In practice, codimension zero families of fixed points will have corners. For example, in Figure \ref{fig:v3} they are the quadrants $\bar{A_{j}^{\circ}}$, see Proposition \ref{prop:monodromy}. The following lemma allows to get rid of  these corners.%Nonetheless, we can avoid them using the following lemma.

\begin{lema}
\label{lem:cornerelimination}
Let $(M,\lambda,\phi)$ be an abstract contact open book, and let $B\subseteq M$ be a codimension zero family of fixed points. Then there is an abstract contact open book $(M,\lambda,\phi_1)\sim (M,\lambda,\phi)$ such that $\Fix \phi_1=B_1\sqcup (\Fix\phi\setminus B)$, where $B_1$ is a codimension zero family of fixed points, which is a manifold with boundary (but no corners), and the triple $(B_1,\d^{+}B_1,\d^{-}B_1)$ is homeomorphic to $(B,\d^{+}B,\d^{-}B)$.
\end{lema}
\begin{proof}
Let $H_{B}\colon N_{B}\to \R$ be the time independent Hamiltonian from Definition \ref{def:0codim}. Then $H_{B}|_{N_B\setminus B}$ has no critical points, so $H_{B}\colon N_B\setminus B\to (-\delta,\delta)$ is a submersion for some $\delta>0$ (if $\d^{-}B$ or $\d^{+}B$ is empty, the image $(-\delta,\delta)$ should be replaced by $[0,\delta)$ or $(-\delta,0]$, respectively). By assumption, the Hamiltonian flow $\psi_{s}^{H_B}$ has no $1$-periodic orbits in $N_B$. Reducing $\delta$ (hence shrinking $N_{B}$) we can assume that it has no $t$-periodic orbits for any $t\in (0,4]$.

Define a smooth family of functions $\rho_t:[-\delta,\delta]\to [-\delta,\delta]$, parametrized by $t\in [0,1]$, such that $\rho_0=\id$, $\rho_t$ vanishes identically in $[-\frac{1}{4}t\delta,\frac{1}{4} t\delta]$, equals the identity outside $[-\frac{1}{2}\delta,\frac{1}{2}\delta]$ and is strictly increasing away from $[-\frac{1}{4}t\delta,\frac{1}{4} t\delta]$ with derivative bounded by $4$. For each fixed $t\in [0,1]$ define a time independent Hamiltonian $H^{t}\de  \rho_t\comp H_{B}\colon N_{B}\to (-\delta,\delta)$.
\smallskip

Let $\phi_{t}\de \psi_{1}^{H^t}\colon N_{B}\to N_{B}$ be its time one flow. Recall that $\phi|_{N_B}$ is the time one flow of the Hamiltonian $H_{B}=H^{0}$, so $\phi_0=\phi|_{N_{B}}$, and $\phi_{t}$ agrees with $\phi$ off  $H_{B}^{-1}([-\frac{1}{2}\delta,\frac{1}{2}\delta])\subsetneq N_B$, so we can extend $\phi_{t}$ to an exact symplectomorphism $\phi_{t}\colon M\to M$, equal to $\phi$ away from $N_{B}$. In particular, $\phi_{t}=\id$ near $\d M$. Thus $(M,\lambda,\phi_{t})$ is an isotopy of abstract contact open books, and $\Fix\phi_{t}\setminus N_{B}=\Fix\phi \setminus B$.

Put $B_t=H_{B}^{-1}([-\frac{1}{4}t\delta,\frac{1}{4} t\delta])$. Because $\pm\frac{1}{4}t\delta$ are regular values of $H_{B}$, the subset $B_t\subseteq M$ is a submanifold with boundary, of codimension zero. We claim that $B_t=\Fix\phi_{t}|_{N_B}$. Clearly, $B_t\subseteq \Fix\phi_t$. Outside $B_t$, the Hamiltonian vector field associated with $H^t$ coincides with the Hamiltonian vector field of $H_{B_i}$, multiplied by the derivative of $\rho_t$, which is bounded by $4$. The claim follows because, by assumption, the time $s$ Hamiltonian flow $\psi^{H_{B}}_s$ has no fixed points in $N_{B}\setminus B$ for $s\in (0,4]$.

Therefore, $\Fix \phi_{t}=B_{t}\sqcup (\Fix \phi\setminus N_{B})$, and $B_{t}$ is a codimension zero family of fixed points, without corners. It remains to see that $(B_{t},\d^{+}B_{t},\d^{-}B_{t})$ is homeomorphic to $(B,\d^{+}B,\d^{-}B)$, for $t>0$ small enough. Write $N_{B}=N_{B}^{+}\sqcup N_{B}^{-}\sqcup B$, where $H_{B}>0$ on $N_{B}^{+}$ and $H_{B}<0$ on $N_{B}^{-}$. Define a smooth vector field $X_{1}$ as $-\nabla H_{B}$ on $N_{B}^{+}$, $\nabla H_{B}$ on $N_{B}^{-}$, and $0$ on $B$. Fix a vector field $X_0$ on $N_{B}$, pointing inwards $\d^{+} B \sqcup \d^{-}B$, see \cite[Appendice, \sec 2]{corners}. Since $X_{1}$ vanishes on $B$, we can choose $t>0$ small enough so that $X\de X_0+X_1$ does not vanish on a neighborhood $N$ of $\bar{B_{t}\setminus B}$, and points inwards both $\d^{+}B_{t}\sqcup \d^{-}B_{t}$ and $\d^{+}B\sqcup \d^{-}B$. As in \cite[Appendice, Proposition 6.1]{corners}, we conclude that both $N\cap B_{t}$ and $N\cap B$ are homeomorphic to $(\d^{+}B_{t}\sqcup \d^{-}B_{t})\times [0,\infty)$, and we can extend this homeomorphism to the required homeomorphism of triples.
\end{proof}

\section{The McLean spectral sequence in fixed point Floer homology}\label{sec:spectral-sequence}

In this section, we introduce fixed point Floer homology $\HF_{*}(\phi,+)$ of an abstract contact open book $(M,\lambda,\phi)$, following Uljarevic and McLean [74, 49]. This definition is a variant of [20], cf.\ [39]. It was first introduced in a slightly different way in Seidel's thesis [64], see \cite[\sec 4]{Seidel_more}.

After some preparations in Section \ref{sec:completions}, we introduce Floer complexes in Section \ref{sec:floer}. In Proposition \ref{prop:spectral-sequence}, we construct a spectral sequence converging to $\HF_{*}(\phi,+)$ in case $\Fix\phi$ is a disjoint union of codimension zero families of fixed points. This spectral sequence is a slight generalization of the axiom \cite[(HF3)]{McLean}, proved in Appendix C of loc.\ cit. Our arguments are essentially the same, but we give a less compressed presentation for convenience of those readers who are not experts in Floer theory.

\subsection{Completions of Liouville domains}\label{sec:completions}

A \emph{contact manifold} $(N,\alpha)$ is a compact manifold $N$ of dimension $2k-1$ with a $1$-form $\alpha\in \Omega^{1}(N)$ such that $\alpha\wedge (d\alpha)^{k-1}\neq 0$. The \emph{Reeb vector field} $R_{\alpha}$ is defined by $\alpha(R_{\alpha})=1$, $d\alpha(R_{\alpha},\sdot)=0$.
\smallskip

Let $(M,\lambda)$ be a Liouville domain. Then $(\d M,\lambda|_{\d M})$ is a contact manifold. For $\eta>0$, the backwards flow of the Liouville vector field $X_\lambda$ defines a diffeomorphism onto its image
\begin{equation}
\label{eq:necktriv}
\Psi\colon (-\eta,0]\times\partial M\to M
\end{equation}
such that $\Psi^*\lambda=e^r\lambda|_{\partial M}$, where $r$ is a coordinate on $(-\eta,0]$. The {\em completion} of $(M,\lambda)$, see \cite[p.\ 239]{CE_from-Stein-to-Weinstein} or \cite[\sec 1.1]{Uljarevic}, is a manifold $\hat{M}=(M\sqcup (-\eta,\infty)\times\partial M)/_{\sim}$, where $\sim$ identifies $(-\eta,0]\times\partial M$ with its image under $\Psi$, together with a Liouville form  defined as  $\lambda$ in $M$ and  $e^r\lambda|_{\partial M}$ in the \emph{cylindrical end} $(-\eta,\infty)\times\partial M$. Abusing notation, we denote this $1$-form by $\lambda$, too.

Let $\frac{\d}{\d r}$ be the vector field corresponding to the coordinate $r$ of the cylindrical end $(-\eta,\infty)\times \d M\subseteq \hat{M}$. The restriction $\lambda|_{\{r\}\times\partial M}$ is a contact form, and we denote by $R_\lambda$ its Reeb vector field. Therefore, $R_\lambda$ is a vector field on $(-\eta,\infty)\times\partial M\subseteq \hat M$, which is tangent to the level sets of the coordinate $r$.

An almost complex structure $J$ on $M$ or $\hat{M}$ is {\em cylindrical} if there is $\epsilon>0$ such that for any $r\geq -\epsilon$, the restriction $J|_{\{r\}\times \d M}$ leaves invariant the contact distribution $\ker(\lambda|_{\{r\}\times M})$ and satisfies $J(\frac{\d}{\d r})=R_\lambda$. This notion coincides with \cite[Definition 4.1]{McLean} and \cite[(2.7), (2.8)]{Uljarevic}.

Fix $a\in \R$. A (time-dependent) Hamiltonian $H_t\colon M\to\RR$ is of {\em slope} $a$ if there exists $\epsilon>0$ such that $H_t\comp\Psi(r,y)=ar$ for $r\geq -\epsilon$, where $\Psi$ is the trivialization~(\ref{eq:necktriv}), see \cite[p.\ 977]{McLean} or \cite[(2.6)]{Uljarevic}. A Hamiltonian of slope $a$ is extended to the completion $\hat{M}$ by the formula $H_t(r,y)=ar$ at $(-\epsilon,\infty)\times \partial M$. At the locus where $H_t(r,y)=ar$ the Hamiltonian flow at time $1$ coincides with the Reeb flow of $R_{\lambda}$ at time $a$. A nonzero slope $a$ is {\em small} if $|a|$ is strictly smaller than the minimal period of a Reeb orbit. In this case, $\psi^{H}_{1}$ has no fixed points in the cylindrical end of $\hat{M}$.

\subsection{The Floer complexes}\label{sec:floer}

Let $(M,\phi,\lambda)$ be a graded abstract contact open book, see Definition \ref{def:acob} and Section \ref{sec:graded-acobs}. We now define the \emph{$\phi$-twisted loop Floer complex} $\CFlp_{*}$ and the \emph{$\check{\phi}$-fixed point Floer complex} $\CFpt_{*}$, using the Floer Cauchy--Riemann equation in a perturbed  \eqref{eq:floerequlj} and unperturbed \eqref{eq:floereqmclean} version, respectively. The first setting is adapted from [74], and mirrors [20] in the compact case, which in turn specifies to the classical Floer theory for $\phi=\id$. The second one is used in [49]: here $\check{\phi}$ is a small Hamiltonian perturbation of $\phi$. It is known how to pass from one setting to the other: we outline the method in Section \ref{sec:floerequivalence}. In particular, both complexes have the same homology $\HF_{*}(\phi,+)$.  Nonetheless, to derive the spectral sequence in Proposition \ref{prop:spectral-sequence}, it will be convenient to use both approaches at different parts of the argument. %This is the reason why we explain both approaches.

\subsubsection{Periodic almost complex structures and Hamiltonians}

Let $(M,\lambda,\phi)$ be a graded abstract contact open book. Extending $\phi$ to the identity in $[0,\infty)\times \d M$ we obtain a compactly supported exact symplectomorphism $\hat M\to\hat M$, which we denote by $\phi$, too.

A family of almost complex structures $J_t$ on $M$ or $\hat{M}$, smoothly parametrized by $t\in \R$, is {\em $\phi$-periodic} if $J_{t+1}=\phi^*J_t$, see \cite [(2.5)]{Uljarevic}. A (time dependent) Hamiltonian $H_{t}$ is {\em $\phi$-periodic} if it satisfies the equality $H_{t+1}=H_t\comp\phi$, see \cite [(2.4)]{Uljarevic}. A {\em $\phi$-twisted loop} in $M$ is a path $\gamma:\RR\to M$ such that $\phi(\gamma(t+1))=\gamma(t)$, see \cite[\sec 2.2]{Uljarevic}.

We warn the reader that in \cite[\sec 2.2]{Dostoglou_Salamon} the periodicity conditions are defined in an opposite way, i.e.\ $\gamma(t+1)=\phi(\gamma(t))$, etc. Also, the symbols $t,s$  play the opposite roles in [20] than in the discussion below, which follows [74].

%\subsubsection{The Conley--Zehnder index of a Hamiltonian twisted loop}
Let $H$ be a $\phi$-periodic Hamiltonian. It satisfies a basic equality
\begin{equation}\label{eq:basic}
\psi_{t+1}^{H}=\phi^{-1}\circ \psi_{t}^{H}\circ \phi\circ \psi_{1}^{H},
\end{equation}
cf.\  \cite[p.\ 586]{Dostoglou_Salamon}.
To prove \eqref{eq:basic}, note that $\psi_{t+1}^{H}=\psi_{t}^{H'}\circ \psi_{1}^{H}$, where $H_{t}'(x)=H_{t+1}(x)$. By $\phi$-periodicity, $H_{t}'=H_{t}\circ \phi$, so $\psi_{t}^{H'}=\phi^{-1}\circ \psi_{t}^{H}\circ \phi$, see \cite[Exercise 3.1.14(iii)]{McDuff_Salamon}. This proves \eqref{eq:basic}.
\smallskip

We denote by $\cP_{\phi,H}$ the space of $\phi$-twisted loops which are \emph{Hamiltonian}, i.e.\ of the form $t\mapsto \psi_{t}^{H}(x)$ for some $x\in M$. %satisfy $\gamma(t)are integral curves of $H$.
We have a one-to-one correspondence
\begin{equation}\label{eq:corresponence_Fix-P}
\cP_{\phi,H}\longleftrightarrow \Fix(\phi \circ \psi_{1}^{H})
\quad\mbox{given by}\quad
\gamma\mapsto \gamma(0).
\end{equation}
Indeed, if $\gamma\in \cP_{\phi,H}$ then $(\phi\circ\psi_{1}^{H})(\gamma(0))=\phi(\gamma(1))=\gamma(0)$, so $\gamma(0)\in \Fix \phi\circ\psi_{1}^{H}$. For the converse, let $x\in \Fix (\phi\circ \psi_{1})$ and
define $\gamma(t)=\psi_{t}^{H}(x)$. We have %Using \eqref{eq:basic}, we compute
\begin{equation*}
\phi(\gamma(t+1))=\phi\circ \psi_{t+1}^{H}(x)\overset{\mbox{\tiny{\eqref{eq:basic}}}}{=}\phi\circ\phi^{-1}\circ \psi_{t}^{H}\circ \phi \circ \psi_{1}^{H}(x)=
\psi_{t}^{H}\circ \phi\circ \psi_{1}^{H}(x)=\psi_{t}^{H}(x)=\gamma(t),
\end{equation*}
where the first and last equality follow from the definition of $\gamma$, and the second-to-last holds because $x\in \Fix (\phi\circ \psi_1^H)$. Thus $\gamma$ is a $\phi$-twisted loop, as needed.
\smallskip

Note that if a $\phi$-periodic Hamiltonian $H$ has small negative (or positive) slope, then all fixed points of $\phi\comp \psi^{H}_1$ are away from $\partial M$, and away from the cylindrical end of of $\hat{M}$.

\subsubsection{The Conley--Zehnder index of a Hamiltonian twisted loop}\label{sec:CZ-loop}

Fix a Hamiltonian $\phi$-twisted loop $\gamma\in \cP_{\phi,H}$. By \eqref{eq:corresponence_Fix-P}, the point $\gamma(0)$ is fixed by the symplectomorphism $\phi\circ \psi_{1}^{H}$. The latter has a unique grading, induced from the grading of $\phi$ by the isotopy $(\psi_{t}^{H})$. We define $\CZ(\gamma)$ as the Conley--Zehnder index of $\gamma(0)\in \Fix(\phi\circ\psi_{1}^{H})$, see Section \ref{sec:CZ}.

This definition lifts the relative Conley--Zehnder index introduced in \cite[Definition 2.10]{Uljarevic} to an absolute one. To see this, let us recall how $\CZ$ was defined in Section \ref{sec:CZ}. Since $\gamma$ is an integral curve of the Hamiltonian flow $\psi^{H}_{t}$, for any time $t\in [0,1]$ the differential of $\psi^{H}_{t}$ at time $t$ induces a mapping $\Fr(M)_{\gamma(0)}\to \Fr(M)_{\gamma(t)}$, which lifts to a mapping $\tilde{\psi}^{H}_t\colon \tilde{\Fr}(M)_{\gamma(0)}\to\tilde{\Fr}(M)_{\gamma(t)}$. The grading of $\phi$ defines a mapping $\tilde{\beta}_{\gamma(1)}\colon \tilde{\Fr}(M)_{\gamma(1)}\to\tilde{\Fr}(M)_{\gamma(0)}$. So, composing we obtain a mapping
\begin{equation*}
\tilde\beta_{\gamma(1)}\circ \tilde{\psi}^{H}_{1}\colon \tilde{\Fr}(M)_{\gamma(0)}\to \tilde{\Fr}(M)_{\gamma(0)}.
\end{equation*}
which gives rise to a homotopy class of paths in the symplectic group relative to its ends as in Section \ref{sec:CZ}. The Conley--Zehnder index $\CZ(\gamma)$ is now the Maslov index of this path.

Given two frames $f_0\in \Fr(M)_{\gamma(0)}$ and $f_1\in \Fr(M)_{\gamma(1)}$ such that %$\tilde{\beta}_{\gamma(1)}(f_1)=f_0$,
any homotopy class (relative to its ends) of frames $f_t\in \Fr(M)_{\gamma(t)}$ connecting $f_0$ and $f_1$ determines a map $\tilde{\Fr}(M)_{\gamma(1)}\to\tilde{\Fr}(M)_{\gamma(0)}$. Let us call the homotopy class {\em admissible} if the map equals $\tilde{\beta}_{\gamma(1)}$.

Fix $\gamma_1\in \cP_{\phi,H_{1}}$, $\gamma_2\in \cP_{\phi,H_{2}}$ for two Hamiltonians $H_1$, $H_2$, and fix $u\colon \RR\times\RR\to M$ satisfying $\phi(u(s,t+1))=u(s,t)$, $u(s,t)=\gamma_1(t)$ for $s\leq 0$ and $u(s,t)=\gamma_2(t)$ for $s\geq 1$. Then the grading  induces, via the choice of a homotopy of admissible classes of framings along $u(s,\sdot)$ for any $s$, a symplectic trivialization $T_{s,t}$ of $u^*TM$ such that $\phi^*T_{s,t+1}=T_{s,t}$. Now, the relative grading of $(\gamma_1,\gamma_2)$ is defined in \cite[Definition 2.10]{Uljarevic} as the difference between the Maslov indices $\mu(\Psi^2)-\mu(\Psi^1)$, where $\Psi^{i}\colon t\mapsto T_{i,t}^{-1}\circ (\psi_{t}^{H})^{*}T_{i,0}\in \Sp(2n)$. The path  $\Psi^{i}$ represents the homotopy class used above to define $\CZ(\gamma_{i})$, as claimed.

\subsubsection{The $\phi$-twisted loop Floer complex}\label{sec:perturbed} Now, we introduce the fixed point Floer homology $\HF_{*}(\phi,+)$ using $\phi$-twisted loops, following [74].

Recall from Definition~\ref{def:acob} that associated with an abstract contact open book $(M,\lambda,\phi)$ we have an action function $\ac\colon M\to\RR$, defined (up to a constant) by $\phi^{*}\lambda-\lambda=-d\ac$. We fix a $\phi$-periodic Hamiltonian $H$ of small \emph{negative} slope. Taking $H$ of small \emph{positive} slope would lead to a definition of another Floer homology group, $\HF_{*}(\phi,-)$, which is not relevant for us: hence for the sake of clarity, we focus on $\HF_{*}(\phi,+)$, and leave the analogous discussion of $\HF_{*}(\phi,-)$ to the reader. The relation between $\HF_{*}(\phi,+)$ and $\HF_{*}(\phi,-)$ is outlined in \cite[Remark 4.1]{Seidel_more}.

The \emph{action functional} on the space of $\phi$-twisted loops is given in \cite[Definition 2.6]{Uljarevic} by the formula
\begin{equation}
\label{eq:actiondef}
\cA_{\phi,H}(\gamma)\de -\int_0^1(\gamma^*\lambda+H_t(\gamma)\, dt) + \ac(\gamma(1)).
\end{equation}
Note that if $\gamma\in \cP_{\phi,H}$ is a constant path, and $H$ is time-independent, then the image of $\gamma$ is a fixed point of $\phi$, say $p$, and $\cA_{\phi,H}(\gamma)=-H(p)+\ac(p)$.

The set of critical points of $\cA_{\phi,H}$ equals exactly $\cP_{\phi,H}$, see \cite[Lemma 2.7]{Uljarevic}. A $\phi$-periodic Hamiltonian $H$ is {\em nondegenerate} if $\det(D(\phi\comp\psi^{H}_1)(x)-\id)\neq 0$ at any fixed point $x$ of $\phi\comp\psi^{H_t}_1$, see \cite[Definition 2.5]{Uljarevic}. If a $\phi$-periodic Hamiltonian $H$ is nondegenerate then the set $\cP_{\phi,H}$ is finite.
\smallskip

Let $J_t$ be a family of cylindrical, $(d\lambda)$-compatible and $\phi$-periodic almost complex structures on $\hat{M}$.  Fix $\gamma_{-},\gamma_{+}\in \cP_{\phi,H}$. A \emph{Floer trajectory} between $\gamma_{-}$ and $\gamma_{+}$ is a smooth map $u\colon \RR^2\to \hat M$ satisfying
\begin{equation}
\label{eq:boundulj}
\phi(u(s,t+1))=u(s,t),\quad
\lim_{s\rightarrow -\infty} u(s,t)=\gamma_{-}(t),\quad
\lim_{s\rightarrow +\infty} u(s,t)=\gamma_{+}(t)
\end{equation}
and the \emph{perturbed Floer Cauchy--Riemann equation}:
\begin{equation}
\label{eq:floerequlj}
\frac{\partial u}{\partial s}+J_t(u)\left(\frac{\partial u}{\partial t}-X^{H}(u)\right)=0,
\end{equation}
see \cite[(2.10)]{Uljarevic}. Whenever additional precision is needed, we refer to such a Floer trajectory as a \emph{$\CFlp_{*}(\phi,H_t,J_t)$-Floer trajectory from $\gamma_{-}$ to $\gamma_{+}$}. Note that if $u(s,t)$ is such a Floer trajectory then so is $u(c+s,t)$ for any $c\in\RR$. So, the additive group $\RR$ acts on the Floer trajectories.

\begin{definition}[{\cite[2.5]{Uljarevic}}]
\label{def:perturbedregularpair}
A pair $(H_t,J_t)$ is {\em regular for $\phi$} if $H_t$ is nondegenerate and the linearization of the operator \eqref{eq:floerequlj} for every Floer trajectory is surjective.
\end{definition}
It is known that a generic pair $(H_t,J_t)$ is regular. In particular, regular pairs exist, cf.\ \cite[Theorem 8.1.1]{AD_book} or \cite[p.\ 316]{Seidel_Fukaya_structures_II}.

The \emph{$\phi$-twisted loop Floer complex} associated with a graded abstract compact open book $(M,\lambda,\phi)$ and a regular pair $(H_t,J_t)$ is the $\ZZ_2$-vector space $\CFlp_{*}(\phi,H_t,J_t)$ generated by the Hamiltonian $\phi$-twisted loops, graded by $-\CZ$, whose differential $\d$ is defined as follows. Given $\gamma_+,\gamma_-\in \cP_{\phi,H}$ such that $\CZ(\gamma_+)=\CZ(\gamma_-)+1$, the coefficient of $\gamma_+$ in $\partial(\gamma_-)$ is the number (mod $2$) of Floer trajectories between $\gamma_{-}$ and $\gamma_{+}$, up to the additive action of $\RR$. By \cite[Theorem~2.18]{Uljarevic}, this differential is well defined and its square is zero, so $\CFlp_{*}$ is indeed a chain complex. By \cite[Corollary 2.22 and Lemma 2.25]{Uljarevic}, the homology groups of this complex are independent of the choice of a regular pair $(H_t,J_t)$, up to unique isomorphism, as long as $H$ remains of small negative slope. The reader should note that we have improved the relative grading used by Uljarevic to the absolute grading defined in Section \ref{sec:CZ-loop} using the grading of $\phi$ as in [49]; the proofs of [74] work in this setting.

As a consequence of the above independence, we can associate to each graded abstract contact open book $(M,\lambda,\phi)$ a $\Z$-graded Floer homology group $\HF_{*}(\phi,+)$, given by $H$ of small negative slope. The sign \enquote{$+$} comes from the notation of [49], where, as explained in Section \ref{sec:Hamiltonian}, Hamiltonians have opposite signs. Taking $H$ of small \emph{positive} slope would lead to $\HF_{*}(\phi,-)$.

\subsubsection{The action filtration} \label{sec:action-filtration}
Let $\CFlp_{*}(\phi,H_t,J_t)$ be the $\phi$-twisted loop Floer complex introduced in Section \ref{sec:perturbed} for a regular pair $(H_t,J_t)$. For any $a\in \R$, let $F^{\leq a}  \CFlp_{*}(\phi,H_t,J_t)$ be the subspace of $\CFlp_{*}(\phi,H_t,J_{t})$ spanned by those Hamiltonian $\phi$-twisted loops $\gamma\in \cP_{\phi,H}$ whose action satisfies $\cA_{\phi,H}(\gamma)\leq a$. It is a subcomplex of $\CFlp_{*}(\phi,H_t,J_{t})$, see \cite[\sec 2.6]{Uljarevic}. To see this, recall that for $\gamma_{+},\gamma_{-}\in \cP_{\phi,H}$, the coefficient of $\gamma_{+}$ in the expression of $\d(\gamma_{-})$ is the count (mod $2$) of the Floer trajectories connecting $\gamma_{-}$ with $\gamma_{+}$. By  \cite[p.\ 867]{Uljarevic}, these trajectories are anti-gradient flow lines of the action functional $\cA_{\phi,H}$. In particular, the differential $\d$ decreases the value of the action, so it maps $F^{\leq a}  \CFlp_{*}(\phi,H_t,J_t)$ to itself, as claimed.

\subsubsection{The $\check{\phi}$-fixed point Floer complex}
\label{sec:unperturbed}
Now we review the definition of Floer homology given in [49]. In Section \ref{sec:floerequivalence} we will see that it is equivalent to the one recalled in Section \ref{sec:perturbed}.

Let again $(M,\lambda,\phi)$ be a graded abstract compact open book. Consider a $\cC^{2}$-small $\phi$-periodic Hamiltonian $H$ which is of small negative slope. Assume that all fixed points of $\check{\phi}\de \phi\comp\psi^{H}_1$ are nondegenerate (hence there is finitely many such points). Let $J_t$ be a family of cylindrical, $(d\lambda)$-compatible and $\check{\phi}$-periodic almost complex structures on $M$. For $p_{-},p_{+}\in \Fix \check{\phi}$, a \emph{Floer trajectory} (or, more precisely: a \emph{$\CFpt_{*}(\check{\phi},J_{t})$-Floer trajectory}) from $p_{-}$ to $p_{+}$ is a smooth map $v\colon \RR^2\to M$ satisfying
\begin{equation}
\label{eq:boundmclean}
\check{\phi}(v(s,t+1))=v(s,t),\quad
\lim_{s\rightarrow -\infty} v(s,t)=p_{-},\quad
\lim_{s\rightarrow +\infty} v(s,t)=p_{+}
\end{equation}
and the \emph{unperturbed Floer Cauchy--Riemann equation}:
\begin{equation}
\label{eq:floereqmclean}
\frac{\partial v}{\partial s}+J_t(v)\left(\frac{\partial v}{\partial t}\right)=0,
\end{equation}
see \cite[Definition 4.3]{McLean}.

We define the \emph{$\check{\phi}$-fixed point Floer complex $\CFpt_{*}(\check{\phi},J_t)$} as the $\ZZ_2$-vector space freely generated by the fixed points of $\check{\phi}$, graded by the minus Conley--Zehnder index, and equipped with the following differential: given $p_+,p_-\in \Fix \check{\phi}$ such that $\CZ(p_+)=\CZ(p_-)+1$, the coefficient of $p_+$ in $\partial(p_-)$ is the number of Floer trajectories between $p_{-}$ and $p_{+}$, up to the additive action of $\RR$ in the $s$ variable of $u$. As we will see in Section \ref{sec:floerequivalence} below, for generic choice of $J_t$ and $H$, this differential is well defined, its square is zero and the resulting homology groups coincide with  $\HF_{*}(\phi,+)$ introduced in Section \ref{sec:perturbed} above. In particular, they do not depend on the choice of the almost complex structure $J_t$, and of the Hamiltonian $H$ of small negative slope, cf.\ \cite[p.\ 980]{McLean}.

\subsubsection{Equivalence of the two Floer complexes}\label{sec:floerequivalence} Here we explain the equivalence of Uljarevic's and McLean's definitions of the Floer complex, recalled in Sections \ref{sec:perturbed} and \ref{sec:unperturbed} above.

Let $H$ be a $\phi$-periodic Hamiltonian and let $(J_t)$ be a family of cylindrical, $(d\lambda)$-compatible, $\phi$-periodic almost complex structures such that $(H_t,J_t)$ is regular for $\phi$, see Definition \ref{def:perturbedregularpair}. As in  Section~\ref{sec:unperturbed} put $\check{\phi}\de\phi\comp\psi^{H}_1$. We claim that there is an isomorphism of chain complexes $\CFlp_{*}(\phi,H_{t},J_{t})\cong \CFpt(\check{\phi},\tilde{J}_{t})$ for a suitable cylindrical, $(d\lambda)$-compatible and $\check{\phi}$-periodic family $\tilde{J}_{t}$.

The correspondence \eqref{eq:corresponence_Fix-P}, together with our definition of $\CZ(\gamma)$, give a natural identification of the underlying graded $\Z_{2}$-vector spaces. For the equality of differential we use the following, known argument from \cite[Proposition 8.6.1]{AD_book}.

Fix $\gamma_-,\gamma_+\in \cP_{\phi,H}$, and let $u:\RR^2\to \hat{M}$ be a $\CFlp_{*}(\phi,H_{t},J_{t})$-Floer trajectory from  $\gamma_{-}$ to $\gamma_{+}$. Let $p_{-},p_{+}$ be the corresponding fixed points of $\check{\phi}$. By \cite[Lemma 2.15]{Uljarevic}, the image of $u$ is contained in $M$. Define $v(s,t):=(\psi^{H}_t)^{-1}(u(s,t))$ and
\begin{equation}
\label{eq:Jtilde}
%v(s,t)\de (\psi^{H}_t)^{-1}(u(s,t))\quad\mbox{and}\quad
\tilde{J}_t\de (\psi^{H}_t)^*J_t=D(\psi^{H}_t)^{-1}\circ J_t\circ D(\psi^{H}_t).
\end{equation}
We claim that $v$ is a $\CFpt_{*}(\check{\phi},\tilde{J}_{t})$-Floer trajectory from $p_{-}$ to $p_{+}$. The proof of \cite[Proposition 8.6.1]{AD_book} precisely shows that $v$ satisfies the unperturbed Cauchy--Riemann Floer equation~\eqref{eq:floereqmclean}, with $J_t$ replaced by $\tilde{J}_t$. Clearly, $\lim_{s\rightarrow \pm\infty} v(s,t)=p_{\pm}$. It remains to show that $\check{\phi}(v(s,t+1))=v(s,t)$ and that $\tilde{J}_{t}$ is a $\check{\phi}$-periodic family of cylindrical almost complex structures. This follows by elementary computation, using the basic equality \eqref{eq:basic}. Indeed, \eqref{eq:basic} and definition of $\check{\phi}$ give
\begin{equation*}
\check{\phi}\circ (\psi_{t+1}^{H})^{-1}=\phi\circ \psi_1^H\circ( \phi^{-1}\circ \psi_t^H\circ\phi\circ\psi_1^H)^{-1}=
\end{equation*}
Hence by definition of $v$, we have
\begin{equation*}
\check{\phi}(v(s,t+1))=\check{\phi}\circ (\psi_{t+1}^{H})^{-1}(u(s,t+1))=(\psi_{t}^{H})^{-1}\circ \phi(u(s,t+1))
\overset{\mbox{\tiny{\eqref{eq:boundulj}}}}{=}
\end{equation*}
which proves the periodicity condition \eqref{eq:boundmclean}. To see that $\tilde{J}_t$ is $\check{\phi}$-periodic, we write
\begin{equation*}
\tilde{J}_{t+1}=(\psi_{t+1}^{H})^{*}J_{t+1}
\overset{\tiny{\mbox{\eqref{eq:basic}}}}{=}
(\phi^{-1}\circ \psi_{t}^{H}\circ \phi\circ \psi_{1}^{H})^{*}J_{t+1}
\overset{\tiny{(*)}}{=}
(\psi_{t}^{H}\circ \phi\circ \psi_{1}^{H})^{*}J_{t}
(\phi\circ \psi_{1}^{H})^{*}\tilde{J}_{t}
\check{\phi}^{*}\tilde{J}_t,
\end{equation*}
as needed. The equality $(*)$ above holds by the $\phi$-periodicity condition $J_{t+1}=\phi^{*}J_{t}$.

Since the pair $(H_t,J_t)$ is regular for $\phi$, the linearized Floer equation for $u$ is surjective, so the same happens for $v$ and the family of almost complex structures $\tilde{J}_t$. This way we get a one to one correspondence between the $\CFlp_{*}(\phi,H_t,J_t)$-Floer trajectories from $\gamma_{-}$ to $\gamma_{+}$, and $\CFpt_{*}(\check{\phi}_t,\tilde{J}_t)$-Floer trajectories from $p_{-}$ to $p_{+}$, modulo the $\R$-action. Hence the differentials in both $\CFlp_{*}(\phi,H_t,J_t)$ and $\CFpt_{*}(\check{\phi},\tilde{J}_t)$ coincide. In particular, the results of [74] quoted in Section \ref{sec:perturbed} can be used to prove that both differentials are well defined and square to zero. We conclude that the complexes $\CFlp_{*}(\phi,H_t,J_t)=\CFpt_{*}(\check{\phi},\tilde{J}_t)$ constructed in Sections \ref{sec:perturbed} and \ref{sec:unperturbed}, respectively, are naturally isomorphic. In particular, both constructions give the same Floer homology groups $\HF_{*}(\phi,+)$.% and $\HF_{*}(\phi,-)$.

\subsubsection{Invariance by isotopy of abstract contact open books}
The following result, crucial for us, was proved in \cite[Lemma B.3]{McLean}. We include a proof for the convenience of the reader.
\begin{prop}
\label{prop:isotopy_invariance}
Let $\{(M_t,\lambda_t,\phi_t)\}_{t\in [0,1]}$ be a graded isotopy of abstract contact open books. Then the groups $\HF_*(\phi_t,+)$ and $\HF_{*}(\phi_t,-)$ are independent of $t$.
\end{prop}
\begin{proof}
Let $\pi\colon M\to [0,1]$ be the total space of the isotopy, and let $\phi\colon M\to M$ be the diffeomorphism which restricts to $\phi_t$ on each fiber, see Definition \ref{def:acob}.

Given such an isotopy $(M,\pi,\lambda)$ of Liouville domains, in \cite[\sec 9.1]{Keating}, an isotopy of completions $(\hat{M},\pi,\lambda)$ is constructed. Since $\phi$ is fiberwise compactly supported, it extends to a diffeomorphism $\phi\colon \hat{M}\to\hat{M}$ which respects fibers and at each fiber is a compactly supported exact symplectomorphism. By \cite[\sec 9.1]{Keating}, $\pi\colon \hat{M}\to [0,1]$ admits a trivialization by an exact symplectomorphism, i.e.\ there is  a diffeomorphism $\Phi:\hat{M_0}\times [0,1]\to\hat{M}$ such that $\Phi^*\lambda_t=\lambda_0+dG_t$, where $G_t$ is the restriction to $M_0\times\{t\}$ of a smooth function $G\colon M\times [0,1]\to\RR$. So, given an isotopy of abstract contact open books $(M,\pi,\lambda,\phi)$, we have an isotopy $(\hat{M}_0\times [0,1],\pr_{[0,1]},\lambda,\phi)$ of their completions, such that $\lambda_t=\lambda_0+dG_t$, and $\phi$ is fiberwise an exact compactly supported symplectomorphism. If the isotopy $(M,\pi,\lambda,\phi)$ is graded, then this grading is inherited to $(\hat{M}_0\times [0,1],\pr_{[0,1]},\lambda,\phi)$.

By construction, the Floer homology of $(\hat{M}_0,\lambda_0+dG_t,\phi_t)$ coincides with the Floer homology of $(\hat{M}_0,\lambda_0,\phi_t)$. The result now follows from \cite[Theorem 2.34]{Uljarevic}.
\end{proof}

\subsection{The McLean spectral sequence}

The remaining part of this section is devoted to the proof of the following result, which is a (slightly more general) version of the axiom (HF3) from \cite[p.\ 980]{McLean}, proved in Appendix C of loc.\ cit; see Remark \ref{rem:McLean-HF3} for comparison. Our proof follows the same path, but we provide a more detailed exposition for the convenience of the reader.

\begin{prop}\label{prop:spectral-sequence}
Let $(M,\lambda,\phi)$ be a graded abstract contact open book. Assume that
\begin{equation*}
\Fix \phi=\bigsqcup_{i=1}^{N} B_{i},
\end{equation*}
where each $B_{i}$ is a codimension zero families of fixed points, see Definition \ref{def:0codim}. Choose an action $\ac\colon M\to \R$, see \eqref{eq:exact-symplectiomorphism}, and a function $\iota\colon \{1,\dots, N\}\to \Z$ with the following property: the function $\hat{\iota}$ from the discrete set $\ac(\bigsqcup_{i=1}^{N}B_{i})\subseteq \R$ to $\Z$, defined by $\hat{\iota}(\ac(B_{i}))=\iota(i)$, is strictly increasing. Assume further that one of the following holds
\begin{enumerate}[(i)]
\item\label{item:ss-H1} $H^{1}(M,\d M;\R)=0$, or
\item\label{item:ss-bd} for every $i$, the codimension zero family $B_i$ has either $\d^{-}B_{i}=\emptyset$ or $\d^{+}B_i=\emptyset$, see \eqref{eq:boundarysplitting1}.
\end{enumerate}
Then there is a spectral sequence $E_{p,q}^{r}$ converging to $\HF_{*}(\phi,+)$, whose first page equals
\begin{equation}\label{eq:spectral-sequence-first-page}
E_{p,q}^{1}=\bigoplus_{\{i: \iota(i)=p\}} H_{d+p+q+\CZ(B_i)} (B_{i},\d^{+}B_{i};\Z_2),
\end{equation}
where $d=\tfrac{1}{2}\dim M$. Moreover, there is an analogous spectral sequence converging to the Floer homology $\HF_{*}(\phi,-)$, whose first page is as above with $\d^{+}B_i$ replaced by $\d^{+}B_i\sqcup \d^{M}B_i$.
\end{prop}

\begin{remark}\label{rem:graph}
The technical assumptions \ref{item:ss-H1} or \ref{item:ss-bd} are introduced to simplify the construction below, and most likely can be omitted. Nonetheless, working under either of these assumptions is sufficient for our purposes. Indeed, as we will see in Proposition \ref{prop:monodromy}\ref{item:B_i-boundary}, if $\phi$ is the \enquote{radius zero} symplectic monodromy, given by the flow of the vector field \eqref{eq:monodromy-vector-field}, then all its codimension zero families of fixed points satisfy $\d^{-}B_i=\emptyset$, so condition \ref{item:ss-bd} holds. Moreover, in the proof of Theorem \ref{theo:Zariski} the Liouville domain $M$ will be (diffeomorphic to) the Milnor fiber of an isolated hypersurface singularity $f\colon (\C^{n},0)\to (\C,0)$, so for $n\geq 3$ it will also satisfy $H^1(M,\d M;\R)=0$, i.e.\ condition \ref{item:ss-H1}.%, see Examples \ref{ex:Milnor-fibration} and \ref{ex:mu_constant}.

We note that each of these assumptions can be replaced by the following, weaker one. Consider a directed graph $\Gamma$ whose vertices are connected components of $M\setminus \bigsqcup_{i}B_i$, and for two such connected components, say $C_1$, $C_2$, there is an edge from $C_1$ to $C_2$ whenever there is an $i$ such that the closure $\bar{C}_1$ meets $\d^{-} B_{i}$, and the closure $\bar{C}_2$ meets $\d^{+}B_{i}$. Now, we assume that the graph $\Gamma$ has no loops.
\end{remark}
%	\begin{example}\label{ex:cusp-by-hand}
%		Let $\C^2\ni (x,y)\mapsto x^2-y^3\in \C$ be an ordinary cusp, let $F_{z}=f^{-1}(z)\cap \B_{4\epsilon}$ for some  $\epsilon>0$, $z\in \D_{\delta}^{*}$ for $0<\delta\ll 1$ be its Milnor fiber, and let $\phi\colon M\to M$ be a symplectic monodromy, introduced in Example \ref{ex:Milnor-fibration} for the standard area form $\omega\std$ on $\C^2$. We will now compute $\HF_{*}(\phi^6,+)$.
%		First, we need to find a suitable representative of $\phi$. We will construct it by hand, starting from the natural map $\Phi_{t}(x,y)= (e^{2\pi\imath\cdot t}x,e^{\frac{2\pi \imath}{3}\cdot t}y)$. In Example \ref{ex:cusp} we will see another computation, using the radius zero monodromy coming from the A'Campo space.
%		Each $\Phi_{t}|_{F_{\delta}}\colon F_{\delta}\to F_{\delta\cdot \exp(2\pi\imath \cdot t)}$ preserves the area, so $f^{-1}(\d D_{\delta})\times [0,1]\ni (x,y,t)\mapsto \Phi_{t}(x,y)\in f^{-1}(\d \D_{\delta})$ is a symplectic monodromy trivialization, see Definition \ref{def:sympmonod}. However, $\Phi_1$ is not compactly supported. To remedy this, we will modify $\Phi_{t}$, as follows. Fix smooth functions $\rho\colon [0,4\epsilon]\to [0,\delta]$, $\mu\colon [2\epsilon,4\epsilon]\to [0,1]$ such that $\rho|_{[0,\epsilon]}=0$, $\rho_{[2\epsilon,4\epsilon]}=1$; $\mu(2\epsilon)=1$, $\mu|_{[3\epsilon,4\epsilon]}=1$ and $\rho|_{(\epsilon,2\epsilon)}$ and $-\mu|_{(2\epsilon,3\epsilon)}$ are strictly increasing. For $z\in \C^{2}\setminus f^{-1}(0)$ let $\tau_{z}$ be the time $\rho(|z|)$ flow of the symplectic lift of the unit vector field on the ray $\{c\cdot f(z): c\geq 0\}\subseteq \C$. Eventually, define $\Phi'_{t}(z)=\tau_{z}\circ \Phi_{\mu(|z|)\cdot t}\circ \tau_{z}^{-1}$. Put $C=\tau_{\delta}(f^{-1}(0)\cap \B_{4\epsilon,3\epsilon})$, $\d_{\mathrm{out}} C=\tau_{\delta}(f^{-1}(0)\cap \d \B_{4\epsilon})$ and let $F$ be the connected component of $f^{-1}(\delta)\setminus \d_{\mathrm{out}}C$ containing $\Int C$. Then $F\times [0,1]\ni (z,t)\mapsto \Phi'_{t}(z)\in f^{-1}(\d \D_{\delta})$ is a symplectic monodromy trivialization relative to the collar trivialization $\psi\colon C\times \S^1\to f^{-1}(\d \D_{\delta})$ constructed in Setting \ref{basic-setting}. Thus $\phi\de \Phi'_{1}$ is as in Example \ref{ex:Milnor-fibration}.
%		For every $z\in \C^{*}$, we have $\Phi_{6}|_{f^{-1}(z)}=\id$, and $\Phi_{t}|_{f^{-1}(z)}$ has no fixed points for $t\in (0,6)$. As a consequence, we have $\Fix \phi=B\sqcup C$, where $B$ is the connected component of $f^{-1}(\delta)\cap \tau_{\delta}(f^{-1}(0)\cap \d \B_{2\epsilon})$ not containing $C$.
%	\end{example}

\begin{remark}\label{rem:McLean-HF3}
The original McLean spectral sequence introduced in  \cite[(HF3)]{McLean} converges to Floer cohomology. Its
first page is $E_{1}^{p,q}=\bigoplus_{\{i: \iota(i)=p\}} H_{d-p-q-\CZ(B_i)} (B_{i};\Z)$. In the setting of [49], this first page (tensored by $\Z_2$) is dual to the first page \eqref{eq:spectral-sequence-first-page}.

To see this, we recall from Remark \ref{rem:McLean-has-no-d+} that in [49] each codimension zero family $B_{i}$ has by assumption $\d^{-}B_{i}=\emptyset$. Moreover, \cite[(HF3)]{McLean} is formulated for a small slope deformation of $\phi$, which forces $\d^{M}B_i=\emptyset$. Thus each $B_i$ in loc.\ cit.\ has $\d B_{i}=\d^{+}B_{i}$, so the group $H_{d-p-q-\CZ(B_i)}(B_{i})$ is isomorphic, by Lefschetz duality, to $H^{d+p+q+\CZ(B_i)}(B_{i},\d^{+}B_{i})$. By the universal coefficient theorem, the latter is dual to $H_{d+p+q+\CZ(B_i)}(B_i,\d^{+}B_i)$, which is the group in \eqref{eq:spectral-sequence-first-page}%, as in \eqref{eq:spectral-sequence-first-page}.}%
, as claimed.

We have chosen the $\Z_2$ coefficients for convenience, since the same choice is made in our references [74, 4]. It is enough for most applications, including the proof of Theorem \ref{theo:Zariski}.
\end{remark}

For the proof of Proposition~\ref{prop:spectral-sequence}, the general idea is the following: choose a Hamiltonian perturbation of $\phi$ such that the fixed points become non-degenerate and contained in the union $\bigsqcup_i B_i$. Prove that any Floer trajectory connecting two fixed points contained in the same component $B_i$ is contained in $B_i$ too. Then, since $\phi|_{B_i}$ equals the identity, following the technique of the classical proof that Hamiltonian Floer homology coincides with ordinary homology, it is possible to identify the pieces of the first page of the spectral sequence. For the precise implementation of this scheme a couple of technicalities need to be taken into account.

A confinement lemma \cite[Lemma C.2]{McLean} of pseudo-holomorphic curves due to McLean is used: it allows to confine Floer trajectories in a small neighborhood $N_{B_i}$ of $B_i$ rather than in $B_i$. Since the almost complex structure $J_t$ needs to be $\phi$-periodic, and $\phi|_{N_{B_i}\setminus B_i}$ is not the identity, it is not possible to choose $J_t$ independent of time in $\phi|_{N_{B_i}\setminus B_i}$. On the other hand, for the proof of the classical fact that Hamiltonian Floer homology coincides with ordinary homology, an almost complex structure independent of time is used in order to prove that Floer trajectories associated with small time independent Hamiltonians are in fact Morse trajectories. In order to deal with this we will produce first a Hamiltonian deformation of $\phi$ whose fixed points is a disjoint union of codimension zero families of fixed points $B'_i$, and each $B'_i$ is strictly contained in the interior of $B_i$. Then a comparably smaller second deformation will produce non-degenerate fixed points inside $\bigsqcup_i B'_i$, and such that any Floer trajectory connecting two fixed points in the same $B'_i$ is contained in $B'_i$. This way, a $\phi$-periodic complex structure independent of $t$ in $\bigsqcup_i B_i$ will suffice to identify the groups appearing in the first page of the spectral sequence. In the rest of this section we carry out the proof following the program just hinted.

\subsubsection{Technical preparations} Our first aim is to provide a small Hamiltonian perturbation of $\phi$, which is well suited to apply the confinement lemma \cite[Lemma C.2]{McLean}; and a similar perturbation of the identity, which we will later use to compare the Floer complex with pieces of the Morse complex.

\paragraph{Step 1}\label{step:1}
By Lemma \ref{lem:cornerelimination}, we can and do assume that each $B_i$ is a connected codimension zero submanifold of $M$ with boundary, i.e.\ it does not have corners.
\smallskip

Choose a collar structure $\Psi:(-\eta,0]\times \partial M\to M$ near the boundary of $M$ such that  $\Psi^*\lambda=e^r\lambda|_{\partial M}$, see \eqref{eq:necktriv}, and $\phi=\id$ in the image of $\Psi$. Choose a small negative slope $a$ and consider a smooth decreasing function $\sigma\colon [-\eta,0]\to \RR$ that is constant near $-\eta$ and equal to the linear function $r\mapsto ar$ near $0$. Define a time-independent Hamiltonian $H$ by  $H=\sigma\circ \pr_{(-\eta,0]}\circ \Psi^{-1}$ in the image of $\Psi$, and $H=\sigma(-\eta)$ otherwise. It has small negative slope; and is $\phi$-periodic since it is constant whenever $\phi\neq \id$. Let $\phi'\de \phi\circ \psi_{1}^{H}$ be the corresponding small negative slope deformation $\phi'$ of $\phi$. Now, $\Fix(\phi')=\bigsqcup_{i=1}^{N}B_{i}''$, where each $B_{i}''$ is a codimension zero family of fixed points, contained in $B_{i}$ in such  way that $\d B_{i}''$ does not intersect $\partial M$ anymore. In terms of the decomposition \eqref{eq:boundarysplitting1}, we have
\begin{equation*}
%	\label{eq:boundarysplitting2}
\partial B_i''=\partial^+ B_i''\sqcup\partial^- B_i'',
\end{equation*}
$\d^{+} B_{i}''=\d^{+} B_{i}$, and the triple $(B_{i}'',\d^{+}B_{i}'',\d^{-}B_{i}'')$ is homeomorphic to $(B_{i},\d^{+}B_{i},\d^{-}B_{i}\sqcup \d^{M} B_i)$.

\begin{notation}\label{not:B=B''}
In the remaining part of the proof we will abuse notation and rename $B''_i=B_i$. This way, $\partial B_i=\partial^+ B_i\sqcup\partial^- B_i$.
\end{notation}

\begin{remark}\label{rem:d-B}
In terms of Notation \ref{not:B=B''}, %the technical
assumption \ref{item:ss-bd} of Proposition \ref{prop:spectral-sequence} implies the following. Take $i\in \{1,\dots, N\}$ such that $\d^{+}B_{i}\neq \emptyset$. Then there is a collar structure $\Psi\colon (-\eta,0]\times \d M \to M$, and a level $c\in (-\eta,0)$, such that every connected component of $\d^{-}B_{i}$ is a connected component of $\Psi(\{c\}\times \d M)$; and $\d^{+}B_i$ is disjoint from the collar $\Psi((-\eta,0]\times \d M)$. Hence if some connected component of $\bar{M\setminus B_i}$, say $C$, meets $\d^{-}B_i$, then $C$ is contained in a collar near $\d M$; in particular $C$ does not meet $\d^{+}B_i$.
%	Assume $H^{1}(M,\d M;\R)\neq 0$. Then in terms of Notation \ref{not:B=B''}, the technical assumption of Proposition \ref{prop:spectral-sequence} implies the following: for all $i\in \{1,\dots, N\}$, either $\d^{+}B_{i}=\emptyset$, or each connected component of $\d^{-}B_{i}$ meets exactly one connected component of $M\setminus B_i$.
\end{remark}

For the next steps we need the following notion.

\begin{definition}
\label{def:growcompat}
Let $F,G\colon U\to\RR$ be smooth functions on an open subset $U\subset M$. We say that $F$ and $G$ {\em grow compatibly in $U$} if the level sets of $F$ coincide with the level sets of $G$, and there is a smooth positive function $\sigma\colon U\to (0,\infty)$, constant on each of those level sets, such that $dF=\sigma\cdot d G$.
\end{definition}

\paragraph{Step 2}\label{step:2}
Let $H_{B_i}\colon N_{B_i}\to\RR$ be the time independent Hamiltonian associated to a codimension zero family $B_i$ of fixed points of $\phi'$, see Figure \ref{fig:FGH}. Choose a compact collar neighborhood $C_{B_i}$ of $\partial B_i$ contained in $N_{B_i}$ such that the restriction
$$H_{B_i}|_{\overline{C_{B_i}\setminus B_i}}:\overline{C_{B_i}\setminus {B_i}}\to\RR$$
is topologically locally trivial. We claim that there is a smooth function $H'_{i}\colon M\to \RR$ such that
\begin{enumerate}
\item\label{item:H_i-top-triv} the restriction $H_{i}'|_{C_{B_{i}}}$ is topologically locally trivial and a submersion at the interior of $C_{B_i}$,
\item\label{item:H_i-compatible}
the functions $H'_{i}$ and $H_{B_{i}}$ grow compatibly in the interior of $C_{B_{i}}\setminus B_{i}$, see Definition \ref{def:growcompat},
\item\label{item:H_i-extends} the restriction of $H_{i}'$ to each connected component of $M\setminus C_{B_i}$ is constant,
\item\label{item:H_i-action} the function $H'_i$ is constant in $B'_i\de\overline{B_i\setminus C_{B_i}}$ and in each connected component of $\partial B_i$. Furthermore, we have $\min\{H'_i(x):x\in\d^+B_i\}>H'_i(B'_i)>\max\{H'_i(x):x\in\d^-B_i\}$.
\end{enumerate}
It is easy to produce a function $\tilde{H}_{i}\colon U\to \R$ satisfying properties \ref{item:H_i-top-triv}--\ref{item:H_i-extends} in some neighborhood $U$ of $C_{B_i}$. In order to obtain $H_{i}'$ satisfying \ref{item:H_i-top-triv}--\ref{item:H_i-extends} and defined everywhere on $M$, we consider two cases.

First, we assume that condition \ref{item:ss-H1} holds, that is, $H^{1}(M,\d M;\R)=0$. By property \ref{item:H_i-extends}, the $1$-form  $d\tilde{H}_{i}$ extends by zero to a closed $1$-form on $M$, supported away from $\d M$. Since $H^{1}(M,\d M;\R)=0$, this $1$-form is exact, i.e.\ there is a smooth function $H_{i}'\colon M\to \R$ such that $dH_{i}'=d\tilde{H}_{i}'$ on $U$, and $dH_{i}'=0$ on $M\setminus U$. Thus $H_{i}'$ satisfies properties \ref{item:H_i-top-triv}--\ref{item:H_i-extends}, as needed.

Now, we assume that condition \ref{item:ss-bd} holds. We can choose $\tilde{H}_{i}$ so that it is constant on $B_i\cap \d C_{B_{i}}$. This way, $\tilde{H}_{i}$ extends to a smooth function $B_{i}\cup C_{B_{i}}\to \R$, constant on $B_i'$. Write $\d(B_i\cup C_{B_i})=\d^{+}\sqcup \d^{-}$, where $\d^{+}$ (respectively, $\d^{-}$) lies in the union of connected components of $C_{B_{i}}$ containing $\d^{+}B_i$ (respectively, $\d^{-}B_i$). We can furthermore choose $\tilde{H}_{i}$ so that it is constant on $\d^{+}$ and on $\d^{-}$. If $\d^{+}B_{i}=\emptyset$ then $\d^{+}=\emptyset$, so $\tilde{H}_{i}$ is constant on $\d(B_i\cup C_{B_i})$, and therefore it extends to a smooth function $H_{i}'\colon M\to \R$ satisfying  \ref{item:H_i-top-triv}--\ref{item:H_i-extends}, as needed. Assume $\d^{+}B_{i}\neq \emptyset$. Then by Remark \ref{rem:d-B}, no connected component of $M\setminus (B_{i}\cup \Int C_{B_{i}})$ meets both $\d^{+}$ and $\d^{-}$.  Hence $\tilde{H}_{i}$ is constant on the boundary of each connected component $M\setminus (B_i\cup \Int C_{B_i})$, so again it extends to a smooth function $H_{i}'\colon M\to \R$ satisfying  \ref{item:H_i-top-triv}--\ref{item:H_i-extends}, as required. Property \ref{item:H_i-action} now follows from \ref{item:H_i-top-triv}--\ref{item:H_i-extends}.
\smallskip

\begin{figure}[htbp]
\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-1,-3.5) rectangle (25,3.5);
\begin{scope}
\fill[black!5] (3,-2.5) -- (3,2.5) -- (7,2.5) -- (7,-2.5) -- (3,-2.5);
\fill[black!10] (4,-2.5) -- (4,2.5) -- (6,2.5) -- (6,-2.5) -- (4,-2.5);
\draw[->] (-1,0) -- (10,0);
\draw[->] (-0.5,-3) -- (-0.5,3);
\draw[black!50, dotted] (2,-2.5)--(2,2.5);
\draw[dotted] (3,-2.5)--(3,2.5);
\draw[black!50, dotted] (4,-2.5)--(4,2.5);
\draw[black!50, dotted] (6,-2.5)--(6,2.5);
\draw[dotted] (7,-2.5)--(7,2.5);
\draw[black!50, dotted] (8,-2.5)--(8,2.5);
\draw [decorate, decoration = {calligraphic brace}, very thick] (3,2.7) --  (7,2.7);
\node[above] at (5,2.7) {\small{$B_{i}$}};
\draw [decorate, decoration = {calligraphic brace}, very thick] (6,-2.7) --  (4,-2.7);
\node[below] at (5,-2.7) {\small{$B_{i}'$}};
\draw [decorate, decoration = {calligraphic brace}, very thick] (4,-2.7) --  (2,-2.7);
\node[below] at (3,-2.7) {\small{$C_{B_{i}}$}};
\draw [decorate, decoration = {calligraphic brace}, very thick] (8,-2.7) --  (6,-2.7);
\node[below] at (7,-2.7) {\small{$C_{B_{i}}$}};
\draw[very thick] (3,0.2) -- (3,-0.2);
\node[above] at (3,0.2) {\small{$\d^{-}B_{i}$}};
\draw[very thick] (7,0.2) -- (7,-0.2);
\node[below] at (7,-0.2) {\small{$\d^{+}B_{i}$}};
\draw[thick] (0,-3) to[out=45,in=180] (3,0) -- (7,0) to[out=0,in=-135] (10,3);
\node[right] at(10,3) {\small{$H_{B_{i}}$}};
\draw[thick] (0,-2) -- (2,-2) to[out=0,in=180] (4,0) -- (6,0) to[out=0,in=180] (8,2) -- (10,2);
\node[right] at(10,2) {\small{$H_{i}'$}};
\end{scope}
\begin{scope}[shift={(15,0)}]
\fill[black!5] (3,-2.5) -- (3,2.5) -- (7,2.5) -- (7,-2.5) -- (3,-2.5);
\fill[black!10] (4,-2.5) -- (4,2.5) -- (6,2.5) -- (6,-2.5) -- (4,-2.5);
\draw[->] (-1,0) -- (10,0);
\draw[->] (-0.5,-3) -- (-0.5,3);
\draw[black!50, dotted] (2,-2.5)--(2,2.5);
\draw[dotted] (3,-2.5)--(3,2.5);
\draw[black!50, dotted] (4,-2.5)--(4,2.5);
\draw[black!50, dotted] (6,-2.5)--(6,2.5);
\draw[dotted] (7,-2.5)--(7,2.5);
\draw[black!50, dotted] (8,-2.5)--(8,2.5);
\draw [decorate, decoration = {calligraphic brace}, very thick] (3,2.7) --  (7,2.7);
\node[above] at (5,2.7) {\small{$B_{i}$}};
\draw [decorate, decoration = {calligraphic brace}, very thick] (6,-2.7) --  (4,-2.7);
\node[below] at (5,-2.7) {\small{$B_{i}'$}};
\draw [decorate, decoration = {calligraphic brace}, very thick] (4,-2.7) --  (2,-2.7);
\node[below] at (3,-2.7) {\small{$C_{B_{i}}$}};
\draw [decorate, decoration = {calligraphic brace}, very thick] (8,-2.7) --  (6,-2.7);
\node[below] at (7,-2.7) {\small{$C_{B_{i}}$}};
\draw[very thick] (3,0.2) -- (3,-0.2);
\node[above] at (3,0.2) {\small{$\d^{-}B_{i}$}};
\draw[very thick] (7,0.2) -- (7,-0.2);
\node[below] at (7,-0.2) {\small{$\d^{+}B_{i}$}};
\draw[thick] (0,-1) -- (2,-1) to[out=0,in=180] (4,0) -- (6,0) to[out=0,in=180] (8,1) -- (9,1);
\node[right] at(9,1) {\small{$H_{i}''$}};
\draw[thick] (0,0) -- (3,0) to[out=0,in=180] (4.5,1) to[out=0,in=180] (5.5,-1) to[out=0,in=180] (7,0) -- (9,0);
\node[below] at(9,0) {\small{$F_{i}$}};
\end{scope}
\end{tikzpicture}
\caption{Auxiliary functions $H_{i}'$, $H_{i}''$ and $F_{i}$ introduced in Steps \protect\hyperref[step:2]{2} and  \protect\hyperref[step:3]{3}.}
\label{fig:FGH}
\end{figure}

Since the restrictions of $H'_{i}$ and $H_{B_i}$ to $\Int C_{B_i}\setminus B_i$ grow compatibly, there is a function $g\colon \Int C_{B_i}\setminus B_{i}\to (0,\infty)$, constant on the level sets of $H'_{i}$, such that $gX^{H_i'}=X^{H_{B_i}}$. Since $X^{H_{B_i}}$ is zero on $B_i$, and $X^{H_i'}$ does not vanish on $\d B_i$, the function $g$ extends by zero to a smooth function $N_{B_{i}}'\to [0,\infty)$, where $N_{B_{i}}'\de B_{i}\cup \Int C_{B_{i}}$.

Fix $r_i>0$ and put $H''_i=(r_i+g)H'_{i}$. Then $H''_i\colon N_{B_{i}}'\to\RR$ is a time independent Hamiltonian such that we have the equality $\psi^{H''_i}_1=\psi^{H_{B_i}}_1\comp\psi^{r_iH'_i}_1$. Indeed, at a level set of $H'_i$ the composition on the right hand side moves a point first time $r_i$ along an integral curve of $X^{H'_i}$, and then time $1$ along an integral curve of $X^{H_{B_i}}$. Moving time $1$ along an integral curve of $X^{H_{B_i}}$ is the same than moving time $g$ along an integral curve of $X^{H'_i}$ (recall that $g$ is constant on those integral curves), so the total effect is moving time $r_i+g$ along an integral curve of $X^{H'_i}$.

We claim that if $r_i$ is positive and small enough the set of fixed points of $\psi^{X^{H''_i}}_1$ in $N_{B_{i}}'$ is equal to $B'_i$. Indeed, $B'_i$ is obviously contained in the fixed point set of $\psi^{H''_i}_1$. To check the opposite inclusion we notice that, since the time $1$ flow of $X^{H_{B_i}}$ coincides with the time $1$ flow of $X^{gH'_i}$, and has no fixed points in $C_{B_i}\setminus B_i$, there exists a positive $r_i$ such that the time $1$ flow of $X^{(r_i+g)H'_i}$ has no fixed points in $C_{B_i}$, either. We fix such $r_i$ for the rest of the section.

%Let $B_1$, ..., $B_N$ be the collection of codimension zero family of fixed points of $\phi'$ (see Step 1).
Define $H'\de \sum_{i=1}^Nr_iH'_i$. The composition $\phi'':=\phi'\comp\psi_1^{H'}$ is an exact symplectomorphism whose fixed point set is the disjoint union of codimension zero families of fixed points $\bigsqcup_{i=1}^N B'_i$.  For every $i$ we have constructed a Hamiltonian $H''_i\colon N_{B_{i}}'\to\RR$ such that $\phi''|_{N_{B_{i}}'}$ is equal to the time $1$ flow $\psi^{H''_i}_1$.
\smallskip

We note that, under the weaker assumption of Remark \ref{rem:graph}, it might not be possible to construct individual functions $H_{i}'\colon M\to \R$. However, it is equally easy to construct a global function $H'\colon M\to \R$ and local $H_{i}''\colon N_{B_{i}}\to \R$; and this is all we need in the sequel. We leave the details to the reader.

\paragraph{Step 3}\label{step:3}
For each $i$ choose a function $F_{i}:B_i\to\RR$ such that
\begin{enumerate}
\item\label{item:F_i-extends} $F_i$ admits a smooth extension $F_i:M\to\RR$ by a constant function equal to $0$ in $M\setminus B_i$,
\item\label{item:F_i-no-crit} the restriction of $F_i$ to $\Int(C_{B_i}\cap {B}_i)$  has no critical points,
\item\label{item:F_i-compatible} the functions $F_i$ and $H_{i}''$ grow compatibly in $\Int(C_{B_i}\cap {B}_i)$,
\item\label{item:F_i-Morse} the critical points of $F_i|_{B_{i}}$ are of Morse type,
\item\label{item:F_i-Fix} for every $t\in (0,1]$, the only $t$-periodic orbits of the Hamiltonian flow of $F_{i}$ at the interior of $B_i$ are constant and equal to the Morse points.
\end{enumerate}
It is easy to produce a function with properties \ref{item:F_i-extends}--\ref{item:F_i-compatible}. A small generic perturbation of such a function which does not modify it at $\Int(C_{B_i}\cap {B}_i)$ achieves property \ref{item:F_i-Morse}. Property \ref{item:F_i-Fix} is achieved multiplying any function with properties \ref{item:F_i-extends}--\ref{item:F_i-Morse} by a sufficiently small positive constant, as a consequence of \cite[Proposition 6.1.5]{AD_book}.

Define $F:=\sum_{i=1}^N F_{i}$. For $\delta>0$ small enough define $\phi_\delta''' \de \phi''\comp\psi_1^{\delta F}$. In the same way that we proved above the existence of a time independent Hamiltonian $H''_i$ satisfying  $\psi^{H''_i}_1=\psi^{H_{B_i}}_1\comp\psi^{r_iH'_i}_1$, here we prove that there exists a time independent Hamiltonian $H_\delta$ such that $\phi'''_\delta=\phi\comp\psi_1^{H_\delta}$. So, we have that $\phi_\delta'''$ is a negative slope Hamiltonian perturbation of $\phi$. Moreover, if $\delta>0$ is small enough, that slope is small, too, and the fixed point set of $\phi_\delta'''$ is the disjoint union of the Morse points of $F_i$ at the interior of $B'_i$. Indeed, by property \ref{item:F_i-Fix} of $F_i$ the fixed points of $\phi_\delta'''$ contained in $B_i'$ are the Morse points of $F_i$. Outside $\bigsqcup_{i} N_{B_i}'$, $\phi_\delta'''$ coincides with $\phi'$, so it has no fixed points there. It remains to show that $\phi_\delta'''$ has no fixed points in $\bigsqcup_{i} C_{B_i''}$. This follows, if $\delta>0$ is small enough, by the same arguments that proved above that if $r_i>0$ is small enough the set of fixed points of $\psi^{H''_i}_1$ is equal to $B'_i$.

\paragraph{Step 4}\label{step:4}
In the three steps described up to now we have provided a Hamiltonian perturbation of $\phi$ that is well suited to analyze McLean's spectral sequence. In the next one we run a similar procedure for the identity symplectomorphism in $M$, with the later aim of producing a spectral sequence similar to McLean's. To prove Proposition \ref{prop:spectral-sequence}, we will compare the first pages of both spectral sequences.

Consider a smooth function $G\colon M\to\RR$ with the following properties:
\begin{enumerate}
\item we have $G|_{N_{B_i}'}=H''_i$,
\item\label{item:G-aviods} the restriction of $G$ to $M\setminus (\bigsqcup_{i} N_{B_i}')$ is a Morse function, and moreover the set of values of $G$ at the Morse points of $G|_{M\setminus (\bigsqcup_{i} N_{B_i}')}$ is disjoint from the finite set $\{G(B'_i)\}_{i=1}^{N}$.
\end{enumerate}

We consider the exact symplectomorphism $\psi'''_{\delta}:=\psi^{\delta G}_1\comp \psi_1^{\delta F}$ for $\delta>0$. As before, we prove that there is a time independent Hamiltonian $K_{\delta}$ such that $\psi'''_{\delta}=\psi^{K_{\delta}}_1$, so that $\psi'''_{\delta}$ is a small negative slope Hamiltonian deformation of $\id_M$. Moreover, $K_{\delta}$ is a Morse function in $M$ which satisfies $K_{\delta'}=\frac{\delta'}{\delta}K_{\delta}$. Therefore, by choosing $\delta$ small enough we may assume that the fixed points $\psi'''_{\delta}$ coincide with the Morse points of $K_{\delta}$ \cite[Proposition 6.1.5]{AD_book}. The function $K_{\delta}$ satisfies also the following properties:
\begin{enumerate}\setcounter{enumi}{2}
\item\label{item:K-boundaries} $\min\{K_\delta(x):x\in\d^+B_i\}>\max\{K_\delta(x):x\in\d^-B_i\}$.
\item\label{item:K-compatible} The functions $K_\delta$ and $H''_i$ grow compatibly in $\Int(C_{B_i}\cap B_i)$.
\end{enumerate}
For \ref{item:K-boundaries} notice that since $F_i|_{\partial B_i}\equiv 0$, we have $K_\delta|_{\d B_i}=\delta G|_{\d B_i}=\delta H''_i|_{\d B_i}=\delta (r_i+g)H'_i |_{\d B_i}=\delta r_iH'_i |_{\d B_i}$. Hence property \ref{item:H_i-action} of Step \hyperref[step:2]{2} implies \ref{item:K-boundaries}. Property \ref{item:K-compatible} is a consequence of property \ref{item:F_i-compatible} of Step \hyperref[step:3]{3}.

\paragraph{Step 5}\label{step:5}
Let $J_t,J'$ be cylindrical $\omega$-compatible almost complex structures such that $J_t$ is $\phi$-periodic, $J'$ is time-independent and they have the following additional properties:
\begin{enumerate}
\item\label{item:J'=Jt} $J_t$ is independent of $t$ in $\bigsqcup_{i} B_i$ and we have the equality %coincidence
\begin{equation}
\label{eq:coincidenceJ}
J_t|_{\bigsqcup_{i} B_i}=J'|_{\bigsqcup_{i} B_i}.
\end{equation}
\item\label{item:J'-Morse} the pair $(K_{\delta},J')$ is Morse-Smale for any $\delta>0$ (that is, the linearized Morse operator at each gradient line connecting two Morse points of $K_{\delta}$ is surjective, see \cite[Theorem 10.1.5]{AD_book}),% for more details),
\item\label{item:J-Morse} for each $i$ the pair $(F|_{B'_i},J|_{B'_i})$ is Morse-Smale at the interior of $B'_i$
\end{enumerate}
In order to construct $J_t$ and $J'$ we choose, following \cite[\sec 10]{AD_book}, a time independent $J'$ satisfying property \ref{item:J'-Morse} and then choose any cylindrical $\omega$-compatible and $\phi$-periodic $J_t$ satisfying property \ref{item:J'=Jt}. Property \ref{item:J-Morse} holds because $F|_{B'_i}$ is proportional by a non-zero constant factor to $K_{\delta}|_{B'_i}$.

\subsubsection{The two Floer complexes}

Having fixed $H_\delta$ (for $\delta>0$ small enough) and $J_t$ with the properties of the previous section, following \cite[\sec 8]{AD_book} we find that there exists a time dependent perturbation $\check{H}_{\delta,t}$ of the Hamiltonian $H_\delta$, which is as close as we wish to $H_\delta$, does not modify $H_\delta$ in a neighborhood of the fixed points of $\phi'''_\delta$ and in a neighborhood of $\partial M$, and such that $(\check{H}_{\delta,t},J_t)$ is regular for $\phi$, see Definition~\ref{def:perturbedregularpair}. Then the $\phi$-twisted loop Floer complex $\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$ introduced in Section \ref{sec:perturbed} is well defined and, by Proposition~\ref{prop:isotopy_invariance}, its homology is $\HF_*(\phi,+)$.

We define $\check{\phi}_\delta:=\phi\comp\psi_1^{\check{H}_{\delta,t}}$ and $\tilde{J}_{t}\de (\psi_1^{\check{H}_{\delta,t}})^{*}J_{t}$, see formula~\eqref{eq:Jtilde}. Then the $\check{\phi}_{\delta}$-fixed point Floer complex $\CFpt_{*}(\check{\phi}_\delta,\tilde{J}_t)$ introduced in Section \ref{sec:unperturbed} is well defined and, as we have seen in Section~\ref{sec:floerequivalence}, its homology is $\HF_{*}(\phi,+)$, too.

Similarly, there is a time dependent perturbation $\check{K}_{\delta,t}$ of $K_\delta$ which does not modify $K_{\delta}$ near the critical points of $G$, such that $(\check{K}_{\delta,t},J')$ is regular for the identity, see Definition \ref{def:perturbedregularpair}. Then the $\id$-twisted loop Floer complex $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$ is well defined, and its homology is $\HF_*(\id,+)$.%, see [74].

As before, putting $\tilde{J}_t'=(\psi_1^{\check{K}_{\delta,t}})^{*}J'$, we see that the $\psi_1^{\check{K}_{\delta,t}}$-fixed point Floer complex $\CFpt_{*}(\psi_1^{\check{K}_{\delta,t}},\tilde{J}'_t)$ is well defined and computes the same homology as $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$.% by Section~\ref{sec:floerequivalence}. Here $\tilde{J}'_t$ is the time-dependent almost complex structure derived from $J'$ by the procedure of Section~\ref{sec:floerequivalence}.

The perturbation procedure described in \cite[\sec 8]{AD_book} makes clear that the perturbations $\check{H}_{\delta,t}$ and $\check{K}_{\delta,t}$ can be chosen so that they coincide in $\bigsqcup_{i} B_i$, that is,
\begin{equation}
\label{eq:coincidenceH}
\check{K}_{\delta,t}|_{\bigsqcup_{i}  B_i}=\check{H}_{\delta,t}|_{\bigsqcup_{i}  B_i}.
\end{equation}
The idea is to replace the space of perturbations of the Hamiltonian, denoted by $\cC^\infty_\epsilon(H_0)$ in \cite[\sec 8]{AD_book}, by a Banach space of perturbations $\cC^\infty_\epsilon(H_\delta,K_\delta)$ whose elements are pairs $(h,k)$ where $h:M\times\RR\to \RR$ is a $\phi$-periodic time dependent Hamiltonian, $k:M\times\RR\to \RR$ is a periodic time dependent Hamiltonian and $h$ and $k$ coincide in  $\bigsqcup_{i}B_i$. Then it is possible to prove like in \cite[\sec 8]{AD_book} that the locus of $(h,k)$ such that $(H_\delta+h,J_t)$ and $(K_\delta+k,J')$ are both regular pairs is dense in $\cC^\infty_\epsilon(H_\delta,K_\delta)$.

\subsubsection{The McLean spectral sequence} We now use the action filtration introduced in Section \ref{sec:action-filtration} to define the McLean spectral sequence.

Recall that the action function $\ac$ of $\phi$ is constant in each codimension zero family $B_i$  of fixed points of $\phi$. We have denoted its value by $\ac(B_i)$. Choose a positive number $\xi$ such that $\xi<\tfrac{1}{3}\min\{|\ac(B_i)-\ac(B_j)|:\ac(B_i)\neq \ac(B_j)\}$.
Let $\iota$ be the indexing function from the statement of Proposition \ref{prop:spectral-sequence}. For an integer $p$ put $I_{p}=\iota^{-1}(p)$ and
$a_p\de \max\{\ac(B_{i}):\iota(i)\leq p\}$. Then for every $i\in I_{p}$ we have $\ac(B_{i})=a_{p}$. Put $b_{p}=a_{p}+\xi$. Clearly, $b_p\leq b_q$ if $p\leq q$.

Put $F_{p}\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)\de F^{\leq b_p}\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$, where the right hand side is defined in  Section \ref{sec:action-filtration}. This is an increasing filtration  by subcomplexes of $\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$. The \emph{McLean spectral sequence} $E_{p,q}^{r}$ is the spectral sequence associated with this filtration. We have $E_{p,q}^{r}\Rightarrow \HF_{p+q}(\phi,+)$.
\smallskip

Recall from Section \ref{sec:perturbed} that the complex $\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$ is generated as a graded vector space by the set $\cP_{\phi,\check{H}_{\delta,t}}$ of Hamiltonian $\phi$-twisted loops. Since we have chosen $\delta$ and $\check{H}_{\delta,t}$ small, the image of every $\gamma\in\cP_{\phi,\check{H}_{\delta,t}}$ is contained in some $B'_{i}$.

We have that $\phi'''_\delta|_{B'_i}$ is equal to the time $1$ Hamiltonian flow of $H_\delta|_{B'_i}$, which is equal to $\delta F_i|_{B'_i}$. Then, by property \ref{item:F_i-Fix} of $F_i$, the only $1$-periodic orbits of the Hamiltonian flow of $H_\delta|_{B'_i}$ are constant and equal to the Morse points of $F_i$. According to the procedure used to define the small perturbation $\check{H}_{\delta,t}$ following \cite[\sec 8]{AD_book}, we have that the only $\phi$-twisted Hamiltonian loops for the Hamiltonian $\check{H}_{\delta,t}$ which are contained in $B'_i$ are the same as those of $H_\delta|_{B'_i}$. Hence each $\gamma\in \cP_{\phi,\check{H}_{\delta,t}}$ is constant and equal to a Morse point of $F_i$ in $B_{i}'$, for some index $i$. Formula \eqref{eq:actiondef} shows that the action $\mathcal{A}_{\phi,\check{H}_{\delta,t}}$ of such point  belongs to $(\ac(B_{i})-\xi,\ac(B_{i})+\xi)$. We thus get an isomorphism of graded $\Z_{2}$-vector spaces
\begin{equation}\label{eq:first-page}
F_{p}\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)/F_{p-1}\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)\cong\bigoplus_{i\in I_{p}}C^i_{*},
\end{equation}
where $C^i_*$ is a subspace of $\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$ generated by the Morse points of $F_i|_{B'_i}$. Lemma \ref{lem:confinement} below implies that each $C^{i}_{*}$ is in fact a subcomplex of % and whose differential is induced by the Floer differential of
$F_{p}\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)/F_{p-1}\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$, hence \eqref{eq:first-page} is an isomorphism of chain complexes.
\smallskip

In order to compute the first page of McLean's spectral sequence we will compare it with the first page of a similar spectral sequence associated with a filtration in the $\id$-twisted loop Floer complex $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$ which we introduce now. Let $a'_1<\dots <a'_{m}$ be the critical values of $G$; notice that some of these values are equal to $G(B_{i}')$. Let $I_{j}'$ be the set of those indices $i$ such that $a_{j}'=G(B_{i}')$.

Since $\delta$ is as small as we wish, we have that $K_\delta$ and $G$ are as close as we wish in the maximum norm. Therefore we may assume that the critical values of $K_\delta$ belong to $\bigsqcup_{i=1}^{m}(a'_i-\xi',a'_i+\xi')$, for a fixed $\xi'>0$ such that $\xi'<\tfrac{1}{3}\min\{|a'_i-a'_j|:i\neq j\}$.

Fix $b'_1<\dots <b'_{m}$ such that $b'_{m}>a'_{m}+\xi'$ and $b'_i\de \frac{1}{2}(a'_i-a'_{i-1})$ for $i<m'$. We define an increasing filtration by subcomplexes of $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$ by $F_j\CFlp_{*}(\id,\check{K}_{\delta,t},J')\de F^{\leq b'_j}\CFlp_{*}(\id,\check{K}_{\delta,t},J_t)$, where as in Section \ref{sec:action-filtration}, $F^{\leq b'_j}\CFlp_{*}(\id,\check{K}_{\delta,t},J')$ is the subcomplex generated by Hamiltonian $\id$-twisted loops whose action is bounded from above by $b'_j$.

As before, since the perturbations $K_{\delta}$ and $\check{K}_{\delta,t}$ are small, we conclude that the set of Hamiltonian $\id$-twisted loops $\cP_{\id,\check{K}_{\delta,t}}$ coincides with the set of Morse points of $K_{\delta}$. Since the action function for the identity symplectomorphism can be taken vanishing identically, Formula~\eqref{eq:actiondef} shows that the action $\mathcal{A}_{\id,\check{K}_{\delta,t}}$ of such a point $x$ belongs to $(a'_{j}-\xi',a'_{j}+\xi')$ for a certain index $j$. If $I'_{j}\neq \emptyset$, property \ref{item:G-aviods} of $G$ shows that $x$ is a critical point of $F_{i}|_{B_{i}'}$ for some $i\in I_{j}'$. We conclude that, whenever $I_{j}'\neq\emptyset$, we have an isomorphism of graded $\Z_{2}$-vector spaces
\begin{equation}\label{eq:first-page_D}
F_{j}\CFlp_{*}(\id,\check{K}_{\delta,t},J')/F_{j-1}\CFlp_{*}(\id,\check{K}_{\delta,t},J')\cong\bigoplus_{i\in I_{j}'}D^i_{*},
\end{equation}
where $D^i_*$ is a subspace of $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$ generated by the Morse points of $F_i|_{B'_i}$. As before, Lemma \ref{lem:confinement} shows that \eqref{eq:first-page_D} is an isomorphism of chain complexes.

We claim that we have an isomorphism of chain complexes
\begin{equation}\label{eq:C=D}
\end{equation}
By definition, both underlying $\Z_2$-vector spaces are spanned by the Morse points of $F_{i}|_{B_{i}'}$, so they coincide. The grading of such a point, say $x$, in $C_{*}^{i}$ is $-\CZ_{\phi\circ \psi}(x)$, where $\psi$ is the time one flow of $\check{H}_{\delta,t}$. By \eqref{eq:coincidenceH}, $\psi$ coincides in $B_i$ with the time one flow of $\check{K}_{\delta,t}$. Hence the grading of $x$ in $D_{*}^{i}$ is $-\CZ_{\psi}(x)$.
Lemma \ref{lem:CZ-Hamiltonian}\ref{item:CZ_phi+Morse} shows that  $\CZ_{\phi\circ \psi}(x)=\CZ_{\phi}(x)+\CZ_{\psi}(x)=\CZ(B_i)+\CZ_{\psi}(x)$. Thus the grading of $x$ in $C_{*}^{i}$ equals the grading of $x$ in $D_{*}^{i}$ minus $\CZ(B_i)$, which gives the required degree shift.

It remains to  show that the differentials in complexes $D^{i}_{*}$ and $C^{i}_{*}$ are equal. They are computed by counting, respectively, $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$- and $\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$- Floer trajectories connecting the Morse points of $F_{i}|_{B_{i}'}$. By the confinement Lemma \ref{lem:confinement} below, these trajectories are contained in $B_{i}$. There, we have equalities $\phi|_{B_i}=\id_{B_i}$, $J_t|_{B_i}=J'|_{B_i}$, see~\eqref{eq:coincidenceJ}, and $\check{K}_{\delta, t}|_{B_i}=\check{H}_{\delta, t}|_{B_i}$, see~\eqref{eq:coincidenceH}. Hence the Floer trajectories counted by both differentials coincide, and we get isomorphism \eqref{eq:C=D}, as needed.
\smallskip

In the above proof, we have used the following confinement lemma for pseudo-holomorphic curves due to McLean, see \cite[Lemma C.2]{McLean}.

\begin{lema}
\label{lem:confinement}
Fix any index $p$. If $\delta>0$ and the perturbation $\check{H}_{\delta,t}$ are chosen small enough, for any two Morse points $q_1,q_2$ of $F$ which are contained in $\bigsqcup_{i\in I_p}B'_i$, we have that any  $\CFlp_{*}(\phi,\check{H}_{\delta,t},J_t)$-Floer trajectory from $q_1$ to $q_2$ %for the pair $(\check{H}_{\delta,t},J_t)$
has image completely contained in $\bigsqcup_{i\in I_p}B_i$.

Similarly, if $\delta>0$ and the perturbation $\check{K}_{\delta,t}$ are chosen small enough, for any two Morse points $q_1,q_2$ of $F$ which are contained in $\bigsqcup_{i\in I'_{p}}B'_i$, any $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$-Floer trajectory from $q_1$ to $q_2$ %for the pair $(\check{K}_{\delta,t},J')$
has image completely contained in $\bigsqcup_{i\in I'_{p}}B'_i$. %Notice that the confinement property is stronger in this case.
\end{lema}
\begin{proof}
For the first case the Lemma coincides exactly with \cite[Lemma C.2]{McLean}, applied to $\phi''$, where $\bigsqcup_{i\in I_{p}}B'_i$ plays the role of codimension zero family $B$ of fixed points and $\bigsqcup_{i\in I_{p}}B_i$ is the neighborhood where $\phi'$ is Hamiltonian, except that loc.\ cit.\ is formulated for $\CFpt_{*}$-Floer trajectories. The case of $\CFlp_{*}$-Floer trajectories is reduced to McLean's setting by the procedure described in Section \ref{sec:floerequivalence}. Moreover, in loc.\ cit.\ $B$ is assumed to be connected, which is not the case for $\bigsqcup_{i\in I_{p}}B'_i$. However, what is actually used in the proof is not the connectedness of $B$, but the fact that the action function $\ac$ is constant on $B$: this is satisfied by $\bigsqcup_{i\in I_{p}}B'_i$ by definition of $I_p$.

For the second case notice that by \cite[Proposition 10.1.9]{AD_book}, if $\delta$ is small enough the solutions of the perturbed Floer Cauchy--Riemann equation \eqref{eq:floerequlj} for $(K_\delta,J')$, i.e.\  $\CFlp_{*}(\id,K_{\delta},J')$-Floer trajectories, are time-independent, and hence equal to gradient trajectories for $-K_\delta$. By Property \ref{item:K-boundaries} of Step \hyperref[step:4]{4} a gradient trajectory starting and ending in $B'_i$ can not escape $B_i$. Moreover, by Property \ref{item:K-compatible} of Step \hyperref[step:4]{4}, a gradient trajectory cannot escape $B'_i$ either, and since the critical points of $K_\delta$ are contained at the interior of $B'_i$ we conclude that all gradient trajectories are contained at the interior of $B'_i$. This proves the confinement  of $\CFlp_{*}(\id,K_{\delta},J')$-Floer trajectories. Now if the perturbation $\check{K}_{\delta,t}$ is chosen small enough, confinement of $\CFlp_{*}(\id,\check{K}_{\delta,t},J')$-Floer trajectories follows from Gromov compactness.
\end{proof}

\begin{remark}
The proof of \cite[Lemma C.2]{McLean} uses a version of Gromov compactness due to Fish \cite[Theorem 3.1]{Fish}. The latter is a substantial generalization of [42]. However, the proof of \cite[Lemma C.2]{McLean} remains valid with the reference to [29] replaced by [42].
\end{remark}

\subsubsection{Computation of the first page}\label{sec:computation}
To prove Proposition \ref{prop:spectral-sequence}, we need to compute the homology of the complex $C_{*}^{i}$ appearing in the formula \eqref{eq:first-page}. Using the isomorphism \eqref{eq:C=D},  it is enough to compute the homology of the complex $D_{*}^{i}$, defined in \eqref{eq:first-page_D}.

We need to show that the homology of $D_{*}^{i}$ is the relative homology of $(B_{i}, \d^{+}B_{i})$, up to a degree shift. For this we follow the proof that $\HF_*(\id_M,+)=H_{*-d}(M,\ZZ_2)$ via continuation maps in Floer theory. In [74] it is remarked that our case (in which $M$ is a Liouville domain) is treated like in the classical proof that Hamiltonian Floer homology for compact $M$ with vanishing second homotopy group, equals ordinary homology shifted by the dimension. This proof is fully explained in [4].

We recall the needed details here. In \cite[\sec 10]{AD_book} it is proved that since the pair $(K_{\delta}, J')$ is Morse-Smale, the Morse complex of $(K_{\delta}, J')$ is well defined. Since we have $\xi K_\delta=K_{\xi\delta}$, if $\delta$ is chosen small enough, the $\CFlp_{*}(\id,K_{\delta}, J')$-Floer trajectories are independent of $t$, see \cite[Proposition 10.1.9]{AD_book}. Hence the pair $(K_{\delta}, J')$ is regular for $\id$. The $\id$-twisted Floer complex $\CFlp_{*}(\id,K_{\delta}, J')$ is well defined and coincides with the Morse complex of $(K_{\delta}, J')$.

In \cite[\sec 11]{AD_book}, using Floer's continuation maps, a quasi-isomorphism
\begin{equation}
\label{eq:continuationmap}
\Phi\colon \CFlp_{*}(\id,K_{\delta}, J')\to  \CFlp_{*}(\id,\check{K}_{\delta, t},J')
\end{equation}
is defined. The action filtration $F_{j} \CFlp_{*}(\id,K_{\delta}, J')$ is defined for the complex $ \CFlp_{*}(\id,K_{\delta}, J')$ in a similar way as above. We claim that the quasi-isomorphism~(\ref{eq:continuationmap}) can be defined in such a way that it respects the action filtration in both sides. To prove the claim we recall some ingredients involved in the construction of the continuation quasi-isomorphism. First we define continuation data $L:M\times\RR\times\RR\to\RR$, where $L$ is a function such that ${L}(p,t,s)=K_{\delta}(p)$ for $s\leq -1$ and  ${L}(p,t,s)\de \check{K}_{\delta,t} (p)$ for $s\geq -1$. Then a small generic perturbation $\tilde{L}:M\times\RR\times\RR\to\RR$ is constructed, without modifying outside $s\in [-1,1]$, and from it $\Phi$ is constructed by counting pseudo-holomorphic curves $u\colon \R\times\S^1\to M\times\R$. The point is that the pseudo-holomorphic curves satisfy the energy bound provided in \cite[Proposition 11.1.2]{AD_book}. If a pseudo-holomorphic curve $u$ connects $q_1$ and $q_2$ then its energy is bounded by
\begin{equation*}
\cA_{\id,K_{\delta}}(q_1)-\cA_{\id,\check{K}_{\delta, t}}(q_2)+c,
\end{equation*}
where $c$ is a constant determined in \cite[p.\ 387]{AD_book}: it is enough that $c$ bounds $\partial\tilde{L}/\partial s$. Choosing $K_{\delta}$ and its perturbation $\check{K}_{\delta, t}$ small, the perturbation $\tilde{L}$ can be chosen small, too. So we can set $c$ as small as we wish, and doing so makes it  clear that $\Phi$ preserves the action filtration.

As a consequence we obtain a quasi-isomorphism
\begin{equation*}%\label{eq:graded-pieces}
F_{j}\CFlp_{*}(\id,K_{\delta},J')/F_{j-1}\CFlp_{*}(\id,K_{\delta},J')\to F_{j}\CFlp_{*}(\id,\check{K}_{\delta,t},J')/ F_{j-1}\CFlp_{*}(\id,\check{K}_{\delta,t},J').
\end{equation*}
Assume $I'_{j}\neq\emptyset$. The complex on the right-hand side equals $\bigoplus_{i\in I_{j}'} D^{i}_{*}$, see \eqref{eq:first-page_D}. The complex on the left-hand side is defined analogously, as a vector space spanned by those critical points of $K_{\delta}$ whose action lies in the interval $(a_j'-\xi,a_j'+\xi)$. They are precisely the Morse points of $K_{\delta}$ contained in $\bigsqcup_{i\in I_{j}'}B_{i}'$. Since $K_{\delta}$ and $J'$ are time-independent, the Floer equation \eqref{eq:floerequlj} used to define the differential in $\CFlp_{*}$ reduces to the Morse equation used to define the differential in the Morse complex for $(-K_{\delta},J')$, see \cite[\sec 10.1]{AD_book} or \cite[Theorems 7.1 and 7.3]{Salamon_Zehnder}: note that the sign of the Hamiltonian gets switched as explained in Section \ref{sec:Hamiltonian}. Therefore, $D_{*}^{i}$ corresponds to a piece of the Morse complex for $(-K_{\delta},J')$ generated by the Morse points contained in $B_{i}$. This piece computes the relative homology of $(B_{i},\d^{+}B_i)$. Indeed, in the decomposition \eqref{eq:boundarysplitting1}, the gradient of $-K_{\delta}$ points inwards $\d^{+}B$ and outwards $\d^{-}B_i$, so %, and outwards $\d^{M}B_i$ since $K_{\delta}$ has negative slope:
the claim follows from \cite[(2.2)]{Rot-thesis}. Using Lemma \ref{lem:CZ-Hamiltonian}\ref{item:CZ_Morse} to compute the grading, we get
\begin{equation*}
\end{equation*}
cf.\ \cite[p.\ 834]{Seidel_Dehn-twist}.
Together with the formulas \eqref{eq:first-page} and \eqref{eq:C=D}, this gives
\begin{equation*}
E_{p,q}^{1}=\bigoplus_{i\in I_p} H_{p+q}(C_{*}^{i})=
\bigoplus_{i\in I_p} H_{p+q+\CZ(B_i)}(D^{i}_{*})=
\bigoplus_{i\in I_p} H_{p+q+\CZ(B_i)+d}(B_{i},\d^{+}B_{i}),
\end{equation*}
as claimed in Proposition \ref{prop:spectral-sequence}.

To get the analogous spectral sequence converging to $\HF_{*}(\phi,-)$, we need to replace the small negative slope by a small positive one. This way, in Step \hyperref[step:1]{1} $\d^{M}B_i$ becomes a part of $\d^{+}B_i$, see Notation \ref{not:B=B''}, so %the gradient of $K_{\delta}$ points outwards $\d^{M}B_i$, and
at the end of the day we get relative homology of $B_i$ modulo $\d^{+}B_{i}\sqcup \d^{M}B_i$, as claimed.

\section{The A'Campo abstract contact open book}\label{sec:monodromy}

In this section, we generalize McLean spectral sequence \cite[Theorem 1.2]{McLean}, which converges to the fixed point Floer homology of the monodromy. We will formulate it in Proposition \ref{prop:spectral-sequence-monodromy} after some preparations. To prove it, we will phrase Proposition \ref{prop:omega} in the language of graded abstract contact open books. This way, in Section \ref{sec:isotopy-to-radius-zero} we will get an isotopy between the natural abstract contact open book, constructed at (any small) positive radius in Section \ref{sec:basic-monodromy} (see Example \ref{ex:Milnor-fibration}), and the one at radius zero, given by the flow of \eqref{eq:monodromy-vector-field}, which has good dynamical properties just like the topological A'Campo model [3], or McLean model resolution [49], see Remark \ref{rem:McLean-5.41}. We summarize those properties in Proposition \ref{prop:monodromy}, which is the main result of this section. In particular, we will see that the radius zero monodromy satisfies the assumptions of Proposition \ref{prop:spectral-sequence}, which provides the required spectral sequence.

\subsection{The generalized McLean spectral sequence for isolated singularities}\label{sec:monodromy-spectral-sequence}
%\subsection{The statement of the spectral sequence}\label{sec:monodromy-dynamics}

We work with the monodromy graded abstract contact open book constructed in Example \ref{ex:typical} for a single function (without parameters). Let us briefly recall its definition.

Let $Z$ be a Stein manifold and let $\varrho\colon Z\to \R$ be an exhaustive strictly plurisubharmonic function. Consider the Liouville form $\lambda_Z:=-d^{c}\varrho$, see Section \ref{sec:symplectic-intro}. Let $Y\subseteq Z$ be a closed analytic subset whose all singularities are isolated, and, in case $\dim_{\C}Y=2$, their links are rational homology spheres. Let
\begin{equation*}
f\colon Y\to \C
\end{equation*}
be a holomorphic function such that $\Sing Y\subseteq f^{-1}(0)$. Assume that $0\in \C$ is the only critical value of $f$, and all singularities of  $f^{-1}(0)$ are isolated. Then $f|_{f^{-1}(\D_{\delta}^{*})}$ is a submersion for some $\delta>0$.

We choose regular values $\xi_{V}<\xi_{W}<\xi_{U}$ of $\varrho|_{Y}$, and put $U_Y=\varrho^{-1}(-\infty,\xi_U)\cap Y$, $V_Y= \varrho^{-1}(-\infty,\xi_V)\cap Y$ and $W_Y= \varrho^{-1}(-\infty,\xi_W)\cap Y$. Now,  Example \ref{ex:typical} (with $\epsilon=\eta=0$) provides a subset $N\subseteq Y$ such that $f^{-1}(0)\cap \bar{N}=f^{-1}(0)\cap \bar{W}_Y$, and the restriction $f|_{N}\colon N\to \D_{\delta}^{*}$ is a Liouville fibration, whose monodromy yields abstract contact open books
%Note that $\psi$ satisfies the assumptions \eqref{eq:ass-cohtrivial} and \eqref{eq:ass-exact} of Setting \ref{basic-setting}. Indeed, if $n=2$, these conditions hold by Remark \ref{rem:plane-curves}. If $n\geq 4$ then they hold because $N_{z}\setminus \d N_{z}$ is a Stein manifold of dimension $n-1\geq 3$, so we have the vanishing $H^{i}(N_{z},\d N_{z};\R)=H_{2n-2-i}(N_{z}\setminus \d N_{z};\R)=0$ for $i\in \{1,2\}$ required by equation \eqref{eq:vanishing}. Therefore, choosing  in Setting \ref{basic-setting} a standard family of loops $\Upsilon\colon (0,\delta]\times \S^1\ni (r,\theta)\mapsto r\theta\in \D_{\delta}^{*}$, we obtain, for each $z\in \D_{\delta}^{*}$, a monodromy abstract contact open book
\begin{equation}\label{eq:monodromy-acob}
\end{equation}
see \eqref{eq:basic-acob-example}. Note that the fiber $N_{z}$ is diffeomorphic to $f^{-1}(z)\cap \bar{W}_Y$. If $Y=\C^n$, $\varrho(z)=\tfrac{\pi}{2}||z||^2$, and $W_Y$ is a Milnor ball, then \eqref{eq:monodromy-acob} is the abstract contact open book constructed in Example \ref{ex:Milnor-fibration}.

Note also that in the formula \eqref{eq:monodromy-acob} we abuse the notation and write $\lambda_{Z}$ instead of its restriction $\lambda_{Z}|_{N_{z}}$. We will do so several times in this section, to keep the formulas concise.%, we will abuse notation this way several times in this section.
\smallskip

We assume furthermore that
\begin{equation}\label{eq:c1-assumption}
c_{1}(Y\setminus f^{-1}(0))=0.
\end{equation}

Then, as explained in Remark \ref{rem:c1}\ref{item:c1_Kahler}, a nonvanishing section of $K_{Y\setminus f^{-1}(0)}$ endows the abstract contact open book \eqref{eq:monodromy-acob} with a grading. Its graded isotopy type does not depend on $z\in \D_{\delta}^{*}$.
%key technical ingredient in the proof of Theorem \ref{theo:Zariski}.
\smallskip

To state our results, we need to fix some additional data. Let $h'\colon Z'\to Z$ be a log resolution of $(f,Y)$. That is, $h'$ is a proper modification such that, if $X$ denotes the  proper transform of $Y$, then $h\de h'|_X\colon X\to Y$ is a resolution of $f$. Put $D=(f\circ h)^{-1}(0)$. Then $X$ is a smooth complex manifold, and $D\redd$ is snc. We assume that $h$ is an isomorphism away from $D$ and near the preimage of the collar $\bar{U}_Y\setminus V_Y$. We write the irreducible decomposition of $D$ as
\begin{equation*}
D=\sum_{i\in \cP} D_{i}+\sum_{i\in \cE} m_{i}D_{i},
\end{equation*}
where the first sum runs through the components of the proper transform of $f^{-1}(0)$, and the second one through exceptional components of $h$. %In other words, $h^{-1}_{*}(f^{-1}(0))=\sum_{i\in \cP} D_{i}$ and $\Exc h=\sum_{i\in \cE} D_{i}$.
For $i\in \cP$ we put $m_{i}=1$.

Fix an integer $m\geq 1$ and assume that the log resolution $h$ is \emph{$m$-separating}, that is,
\begin{equation}\label{eq:m-separatedness}
\mbox{if } i\neq j \mbox{ and } D_{i}\cap D_{j}\neq \emptyset \mbox{ then } m_{i}+m_{j}>m.
\end{equation}
It can always be achieved by composing $h'$ with further blowups, see \cite[Lemma 2.9]{BBLN_contact-loci}.

Since $Y$ is Stein, we can fix an ample divisor $H$ on $X$ such that
\begin{equation}\label{eq:H}
H=\sum_{i\in \cE}b_{i}D_{i}
\end{equation}
for some negative integers $b_{i}$. For $i\in \cP$ we put $b_i=0$.
\smallskip

We will also need some additional notation. First, by assumption \eqref{eq:c1-assumption}, we can write the canonical bundle of $X$ as
\begin{equation}\label{eq:K}
K_{X}=\sum_{i\in \cP\cup \cE} a_{i}D_{i},
\end{equation}
for some \emph{discrepancies} $a_{i}\in \Z$. Note that if $c_{1}(Y)=0$ then we have $a_{i}=0$ for $i\in \cP$. We put
\begin{equation}\label{eq:Sm}
S_{m}=\{i\in \cP\cup \cE : m_{i}|m\},\quad
S_{m,p}=\{i\in S_{m}: p=\frac{m}{m_i}b_{i}\}.
\end{equation}
%so $S_{m}$ indexes those components of $D_i$ whose multiplicities divide the fixed number $m$, and we have a decomposition $S_{m}=\bigsqcup_{p}S_{m,p}$.
Note that $\cP\subseteq S_{m,0}$.

Fix $i\in \cP\cup \cE$. Recall from \eqref{eq:stratification} the notation $X_{i}^{\circ}=D_{i}\setminus (D-D_{i})$. Put $D_{i}^{\circ}= X_{i}^{\circ}\cap \bar{W}_X$, where $W_X=h^{-1}(W_Y)$. We now introduce a natural $m_i$-fold covering
\begin{equation*}
\nu_{i}\colon B_{i}^{\circ}\to D_{i}^{\circ},
\end{equation*}
following \cite[\sec 2.3]{Denef_Loeser-Lefshetz_numbers}. It can be thought of as a piece of the fiber $f^{-1}(\delta)\cap \bar{W}_Y$ which \enquote{corresponds to} the component $D_i$. To define $\nu_i$, fix any holomorphic chart $G_{X}\subseteq X$ around a point of $X_{i}^{\circ}$, and let $\{z_{i}=0\}$ be a local equation of $D_{i}$ in $G_{X}$. Then  $f|_{G_{X}}=\mu\cdot z_{i}^{m_i}$ for some $\mu\in \cO_{X}^{*}(G_{X})$. Define $\tilde{G}_{X}=\{(z,x)\in \C\times (G_{X}\cap D_{i}^{\circ}): \mu(x)\cdot  z^{m_{i}}=1 \}$. Gluing these charts, one gets a topological covering $\nu_{i}\colon B_{i}^{\circ}\to D_{i}^{\circ}$ with Galois group $\Z_{m_i}$, as needed.

Note that if $i\in \cE$, i.e.\ $D_i$ is contracted by $h$, then $\nu_i$ is exactly the $m_i$-fold covering of $X_{i}^{\circ}$ considered in \cite[\sec 2.3]{Denef_Loeser-Lefshetz_numbers} or \cite[Theorem 1.2]{McLean}. If $i\in \cP$, i.e.\ $D_{i}$ is a proper transform of some component of $f^{-1}(0)$, then $\nu_i$ is an isomorphism; and $B_{i}^{\circ}$ can be identified with a submanifold $X_{i}^{\circ}\cap \bar{W}_{X}$ of $X_{i}^{\circ}$, with boundary $D_i\cap \d \bar{W}_{X}$. In other words, if $i\in \cP$ then $B_{i}^{\circ}$ is diffeomorphic to $(h(D_i)\cap \bar{W}_Y)\setminus \operatorname{Bs} h^{-1}$, where $\operatorname{Bs} h^{-1}$ is the locus where $h$ is not an isomorphism.

With this notation at hand, we can formulate the first result of this section. We denote by $H^{BM}_{*}$ the Borel--Moore homology with coefficients in $\Z_2$.

\begin{prop}\label{prop:spectral-sequence-monodromy}
There is a spectral sequence $E_{p,q}^{r}$ converging to $\HF_{*}(\phi_{z}^{m},+)$ whose first page equals
\begin{equation}\label{eq:spectral-sequence-monodromy}
E_{p,q}^{1}=\bigoplus_{i\in S_{m,p}} H^{BM}_{n-1+p+q+2\frac{m}{m_i}(a_i+1)-2m}(B_{i}^{\circ}).
\end{equation}
\end{prop}

Proposition \ref{prop:spectral-sequence-monodromy} will be proved in two steps. First, in Section \ref{sec:isotopy-to-radius-zero}, we will use the lift of the radial vector field on $\C_{\log}$ to isotope the abstract contact open book \eqref{eq:monodromy-acob} to the abstract contact open book $(F,\lambda,\phi)$ at radius zero, i.e.\ at the boundary of the A'Campo space; where $\phi$ is the time one flow of the vector field \eqref{eq:monodromy-vector-field}. Next, in Section \ref{sec:monodromy-dynamics} we will identify $B_{i}^{\circ}$ with the intersections $F\cap A_{i}^{\circ}$, see \eqref{eq:stratification-pullback}, which, as we will see, are codimension zero families of fixed points of $\phi$ whenever $m_i|m$. The $m$-separating condition \eqref{eq:m-separatedness} will guarantee that $\phi$  has no more fixed points. %In Proposition \ref{prop:monodromy}\ref{item:B_i-action},\ref{item:B_i-CZ} we will compute the action and the Conley--Zehnder index for these families.
Once this is done, Proposition \ref{prop:spectral-sequence-monodromy} will follow from Proposition \ref{prop:spectral-sequence}.

\begin{remark}\label{rem:chosen_H}
The spectral sequence \eqref{eq:spectral-sequence-monodromy} depends on the choice of the $m$-separated resolution $h$ and on the ample divisor $H$. Since the limit $\HF_{*}(\phi^{m},+)$ does not depend on $h$ and $H$, flexibility of these choices suggests that some Floer differentials in \eqref{eq:spectral-sequence-monodromy} should be zero.

One reason for such vanishing can be seen as follows. Since the action functional decreases along the Floer trajectories, see Section \ref{sec:action-filtration}, if we have an inequality of actions  $\ac(B_{i}^{\circ})<\ac(B_{j}^{\circ})$ then there are no Floer trajectories from $B_{i}^{\circ}$ to $B_{j}^{\circ}$. Hence any differential from (a sub-quotient of) $H^{BM}_{*}(B_{i}^\circ)$ to $H^{BM}_{*}(B_{j}^{\circ})$ vanishes. By Proposition \ref{prop:monodromy}\ref{item:B_i-action}, the inequality $\ac(B_{i}^{\circ})<\ac(B_{j}^{\circ})$ is equivalent to $\frac{b_i}{m_i}<\frac{b_j}{m_j}$.

For example, let $f$ be an irreducible plane curve germ. Then the dual graph of $D$ is a tree, with the proper transform of $f^{-1}(0)$ as a root. Following the resolution process it is easy to see that, for any $H$, the quotients $\frac{b_i}{m_i}$ decrease as we move away from the root. Thus if $D_i$ lies closer to the root than $D_j$, all differentials in \eqref{eq:spectral-sequence-monodromy} from $H_{*}^{BM}(B_i^{\circ})$ to $H^{BM}_{*}(B_{j}^{\circ})$ vanish.  However, in the opposite situation, that is if $D_{j}$ lies closer to the root than $D_i$, the ampleness condition is restrictive enough so that we cannot guarantee the vanishing of differentials from $H^{BM}_{*}(B_{i}^{\circ})$ to $H^{BM}_{*}(B_{j}^{\circ})$, see Example \ref{ex:cusp}.

This result is compatible with the corresponding one concerning topology of the $m$-th contact locus $\mathscr{X}_{m}$ of $f$. Recall that \cite[Theorem 1.1]{BBLN_contact-loci} provides a spectral sequence converging to $H^{*}_{c}(\mathscr{X}_m)$ whose page is dual (up to a shift) to \eqref{eq:spectral-sequence-monodromy}, which  leads to the \emph{Arc-Floer conjecture} $\HF_{*}(\phi^{m},+)\cong H^{*}_{c}(\mathscr{X}_{m})$, see Conjecture 1.5 loc.\ cit. The differentials $H_{*}(B_j)\to H_{*}(B_{i})$ in the above spectral sequence count, roughly speaking, families of arcs lifting to $D_{i}$, whose limit lifts to $D_j$. In case of irreducible plane curves, the proof of \cite[Theorem 3.4]{JaviEdu_conctact-loci} shows that there are no such families if $D_i$ lies closer to the root than $D_j$, so the corresponding differential, again, is zero. But, in the opposite situation, if $D_j$ lies closer to the root than $D_i$, the differential can be non-zero.
\end{remark}

\begin{example}\label{ex:cusp}
Let $f$ be a simple cusp $x^{2}-y^{3}$. By \cite[p.\ 26]{JaviEdu_conctact-loci} we have $\HF_{*}(\phi^{6},+)=H_{*-1}^{BM}(B^{\circ})$, where $B^{\circ}$ is the entire Milnor fiber, i.e.\ a punctured torus. Thus $\HF_{*}(\phi^{6},+)$ equals $\Z_2$ in degrees $2$ and $3$, and $0$ in the remaining degrees. Let $X\to \C^{2}$ be the minimal log resolution of $f$; it is $6$-separated. We now describe the spectral sequence \eqref{eq:spectral-sequence-monodromy} for various choices of ample divisors $H$ on $X$.

Write $D=D_0+2D_1+3D_2+6D_3$, where $D_0$ is the proper transform of $f^{-1}(0)$, $D_3$ is the component of $\Exc h$ meeting $D_0$, and $D_1,D_2$ are tips of $\Exc h$. Then $B_{1}^{\circ}$ and $B_{2}^{\circ}$ are disjoint unions of $3$ and $2$ disks, respectively; and $B_{3}^{\circ}$ is a torus with six disks removed. Since $K_{X}=D_{1}+2D_{2}+4D_{3}$, by Proposition \ref{prop:monodromy}\ref{item:B_i-CZ} we have $\CZ(B_{1}^{\circ})=\CZ(B_{2}^{\circ})=0$ and $\CZ(B_{3}^{\circ})=-2$. A direct computation shows that a divisor $H=\sum_{i=1}^{3}b_{i}D_{i}$ is ample if and only if $\frac{b_1}{m_1},\frac{b_2}{m_2}<\frac{b_3}{m_3}$, see Remark \ref{rem:chosen_H}, and $2\frac{b_{1}}{m_{1}}+3\frac{b_2}{m_2}>6\frac{b_{3}}{m_{3}}$.

Choose $H$ so that $\frac{b_{1}}{m_{1}}=\frac{b_{2}}{m_{2}}=\frac{-13}{6}$ and $\frac{b_{3}}{m_{3}}=\frac{-12}{6}$ (strictly speaking, $H$ is a $\Q$-divisor, but it is easy to see that \eqref{eq:spectral-sequence-monodromy} works  as long as $6\cdot \frac{b_i}{m_i}\in \Z$). Then the nonzero entries of the $E^{1}$ page of \eqref{eq:spectral-sequence-monodromy} are $E^{1}_{-13,14}=H^{BM}_{2}(B_{1}^{\circ})\oplus H^{BM}_{2}(B_{2}^{\circ})=\Z_{2}^{5}$, $E^{1}_{-12,14}=H^{BM}_{1}(B_{3}^{\circ})=\Z_{2}^{6}$, and $E^{1}_{-12,15}=H^{BM}_{2}(B_{3}^{\circ})=\Z_{2}$. In particular, in agreement with Remark \ref{rem:chosen_H}, there is no differential from $H_{*}^{BM}(B_{1}^{\circ})$ or $H_{*}^{BM}(B_{2}^{\circ})$ to $H_{*}^{BM}(B_{3}^{\circ})$. On the other hand, there is a non-trivial differential in the opposite direction. Indeed, since we know that the limit $\HF_{*}(\phi^{6},+)$ is $0$ in degree $1$,  we see that the differential $E^{1}_{-12,14}\to E^{1}_{-13,14}$ is surjective, and the spectral sequence degenerates on the second page.

Similarly, choose $H$ so that  $\frac{b_{1}}{m_{1}}=\frac{-13}{6}$, $\frac{b_{2}}{m_{2}}=\frac{-14}{6}$ and $\frac{b_{3}}{m_{3}}=\frac{-12}{6}$. Then the nonzero entries on the $E^{1}$-page are $E^{1}_{-14,15}=H^{BM}_{2}(B_{2}^{\circ})=\Z_{2}^{2}$, $E^{1}_{-13,14}=H^{BM}_{2}(B_{1}^{\circ})=\Z_{2}^{3}$; and as before  $E^{1}_{-12,14}=H^{BM}_{1}(B_{3}^{\circ})=\Z_{2}^6$, $E^{1}_{-12,15}=H^{BM}_{2}(B_{3}^{\circ})=\Z_{2}$. The only nonzero differential on $E^1$ is $E^{1}_{-12,14}\onto E^{1}_{-13,14}$; on $E^{2}$ it is $E^{2}_{-12,14} \onto E^{2}_{-14,15}$ and on $E^{3}$ the spectral sequence degenerates. These differentials count Floer trajectories from $B_{3}^{\circ}$ to $B_{1}^{\circ}$ and $B_{2}^{\circ}$; there are no Floer trajectories from $B_{1}^{\circ}\sqcup B_{2}^{\circ}$ to $B_{3}^{\circ}$.

For $\frac{b_{1}}{m_{1}}=\frac{-14}{6}$, $\frac{b_{2}}{m_{2}}=\frac{-13}{6}$, $\frac{b_{3}}{m_{3}}=\frac{-12}{6}$, we get an analogous spectral sequence, with the roles of $D_1$ and $D_2$ interchanged. 	For $\frac{b_{1}}{m_{1}}=\frac{-15}{6}$, $\frac{b_{2}}{m_{2}}=\frac{-13}{6}$, $\frac{b_{3}}{m_{3}}=\frac{-12}{6}$, the spectral sequence degenerates on $E^{4}$.

% For the point $(-15,-13)$, one gets a spectral sequence degenerating on the fourth page. Still, the only non-trivial differential counts Floer trajectories from $B_{3}^{\circ}$ to $B_{1}^{\circ}$ and $B_{2}^{\circ}$, never the other way round.
\end{example}

\subsection{Isotopy to radius zero}\label{sec:isotopy-to-radius-zero}

We will now construct the \enquote{radius zero} graded abstract contact open book $(F,\lambda,\phi)$, and show that it is isotopic to our original one \eqref{eq:monodromy-acob}. We keep the notation and assumptions from Section \ref{sec:monodromy-spectral-sequence}.

Write $\tilde{f}\de f\circ h$. Put $\tilde{\lambda}_{Y}=-d^{c}(\varrho\circ h)\in \Omega^{1}(X)$, i.e.\ $\tilde{\lambda}_{Y}$ is the suitable restriction of the pullback of the $1$-form $\lambda_{Z}$.
Since $h|_{X\setminus D}$ is an isomorphism, the form $\tilde{\lambda}_{Y}$ makes $\tilde{f}|_{h^{-1}(N)}$ a Liouville fibration. Moreover, since $h$ is an isomorphism near the collar, the collar trivialization chosen in Section \ref{sec:monodromy-spectral-sequence} lifts to $X$. In particular, for each $z\in \D_{\delta}^{*}$ we have a graded abstract contact open book  $(\tilde{N}_{z},\tilde{\lambda}_{Y},\tilde{\phi}_{z})\de (h^{-1}(N_z),\tilde{\lambda}_{Y},h^{-1}\circ\phi_{z}\circ h)$ isomorphic to \eqref{eq:monodromy-acob}. We will now modify the form $\tilde{\lambda}_{Y}|_{X\setminus D}$ to a form $\lambda_{X}$ such that $d\lambda_{X}$ extends to a K\"ahler form on $X$.

Recall that, in formula \eqref{eq:H}, we have fixed an ample divisor $H$. Fix an integer $b>0$ such that $bH$ is very ample, and let $\jmath\colon X\into \P^{\dim |bH|}$ be the embedding given by $|bH|$. Let $\tfrac{1}{4\pi} \omega_{\mathrm{FS}} \in \Omega^{2}(X)$ be the pullback of the Fubini-Study form through $\jmath$. The basis of $H^{0}(\cO_{X}(bH))$ used to define $\jmath$ gives a hermitian metric on $\cO_{X}(bH)$, see \cite[Example 4.1.2(i)]{Huybrechts}.  Let $||\sdot ||$ be the corresponding norm. Choose a section $s$ of $\cO_{X}(bH)$ which is supported on $H$, i.e.\ with a pole of order $-bb_{i}>0$ along $D_{i}$ for each $i\in \cE$, and no zeros or poles elsewhere.
Then by \cite[Examples 4.3.9(iii) and 4.3.12]{Huybrechts}, we have $\omega_{\mathrm{FS}}=2\imath\cdot \bar{\d}\d\log ||s||$. Since $d^{c}=\imath(\d-\bar{\d})$, see Section \ref{sec:symplectic-intro}, we have %$\bar{\d}\d=\frac{i}{2}dd^{c}$
$dd^{c}=2\imath\cdot \bar{\d}\d$, so
\begin{equation*}
\omega_{\mathrm{FS}}=d\lambda_{\mathrm{FS}},\quad \mbox{where}\quad
\lambda_{\mathrm{FS}}\de d^{c}\log ||s||\in \Omega^{1}(X\setminus \Exc h).
\end{equation*}
We remark that the $1$-form $\lambda_{\mathrm{FS}}$ is, up to scaling, the same form as the one used in \cite[Example 5.14]{McLean}: the sign difference results from varying definitions of $d^{c}$, explained in Section \ref{sec:symplectic-intro}.
\smallskip

For $t\geq 0$ put $\lambda_t=\tilde{\lambda}_{Y}+t\cdot \lambda_{\mathrm{FS}}\in \Omega^{1}(X\setminus \Exc h)$, and put  $\omega_{t}\de d\lambda_{t}$. Then $\omega_{t}$ is a K\"ahler form on $X\setminus D$ for every $t\geq 0$, and on $X$ for every $t> 0$. In particular, since $f$ is holomorphic, these forms are fiberwise symplectic over $\D_{\delta}^{*}$. Now, the map $(\tilde{f},\pr_{[0,t_0]})\colon X\times [0,t_0]\to \C\times [0,t_0]$ fits into Setting \ref{basic-setting}, with $P=[0,t_0]$, $P'=\{0\}$. Hence after possibly shrinking $t_0>0$, we get a subset $M\subseteq X\times [0,t_0]$ such that the above map restricted to $M\setminus (D\times [0,t_0])$ is a Liouville fibration over $\D_{\delta}^{*}\times [0,t_0]$, with a collar trivialization which extends over $\D_{\delta}\times [0,t_0]$, and agrees over $\D_{\delta}\times \{0\}$ with the one chosen in Section \ref{sec:monodromy-spectral-sequence} to define \eqref{eq:monodromy-acob}. Taking for $\Upsilon$ the standard family of loops, the construction in Section \ref{sec:basic-setting} provides monodromy graded abstract contact open books $(M_{z,t},\lambda_{t},\phi_{z,t})$, see formula \eqref{eq:basic-acob}. %Note that in case $\dim_{\C}Y=2$ we use the assumption on the links of $Y$ to apply Remark \ref{rem:exact} making $\phi_{z,t}$ exact; and we use the vanishing vanishing \eqref{eq:vanishing} to endow $(M_{z,t},\lambda_{t},\phi_{z,t})$ with a grading.

Since $(M_{z,0},\lambda_{0},\phi_{z,0})=(\tilde{N}_{z},\tilde{\lambda}_{Y},\tilde{\phi}_{z})$, we infer that for every $z\in \D_{\delta}^{*}$ and every $t\in [0,t_0]$, the abstract contact open book $(M_{z,t},\lambda_{t},\phi_{z,t})$ is graded isotopic to \eqref{eq:monodromy-acob}.

Put $\lambda_{X}=\lambda_{t_0}$, $\omega_{X}=d\lambda_{X}$. Then $\omega_{X}$ extends to a K\"ahler form on $X$. Let $A$ be the A'Campo space associated to the function $\tilde{f}\colon X\to \C$, with natural smooth maps $\pi\colon A\to X$ and $\tilde{f}_{A}\colon A\to \C_{\log}$ as in Proposition \ref{prop:AXsmooth}, see diagram \eqref{eq:AX-diagram}.

We note that the pullback $\pi^{*}\lambda_{X}$ extends to a $1$-form on $A$, i.e.\ assumption \eqref{eq:lambda_extends} of Setting \ref{basic-setting2} holds. To see this, we give a local description of $\lambda_{X}$, which will be useful in the proof of Proposition \ref{prop:monodromy}\ref{item:B_i-action}, too. Choose a fine chart $U_{X}$ (see Definition \ref{def:finechartz}) with associated index set $S$ (see \eqref{eq:index-set}), and a trivialization of $\cO_{X}(bH)|_{U_X}$. In this trivialization, we can write our section $s$ as $\xi' \cdot \prod_{i\in S}z_{i}^{bb_{i}}$ for some nonvanishing holomorphic function $\xi'$. The formula for $||\sdot||$ from \cite[Example 4.1.2(i)]{Huybrechts} shows that $\log ||s||=\sum_{i\in S}bb_{i}\log r_{i}+\xi$ for some $\xi\in \cC^{\infty}(U_X)$. Thus $\lambda_{\mathrm{FS}}|_{U_X}=d^{c}\log||s||=-\sum_{i\in S}bb_id\theta_i+d^{c}\xi$, see Example \ref{ex:J}. Since $\tilde{\lambda}_{Y}\in \Omega^{1}(X)$, we get
\begin{equation}\label{eq:lambdaA-locally}
\pi^{*}\lambda_{X}|_{U_X}=-\sum_{i\in S}t_0bb_i\, d\theta_{i}+\pi^{*}\beta\quad\mbox{for some } \beta\in \Omega^{1}(U_X).
\end{equation}
Because the functions $\theta_{i}$ are coordinates of each chart \eqref{eq:AC-chart} covering $\pi^{-1}(U_X)$, the formula \eqref{eq:lambdaA-locally} shows that $\pi^{*}\lambda_{X}$ extends to a smooth form on $A$, as claimed.

We are now in Setting \ref{basic-setting2}. Let $F=\tilde{f}_{A}^{-1}(0,1)\cap \pi^{-1}(\bar{W}_{X})$ be a radius-zero fiber, and for $\delta',\epsilon>0$ let $\lambda_{A}^{\delta',\epsilon}\in \Omega^{1}(A)$ be the $1$-form defined in \eqref{eq:lambdaACdelta}. By Corollary \ref{cor:isotopyisotopy-to-radius-zero}, there is an arbitrarily small  $\epsilon_0>0$ such that, for any $\delta'>0$, the abstract contact open book $(M_{z,t_0},\lambda_{t_0},\phi_{z,t_{0}})$ constructed above is isotopic to
\begin{equation}\label{eq:radius-zero-acob}
\end{equation}
where $\lambda=\lambda_{A}^{\delta',\epsilon_0}|_{F}$, and $\phi\colon F\to F$ is the time-one flow of the monodromy vector field \eqref{eq:monodromy-vector-field}.
%By Proposition  \ref{prop:omega}\ref{item:omega-symplectic}, there is $\epsilon_0>0$ and a neighborhood $U'$ of $U\cap \d A$ such that for every $\delta'>0$ and $\epsilon\in (0,\epsilon_0]$, the form $\omega_{A}^{\delta',\epsilon}=d\lambda_{A}^{\delta',\epsilon}$ defined in \eqref{eq:omegaACdelta} is fiberwise symplectic in $U'$. Shrinking $\delta>0$ if needed, we can assume that $\tilde{f}^{-1}(\D_{\delta})\subseteq U'$. %Since the Liouville vector field of $\lambda^{0,\delta'}_{A}=\pi^{*}\lambda_{X}$ points outwards $(\tilde{f}\circ \pi)^{-1}(0) \cap \d W_{X}$, we can shrink $\delta,\epsilon_0>0$ so that the Liouville vector field of $\lambda^{\epsilon,\delta'}_{A}$ points outwards $(\tilde{f}\circ \pi)^{-1}(z) \cap \d W_{X}$ for every $\epsilon\in [0,\epsilon_0]$ and $z\in \D_{\delta}$.
%By Remark \ref{rem:vanishing_restriction}, for every $\epsilon\geq 0$, there is a $1$-form $\lambda_{X}^{\delta',\epsilon}\in \Omega^{1}(W'_{X}\setminus (V_{X}\cap f^{-1}(0)))$ such that, over each ray $R_{\theta}\subseteq \D_{\delta}$, we have the equality  $\lambda_{A}^{\delta',\epsilon}=\pi^{*}\lambda_{X}^{\delta',\epsilon}$. Once again, the map $(\tilde{f},\pr_{[0,\epsilon_0]})\colon X\times [0,\epsilon_0]\to \C\times [0,\epsilon_0]$ fits into Setting \ref{basic-setting}. Thus shrinking $\delta,\epsilon_0\geq 0$ if needed, we get, for all $z\in \D_{\delta}^{*}$ and all $\epsilon\in [0,\epsilon_0]$, graded isotopic abstract contact open books $(F_{z,\epsilon}',\lambda_{X}^{\delta',\epsilon},\phi_{z,\epsilon}')$.  Since $(F_{z,0}',\lambda_{X}^{\delta',0},\phi_{z,0}')=(F_{z,t_0},\lambda_{t_0},\phi_{z,t_0})$, we conclude that they are graded isotopic to our basic one, \eqref{eq:monodromy-acob}.
%The map $\pi\colon A\to X$ restricted to the intersection of $\pi^{-1}(U_X\setminus V_X)$ with the preimage of each ray in $\C_{\log}$ is a diffeomorphism. Hence the collar trivialization used to construct the family $(F_{z,\epsilon_0}',\lambda_{X}^{\delta',\epsilon_0},\phi_{z,\epsilon_0}')$ lifts to a collar trivialization considered in Setting \ref{basic-setting2}. The procedure described there provides an isotopy between the above abstract contact open books and the abstract contact open book
%	(F,\lambda,\phi),
Moreover, this isotopy endows \eqref{eq:radius-zero-acob} with a grading.  We conclude that the natural monodromy abstract contact open book \eqref{eq:monodromy-acob} is graded isotopic to the radius-zero abstract contact open book \eqref{eq:radius-zero-acob}. The properties of the latter are listed in Proposition \ref{prop:monodromy} below.

\subsection{Dynamics at radius zero}\label{sec:monodromy-dynamics}

In Section \ref{sec:isotopy-to-radius-zero} above, we have constructed an isotopy joining the natural monodromy abstract contact open book \eqref{eq:monodromy-acob} with an abstract contact open book $(F,\lambda,\phi)$ defined after the formula  \eqref{eq:radius-zero-acob}. %Here $F$ is a fiber at radius zero, i.e.\ in the boundary of the A'Campo space.
Dynamical properties of $\phi$ are summarized in the Proposition \ref{prop:monodromy}, which is a key step in the proof of Theorem \ref{theo:Zariski}.  We will state it using notation from Sections \ref{sec:monodromy-spectral-sequence} and \ref{sec:isotopy-to-radius-zero}, which we now briefly recall.% what is needed to state our result.

The fiber $F$ in \eqref{eq:radius-zero-acob} is the intersection of the radius-zero fiber $\tilde{f}_{A}^{-1}(0,1)$ with the preimage $\bar{W}$ of the sub-level set $\bar{W}_Y$ of a strictly plurisubharmonic function $\varrho\colon Y\to \R$ chosen in the beginning of Section \ref{sec:monodromy-spectral-sequence}. This sub-level set should be thought of as a compact \enquote{ball} around the singularities of $f^{-1}(0)$ and $Y$, not necessarily a small one. The subset $A_{i}^{\circ}\subseteq \d A$ is a  piece of the \enquote{radius-zero tube} $\d A$ lying over $D_{i}\setminus (D-D_{i})$, see the formula  \eqref{eq:stratification-pullback}. The manifold $B_{i}^{\circ}$ was introduced in Section \ref{sec:monodromy-spectral-sequence} as follows: if $D_{i}\subseteq \Exc h$ then $\nu_{i}\colon B_{i}^{\circ}\to  D_{i}\setminus (D-D_{i})$ is the $m_i$-fold covering described in \cite[\sec 2.3]{Denef_Loeser-Lefshetz_numbers}; and if $D_{i}$ is a proper transform of a component $C$ of $f^{-1}(0)$ then $B_{i}^{\circ}=\bar{W}_{Y}\cap C\setminus \operatorname{Bs} h^{-1}$, which should be thought of as the smooth part of $C$ intersected with the closed \enquote{ball} $\bar{W}_Y$. The integer $a_{i}$ is the discrepancy of $D_{i}$, see \eqref{eq:K} and $b_{i}$ is the coefficient of $D_i$ in the chosen ample divisor $H$, see \eqref{eq:H}: in particular, $b_i<0$ if $D_i\subseteq \Exc h$ and $b_i=0$ otherwise. The integer $b>0$ is chosen so that $bH$ is very ample; and $t_0,\epsilon_0>0$ are arbitrarily small positive numbers chosen in Section \ref{sec:isotopy-to-radius-zero} in such a way that $\epsilon_0$ can be taken arbitrarily small compared to $t_0$.

\begin{prop}\label{prop:monodromy}
Let $(F,\lambda,\phi)$ be the graded abstract contact open book \eqref{eq:radius-zero-acob}. Then %the fixed point set of $\phi^{m}$ is
\begin{equation}\label{eq:fixed-points}
\Fix \phi^{m}=\bigsqcup_{m_{i}|m}B_{i},
\quad \mbox{where} \quad
B_{i}= \overline{A_{i}^{\circ}}\cap F.
\end{equation}
%see Section \ref{sec:AX-def} for definition of $A_{i}^{\circ}\subseteq A$.
Moreover, for each $i$ such  that $m_{i}|m$, every connected component of $B_{i}$ is a codimension zero family of fixed points of $\phi^m$, see Definition \ref{def:0codim}. Write $\d B_i=\d^F B_i\sqcup \d^{+} B_i \sqcup \d^{-}B_{i}$, where $\d^{F}B_i=F\cap \d B_i$ and $\d^{\pm}B_{i}$ is the disjoint union of $\d^{\pm} B_{i}'$ for all connected components $B_{i}'$ of $B_{i}$, see formula \eqref{eq:boundarysplitting1}. Then the following holds.
\begin{enumerate}
\item \label{item:B_i-boundary} We have $\d^{-} B_i=\emptyset$ and $\partial^F B_i= B_{i}\cap\partial \bar{W}$.
\item \label{item:B_i-covering} There is a diffeomorphism $\kappa_i\colon B_{i}\setminus \d^{+}B_{i}\to B_{i}^{\circ}$ such that $\nu_{i}\circ \kappa_i=\pi|_{F}$.
\item \label{item:B_i-CZ} The Conley--Zehnder index of each point in $B_{i}$ equals $\CZ(B_i)=2\frac{m}{m_{i}}(a_{i}+1)-2m$.
\item \label{item:B_i-action} We can choose the action function $\ac\colon F\to \R$, see \eqref{eq:exact-symplectiomorphism}, % i.e.\ a function satisfying $\lambda-\phi^{*}\lambda=d\ac$, such that
so that $\ac|_{B_i}=(t_0b\cdot b_{i}+\epsilon_0)\frac{m}{m_i}$.
\end{enumerate}
\end{prop}

\begin{proof}
Recall that $\phi^{m}\colon F\to F$ is the time $m$ flow of the symplectic lift of the unit angular vector field on $\S^1=\d \C_{\log}$. Fix a fine chart $U_{X}\subseteq X$, i.e.\ one satisfying the statement of Lemma \ref{lem:smoothchart}, let $U=\pi^{-1}(U_X)$, and let $S$ be the associated index set \eqref{eq:index-set}. In local coordinates \eqref{eq:AC-chart} on $U$, this lift is given by the formula \eqref{eq:monodromy-vector-field}. Therefore, $\phi^{m}|_{U}$ is given by
\begin{equation}\label{eq:monodromy}
\phi^{m}\colon \theta_{j}\mapsto \theta_{j}+m \ell_{j},
\quad\mbox{where}\quad
\ell_{j}=\frac{\zeta(u_j)}{\sum_{i\in S}m_{i}\zeta(u_i)}
\end{equation}
In other words, $\phi$ is a rotation by an angle $2\pi m\ell_{j}$ about the $j$-th axis, see Notation \ref{not:S1}. Note that by Lemma \ref{lem:intro}\ref{item:intro-u=ubar}, the values of $\ell_j$ do not depend on the choice of the chart.

To prove the equality \eqref{eq:fixed-points}, we use the $m$-separatedness assumption \eqref{eq:m-separatedness}, cf.\  \cite[p.\ 242]{A'Campo}.

First, we claim that $\Fix \phi^{m}\subseteq \bigsqcup_{i\in \cP\cup\cE} B_{i}$. To prove this, take $x\in F$ such that $x\not\in \bar{A}_{i}^{\circ}$ for any $i\in \cP\cup\cE$. Put $I=\{i\in S: w_i(x)>0\}$. By Proposition \ref{prop:AX-topo}\ref{item:top-S1-bundle}, $x\in \bar{A}_{I}^{\circ}$, so $\# I\geq 2$ by assumption. Choose $j\in I$ such that $w_i(x)\geq w_j(x)$ for all $i\in I$.
Since both functions $\eta$ and $\zeta$ are increasing, we infer that $\zeta(u_i(x))\geq \zeta(u_j(x))>0$ for all $i\in I$, so $\frac{\zeta(u_i(x))}{\zeta(u_j(x))}\geq 1$ if $i\in I$ and $\frac{\zeta(u_i(x))}{\zeta(u_j(x))}=0$ if $i\not\in I$. Hence
\begin{equation*}
0<m\ell_{j}(x)=m\left(\sum_{i\in S}m_{i}\frac{\zeta(u_i(x))}{\zeta(u_j(x))}\right)^{-1}\leq m\left(\sum_{i\in I} m_{i}\right)^{-1}\overset{\mbox{\tiny{\eqref{eq:m-separatedness}}}}{<} 1,
\end{equation*}
%where the second-to-last equality follows from \eqref{eq:m-separatedness}.
In particular, $m\ell_{j}(x)\not\in \Z$. Formula \eqref{eq:monodromy} shows that $x\not\in \Fix\phi^{m}$, as claimed.

Therefore,  $\Fix \phi^{m}\subseteq \bigsqcup_{i\in \cP\cup\cE} B_{i}$. Take $x\in B_{i}$. By Proposition \ref{prop:AX-topo}\ref{item:top-S1-bundle} we have $w_{i}(x)=1$ and $w_{j}(x)=0$ for all $j\neq i$. Thus  $m\ell_{i}=\tfrac{m}{m_i}$ and $\ell_{j}=0$ for $j\neq i$, so formula \eqref{eq:monodromy} shows that $x\in \Fix\phi^{m}$ if and only if $m_i|m$. This ends the proof of \eqref{eq:fixed-points}.
\smallskip

Now, we claim that each $B_{i}$ is a codimension zero family of fixed points, see Definition \ref{def:0codim}. Say that $i=1$. First, we claim that $B_{1}\subseteq F$ is a codimension zero submanifold with corners.

Fix a point $x\in B_1$, let $U_{X}$ be a fine chart around $\pi(x)$, let $S$ be its index set \eqref{eq:index-set}, and let $U=\pi^{-1}(U_X)$. Recall that by Lemma \ref{lem:intro}\ref{item:intro-u=ubar}, we have $\bar{u}_{j}|_{U\cap F}=u_{j}|_{U\cap F}$ for $j\in S$, %and therefore $\bar{v}_{j}|_{U\cap F}=v_{j}|_{U\cap F}$ for all $j\in S$,
where $\bar{u}_j$ %, $\bar{v}_j$
are global functions introduced in \eqref{eq:def-vbar-ubar-mu}, and $u_j$ are the local ones, introduced in \eqref{eq:def-w_i-u_i}. By Proposition \ref{prop:AX-topo}\ref{item:top-S1-bundle}, we have $\bar{A}_{1}^{\circ}\cap U=\{u_{1}=1\}$. 	In particular, $\bar{A}_{1}^{\circ}\cap U$ is contained in the open subset $U_{1}\subseteq U$ introduced in \eqref{eq:Ui}.  There, we have a smooth coordinate system $(g,(\bar{v}_{i})_{i\in S\setminus \{1\}},\rest)$, see \eqref{eq:AC-chart}, where $U_1\cap \d A=\{g=0\}$.

By Lemma \ref{lem:intro}\ref{item:intro-sum}, we have $\bar{A}_{1}^{\circ}\cap U=\{u_1=1\}=\{u_{i}=0: i\in S\setminus \{1\}\}$. For $x\in U\cap \d A$, we have $u_{i}(x)=0$ if and only if $\bar{v}_{i}(x)\geq 0$. Indeed, if $u_i(x)=0$ then by Lemma \ref{lem:intro}\ref{item:intro-independence} $u_{i}^{p}(x)=0$ for every $p\in R$ such that $\tau^{p}(x)\neq 0$, which by \eqref{eq:def-vbar-ubar-mu} implies that $\bar{v}_{i}(x)\geq 0$. Conversely, if $\bar{v}_{i}(x)\geq 0$ then $v_{i}^{p}(x)\geq 0$ for some $p\in R$, which by Lemma \ref{lem:Ui-simple}\ref{item:Ui-cap-dA} implies that $u_{i}^{p}(x)=0$, so again by  Lemma \ref{lem:intro}\ref{item:intro-independence} we get $u_{i}(x)=0$. We conclude that the subset $\bar{A}_{1}^{\circ}\cap U_1\subseteq \d A\cap U_1$ is given by the inequalities $\{\bar{v}_{i}\geq 0: i\in S\setminus \{1\}\}$, see Figure \ref{fig:v3}.
The fiber $F\cap U_1$ is given in $\d A\cap U_1$ by $\sum_{i\in S}m_{i}\theta_{i}=0$. Therefore, $B_{1}\subseteq F$ is a codimension zero submanifold with corners, as claimed.
\smallskip

Now, according to Definition \ref{def:0codim}, we need to find a neighborhood $N_1$ of $B_1$, and a time independent Hamiltonian $H_{1}\colon N_1\to \R$ such that $\phi|_{N_1}=\psi_{1}^{H_1}$. Define $N_{1}\subseteq F$ as the union of all the sets $U_{1}\cap F$, where $U_1$ is the set \eqref{eq:Ui} defined for an adapted chart $U_X$, and $U_X$ ranges through all fine charts meeting $D_1$. We have seen that $N_{1}$ is a neighborhood of $B_{1}$ in $F$. We claim that
\begin{equation*}
H_{1}\de \tfrac{m}{m_1}\epsilon_0(\bar{v}_{1}+1)|_{N_1}\colon N_{1}\to [0,\infty)
\end{equation*}
is the required time-independent Hamiltonian.

To prove this, once again we restrict to a chart $U_{1}$ with coordinates \eqref{eq:AC-chart}. There, the monodromy $\phi^{m}$ is the time one-flow of a vector field $mX_{\theta}$, where $X_{\theta}$ is the symplectic lift of $\tfrac{\d}{\d \theta}$, given by the formula \eqref{eq:monodromy-vector-field}. Recall that $\tfrac{\d}{\d \theta_1}$ is the coordinate vector field of the chart \eqref{eq:AC-chart}. Put
\begin{equation*}
X_1\de mX_{\theta}-\tfrac{m}{m_1}\tfrac{\d}{\d \theta_{1}}.
\end{equation*}
We claim that $X_1$ is the Hamiltonian vector field $X^{H_1}$. Since $\theta_{*}(X_{\theta})=\tfrac{\d}{\d \theta}$ by definition of $X_{\theta}$, we have $\theta_{*}(X_1)=0$, so $X_1$ is tangent to the fiber $F$. Because $X_{\theta}$ is orthogonal to $F$, in the tangent space to $F$ we have
\begin{equation*}
\omega\AC^{\delta',\epsilon_0}(X_1,\sdot)=\omega\AC^{\epsilon_0}(-\tfrac{m}{m_1}\tfrac{\d}{\d\theta_1},\sdot )=\tfrac{m}{m_1}\epsilon_0 d\bar{v}_1=dH_{1},
\end{equation*}
where the second equality holds by Lemma \ref{lem:TdA}\ref{item:omega-dtheta}. Thus $X_{1}=X^{H_1}$, as claimed.

To verify the conditions of Definition \ref{def:0codim}, we need to check that $\phi^{m}|_{N_1}=\psi^{H_1}_{1}$. Recall that $\phi^{m}$ is the time one flow of $mX_{\theta}$, so it differs from $\psi^{H_1}_{1}$ by a translation by $\tfrac{m}{m_1}$ in the $\theta_{1}$-direction. Since $\tfrac{m}{m_1}\in \Z$, these time one flows are equal, as needed.
\smallskip

It remains to verify the properties \ref{item:B_i-boundary}--\ref{item:B_i-action}. Part \ref{item:B_i-boundary} is clear since the Hamiltonian $H_{i}$ defined above for each codimension zero family $B_i$ satisfies $H_i=\frac{m}{m_1}\epsilon_0(\bar{v}_{i}+1)\geq 0$ and $H_{i}|_{B_{i}}\equiv 0$.
\smallskip

\ref{item:B_i-covering}
Locally on $A_{i}^{\circ}$, the fiber $F$ is given by $\{m_i\theta_{i}=0\}$, and the restriction of  $\pi$ collapses the angular coordinate $\theta_{i}$. This yields the required covering, cf.\ Remark \ref{rem:vanishing_restriction}.
\smallskip

\ref{item:B_i-CZ} Say $i=1$. Recall that the Conley--Zehnder index is constant in a codimension zero family of fixed points, so it is enough to compute it for a single point $x\in F\cap A_{1}^{\circ}$. Choose a fine a chart $U_{X}\subseteq X$ around $\pi(x)$ whose associated index set is $\{1\}$, i.e.\ $U_{X}$ does not meet $D\setminus D_1$. The smooth coordinates \eqref{eq:AC-chart} on $U\cap \d A$ are $(\theta_{1},z_{i_2},\dots,z_{i_n})$: we have skipped $g$ since it cuts out $\d A$. We will compute $\CZ(x)$ using \cite[Lemma A.8]{McLean}. To do this, recall that the grading of $(F,\lambda,\phi)$ is induced by isotopy from the one on $(M_{z,t_0},\lambda_z,\phi_{z,t_0})$. The latter is given by a section $s\vert$ of the relative canonical bundle $K\vert$, such that $s\vert\otimes d\tilde{f}$ maps under \eqref{eq:K-vert} to the section $s_{K}$ of $K_{X\setminus D}$ chosen in Section \ref{sec:basic-grading}. In this case, $s_{K}$ is simply the pullback of the holomorphic volume form on $\C^{n}$, see Example \ref{ex:Milnor-fibration}.

Since $s_{K}$ has zero of order $a_1\in \Z$ along $D_1$, we can choose the coordinates $(z_{i_2},\dots,z_{i_{n}})$ of our adapted chart $U_X$ so that $s_{K}|_{U_X}=z_{1}^{a_1}\, dz_{1}\wedge dz_{i_2}\wedge \dots \wedge dz_{i_n}$. We have $df|_{U_X}=m_1z_{1}^{m_{1}-1}\,dz_{1}$, so $s\vert|_{U_{X}}=\frac{1}{m_1}z_{1}^{a_{1}-m_{1}+1}\, dz_{i_2}\wedge \dots \wedge dz_{i_n}$. %Therefore, the induced grading of $(F,\lambda,\phi)$ corresponds under \eqref{eq:gradings-trivializations} to $\frac{1}{m_1}e^{(a_{1}-m_{1}+1)\cdot 2\pi\imath \theta_{1}}\, dz_{i_2}\wedge \dots \wedge dz_{i_n}$.
By \eqref{eq:monodromy}, the symplectomorphism $\phi^{m}$ is given by $\theta_1\mapsto \theta_1+m\ell_1$, where $\ell_1=\frac{1}{m_1}$. It follows that the winding number used in \cite[Lemma A.8]{McLean} equals $-(a_{1}-m_{1}+1)\cdot \frac{m}{m_{1}}=-\frac{m}{m_1}(a_1+1)+m$, so $\CZ(x)=2\frac{m}{m_{1}}(a_{1}+1)-2m$, as needed.
\smallskip

\ref{item:B_i-action}
For every $i\in \cP\cup \cE$, choose a point $x_{i}\in A_{i}^{\circ}\cap F$. Recall that the function $\ac$ is defined up to an additive constant, and is constant in each $B_i$. Therefore, to prove \ref{item:B_i-action} it is sufficient to show that for every $i,j\in \cP\cup \cE$ we have
\begin{equation*}
\ac(x_j)-\ac(x_i)=(t_0bb_{j}+\epsilon_0)\frac{m}{m_j}-(t_0bb_{i}+\epsilon_0)\frac{m}{m_i}.
\end{equation*}
By definition, $\ac$ satisfies $(\phi^m)^{*}\lambda-\lambda=-d\ac$. Therefore, it is enough to prove that for some path $\gamma\colon [0,1]\to F$ such that $\gamma(0)=x_i$, $\gamma(1)=x_j$, we have
\begin{equation}\label{eq:action-computation}
\int_{\gamma}(\phi^{m})^{*}\lambda-\lambda=
\end{equation}
We can choose $\gamma$ in such a way that it avoids triple intersections, i.e.\ $\gamma([0,1])\cap \bigsqcup_{\#I\geq 3} A_{I}^{\circ}=\emptyset$. Then $\gamma$ can be split into segments with images lying over adapted charts whose associated index sets \eqref{eq:index-set} have at most two elements. Thus considering each piece of $\gamma$ separately, it is sufficient to prove \eqref{eq:action-computation} for $\gamma\colon [0,1]\to U$, where $U$ is the preimage of an adapted chart $U_{X}$ with index set $S\de \{i,j\}$.

%First, we need a local description of the form $\lambda_{X}=\tilde{\lambda}_{Y}+t_0\lambda_{\mathrm{FS}}$, introduced in Section \ref{sec:isotopy-to-radius-zero}. Fix a fine chart $U_X\subseteq X$, and let $S$ be its associated index set, see Definition \ref{def:adapted-chart}.
%	We will now describe the restriction $\lambda_{X}|_{U_X}$. Recall from Section \ref{sec:isotopy-to-radius-zero} that $\lambda_{X}=\tilde{\lambda}_{Y}+t_0\lambda_{\mathrm{FS}}$, where $\tilde{\lambda}_{Y}$ is a smooth form on $X$, and $\lambda_{\mathrm{FS}}=d^{c}\log ||s||$, where $s$ is a section of $\cO_{X}(bH)$ with poles of order $-bb_l$ along $D_l$ and no zeros or poles elsewhere.
%	 Thus inside $U_{X}$, we have $\log||s||=\log(\prod_{l\in S}|\xi z_{l}^{bb_{l}}|)=\sum_{l\in S} bb_{l}\log r_{l}+\log |\xi|$ for some nonvanishing holomorphic function $\xi$. Since $d^{c}r_{l}=-r_{l}\, d\theta_{l}$, see Example \ref{ex:J}, we have $\lambda_{\mathrm{FS}}|_{U_X}=d^{c}\log||s||=-\sum_{l\in S} bb_{l}\, d\theta_{l}+\beta'$ for some smooth $1$-form $\beta''\in \Omega^{1}(U_X)$. Therefore,

By the formula \eqref{eq:lambdaA-locally}, we have $\pi^{*}\lambda_{X}|_{U_{X}}=-\sum_{l\in S} t_0bb_{l}\, d\theta_{l}+\pi^{*}\beta'$ for some $\beta'\in \Omega^{1}(U_X)$. Using Lemma \ref{lem:d-theta} and \ref{lem:TdA}\ref{item:ort-Z-vj}, we conclude that $\lambda|_{U\cap F}=\sum_{l\in S}(-t_0bb_{l}+\epsilon_0 \bar{v}_l)\, d\theta_{l}+\pi^{*}\beta$ for some $\beta\in \Omega^{1}(U_X)$. The formula \eqref{eq:monodromy} shows that $\pi|_{F}\circ \phi=\pi|_{F}$, so $(\phi^{m})^{*}(\pi^{*}\beta)-(\pi^{*}\beta)=0$. Now by \eqref{eq:monodromy} we have
\begin{equation}\label{eq:action-integral}
\int_{\gamma} (\phi^{m})^{*}\lambda-\lambda = \int_{\gamma} \textstyle\sum_{l\in S}(-t_0bb_{l}+\epsilon_0 \bar{v}_l)m\, d\ell_{l}.
\end{equation}
If $S=\{i\}$ then the image of $\gamma$ is contained in $A_{i}^{\circ}$, where by \eqref{eq:monodromy} the function $\ell_{i}$ is constant. Hence the above integral is zero, which proves \eqref{eq:action-computation} in this case.

Assume that $S=\{i,j\}$ for $i\neq j$. We claim that, along $\gamma$, we have the equality
\begin{equation}\label{eq:inntegration-by-parts}
\bar{v}_i d\ell_i+\bar{v}_j d\ell_j=d(\bar{v}_i\ell_i+\bar{v}_j\ell_j).
\end{equation}
Inside $A_{i}^{\circ}$, the functions $\bar{v}_i$, $\ell_i$ are constant, and $\ell_j=0$, so $\bar{v}_{j}\ell_j=0$ is constant, too. Hence both forms are zero in $A_{i}^{\circ}$, and by symmetry the same holds in $A_{j}^{\circ}$. It remains to prove the equality \eqref{eq:inntegration-by-parts} in $A^{\circ}_{i,j}$.

Fix $l\in \{i,j\}$. By Lemma \ref{lem:intro}\ref{item:intro-u=ubar}, in $A_{i,j}^{\circ}$ we have $\bar{v}_l=-u_{l}$. Recall that, by definition \eqref{eq:def-w_i-u_i} of $u_l$ we have $w_{l}=\eta^{-1}(u_{l})$, so $dw_{l}=\zeta(u_l)\, d u_l$ by definition of $\zeta$. We conclude that $\zeta(u_l)d\bar{v}_l=-\zeta(u_l)du_l=-dw_l$. By Lemma  \ref{lem:intro}\ref{item:intro-sum} we have $w_i+w_j=1$, so  $\zeta(u_i)d\bar{v}_i+\zeta(u_j)d\bar{v}_j=-dw_i-dw_j=0$. Hence $\ell_id\bar{v}_i+\ell_j d\bar{v}_j=0$ by definition \eqref{eq:monodromy} of $\ell_i$, $\ell_j$. This proves \eqref{eq:inntegration-by-parts}.

Now, the equality \eqref{eq:inntegration-by-parts} implies that the integral \eqref{eq:action-integral} is equal to the difference of values of $\sum_{l\in S}(-t_0bb_l+\epsilon_0 \bar{v}_l)m\ell_l$ at the endpoints $x_j$, $x_i$ of $\gamma$. Since $x_j\in A_{j}^{\circ}$, we have $\ell_{i}(x_j)=0$, and similarly $\ell_{j}(x_i)=0$. Moreover, $\bar{v}_{j}(x_j)=\bar{v}_i(x_i)=-1$ and $\ell_{j}(x_j)=\frac{1}{m_j}$, $\ell_{i}(x_i)=\frac{1}{m_i}$. Thus
\begin{equation*}
\int_{\gamma} (\phi^{m})^{*}\lambda-\lambda =
\end{equation*}
as claimed in the formula \eqref{eq:action-computation}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:spectral-sequence-monodromy}]
The monodromy abstract contact open book \eqref{eq:monodromy-acob} is graded isotopic to \eqref{eq:radius-zero-acob}. Hence by Proposition \ref{prop:isotopy_invariance}, their fixed point Floer homology groups are the same, i.e.\ $\HF_{*}(\phi_z,+)=\HF_{*}(\phi,+)$. By Proposition \ref{prop:monodromy}, $\phi$ satisfies the assumptions of Proposition \ref{prop:spectral-sequence}, so we get a spectral sequence converging to $\HF_{*}(\phi,+)$, whose first page $E_{p,q}^{1}$ is given by \eqref{eq:spectral-sequence-first-page}.

By Proposition \ref{prop:monodromy}\ref{item:B_i-action}, we can choose the action function $\ac$ for $\phi$ in such a way that $\ac|_{B_{i}}=(t_0 bb_i+\epsilon_0)\frac{m}{m_i}$ for every $i\in S_{m}$. Moreover, the number $\epsilon_0>0$ can be chosen arbitrarily small compared to $t_0 b>0$. Therefore, we can choose the indexing function $\iota \colon S_{m} \to \Z$ in Proposition \ref{prop:spectral-sequence} to be $\iota(i)=\frac{m}{m_i}b_{i}$. Hence the indexing set $\{i: \iota(i)=p\}$ in \eqref{eq:spectral-sequence-first-page} is the set $S_{m,p}$ defined in \eqref{eq:Sm}. Substituting to \eqref{eq:spectral-sequence-first-page} the formula $\CZ(B_i)=2\frac{m}{m_i}(a_i+1)-2m$ from Proposition \ref{prop:monodromy}\ref{item:B_i-CZ}, we get %a spectral sequence converging to $\HF_{*}(\phi,+)$, whose first page equals
\begin{equation*}
E_{p,q}^{1}=\bigoplus_{i\in S_{m,p}}
H_{n-1+p+q+2\frac{m}{m_i}(a_i+1)-2m}(B_{i},\d^{+} B_{i};\Z_2).
\end{equation*}
By definition, $B_{i}$ is compact, and by Proposition \ref{prop:monodromy}\ref{item:B_i-covering}, the manifold $B_{i}\setminus \d^{+}B_{i}$ is diffeomorphic to $B_{i}^{\circ}$. Thus $H_{*}(B_i,\d^{+}B_{i};\Z_2)=H_{*}^{BM}(B_{i}^{\circ})$, as required by the formula \eqref{eq:spectral-sequence-monodromy}.
\end{proof}

Remarks \ref{rem:McLean-5.41}, \ref{rem:degree-shift} and \ref{rem:McLean} below compare our results with the ones of  [49].

\begin{remark}%[{Comparison with [49]}]
\label{rem:McLean-5.41}
Assume that $Y=\C^n$, $h\colon X\to \C^{n}$ is an $m$-separating resolution of an isolated singularity $0\in f^{-1}(0)$, and $W_Y=\B_{\epsilon}$ is a Milnor ball. Then, Proposition \ref{prop:monodromy} shows that the radius zero monodromy satisfies almost the same dynamical properties as McLean's \emph{model resolution}, see \cite[Theorem 5.41]{McLean}.

The key difference is that our monodromy has codimension zero family of fixed points corresponding to the proper transform of $f^{-1}(0)$. If $W_Y$ is a Milnor ball, they can be avoided by a small perturbation near the boundary, as done in loc.\ cit. However, to prove Theorem \ref{theo:Zariski} we will need to apply Proposition \ref{prop:monodromy} in more generality (namely, keeping the radius of the ball fixed within the family, see Example \ref{ex:mu_constant}).

Note that Proposition \ref{prop:monodromy} is much more general: for example, it holds for an embedded resolution of $f$ with multiple isolated singularities in $f^{-1}(0)$. Moreover, with minor modifications one can formulate a result analogous to Proposition \ref{prop:monodromy}\ref{item:B_i-boundary},\ref{item:B_i-covering} for a degeneration of projective varieties. In this setting, the Conley--Zehnder index computation in Proposition \ref{prop:monodromy}\ref{item:B_i-CZ} can be carried out, too, provided $\phi$ admits a grading: this happens when fibers are Calabi--Yau.

Eventually, we note that the abstract contact open book in Proposition \ref{prop:monodromy} is actually isotopic to the symplectic monodromy of $f$ (defined in a natural way in Example \ref{ex:typical}). Meanwhile, the model resolution constructed in \cite[Example 5.14]{McLean} just shares the same associated graded contact pair, which in this case is $(\d\B_{\epsilon},\d \B_{\epsilon} \cap f^{-1}(0))$.% the embedded contact type of the link.}
\end{remark}

\begin{remark}\label{rem:degree-shift}
The formula for Conley--Zehnder index in Proposition \ref{prop:monodromy}\ref{item:B_i-CZ} differs from the one in  \cite[Theorem 5.41(3)]{McLean} by $2m$.

This shift most likely arises due to a mistake in the computation carried out in \cite[p.\ 1025]{McLean}. Let us indicate a possible source of this mistake. The Conley--Zehnder index is computed in loc.\ cit.\ from the winding number associated to a trivialization of the vertical canonical bundle $K\vert$, using Lemma A.8 loc.\ cit. At the bottom of p.\ 1025 loc.\ cit., it is claimed that this winding number is the same as the one for the trivialization of $K_{X\setminus D}$. To be more precise, loc.\ cit.\ uses an analogue of canonical bundles on the model resolutions, but these  two coincide locally, when $X$ is a chart adapted to $f$.

The claimed equality is false. To see this, take $X=\C^{2}$ with coordinates $(z_1,z_2)$, let $f(z_1,z_2)=z_1$, and let $dz_1\wedge dz_2$ be a trivialization of $K_{X\setminus D}$. As in the formula \eqref{eq:K-vert},  write  $K_{X\setminus D}=K\vert \otimes f^{*}K_{\C^{*}}$. The trivialization of $f^{*}K_{\C^{*}}$ is given by $df=dz_{1}$, so the compatible trivialization of $K\vert$ is $dz_2$. To compute the winding number, as in the proof Proposition \ref{prop:monodromy}\ref{item:B_i-CZ}, we pull back these forms by $\Phi_{m\theta}^{-1}\colon (z_1,z_2)\mapsto (e^{-2\pi\imath\cdot m\theta}z_1,z_2)$, and trace the path in $\C^{*}$ made by the leading coefficient. For $K_{X\setminus D}$, the pullback is $e^{-2\pi\imath \cdot m\theta} dz_1\wedge dz_2$, so the winding number is $-m$. In turn, for $K\vert$, the pullback is $dz_2$, so the winding number is $0$. Hence \cite[Lemma A.8]{McLean} shows that to get the correct Conley--Zehnder index, one needs to subtract $2m$ from the one computed in loc.\ cit. %This explains the $2m$ shift.
\end{remark}
\begin{remark}\label{rem:McLean}
Let $Y=\C^n$, let $f\colon \C^n\to \C$ be a holomorphic function with an isolated critical point $0\in \C^n$, and let $\bar{U}=\B_{\epsilon'}$, $\bar{W}=\B_{\epsilon}$ be Milnor balls. Then Proposition \ref{prop:spectral-sequence-monodromy} provides a spectral sequence converging to Floer homology of the symplectic monodromy of the Milnor fibration \eqref{eq:milfibtbintro} in the tube, introduced in Example \ref{ex:Milnor-fibration}.

In this case, the term of \eqref{eq:spectral-sequence-monodromy} corresponding to the proper transform of $f^{-1}(0)$ vanishes. To see this, note that $\bigsqcup_{i\in \cP} B_{i}^{\circ}=f^{-1}(0) \cap \B_{\epsilon}^{*}$. By the conical structure theorem \cite[2.10]{Milnor}, $f^{-1}(0) \cap \B_{\epsilon}$ is a cone over $f^{-1}(0)\cap \d \B_{\epsilon}$, so $H_{*}^{BM}(B_{i}^{\circ})=0$ for every $i\in \cP$.

Therefore, in this case the first page \eqref{eq:spectral-sequence-monodromy} is similar to the one in  \cite[Theorem 1.2]{McLean}, cf.\ Remark \ref{rem:McLean-HF3}. The spectral sequence in loc.\ cit.\  converges to the Floer cohomology of the symplectic monodromy of the Milnor fibration in the sphere, i.e.\ $f/|f|\colon \B_{\epsilon}\setminus f^{-1}(0)\to \S^1$.
\end{remark}

\begin{remark}