the choice of $\widetilde{\phi}_h \in \Cont(V,\xi)$, and therefore we will make the computation for $h = 0$. The symplectic lift $\widetilde{H} : \mathbb{C}^n \times [0,1] \to \mathbb{R}$ can be taken as $\widetilde{H} = 0$, and the resulting generating functions $\mathcal{F}_{\lambda}^{(N)}$ are symply the quadratic functions $\mathcal{G}_{\lambda}^{(N)}$ generating the $\mathbb{K}$-action. In particular, they are $\mathbb{T}^n$-invariant, which makes it possible to compute the $\mathbb{K}$-equivariant and $\mathbb{K}_0$-equivariant cohomology groups
\begin{equation}
H_{\mathbb{K}}^*(G_N^-(\nu), \partial G_N^-(\nu)), \quad \text{and} \quad H_{\mathbb{K}_0}^*(G_N^-(\nu), \partial G_N^-(\nu)).
\end{equation}
For the sake of readability, we denote these groups by $H_{\mathbb{K}}^*(G_N(\nu))$ and $H_{\mathbb{K}_0}^*(G_N(\nu))$ respectively, keeping in mind that we are dealing with relative cohomology groups. They are related by the following so-called \textit{Gysin sequence}
\begin{equation}
... \longrightarrow H_{\mathbb{K}}^*(G_N^-(\nu)) \overset{eu}{\longrightarrow} H_{\mathbb{K}}^{*+2}(G_N^-(\nu)) \overset{\pi^*}{\longrightarrow} H_{\mathbb{K}_0}^{*+2}(G_N^-(\nu)) \overset{\int}{\longrightarrow} H_{\mathbb{K}}^{*+1}(G_N^-(\nu)) \longrightarrow ...,
\end{equation}
where $eu$ denotes the multiplication by the Euler class of the $S^1$-bundle 
\begin{equation}
\pi : (G_N^-(\nu))_{\mathbb{K}_0} \overset{\mathbb{K} / \mathbb{K}_0}{\longrightarrow} (G_N^-(\nu))_{\mathbb{K}},
\end{equation}
$\pi^*$ is the pull-back induced in relative equivariant cohomology, and $\int$ denotes the fiber integration. Notice that there is a natural inclusion
\begin{equation}
(G_N^-(\nu), \partial G_N^-(\nu)) \subset (\mathcal{G}_N^-(\nu), \partial \mathcal{G}_N^-(\nu)),
\end{equation}
and the pair on the right handside is $\mathbb{K}$-equivariantly homotopic to $(\Gamma_N(\nu), \partial \Gamma_N(\nu))$ (since $\mathcal{G}_{\lambda}^{(N)}$ is homogeneous). Furthermore, the torus $\mathbb{K}$ acts on $(\Gamma_N(\nu), \partial \Gamma_N(\nu))$ trivially, and therefore the $\mathbb{K}$-equivariant cohomology groups $H_{\mathbb{K}}^*(\Gamma_N(\nu), \partial \Gamma_N(\nu))$ are just a free $H_{\mathbb{K}}^*(\pt)$ module of rank $1$. The same holds for the $\mathbb{K}_0$-equivariant cohomology groups of course, which are a free module of rank $1$ over $H_{\mathbb{K}_0}^*(\pt)$. We denote them by $H_{\mathbb{K}}^*(\Gamma_N(\nu))$ and $H_{\mathbb{K}_0}^*(\Gamma_N(\nu))$ respectively. Along with the Gysin sequence, this induces a commutative diagram
\begin{equation}
\begin{tikzcd}
... \arrow{r} & H_{\mathbb{K}}^*(G_N(\nu)) \arrow{r}{eu} & H_{\mathbb{K}}^{*+2}(G_N(\nu)) \arrow{r}{\pi^*} & H_{\mathbb{K}_0}^{*+2}(G_N(\nu)) \arrow{r}{\int} & H_{\mathbb{K}}^{*+1}(G_N(\nu)) \arrow{r} & ...\\
... \arrow{r} & H_{\mathbb{K}}^*(\Gamma_N(\nu)) \arrow{r}{eu} \arrow{u} & H_{\mathbb{K}}^{*+2}(\Gamma_N(\nu)) \arrow{r}{\pi^*} \arrow{u} & H_{\mathbb{K}_0}^{*+2}(\Gamma_N(\nu)) \arrow{r}{\int} \arrow{u} & H_{\mathbb{K}}^{*+1}(\Gamma_N(\nu)) \arrow{r} \arrow{u} & ...,
\end{tikzcd}
\end{equation}
where we have kept the same notations as above for the maps in the Gysin sequence associated with the $S^1$-bundle \begin{equation}
\pi : B \mathbb{K}_0 \overset{\mathbb{K}_0 / \mathbb{K}}{\longrightarrow} B \mathbb{K}.
\end{equation}
Recall that there is a natural isomorphism $H_{\mathbb{T}^n}^*(\pt) \simeq \mathbb{C}[u_1,...,u_n]$. If $I$ denotes the ideal generated by polynomials vanishing on the complexification $\mathfrak{k}$, the Chern-Weil isomorphism induces an isomorphism 
\begin{equation}
H_{\mathbb{K}}^*(\pt) \simeq \mathbb{C}[u_1,...,u_n] / I,
\end{equation}
This is nothing but the ring of regular functions on $\mathfrak{k} \otimes \mathbb{C}$. In particular, $p \in \mathbb{C}[u_1,...,u_n] / I$. In fact, $p$ is the Euler class of the $S^1$-bundle
\begin{equation}
B \mathbb{K}_0 \overset{\mathbb{K}_0 / \mathbb{K}}{\longrightarrow} B \mathbb{K},
\end{equation}
since it is the generator of the cohomology $H^*(B (\mathbb{K}_0 / \mathbb{K}))$. In particular, the lower left homomorphism $H_{\mathbb{K}}^*(\Gamma_N(\nu)) \to H_{\mathbb{K}}^*(\Gamma_N(\nu))$ is simply the multiplication by $p$. 

Let us now investigate what happens to this diagram in the limit $N \to \infty$. It is shown in \cite{Giv1995} section $6$, that the maps $H_{\mathbb{K}}^*(\Gamma_N(\nu)) \to H_{\mathbb{K}}^*(G_N(\nu))$ give rise to a homomorphism of $\mathbb{C}[u_1,...,u_n] / I$-modules
\begin{equation}
\mathcal{H}_{\mathbb{K}}^*(\Gamma(\nu)) \to \mathcal{H}_{\mathbb{K}}^*(G(\nu)),
\end{equation}
where $\mathcal{H}_{\mathbb{K}}^*(G(\nu))$ stands for the limit 
\begin{equation}
\mathcal{H}_{\mathbb{K}}^*(G(\nu)) := \underset{N \to \infty}{\lim} H_{\mathbb{K}}^{*+4nN}(G_N^-(\nu), \partial G_N^-(\nu)),
\end{equation}
and $\mathcal{H}_{\mathbb{K}}^*(\Gamma(\nu))$ is naturally isomorphic to the quotient
\begin{equation}
\mathcal{R} := \mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}] / I \mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}],
\end{equation} 
which is nothing but the ring of regular functions on the complexified torus $\mathbb{K}_{\mathbb{C}}$.
In fact, Givental shows that this homomorphism has trivial cokernel, and that its kernel is given by the projection to $\mathcal{R}$ of the following $\mathbb{C}[u_1,...,u_n]$-submodule of $\mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}]$:
\begin{equation}
J^*(\nu) := \langle u_1^{m_1+a_1}...u_n^{m_n+a_n} : m \in \mathfrak{k}_{\mathbb{Z}}, \quad \iota(m)=(m_1,...m_n) \in \mathfrak{k}_{\mathbb{Z}}, \quad p(m) = \nu, \quad a_i > 0 \rangle.
\end{equation}
Let us denote by $\mathcal{J}_{\mathbb{K}}^*(\nu)$ this projection. If $\nu \leq \nu'$, there is an inclusion 
\begin{equation}
\mathcal{J}_{\mathbb{K}}^*(\nu') \subset \mathcal{J}_{\mathbb{K}}^*(\nu),
\end{equation}
and in the limit $\nu \to \infty$, this module becomes trivial. In particular, the homomorphism
\begin{equation}
\underset{\nu \to \infty}{\lim} \mathcal{H}_{\mathbb{K}}(\Gamma(\nu)) \to \underset{\nu \to \infty}{\lim} \mathcal{H}_{\mathbb{K}}^*(G(\nu))
\end{equation}
is an isomorphism, and the left handside is nothing but $\mathcal{R}$. Similarly as for the groups $HF^*(V)$, one can show that the limit $\underset{\nu \to \infty}{\lim} \mathcal{H}_{\mathbb{K}}^*(G(\nu))$ is independent of the choice of Hamiltonian, and defines cohomology groups $HF^*(M)$. The Gysin sequence is functorial, and the maps involved commute with the limits in $N \to \infty$ and $\nu \to \infty$. Since direct limits are exact functors, we obtain a Gysin sequence at the limit
\begin{equation}
\begin{tikzcd}
... \arrow{r} & HF^*(M) \arrow{r}{eu} & HF^{*+2}(M) \arrow{r}{\pi^*} & HF^{*+2}(V) \arrow{r}{\int} & HF^{*+1}(M) \arrow{r} & ...\\
... \arrow{r} & \mathcal{R} \arrow{r}{eu} \arrow{u} & \mathcal{R} \arrow{r}{\pi^*} \arrow{u} & \underset{\nu \to \infty}{\lim} \mathcal{H}_{\mathbb{K}_0}^{*+2}(\Gamma(\nu)) \arrow{r}{\int} \arrow{u} & \mathcal{R} \arrow{r} \arrow{u} & ...,
\end{tikzcd}
\end{equation}
where $\mathcal{H}_{\mathbb{K}_0}^{*+2}(\Gamma(\nu)) := \underset{N \to \infty}{\lim} H_{\mathbb{K}_0}^*(\Gamma_N(\nu))$.
The first, second, and fourth vertical maps are isomorphisms, and therefore 
\begin{equation}
HF^*(V) \simeq \underset{\nu \to \infty}{\lim} \mathcal{H}_{\mathbb{K}_0}^*(\Gamma(\nu)).
\end{equation}
Since the multiplication by the Euler class commutes with the limits in $N \to \infty$ and $\nu \to \infty$, the lower horizontal map
\begin{equation}
\mathcal{R} \overset{eu}{\longrightarrow} \mathcal{R}
\end{equation}
is nothing but the multiplication by $p$ ($\mathcal{R}$ is a $H_{\mathbb{K}}^*(\pt)$-modules). The complexified torus $\mathbb{K}_{\mathbb{C}}$ is an irreductible algebraic variety, and therefore the ring of regular functions on $\mathbb{K}_{\mathbb{C}}$ does not admit any zero divisor. This means that the above homomorphism is injective. Moreover, any regular function on $\mathfrak{k}_0 \times \mathbb{C}$ can be extended to $\mathfrak{k} \otimes \mathbb{C}$, and therefore the pull-back $\pi^* : H_{\mathbb{K}}^*(\pt) \to H_{\mathbb{K}_0}^*(\pt)$ is surjective. At the limit, $\pi^*$ remains surjective, so that we obtain
\begin{equation}
HF^*(V) \simeq \mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}] / I_0 \mathbb{C}[u_1,...,u_n,u_1^{-1},...,u_n^{-1}],
\end{equation}
where $I_0$ denotes the ideal of $\mathbb{C}[u_1,...,u_n]$ generated by polynomials which vanish on $\mathfrak{k}_0$. This is the $\mathbb{C}[u_1,...u_n] / I_0$-module of regular functions on the complexified torus $\mathbb{K}_{0\mathbb{C}}$. In particular, it is trivial if and only if $\mathfrak{k}_0 = 0$, that is $\mathfrak{k} = \mathbb{R}$, or in other words, $M \simeq \mathbb{C} P^n$. 
\end{comment}


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