In the tangent bundle $T(U\setminus\partial A)$, we consider the maximum norm $||\sdot||$ with respect to the coefficients of the basis $((\tilde{\nu}_i^{2k})_{i=1}^{2k},(\xi_i^{2k})_{i=1}^{2(n-k)})$. The diagonal entries of $A$ are equal to $\tilde{\omega}_{E}^{J}(\tilde{\nu}_{i}^{2k},\tilde{\nu}_{i}^{2k})$, so by property \ref{lem:matrix}\ref{item:a_ii}, they are bounded from below by $\tfrac{1}{4}$. It follows that %	Property \ref{lem:matrix}\ref{item:a_ii}, which is kept by the diagonalization, implies that
\begin{equation}\label{eq:constant-nu}
\tilde{\omega}_E^J(\nu ,\nu )\geq\tfrac{1}{4} ||\nu ||^2.
\end{equation}
Property \ref{lem:matrix}\ref{item:P} asserts that the functions $\tilde{\omega}_{E}^{J}(\xi_{i}^{2k},\xi_{j}^{2k})$, which are entries of the matrix $P$, are bounded. Thus there is a constant $K>0$ such that, uniformly on $U\setminus \d A$, we have the inequality
\begin{equation}\label{eq:constant-xi}
|\tilde{\omega}_E^J(\xi ,\xi )|\leq K ||\xi ||^2.
\end{equation}

%	Now, we fix $\epsilon>0$.
Recall that the form $\omega\AC^{\epsilon}$ was defined in  \eqref{eq:omegaAC} as 	$\pi^{*}\omega_{X}+\epsilon\omega_{E}$. Hence by Lemma \ref{lem:omegaAC-sym}, for every $\tau>0$ we have the equality of restrictions
\begin{equation*}
\quad\mbox{where}\quad
(\tilde{\omega}\AC^{\epsilon})^{J}= \pi^{*}\omega_{X}^{J}+\epsilon\tilde{\omega}_E^{J}.
\end{equation*}
We need to show that there is $\epsilon_0>0$ and a neighborhood $V_X$ of $x_0$ such that whenever $\epsilon\in [0,\epsilon_0]$, %and $y\in \pi^{-1}(V_X)$,
we have the inequality $(\tilde{\omega}\AC^{\epsilon})^{J}(v,v)>0$ for every nonzero $v\in T(\pi^{-1}(V_{X})\setminus \d A)$.

We split $v=\nu +\xi $ as above. Consider the case $||\nu ||> 2K^{\frac{1}{2}}||\xi ||$. Since the form $\omega_{X}^{J}$ is positive definite, we have $(\pi^{*}\omega_{X})^{J}(v,v)\geq 0$, so  $(\tilde{\omega}\AC^{\epsilon})^{J}(v,v)\geq \epsilon\tilde{\omega}_{E}^{J}(v,v)$. Substituting the inequalities \eqref{eq:constant-nu} and \eqref{eq:constant-xi} to \eqref{eq:omega-splitting}, we get $\tilde{\omega}_E^J(v,v)\geq \frac{1}{4}||\nu ||^2-K||\xi ||^2>0$ by assumption. Thus  $(\tilde{\omega}\AC^{\epsilon})^{J}(v,v)>0$, as needed.

Hence we can assume
\begin{equation}
\label{eq:assumption}
||\nu ||\leq 2K^{\frac{1}{2}}  ||\xi ||.
\end{equation}
%	Note that since we assume $v\neq 0$, the assumption \eqref{eq:assumption} implies that $\xi \neq 0$.
Using \eqref{eq:omega-splitting}, we expand the equality $(\tilde{\omega}\AC^{\epsilon})^{J}(v,v)=(\pi^*\omega_{X})^J(v,v)+\epsilon\tilde{\omega}_{E}^{J}(v,v)$ as
\begin{equation}
\label{eq:quadraticexpansion}
(\tilde{\omega}\AC^{\epsilon})^J(v,v)=(\pi^*\omega_X)^J(\nu ,\nu )+2(\pi^*\omega_X)^J(\nu ,\xi )+(\pi^*\omega_X)^J(\xi ,\xi )+\epsilon\tilde{\omega}_E^J(\nu ,\nu )+\epsilon\tilde{\omega}_E^J(\xi ,\xi ).
\end{equation}
The inequality \eqref{eq:constant-nu} implies that $\tilde{\omega}_{E}^{J}(\nu ,\nu )\geq 0$, so since $(\pi^*\omega_X)^J$ is positive definite, for $v\neq 0$ we have
\begin{equation*}
(\pi^*\omega_X)^J(\nu ,\nu )+\tfrac{1}{3}(\pi^*\omega_X)^J(\xi ,\xi )+\epsilon\tilde{\omega}_E^J(\nu ,\nu )>0.
\end{equation*}
We group the remaining summands of the expansion~\eqref{eq:quadraticexpansion} as follows:
\begin{equation*}
\begin{split}
B&\de \tfrac{1}{3}(\pi^*\omega_X)^J(\xi ,\xi )+2(\pi^*\omega_X)^J(\nu ,\xi ),\\
C &\de \tfrac{1}{3}(\pi^*\omega_X)^J(\xi ,\xi )+\epsilon\tilde{\omega}_E^J(\xi ,\xi ).
\end{split}
\end{equation*}

Let $||\sdot ||_X$ be a norm in the tangent bundle $T \bar{U}_X$. By property \ref{lem:matrix}\ref{item:xi-iso}, there is a constant $K_0>0$ such that $||\pi_*\xi ||_X\geq K_0||\xi ||$. By positive definiteness of $\omega_X^J$ there exists a constant $K_1'>0$ such that $\omega_X^J(\eta ,\eta )\geq K_{1}'||\eta ||_{X}^2$ for every $\eta$ tangent to $\bar{U}_{X}$. Putting $K_{1}=K_{1}'K_{0}^2$ we get
\begin{equation}\label{eq:C-bound}
(\pi^*\omega_X)^J(\xi ,\xi )\geq K_1||\xi ||^2.
\end{equation}
There are also constants $K_{2}',K_{2}''>0$ such that $|(\pi^{*}\omega_{X})^{J}(\nu ,\xi )|\leq  K_{2}'\cdot ||\pi_{*}\nu ||_{X}\cdot ||\pi_{*}\xi ||_{X}$ and $||\pi_{*}\xi ||_{X}\leq K_{2}'' ||\xi ||$, so putting $K_{2}=K_{2}'K_{2}''$ we get $|(\pi^*\omega_X)^J(\nu ,\xi )|\leq K_2||\xi ||\cdot||\pi_*\nu ||_X$.  This inequality together with \eqref{eq:C-bound} implies that
\begin{equation}\label{eq:B-bound}
B\geq \tfrac{1}{3}K_{1}||\xi ||^{2}-2K_{2} ||\xi ||\cdot||\pi_*\nu ||_X.
\end{equation}

Recall that $\tilde{\nu}_{i}^{2k}=\sigma_{i}^{-1/2}\nu_{i}^{2k}$, so by  property~\ref{lem:matrix}\ref{item:vectorsmall}, we have $\tilde{\nu}_{i}^{2k}\rightarrow 0$ as $r_{\bar{i}}\rightarrow 0$. Hence there is a continuous function $\kappa\colon [0,\infty)^k\to[0,\infty)$ such that $\kappa(0)=0$ and $||\pi_*\nu ||_X\leq \kappa(r_1,\dots,r_k)||\nu ||$. Together with the assumption \eqref{eq:assumption}, this implies that $||\pi_{*}\nu ||_{X}\leq 2K^{\frac{1}{2}} \kappa(r_1,\dots,r_k) ||\xi ||$. Substituting this inequality to \eqref{eq:B-bound}, we get
\begin{equation*}
%\tfrac{1}{3}K_1||\xi ||^2-2K_2||\xi ||\cdot ||\pi_*\nu ||_X=
%||\xi ||\cdot (\tfrac{1}{3}K_1||\xi ||-2K_2||\pi_*\nu ||_X)\geq \\
%\geq||\xi ||\cdot (\tfrac{1}{3}K_1||\xi ||-2K_2\kappa(r_1,\dots,r_k)\cdot ||\nu ||)
\geq
%\tfrac{1}{3}K_{1}||\xi ||^{2}-2K_{2} ||\xi ||\cdot 2K^{\frac{1}{2}} \kappa(r_1,\dots,r_k) ||\xi ||
||\xi ||^2(\tfrac{1}{3}K_1-4K^{\frac{1}{2}}K_2\cdot \kappa(r_1,\dots,r_k)).
\end{equation*}
Since $r_{1}, \dots,r_k$ vanish at $x_0$, we can shrink the neighborhood $U_X$ of $x_0$ so that $\kappa(r_1,\dots,r_k)<\frac{\frac{1}{3}K_1}{4K^{\frac{1}{2}}K_2}$. After this shrinking of $U_X$, we get $B\geq 0$. Notice that this shrinking is independent of $\epsilon$.
\smallskip

The inequalities \eqref{eq:C-bound} and \eqref{eq:constant-xi} imply that
\begin{equation*}
C\geq %\tfrac{1}{3}K_{1}||\xi ||^{2}-\epsilon K ||\xi ||^{2}=
(\tfrac{1}{3}K_{1}-\epsilon K)\cdot ||\xi ||^{2}.
\end{equation*}
Choose $\epsilon_0\de \frac{1}{3}\frac{K_1}{K}$. Then for every $\epsilon\in [0,\epsilon_0]$ we have $C\geq 0$, so $(\tilde{\omega}\AC^{\epsilon})^J(v,v)>0$, as needed.
\end{proof}

\subsection{Non-degeneracy on \texorpdfstring{$\d A$}{dA}}\label{sec:lifts}

In Proposition \ref{prop:positivity} we have shown that the forms $\omega\AC^{\epsilon}$ and $\omega\AC^{\delta,\epsilon}$ are fiberwise nondegenerate near $\d A$. Proposition \ref{prop:lifts} below, which we will prove in this section, shows that the same holds on $\d A$. Recall from Proposition \ref{prop:AXsmooth}\ref{item:AX-gsmooth} that $\theta\colon A\to \S^1$  is a submersion extending $f/|f|\colon X\setminus D\to \S^1$ to $\d A$.

\begin{prop}%[Monodromy vector field]
\label{prop:lifts}
%Fix a K\"ahler form $\omega_{X}\in \Omega^{2}(X)$.
Fix $x_0\in D$. There is an $\epsilon_0>0$, and a fine chart $U_X$ containing $x_0$, such that for every $\epsilon\in (0,\epsilon_0]$ and any $\delta>0$, the restriction of each of the forms $\omega\AC^{\epsilon}$ and $\omega\AC^{\delta,\epsilon}$  to $U\cap \d A$ is nondegenerate on every fiber of $\theta|_{U\cap \d A}$, where $U=\pi^{-1}(U_X)$. Moreover, the symplectic lift to $U\cap \d A$ of the unit angular vector field on $\S^1$ %=\{0\}\times \S^1$
is independent of the choice of the form $\omega\AC^{\epsilon}$ and $\omega\AC^{\delta,\epsilon}$ defining the symplectic connection, and is given by the formula \eqref{eq:monodromy-vector-field}.
\end{prop}

We notice that at $\partial A$ the forms $\omega\AC^{\epsilon}$ and $\omega\AC^{\delta,\epsilon}$ coincide, so it is enough to deal with the former one.

As in Section \ref{sec:positivity}, we fix $x_{0}\in X_{S}^{\circ}$ for some $S\neq \emptyset$ and order the components of $D$ so that $S=\{1,\dots, k\}$. We fix a fine chart $U_{X}$ around $x_0$ with index set $S$; and put $U=\pi^{-1}(U_X)\subseteq A$.

Let us recall the definition of the smooth structure on $U$. In \eqref{eq:Ui} we have defined an open covering $U=\bigcup_{i=1}^{k}U_i$ by $U_i=\{w_{i}>\frac{1}{n+1}\}$. The smooth coordinate chart \eqref{eq:AC-chart} on $U_1$ is
\begin{equation*}
(g,\bar{v}_2,\dots,\bar{v}_k,\theta_1,\dots,\theta_k,z_{i_{k+1}},\dots, z_{i_n})\colon U_1\to [0,1)\times \R^{k-1}\times (\S^1)^{k}\times \C^{n-k},
\end{equation*}
where $\{i_{k+1},\dots, i_{n}\}$ is some subset of $\Z_{>N}$. The charts on $U_2,\dots, U_k$ are defined analogously. The vector fields
\tfrac{\d}{\d\theta_{1}},\dots,\tfrac{\d}{\d \theta_{k}}
on $U_1$ used in formula \eqref{eq:monodromy-vector-field} are defined as the ones dual to $d\theta_{1},\dots,d\theta_{k}$ with respect to the basis of $T^{*}U_1$ given by the above chart \eqref{eq:AC-chart}.
\smallskip

In the proof of Proposition \ref{prop:lifts}, we will treat separately the subsets $U_{1}\cap A_{I}^{\circ}$ lying over the strata $U_{X}\cap X_{I}^{\circ}$, see \eqref{eq:stratification-pullback}. Therefore, it will be convenient to use slightly different charts over each stratum, which we introduce in formula \eqref{eq:coordinates-on-stratum} below. For the deepest stratum $X_{S}^{\circ}$, formula \eqref{eq:coordinates-on-stratum} agrees with \eqref{eq:AC-chart}.

\begin{lema}\label{lem:coordinates-on-stratum}
Fix a subset $I\subseteq S$ containing $1$. Put $U_{I}^{\circ}\de U_1\cap A_{I}^{\circ}$, and reorder the components of $D$ so that $I=\{1,\dots, l\}$. Then, after possibly shrinking $U_X$, the following hold.
\begin{enumerate}
\item\label{item:coordinates-on-stratum} %Let $x_{2j-1}$, $x_{2j}$ be the real and imaginary part of $z_{i_j}$ for $j\in \{1,\dots,n-k\}$. Then t
For $i\in \{l+1,\dots, k\}$, put $s_{i}=\log r_{i}$, see \eqref{eq:coordinates-sx}. Then the map
\begin{equation}\label{eq:coordinates-on-stratum}
(\bar{v}_2,\dots,\bar{v}_{l},s_{l+1},\dots,s_{k},\theta_{1},\dots,\theta_k,z_{i_{k+1}},\dots,z_{i_{n}})\colon U_{I}^{\circ}\to \R^{k}\times (\S^1)^{k}\times \C^{n-k}.
\end{equation}
is a smooth coordinate chart on a neighborhood of $U_{I}$ in $\d A$.
\item\label{item:coordinate-vector-fields-on-stratum} The vector fields $
\tfrac{\d}{\d\theta_{1}},\dots,\tfrac{\d}{\d \theta_{k}}
$ are dual to $d\theta_1,\dots,d\theta_k$ with respect to the basis of $T^{*}U_{I}$ given by coordinates \eqref{eq:coordinates-on-stratum}.
\item\label{item:splitting-on-stratum} Define vector fields $\tfrac{\d}{\d \bar{v}_2},\dots,\tfrac{\d}{\d \bar{v}_l}$ as dual to $d\bar{v}_2,\dots,d\bar{v}_l$ in the above basis. % with respect to the above basis.
We have a splitting
\begin{equation*}
T U_{I}^{\circ}=\cV_{I}\oplus \Theta_{I}\oplus \cZ_{I},
\quad
\mbox{where }
\cV_{I}=\sspan\{\tfrac{\d}{\d \bar{v}_{2}},\dots, \tfrac{\d}{\d \bar{v}_l}\},\  \Theta_{I}=\sspan\{\tfrac{\d}{\d \theta_{1}},\dots, \tfrac{\d}{\d \theta_l}\},
\end{equation*}
%			where
%			$\cV_{I}=\sspan\{\tfrac{\d}{\d \bar{v}_{2}},\dots, \tfrac{\d}{\d \bar{v}_l}\}$; $\Theta_{I}=\sspan\{\tfrac{\d}{\d \theta_{1}},\dots, \tfrac{\d}{\d \theta_l}\}$,
and $\cZ_{I}$ is spanned by the vector fields dual to $ds_{l+1},\dots,ds_{k}$, $d\theta_{l+1},\dots, d\theta_{k}$ and to the (real and imaginary parts of) $dz_{i_{k+1}},\dots, dz_{i_{n}}$, with respect to the basis of $T^{*}U_{I}$ given by the chart  \eqref{eq:coordinates-on-stratum}. \\
We have $\pi_{*}|_{\cV_{I}\oplus \Theta_{I}}=0$, and $\pi_{*}|_{\cZ_{I}}$ maps $\cZ_{I}$ isomorphically onto the tangent bundle to $X_{I}^{\circ}\cap \pi(U_1)$.
\end{enumerate}
\end{lema}
\begin{proof}
\ref{item:coordinates-on-stratum} On the stratum $U_{X}\cap X_{I}^{\circ}$, we have a smooth coordinate system $(z_{l+1},\dots,z_{k},z_{i_1},\dots,z_{i_{n-k}})$. Since the coordinates $z_{l+1},\dots,z_{k}$ do not vanish on $U_{X}\cap X_{I}^{\circ}$, we can replace each of them by polar coordinates $(r_i,\theta_{i})$, or $(s_i,\theta_i)$. Since the map $\pi$ is smooth, the pullback of each $s_i$ and $\theta_i$ to $U \cap A_{I}^{\circ}$ is smooth, too. Thus the map \eqref{eq:coordinates-on-stratum} is smooth. To see that it defines a system of coordinates, it is enough to see that the block $[\frac{\d s_{i}}{\d \bar{v}_{j}}]_{l<i,j\leq k}$ of its Jacobian matrix is invertible. By definition \eqref{eq:def-r_i-t_i} of $t_{i}$, we have $s_{i}=-(m_it_i)^{-1}$, and by Lemma \ref{lem:Ui-simple}\ref{item:Ui-cap-dA}, on $U_{I}^{\circ}$ we have $t_{i}=v_i$. Thus it is enough to show that the matrix  $[\frac{\d v_{i}}{\d \bar{v}_{j}}]_{l<i,j\leq k}$ is invertible. By Lemma \ref{lem:dvbar}\ref{item:tame_1-bar}, as we approach $x_0$ we have $d(v_{i}-\bar{v}_{i})\rightarrow 0$, so our matrix approaches the identity; in particular it is invertible once $U_X$ is small enough.

\ref{item:coordinate-vector-fields-on-stratum} %We need to show that the vector fields $\tfrac{\d}{\d \theta_1},\dots,\tfrac{\d}{\d \theta_{k}}$, defined above using coordinates \eqref{eq:AC-chart}, are dual to $d\theta_1,\dots,d\theta_k$ with respect to the coordinates \eqref{eq:coordinates-on-stratum}, too.
%Denote the maps \eqref{eq:AC-chart} and \eqref{eq:coordinates-on-stratum} by $\varphi$ and $\varphi_{I}$, respectively.
The coordinates of $\tfrac{\d}{\d \theta_{i}}$ in our basis of $TU_{I}^{\circ}$ is given by the $(k+i)$-th column of the above Jacobian matrix.
%of $\varphi_{I}\circ \varphi^{-1}$.
Its entries are $\frac{\d \bar{v}_j}{\d \theta_i}=0$, $\frac{\d s_j}{\d \theta_i}=0$, $\frac{\d \theta_{j}}{\d \theta_{i}}=\delta_{i}^{j}$ and $\frac{\d z_j}{\d \theta_{i}}=0$, as needed.

%\ref{item:splitting-on-stratum} Follows directly from the definition.

\ref{item:splitting-on-stratum} %\sloppy
The map $\pi$ collapses $A_{I}^{\circ}$ onto $X_{I}^{\circ}$. In local coordinates \eqref{eq:coordinates-on-stratum} on $U_{I}^{\circ}$ and $(s_{l+1},\dots,s_{k},\allowbreak\theta_{l+1},\dots,\theta_{k},\allowbreak z_{i_1},\dots,z_{i_{n-k}})$ on its image in $X_{I}^{\circ}$, this map is a projection
\begin{equation*}
(s_{l+1},\dots,s_{k},\theta_{l+1},\dots,\theta_{k}, z_{i_1},\dots,z_{i_{n-k}}).		 \end{equation*}
Thus $\pi_{*}|_{\cV_{I}\oplus \Theta_{I}}=0$, and $\pi_{*}|_{\cZ_{I}}$ is an isomorphism onto its image, as needed.
\end{proof}

\begin{lema}\label{lem:TdA}
On $U_{I}^{\circ}$, the following hold.
\begin{enumerate}
\item\label{item:ort-Z} For every $\beta\in \Omega^{*}(U_X \cap X_{I}^{\circ})$ we have $\pi^{*}\beta|_{\mathcal{V}_{I}\oplus \Theta_{I}}=0$.
\item\label{item:ort-Z-vj} For every $j>l$ we have $\bar{v}_{j}=\bar{t}_{j}$ for some $\bar{t}_{j}\in \cC^{\infty}(U_X\cap X_{I}^{\circ})$. In particular, $d\bar{v}_j|_{\mathcal{V}_I\oplus \Theta_I}=0$.
\item\label{item:ort-alpha} For every $i\in \{1,\dots, k\}$, $j\in \{1,\dots, l\}$ we have $\alpha_{i}|_{\mathcal{V}_{I}}=0$ and $\alpha_{i}(\tfrac{\d}{\d \theta_{j}})=\delta_{i}^{j}$.
\item\label{item:dv1} We have $d\bar{v}_{1} =-\sum_{i=2}^{l}\frac{\zeta(u_i)}{\zeta(u_1)}\, d\bar{v}_{i}$, where $\zeta=(\eta^{-1})'\colon [0,\infty)\to[0,\infty)$ is as in \eqref{eq:monodromy-vector-field}.
\item\label{item:dvi} For every $i\in \{1,\dots, l\}$ we have $d\bar{v}_{i}|_{\Theta_{I}\oplus \cZ_{I}}=0$.
\item\label{item:omega-dtheta}
For every $i\in \{1,\dots, l\}$ we have
$\omega\AC^{\epsilon}(\sdot, \frac{\d}{\d\theta_{i}})=\epsilon\, d\bar{v}_{i}$.
\end{enumerate}
\end{lema}
\begin{proof}
\ref{item:ort-Z} By Lemma \ref{lem:coordinates-on-stratum}\ref{item:splitting-on-stratum}, we have $\pi_{*}|_{\mathcal{V}_{I}\oplus \Theta_{I}}=0$, so $\pi^{*}\beta|_{\mathcal{V}_{I}\oplus \Theta_{I}}=0$.

\ref{item:ort-Z-vj} Recall that $\bar{v}_{j}=\sum_{p}\tau^{p}v_{j}^{p}$, where each $\tau^{p}$ is a  smooth function on $X$; and $v_{j}^{p}=0$ for those $p\in R$ such that $U_X^p\cap D_j=\emptyset$. Take $p\in R$ such that $U_X^p\cap D_j\neq \emptyset$. Since $j\not\in I$, Lemma \ref{lem:intro}\ref{item:intro-w-zero} and definition \eqref{eq:def-v_i} of $v_{j}^{p}$ imply that $v_{j}^{p}|_{U_{I}^{\circ}\cap U^{p}}=t_{j}^{p}|_{U_{I}^{\circ}\cap U^{p}}$. Again since $j\not \in I$, the restriction $t_{j}^{p}|_{U_{I}^{\circ}\cap U^p}$ is a pullback of a smooth function on $(U_{X}\cap X_{I}^{\circ})\cap U_{X}^{p}$. We conclude that $\sum_{p}\tau^{p}t_{j}^{p}$ is a smooth function on $U_X\cap X_{I}^{\circ}$, whose pullback to $U_{I}^{\circ}$ equals $\bar{v}_{j}$. This proves the first statement of \ref{item:ort-Z-vj}; the second follows from \ref{item:ort-Z}.

\ref{item:ort-alpha} By Lemma \ref{lem:d-theta}\ref{item:alpha-dtheta}, we have $\alpha_{i}=d\theta_{i}+\pi^{*}\beta_{i}$ for some $\beta_{i}\in \Omega^{1}(U_X)$. Hence part  \ref{item:ort-Z} implies that $\alpha_{i}|_{\mathcal{V}_{I}\oplus \Theta_{I}}=d\theta_{i}|_{\mathcal{V}_{I}\oplus \Theta_{I}}$. Now \ref{item:ort-alpha} follows from the definition of the coordinate vector fields $\tfrac{\d}{\d \theta_j}$.

\ref{item:dv1} Note first that the restriction $u_{1}|_{U_1}$ is nonzero by definition \eqref{eq:Ui} of $U_1$, so each summand in \ref{item:dv1} is well defined. By continuity, it is enough to prove \ref{item:dv1} at each point $x\in \Int_{\d A} U_{I}^{\circ}$.

Fix $i\in \{1,\dots, l\}$. Since $x\in \Int_{\d A} U_{I}^{\circ}$, we have $t_{i}=0$ on some neighborhood of $x$ in $\d A$. There, $\bar{v}_{i}=-\bar{u}_i$ by definition \eqref{eq:def-vbar-ubar-mu} of $\bar{v}_{i}$; so $\bar{v}_i=-u_i$ by Lemma \ref{lem:intro}\ref{item:intro-u=ubar}. The formula \eqref{eq:def-w_i-u_i} gives $u_{i}=\eta(w_i)$, so $w_{i}=\eta^{-1}(u_i)$, and therefore $dw_{i}=\zeta(u_i)du_i$ by definition of $\zeta$. It follows that on $T_{x}U_{I}^{\circ}$ we have $dw_{i}=-\zeta(u_i) d\bar{v}_{i}$.%, for all $i\in \{1,\dots, k\}$.

In particular, $d\bar{v}_{1}=-\tfrac{1}{\zeta(u_1)}dw_{1}$. By Lemma \ref{lem:intro}\ref{item:intro-sum} we have $w_{1}=-\sum_{i=2}^{l}w_{i}$, so
\begin{equation*}
\end{equation*}
as needed. %The formula extends to the whole $A_{S}^{\circ}$ by continuity.

\ref{item:dvi} For $i\in \{2,\dots, l\}$ we have $d\bar{v}_{i}|_{\Theta_{I}\oplus \cZ_{I}}=0$ by definition of $\Theta_{I}\oplus\cZ_{I}$, see Lemma \ref{lem:coordinates-on-stratum}\ref{item:splitting-on-stratum}. The remaining equality $d\bar{v}_1|_{\Theta_{I}\oplus \cZ_{I}}=0$ follows from \ref{item:dv1}.

\ref{item:omega-dtheta} By \ref{item:ort-Z}, we have $\pi^{*}\omega_{X}(\sdot,\tfrac{\d}{\d \theta_i})=0$, so $\omega\AC^{\epsilon}(\sdot,\tfrac{\d}{\d \theta_i})=\epsilon \omega_{E}(\sdot,\tfrac{\d}{\d \theta_i})$. By definition \eqref{eq:omegaE} of $\omega_{E}$, we have $\omega_{E}|_{\partial A}=\sum_{j=1}^{N}d\bar{v}_{j}\wedge \alpha_{j}+\sum_{j=1}^{N}\bar{v}_{j}d\alpha_{j}$. By Lemma \ref{lem:d-theta}, the $1$-forms $\alpha_j$ for $j>l$, and the $2$-forms $d\alpha_{j}$ for all $j$ are pullbacks of smooth forms from $U_X$. Thus parts \ref{item:ort-Z} and \ref{item:ort-Z-vj} imply that $(d\bar{v}_{j}\wedge \alpha_{j})(\sdot, \tfrac{\d}{\d \theta_{i}})=0$ for $j>l$ and $d\alpha_{j} (\sdot,\tfrac{\d}{\d \theta_i})=0$ for all $j$. We conclude that
\begin{equation*}
\omega\AC^{\epsilon}\left(\sdot,\frac{\d}{\d \theta_i}\right)=
\epsilon \sum_{j=1}^{l}\alpha_{j}\left(\frac{\d}{\d\theta_i}\right)\cdot d\bar{v}_{j}-d\bar{v}_{j}\left(\frac{\d}{\d\theta_{i}}\right)\cdot \alpha_j=
%\epsilon d\bar{v}_{i}-\epsilon d\bar{v}_{1}(\tfrac{\d}{\d \theta_{i}})\alpha_1=
\epsilon d\bar{v}_i,
\end{equation*}
where in the second equality we have used identities $\alpha_{j}(\tfrac{\d}{\d\theta_i})=\delta_{i}^{j}$ and $d\bar{v}_{j}(\tfrac{\d}{\d \theta_{i}})=0$ from \ref{item:ort-alpha}, \ref{item:dvi}.
\end{proof}

\begin{lema}\label{lem:Z}
We can shrink $U_{X}$ so that the following holds. There is $\epsilon_0>0$ such that for every $\epsilon\in [0,\epsilon_0]$, the restriction of the form $\omega_{A}^{\epsilon}$ to the subbundle $\cZ_{I}$ is nondegenerate.
\end{lema}
\begin{proof}
By Lemma \ref{lem:coordinates-on-stratum}\ref{item:splitting-on-stratum}, the differential of $\pi$ maps $\cZ_{I}$ isomorphically onto the tangent bundle to the stratum $X_{I}^{\circ}\cap \pi(U_1)$. In particular, the standard almost complex structure $J$ on $X$ pulls back to an almost complex structure on $\cZ_{I}$. We denote this pullback by $J$, too. Now it is enough to prove that, after possibly shrinking $U_{X}$, there is an $\epsilon_0>0$ such that for all $\epsilon\in [0,\epsilon_0]$, the symmetric part of the bilinear form $\cZ_{I}\times \cZ_{I}\ni (v,w)\mapsto \omega_{A}^{\epsilon}(v,Jw)\in \R$ is positive definite.

By Lemma \ref{lem:TdA}\ref{item:dvi}, we have $d\bar{v}_{i}|_{\cZ_{I}}=0$ for all $i\in \{1,\dots, l\}$. Hence %the restriction of $\omega_{E}$ to $\cZ_{I}$ equals:
\begin{equation*}
\omega_{E}|_{\cZ_{I}}=\sum_{i=l+1}^{k}d\bar{v}_{i}\wedge \alpha_{i}+\check{\omega}_{1},
\end{equation*}
where $\check{\omega}_{1}\de \sum_{j=k+1}^{N}d\bar{v}_{j}\wedge \alpha_{j}+\sum_{j=1}^{N}\bar{v}_{j}d\alpha_{j}$. By Lemma \ref{lem:d-theta}\ref{item:dvbar-rest-fiberwise},\ref{item:alpha-rest},\ref{item:dalpha}, the $1$-forms $d\bar{v}_{j}$, $\alpha_{j}$ for $j>k$; and the $2$-forms $d\alpha_j$ for all $j$, are pullbacks of smooth, bounded forms on $U_{X,I}^{\circ}\de U_X\cap X_{I}^{\circ}$. By Lemma \ref{lem:TdA}\ref{item:ort-Z-vj}, the functions $\bar{v}_{j}$ for $j>l$ are pullbacks of smooth functions on $U_{X,I}^{\circ}$. Clearly the functions $\bar{v}_{j}$ are bounded. We conclude that the form $\check{\omega}_1$ is bounded in the natural coordinates of $U_{X,I}^{\circ}$.

By Lemma \ref{lem:dvbar}\ref{item:tame_1-bar}, for every $i\in \{l+1,\dots, k\}$, the restriction of $d\bar{v}_{i}$ to $T U_{I}^{\circ}$ equals $(1+c_{i}t_{i})dv_{i}+\sigma_{i}\gamma_{i}$, where $c_{i}$ (respectively, $\gamma_{i}$) is a pullback of a smooth, bounded function (respectively, $1$-form) on $U_{X,I}^{\circ}$; and $\sigma_{i}$ is the function defined in \eqref{eq:rho-sigma}. Moreover, since $u_{i}|_{U_{I}^{\circ}}=0$, we have $dv_{i}=dt_{i}=t_{i}^{2}m_{i}\, ds_{i}$ by Lemma \ref{lem:nu-j}\ref{item:dt-ds}, and $t_{i}^{2}=\sigma_{i}$ by definition \eqref{eq:rho-sigma} of $\sigma_{i}$. Eventually, by Lemma \ref{lem:d-theta}\ref{item:alpha-dtheta}, we have $\alpha_{i}=d\theta_{i}+\pi^{*}\beta_{i}$ for some bounded $\beta_{i}\in \Omega^{1}(U_{X})$. As in the proof of Lemma \ref{lem:omegaAC-sym}, we conclude that
\begin{equation*}
\omega_{E}|_{\cZ_{I}}=\sum_{i=l+1}^{k} \sigma_{i}(m_{i}(1+c_i t_i)ds_{i}\wedge d\theta_{i}+\gamma_{i}\wedge d\theta_{i}+m_{i}(1+c_i t_i)ds_{i}\wedge \check{\beta}_{i})+\omega_{1},
\end{equation*}
for some smooth, bounded $c_i\in \cC^{\infty}(U_{X,I}^{\circ})$, $\gamma_{i},\check{\beta}_{i}\in \Omega^{1}(U_{X,I}^{\circ})$ and $\omega_{1}\in \Omega^{2}(U_{X,I}^{\circ})$.

This is precisely formula \eqref{eq:omegaAC-mu} with two modifications. First, the sum starts at $i=l+1$ instead of $1$. Second, the functions and forms which in \eqref{eq:omegaAC-mu} are bounded from $X$, i.e.\ bounded in the natural coordinates of $U_X\cap X_{\emptyset}^{\circ}$, are now bounded in the natural coordinates of $U_X\cap X_{I}^{\circ}$.

The remaining part of the proof follows by repeating the proof of Proposition \ref{prop:positivity}, given in Section \ref{sec:positivity}, starting from formula \eqref{eq:omegaAC-mu}. The only difference is that now the computations take place over the stratum $X_{I}^{\circ}$ instead of $X_{\emptyset}^{\circ}$. In particular, instead of vector fields $\nu_{1},\dots,\nu_{2k}$ on $U_{\emptyset}^{\circ}$, which were dual to $ds_1,\dots,d s_k$, $d\theta_1,\dots,d\theta_k$ with respect to the coordinates \eqref{eq:coordinates-sx}, we now use vector fields which are dual to $ds_{l+1},\dots,d s_{k}$, $d\theta_{l+1},\dots,d\theta_{k}$ with respect to the coordinates \eqref{eq:coordinates-on-stratum}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:lifts}]
The point $x_0\in D$ lies in a stratum $X_{S}^{\circ}$ for some nonempty $S\subseteq \{1,\dots, N\}$, see \eqref{eq:stratification}. Choose a fine chart $U_X$ around $x_0$, whose associated index set \eqref{eq:index-set} equals $S$. The preimage $U=\pi^{-1}(U_X)$ admits a finite covering $U=\bigcup_{i\in S}U_i$, where each $U_i$ is the domain of a smooth chart \eqref{eq:AC-chart}. Therefore, it is enough to prove the following. For every $i_0\in S$, we can shrink the chart $U_{X}$ around $x_0$, and find a  positive number $\epsilon_0>0$ such that for every $\epsilon\in (0,\epsilon_0]$, the form $\omega_{A}^{\epsilon}$ is fiberwise nondegenerate on $U_{i_0}$; and the symplectic lift of $\frac{\d}{\d \theta}$ to $U_{i_0}$ is given by the formula \eqref{eq:monodromy-vector-field}.

As before, we reorder the components of $D$ so that $S=\{1,\dots, k\}$ and $i_0=1$. We can assume that $U_X$ and $\epsilon_0>0$ are small enough so that the statement of Lemma \ref{lem:Z} holds for all subsets $I\subseteq U_X$ containing $1$. We fix one such subset, say $I=\{1,\dots, l\}$, and use coordinates \eqref{eq:coordinates-on-stratum} on $U_{I}^{\circ}=U_1\cap A_{I}^{\circ}$. Also, we fix an $\epsilon\in (0,\epsilon_0]$.

Fix a point $x\in U_I^{\circ}$. Let $F$ be the fiber of $\theta$ passing through $x$. Since the fine chart $U_{X}$ is by definition adapted to $f$, Definition \ref{def:adapted-chart}\ref{item:f-locally} together with the formula \eqref{eq:def-theta_i} for $\theta_{i}$ show that
\begin{equation*}
\theta|_{U_I^{\circ}}=m_1\theta_{1}+\dots+m_{k}\theta_k,
\end{equation*}
where we use additive notation \ref{not:S1} on $\S^1$. Hence
\begin{equation}\label{eq:TF}
T_{x}F=\ker \left[\sum_{i=1}^{k} m_{i} d\theta_{i}\right]\subseteq T_{x}U_I^{\circ}.
\end{equation}
Assume that a nonzero vector $\nu\in T_{x}U_I^{\circ}$ satisfies $\omega\AC^{\epsilon}(\nu,\nu')=0$ for all $\nu'\in T_{x}F$. To prove non-degeneracy of $\omega\AC^{\epsilon}|_{T_{x}F}$, we need to show that $\nu\not\in T_{x}F$.

Let $\nu^{\theta}$ be the vector given by the formula \eqref{eq:monodromy-vector-field}. We note that for $i\in \{l+1,\dots, k\}$, we have $u_i=0$, so the sum in \eqref{eq:monodromy-vector-field} actually runs from $i=1$ to $l$. For every $j\in \{1,\dots, k\}$, we have $d\theta_{j}(\nu^{\theta})=(\sum_{i=1}^{l}m_{i}\zeta(u_i))^{-1}\cdot \zeta(u_j)\geq 0$, with strict inequality for $j=1$. Hence formula \eqref{eq:TF} gives $\nu^{\theta}\not\in T_{x}F$. Thus to show that $\nu\not\in T_{x}F$, it is enough to show that $\nu$ is proportional to \eqref{eq:monodromy-vector-field}.

Using coordinate vector fields $\tfrac{\d}{\d \bar{v}_i}$ introduced in Lemma \ref{lem:coordinates-on-stratum}\ref{item:splitting-on-stratum}, we can write
\begin{equation}\label{eq:ort}
\nu=\sum_{i=2}^{l} a_{i}\frac{\d}{\d \bar{v}_{i}}+\sum_{i=1}^{l}b_{i}\frac{\d}{\d \theta_{i}}+\xi,\quad\mbox{for some } a_i,b_i\in \R\mbox{ and } \xi\in \mathcal{Z}_{I}.
\end{equation}
First, we claim that all coefficients $a_2,\dots, a_{l}$ in \eqref{eq:ort} are zero.

For $i\in \{2,\dots,l\}$ put $\nu^\theta_{i}=\tfrac{1}{m_{i}}\tfrac{\d}{\d \theta_{i}}-\tfrac{1}{m_{1}}\tfrac{\d}{\d \theta_{1}}\in T_{x}U_{I}^{\circ}$. Formula \eqref{eq:TF} shows that
$\nu_{i}^{\theta}\in T_{x}F$, so $\omega\AC^{\epsilon}(\nu,\nu^{\theta}_i)=0$ by definition of $\nu$. Using Lemma \ref{lem:TdA}\ref{item:omega-dtheta}, we compute
\begin{equation*}
0=\omega\AC^{\epsilon}(\nu,\nu^\theta_{i})= \tfrac{1}{m_{i}} \omega\AC^{\epsilon}(\nu, \tfrac{\d}{\d \theta_{i}})- \tfrac{1}{m_{1}}\omega\AC^{\epsilon}(\nu,\tfrac{\d}{\d \theta_{1}})=\tfrac{1}{m_{i}}\cdot \epsilon d\bar{v}_i(\nu)-\tfrac{1}{m_{1}}\cdot  \epsilon d\bar{v}_{1}(\nu)=\epsilon(\tfrac{a_i}{m_i}-\tfrac{1}{m_{1}}d\bar{v}_{1}(\nu)),
\end{equation*}
where the last equality follows from definition of the coordinate vector fields. Thus for all $i\in \{2,\dots, l\}$, the numbers $\tfrac{a_i}{m_{i}}$ are equal to the same number $a\de \tfrac{1}{m_{1}}d\bar{v}_{1}(\nu)$.

Suppose $a\neq 0$. Then $\tfrac{a_i}{a}=m_i$. Since $d\bar{v}_{1}|_{\Theta_{I}\oplus \cZ_{I}}=0$ by Lemma \ref{lem:TdA}\ref{item:dvi}, the formula \eqref{eq:ort} for $\nu$ gives
\begin{equation*}
m_1=\frac{1}{a}d\bar{v}_1(\nu)=\frac{1}{a}\sum_{i=2}^{l}a_{i} d\bar{v}_{1}\left(\frac{\d}{\d \bar{v}_i}\right)=-\sum_{i=2}^{l} m_{i}\frac{\zeta(u_i)}{\zeta(u_1)}\leq 0,
\end{equation*}
where for the last equality we used the fact that $\tfrac{a_i}{a}=m_i$, and the formula for $d\bar{v}_1$ from Lemma \ref{lem:TdA}\ref{item:dv1}. This is a contradiction. Therefore, $a=0$, so $a_{i}=am_i=0$ for all $i\in \{2,\dots, l\}$, as claimed.
\smallskip

Now, we claim that the vector $\xi\in\cZ_{I}$ in \eqref{eq:ort} is zero. Fix any $\xi'\in \mathcal{Z}_{I}$. Then $\xi'\in T_{x}F$ by \eqref{eq:TF}, so $\omega\AC^{\epsilon}(\nu,\xi')=0$ by definition of $\nu$. By Lemma \ref{lem:TdA}\ref{item:omega-dtheta}, for any $i\in \{1,\dots,l\}$ we have $\omega\AC^{\epsilon}(\tfrac{\d}{\d \theta_{i}},\xi')=-\epsilon d\bar{v}_{i}(\xi')=0$, where the last equality follows from Lemma \ref{lem:TdA}\ref{item:dvi}. Since we have shown that $a_2=\dots=a_{l}=0$, we infer from \eqref{eq:ort} that $0=\omega\AC^{\epsilon}(\nu,\xi')=\omega\AC^{\epsilon}(\xi,\xi')$, for all $\xi'\in \cZ_{I}$. Since by Lemma \ref{lem:Z}, the restriction $\omega\AC^{\epsilon}|_{\cZ_{I}}$ is nondegenerate, we get $\xi=0$, as claimed.

Therefore, $\nu=\sum_{i=1}^{l}b_{i}\tfrac{\d}{\d \theta_{i}}$ for some $b_1,\dots,b_{l}\in \R$. Since for all $j\in \{2,\dots, l\}$ we have $\tfrac{\d}{\d \bar{v}_{j}}\in T_{x}F$, we compute using Lemma \ref{lem:TdA}\ref{item:omega-dtheta} that
\begin{equation*}
0=\omega\AC^{\epsilon}\left(\frac{\d}{\d \bar{v}_{j}},\nu\right)=
\sum_{i=1}^{l}b_{i}\omega\AC^{\epsilon}\left(\frac{\d}{\d \bar{v}_{j}},\frac{\d}{\d \theta_{i}}\right)=
\epsilon\sum_{i=1}^{l}b_{i}d\bar{v}_{i}\left(\frac{\d}{\d\bar{v}_{j}}\right)=
\epsilon\left(b_{j}-b_{1}\frac{\zeta(u_j)}{\zeta(u_1)}\right),
\end{equation*}
where the last equality follows from Lemma \ref{lem:TdA}\ref{item:dv1}. We conclude that for all $j\in \{1,\dots, l\}$ we have $b_{j}=\zeta(u_{j})b$ for some $b\in \R$. This shows that $\nu$ is proportional to \eqref{eq:monodromy-vector-field}, as claimed. In particular, the form $\omega_{A}^{\epsilon}$ is non-degenerate at $x$.

It remains to show that if $\nu$ is normalized by $\theta_{*}(\nu)=\tfrac{\d}{\d \theta}$ then $\nu$ equals \eqref{eq:monodromy-vector-field}. In this case, we have
\begin{equation*}
\frac{\d}{\d\theta}=\theta_{*}(\nu)=\sum_{i=1}^{l}b_{i}m_{i} \cdot \frac{\d}{\d\theta}=b\sum_{i=1}^{l} \zeta(u_{i})m_{i}\cdot \frac{\d}{\d\theta},
\end{equation*}
hence $b=(\sum_{i=1}^{l}\zeta(u_{i})m_{i})^{-1}$, as needed.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:omega}]
Fix $x\in \d A$. By Propositions \ref{prop:positivity} and \ref{prop:lifts}, there is an $\epsilon_{x}>0$ and an open neighborhood $V_{x}$ of $x$ in $A$, such that, for any $\delta>0$, each of the forms $\omega\AC^\epsilon$ and $\omega\AC^{\delta,\epsilon}$ is fiberwise nondegenerate on $V_{x}$ for every $\epsilon\in (0,\epsilon_{x}]$, and on $V_{x}\setminus \d A$ for every $\epsilon\in [0,\epsilon_x]$. Since $\d A \cap \bar{W}$ is compact, choosing a finite sub-cover from $\{V_{x}\}$ we get an $\epsilon_0>0$ and a neighborhood $W'$ of $\d A\cap \bar{W}$, such that, again, each of the forms $\omega\AC^\epsilon$ and $\omega\AC^{\delta,\epsilon}$ is fiberwise nondegenerate on $W'$ for all $\epsilon\in (0,\epsilon_0]$. This proves part \ref{item:omega-symplectic}, part \ref{item:omega-lift} follows from Proposition \ref{prop:lifts}.
\end{proof}

We end this section with a simple example illustrating Definition \ref{def:AX} of the A'Campo space, and the  monodromy  at radius zero given by Proposition \ref{prop:omega}\ref{item:omega-lift}. In Proposition \ref{prop:monodromy} we will describe the latter more precisely, in terms of the associated abstract contact open book.

\begin{example}\label{ex:monodromy}
\begin{figure}[ht]
\begin{subfigure}{\textwidth}
\centering
\begin{tikzcd}[column sep=large]
\begin{tikzpicture}[scale=0.7]
\path[use as bounding box] (-3,-1) rectangle (4,5.3);
\draw (-2,5) ellipse (1 and 0.3);
\draw (-1,5) -- (-1,3) to [out=-90, in=180] (1,1) to (3,1);
\draw (-3,5) -- (-3,3) to [out=-90, in=180] (1,-1) to (3,-1);
%\draw[very thick] (-2,4.7) -- (-2,2.8) to [out=-90, in=180] (-0.4,1.6);
%\draw[very thick, dashed] (-0.4,1.6) to [out=-90, in=180] (1.2,0) -- (2.7,0);
\draw[dashed] (-1,3) arc(0:180:1 and 0.3);
\draw (-3,3) arc(180:360:1 and 0.3);
\draw (1,1) arc(90:270:0.3 and 1);
\draw[dashed] (1,-1) arc(-90:90:0.3 and 1);
%\draw[very thick] (2.7,0) -- (3.3,0);
\draw (3,0) ellipse (0.3 and 1);
\node[below right] at (-1.1,4.5) {\small{$A_{1}^{\circ}$}};
\node[below right] at (-0.7,2.4) {\small{$A_{1,2}^{\circ}$}};
\node[above right] at (1.5,1) {\small{$A_{2}^{\circ}$}};
\end{tikzpicture}
\ar[r]
\ar[d]
\ar[rd, "\pi", xshift=10]
\begin{tikzpicture}[scale=0.7]
\path[use as bounding box] (-3,-1) rectangle (4,5.3);
\draw[fill] (0,0) circle [radius=0.1];
\draw[very thick] (-2,2) to [out=-90,in=180] (0,0);
\draw[fill] (-2,2) circle [radius=0.1];
\draw (-3,5) -- (-2,2) -- (-1,5);
%\draw[very thick] (-2,2) -- (-2,4.7);
\draw (-2,5) ellipse (1 and 0.3);
\draw (3,1) -- (0,0) -- (3,-1);
%\draw[very thick, dashed] (0,0) -- (2.7,0);
%\draw[very thick] (2.7,0) -- (3.3,0);
\draw (3,0) ellipse (0.3 and 1);
\node at (-0.8,4) {\small{$\Gamma_{1}^{\circ}$}};
\node at (2,1) {\small{$\Gamma_{2}^{\circ}$}};
%\node [below left] at (0,0) {\small{$[1:0]$}};
%\node [below left] at (-2,2) {\small{$[0:1]$}};
\node at (-1.9,0.4) {\small{$\Gamma_{1,2}^{\circ}$}};
\draw[->] (-1.6,1.4) to [out=-70,in=160] (-0.6,0.4);
\node at (-1.1,0.9) {\tiny{$\upsilon$}};
%\node at (1.8,2.3) {\small{$\upsilon=\frac{2\zeta(u_2)}{\zeta(u_1)+2\zeta(u_2)}$}};
%\draw[->, gray] (0.1,2.4) to[out=180,in=0] (-0.9,1);
\end{tikzpicture}
\ar[d]
\\
\begin{tikzpicture}[scale=0.7]
\path[use as bounding box] (-2.5,-1) rectangle (4.5,3.5);
\draw (0,3) ellipse (1 and 0.3);
\draw (-1,3) -- (-1,-1) -- (3,-1);
\draw (1,3) -- (1,1) -- (3,1);
\draw (-1,-1) to [out=60, in=210] (1,1);
\draw[dashed] (-1,-1) to [out=30, in=-120] (1,1);
%\draw[very thick] (0,2.7) -- (0,0.3);
%\draw[very thick, dashed] (0.3,0) -- (2.7,0);
%\draw[very thick] (2.7,0) -- (3.3,0);
\draw (3,0) ellipse (0.3 and 1);
\node at (2,2.8) {\small{$(X_{1}^{\circ})_{\log}$}};
\node at (3,1.3) {\small{$(X_{2}^{\circ})_{\log}$}};
\end{tikzpicture}
\ar[r]
\begin{tikzpicture}[scale=0.7]
\path[use as bounding box] (-2.5,-1) rectangle (4.5,3.5);
\draw[fill] (0,0) circle [radius=0.1];
\draw (-1,3) -- (0,0) -- (1,3);
%\draw[very thick] (0,0) -- (0,2.7);
\draw (0,3) ellipse (1 and 0.3);
\draw (3,1) -- (0,0) -- (3,-1);
%\draw[very thick, dashed] (0,0) -- (2.7,0);
%\draw[very thick] (2.7,0) -- (3.3,0);
\draw (3,0) ellipse (0.3 and 1);
\node at (1.2,2) {\small{$D_1$}};
\node at (2.1,1) {\small{$D_2$}};
%			\node at (0,-0.6) {\small{$X_{1,2}^{\circ}$}};
\end{tikzpicture}
\end{tikzcd}
\caption{Radius-zero fibers in $A$, $\Gamma$, $X_{\log}$ and $X=\C^2$, see diagram \protect\eqref{eq:AX-diagram}.}
\label{fig:construction}
\end{subfigure}
\medskip

\begin{subfigure}{\textwidth}
\centering
\begin{tikzcd}[column sep=large]
\begin{tikzpicture}[scale=0.7]
\path[use as bounding box] (-4,-1) rectangle (3,5.5);
\draw (-2,5) ellipse (1 and 0.3);
\draw (-1,5) -- (-1,3) to [out=-90, in=180] (1,1) to (3,1);
\draw (-3,5) -- (-3,3) to [out=-90, in=180] (1,-1) to (3,-1);
\draw[very thick] (-2,4.7) -- (-2,2.8) to [out=-90, in=180] (1,0) -- (2.7,0);
\draw[dashed] (-1,3) arc(0:180:1 and 0.3);
\draw (-3,3) arc(180:360:1 and 0.3);
\draw (1,1) arc(90:270:0.3 and 1);
\draw[dashed] (1,-1) arc(-90:90:0.3 and 1);
\draw (3,0) ellipse (0.3 and 1);
%			\node[below right] at (-1,5) {\small{$A_{1}^{\circ}$}};
%			\node[below right] at (-1,3) {\small{$A_{1,2}^{\circ}$}};
%			\node[above right] at (1,1) {\small{$A_{2}^{\circ}$}};
\end{tikzpicture}
\ar[r, "\phi", yshift=50]
\begin{tikzpicture}[scale=0.7]
\path[use as bounding box] (-4,-1) rectangle (7,5.5);
\draw (-2,5) ellipse (1 and 0.3);
\draw (-1,5) -- (-1,3) to [out=-90, in=180] (1,1) to (3,1);
\draw (-3,5) -- (-3,3) to [out=-90, in=180] (1,-1) to (3,-1);
\draw[very thick] (-2,4.7) -- (-2,2.8) to [out=-90, in=180] (-0.4,1.6);
\draw[very thick, dashed] (-0.4,1.6) to [out=-90, in=180] (1.2,0) -- (2.7,0);
\draw[dashed] (-1,3) arc(0:180:1 and 0.3);
\draw (-3,3) arc(180:360:1 and 0.3);
\draw (1,1) arc(90:270:0.3 and 1);
\draw[dashed] (1,-1) arc(-90:90:0.3 and 1);
\draw[very thick] (2.7,0) -- (3.3,0);
\draw (3,0) ellipse (0.3 and 1);
\node[below right] at (-1,5) {\small{$A_{1}^{\circ}\colon (\theta_1,\theta_2)\mapsto (\theta_1,\theta_2)$}};
\node[below right] at (-1,3) {\small{$A_{1,2}^{\circ}\colon (\theta_1,\theta_2)\mapsto (\theta_{1}-\upsilon,\theta_2+\tfrac{1}{2}\upsilon)$}};
\node[above right] at (1,1) {\small{$A_{2}^{\circ} \colon (\theta_1,\theta_2) \mapsto (\theta_1,\theta_2+\tfrac{1}{2})$}};
\end{tikzpicture}
\end{tikzcd}
\caption{Monodromy at radius zero, see Propositions \ref{prop:omega}\ref{item:omega-lift} and \ref{prop:monodromy}.}
\label{fig:monodromy}
\end{subfigure}
\caption{Example \protect\ref{ex:monodromy}: the A'Campo space for the function $f(z_1,z_2)=z_1z_2^2$.}
\label{fig:example}
\end{figure}
Take $f\colon X=\C^{2}\to \C$ given by $f(z_1,z_2)=z_1z_2^2$.
Then $f^{-1}(0)=D_{1}+2D_{2}$, where $D_{i}=\{z_{i}=0\}$. In the bottom-right of Figure \ref{fig:construction}, the disks $D_{1},D_{2}$ meeting at the origin are shown as the vertical and horizontal cone, respectively. The fiber $f_{\log}^{-1}(0,1)\subseteq X_{\log}$ is shown in the bottom-left. The restriction of the map $X_{\log}\to X$ to this fiber is an isomorphism in the vertical cylinder, is a double cover in the horizontal one, and collapses the middle circle to the origin. Hence a natural monodromy at radius zero in the Kato--Nakayama space $X_{\log}$ is not continuous: indeed, in this example it is the identity in $(X_{1}^{\circ})_{\log}$, and a  half-twist in $(X_{2}^{\circ})_{\log}$.

The top-right picture in Figure \ref{fig:construction}  is the fiber over $0$ in $\Gamma=\operatorname{graph}(\mu)$: there, the origin of $\C^2$ is replaced by a $1$-simplex $\Gamma_{1,2}^{\circ}$. Eventually, in the fiber $f_{A}^{-1}(0,1)\subseteq A$, shown in the top-left, we multiply this simplex by the middle circle of $f_{\log}^{-1}(0,1)$. The radius-zero monodromy $\phi$ in $A$ interpolates between the identity in $A_{1}^{\circ}$ and half-twist in $A_{2}^{\circ}$, using a parameter of the $1$-simplex $\Gamma_{1,2}^{\circ}$, call it $\upsilon\in [0,1]$. Figure \ref{fig:monodromy} shows how the bold ruling is transformed by $\phi$. A precise formula for the parameter $\upsilon$ can be obtained from equation \eqref{eq:monodromy-vector-field}: in this example, we have  $\upsilon=\frac{2\zeta(u_2)}{\zeta(u_1)+2\zeta(u_2)}$.
%It is shown at the bottom-right of Figure \ref{fig:monodromy}, as follows. The axes $D_{1},D_{2}$ are  disks meeting at the origin, shown as the vertical and horizontal cone, respectively. The bold line is the image of the \enquote{front} ruling under the monodromy: it gets is flipped to the back of $D_1$, and remains untouched on $D_{2}$.
%	The remaining pictures in Figure \ref{fig:construction} are radius-zero fibers in $\Gamma$ (top-right), $X_{\log}$ (bottom-left) and $A$ (top-left). In $\Gamma$, we replace the origin by a $1$-simplex $\Gamma_{1,2}^{\circ}$, with coordinates $[w_1:w_2]$, $w_1+w_2=1$. In $X_{\log}$, we introduce angular coordinates $\theta_{1}$, $\theta_{2}$ satisfying $2\theta_{1}+\theta_{2}=0$, i.e.\  we replace the origin by a circle. There, we can define a the monodromy is \emph{not continuous}.  Eventually, in $A$ we multiply this circle by $\Gamma_{1,2}^{\circ}$ and make the monodromy interpolate between the two pieces, using a parameter $\upsilon\in [0,1]$ on $\Gamma_{1,2}^{\circ}$.
\end{example}

\section{Symplectic monodromy and abstract contact open books}\label{sec:acobs}

We now recall further notions concerning Liouville domains and their fibrations.  For a general introduction, see [13, 52, 30, 33, 55, 67] and references therein. Our aim is to establish a natural setting \ref{basic-setting} where we can produce or extend symplectic monodromies.

\subsection{Hamiltonian vector fields}\label{sec:Hamiltonian}

Let $(M,\omega)$ be a symplectic manifold. A \emph{Hamiltonian} is a smooth function $H\colon M\times \R\ni (x,t)\mapsto H_{t}(x)\in \R$. A \emph{time-independent Hamiltonian} is a smooth function $H\colon M\to \R$, which we view as a Hamiltonian $M\times \R\ni (x,t)\mapsto H(x)\in \R$. In other words, our Hamiltonians are always time-dependent, unless explicitly stated otherwise.

Fix a Hamiltonian $H\colon M\times \R\to \R$. Since $\omega$ is non-degenerate, there exists a unique vector field $X^{H}$, called a \emph{Hamiltonian vector field} of $H$, satisfying the equality
\begin{equation*}
\omega(X^{H},\sdot)=dH.
\end{equation*}
The \emph{Hamiltonian flow} of $H$ is a family of symplectomorphisms $\psi_{t}^{H}\colon M\to M$, parametrized by $t\in \R$, defined by the formulas
\begin{equation*}
\psi^{H}_{0}=\id_{M},\quad \tfrac{d}{dt}\psi_{t}^{H}=X^{H}.
\end{equation*}
Our definition of $X^{H}$ coincides with the one used in [74, 52, 20, 55]. However, [4, 49] use the opposite one, namely $\omega(\sdot, X^{H})=dH$: thus for a given Hamiltonian $H$, our vector field $X^{H}$ and flow $\psi^{H}_{t}$ are equal to $-X^{H}$ and $(\psi^{H}_{t})^{-1}$ from [4, 49].

So, when we quote [74] we need no adaptations. However, we will quote results from [4, 49] too. To avoid sign mistakes, whenever we do this, we replace $H$ in loc.\ cit.\ by $-H$: this way, we get the same vector field $X^{H}$ and flow $\psi^{H}_{t}$ as in loc.\ cit; so no further modifications are needed. In particular, to get the same Floer Cauchy--Riemann equation \eqref{eq:floereqmclean} as used in \cite[Definition 4.3]{McLean}, we will need to perturb $\phi$ by an opposite Hamiltonian, so we will chose it \emph{negative} near $\d M$, despite the symbol \enquote{$+$}; see Section \ref{sec:perturbed} for details.

\subsection{Liouville and symplectic fibrations}\label{sec:Liouville-fibrations}

\begin{notation}\label{not:fibers}
Fix a map $f\colon X\to Y$. If $f$ is clear from the context, we will use the following, simplified notation for its fibers. For $y\in Y$, we put $X_{y}=f^{-1}(y)$. Given a form $\lambda\in \Omega^{*}(X)$, we write $\lambda_{y}\de \lambda|_{X_y}$ for the restriction of $\lambda$ to a fiber. For subsets $U\subseteq X$, $V\subseteq Y$ we put $U|_{V}\de U\cap f^{-1}(V)$.
\end{notation}

A subset $B$ of a manifold $M$ is a \emph{codimension zero submanifold with corners} if for every point $p\in B$ there is a coordinate chart $\varphi\colon U\to\RR^d$ of $M$ around $p$ such that $\varphi(B\cap U)$ is an open subset of $\RR_{\geq 0}^k\times\RR^{d-k}$ for some $k\geq 0$. A \emph{manifold with corners} is a topological space which embeds into a smooth manifold as a codimension zero manifold with corners. This definition is compatible with the one given in \cite[Appendice]{corners} by Proposition 3.1 loc.\ cit. We say that a map (or a form) is smooth on $B$ if it extends to a smooth map (or form) on some neighborhood of $B$ in the ambient manifold $M$.

For example, the closure of each $A_{i}^{\circ}$ in Figure \ref{fig:v3} is a codimension zero submanifold with corners of $\d A$, cf.\ Proposition \ref{prop:monodromy}. Another example is the total space of the Milnor fibration
\begin{equation}
\label{eq:milfib1}
f\colon \B_{\epsilon}\cap f^{-1}(\D_{\delta}^{*})\to \D_{\delta}^{*},
\end{equation}
where $f\colon \C^n\to \C$ is a holomorphic function with an isolated critical point at $0\in \C^n$.%, and $\epsilon,\delta>0$ are small.
\smallskip

Given a smooth map $f\colon M\to B$ between manifolds with corners, we split the boundary $\partial M$ as:
\begin{equation*}
\d M=\d\vert M\cup \d\hor M, \quad\mbox{where}\quad \d\vert M=f^{-1}(\d B)\cap \d M\quad\mbox{and}\quad \d\hor M=\overline{\d M\setminus \d\vert M}.
\end{equation*}
We say that $f$ is a \emph{fibration} if it is smoothly locally trivial, i.e.\ every $b\in B$ has a neighborhood $U$ and a diffeomorphism $\varphi\colon U\times f^{-1}(p)\to f^{-1}(U)$ such that $f\circ\varphi=\pr_{U}$. By the Ehressmann lemma, see e.g.\ \cite[6.2.10]{handbook}, a proper map $f$ is a fibration if both $f$ and $f|_{\d\hor M}$ are surjective and submersive.

Let $\lambda\in\Omega^1(M)$ be a $1$-form. The pair $(f,\lambda)$ is called a {\em Liouville fibration} if $f\colon M\to B$ is a fibration and $(M_b,\lambda|_{M_b})$ is a Liouville domain for each $b\in B$, see Section \ref{sec:symplectic-intro}. The similar notion of a \emph{symplectic fibration} is defined replacing $\lambda$ by a closed, fiberwise symplectic form $\omega$, see Section \ref{sec:symplectic-intro}. A Liouville fibration induces a symplectic fibration by taking $\omega=d\lambda$.

Note that in the definition of a symplectic fibration we require that the form $\omega\in \Omega^{2}(M)$ is \emph{closed}, and consider it a part of the data. This definition, while convenient for our purposes, is \emph{not} a standard one. For example, in  \cite[\sec 4.2]{Oh_1}, what we call a symplectic fibration is called a \emph{Hamiltonian fibration} with a choice of a coupling form, cf.\ \cite[\sec 6.4]{McDuff_Salamon}. We impose this more restrictive condition since it holds in all our applications, and guarantees that the flows of symplectic lifts are symplectomorphisms.%. Meanwhile, with a usual definition additional conditions are needed, see \cite[\sec 4.1]{Oh_1} for a discussion.}

%%%%%%%%%%% ! ! ! ! ! ! ! ! ! ! ! %%%%%%%%%%%%%%%%%%%%%%%%%%%
%\red{Our \emph{symplectic fibrations are \emph{Hamiltonian fibrations} in \cite[Definition 4.2.1]{Oh_1}. In \cite[Definition 4.1.1]{Oh_1}, a \emph{symplectic fibration} is a fibration with symplectic fibers, see exercise 4.1.3 loc.\ cit. Then, a symplectic connection is the one with symplectic holonomy (Def.\ 4.1.5). By Cartan formula, for a Hamiltonian fibration any symplectic connection in our sense has symplectic holonomy (Proposition 4.1.6). In fact, in this case holonomy is Hamiltonian (Theorem 4.2.2)}}

\subsection{Definition of a symplectic monodromy}

In this section we will modify the Milnor fibration \eqref{eq:milfib1} so that its monodromy becomes a compactly supported symplectomorphism. Note that such a modification is needed: symplectomorphisms preserve volume, and there is no reason why two fibers of \eqref{eq:milfib1} should have the same volume. The following notion, introduced by McLean in \cite[Definition 3.12]{McLean}, abstracts the properties of symplectic monodromy that we need.

\begin{definition}\label{def:acob}
Let $(M,\lambda)$ be a Liouville domain. We say that $(M,\lambda,\phi)$ is an \emph{abstract contact open book} if $\phi\colon M\to M$ is a diffeomorphism that is {\em compactly supported}, i.e.\ satisfies $\phi|_{C}=\id_{C}$ for some neighborhood $C$ of $\d M$, and {\em exact}, that is:
\begin{equation}\label{eq:exact-symplectiomorphism}
\phi^{*}\lambda-\lambda=-d \ac
\end{equation}
for some function $\ac \colon M\to \R$, called \emph{action} of $\phi$. Note that the formula \eqref{eq:exact-symplectiomorphism} defines $\ac$ uniquely, up to an additive constant. Moreover, it implies that $\phi$ is a symplectomorphism.

Let $M$ be a smooth manifold, let $f\colon M\to [0,1]$ be a smooth map, and let $\lambda\in \Omega^{1}(M)$ be a $1$-form such that $(f,\lambda)$ is a Liouville fibration. Let $\phi\colon M\to M$ be a diffeomorphism which preserves each fiber of $f$ and is equal to the identity close to $\d\hor M$. Put $\phi_{t}=\phi|_{M_t}$,  see Notation \ref{not:fibers}. If for every $t\in [0,1]$ the triple $(M_t,\lambda_t,\phi_t)$ is an abstract contact open book, we say that the family $\{(M_t,\lambda_t,\phi_t)\}_{t\in [0,1]}$ is an \emph{isotopy of abstract contact open books}. In this case, we say that the abstract contact open books $(M_t,\lambda_t,\phi_t)$ are \emph{isotopic}, and write $(M_t,\lambda_t,\phi_t)\sim (M_s,\lambda_s,\phi_s)$ for $t,s\in [0,1]$.
\end{definition}

Clearly, if $(M,\lambda,\phi)$ is an abstract contact open book, then so is $(M,\lambda,\phi^m)$ for any iterate $\phi^m$ of $\phi$; and if $(M_0,\lambda_0,\phi_0)\sim (M_1,\lambda_1,\phi_1)$ then $(M_0,\lambda_0,\phi_0^m)\sim (M_1,\lambda_1,\phi_1^m)$ for every $m\geq 1$.
\smallskip

In order to turn monodromies into abstract open books and compare their isotopy classes, we will need the following notion.

\begin{definition}
\label{def:collartriv}
Let $f\colon M\to B\times P$ be a fibration from a manifold with corners to a product of %$P$ is a manifold and $B$ is a
manifolds with boundary. Fix $b_0\in B$ and let $M_{b_{0}}\de (\pr_{B}\circ f)^{-1}(b_0)$, see Notation \ref{not:fibers}.
A \emph{$P$-fiberwise  trivialization} of $f$ is a diffeomorphism
\begin{equation}
\label{eq:triv}
\psi\colon M_{b_0}\times B\to M
\end{equation}
such that for all points $x\in M_{b_{0}}$, and all $b\in B$, we have
\begin{equation*}
\end{equation*}
A \emph{$P$-fiberwise collar trivialization} of $f$ is an open neighborhood $C$ of $\d\hor M$ in $M$, together with a $P$-fiberwise trivialization of $f|_C$.

Let $\omega\in \Omega^2(M)$ be a $2$-form such that $(f,\omega)$ is a symplectic fibration. A $P$-fiberwise trivialization \eqref{eq:triv} is called {\em $(B\times P)$-fiberwise symplectic} if for any $(b,p)\in B\times P$ we have %the equality
\begin{equation}
\label{eq:simpcolltriv}
\psi^*(\omega_{b,p})=\omega_{b_{0},p},
\end{equation}
where $\omega_{b,p}$ is the restriction of $\omega$ to the fiber $M_{b,p}=f^{-1}(b,p)$.
\end{definition}

\begin{definition}
\label{def:sympmonod}
Let $f\colon M\to \S^1$ and $\omega\in\Omega^2(M)$ be such that $(f,\omega)$ is a symplectic fibration. A {\em symplectic monodromy trivialization} is a smooth map $\Phi:M_1\times [0,1]\to M$ such that  $f(\Phi(x,t))=e^{2\pi\imath t}$ for all $(x,t)\in M\times [0,1]$, and such that for all $t\in [0,1]$, the restriction $\Phi_t:=\Phi|_{M_1\times\{t\}}$ defines a symplectomorphism from $(M_1,\omega_1)$ to $(M_\theta,\omega_\theta)$, where $\theta=e^{2\pi\imath t}$, $\omega_\theta=\omega|_{M_\theta}$, and $\Phi_{0}=\id_{M_1}$. The corresponding {\em symplectic monodromy} is the symplectomorphism $\phi\de\Phi_1$.

Assume that $\omega=d\lambda$, and that $(f,\lambda)$ is a Liouville fibration. Fix an $\S^1$-fiberwise symplectic collar trivialization $\psi\colon C\times \S^1\to M$ of $f$. A {\em symplectic monodromy trivialization relative to $\psi$} is a symplectic monodromy trivialization $\Phi$ satisfying $\Phi(x,t)=\psi(x, e^{2\pi\imath t})$ for all $(x,t)\in C\times [0,1]$. A {\em symplectic monodromy relative to $\psi$} is the symplectomorphism $\phi:=\Phi_1$ for a symplectic monodromy trivialization relative to $\psi$. Observe that in this case $\phi$ is the identity close to the boundary $\partial M_1$, so if $\phi$ is exact, then $(M_1,\lambda_1,\phi)$ is an abstract contact open book.
\end{definition}

\begin{remark}
\label{rem:mondunique}
Fix $f\colon M\to \S^1$ and $\omega\in\Omega^2(M)$ such that $(f,\omega)$ is a symplectic fibration. If a symplectic monodromy exits, then it is unique as an element of the group of symplectomorphisms up to a symplectic isotopy. Indeed,
given two symplectic monodromy trivializations $\Phi$ and $\Phi'$, the needed isotopy is $\Phi'_t\comp\Phi_{t}^{-1}\comp\Phi_1$.

%Similarly, f
Fix $f\colon M\to \S^1$ and $\lambda\in\Omega^1(M)$ such that $(f,\lambda)$ is a Liouville fibration. Fix an  $\S^1$-fiberwise symplectic collar trivialization $\psi$ of $f$. Then as before, if a symplectic monodromy relative to $\psi$ exists, it is unique in the group of compactly supported symplectomorphisms of $\partial M_1$, up to a compactly supported symplectic isotopy. In particular, if every symplectic monodromy $\phi$ relative to $\psi$ is exact, then the isotopy class of abstract contact open books $(M_1,\lambda_1,\phi)$ does not depend on the choice of $\phi$. %the symplectic monodromy relative to $\psi$.
\end{remark}

We now distinguish a class of \emph{cohomologically trivial} collar trivializations. In Proposition \ref{prop:sufficientmonod} we will use this technical condition to construct symplectic monodromy via a Moser argument.% \cite[\sec 3.2]{McDuff_Salamon}.}

%The next proposition provides sufficient conditions for constructing a symplectic monodromy.

Let $f\colon M\to B\times P$ as in Definition~\ref{def:collartriv} and let $\lambda\in\Omega^1(M)$ be such that $(f,\lambda)$ is a Liouville fibration. Put $\omega=d\lambda$. Assume that we are given a $P$-fiberwise collar trivialization
\begin{equation*}
\psi\colon C_{b_0}\times B\to C,
\end{equation*}
that is $(B\times P)$-fiberwise symplectic, i.e.\ satisfies condition \eqref{eq:simpcolltriv}. For any $b\in B$ choose an open neighborhood $U_b$ of $b$ in $B$ and a $P$-fiberwise local trivialization
\begin{equation*}
\Theta:M_{b}\times U_b\to f^{-1}(U_b\times P)
\end{equation*}
which is induced by $\psi$ on the collar $C$, that is, such that for any $x\in M_{b}\cap C$ and any $b'\in U_{b}$ we have
\begin{equation*}
\Theta(x,b')=\psi(\pr_{C_{b_0}}(\psi^{-1}(x)),b').
\end{equation*}
%where $C_{b_0}=(\pr_{B}\circ f)^{-1}(b_0) \cap C$, and $\pr_{C_{b_0}}\colon C_{b_0}\times B\to C_{b_0}$ is the projection, see Notation \ref{not:fibers}.

For every $p\in P$ the form $\Theta^*\omega$ can be interpreted as a family of symplectic forms on the fiber $M_{b,p}$ parametrized by $U_b$; for any $b'\in U_b$, denote by $(\Theta^*\omega)_{b',p}\in\Omega^2(M_{b,p})$ the corresponding form.
Since the collar trivialization $\psi$ is $(B\times P)$-fiberwise symplectic, for every $b,b'\in U_b$ we have the equality of restrictions
hence the difference
\begin{equation}\label{eq:eta_Theta}
\eta^{\Theta}_{b,b';p}\de (\Theta^*\omega)_{b,p}-(\Theta^*\omega)_{b',p}
\end{equation}
vanishes in a neighborhood of the boundary $\d M_{b,p}$. Thus we obtain a relative cohomology class $[\eta^{\Theta}_{b,b';p}]\in H^2(M_{b,p},\d M_{b,p};\RR)$. We claim that this cohomology class does not depend on the choice of the trivialization $\Theta$, and hence denote it by  $[\eta_{b,b';p}]$.

To prove the claim, we choose any class $[z]\in H_2(M_{b,p},\d M_{b,p};\RR)$. We have $\eta^{\Theta}_{b,b';p}(z)=\omega(\Theta(z,b))-\omega(\Theta(z,b'))$. Hence, if $\Theta'$ is a different trivialization we have
Since $\Theta$ and $\Theta'$ coincide close to $\d M_{p,b}$, the chain $\Theta(z,b)-\Theta'(z,b)$ is closed. Therefore, since $\omega$ is exact we have the vanishing $\omega(\Theta(z,b)-\Theta'(z,b))=0$. The second summand vanishes for the same reasons. This proves the claim.

\begin{definition}
\label{def:cohtriv}
Fix a Liouville fibration $(f,\lambda)$ and a $(B\times P)$-fiberwise symplectic collar trivialization $\psi$ as in the discussion above. Fix $p\in P$. We say that $(f,\lambda)$ is {\em cohomologically trivial with respect to $\psi$ at $p$} if for every $b\in B$ and a trivialization $\Theta$ as above, the cohomology class $[\eta_{p;b,b'}]$ vanishes in $H^2(M_{b,p},\d M_{b,p};\RR)$ for any $b'\in U_b$.
\end{definition}

\begin{prop}
\label{prop:sufficientmonod}
Let $f\colon M\to \S^1\times P$ be as in Definition~\ref{def:collartriv} and let $\lambda\in\Omega^1(M)$ be such that $(f,\lambda)$ is a Liouville fibration. Let $\psi\colon C_{1}\times \S^1\to C$ be its $P$-fiberwise collar trivialization which is $(\S^1\times P)$-fiberwise symplectic.
Assume that $(f,\lambda)$  is cohomologically trivial with respect to $\psi$ at every point $p\in P$, see Definition \ref{def:cohtriv}. Then there exists an open covering $\{Q_i\}_{i\in I}$ of $P$, and, for each $i\in I$, a smooth map
\begin{equation*}
\Theta^i:M_1|_{Q_i}\times [0,1]\to M|_{Q_i},
\end{equation*}
such that for every $(x,t)\in M_1|_{Q_i}\times [0,1]$ we have $(\pr_P\circ f\circ \Theta^i)(x,t)=(\pr_P\circ f)(x)$, and for every $p\in Q_i$ the restriction $\Theta^{i}|_{M_{1,p}\times [0,1]}$ is a symplectic monodromy trivialization relative to the $\S^1$-fiberwise symplectic collar trivialization $\psi|_{C_{1,p}\times\S^1}$.

For $i\in I$ and $p\in Q_i$, denote by $\phi_{p}^{i}:M_{1,p}\to M_{1,p}$ the associated symplectic monodromy. Assume that $P$ is connected, and for every $i\in I$, $p\in Q_i$ the symplectomorphism $\phi^{i}_{p}$ is exact. Then for every $i\in I$, $p\in Q_i$, the triple $(M_{1,p},\lambda_{1,p},\phi_{p}^{i})$ is an abstract contact open book, whose isotopy type does not depend on the choice of $i\in I$, $p\in Q_i$.
\end{prop}

\begin{proof}
For the first assertion, we will use a parametrized Moser argument. Put $\omega=d\lambda$.

Since $f$ is a fibration and admits a $P$-fiberwise collar trivialization $\psi$, there exists a smooth map
\begin{equation*}
\Theta':M_1\times [0,1]\to M
\end{equation*}
such that we have  $(\pr_P\circ f\circ \Theta')(x,t)=(\pr_P\circ f)(x)$ for any $(x,t)\in M_1\times [0,1]$, and $\Theta'(x,t)=\psi(x,e^{2\pi\imath t})$ for any $(x,t)\in C\cap M_1\times [0,1]$. We view the pullback $\omega':=(\Theta')^*\omega$ as a family of closed $2$-forms in $\Omega^2(M_1)$ parametrized by $t\in [0,1]$ which are symplectic fiberwise over $P$. We denote by $\omega'_t$ the restriction of $\omega'$ to $M_1\times\{t\}$. Since $\psi$ is $(\S^1\times P)$-fiberwise symplectic, $\Theta'$ can be chosen so that the restriction $\omega'_t|_{C_{1,p}}$ is independent of $t$ for any $p\in P$. We now follow the strategy of the proof of \cite[Theorem 3.2.4]{McDuff_Salamon}, taking care of the fact that our construction is parametrized over $P$ and compatible with the collar trivialization.

Since the assertion is local in the base $P$, we may assume that $M_1$ is of the form $M_1=P\times N_1$. Then we can view  $\omega'_t$ as a family of closed forms $\omega'_{t,p}\in\Omega^{2}(N_1)$ which are independent of $t$ near $\partial N_1$, for each fixed $p\in P$. Since $(f,\lambda)$ is cohomologically trivial with respect to $\psi$, the relative cohomology class $[\omega'_{t,p}-\omega'_{t',p}]$ vanishes in $H^{2}(N_1,\partial N_1,\RR)$ for every fixed $p\in P$, and every $t,t'\in [0,1]$. Therefore, the class  $[\frac{d}{dt}\omega'_{t,p}]$ vanishes in the same group. This means that for any $p\in P$ and any $t\in [0,1]$ there exists a $1$-form $\sigma_{t,p}\in\Omega^1(N_1)$ which vanishes near $\d N_1$ and satisfies $d\sigma_{t,p}=\frac{d}{dt}\omega'_{t,p}$.

We will produce such a family of $\sigma_{t,p}$ smoothly depending on $t$ and $p$. For this, out of the two methods suggested in the proof of \cite[Theorem 3.2.4]{McDuff_Salamon} to construct $\sigma_{t,p}$, we adapt the first one, mimicking the proof of the compactly supported Mayer-Vietoris sequence in de Rham cohomology.

To be precise: denote by $\Omega_c^*(N_1\times [0,1]\times P)|_{[0,1]\times P}$ the {chain} complex formed by families $\xi_{t,p}$ of compactly supported forms in $N_1\setminus\d N_1$ smoothly depending on $(t,p)$. Let $H_c^*(N_1\times [0,1]\times P)|_{[0,1]\times P}$ be its cohomology. Define a homomorphism
\begin{equation*}
%	\label{eq:deRhamiso}
\gamma\colon H_c^*(N_1\times [0,1]\times P)|_{[0,1]\times P}\to H_c^*(N_1\setminus\d N_1)\otimes\cC^\infty([0,1]\times P)
\end{equation*}
as follows: choose {a basis} $[\Gamma_{1}],\dots,[\Gamma_{r}]$ {of} $H_{*}(N_1,\d N_1)$ and define
\begin{equation*}
\gamma\left([\xi_{t,p}]\right) \de \sum_{j=1}^r\left(\int_{\Gamma_{j}}\xi_{t,p}\right)\cdot \left[\Gamma_{j}\right]^*,
\end{equation*}
where $\{[\Gamma_{j}]^*\}_{j=1}^r$ is the {basis of $H_{c}^{*}(N_1\setminus \d N_1)$ dual to} $\{[\Gamma_{j}]\}_{j=1}^r$.

In order to produce the needed family it is enough to prove that $\gamma$ is an isomorphism. This is proved by induction on the minimal number of open subsets of a good cover of $N_1$, where a good cover is a cover by contractible open subsets such that any finite intersection is contractible. Existence of good covers is established in \cite[Theorem 5.1]{Bott_Tu}. The initial step of the induction follows the proof of the compactly supported Poincar\'e Lemma \cite[Propositions 4.6, 4.7]{Bott_Tu}. The inductive step follows the Mayer-Vietoris argument of \cite[\sec I.5]{Bott_Tu}.

After the smooth family $\sigma_{t,p}$ has been constructed, applying Moser trick to the family $\sigma_{t,p}$ as in \cite[\sec 3.2]{McDuff_Salamon} we produce a family of diffeomorphisms $\Theta''_{t,p}\colon N_1\to N_1$ that are equal to the identity near $\partial N_1$ and such that $(\Theta''_{t,p})^*\omega'_{t,p}= \omega'_{0,p}$ for any $t\in [0,1]$ and $p\in P$. Then the required smooth map $\Theta$ is defined by $\Theta_{t,p}:=\Theta'_{t,p}\comp \Theta''_{t,p}$.

For the second assertion, we note that for any $p\in Q_i$ the monodromy diffeomorphism $\phi_{p}^{i}$ induced by $\Theta^i$ is compactly supported, so if it is exact, then  $(M_{1,p},\lambda_{1,p},\phi_{p}^{i})$ is an abstract contact open book. Clearly, $(M_{1,p},\lambda_{1,p},\phi_{p}^{i})\sim (M_{1,q},\lambda_{1,q},\phi_{q}^{i})$ if $p,q\in Q_{i}$ and $Q_i$ is connected. If $p\in Q_i\cap Q_{j}$ then since $\phi_{p}^{i}$ and $\phi_{p}^{j}$ are both symplectic monodromies relative to the same collar trivialization, we have $(M_{1,p},\lambda_{1,p},\phi_{p}^{i})\sim (M_{1,p},\lambda_{1,p},\phi_{p}^{j})$ by Remark~\ref{rem:mondunique}.
\end{proof}

\begin{definition}\label{def:monodromy_acob}
Let $f\colon M\to P\times \S^1$ be a Liouville fibration, with a connected $P$, and let $\psi$ be its fiberwise symplectic collar trivialization.
The abstract contact open book from Proposition \ref{prop:sufficientmonod} will be called the \emph{monodromy abstract contact open book} of $f,\psi$. It is unique up to an isotopy, and exists if $\psi$ is cohomologically trivial and the monodromy $\phi$ is exact.
\end{definition}

\subsection{Grading and Conley--Zehnder index}

In order to define a grading of the Floer complex, we need to consider \emph{graded} abstract contact open books. We now recall basic notions leading to their definition, following \cite[Appendix A]{McLean}.

\subsubsection{Graded principal bundles} Let $G$ be a Lie group with universal cover $\tilde{G}$, and let $p\colon F\to M$ be a principal $G$-bundle. A \emph{grading} of $F$ is a principal $\tilde{G}$ bundle $\tilde{F}\to M$ together with an isomorphism $\alpha\colon \tilde{F}\times_{\tilde{G}} G \to F$. Thus there is a natural covering map $\alpha'\colon\tilde{F}\to F$. An isotopy between two gradings $\alpha_0$, $\alpha_1$ is a principal $G$-bundle over $M\times [0,1]$, together with a grading which restricts to $\alpha_i$ over $M\times \{i\}$ for $i\in \{0,1\}$. Choose a base point $f\in F$. Then by \cite[Lemma A.3]{McLean}, there is a one-to-one correspondence
\begin{equation}\label{eq:gradings-subgroups}
\begin{split}
&\{\mbox{gradings of }p\colon F\to M\}/_{\mbox{\tiny{isotopy}}}
\longleftrightarrow \\
\{N\vartriangleleft \pi_{1}(F,f): p_{*}|_{N}\colon N\to \pi_{1}(M,p(f))\mbox{ is an isomorphism}\}
\end{split}
\end{equation}
sending a grading $(\tilde{p}\colon \tilde{F}\to M,\alpha)$ to  $\alpha'_{*}(\pi_{1}(\tilde{F},\tilde{f}))\vartriangleleft \pi_{1}(F,f)$, where $\tilde{f}\in (\alpha')^{-1}(f)$.% is the preimage of $f$ by the isomorphism $\alpha$ given by the grading.

\subsubsection{Graded symplectic manifolds} \label{sec:graded-manifolds}
Let $(M,\omega)$ be a symplectic manifold of dimension $2d$. A \emph{grading} of $M$ is a grading of its symplectic frame bundle $\Fr(M)\to M$, which is a principal $\Sp(2d)$-bundle, see \cite[Definition A.6]{McLean}. Choose an $\omega$-compatible almost complex structure $J$, so that $TM$ becomes a complex vector bundle of rank $d$. Let $K^{*}_{M}\de \Lambda^{d} TM$ be the corresponding anti-canonical bundle. The natural maps
\begin{equation*}
\begin{tikzcd}
\Sp(2d) & \ar[l, hook'] U(d) \ar[r, "\det"] & \S^1
\end{tikzcd}
\end{equation*}
induce isomorphisms between fundamental groups, hence \eqref{eq:gradings-subgroups} gives a one-to-one correspondence between gradings of $M$ and gradings of the $\S^{1}$-bundle of $K^{*}_{M}$. By \cite[Lemma A.5]{McLean}, the latter are in one-to-one correspondence with homotopy classes of $\cC^{\infty}$ trivializations of $K^{*}_{M}$, or, equivalently, of $K_{M}\de \Lambda^{d} T^{*}M$. Recall that $K_M$ is a complex line bundle, so its $\cC^{\infty}$ trivializations are given by nonvanishing $\cC^{\infty}$ sections. To summarize, we have a one-to-one correspondence
\begin{equation}\label{eq:gradings-trivializations}
\{\mbox{gradings of } M\}/_{\mbox{\tiny{isotopy}}}
\longleftrightarrow
\{\mbox{nonvanishing smooth sections of }K_{M} \}/_{\mbox{\tiny{homotopy}}}.
\end{equation}
Likewise, given a symplectic fibration $M\to P$ with a fiberwise symplectic form $\omega$ and an $\omega$-compatible almost complex structure $J$ of the relative tangent bundle, let $F\to M$ be the vertical symplectic frame bundle, and let $K\vert$ be the relative canonical bundle. We have a one-to-one correspondence
\begin{equation}\label{eq:gradings-trivializationsfiberwise}
\{\mbox{gradings of } F\}/_{\mbox{\tiny{isotopy}}}
\longleftrightarrow
\{\mbox{nonvanishing smooth sections of }K\vert \}/_{\mbox{\tiny{homotopy}}}.
\end{equation}

\subsubsection{Graded symplectomorphisms and graded abstract contact open books}\label{sec:graded-acobs}

Let $\phi\colon M\to M$ be a symplectomorphism. Let $\beta\colon \Fr(M)\to \phi^{*}\Fr(M)$ be the induced automorphism of the symplectic frame bundle. A \emph{grading} of $\phi$ is an isomorphism $\tilde{\beta}\colon \tilde{\Fr}(M)\to\phi^{*}\tilde{\Fr}(M)$ which lifts $\beta$ via the grading isomorphism $\alpha\colon \tilde{\Fr}(M)\times_{\tilde{\Sp}(2d)} \Sp(2d) \to \Fr(M)$, i.e.\ satisfies $\tilde{\beta}\times_{\tilde{\Sp}(2d)}\Sp(2d)=\beta$.

A \emph{grading} of an abstract contact open book $(M,\lambda,\phi)$ is a grading of the symplectic manifold $(M,d\lambda)$ and a compatible grading of the symplectomorphism $\phi\colon M\to M$. An isotopy of abstract contact open books $\{(M_t,\lambda_t,\phi_t)\}_{t\in [0,1]}$ is \emph{graded} if there is a family $\tilde{\beta_t}\colon \tilde{\Fr}(M_t)\to \phi_t^{*}\tilde{\Fr}(M_t)$ of gradings of $\phi_t$ depending smoothly on $t$.

\begin{lema}
\label{lem:liftgradings}
Let $f\colon M\to \S^{1}\times P$ be a symplectic fibration with a closed fiberwise symplectic form $\omega$, and let $F\to M$ be the vertical symplectic frame bundle. Let $\Phi\colon M_1\times [0,1]\to M$ be a map that preserves fibers over $P$, and is a symplectic monodromy trivialization fiberwise over $P$. Any grading $\tilde{F}\to M$ of $F$ induces a grading $\tilde{\beta}_p$ of the symplectic monodromy $\phi_p\colon M_{1,p}\to M_{1,p}$ defined by $\Phi$, which depends smoothly on $p$.
\end{lema}
\begin{proof}
By the symplectic monodromy construction, given any symplectic monodromy $\phi\colon M_1\to M_1$ there exists a closed fiberwise symplectic form $\omega'\in\Omega^2(M)$ such that $\omega'_{t,p}=\omega_{t,p}$ for all $(t,p)\in \S^1\times P$ and $\phi$ is the time $1$ flow of the lifting of the unit vector field $(0,\frac{\d}{\d\theta})$ on $\S^1\times P$ by the symplectic connection associated with $\omega'$. Indeed, let $\bar{M}$ be the mapping cylinder of $\phi$, i.e.\ $\bar{M}=M_{1}\times [0,1]/_{\sim}$ with $(x,1)\sim (\phi(x),0)$. We have an induced map $\bar{M}\ni (x,t)\mapsto (e^{2\pi\imath t},\pr_{P}\circ f(x))\in \S^1\times P$, and a diffeomorphism $\bar{\Phi}\colon \bar{M}\ni (x,t)\mapsto \Phi(x,t)\in M$. Since $\phi^{*}\omega_{1}=\omega_{1}$, the form $\pr_{M_1}^{*}\omega_{1}$ descends to a fiberwise symplectic form $\bar{\omega}$ on $\bar{M}$. Put $\omega'=(\bar{\Phi}^{-1})^{*}\bar{\omega}$. Since $\Phi$ is fiberwise a symplectic monodromy trivialization, its restrictions $\Phi_{t,p}\colon M_{1,p}\to M_{t,p}$ are symplectomorphisms. It follows that $\omega'_{t,p}=\omega_{t,p}$. The symplectic lift of $(0,\tfrac{\d}{\d \theta})$ from $P\times \S^1$ to $M$ is $\bar{\Phi}_{*}\frac{\d}{\d t}$, and its time one flow is $\phi$, as needed.

Since the statement only concerns fibers we may assume $\omega=\omega'$ for the rest of the proof. The symplectic connection defines a lift $\mathcal{X}$ of the vector field $(0,\frac{\d}{\d\theta})$ to $F$, and the automorphism  $\beta\colon F\to F$ induced by $\phi$ is the time $1$ flow of $\mathcal{X}$. Since $\tilde{F}$ is a covering space of $F$, there is a unique lifting of $\mathcal{X}$ to a vector field $\mathcal{Y}$ in $\tilde{F}$, and $\tilde{\beta}$ is the time $1$ flow of $\mathcal{Y}$.
\end{proof}

\begin{cor}
\label{cor:gradedunicitymon}
Given a symplectic fibration $M\to\S^{1}$ and a grading $\tilde{F}\to M$ of the vertical symplectic frame bundle, any symplectic monodromy inherits a grading, and the graded symplectic monodromy is unique up to graded isotopy. Furthermore, in the situation of Proposition~\ref{prop:sufficientmonod}, a grading of the vertical symplectic frame bundle of $M\to P\times\S^{1}$ induces a grading of each symplectic monodromy $\phi_p$, and of each isotopy connecting them.
\end{cor}

\subsubsection{Conley--Zehnder index}\label{sec:CZ}

Let $\phi\colon M\to M$
be a graded symplectomorphism. Let $x\in M$ be a fixed point of $\phi$. The grading $\tilde{\beta}\colon \tilde{\Fr}(M)\to \phi^{*}\tilde{\Fr}(M)$ sends the fiber $\tilde{\Fr}(M)_{x}$ over $x$ to itself. Its restriction $\tilde{\beta}_{x}$ is thus an element of $\tilde{\Sp}(2d)$, well defined up to a conjugation. It can be viewed as a homotopy class (relative to the ends) of a path $\gamma$ of symplectic matrices, starting from $\id$; well defined up to a conjugation. The \emph{Conley--Zehnder index of $x$}, denoted by $\CZ_{\phi}(x)$, is the Maslov index of $\gamma$, see \cite[\sec 4]{RS_CZ-index} for its definition. This number depends only on the conjugacy class of this path. Indeed, by \cite[Theorem 4.1]{RS_CZ-index} the Maslov index is additive under concatenation of paths, so it is invariant under conjugation. In Section \ref{sec:spectral-sequence} we will need the following, known result.
%We will now give a formula for a Conley--Zehnder index of a symplectic monodromy with good properties. Let $f\colon M\to \S^1$ be a symplectic fibration admitting a symplectic monodromy trivialization $\Phi\colon M_1\times [0,1]\to M$. Let $\phi=\Phi_{1}\colon M_1\to M_1$ be the symplectic monodromy, see Definition \ref{def:sympmonod}. Let $J$ be
%an almost complex structure on the vertical tangent bundle $\ker f_{*}$, which is fiberwise compatible with the fiberwise symplectic form. % compatible with $\omega|_{M_{\theta}}$.
%Then $\ker f_{*}$ is a complex vector bundle of rank $d\de\tfrac{1}{2}\dim M_1$. Let $K\vert\de \Lambda^{d}(\ker f_{*})^{\vee}$ be the relative canonical bundle. Assume that $K\vert$ admits a nonvanishing section $s$. By \eqref{eq:gradings-trivializationsfiberwise} it determines a grading of the vertical symplectic frame bundle $F$. As in Section \ref{sec:graded-acobs}, we get this way a grading of $\phi$, too.
%Let $x\in M_1$ be a fixed point of $\phi$. Assume that the differential of $\phi$ at $x$ is the identity. Let $K\vert\ni z\mapsto (y,\hat{s}_{y}(z))\in M\times \C$ be the trivialization of $K\vert$ given by the section $s$, so for each $y\in M$, the map $\hat{s}_{y}\colon (K\vert)_{y}\to \C$ is an isomorphism between the fiber of $K\vert$ over $y$ with $\C$. Let $w\in \Z$ be the winding number of the path $[0,1]\ni t\mapsto \hat{s}_{\Phi_{t}(x)} \circ (\Phi^{-1}_{t})^{*} \circ \hat{s}_{x}^{-1}\in \Aut(\C,0)=\C^{*}$. Then by \cite[Lemma A.8]{McLean}, the Conley--Zehnder index of $x\in \Fix\phi$ with respect to the above grading equals
%	\CZ(x)=-2w.
%We recall how Conley--Zehnder index changes under a small Hamiltonian deformation.

\begin{lema}\label{lem:CZ-Hamiltonian}
Let $(M,\omega)$ be a graded symplectic manifold of dimension $2d$. Let $H\colon M\to \R$ be a time-independent Hamiltonian, and let $\psi^{H}_{1}\colon M\to M$ be its time-one flow, see Section \ref{sec:Hamiltonian}. Let $x$ be a Morse point of $H$, of Morse index $\Ind(x)$. Assume furthermore that $H$ is $\cC^{2}$-small at $x$, more explicitly, $||\operatorname{Hess}_{x}(H)||<2\pi$. Then the following hold.
\begin{enumerate}
\item \label{item:CZ_Morse} The point $x$ is a fixed point of $\psi^{H}_{1}$ of Conley--Zehnder index $d-\Ind(x)$.
\item \label{item:CZ_phi+Morse} Let $\phi\colon M\to M$ be a graded symplectomorphism such that $x$ is a fixed point of $\phi$. Then $x$ is a fixed point of $\phi\circ \psi_{1}^{H}$, of Conley--Zehnder index $\CZ_{\phi}(x)+d-\Ind(x)$.
\end{enumerate}
\end{lema}
\begin{proof}
Part \ref{item:CZ_Morse} is proved in \cite[Corollary 7.2.2]{AD_book}. Part \ref{item:CZ_phi+Morse} follows from \ref{item:CZ_Morse} since the Maslov index of a concatenation of paths is a sum of their Maslov indices \cite[Theorem 4.1]{RS_CZ-index}.
\end{proof}
%We now recall need to see how the Conley--Zehnder index changes under a small Hamiltonian deformation. Let again
% Then $x$ is a fixed point of $\psi^{H}_1$.  Now by \cite[Corollary 7.2.2]{AD_book}, the Conley--Zehnder index of $x\in \Fix\psi^{H}_{1}$ equals
%	\CZ(x)=d-\Ind(x).
%Note that the formula in loc.\ cit.\ differs from \eqref{eq:CZ_Morse} by a sign. As explained in Section \ref{sec:Hamiltonian}, this is because we use the opposite sign convention for definition of the Hamiltonian vector field. Therefore, to get the same $\psi_{1}^{H}$ we need to apply loc.\ cit.\ to $-H$, and the Morse index of $-H$ at $x$ is $2d-\Ind(x)$.
%Now, , so $x\in \Fix\check{\phi}$, too. Since the Maslov index of a concatenation of paths is a sum of their Maslov indices, see \cite[Theorem 2.3]{RS_CZ-index}, the Conley--Zehnder index $\CZ_{\check{\phi}}(x)$ of $x$ as a fixed point of $\check{\phi}$ is the sum of the Conley--Zehnder indices of $x$ as a fixed point of $\phi$ and $\psi^{H}_{1}$, so by \eqref{eq:CZ_Morse}
%	\CZ_{\check{\phi}}(x)=\CZ_{\phi}(x)+\CZ_{\psi_{1}^{H}}(x)=\CZ_{\phi}(x)+d-\Ind(x).

\subsection{Constructing symplectic monodromies}\label{sec:basic-setting}

The aim of this section is to give a general construction of symplectic monodromy for families of degenerations of complex manifolds over a punctured disc. Later, we will apply it to a $\mu$-constant family of isolated hypersurface singularities, see Example \ref{ex:mu_constant}; and a log resolution of a single isolated hypersurface singularity, see Section \ref{sec:isotopy-to-radius-zero}. Now, we work in the following general Setting \ref{basic-setting}.

\begin{setting}[see Figure \ref{fig:collars}]\label{basic-setting}
Let $X$ be a complex variety (possibly singular). Let $P$ be a compact domain in a manifold, and let $(f_{p})_{p\in P}\colon X\to \C$ be a smooth family of regular maps. Put
\begin{equation*}
F:X\times P\ni (x,p)\mapsto (f_{p}(x),p)\in \C\times P.
\end{equation*}
Assume that for some $\delta>0$, the restriction of $F$ over $\D_{\delta}^{*}\times P$ is a submersion. In  particular, all fibers $f_{p}^{-1}(z)$ for $z\in \D_{\delta}^{*}$, $p\in P$ are smooth and contained in the smooth part $X\reg$ of $X$.  Let $U\subseteq X$ be an open subset whose closure is compact and whose boundary is a manifold, and such that for every $p\in P$, we have $f_{p}^{-1}(0)\trans \d \bar{U}$. Assume that we have open subsets $V,W\subseteq U$ with the same transversality property, and such that $\overline{V}\subset W\subset\overline{W}\subset U$.
Put $C'=((\bar{U}\setminus V)\times P)\cap F^{-1}(\D_{\delta}\times P)$. As usual, we write $C'_{z,p}=(\bar{U}\setminus V)\cap f_{p}^{-1}(z)$ for $z\in \D_{\delta}$, $p\in P$, see Notation \ref{not:fibers}.

Assume that $\pr_{X}(C')\subseteq X\reg$, and that the restriction $F|_{C'}\colon C'\to \D_{\delta}\times P$ is a submersion. Thus for every $p\in P$, the fiber $C'_{0,p}$ is smooth  and contained in $X\reg$. Furthermore, we assume that the restriction of $F$ gives a locally trivial fibration of pairs
\begin{equation}
\label{eq:collarfibrationaux1}
F\colon (C',(\d \bar{W}\times P) \cap C')   \to\D_\delta\times P.
\end{equation}
By Ehressmann Lemma, the latter condition can always be achieved by shrinking $\delta>0$. Now
for each $(z,p)\in\D_\delta\times P$, each connected component of $C'_{z,p} \setminus \d \bar{W}$ cannot meet $\d \bar{V}$ and $\d \bar{U}$ at the same time. %We call the ones meeting $\d \bar{V}$ the {\em inner components of the fibration~\eqref{eq:collarfibrationaux1} over $(z,p)$}.
\smallskip

Let $\lambda\in \Omega^{1}(C' \cup F^{-1}(\D_{\delta}^{*}\times P))$ be a $1$-form whose restriction to each collar $C'_{0,p}$ and to each smooth fiber $f_{p}^{-1}(z)\cap \bar{U}$ for $z\in \D_{\delta}^{*}$, $p\in P$ is Liouville. Furthermore, we assume that for some $p_0\in P$, the  Liouville vector field of $\lambda|_{C_{0,p_0}'}$ points outwards $f^{-1}_{p_0}(0)\cap \d \bar{W}$.
\end{setting}

\begin{figure}
\begin{tikzpicture}[scale=0.7]
%	\path[draw, use as bounding box] (-6,-7.6) rectangle (8,4);
\path[draw, name path=c] (-5,2.5) to [out=0,in=180] (-4,2) to [out=0,in=180] (-3,2.5);
\path[draw, name path=d] (-5,1.5) to [out=0,in=180] (-4,1) to [out=0,in=180] (-3,1.5);
\tikzfillbetween[ of=c and d ] {pattern=vertical lines, pattern color=black!15};
\path[draw, name path=c] (1,2.5) to [out=0,in=180] (2,2) to [out=0,in=180] (3,2.7);
\path[draw, name path=d] (1,1.5) to [out=0,in=180] (2,1) to [out=0,in=180] (3,1.7);
\tikzfillbetween[ of=c and d ] {pattern=vertical lines, pattern color=black!15};;
\draw (-6,0) -- (8,0);
\node[below left] at (8,0) {$\d \bar{V}\times P$};
\draw (-6,2) -- (8,2);
\node[below left] at (8,2) {$\d \bar{W}\times P$};
\draw (-6,3.5) -- (8,3.5);
\node[below left] at (8,3.5) {$\d \bar{U}\times P$};
%    \draw[very thick] (-4,4) -- (-4,0) to[out=-90,in=0] (-4.7,-2) to[out=0,in=90]  (-4,-4);
%    \draw[ultra thick] (-3,4) -- (-3,0) to[out=-90,in=90] (-3.4,-2) to[out=-90,in=90]   (-3,-4);
%    \draw (-5,4) -- (-5,0) to[out=-90,in=90] (-5.4,-2) to[out=-90,in=90]   (-5,-4);
\draw[->] (-4,-4.4) -- (-4,-5.8);
\node[right] at (-4,-5.2) {\small{$f_{p_0}$}};
\draw (-4,-6.6) ellipse (1 and 0.5);
\filldraw (-4,-6.6) circle (1.5pt);
\filldraw (-3,-6.6) circle (2pt);
\filldraw (-4,2) circle (2pt);
\filldraw (-4,1) circle (2pt);
\filldraw (2,2) circle (2pt);
\filldraw (2,1) circle (2pt);
\node[right] at (-3,-6.6) {\small{$\delta$}};
\draw[->] (2,-4.4) -- (2,-5.8);
\node[right] at (2,-5.2) {\small{$f_{p}$}};
\draw (2,-6.6) ellipse (1 and 0.5);
\filldraw (2,-6.6) circle (1.5pt);
\filldraw (3,-6.6) circle (2pt);
\node[right] at (3,-6.6) {\small{$\delta$}};
\node at (-5.2,-6.2) {\small{$\D_\delta$}};
\draw[line width=2.5] (-4,2)--(-4,1);
\draw[line width=2.5] (2,2)--(2,1);
\draw[decoration=snake, decorate, segment length=30, dashed] (-3,2.5) -- (1,2.5);
\draw[decoration=snake, decorate, segment length=30, dashed] (-3,1.5) -- (1,1.5);

\draw[decoration=snake, decorate, segment length=30, dashed] (-6,2.5) -- (-5,2.5);
\draw[decoration=snake, decorate, segment length=30, dashed] (-6,1.5) -- (-5,1.5);

\draw[decoration=snake, decorate, segment length=30, dashed] (3,2.7) -- (5,2.5);
\draw[decoration=snake, decorate, segment length=30, dashed] (3,1.7) -- (5,1.5);
\node[left] at (-6.3,2) {$C$};
\draw [decorate, decoration = {calligraphic brace}, very thick] (-6.2,1.5) --  (-6.2,2.5);
\node[left] at (-6.3,0.8) {\small{$C_{0,p_0}$}};
\draw [->] (-6.3,0.8) to[out=0,in=180] (-4.2,1.6);
\draw [decorate, decoration = {calligraphic brace}, very thick] (-2.7,2.5) --  (-2.7,-4);
\node[right] at (-2.7,-0.9) {$N_{\delta,p_0}$};
\draw [decorate, decoration = {calligraphic brace}, very thick] (3.3,2) --  (3.3,-4);
\node[right] at (3.4,-1.1) {$\bar{W}_{\delta,p}$};
\draw [decorate, decoration = {calligraphic brace}, very thick] (8.2,3.5) --  (8.2,0);
\node[right] at (8.3,1.75) {\small{$C'$}};

\draw[very thick] (2,4) -- (2,0) to[out=-90,in=0] (1.7,-1) to[out=0,in=90] (2,-2) -- (2,-2.5) to[out=-90,in=0] (1.7,-3) to[out=0,in=90]  (2,-4);
\draw[ultra thick] (3,4) -- (3,0) to[out=-90,in=90] (2.6,-1.2) to[out=-90,in=90] (2.8,-2) to[out=-90,in=90] (2.6,-2.8) to[out=-90,in=90] (3,-4);
\draw (1,4) -- (1,0) to[out=-90,in=90] (0.6,-1.2) to[out=-90,in=90] (0.8,-2) to[out=-90,in=90] (0.6,-2.8) to[out=-90,in=90] (1,-4);

\draw[very thick] (-4,4) -- (-4,0) to[out=-90,in=0] (-4.3,-1.3) to[out=0,in=90] (-4,-2) -- (-4,-2.3) to[out=-90,in=0] (-4.3,-2.7) to[out=0,in=90]  (-4,-4);
\draw[ultra thick] (-3,4) -- (-3,0) to[out=-90,in=90] (-3.4,-1.4) to[out=-90,in=90] (-3.2,-2) to[out=-90,in=90] (-3.4,-2.6) to[out=-90,in=90] (-3,-4);
\draw (-5,4) -- (-5,0) to[out=-90,in=90] (-5.4,-1.4) to[out=-90,in=90] (-5.2,-2) to[out=-90,in=90] (-5.4,-2.6) to[out=-90,in=90] (-5,-4);

\draw (-6,-7.5)-- (8,-7.5);
\node[above left] at (8,-7.5) {$P$};
\draw (-4,-7.6) -- (-4,-7.4);
\node[below] at (-4,-7.5) {\small{$p_0$}};
\draw (2,-7.6) -- (2,-7.4);
\node[below] at (2.2,-7.5) {\small{$p\neq p_0$}};

\end{tikzpicture}
\caption{Producing collar trivializations in Setting \protect\ref{basic-setting}.}
\label{fig:collars}
\end{figure}

The reader may have in mind the following example: the ambient space is $X=\C^{n}$, the parameter space $P$ is, say, $\D_{\eta}$; the map $F=(f_t,t)\colon \C^n\times \D_\eta\to\CC\times\D_\eta$ is a family of holomorphic functions with isolated singularities and $V\subset W\subset U\subset \C^n$ are concentric balls, such that the only critical value of $f_t$ is $0$, and $f_t^{-1}(0)$ is smooth in $\bar{U}\setminus V$; and $\lambda$ is the standard form $\lambda=\pi \cdot \sum_{i}r_{i}^{2}\, d\theta_{i}$ from Example \ref{ex:J}.
\smallskip

The remaining part of Section \ref{sec:basic-setting} is organized as follows. In Section \ref{sec:basic-Liouville} we show how to adjust $\d W$ within $U\setminus V$ so that $F|_{W\times P}$ becomes a Liouville fibration over $\D^{*}_{\delta}\times P$, with a cohomologically trivial collar trivialization, see Definitions \ref{def:collartriv} and \ref{def:cohtriv}. This may require shrinking $\delta>0$ or the parameter space $P$. Next, in Section \ref{sec:basic-monodromy} we observe that under a mild assumption \eqref{eq:ass-exact}, the monodromy of this Liouville fibration, see Definition \ref{def:sympmonod}, makes the fibers into isotopic abstract contact open books, see Definition \ref{def:acob}. Next, in Section \ref{sec:basic-grading}, we endow those abstract contact open books with a grading, under another mild assumption \eqref{eq:vanishing-c1}. In Remarks \ref{rem:exact} and \ref{rem:c1} we comment that these additional  assumptions \eqref{eq:ass-exact} and \eqref{eq:vanishing-c1} are satisfied in most interesting cases. Eventually, in Section \ref{sec:basic-radius-zero} we extend each fibration to the A'Campo space constructed in Section \ref{sec:ACampo}, and,  under yet another assumption \eqref{eq:lambda_extends}, we use forms introduced in Section \ref{sec:symplectic} to get isotopic abstract contact open books at radius zero. In Section \ref{sec:basic-applications} we gather basic applications of this construction which will be used later.% relevant later in the article.}

\subsubsection{Constructing a Liouville fibration with a collar trivialization in Setting \ref{basic-setting}}\label{sec:basic-Liouville} \

We keep the notation and assumptions of Setting \ref{basic-setting}.
Let $(r,\theta)$ be the polar coordinates in the disc $\D_\delta$. The radial vector field $\frac{\d}{\d r}$ in $\D_\delta^{*}$ does not extend to a smooth vector field at the origin, but it does extend if restricted to a ray  $R_{\theta}\subseteq \D_{\delta}$ of constant argument $\theta\in \S^1$. %Recall that $F|_{C'}\colon C'\to \D_{\delta}\times P$, with a form $d\lambda$, is a symplectic fibration.
Let $\mathcal{X}_{\theta}$ be the symplectic lift of $(\frac{\d}{\d r})|_{R_{\theta}}$ to $F^{-1}(R_{\theta}\times P)\cap C'$.

Let $C_0$ be a compact neighborhood of $F^{-1}(\{0\}\times P)\cap(\d \bar{W}\times P)$ in $F^{-1}(\{0\}\times P)\cap (\bar W\setminus \bar V\times P)$. Its slice $C_{0,p}$ over $p\in P$ is a compact collar neighborhood of $\d \bar{W}\cap f_{p}^{-1}(0)$ in $\bar{W}\cap f_{p}^{-1}(0)$. Let $P'\subseteq P$ be a compact subdomain such that all  integral curves of $\mathcal{X}_\theta$ with initial points in $C_{0}\cap F^{-1}(\{0\}\times P')$ exist for all times $t\in [0,\delta]$ and are contained in $C'$; one can always take $P'=\emptyset$.
Then after shrinking $P$ around $P'$, or after shrinking $\delta>0$ in case $P'=\emptyset$, we get the same property for $C_0\cap F^{-1}(\{0\}\times P)$. Here we use compactness of $C_0$ and $\D_\delta$ in the first case; and of $C_0$ and $P$ in the second case. This way, for each $\theta\in \S^1$ we obtain a smooth flow
\begin{equation*}
\varphi_{\theta}\colon C_0\times [0,\delta]\ni ((x,p),t)\mapsto (\varphi_{t\cdot \theta,p}(x),p)\in X\times P.
\end{equation*}
The above formula defines, for each $z\in \D_{\delta}$, $p\in P$, a map $\varphi_{z,p}\colon C_{0,p}\to C_{z,p}'$. More explicitly, we put $\varphi_{0,p}=\id_{C_{0,p}}$ and $\varphi_{z,p}(x)=\varphi_{z/|z|}((x,p),|z|)$ for $z\in \D_{\delta}^{*}$. Since the flow $\varphi_{\theta}$ is defined by a symplectic connection, each map $\varphi_{z,p}$ is a symplectomorphism onto its image. Put
\begin{equation*}
C\de \bigcup_{\theta\in\S^1}\varphi_{\theta}(C_0\times [0,\delta]),\quad
\d_{\mathrm{out}}C\de \bigcup_{\theta\in\S^1}\varphi_{\theta}((C_0\cap (\d\bar{W}\times P))\times [0,\delta])
\end{equation*}
Note that $C_{0}=F^{-1}(\{0\}\times P)\cap C$. Now $F|_{C}\colon C\to \D_{\delta}\times P$ is a symplectic fibration, and
\begin{equation}\label{eq:collar-triv-center}
\psi_{0}\colon C_{0}\times \D_{\delta}\ni ((x,p),z)\mapsto (\varphi_{z,p}(x),p)\in C
\end{equation}
is a $P$-fiberwise trivialization of $F|_{C}$ which is $(\D_{\delta}\times P)$-fiberwise symplectic.
%Then, for a fixed $z_0\in \D_{\delta}^{*}$, the map
%	\label{eq:asscolltriv1}
%	\psi\colon C_{z_0}\times\D_{\delta}^{*}\ni ((x,p),z)\mapsto (\varphi_{p,z}(\varphi_{p,z_0}^{-1}(x)),p)\in C
%is a $P$-fiberwise trivialization which is $(\D_\delta^{*}\times P)$-fiberwise symplectic. %On the other hand, after possibly shrinking $\delta>0$ the map
The map
\begin{equation}
\label{eq:collarfibrationaux1red}
F\colon (C', \d_{\mathrm{out}}C )   \to\D_\delta\times P
\end{equation}
is a locally trivial fibration of pairs, which coincides with~\eqref{eq:collarfibrationaux1} over $\{0\}\times P$. In particular, it also has the property that for each $(z,p)\in\D_\delta\times P$, each connected component of $C'_{z,p}\setminus \d_{\mathrm{out}}C_{z,p}$ cannot meet $\d \bar{V}$ and $\d \bar{U}$ at the same time. We call those meeting $\d \bar{V}$ the {\em inner components of the fibration~\eqref{eq:collarfibrationaux1red} over $(z,p)$}.
\begin{equation}
\label{eq:Liouv_fib1}
F|_N\colon N\to \D_\delta^*\times P
\end{equation}
as the symplectic fibration whose fiber over $(z,p)$ is the union of $\overline{V}_{z,p}$ and the inner components of the fibration~\eqref{eq:collarfibrationaux1red} over $(z,p)$.
Note that, by Ehresmann Lemma, the fiber $N_{z,p}=F^{-1}(z,p) \cap N$ is diffeomorphic to $\bar{W}_{z,p}\de f_{p}^{-1}(z)\cap \bar{W}$, and its diffeomorphism type does not depend on $(z,p)\in \D_{\delta}^{*}\times P$.

Using \eqref{eq:collar-triv-center}, we get a collar trivialization of \eqref{eq:Liouv_fib1}, as follows. Fix a point $z_{0}\in \D_{\delta}^{*}$, let $C_{z_0}=F^{-1}(\{z_0\}\times P)\cap C$, and let $\hat{\varphi}_{z_0}\colon C_{z_0}\ni (x,p)\mapsto (\varphi_{z_0,p}^{-1}(x),p)\in C_0$ be the fiberwise identification of $C_{z_0}$ with $C_{0}$ via the trivialization \eqref{eq:collar-triv-center}. Now, the map
\begin{equation}
\label{eq:asscolltriv1}
\psi\de \psi_0\circ (\hat{\varphi}_{z_0}\times \id_{\D_{\delta}^{*}})\colon C_{z_0}\times\D_{\delta}^{*}\to C
%\ni ((x,p),z)\mapsto (\varphi_{p,z}(\varphi_{p,z_0}^{-1}(x)),p)\in C
\end{equation}
is a $P$-fiberwise collar trivialization of \eqref{eq:Liouv_fib1}, which is $(\D_\delta^{*}\times P)$-fiberwise symplectic.

\begin{lema}\label{lem:cohtrivial}
The collar trivialization \eqref{eq:asscolltriv1} is  cohomologically trivial at every $p\in P$.
\end{lema}
\begin{proof}
By Definition \ref{def:cohtriv}, \eqref{eq:asscolltriv1} is cohomologically trivial at $p\in P$ if for every $z\in \D_{\delta}^{*}$, and every $z'\in \D_{\delta}^{*}$ sufficiently close to $z$, the class $[\eta_{z,z';p}]\in H^{2}(N_{z,p},\d N_{z,p};\R)$ is zero. This class was defined in the formula \eqref{eq:eta_Theta}, as follows. Let $U_{z}\subseteq \D_{\delta}^{*}$ be a neighborhood of $z$ such that the restriction  $F|_{N\cap F^{-1}(U_z\times P)}$ admits a $P$-fiberwise trivialization $\Theta$ which agrees with $\psi$ in the collar. For $z'\in U_{z}$ let $\Theta_{z',p}\colon N_{z,p}\to N_{z',p}$ be the corresponding map identifying the fibers. Then $[\eta_{z,z';p}]$ is the class of $\omega_{z,p}-\Theta_{z',p}^{*}\omega_{z',p}$.

We need to show that, for every $\Delta\in H_{2}(N_{z,p},\d N_{z,p};\R)$, the integral
\int_{\Delta}\omega_{z,p}-\Theta^{*}_{z',p}\omega_{z',p}
vanishes. By Stokes theorem, this is equivalent to vanishing of
\int_{\Gamma} \lambda_{z,p}-\Theta_{z',p}^{*}\lambda_{z',p}
for every $1$-cycle $\Gamma$ on $\d N_{z,p}$.

The condition that $\Theta$ agrees with $\psi$ in the collar means that $\Theta_{z',p}|_{C_{z',p}}= \psi_{z,p}\circ \psi^{-1}_{z',p}$, where $\psi_{z,p}\colon C_{z_0,p}\to C_{z,p}$ is the symplectomorphism induced by $\psi$. Using the formula \eqref{eq:asscolltriv1} defining $\psi$, we see that $\psi_{z,p}=\varphi_{z,p}\circ \varphi_{z_0,p}^{-1}$, so $\Theta_{z',p}|_{C_{z',p}}=\varphi_{z',p}\circ \varphi_{z,p}^{-1}$. In consequence, to prove the lemma it is enough to show the vanishing of the integral $\int_{\Gamma} \lambda_{0,p}-\varphi_{z,p}^{*}\lambda_{z,p}$ for every $1$-cycle $\Gamma$ on $\d \bar{W}_{0,p}$.

Fix such a $1$-cycle $\Gamma$, and an argument $\theta\in \S^1$. Consider a function $\tau\colon [0,\delta]\ni r \mapsto \int_{\Gamma} \lambda_{0,p}-\varphi_{r\theta,p}^{*}\lambda_{r\theta,p}$. We need to prove that $\tau\equiv 0$. Since  $\varphi_{0,p}=\id_{\bar{W}_{0,p}}$, we have $\tau(0)=0$. It remains to prove that $\tau'(r)=0$ for all $r\in [0,\delta]$. Because $\varphi_{t\theta,p}$ is a time $t$ flow of $\cX_{\theta}$, we have $\tau'(r) =-\int_{\Gamma}\frac{d}{d t}\varphi_{t\theta,p}^{*}\lambda_{t\theta,p}|_{t=r}=-\int_{\varphi_{r\theta,p}(\Gamma)}L_{\cX_{\theta}}\lambda_{r\theta,p}$. By Cartan formula we have $L_{\cX_{\theta}}\lambda_{r\theta,p}=d(\lambda_{r\theta,p}(\cX_{\theta}))+\omega_{r\theta,p}(\cX_{\theta},\sdot)$. The restriction of $\omega_{r\theta,p}(\cX_{\theta},\sdot)$ to $\d N_{r\theta,p}$ is zero, because $\cX_{\theta}$ is $\omega$-orthogonal to the fibers of $F$. Hence the form $L_{\cX_{\theta}}\lambda_{r\theta,p}$ is exact, so its integral along a closed cycle is zero by Stokes theorem. Thus $\tau'(r)=0$, as needed.
\end{proof}

\begin{remark}\label{rem:cohtriv}
Consider a special case when $W_{z,p}$ is Stein manifold of dimension $n-1$. This happens e.g.\ if $W$ is a ball in $\C^n$, see Example \ref{ex:Milnor-fibration}. For $n\geq 4$ the group
$H^{2}(N_{z,p},\d N_{z,p})=H_{2n-4}(N_{z,p})=H_{2n-4}(W_{z,p})$ vanishes, so the cohomological triviality proved in Lemma \ref{lem:cohtrivial} holds automatically.

In case $n=2$ (e.g.\ for plane curves), Lemma \ref{lem:cohtrivial} reduces to the observation that all fibers $N_{z,p}$ have the same symplectic area. The latter holds because, by construction, the integral curves of $-\cX_{z/|z|}$ starting in $N_{z,p}$ do not escape the collar $C$, hence in time $|z|$ they map $N_{z,p}$ to the central fiber.
\end{remark}

We now continue the general construction, and upgrade the symplectic fibration \eqref{eq:Liouv_fib1} to a Liouville one. Assume that there is a nonempty compact subdomain $P_0\subseteq P$, and a number $\delta_0\in [0,\delta]$, such that the Liouville vector field $X_{\lambda}$ of $\lambda$ points outwards $\d_{\mathrm{out}}C_{z,p}$ for all $p\in P_0$ and $z\in \D_{\delta_0}$. Recall that we have assumed it for $\d_{\mathrm{out}}C_{0,p_0}=f_{p_0}^{-1}(0)\cap \d\bar{W}$, so we can always take $P_0=\{p_0\}$, $\delta_0=0$; nonetheless, sometimes other choices will be convenient, too. Since $P_0$ and $\D_{\delta_0}$ are compact, after possibly
%replacing $P$ with some neighborhood of $P_0$, and $\delta$ with some number in $(\delta_0,\delta]$,
shrinking $P$ around $P_0$ and $\delta>\delta_0$, we can assume that $X_{\lambda}$ points outwards $ \d_{\mathrm{out}}C_{z,p}$  for all $(z,p)\in \D_{\delta}\times P$. Now the restriction \eqref{eq:Liouv_fib1} is  a Liouville fibration, with a cohomologically trivial collar trivialization \eqref{eq:asscolltriv1}.

\begin{remark}\label{rem:Seidel}
In the language of \cite[\sec 1.1]{Seidel_exact-sequence}, cf.\ \cite[Remark 15.2]{Seidel_Fukaya-categories}, the restriction $F|_{N}$ is %a Liouville fibration admitting a collar trivialization: such fibrations are called
an \emph{exact symplectic fibration}. If $X$ is smooth then the restriction $F|_{\bar{N}}$, i.e.\ $F|_{N}$ together with the singular fiber, is an \emph{exact symplectic fibration with singularities} in the sense of \cite[\sec 15a]{Seidel_Fukaya-categories}.
\end{remark}

\subsubsection{Constructing symplectic monodromy in Setting \ref{basic-setting}}\label{sec:basic-monodromy}
%Having constructed a cohomologically trivial collar trivialization, we proceed to describe the associated monodromy abstract contact open books, see Definition \ref{def:monodromy_acob}.
Our next goal is to introduce a monodromy making the fibers of \eqref{eq:Liouv_fib1} into an abstract contact open books.

To this end, we fix a family of loops $\Upsilon$ in $\D_{\delta}^{*}$. More precisely, we choose a domain $Q$ and a smooth map $\Upsilon\colon Q\times\S^1 \to \D_{\delta}^{*}$. The reader may keep in mind the example $\Upsilon\colon (0,\delta]\times\S^1\ni (r,\theta)\mapsto r\theta\in \D_{\delta}^{*}$. Define $N'$ as the fibered product of the Liouville fibration $N\to \D_{\delta}^{*}\times P$ given by \eqref{eq:Liouv_fib1} and the map $\Upsilon\times \id_{P}\colon Q\times \S^1\times P\to \D_{\delta}^{*}\times P$. Then the induced map $N'\to Q\times \S^{1}\times P$ together with the pullback of $\lambda|_{N}$ form a Liouville fibration, which inherits the fiberwise symplectic collar trivialization \eqref{eq:asscolltriv1} constructed above. By Lemma \ref{lem:cohtrivial}, the latter is cohomologically trivial.

By Proposition \ref{prop:sufficientmonod}, for every $q\in Q$, $p\in P$, and $\theta\in \S^1$ we have a symplectic monodromy trivialization $\Theta_{q,p}\colon N'_{q,\theta,p}\times [0,1]\to N_{q,p}'$, see Definition \ref{def:sympmonod}. %For $\theta\in \S^{1}$, the fiber $N'_{q,\theta,p}$ can be identified with $N_{z,p}$, where $z=\Upsilon(q,\theta)\in \D_{\delta}^{*}$. %Let $\phi_{z,p}$ be the associated symplectic monodromy.
Now, we assume that
\begin{equation}\label{eq:ass-exact}
\mbox{each associated symplectic monodromy } \phi_{q,\theta,p}\colon N'_{q,\theta,p}\to N'_{q,\theta,p} \mbox{ is exact}.
\end{equation}
If the condition \eqref{eq:ass-exact} holds then each triple
\begin{equation} \label{eq:basic-acob}
\end{equation}
is a \emph{monodromy abstract contact open book} of $\Theta_{q,p}$, see Definition \ref{def:monodromy_acob}. Its isotopy type does not depend on the choice of $q,\theta,p$. The Liouville domain $(N'_{q,\theta,p},\lambda_{q,\theta,p})$ is isomorphic to the fiber $(N_{z,p},\lambda_{z,p})$ for $z=\Upsilon(q,\theta)$. Its underlying manifold $N_{z,p}\subseteq f_{p}^{-1}(z)$ is diffeomorphic to $\bar{W}_{z,p}=f_{p}^{-1}(z)\cap \bar{W}$. %Clearly, the monodromy $\phi_{q,\theta,p}$ depends on the chosen loop.

\begin{remark}\label{rem:exact} We now gather some natural examples when condition \eqref{eq:ass-exact} is satisfied, so the construction above yields monodromy abstract contact open books \eqref{eq:basic-acob}.
\begin{parlist}
\item\label{item:exact_H1} Clearly, \eqref{eq:ass-exact} holds if for some (hence, any) $z\in \D_{\delta}^{*}$, $p\in P$ we have $H^{1}(\bar{W}_{z,p},\d \bar{W}_{z,p};\R)=0$.
\item\label{item:exact_Stein} Assume that $W_{z,p}$ is a Stein manifold of dimension $n-1\geq 2$. Then by Lefschetz duality  $H^{1}(\bar{W}_{z,p},\d \bar{W}_{z,p})=H_{2n-3}(W_{z,p})=0$ since $2n-3>n-1$, so condition \eqref{eq:ass-exact} holds by \ref{item:exact_H1}.
\item \label{item:exact_link} Let $L_{p}\de \Int (C_{p}\cup N_{p})\subseteq X\reg\setminus \Sing f_{p}^{-1}(0)$ be the \enquote{link} of $\Sing f_{p}^{-1}(0)\subseteq X$. We claim that if $H^{1}(L_{p};\R)=0$ then condition \eqref{eq:ass-exact} holds.

Indeed, the symplectic monodromies $\phi_{q,\theta,p}$ constructed above, together with $\id_{C_p}$, glue to a symplectomorphism $\phi\colon L_{p}\to L_{p}$. By assumption, the closed form $\phi^{*}\lambda_{p}-\lambda_{p}$ is exact. Hence its restricted pullbacks $\phi_{q,\theta,p}^{*}\lambda_{q,\theta,p}-\lambda_{q,\theta,p}$ are exact, too, i.e.\ the condition \eqref{eq:ass-exact} holds, as claimed.

\item \label{item:exact_link-example} Part \ref{item:exact_link} can be applied in the following situation. Let $X$ be an algebraic surface with an isolated singularity, whose link $L$ is a rational homology sphere. Take $U$ small enough so that each $L_{p}$ defined above is homotopically equivalent to $L$. Now, we have $H^{1}(L_{p};\R)=0$, so condition \eqref{eq:ass-exact} holds by \ref{item:exact_link}.

\item \label{item:exact_more-links} More generally, assume that $X$ is an algebraic surface whose all singularities have rational homology sphere links, and that $\lambda_{X}$ is defined everywhere on $X\reg$: for example, $X=\C^2$ and $\lambda=\lambda\std$. Using \ref{item:exact_link-example} we first construct compactly supported, exact monodromies around each singularity of $X$ and $f^{-1}(0)$ as above; and then we extend them by identity to the remaining part of the fiber. %Thus once again we obtain monodromy abstract contact open books \eqref{eq:basic-acob}.

In case $P=\{\pt\}$, the same construction works for $f$ replaced by $f\circ h$, where $h$ is a log resolution restricting to an isomorphism away from the zero fiber and in the collar.
\end{parlist}
\end{remark}

\subsubsection{Grading in Setting \ref{basic-setting}.}\label{sec:basic-grading}
Recall that under the assumptions of Setting \ref{basic-setting} and \eqref{eq:ass-exact}, we have constructed monodromy  abstract contact open books \eqref{eq:basic-acob}. Now, we endow them with a grading.

Fix $p\in P$. Let $T_{p,\mathrm{vert}}\de \ker (f_{p})_{*} \subseteq T(X\setminus f_{p}^{-1}(0))$ be the vertical tangent bundle. Fix an almost complex structure $J_{p}$ on $T_{p,\mathrm{vert}}$ which is compatible with the fiberwise symplectic form $d\lambda_{p}$, and assume
\begin{equation}\label{eq:vanishing-c1}
\end{equation}
This first Chern class depends only on $d\lambda_{p}$, not on $J_{p}$, see \cite[Defintion 4.1.4]{McDuff_Salamon}.
Let  $K_{p,\mathrm{vert}}$ be the top exterior power of $T_{p,\mathrm{vert}}^{*}$. Condition \eqref{eq:vanishing-c1} means that $K_{p,\mathrm{vert}}$ admits a nonvanishing $\cC^{\infty}$ section. By the correspondence \eqref{eq:gradings-trivializationsfiberwise}, its pullback induces a grading of the vertical symplectic frame bundle of the Liouville fibration $N'\to Q\times \S^1\times P$. By Corollary~\ref{cor:gradedunicitymon}, this grading makes each triple \eqref{eq:basic-acob} a graded abstract contact open book, whose graded isotopy class does not depend on the choice of $q,\theta$. Moreover, grading for one $p\in P$ determines uniquely the grading for all $p\in P$.

\begin{remark}\label{rem:c1} Like in Remark \ref{rem:exact}, we gather some examples where condition \eqref{eq:vanishing-c1} holds, so the monodromy abstract contact open books \eqref{eq:basic-acob} are naturally graded.
\begin{parlist}
\item\label{item:c1_H1} Clearly, \eqref{eq:vanishing-c1} holds if for some (hence, any) $z\in \D_{\delta}^{*}$, $p\in P$ we have $H^{2}(W_{z,p};\R)=0$.
\item\label{item:c1_Kahler} Assume that $d\lambda_{p}$ is K\"ahler. Then we can take for $J_{p}$ the standard complex structure. Denoting by $K_{X_{p}^{*}}$ the canonical bundle of the complex manifold $X_{p}^{*}\de X\setminus f_{p}^{-1}(0)$, we get an isomorphism
\begin{equation}\label{eq:K-vert}
K_{X_{p}^{*}}=K_{p,\mathrm{vert}} \otimes f_{p}^{*}(T^{*}\D_{\delta}^{*}).
\end{equation}
Thus if $c_{1}(X_{p}^{*})=0$ then condition \eqref{eq:vanishing-c1} holds, and every $\cC^{\infty}$ section of $K_{X_{p}^{*}}$ gives rise to a grading of the monodromy abstract contact open books \eqref{eq:basic-acob}. %In this case, Conley--Zehnder index of their fixed points can be explicitly computed using \cite[Lemma A.8]{McLean}; an example of such computation will be given in Proposition \ref{prop:monodromy}\ref{item:B_i-CZ}.
\end{parlist}
\end{remark}

\subsubsection{Summary} Assume that in Setting \ref{basic-setting}, conditions \eqref{eq:ass-exact} and \eqref{eq:vanishing-c1} hold. Then (after possibly shrinking $\delta>0$ or $P$) the above procedure, say with $\Upsilon\colon (0,\delta]\times\S^1\ni (r,\theta)\mapsto r\theta\in \D_{\delta}^{*}$ defines, for all $(r,\theta,p)\in (0,\delta]\times \S^1\times P$, \emph{monodromy graded abstract open books} \eqref{eq:basic-acob}, all graded isotopic to each other. The underlying space $N_{r,\theta,p}$ of \eqref{eq:basic-acob} is diffeomorphic to the fiber $\bar{W}_{r\cdot \theta,p}$.

Moreover, conditions \eqref{eq:ass-exact} and \eqref{eq:vanishing-c1} hold e.g.\ if $X$ is a Stein manifold with trivial canonical bundle, and $d\lambda$ is an exact K\"ahler form on $X$; e.g.\ $(X,\lambda)=(\C^{n},\lambda\std)$, see Remarks \ref{rem:exact}\ref{item:exact_Stein},\ref{item:exact_more-links} and \ref{rem:c1}\ref{item:c1_Kahler}.

\subsubsection{Extending monodromies to radius zero.}\label{sec:basic-radius-zero}

To get monodromy abstract contact open book \enquote{at radius zero}, we need to perform the above construction in Setting \ref{basic-setting2}, which slightly modifies Setting \ref{basic-setting}, replacing base $\C$ with  $\C_{\log}$. Below, we only consider the case which will be needed later, leaving straightforward generalizations (e.g.\ to more general families of loops) for the reader.

For $\delta>0$ we denote by $\D_{\delta,\log}$ the preimage of $\D_{\delta}$ in $\C_{\log}$, and identify $\D_{\delta,\log}\setminus \d \C_{\log}$ with $\D_{\delta}^{*}$.

\begin{setting}\label{basic-setting2}
Let $X$ be a smooth complex manifold, let $f\colon X\to \C$ be a holomorphic function, and let $U_X,W_X,V_X\subseteq X$ and $C'_{X}=\bar{U}_{X}\setminus V_{X}$ be as in Setting \ref{basic-setting} for $P=\{\pt\}$.  	Let $\lambda_X\in \Omega^{1}(C'_{X}\cup f^{-1}(\D_{\delta}^{*}))$ be a $1$-form such that $d\lambda_{X}$ extends to a K\"ahler form on $X$.

Assume that the singular fiber $f^{-1}(0)$ is snc, and let $A$ be A'Campo space for $f$. It is endowed with a smooth structure introduced in Section~\ref{sec:AX-smooth} using some adapted atlas and a partition of unity. Let $\pi\colon A\to X$ and $f_A\colon A\to \C_{\log}$ be the smooth maps as in the diagram \eqref{eq:AX-diagram}, see Proposition \ref{prop:AXsmooth}. Put $U=\pi^{-1}(U_X)$, $W=\pi^{-1}(W_X)$, $C'=\pi^{-1}(C'_X)$. Assume that
\begin{equation}\label{eq:lambda_extends}
\mbox{the pullback }\pi^{*}\lambda_{X}\in \Omega^{1}(A\setminus \d A)\mbox{ extends to a $1$-form on }A.
\end{equation}
Then for every $\delta',\epsilon>0$, formula \eqref{eq:lambdaACdelta} defines a $1$-form  $\lambda_{A}^{\delta',\epsilon}\in \Omega^{1}(A)$. Put $\omega_{A}^{\delta',\epsilon}=d\lambda_{A}^{\delta',\epsilon}$.
\end{setting}

Our goal is to define, in the above Setting \ref{basic-setting2}, symplectic monodromy over $\d \C_{\log}$ with respect to $\omega_{A}^{\delta',\epsilon}$, whose associated abstract contact open book will be graded isotopic with monodromy abstract contact open books \eqref{eq:basic-acob} constructed above over $\d\D_{\delta}$; for $\epsilon,\delta>0$ small enough.
\smallskip

In Section \ref{sec:basic-Liouville} we have seen that for small enough $\delta>0$, formula \eqref{eq:collar-triv-center} defines a $\D_{\delta}$-fiberwise symplectic collar trivialization
\begin{equation*}
\psi_{X}\colon C_{X,0} \times \D_{\delta}\to C_{X},
\end{equation*}
where $C_{X,0}=C_{X}\cap f^{-1}(0)$. Let $f|_{N_{X}}\colon N_{X}\to \D_{\delta}^{*}$ be the corresponding Liouville fibration \eqref{eq:Liouv_fib1}.

In Section \ref{sec:basic-monodromy}, we took a family of loops $\Upsilon$. Now, for simplicity, we take  $\Upsilon\colon (0,\delta]\times \S^{1}\ni (r,\theta)\mapsto r\theta\in \D_{\delta}^{*}$. Assume that the induced monodromies are exact, i.e.\ condition \eqref{eq:ass-exact} holds. Then formula \eqref{eq:basic-acob} gives, for every $r\in (0,\delta]$ and $\theta\in \S^1$, isotopic monodromy abstract contact open books
\begin{equation}\label{eq:basic-acob-standard}
\end{equation}
The Liouville domain $(N_{r,\theta},\lambda_{r,\theta})$ is naturally identified with the fiber $N_{X}\cap f^{-1}(r\theta) $ equipped with the restriction of $\lambda_{X}$. Its  underlying space $N_{r,\theta}$ is diffeomorphic to $\bar{W}_X\cap f^{-1}(z)$ for any $z\in \D_{\delta}^{*}$.
\smallskip

By the assumptions of Setting \ref{basic-setting}, the special fiber $f^{-1}(0)$ is smooth along the collar $C_X$, so the preimage $C\de \pi^{-1}(C_{X})$ is a fibered product of the restriction  $f|_{C_X}\colon C_{X}\to \D_{\delta}$ with the natural map $\D_{\delta,\log}\to \D_{\delta}$. Thus the trivialization $\psi_{X}$ pulls back to a trivialization
\begin{equation*}
\psi_{0}\colon C_{0}\times \D_{\delta,\log}\to C,
\end{equation*}
where $C_0\de C\cap f_{A}^{-1}(0,1)$ is the intersection of $C$ with a radius-zero fiber.

By Proposition \ref{prop:omega}\ref{item:omega-symplectic} there is a neighborhood $U'$ of $\bar{U}\cap \d A$, and an $\epsilon_0>0$ such that for every $\delta'>0$ and $\epsilon\in (0,\epsilon_0]$, the form $\omega_{A}^{\delta',\epsilon}$ is fiberwise symplectic in $U'$. Thus after  shrinking $\delta>0$ %(hence $X$)
if needed, the restriction $f_{A}|_{\bar{U}\cap f_{A}^{-1}(\D_{\delta,\log})}\colon \bar{U}\cap f_A^{-1}(\D_{\delta,\log})\to \D_{\delta,\log}$, together with the form $\omega_{A}^{\delta',\epsilon}$, is a symplectic fibration for every $\delta'>0$ and $\epsilon\in (0,\epsilon_0]$. Fix one $\delta'>0$ and define a family of $1$-forms $\lambda\in \Omega^{1}(A\times (0,\epsilon_0] \cup C\times [0,\epsilon_0])$ by $\lambda_{\epsilon}=\lambda_{A}^{\delta',\epsilon}$. %We fix one $\epsilon\in (0,\epsilon_0]$ and $\delta'\in (0,\delta)$.

By Remark \ref{rem:vanishing_restriction}, there is a $1$-form $\beta_X\in \Omega^{1}(C_{X,0})$ such that $\lambda_{\epsilon}|_{C\cap \d A}=\pi^{*}(\lambda_{X}|_{C_{X,0}}+\epsilon\beta_{X})$. Hence
\begin{equation*}%\label{eq:collar-triv-pullback}
%\psi_0|_{C_0\times \d\D_{\delta,\log}}\times \id_{[0,\epsilon_0]}\colon
C_0 \times \d\D_{\delta,\log}\times [0,\epsilon_0]%\to
\ni (x,\theta,\epsilon)\mapsto
\in
C\times [0,\epsilon_0]
\end{equation*}
is an $[0,\epsilon_0]$-fiberwise trivialization of $f_{A}|_{C\cap \d A}\times \id_{[0,\epsilon_0]}$, which is  $(\d\D_{\delta,\log}\times [0,\epsilon_0])$-fiberwise symplectic with respect to the form $d\lambda$.

Consider the symplectic lift $\mathcal{X}$ of the radial vector field $\tfrac{\d}{\d r}$ from $\D_{\delta,\log}$ to $C\times[0,\epsilon_0]$. Note that, unlike in Section \ref{sec:basic-Liouville}, now we do not need to restrict it to rays. By definition of $C_X$ in Section \ref{sec:basic-Liouville}, the flow lines of $\cX$ starting in $(C\cap \d A)\times\{0\}$ exist in $C'\times \{0\}$ for all times $t\in [0,\delta]$. Since $C\cap \d A$ and $[0,\delta]$ are compact, shrinking $\epsilon_0>0$ if needed, we infer that the same is true for the flow lines starting in $(C\cap \d A)\times [0,\epsilon_0]$. This means that the vector field $\mathcal{X}$ defines a flow
\begin{equation*}
%\varphi \colon
(C\cap \d A)\times [0,\epsilon_0]\times [0,\delta] \ni (x,\epsilon,t) \mapsto (\varphi_{\epsilon,t}(x),\epsilon)\in C' \times [0,\epsilon_0]
\end{equation*}
such that for every $\theta\in \S^{1}$, the restriction of $\varphi_{\epsilon,t}$ to $C \cap f_{A}^{-1}(0,\theta)$ is a symplectomorphism onto its image; with respect to the form $d\lambda_{\epsilon}$. The image of this restriction is contained in $C'\cap f_{A}^{-1}(t,\theta)$.

Thus for some $\tilde{C}\subseteq C'\times [0,\epsilon_0]$, the restriction $(f_{A}\times \id_{[0,\epsilon_0]})|_{\tilde{C}}$ admits a $[0,\epsilon_0]$-fiberwise trivialization
\begin{equation}\label{eq:colltriv2}
C_0\times \D_{\delta,\log}\times [0,\epsilon_0]
\ni
\mapsto
\in \tilde{C}.
\end{equation}
which is  ($\D_{\delta,\log}\times [0,\epsilon_0]$)-fiberwise symplectic. Note that this trivialization restricts to $\psi_0$ over $\epsilon=0$.

%Recall that for $\epsilon>0$, the forms $\lambda_{A}^{\epsilon,\delta'}$ are fiberwise symplectic in $\bar{U}\cap \d A$, too.
Exactly as in Section \ref{sec:basic-Liouville}, we define $\d_{\mathrm{out}}\tilde{C}$ and, using inner components of the locally trivial fibration $f_{A}\times \id_{[0,\epsilon_0]}\colon (\tilde{C},\d_{\mathrm{out}}\tilde{C})\to \D_{\delta,\log}\times [0,\epsilon_0]$, we define a locally trivial fibration
\begin{equation}\label{eq:Liouv_fib2}
(f_{A}\times \id_{[0,\epsilon_0]})|_{\tilde{N}}\colon \tilde{N}\to \D_{\delta,\log}\times [0,\epsilon_0]
\end{equation}
which agrees with $f_{A}|_{\bar{W}\cap \d A}\times \id_{[0,\epsilon_0]}$ over $\d\D_{\delta,\log}\times [0,\epsilon_0]$, and with the Liouville fibration \eqref{eq:Liouv_fib1} over $\D_{\delta}^{*}\times\{0\}$. Moreover, the fibration \eqref{eq:Liouv_fib2} is symplectic over $B\de (\D_{\delta,\log}\times (0,\epsilon_0])\cup(\D_{\delta}^{*}\times [0,\epsilon_0])$. %Denote the corresponding restrictions by $F'\colon \tilde{N}'\to \D_{\delta}^{*}\times [0,\epsilon_0]$ and $F''\colon \tilde{N}''\to \D_{\delta,\log}\times (0,\epsilon_0]$.
\smallskip

By definition of $C_{X}$, the Liouville vector field of $\lambda_{X}$ points outwards $\d_{\mathrm{out}} C_{X}$, hence the one for $\lambda_{0}=\pi^{*}\lambda_{X}$ points outwards $\tilde{N}_{r,\theta,0}$ for every $(r,\theta)\in \D_{\delta,\log}$. Since $\D_{\delta,\log}$ is compact, we can shrink $\epsilon_0>0$ so that the same is true for all $\epsilon\in [0,\epsilon_0]$. Now, the restriction of \eqref{eq:Liouv_fib2} over $B$ is a Liouville fibration, with a  fiberwise collar trivialization obtained from \eqref{eq:colltriv2} by moving the base point inside $\D_{\delta}^{*}$ like in the formula \eqref{eq:asscolltriv1}. Over $\D_{\delta}^{*}\times \{0\}$, this collar trivialization agrees with the one constructed for \eqref{eq:Liouv_fib1} in Section \ref{sec:basic-Liouville}. Exactly the same argument as in Lemma \ref{lem:cohtrivial} shows that the above collar trivialization is cohomologically trivial.
\smallskip

As before, we assume that the symplectic monodromy associated to its restriction over each loop $\S^1\ni \theta\mapsto (r,\theta,\epsilon)\in B$ is exact. This condition is satisfied in the situations described in Remark \ref{rem:exact}. Eventually, we get monodromy abstract contact open books
\begin{equation}\label{eq:basic-acob-Clog}
\end{equation}
for all $(r,\theta,\epsilon)\in B$, all isotopic to each other. Note that for $r>0$ and $\epsilon=0$, the abstract contact open book \eqref{eq:basic-acob-Clog} can be identified with \eqref{eq:basic-acob-standard}, since both are obtained using the same collar trivialization. In turn, for $(r,\theta)=(0,1)\in \d\D_{\delta,\log}$ and $\epsilon=\epsilon_0>0$, the abstract contact open book \eqref{eq:basic-acob-Clog} is equal to
\begin{equation}\label{eq:basic-radius-zero-acob}
\end{equation}
where $F=\bar{W}\cap f_{A}^{-1}(0,1)$ is a radius-zero fiber, $\lambda=\lambda_{A}^{\delta',\epsilon_0}|_{F}$, and $\phi\colon F\to F$ is the time one flow of the monodromy vector field given locally by the formula \eqref{eq:monodromy-vector-field}.
%	 	\smallskip
%	 	Moreover, if $c_{1}(X\setminus f^{-1}(0))=0$ then the natural grading of \eqref{eq:basic-acob} defined in Setting \ref{basic-setting} by a nonvanishing section of $K_{X\setminus f^{-1}(0)}$ induces a grading of  \eqref{eq:basic-radius-zero-acob}, too.

We summarize the above construction in the following corollary, which will be used in Section \ref{sec:isotopy-to-radius-zero}.

\begin{cor}\label{cor:isotopyisotopy-to-radius-zero}
The abstract contact open books \eqref{eq:basic-acob} defined in Section \ref{sec:basic-monodromy} using a family of loops $\Upsilon\colon (0,\delta]\times \S^{1}\ni (r,\theta)\mapsto r\theta\in \D_{\delta}^{*}$, are isotopic to the \enquote{radius-zero} abstract contact open book \eqref{eq:basic-radius-zero-acob}.
\end{cor}

Moreover, if $c_{1}(X\setminus f^{-1}(0))=0$ then the natural grading of \eqref{eq:basic-acob} defined in Remark \ref{rem:c1}\ref{item:c1_Kahler} by a nonvanishing section of $K_{X\setminus f^{-1}(0)}$ induces a grading of  \eqref{eq:basic-radius-zero-acob}, by Corollary \ref{cor:gradedunicitymon}.

\subsubsection{Applications} \label{sec:basic-applications}
A typical situation where the above constructions apply is the following.

\begin{example}
\label{ex:typical}
Let $Z$ be a Stein manifold and let $\varrho\colon Z\to \R$ be an exhaustive strictly plurisubharmonic function. %, see Section \ref{sec:symplectic-intro}. %Consider the Liouville form $\lambda_Z:=-d^{c}\varrho$, see Section \ref{sec:symplectic-intro}.
Let $X\subseteq Z$ be a closed analytic subset whose all singularities are isolated. In case $\dim_{\C}X=2$, assume further that the links of all singularities of $X$ are rational homology spheres, so the assumptions of Remark \ref{rem:exact}\ref{item:exact_Stein} or \ref{item:exact_more-links} hold. Choose regular values $\xi_{V}<\xi_{W}<\xi_{U}$ of $\varrho|_{X}$ such that $\varrho^{-1}[\xi_{V},\xi_{U}]\cap X\subseteq X\reg$, and define $V=\varrho^{-1}(-\infty,\xi_V)$, $W=\varrho^{-1}(-\infty,\xi_W)$, $U=\varrho^{-1}(-\infty,\xi_U)$.% as the corresponding sub-level sets of $\varrho$.

Consider a family of $1$-forms $\lambda_{t}\in \Omega^{1}(X\reg)$ smoothly dependent on $t\in[0,\epsilon]$ for some $\epsilon\geq 0$, such that $\lambda_{0}=-d^{c}\varrho|_{X\reg}$. Recall from Section \ref{sec:symplectic-intro} that the Liouville vector field of $\lambda_0$ is the gradient field of $\varrho$, so it points outwards $\d \bar{W}$.
\smallskip

Let $f_s\colon X\to\CC$ be a family of holomorphic functions with isolated singularities in $X$, holomorphically depending on a parameter $s\in \D_\eta$ for some $\eta\geq 0$. Assume that
\begin{equation}\label{eq:assumption-only-0}
\mbox{the only critical value of }f_s%\mbox{ restricted to } %\varrho^{-1}(-\infty,c)
\mbox{ is }0.
\end{equation}
In particular, all fibers $f_{s}^{-1}(z)$ for $z\neq 0$ are smooth and contained in $X\reg$.
\smallskip

Note that, since the form $d\lambda_{0}=-dd^{c}\varrho$ is K\"ahler, it restricts to a symplectic form on each fiber $f_{s}^{-1}(z)\cap \bar{U}$ for $z\neq 0$. Thus for some $\delta>0$, the restrictions $f_{s}|_{\bar{U}\cap f^{-1}(\D_{\delta}^{*})}$, for all $s\in \D_{\eta}$, are symplectic fibrations with respect to $d\lambda_{0}$. In case $\epsilon>0$, we can shrink $\epsilon>0$ so that the same is true for all $d\lambda_{t}$, $t\in [0,\epsilon]$: here we use compactness of $\bar{U}$, $\D_{\eta}$ and $\D_{\delta}$. Now, our situation fits into Setting \ref{basic-setting}, with $P=\D_{\eta}\times [0,\epsilon]$ and $F\colon X\times P\ni (x,s,t)\mapsto (f_{s}(x),s,t)\in \C\times P$.

The construction in Section \ref{sec:basic-Liouville} shows that we can shrink $\delta>0$ and $\epsilon,\eta>0$ (in case they are nonzero) so that the map $F$ restricts to a Liouville fibration $F|_{N}\colon N\to \D_{\delta}^{*}\times P$ defined in \eqref{eq:Liouv_fib1}, with a collar trivialization \eqref{eq:asscolltriv1}. By construction, the special fiber $\bar{N}_{0,s,t}$ equals $\bar{W}_{0,s,t}$ for all $s\in \D_{\eta}$, $t\in [0,\epsilon]$.

In the construction of Section \ref{sec:basic-monodromy}, we take the standard family of loops $\Upsilon\colon (0,\delta]\times \S^1\ni (r,\theta)\mapsto r\theta\in \D_{\delta}^{*}$. Recall that condition \eqref{eq:ass-exact} holds by Remark \ref{rem:exact}\ref{item:exact_Stein},\ref{item:exact_more-links}. Thus for every $z\in \D_{\delta}^{*},s\in \D_{\eta},t\in[0,\epsilon]$ formula \eqref{eq:basic-acob} defines isotopic abstract contact open books
\begin{equation}\label{eq:basic-acob-example}
\end{equation}
Here $N_{z,s,t}$ is a subset of $\bar{U}\cap f_{s}^{-1}(z)$ diffeomorphic to $\bar{W}\cap f_{s}^{-1}(z)$, and $\lambda_{z,s,t}$ is the restriction of $\lambda_{t}$. %Note that in case $\dim_{\C}X=2$ we have used the assumption on links of $X$ to apply Remark \ref{rem:exact}, which guarantees exactness of the monodromy.

If furthermore $c_{1}(X\reg)=0$ then by Remark \ref{rem:c1}\ref{item:c1_Kahler}, a choice of a nonvanishing section of $K_{X\reg}$ endows all abstract contact open books \eqref{eq:basic-acob-example} with a grading, making them graded isotopic.% to each other.
\end{example}

\begin{remark}
\label{rem:muconstant}
In the situation of Example~\ref{ex:typical}, if $X$ is smooth and near each critical point of $f_0$ the family $f_s$ has constant Milnor number, then condition \eqref{eq:assumption-only-0} is satisfied.
\end{remark}

Example \ref{ex:typical} covers the case of $\mu$-constant families considered in Theorem \ref{theo:Zariski}. We will make it precise in Example \ref{ex:mu_constant} below. To do this, we first recall some known facts about the Milnor fibration.

Let $f\colon (\C^n,0)\to (\C,0)$ be a holomorphic germ with an isolated singularity. By \cite[Corollary 2.9]{Milnor}, there is a radius $\epsilon_0>0$ such that the ordinary sphere $\d \B_{\epsilon}$ of radius $\epsilon$ centered at $0$ is transversal to $f^{-1}(0)$ for any $\epsilon\in (0,\epsilon_0]$. Any $\epsilon \in (0,\epsilon_0]$ is called {\em a Milnor radius}, and $\B_\epsilon$ is called a {\em Milnor ball}.

Fix a Milnor radius $\epsilon>0$ for $f$. There exists $\delta>0$ such that $f|_{\B_\epsilon\cap f^{-1}(\D_\delta^*)}\colon \B_\epsilon\cap f^{-1}(\D_\delta^*)\to \D_\delta^*$ is a locally trivial fibration. Its restriction over $\d \D_\delta$ is, up to a diffeomorphism, independent on the choices \cite[Theorem 2.10]{Milnor}. It is called the \emph{Milnor fibration} of $f$, see formula \eqref{eq:milfibtbintro} in the introduction.

\begin{example}
\label{ex:Milnor-fibration}
Let $f\colon (\C^{n},0)\to (\C,0)$ be a germ of isolated hypersurface singularity. Put $\varrho(z)=\frac{\pi}{2}||z||^2$, and let $\lambda=-d^{c}\varrho$ be the standard Liouville form, see Example \ref{ex:J}. Let $\xi_V<\xi_W<\xi_U$ be Milnor radii for $f$. Then Example \ref{ex:typical} (with $\epsilon=\eta=0$) gives a subset $N\subseteq \B_{\xi_U}$ and a $\delta>0$ such that the restriction $f|_{N}\colon N\to \D_{\delta}^{*}$, is a Liouville fibration, diffeomorphic to $f|_{\B_{\xi_U}\cap f^{-1}(\D_\delta^*)}$. Moreover, for each $z\in \D_{\delta}^{*}$, we get a monodromy abstract contact open book $(N_{z},\lambda_{z},\phi_z)$. Using the standard holomorphic volume on $\C^{n}$, we endow each $(N_z,\lambda_z,\phi_z)$ with a grading.%It is graded by the  standard holomorphic volume form on $\C^{n}$.}

We remark that, if $n\geq 3$, then by \cite[Theorem 6.4]{Milnor} the Milnor fiber is simply connected, so  by the correspondence \eqref{eq:gradings-subgroups} the above grading of $(N_{z},\lambda_{z},\phi_{z})$ %introduced above using a holomorphic volume form on $\C^n$ %, introduced above using triviality of $K_{\C^n}$,
is %in fact
the unique one.
\end{example}

\begin{example}
\label{ex:mu_constant}
More generally, let $\C^{n}\times \D_{\eta}\ni (x,s)\mapsto f_{s}(x)\in \C$ be a $\mu$-constant family of holomorphic functions with isolated singularities. As before, take $\varrho(z)=\frac{\pi}{2}||z||^2$, and $\lambda=-d^{c}\varrho$.

Let $\xi_V<\xi_W<\xi_U$ be Milnor radii for $f_0$. Then for $\delta,\eta>0$ small enough, formula \eqref{eq:basic-acob-example} in  Example \ref{ex:typical} (with $\epsilon=0$) defines graded isotopic abstract contact open books $(N_{z,s},\lambda_{z,s},\phi_{z,s})$ for $z\in \D_{\delta}^{*}$, $s\in \D_{\eta}$. For $s=0$, they agree with the ones constructed in Example \ref{ex:Milnor-fibration} for $f_0$.
\end{example}

%For $s=0$, the graded abstract contact open book $(N_{z,0},\lambda_{z,0},\phi_{z,0})$ is graded isotopic to the one constructed in Example \ref{ex:Milnor-fibration} for the Milnor fibration of $f_0$. However, this might no longer be the case for $s\neq 0$. Indeed, as we have explained in the introduction, the Milnor radii chosen in Example \ref{ex:mu_constant} for $f_0$ are not necessarily Milnor radii for $f_s$, $s\neq 0$. In fact, it is a long-standing, open question if a Milnor radius for $f_s$ can be chosen independently of $s$; even modifying the distance function $\varrho$, see [56].
Note that the Milnor radii chosen in Example \ref{ex:mu_constant} for $f_0$ are not necessarily Milnor radii for $f_s$, $s\neq 0$. In fact, it is a long-standing, open question if a Milnor radius for $f_s$ can be chosen independently of $s$; even modifying the distance function $\varrho$, see [56].

In particular, the graded abstract contact open books produced in Example \ref{ex:mu_constant} for $f_0$, and the analogous ones produced using a Milnor radius for $f_s$, might not be isotopic.
To prove Theorem \ref{theo:Zariski}, we will compare their Floer homology. A crucial ingredient of this comparison will be the fact that they differ by a homologically trivial cobordism. More precisely, the following result is known, see Figure \ref{fig:family}.

\begin{prop}[[46]]
\label{prop:homologtriv}
Let $\C^{n}\times \D_{\eta}\ni (x,s)\mapsto f_{s}(x)\in \C$ be a $\mu$-constant family of isolated hypersurface singularities. Let $\epsilon_0$ a Milnor radius for $f_0$. For any $s$ small enough and any Milnor radius $\epsilon_s<\epsilon_0$ for $f_s$, the cobordism
$\overline{f_s^{-1}(0)\cap \B_{\epsilon_0}\setminus \B_{\epsilon_s}}$
is homologically trivial, that is, we have the vanishing
\begin{equation*}
H_*(\overline{f_s^{-1}(0)\cap \B_{\epsilon_0}\setminus \B_{\epsilon_s}},f_s^{-1}(0)\cap\d \B_{\epsilon_0};\ZZ)=0.
\end{equation*}
\end{prop}

\begin{figure}[htbp]
\begin{tikzpicture}
\path[use as bounding box] (-6,-2.3) rectangle (10,1.5);
\draw[->] (-.7,-2.2) -- (5,-2.2);
\draw (-0.1,-2.1) -- (-0.1,-2.3);
\node[below] at (-0.1,-2.2) {\small{$s=0$}};
\draw (3.9,-2.1) -- (3.9,-2.3);
\node[below] at (3.9,-2.2) {\small{$s\neq 0$}};
\node[below left] at (5,-2.2) {\small{$s$}};
\draw[decoration=snake, decorate, segment length=60, name path=a] (-.7,1.3) -- (5,1.3);
\draw[decoration=snake, decorate, segment length=60, name path=b] (-.7,0.9) -- (5,0.9);
\tikzfillbetween[ of=a and b ] {pattern=vertical lines, pattern color=black!15};
\draw (-0.2,1.4) to [out=-90,in=180] (0.3,0) to [out=180,in=90] (-0.2,-1.4);
\draw[thick] (0,1.4) to [out=-90,in=170] (0.4,0.1) to [out=0,in=0] (0.4,-0.1) to [out=-170,in=90] (0,-1.8);
\draw (0,0) circle (1.2);
\draw (3.8,1.35) to [out=-90,in=145] (3.9,.8)
to [out=-45,in=90] (4,.7) to [out=-90,in=45] (3.9,.6) to [out=-145,in=90] (3.8,.4)
to [out=-90,in=180] (4.2,0) to [out=180,in=90] (3.8,-1.8);
\draw[thick] (4,1.4) to [out=-90,in=145] (4.1,.8)
to [out=-45,in=90] (4.2,.7) to [out=-90,in=45] (4.1,.6) to [out=-145,in=90] (4,.4)
to [out=-90,in=170] (4.3,0.1) to [out=0,in=0] (4.3,-0.1) to [out=-170,in=90] (4,-1.4);
\draw[ultra thick] (4,1.4) to [out=-90,in=145] (4.1,.8)
to [out=-45,in=90] (4.2,.7) to [out=-90,in=45] (4.1,.6) to [out=-145,in=90] (4,.4);
\filldraw (4,1.4) circle (2pt);
\filldraw (4,.35) circle (2pt);
\draw  (4.2,0) circle (.4);
\draw (4.2,0) circle (1.2);
\node at (1,-1.2) {\small{$\B_{\epsilon_0}$}};
\node at (5.2,-1.2) {\small{$\B_{\epsilon_0}$}};
\node at (4.8,-0.4) {\small{$\B_{\epsilon_s}$}};
\draw [ ->, gray] (6,0) to [out=180,in=0] (4.7,0);
\node[right] at (6,0) {\small{Milnor ball for $f_{s}$}};
\draw [ ->, gray] (6,.9) to [out=180,in=0] (4.3,.7);
\node[right] at (6,.9) {\parbox{3.2cm}{\small{homologically trivial cobordism [46]}}};
\draw [ ->, gray] (6,-0.8) to [out=180,in=0] (4.1,-.8);
\node[right] at (6,-0.8) {\small{$f_{s}^{-1}(\delta)$}};
\draw [ ->, gray] (6,-1.7) to [out=180,in=0] (3.9,-1.7);
\node[right] at (6,-1.7) {\small{$f_{s}^{-1}(0)$}};
\draw [ ->, gray] (-2,1.1) to [out=0,in=180] (-0.8,1.1);
\node[left] at (-2,1.1) {\small{collar trivialization}};
\draw [ ->, gray] (-2,-0.8) to [out=0,in=180] (-0.3,-0.8);
\node[left] at (-2,-0.8) {\small{$f_{0}^{-1}(0)$}};
\draw [ ->, gray] (-2,-1.7) to [out=0,in=180] (-0.1,-1.7);
\node[left] at (-2,-1.7) {\small{$f_{0}^{-1}(\delta)$}};
\draw [ ->, gray] (-2,0) to [out=0,in=180] (-1.3,0);
\node[left] at (-2,0) {\small{Milnor ball for $f_{0}$}};
\end{tikzpicture}
\caption{$\mu$-constant family, see Example \ref{ex:mu_constant}}
\label{fig:family}
\end{figure}

\subsection{Codimension zero families of fixed points}\label{sec:0codim}
We end this section by introducing a notion of \emph{codimension zero families of fixed points}, slightly extending the one introduced by McLean in \cite[p.\ 978]{McLean}. In Proposition \ref{prop:spectral-sequence} we will see that if all fixed points of $\phi$ come in such families, then their topology gives information about Floer homology groups $\HF_{*}(\phi,+)$. In Section \ref{sec:monodromy} we will see that this is the case for the \enquote{radius zero} monodromy on the boundary of the A'Campo space.
%For definition of manifolds with corners we refer to Section \ref{sec:Liouville-fibrations} or \cite[Appendice]{corners}.

\begin{definition}%[{\cite[p.\ 978]{McLean}}]
\label{def:0codim}
Let $(M,\lambda,\phi)$ be an abstract contact open book.
A {\em codimension zero family of fixed points} $B\subseteq M$ is a compact, connected codimension zero submanifold of $M$ with corners, such that there exist an open neighborhood $N_B$ of $B$ in $M$ and a time-independent Hamiltonian $H_{B}\colon N_{B}\to \R$ satisfying
\begin{equation*}
B=H_{B}^{-1}(0)=\Fix \phi|_{N_{B}}\quad\mbox{and}\quad \phi|_{N_{B}}=\psi^{H_{B}}_{1}|_{N_{B}}\colon N_{B}\to N_{B}.
\end{equation*}
The action function $\ac$ associated with $(M,\lambda,\phi)$ is constant in $B$, and we denote its value by $\ac(B)$. If $(M,\lambda,\phi)$ is graded then the function $\CZ$ is also constant in $B$, and we write $\CZ(B)$ for its value.
\end{definition}

Put $\d^{M}B=B\cap \d M$. Since $\phi=\id$ near $\d M$, the set $\d^{M}B$ is either empty, or a union of some connected components of $\d M$.

Since $B=H_{B}^{-1}(0)$ is a codimension zero submanifold with corners, at every point $p\in \d B\setminus \d^{M}B$ there is a chart $U_{p}\subseteq N_{B}$ such that $H_{B}|_{U_{p}\setminus B}$ is either positive or negative. Denote by $\d^{+}B$ (resp.\ $\d^{-}B$) the set of those points $p$ such that $H_{B}|_{U_{p} \setminus B}>0$ (resp.\ $<0$). This way, we get a decomposition
\begin{equation}
\label{eq:boundarysplitting1}
\partial B=\partial^M B\sqcup\partial^+ B \sqcup \partial^- B.
\end{equation}
where each factor is a union of connected components of $\d B$.

\begin{remark}\label{rem:McLean-has-no-d+}
Definition \ref{def:0codim} generalizes the one given in \cite[p.\ 978]{McLean}. Namely, a codimension zero family $B$ of fixed points satisfies the definition of loc.\ cit.\ if $\d^{-}B=\emptyset$, i.e.\ if the Hamiltonian $H_{B}$ is nonnegative (we warn the reader that, due to the opposite sign convention explained in Section \ref{sec:Hamiltonian}, the Hamiltonian in loc.\ cit.\ is nonpositive). As a result of this more general definition,  our version of McLean's axiom (HF3), stated in Proposition \ref{prop:spectral-sequence}, involves relative homology. This more general setting is needed to treat the additional fixed point component $B_0$, discussed in Section \ref{sec:intro_summary}.
\end{remark}