g^{\sharp}=\omega^{\sharp}(\sdot, J\sdot),\ g^{\flat}=\omega^{\flat}(\sdot,J\sdot),\ g_{q}=\omega_{q}^{\epsilon}(\sdot,J\sdot) \\
\mbox{and}\quad
g^{\sharp}_{\new}=t\cdot g^{\sharp},\  g^{\flat}_{\new} = t\cdot g^{\flat},
\ g_{q,\new}=t\cdot g_{q}.
\end{split}\end{equation*}

Since $\omega^{\sharp}$ and $\omega^{\flat}$ are fiberwise $J$-compatible, the above forms are fiberwise symmetric, i.e. their images in $T^{*}_{\vert} (A\setminus \d A)$ lie in $S^{2}T^{*}_{\vert} (A\setminus \d A)$. We denote these images by the same letters.

The following hold.

On $A\setminus \d A$, we have fiberwise equalities

\begin{equation}\label{eq:fiberwise-metrics}
g^{\sharp} = \sum_{i=1}^{N} \left(\frac{1}{m_{i}\hat{\sigma}_{i}} (d\hat{v}_{i})^2 + \hat{\sigma}_{i}\cdot m_i\, \hat{\alpha}_{i}^{2}\right)+\check{g}
\quad\mbox{and}\quad
g^{\flat} = \sum_{i\in \cS} \left(\frac{1}{t} \cdot (d\hat{w}_{i})^{2}+t\cdot m_{i}^{2}\, \hat{\alpha}_{i}^{2}\right)+\tilde{g},
\end{equation}

where $\hat{\sigma}=\hat{t}_{i}^2+\hat{t}_{i}\hat{u}_{i}^2$, see formula (<ref>); and $\tilde{g},\check{g}$ are smooth, symmetric $2$-forms on $A$ which are bounded with respect to the natural coordinates of $X$.

*  The sections $g^{\flat}_{\new}$, $g^{\sharp}_{\new}$ and $g_{q,\new}$ of $S^{2}T_{\vert}^{*}(A\setminus \d A)$ extend to smooth sections of $S^{2}T^{*}_{\vert} A$.

*  On $\d A$, we have fiberwise equalities

\begin{equation}\label{eq:limit-metrics}
g^{\sharp}_{\new} = \sum_{i=1}^{N} \frac{1}{m_{i}}\, \eta'(\hat{w}_i) (d \hat{w}_i)^2
\quad\mbox{and}\quad
g^{\flat}_{\new} = \sum_{i\in \cS} (d\hat{w}_{i})^2,
\end{equation}

where $\eta'(\hat{w}_{i}) (d\hat{w}_i)^2$ extends to a smooth form on $\d A$, vanishing at the zero locus of $\hat{w}_i$.

Put $\hat{\alpha}_{i}^{c}=\hat{\alpha}_{i}\circ J$. Definitions (<ref>) give fiberwise equalities

\begin{equation}\label{eq:dcw}
\hat{w}_{i}=-m_{i} t \hat{s}_{i},\quad \mbox{so} \quad d^{c}\hat{w}_{i}=m_{i}t\,\hat{\alpha}_{i}\quad\mbox{and}\quad  \hat{\alpha}_{i}^{c}=-\frac{1}{m_{i}t}\, d\hat{w}_{i}.
\end{equation}

First, we prove all claims about $g^{\flat}$. By formula (<ref>) we have $\omega^{\flat}=-\sum_{i\in \cS} m_{i} d\hat{w}_{i}\wedge  \hat{\alpha}_{i}+\tilde{\omega}$, where $\tilde{\omega}=\sum_{i\in \cS} m_{i}\hat{w}_{i}\, d\hat{\alpha}_{i}$. Substituting formulas (<ref>) we get the second equality of (<ref>), with $\tilde{g}\de \tilde{\omega}(\sdot, J\sdot)$.  By Lemma <ref><ref>, each form $d\hat{\alpha}_{i}$ is a pullback of a smooth $2$-form on $X$; hence $d\hat{\alpha}_{i}(\sdot, J\sdot)$ is a smooth section of $\otimes^{2}T^{*}X$. It follows that $\tilde{g}$ extends to a smooth section of $S^{2}T^{*}A$, bounded in the natural coordinates of $X$, as claimed in <ref>. Multiplying the result by $t$, we get a fiberwise equality

\begin{equation*}
g^{\flat}_{\new}=\sum_{i\in \cS} \left( (d\hat{w}_i)^2+t^2\cdot m_{i}^2\, \hat{\alpha}_i^2\right)+t\cdot \tilde{g}.
\end{equation*}

By Proposition <ref><ref>, the functions $\hat{w}_{i}$ and forms $\hat{\alpha}_{i}$ are smooth, so the right-hand side extends to a smooth section of $S^{2}T^{*}A$, as claimed in <ref>. Substituting $t=0$ we get the second equality of (<ref>).

Now, we prove
the claims about $g^{\sharp}$. Recall from formula (<ref>) that $\hat{v}_{i}=\hat{t}_{i}-\hat{u}_{i}$, where $\hat{u}_{i}=\eta(\hat{w}_{i})=(1-\log(\hat{w}_{i}))^{-1}$. We have the  following equalities, cf. <cit.>:

\begin{equation*}%\label{eq:du}
d\hat{u}_{i}=\eta'(\hat{w}_{i})\, d\hat{w}_{i}=-(1-\log(\hat{w}_{i}))^{-2}\cdot (-\hat{w}_{i}^{-1})\, d\hat{w}_{i}=\hat{u}_i^{2}\hat{w}_{i}^{-1}\, d\hat{w}_{i}=\hat{u}_{i}^{2} \hat{t}_{i}\cdot t^{-1}\, d\hat{w}_i,
\end{equation*}

where in the last equality we used an identity $t=\hat{t}_{i}\hat{w}_{i}$. This identity gives also a fiberwise equality $d\hat{t}_{i}=d(t\hat{w}_{i}^{-1})=-t\hat{w}_{i}^{-2}\, d\hat{w}_{i}=-\hat{t}_{i}^{2}\cdot t^{-1}\, d\hat{w}_{i}$. Combining those two, we get a fiberwise equality

\begin{equation}\label{eq:dv}
d\hat{v}_{i}=-\hat{t}_{i}^{-2}\cdot t^{-1}\, d\hat{w}_{i}-\hat{t}_{i}\hat{u}_{i}^2\cdot t^{-1}\, d\hat{w}_{i}=
-t^{-1}\hat{\sigma}_{i}\, d\hat{w}_{i},
\end{equation}

cf. <cit.>. Using formula (<ref>), we get fiberwise equalities

\begin{equation}\label{eq:dcv}
d^{c}\hat{v}_{i}=-m_{i}\hat{\sigma}_{i}\,\hat{\alpha}_{i}\quad\mbox{and}\quad \hat{\alpha}_{i}^{c}=\frac{1}{m_{i}\hat{\sigma}_{i}}\, d\hat{v}_{i}.
\end{equation}

By definition (<ref>) of $\omega^{\sharp}$, we have $\omega^{\sharp}=\sum_{i=1}^{N} d\hat{v}_{i}\wedge \hat{\alpha}_{i}+\check{\omega}$, where $\check{\omega}=\sum_{i=1}^{N} \hat{v}_{i}\, d\hat{\alpha}_{i}$. Substituting equalities (<ref>) we get the remaining part of <ref>, where like before $\check{g}\de \check{\omega}(\sdot, J\sdot)$ is a smooth section of $S^{2}T_{\vert}^{*}A$, bounded in natural coordinates of $X$. Multiplying by $t$, we get a fiberwise equality

\begin{equation*}
g_{\new}^{\sharp} = \sum_{i=1}^{N} \left(\frac{t}{m_i\hat{\sigma}} (d\hat{v}_{i})^2 + t\hat{\sigma}_{i}\cdot m_i\, \hat{\alpha}_{i}^{2}\right)+t\cdot \check{g}
\overset{\tiny{\mbox{\eqref{eq:dv}}}}{=}\sum_{i=1}^{N}\left(-\frac{1}{m_{i}} d\hat{w}_i\cdot d\hat{v}_i+t\hat{\sigma}_i\cdot m_i\, \hat{\alpha}_i^2\right)+t\cdot\check{g}.
\end{equation*}

Recall that by Proposition <ref><ref> the functions $\hat{v}_i$, $\hat{w}_i$ and the forms $\hat{\alpha}_i$ are smooth. By Lemma <ref> below, the functions $t\hat{\sigma}_i$ are smooth, too, so the right-hand side extends to a smooth, fiberwise symmetric $2$-form on $A$, as claimed in <ref>. Substituting $t=0$, we get

\begin{equation*}
g^{\sharp}_{\new}|_{\d A}=-\sum_{i=1}^{N} \frac{1}{m_{i}} d\hat{w}_i\cdot d\hat{v}_i.
\end{equation*}

It remains to show that $-d\hat{w}_i\cdot d\hat{v}_i$ is a smooth extension of $\eta'(\hat{w}_{i}) (d\hat{w}_i)^2$ to $\d A$, vanishing on $\{\hat{w}_i=0\}$.

The forms $d\hat{w}_{i}$ and $d \hat{v}_i$ are smooth on $\d A$. Clearly, $d\hat{w}_i=0$ on the zero locus of $\hat{w}_i$, so $-d\hat{w}_i\cdot d\hat{v}_i$ is a smooth form on $\d A$, vanishing on $\{\hat{w}_i=0\}$. On the remaining part of $\d A$ we have $\hat{t}_i=t\hat{w}_{i}^{-1}=0$, so $d\hat{v}_i=-d\hat{u}_i=-\eta'(\hat{w}_{i})\, d\hat{w}_i$, which proves the claim.

In the above proof, we used  Lemma <ref> below, which asserts that the functions $t\hat{\sigma}_i$ are smooth. However, the forthcoming applications require only the fact that those functions are continuous on $A$ and vanish on $\d A$, which follows directly from boundedness of each $\hat{\sigma}_i=\hat{t}_i^2+\hat{t}_i\hat{u}_i^2$. Therefore, we give only an outline of the proof of Lemma <ref>, which is similar to the case of functions $\hat{w}_i$ treated in Proposition <ref><ref>. The key idea is that the function $t$ allows to flatten all derivatives of $\hat{\sigma}_i$.

The functions
$t\hat{\sigma}_{i}$ for $i\in \{1,\dots, N\}$ extend to smooth functions on $A$.

We work in the preimage of an adapted chart $U_X$, say with associated index set $\{1,\dots, k\}$; and consider its subset $U_{1}=\{w_{1}>\frac{1}{n+2}\}$; with coordinates (<ref>) given by $(g;\hat{v}_{2},\dots,\hat{v}_k;\rest)$.

Assume first that $i>k$, i.e. $D_i\cap U_X=\emptyset$. Then $\hat{r}_{i}|_{U_{1}}>0$, so $\hat{t}_{i}=-(m_{i}\hat{r}_{i})^{-1}$ is smooth on $U_1$. Thus $\hat{u}_i|_{U_1}=\hat{t}_i|_{U_1}-\hat{v}_i|_{U_1}$ is smooth by Proposition <ref><ref>, and therefore  $\hat{\sigma}_{i}|_{U_1}$ is smooth, too.

Assume now that $i=1$. We have $\hat{w}_{1}|_{U_1}>0$ by definition of $U_1$, so since $\hat{w}_{1}$ and $\eta|_{(0,1]}$ are smooth, so is their composition $\hat{u}_{1}$. By Proposition <ref><ref>, so are the functions $\hat{t}_{1}=\hat{v}_{1}-\hat{u}_1$ and $\hat{\sigma}_1$.

It remains to consider the case $i\in \{2,\dots, k\}$, i.e. when $\hat{v}_i$ is a coordinate of $U_1$. Now $\hat{\sigma}_i$ might not be smooth, but we will show that $t\hat{\sigma}_i$ is. Using the identity $t=t_iw_i$, we write

\begin{equation}\label{eq:tsigma}
t\hat{\sigma}_i=w_i\cdot (t_i\hat{t}_i^2+t_i\hat{t}_i \hat{u}_i^2).
\end{equation}

Recall that in the proof of Proposition <ref><ref> we have introduced a slight modification of the algebra $\cA_{i}$ from <cit.>. The functions $t_{i}$ and $u_{i}$ belong to this algebra. Indeed, in the notation of loc. cit. we have $t_i,u_i\in \cW_i$, $t_i\in \cR_{i}$ by formula (38) and $u_i\in \cW_i$ by Lemma 3.41(a). Now our  modification of $\cA_i$ was done in such a way that the functions $a$ used in loc. cit. are exactly those in Lemma <ref>. Thus Lemma <ref><ref>,<ref> implies that  $\hat{t}_i-t_i\in \cA_i$ and $\hat{u}_i-u_i\in \cA_i$. We conclude that $\hat{t}_i,\hat{u}_i\in \cA_i$, so $t\hat{\sigma}_{i}\in w_i\cdot \cA_i$ by formula (<ref>).

It follows that the function $t\hat{\sigma}_i$ lies in the algebra $\cW_{i}'\cdot \cA_i\cdot \cP$ introduced in p. 32 loc. cit: here $\cW_{i}'$ and $\cP$ are some algebras containing $w_i$ and $1$, respectively. As observed in the proof of Proposition <ref><ref>, the proof of Lemma 3.49 loc. cit. shows that all elements of the algebra $\cW_{i}'\cdot \cA_i\cdot \cP$ are smooth on $U_1$. Thus $t\hat{\sigma}_i$ is smooth on $U_1$; as needed.

§.§ Admissible paths

We have set up a bi-parametric approach to study the degeneration of Kähler manifolds $X_{z}^{q}$, where the first parameter $z$ determines the fiber $X_z$ of the degeneration, and the second one, $q$, determines the Kähler form $\omega_{q}^{\epsilon}$ within the family (<ref>). To prove our main results, we choose some paths in this bi-parametric model. First, we distinguish admissible paths as those used in  Theorem <ref><ref> and Theorem <ref><ref>,<ref>, that is, we pose the following definition.

A smooth path $\gamma=(\gamma_1,\gamma_2)\colon [0,\rho)\to \C_{\log}\times [0,1]$ is admissible if $|\gamma_1(0)|=\gamma_2(0)=0$ and $|\gamma_1(\tau)|>0$, $\gamma_2(\tau)>0$ for $\tau>0$.

Let $(r,\theta)$ be coordinates on $\C_{\log}=[0,\infty)\times \S^{1}$. In our setting, it is natural to replace $r$ by $t=e^{-\frac{1}{|f|}}$, see formula (<ref>): indeed, both the volume $\vol^{\Omega}_{\new}$ and the metrics $g_{q,\new}$, defined to give a finite nonzero limits as $|f|\rightarrow 0$, required a rescaling by a power of $t$, see Sections <ref> and <ref>. So, we endow $\C_{\log}$ with another smooth coordinate system, not compatible with $(r,\theta)$, given by $(t,\theta)=(e^{-\frac{1}{r}},\theta)$.

For any admissible path $\gamma\colon [0,\rho)\to \C_{\log}\times [0,1]$, the restriction $\gamma|_{[0,\rho')}$ for some $\rho'>0$ can be parametrized, in the above coordinates $(t,\theta)$ of $\C_{\log}$, as $\gamma(h)=(h,\theta(h),q(h))$, where $\theta\colon [0,\rho')\to \S^1$ and $q\colon [0,\rho')\to [0,1]$ are continuous functions whose restrictions to $(0,\rho')$ are smooth. Such a parametrization will be called normal. We can now distinguish paths used in Theorem <ref><ref>.

Let $\gamma(h)=(h,\theta(h),q(h))$ be a normal parametrization of an admissible path $\gamma$. We say that $\gamma$ is special if the function $q$ is $\cC^{2}$ and $q'(0)=0$.

[see Figure <ref>]
Let $\gamma=(\gamma_{1},\gamma_{2})\colon [0,\rho)\to \C_{\log}\times [0,1]$ be an admissible path such that for all $\tau\in [0,\rho)$ we have $|\gamma_{1}(\tau)|=\tau$, where the absolute value refers to the natural coordinates $(r,\theta)$ on $\C_{\log}$, see Theorem <ref><ref>. We claim that $\gamma$ is special. Indeed, the normal parametrization of $\gamma$ is $(h,\theta(h),q(h))$, where $\theta(h)$ is the argument of $\gamma_{1}(e^{-\frac{1}{h}})$, and $q(h)=\gamma_{2}(e^{-\frac{1}{h}})$. Now $q$ is smooth since $\gamma_{2}$ is; and $q'(h)=-h^{-2}e^{-\frac{1}{h}}\gamma_{2}(e^{-\frac{1}{h}})\rightarrow 0$ as $h\rightarrow 0$, as needed.

Let $h \mapsto (h,\theta(h),q(h))$ be a normal parametrization of an admissible path. We denote by $X_{h}^{\gamma}$ the fiber $(t,\theta)^{-1}(h,\theta(h))\subseteq A$, equipped with a $2$-form $\omega^{\gamma}_{h}\de \omega^{\epsilon}_{q(h)}$, defined in formula (<ref>). Recall that $\epsilon>0$ is a number fixed at the beginning of this section, sufficiently small for so that Proposition <ref> holds. For a fixed $q>0$, we write $\Psi_{h}^{q}=\Phi_{\eta(h)}^{q}$, so $\Psi_{\bullet}^{q}$ is a reparametrization of the flow (<ref>) in such a way that $\Psi_{h}^{q}$ maps $X_{0}^{\gamma}$ to $X_{h}^{\gamma}$. We define regions $(X_{h}^{\gamma})_{\cS}^{\gen}\subseteq (X_{h}^{\gamma})_{\cS}^{\sm}\subseteq X_{h}^{\gamma}$ as in Section <ref>, i.e. as images of $(X_{0}^{\gamma})_{\cS}^{\gen}$ and $(X_{0}^{\gamma})_{\cS}^{\sm}$ by the flow $\Psi_{h}^{q(h)}$.

Thus on $(X_{h}^{\gamma})^{\sm}_{\cS}$ we have an ivy-like Lagrangian torus fibration (<ref>), with respect to the form $\omega_{h}^{\gamma}$. We write $g^{\gamma}_{h,\new}\de h\cdot \omega_{h}^{\gamma}(\sdot,J\sdot)$ for the rescaled metric on $X_{h}^{\gamma}$.
We denote by $A^{\gamma}\subseteq A$ the union of fibers $X_{h}^{\gamma}$; so $A^{\gamma}$ , and put $A^{\gamma}_{+}\de A^{\gamma}\setminus X_{0}^{\gamma}$.

§.§ The Gromov–Hausdorff limit

With the above notation, we have the following consequence of Lemma <ref>.

Let $\gamma$ be an admissible path. Then for every $h\geq 0$ we have

\begin{equation*}
g^{\gamma}_{h,\new}=(1-q(h))\cdot\epsilon\cdot \sum_{i\in \cS} (d\hat{w}_{i})^{2}+ \tilde{g}_h,
\end{equation*}

where $\tilde{g}_{0}=0$, and for $h>0$, $\tilde{g}_{h}$ is a Riemannian metric on $X_{h}^{\gamma}$. In particular, $g_{0,\new}^{\gamma}=\epsilon \cdot \sum_{i\in \cS} (d\hat{w}_i)^2$.

Let $g_{X}=\omega_{X}(\sdot, J\sdot)$ be the Kähler metric on $X$. Recall that $\omega_{h}^{\gamma}=\omega_{q(h)}^{\epsilon}$, so formula (<ref>) gives

\begin{equation*}%\label{eq:g-s}
g^{\gamma}_{h,\new}= h\cdot \pi^{*}g_{X}+ q(h) \cdot  \epsilon\cdot g^{\sharp}_{\new}+ (1-q(h))\cdot \epsilon\cdot  g^{\flat}_{\new}.
\end{equation*}

Substituting the second equality of (<ref>) we get the required formula, with $\tilde{g}_h\de h\cdot \pi^{*}g_{X}+ q \cdot  \epsilon\cdot g^{\sharp}_{\new}$. Since the path $\gamma$ is admissible, we have $q(h)>0$ for $h>0$, so $\tilde{g}_{h}=h\cdot \omega^{q(h)\cdot \epsilon}_{1}(\sdot, J\sdot)$ is a Riemannian metric by Proposition <ref><ref>; and $q(0)=0$, so $\tilde{g}_{0}=0$, as claimed.

We can now prove the main result of this section.
For a definition and basic properties of the Gromov–Hausdorff convergence we refer to <cit.>.

Let $\gamma$ be an admissible path. For $h>0$, we equip the fiber $X_{h}^{\gamma}$ with a Riemannian metric $\frac{1}{\epsilon}\cdot g^{\gamma}_{h,\new}$. Then we have Gromov–Hausdorff convergence:

\begin{equation*}%\label{eq:GH-limit}
\lim_{h\rightarrow 0} (X_{h}^{\gamma}, X_{h}^{\gamma}\setminus (X_{h}^{\gamma})_{\cS}^{\sm}) =(\Delta_{\cS},\Delta_{\cS}^{\geq 2}),
\end{equation*}

where $\Delta_{\cS}\subseteq \Delta_{D}\subseteq \R^{N}$ is the essential skeleton, see Section <ref>, equipped with the standard metric of $\R^{N}$; and $\Delta_{\cS}^{\geq 2}\subseteq \Delta_{\cS}$ is the union of faces of $\Delta_{\cS}$ of codimension at least $2$.

For $i\in \{1,\dots, N\}$ put $\hat{w}_{i}'=\hat{w}_{i}$ if $i\in \cS$ and $\hat{w}_{i}'=0$ otherwise, and consider a smooth map $\mu=(\hat{w}_{1}',\dots,\hat{w}_{N}')\colon A\to \R^{N}$. It is easy to see that
$\mu(A_{I}^{\circ})=\Delta_{I}$ for every face $\Delta_{I}$ of $\Delta_{\cS}$,  cf. Lemma <ref> or <cit.>, so
$\mu(X_{0}^{\gamma})=\Delta_{\cS}$ and $\mu(X_{0}^{\gamma}\setminus (X_{0}^{\gamma})_{\cS}^{\sm})=\Delta_{\cS}^{\geq 2}$.

For any $h\in [0,\delta)$, the restriction of $\mu$ to a fiber $X_{h}^{\gamma}$ gives maps of compact metric spaces

\begin{equation*}
\mu_{h}\colon X_{h}^{\gamma}\to \Delta_{\cS}\cup \mu(X_{h}^{\gamma}),
\quad\mbox{and}\quad
\mu_{h}^{\geq 2} \colon X_{h}^{\gamma}\setminus (X_{h}^{\gamma})_{\cS}^{\sm}\to \Delta_{\cS}^{\geq 2}\cup \mu(X_{h}^{\gamma}\setminus (X_{h}^{\gamma})_{\cS}^{\sm}).
\end{equation*}

Let $d_{h}$, $d_{\std}$ be the distance functions associated to the metric $\frac{1}{\epsilon}\cdot g_{h,\new}^{\gamma}$ on $X_{h}^{\gamma}$ and to the euclidean metric on $\Delta_{\cS}$, respectively. Lemma <ref> implies that for any two points $x,y\in X_{h}^{\gamma}$ we have $(1-q(h))^{\frac{1}{2}}\cdot d_{\std}(\mu_{h}(x),\mu_{h}(y))\leq d_{h}(x,y)\leq d_{\std}(\mu_{h}(x),\mu_{h}(y))+D_{h}$, where $D_{h}$ is the diameter of $X_{h}^{\gamma}$ with respect to the metric $\tilde{g}_{h}$, hence $D_{h}\rightarrow 0$ as $h\rightarrow 0$. Since $q(h)\rightarrow 0$ as $h\rightarrow 0$, too, we conclude that the distortion of $\mu_h$ and $\mu_{h}^{\geq 2}$, see <cit.>, converges to $0$ as $h\rightarrow 0$.

Continuity of $\mu$ implies that for every $\eta>0$ there is $\delta_0>0$ such that for every $h\in [0,\delta_0]$, each point of $\Delta_{\cS}$ is at distance at most $\eta$ to $\mu(X_{h}^{\gamma})$. This means that $\mu(X_{h}^{\gamma})$ is an $\eta$-net for $\Delta_{\cS}\cup \mu(X_{h}^{\gamma})$, see <cit.>. Similarly, $\mu(X_{h}^{\gamma}\setminus (X_{h}^{\gamma})_{\cS}^{\sm})$ is an $\eta$-net for $\Delta_{\cS}^{\geq 2}\cup \mu(X_{h}^{\gamma}\setminus (X_{h}^{\gamma})_{\cS}^{\sm})$. Shrinking $\delta_0>0$ if needed, we infer that $\mu_{h}$ and $\mu_h^{\geq 2}$ are $\eta$-isometries, see Definition 7.3.27 loc. cit. The required Gromov–Hausdorff  convergence follows from Corollary 7.3.28(2) loc. cit.

§.§ Summary

Before we specialize to the Calabi–Yau case, we summarize the general results obtained so far.

Let $(X,\omega_{X})$ be a Kähler manifold, let $f\colon X\to \C$ be a holomorphic function whose unique singular fiber $f^{-1}(0)$ is snc, and let $D_1,\dots, D_{N}$ be its irreducible components. Choose a subset $\cS\subseteq \{1,\dots, N\}$ so that $\Delta_{\cS}$ has a maximal or a submaximal face, i.e. there is $I\subseteq \cS$ such that $\#I\geq \dim X-1$ and $\bigcap_{i\in I}D_{i}\neq \emptyset$. Moreover, choose an open subset $V\subseteq X$ such that $f|_{\bar{V}}$ is proper, let $W$ be its slight modification as in Setting <ref> (one can take $W=X$ if $f$ is proper), and let $W_{z}=f^{-1}(z)\cap W$. Then there is a $\delta>0$ such that for every $\epsilon>0$ small enough, the following hold.

Kähler potential  For every $z\in \D_{\delta}^{*}$ and every $q\in [0,1]$, the form $\omega_{q}^{\epsilon}|_{W_{z}}$ defined in (<ref>) is Kähler.
Lagrangian fibration For every $z\in \D^{*}_{\delta}$ and every $q\in (0,1]$, formula (<ref>) defines an ivy-like Lagrangian torus fibration of a codimension zero submanifold $(W_{z}^{q})_{\cS}^{\sm}\subseteq W_{z}$ with boundary and corners, with respect to the Kähler form $\omega_{q}^{\epsilon}$.
Gromov–Hausdorff limit  Fix a path $(z,q)\colon [0,\delta) \to \C_{\log}\times [0,1]$ such that $|z(0)|=q(0)=0$ and $z(h),q(h)>0$ for $h>0$. For each $h>0$, let $W_{h}$ be the fiber $W_{z(h)}$ with the Kähler metric given by $\omega_{q(h)}^{\epsilon}$, rescaled so that its diameter is independent of $h$. Then we have Gromov–Hausdorff convergence

\begin{equation*}
\lim_{h\rightarrow 0} (W_{h},W_{h}\setminus W_{h}^{\sm})=(\Delta_{\cS},\Delta_{\cS}^{\geq 2}),
\end{equation*}

where $\Delta_{\cS}$ is the subcomplex of the dual complex of $f^{-1}(0)$ spanned by vertices indexed by $\cS$, equipped with some multiple of the euclidean metric; and $\Delta_{\cS}^{\geq 2}$ is the union of its faces of codimension at least $2$.

Parts <ref>, <ref> and <ref>  are proved in Propositions <ref><ref>, <ref> and <ref>, respectively.

Let $f$ be a model constructed in Proposition <ref>; and let $\Delta_{\cS}$ be the essential skeleton. Now the result follows from Theorem <ref> and the fact that for the canonical choice of $\ll_{\cS}$ described in Remark <ref>, formula  (<ref>) defines a usual Lagrangian torus fibration.

§ THE CALABI–YAU CASE

From now on, we assume that $f^{\circ}$ is a maximal Calabi–Yau degeneration admitting a semi-stable snc model. Recall from Section <ref> that the latter property can always be achieved after a finite base change, which preserves  maximality. We let $f\colon X\to \C$ be the snc model of $f^{\circ}$ constructed in Proposition <ref>. We use notation introduced in Section <ref>.

By Proposition <ref><ref> we have a holomorphic $(n+1)$-form $\Theta$ on $X$ whose divisor is $\sum_{i}(\nu_i-1)D_{i}$ with $\min_{i}\nu_{i}=0$. Thus the subcomplex $\Delta_{\cS}\subseteq \Delta_{D}$ for $\cS=\{i:\nu_{i}=0\}$ is the essential skeleton, see formula (<ref>).   As $f^{\circ}$ is maximal, <cit.> gives $\dim \Delta_{\cS}=n$, i.e. $\Delta_{\cS}$ has a maximal face.

We choose a section $\Omega$ of the vertical canonical bundle given by $\Omega\de \frac{\Theta}{d\log f}$, see formula (<ref>). Its restriction to any smooth fiber is a holomorphic $n$-form, which we denote by $\Omega$, too. We denote by $\vol^{\Omega}$ the induced fiberwise volume form, defined in formula (<ref>). We put

\begin{equation}\label{eq:Omega_new_def}
\Omega_{\new}=t^{n}\cdot \Omega
\quad \mbox{and}\quad \vol^{\Omega}_{\new} = t^{n}\cdot \vol^{\Omega}.
\end{equation}

§.§ Almost all volume is fibered by Lagrangian tori

In Proposition <ref> we have constructed a Lagrangian torus fibration on the subset $(X_{z})_{\cS}^{\sm}$ of the fiber $X_{z}$. Proposition <ref> below shows that as $z\rightarrow 0$, the volume of $(X_{z})_{\cS}^{\sm}$ – and even of its subset $(X_{z})_{\cS}^{\gen}$ – approaches the whole volume of the fiber.  The proof follows a well-known local computation, see e.g. <cit.> or <cit.>, which we perform in the A'Campo space. The key point is that the suitably rescaled fiberwise volume forms (<ref>) extend to finite, nonzero ones at radius zero.

Let $\Delta_{I}$ be a maximal essential face. The following hold.

*  The forms $\Omega_{\new} $ and $\vol_{\new}^{\Omega}$ extend to smooth sections of $\bigwedge^{n}_{\C}T^{*}_{\C,\vert}A$ and $\bigwedge^{2n} T^{*}_{\vert}A$, respectively.

*  The restrictions $\Omega_{\new}|_{\d A}$ and $\vol_{\new}|_{\d A}$ are zero outside of $A_{\cS}^{\gen}$.

*  Let $U_{X}$ be an adapted chart with associated index set $I$, let $U=\pi^{-1}(U_X)$, and let $w_j$, $\theta_j$ be the corresponding functions introduced in (<ref>). Then for some nonvanishing holomorphic function $c\in \cO_{X}^{*}(U_X)$, and every $i\in I$, we have a fiberwise equality on $U$

\begin{equation}\label{eq:Omega_new}
\Omega_{\new}= \pm c \bigwedge_{j\in I\setminus\{i\}} (dw_j-t^2\, \imath d\theta_j).
\end{equation}

*  The number $c_{0}\de c(X_{I})$ does not depend on the maximal essential face $\Delta_{I}$, up to a sign.

*  On $U\cap A_{I}^{\circ}$ we have fiberwise equalities

\begin{equation}\label{eq:volume_on_maximal_face}
\Omega_{\new}=\pm\, c_0 \cdot \bigwedge_{j\in I\setminus \{i\}} dw_{j}
\quad\mbox{and}\quad
\vol^{\Omega}_{\new}=(-1)^{n}\cdot  |c_0|^2 \cdot \bigwedge_{j\in I\setminus \{i\}} (dw_{j}\wedge d\theta_{j}).
\end{equation}

First, we prove the assertions about $\Omega_{\new}$. Choose an adapted chart $U_{X}$ with associated index set $J$, say $J=\{1,\dots, k\}$, fix an index $i\in J$, say $i=1$; and let $U=\pi^{-1}(U_X)$. Recall that $\Omega=\frac{\Theta}{d\log f}$, where $\Theta$ is a holomorphic $(n+1)$-form with zero of order $\nu_j-1$ along each $D_j$. There is a nonvanishing holomorphic function $c\in \cO_{X}^{*}(U_X)$ such that

\begin{equation}\label{eq:Theta_locally}
\Theta|_{U_X}=c\cdot \prod_{j=1}^{k}z_{j}^{\nu_j-1}\, dz_1\wedge\dots \wedge dz_k\wedge \zeta=c\cdot \prod_{j=1}^{k}z_{j}^{\nu_{j}}\, d\log z_1\wedge d\log z_2\wedge \dots \wedge d\log z_{k} \wedge \zeta,
\end{equation}

where $\zeta=dz_{i_{k+1}}\wedge\dots\wedge dz_{i_{n+1}}$ comprises the remaining coordinates of $U_X$. Since $\frac{\d \log f}{\d \log z_{1}}=m_{1}$, we have

\begin{equation*}
\Omega=\frac{\Theta}{d\log f}=
\frac{c}{m_1}\prod_{j=1}^{k}z_{j}^{\nu_j}\, d\log z_2\wedge \dots\wedge d\log z_k\wedge \zeta.
\end{equation*}

Formulas (<ref>) give $\log z_j=s_j+\imath\theta_j=-\frac{1}{tm_j} w_j+\imath\theta_j$, so we have a fiberwise equality

\begin{equation*}
d\log z_j=-\frac{1}{tm_j}\, dw_j+\imath\, d\theta_j=-\frac{1}{tm_j}\left (dw_j-tm_{j}\, \imath d\theta_{j}\right).
\end{equation*}
Substituting it to the above expression for $\Omega$, we get a fiberwise equality

\begin{equation}\label{eq:tOmega_fiberwise}
\Omega=t^{1-k}\cdot \frac{c\cdot (-1)^{k-1}}{m_1\cdot\ldots\cdot m_{k}}\prod_{j=1}^{k} z_j^{\nu_j}\cdot  (dw_2-tm_2\, \imath d\theta_2)\wedge \dots\wedge (dw_k-tm_k\, \imath d\theta_k) \wedge \zeta.
\end{equation}

By Proposition <ref><ref>, we have $\nu_j\geq 0$ for all $j$, with equality if and only if $j\in \cS$. Hence the function $\prod_{j=1}^{k} z_j^{\nu_j}$ is smooth on $U_X$, and equals $0$ on $U_X\cap X_{J}$ unless $J\subseteq \cS$. The function $c$ and the form $\zeta$ are smooth on $U_X$ by definition. Since $w_j,\theta_j$ are smooth in $U\subseteq A$, formula (<ref>) shows that $t^{n}\Omega=\Omega_{\new}$ extends to a smooth section of $\bigwedge^{n}_{\C}T^{*}_{\C,\vert}A$, and $\Omega_{\new}|_{U\cap A_{J}^{\circ}}=0$ unless $k=n+1$ and $J\subseteq \cS$, i.e. unless $\Delta_{J}$ is a maximal, essential face. This proves the first part of <ref> and <ref>. Moreover, if  $\Delta_{J}$ is a maximal, essential face, i.e. $J=\{1,\dots, n+1\}\subseteq \cS$ then $(m_{i},\nu_i)=(1,0)$ for all $i\in J$ by Proposition <ref><ref>,<ref>. Substituting this to formula (<ref>) we get part <ref>. Furthermore, substituting $t=0$ gives the following fiberwise equality on $U\cap \d A$

\begin{equation*}
\Omega_{\new}= c\cdot (-1)^{n} \cdot dw_2\wedge \dots \wedge dw_{n+1}.
\end{equation*}

Note that on $U\cap \d A \setminus \pi^{-1}(D_j)$ we have $dw_j=0$, so on $U\cap \d A\setminus A_{J}^{\circ}$ both sides of the above equality are zero. On $A_{J}^{\circ}$, the function $c$ above is constant, equal to the value of $c\in \cO_{X}^{*}(U_X)$ at the point $X_J$. This proves the first equality in (<ref>).
Now, we prove the assertions about $\vol^{\Omega}_{\new}$. Using formula (<ref>) we get a fiberwise equality

\begin{equation*}\begin{split}
& \Omega\wedge \bar{\Omega}= t^{2-2k} \cdot \frac{|c|^2 }{m_1^2\cdot\ldots\cdot m_{k}^2}\prod_{j=1}^{k} |z_j|^{2\nu_j} \cdot  \bigwedge_{j=2}^{k} (dw_j-tm_j\, \imath d\theta_j) \wedge \zeta \wedge \bigwedge_{j=2}^{k} (dw_j+tm_j\, \imath d\theta_j) \wedge \bar{\zeta}
= \\
=\ & t^{1-k} \cdot \frac{|c|^2 \cdot (-1)^{(k-1)(n+1-k)}\cdot (2\imath)^{k-1}(-1)^{\frac{1}{2}(k-1)(k-2)}}{m_1^2\cdot\ldots\cdot m_{k}^2} \cdot \prod_{j=1}^{k} |z_j|^{2\nu _j}
\cdot \bigwedge_{j=2}^{k} (dw_j\wedge m_j\, d\theta_j) \wedge \zeta\wedge \bar{\zeta},
\end{split}\end{equation*}

where the sign $(-1)^{(k-1)(n+1-k)}$ comes from moving the $(n+1-k)$-form $\zeta$ to the right; and the constant $(2\imath)^{k-1}(-1)^{\frac{1}{2}(k-1)(k-2)}$ comes from multiplying out the $1$-forms. Like before, we conclude that $t^{n}\Omega\wedge \bar{\Omega}$ extends to a smooth section of $\bigwedge^{2n}T^{*}_{\vert} A$, and vanishes on $U\cap \d A$ unless $\Delta_{J}$ is a maximal face of $\Delta_{\cS}$. In the latter case we have $(m_j,\nu_j)=(1,0)$ for all $j\in J$, so the above expression reads as

\begin{equation*}
t^{n}\Omega \wedge \bar{\Omega} = |c|^2 \cdot  (2\imath)^{n}(-1)^{\frac{1}{2}n(n-1)} \cdot (dw_2\wedge d\theta_2)\wedge\dots \wedge (dw_{n+1}\wedge d\theta_{n+1}).
\end{equation*}

which ends the proof of <ref>, <ref> and <ref>.  The remaining part <ref> is shown in the proof of <cit.>, let us recall the argument here for completeness.

Fix two maximal faces $\Delta_{I}$ and $\Delta_{J}$ of $\Delta_{\cS}$. We claim that $c(X_{I})=\pm c(X_J)$. Since the essential skeleton is a pseudomanifold <cit.>, we can assume that $\Delta_{I}$ and $\Delta_{J}$ meet along a common submaximal face $\Delta_{I'}$, see part (3) of the definition of pseudomanifold given in 4.1.2 loc. cit.

Say that $I'=\{2,\dots, n+1\}$, $I=I'\sqcup\{1\}$, $J=I'\sqcup\{n+2\}$. By Proposition <ref><ref>, the stratum $X_{I'}$ is isomorphic to $\P^1$, and $X_{I}$, $X_{J}$ are two points on $X_{I'}$. By Proposition <ref><ref> we have $\nu_{i}=0$ for $i\in \{1,\dots,n+2\}$, so in an adapted chart at a point $p\in X_{I'}$, the local formula (<ref>) defining $c$ reads as

\begin{equation*}
\Theta=c\cdot d\log z_2\wedge \dots \wedge d\log z_{n+1} \wedge \zeta,
\end{equation*}

where up to a sign we have $\zeta=d \log z_1$ if $\{p\}=X_{I}$, $\zeta=d\log z_{n+2}$ if $\{p\}=X_{J}$, and otherwise $\zeta=d z$ for a holomorphic coordinate $z$ on $X_{I'}$. It follows that the logarithmic form $c\cdot \zeta$ is the residue of $\Theta$ along $X_{I'}$; with poles at $X_{I}$ and $X_{J}$ whose residues are equal, respectively, $\pm c(X_{I})$ and $\pm c(X_{J})$. By the Poincaré residue theorem, those residues add up to $0$, so $c(X_{I})=\pm c(X_J)$, as claimed.

We can now prove the title result of this section. Recall that in Proposition <ref>, we have constructed a Lagrangian torus fibration of a region $(X_{z}^{q})_{\cS}^{\sm}\subseteq X_{z}$, which maps the generic region $(X_{z}^{q})_{\cS}^{\gen}$ to the union of maximal faces of $\Delta_{\cS}$.  The following result shows that this region is large.

Fix $q\in (0,1]$, and let $(X_{z}^{q})_{\cS}^{\gen}$ be as in Section <ref>. Then we have

\begin{equation*}
\lim_{|z|\rightarrow 0} \frac{\int_{(X_{z}^{q})_{\cS}^{\gen}} \vol^{\Omega}}{\int_{X_{z}} \vol^{\Omega}}=1.
\end{equation*}

Put $\theta=\frac{z}{|z|}$, $F_{\theta}=f_{A}^{-1}(0,\theta)$.
The sets $(X_{z}^{q})_{\cS}^{\gen}$ and $X_{z}$ are images of $A^{\gen}_{\cS,\theta}$ and $F_{\theta}$ by the flow (<ref>). By Lemma <ref> the form $t^{n}\vol^{\Omega}$ extends to a fiberwise form $\vol_{\new}^{\Omega}$ on $A$, whose restriction to $F_{\theta}$ is zero outside of $A_{\cS,\theta}^{\gen}$. Therefore, we have

\begin{equation*}
\lim_{|z|\rightarrow 0} \int_{(X_{z}^{q})_{\cS}^{\gen}} t^{n} \vol^{\Omega}=
\int_{A_{\cS,\theta}^{\gen}} \vol_{\new}^{\Omega} =
\int_{F_{\theta}} \vol_{\new}^{\Omega}=\lim_{|z|\rightarrow 0} \int_{X_{z}} t^{n} \vol^{\Omega},
\end{equation*}

which ends the proof.

§.§ Asymptotic Ricci flatness

Recall that on each smooth fiber $X_{z}$ of $f^{\circ}$, the cohomology class of $\omega_{X}$ is represented by a unique Ricci-flat form $\omega_{\CY}$, and this condition can be written equivalently as

\begin{equation*}
\tfrac{1}{n!}\omega_{\CY}^{n}=c\cdot \vol^{\Omega}_{\new},
\end{equation*}

for some positive function $c$ on the base $\D_{\delta}^{*}$, see formula (<ref>). The volume form on the right-hand side is rescaled so that the total volume on both sides equals $\int_{X_{z}}\frac{1}{n!}\omega_{X}^{n}$. To see that the appropriate rescaling is indeed of the same order as $\vol_{\new}^{\Omega}$, note that it should integrate to a nonzero number on all fibers in the A'Campo space, including the radius-zero ones. This is true for $\vol_{\new}^{\Omega}$ by Lemma <ref><ref>.

Alternatively, one can use <cit.>: indeed, in the notation of loc. cit. the rescaled volume form should be of order $((\log|q|)^{m} |q|^{2k})^{-1}\, \vol^{\Omega}$, where $q=f$ (so $-\log|q|=t^{-1}$), $m=n$ since $f^{\circ}$ is maximal, and $k=\min_{i}\nu_{i}$, which equals $0$ by Proposition <ref><ref>.

Proposition <ref> below, roughly speaking, asserts that our forms $\omega^{\gamma}$ give some approximation of $\omega_{\CY}$ in the generic region $(X_{z}^{q})^{\gen}_{\cS}$. Of course, due to some choices made in the definition of $\omega^{\gamma}$, we cannot expect an exact equality $\omega^{\gamma}=\omega_{\CY}$ at positive radius (nor off the generic region, see Remark <ref>).

Let $\gamma$ be an admissible path, see Definition <ref>. There is a continuous function $c_{+}\colon A^{\gamma}_{+}\cup (X_{0}^{\gamma})_{\cS}^{\gen}\to \R_{>0}$ such that on each fiber $X_{h}^{\gamma}$ and on $(X_{0}^{\gamma})^{\gen}_{\cS}$, we have we have an equality

\begin{equation}\label{eq:MA_weak}
\tfrac{1}{n!} (\omega_{h}^{\gamma})^{n}=c_{+}\cdot \vol^{\Omega}_{\new}
\end{equation}

and the restriction of $c_{+}$ to $(X_{0}^{\gamma})_{\cS}^{\gen}$ is a positive constant. This constant equals $\epsilon^n\cdot (n+1)\cdot |c_0|^{-2}$, where $\epsilon>0$ is a number fixed in Notation <ref>, and  $c_0\in \C^{*}$ is the number in Lemma <ref><ref>.

By Lemma <ref><ref>,<ref>  the form $\vol^{\Omega}_{\new}$ extends to $X_{0}^{\gamma}$ and restricts to a volume form on $(X_{0}^{\gamma})_{\cS}^{\gen}$, so we have a continuous function $c_{+}\colon A^{\gamma}_{+}\cup (X_{0}^{\gamma})_{\cS}^{\gen}\to \R$ satisfying fiberwise equality (<ref>). By Proposition <ref><ref>, $(\omega_{h}^{\gamma})^{n}$ is a volume form on each fiber $X_{h}^{\gamma}$ for $h>0$, so the function $c_{+}$ is positive there. It remains to prove (<ref>) at radius zero, i.e. prove that on $(X_{0}^{\gamma})_{\cS}^{\gen}$ we have

\begin{equation*}
\tfrac{1}{n!} (\omega_{0}^{\epsilon})^{n}=(\epsilon^n\cdot (n+1)\cdot |c_0|^{-2})\cdot \vol^{\Omega}_{\new}.
\end{equation*}

Recall that $(X_{0}^{\gamma})_{\cS}^{\gen}$ is the union  of pieces $\Int_{\d A} A_{I}^{\circ}\cap X_{0}^{\gamma}$ for all maximal faces $\Delta_{I}$ of the essential skeleton $\Delta_{\cS}$. Thus it is enough to show that the above equality holds fiberwise on each $A_{I}^{\circ}$.

Fix a maximal, essential face $\Delta_{I}$. By Lemma <ref><ref>, on $A_{I}^{\circ}$ we have $\hat{w}_i=w_i$, $\hat{\alpha}_i=d\theta_i$ for $i\in I$ and $\hat{w}_i=0$, $\hat{\alpha}_i=0$ for $i\neq I$. Thus $\omega^{\flat}=-\sum_{i\in I}dw_{i}\wedge d\theta_{i}$ by formula (<ref>). Since $\pi(A_{I}^{\circ})=X_{I}^{\circ}$ is a point, definition (<ref>) of $\omega_{q}^{\epsilon}$ shows that on $A_{I}^{\circ}$ we have $\omega_{0}^{\epsilon}=\epsilon \cdot \omega^{\flat}$, and therefore

\begin{equation*}
\tfrac{1}{n!}(\omega_{0}^{\epsilon})^{n}=(-\epsilon)^{n} \cdot \sum_{i\in I} \bigwedge_{j\in I\setminus \{i\}} (dw_{j}\wedge d\theta_{j})
\overset{\mbox{\tiny{\eqref{eq:volume_on_maximal_face}}}}{=}
(\epsilon^{n} \cdot (n+1)\cdot |c_0|^{-2})\cdot \vol^{\Omega}_{\new},
\end{equation*}

as needed.

The function $c_{+}$ does not extend to the remaining part $X_{0}^{\gamma}\setminus (X_{0}^{\gamma})_{\cS}^{\gen}$. Indeed, on $\d A\setminus A_{\cS}^{\gen}$ the form $\vol^{\Omega}_{\new}$ is zero by Lemma <ref>, while $\omega_{0}^{\gamma}$ has some contributions from pullbacks of forms from $X$, namely: the initial Kähler form $\omega_{X}$, the forms $\beta$ from Lemma <ref><ref>, and the forms $d\hat{\alpha}_i$, cf. Lemma <ref><ref>. In particular, on the subset $R_{i}^{\circ}\cap X_{0}^{\gamma}$, which can be identified with an open subset of $D_{i}\setminus \bigcup_{j\neq i} D_j$, the form $\frac{1}{n!}(\omega^{\gamma}_{0})^n$ restricts to the standard volume form $\frac{1}{n!}\omega_{X}^{n}$.

The following easy consequence of Proposition <ref> will be useful in the next section.

Fix $h>0$. Let $L_{h}\subseteq X_{h}^{\gamma}$ be a fiber of a Lagrangian torus fibration (<ref>), and let $\vol^{g}_{\new}$ be a Riemannian volume form induced by the metric $g_{h,\new}^{\gamma}$. Then we have $\vol^{g}_{\new}|_{L_h}=\sqrt{c_{+}}\cdot |\Omega_{\new}|_{L_{h}}|$.

Fix a point $x\in L_{h}$ and let $e_{1},\dots, e_{n}$ be an orthonormal basis of  $T_{x}L_{h}$. Since $L_{h}$ is Lagrangian with respect to a $J$-compatible form $h\cdot \omega_{h}^{\gamma}$, the collection $e_1,\dots,e_{n}$, $Je_1,\dots, Je_n$ is an orthonormal basis for $T_{x}X_{h}^{\gamma}$ with respect to the metric $g_{h,\new}^{\gamma}=h\cdot \omega_{h}^{\gamma}(\sdot,J\sdot)$ see formulas (1.2) and (1.2)' in [17]. Substituting it to the formula (<ref>) multiplied by $t^{n}$, we get $1=c_{+}\cdot |\Omega_{\new}(e_1,\dots, e_n)|^2$, as needed.

§.§ Generic Lagrangian tori are asymptotically special

We recall the notion of special Lagrangian submanifolds, introduced in [17]. Let $Y$ be a Calabi–Yau manifold, with a holomorphic volume form $\Omega_{Y}$ and a Ricci-flat Kähler form $\omega_{\CY}$, see equation (<ref>). A Lagrangian submanifold $L$ of $Y$ is special of phase $\varpi\in \S^{1}$ if  $\Im (\varpi\cdot \Omega_{Y})|_{L}=0$. Condition (<ref>) guarantees that special Lagrangian submanifolds are calibrated with respect to $\Re (\varpi\cdot \Omega_{Y})$, see <cit.> in particular they minimize volume in their homology class, see Proposition 7.1 loc. cit. The specialty condition makes the geometry of Lagrangians much more rigid, see <cit.>; and indeed the Lagrangian tori in SYZ conjecture are expected to be special, see paragraph <ref> in the introduction. In this section, we prove that for our tori specialty holds asymptotically, in the generic region, as claimed in Theorem <ref><ref>.

To give a precise statement in  Proposition <ref> below, we introduce some notation. Fix an admissible path $\gamma$, see Definition <ref>, and a point  $b\in E_{\cS}$. Consider a family of Lagrangian torus fibers $L_{h}\subseteq X_{h}^{\gamma}$ over $b$. It follows from Lemma <ref> that for $h>0$, the form  $\Omega$ does not vanish on $L_{h}$, so at each point it is a nonzero complex multiple of a volume form on $L_{h}$. This allows us to pose the following definition.

Let $\vol_{h}$ be a volume form on $L_{h}$. Let $c_{h}\colon L_{h}\to \C^{*}$ be the smooth function satisfying

\begin{equation*}
\vol_{h}=c_{h} \cdot \Omega|_{L_h}.
\end{equation*}

We define the phase map as $\varpi_{h}\de \frac{c_{h}}{|c_{h}|}\colon L_{h}\to \S^{1}$.

Clearly, this definition does not depend on the choice of the volume form $\vol_{h}$. We let $L^{\gamma}\subseteq A^{\gamma}$ be the union of fibers $L_{h}$ for all $h\geq 0$; and let $L_{+}=L\setminus L_{0}$, so the fiberwise phase map yields a smooth map $\varpi\colon L_{+}\to \S^1$. We can now state the main result of this section, which implies Theorem <ref><ref>.

Let $\gamma$ be a special admissible path, see Definition <ref>. Let $b$ be an interior point of a maximal, essential face $\Delta_{I}$, and let $L\subseteq A^{\gamma}$ be the family of fibers of the Lagrangian fibration (<ref>) over $b$. Then the phase map $\varpi\colon L_{+}\to\S^1$ introduced in Definition <ref> extends to a continuous map $L\to \S^1$, whose value on $L_0$ is a constant

\begin{equation}\label{eq:asymptotic_specialty}
\varpi|_{L_0}=\pm\, \imath^{n}\cdot \frac{c_0}{|c_0|},
\end{equation}

where $c_0\in \C^{*}$ is the number in formula (<ref>), and the sign depends only on  the face $\Delta_{I}$.

In the generic region $(X_{h}^{\gamma})_{\cS}^{\gen}$ our form satisfies the asymptotic Ricci flatness condition (<ref>), so the asymptotically special Lagrangian tori $L_h$ should asymptotically minimize volume there. And indeed, we will see that the arguments of <cit.> yield the following corollary.

Let $\gamma$ be a special admissible path, let $b$ be an interior point of a maximal face of $\Delta_{\cS}$, and for $h>0$ let $L_{h}\subseteq (X_{h}^{\gamma})_{\cS}^{\gen}$ be the fiber over $b$ of the Lagrangian torus fibration (<ref>). Let $\vol^{g}_{\new}$ be a Riemannian volume form on $L_{h}$ associated to the metric $g_{\new}^{\gamma}$. Then we have a convergence

\begin{equation}\label{eq:volume_minimizing}
\lim_{h\rightarrow 0}\ \frac{\int_{L_h}\vol^{g}_{\new}}{\min_{\Gamma\in [L_h]} \int_{\Gamma}\vol^{g}_{\new}}=1,
\end{equation}

where $[L_{h}]$ is the class of $L_{h}$ in $H_{n}((X_{h}^{\gamma})_{\cS}^{\gen};\R)$.

The remaining part of this section is devoted to the proof of Proposition <ref> and Corollary <ref>.

We work in $(X_{h}^{\gamma})_{I}^{\gen}$ for a fixed maximal essential face $\Delta_{I}$, say $I=\{1,\dots, n+1\}$. By Lemma <ref><ref>, there is an adapted chart $U_{X}$ around the point $X_{I}^{\circ}$ such that for each $i\in I$, the distance function $\hat{r}_{i}$ equals the radial coordinate $r_{i}$ of $U_{X}$. Thus on $U\de \pi^{-1}(U_{X})$ the functions $\hat{w}_i$, $\hat{v}_i$ and the form $\hat{\alpha}_i$ introduced in formula (<ref>) for each $i\in I$ are equal to the functions $w_i$, $v_i$ and the form $d\theta_{i}$ introduced for $U_X$ in formula (<ref>). Moreover, for $i\not\in I$ the forms $d\hat{v}_i$, $d\hat{w}_i$ and $\hat{\alpha}_{i}$ are identically zero. As a consequence, on $U$  formulas  (<ref>) and (<ref>) read as

\begin{equation}\label{eq:omega-easy}
\omega^{\sharp}=\sum_{i\in I} dv_{i}\wedge d\theta_{i}\quad\mbox{and}\quad\omega^{\flat}=\sum_{i\in I} dw_{i}\wedge d\theta_{i}.
\end{equation}

Since the radius-zero  torus $L_{0}$ is a compact subset of $U\cap \Int_{\d A}A_{I}^{\circ}$, we can and do work in a neighborhood $V$ of $L_0$ in $U$ where each function $w_{i}$ is bounded away from $0$. As a consequence of this boundedness, we have the following convenient coordinate system.

The smooth map $(t,\theta;w_1,\dots,w_n,\theta_{1},\dots,\theta_{n})$ endows $V$ with a new smooth structure, fiberwise compatible with the one inherited from $A$. This smooth structure has the following properties.

*  For every $i\in I$, the functions $v_{i}$, $w_{i}$ and $\theta_i$ are smooth.

*  The map $\pi\colon V\to X$ is smooth.

*   For every $q\in [0,1]$, the form $\omega_{q}^{\epsilon}$ is smooth, closed and fiberwise symplectic.

*  The coordinate vector field $\frac{\d}{\d t}$ satisfies $dw_{i}(\frac{\d}{\d t})=0$ and $d\theta_{i}(\frac{\d}{\d t})=0$ for all $i\in I$.

Let $V=\bigcup_{i\in I} V_{i}$ be the covering of $V$ by charts (<ref>). Recall that on $V$ we have $\theta=\sum_{i}\theta_{i}$ and $\hat{v}_{i}=v_{i}$ for all $i\in I$, so the coordinate system  (<ref>) on $V_i$ is equivalent to $(g,\theta,(v_{j})_{j\in I_i},(\theta_{j})_{j\in I_i})$, where $I_i=I\setminus \{i\}$. Since $t=\exp(1-g^{-1})$ is a reparametrization of $g$, it is clear that the map $(t,\theta,(v_{j})_{j\in I_i},(\theta_{j})_{j\in I_i})$ endows $V$ with a smooth structure, fiberwise compatible with the one from $A$.

We claim that it is equivalent to the one in the statement of the lemma. By definition (<ref>) of $v_j$, we have $v_{j}=tw_{j}^{-1}-\eta(w_{j})$, so $dv_{j}=w_{j}^{-1}\, dt-(tw_{j}^{-2}+\eta'(w_j))\, dw_{j}$. The coefficient $tw_{j}^{-2}+\eta'(w_j)$ is smooth and positive since $w_{j}>0$, so we can smoothly change each coordinate $v_j$ for $w_{j}$. Eventually, since $\sum_{j\in I}w_j=1$ and $\sum_{j\in I}\theta_{j}=\theta$, we can change $w_{n+1}$, $\theta_{n+1}$ for $w_{i}$, $\theta_i$, as needed.

Thus the map $(t,\theta;w_1,\dots,w_n,\theta_{1},\dots,\theta_{n})$ gives a smooth structure on the whole $V$, equivalent to $(t,\theta,(v_{j})_{j\in I_i},(\theta_{j})_{j\in I_i})$ for each $i\in I$. This implies property <ref>. We now prove the remaining ones.

<ref> It is enough to show that the coordinate functions $r_{i}\cos(2\pi \theta_i)$, $r_{i}\sin(2\pi \theta_{i})$ are smooth for each $i\in I$.  To see this, recall that $\theta_{i}$ is smooth by <ref>; and  $r_{i}=\exp(-t_{i}^{-1})=\exp(w_{i}t^{-1})$ is smooth because $w_{i}$ is positive and smooth by <ref>.

<ref> Part <ref> and formula (<ref>) show that the forms $\omega^{\flat}$ and $\omega^{\sharp}$ are smooth and closed. By <ref>, the form $\pi^{*}\omega_{X}$ is smooth and closed, too, hence so is $\omega_{q}^{\epsilon}$. Since the new smooth structure on $V$ is fiberwise compatible with the one in $A$, fiberwise non-degeneracy of $\omega_{q}^{\epsilon}$ follows from Proposition <ref>.

<ref> For $i\neq n+1$, this property follows from the definition of the coordinate chart; for $i=n+1$ we use relations $w_{n+1}=1-\sum_{i=1}^{n}w_i$ and $\theta_{n+1}=\theta-\sum_{i=1}^{n}\theta_{i}$.

Let $\cQ\subseteq \cC^{\infty}(V)$ be the ideal of functions $p$ such that for every $k\in \Z$, $t^{k}p$ extends to a smooth function on $V$, with respect to the smooth structure from Lemma <ref>. The following hold.

*  The ideal $\cQ$ is closed under taking partial derivatives.

*  For every smooth form $\beta$ on $U_{X}$ we have $\pi^{*}\beta(\frac{\d}{\d t})\in \cQ$.

*  Let $\nu_{t}^{0}$ be the symplectic lift, with respect to the form $\omega_{0}^{\epsilon}$ and the submersion $(t,\theta)$, of the unit radial vector field on $\C_{\log}$. Then the vector field $\nu_{t}^{0}-\frac{\d}{\d t}$ has coefficients in $\cQ$.

*  Let $\Psi^{0}\colon \d V\times [0,\rho) \to V$ be the flow of $\nu_{t}^{0}$, where $\d V\de V\cap \d A$. Then for every integer $k\geq 0$ there is a smooth map $\psi_{k}\colon \d V\times [0,\rho) \to V$ such that in coordinates of Lemma <ref> we have

\begin{equation*}
\Psi_{h}^0(0,\bar{w};h)=(h,\bar{w})+h^k\cdot \psi_{k}(0,\bar{w};h),
\end{equation*}

where we write $\bar{w}\de (\theta,w_1,\dots,w_n,\theta_1,\dots,\theta_n)$ for the remaining coordinates of a point in $V$.

<ref> Let $\d$ be one of the partial derivatives. For every $p∈$ and $k∈$ we have $t^k·p̣=(̣t^k p)-kt^k-1 p·ṭ$, which is a smooth function by definition of $$, so $p̣∈$, as needed.

\ref{item:Q-beta} Since $$ is an ideal, it is sufficient to check \ref{item:Q-beta} for the coordinate forms $d(r_icos(2πθ_i))$ and $d(r_isin(2πθ_i))$. By \ref{item:Q-closed} it is enough to check that $r_i∈$. As in the proof of Lemma \ref{lem:new-coordinates}\ref{item:V-pi}, we write $r_i=exp(-w_it^-1)$ and use the fact that $w_i$ is bounded from below by a positive number.

\ref{item:Q-lift} Since the vector field $/ṭ-ν_t^0$ is vertical, and the form $ω_0^ϵ$ is fiberwise non-degenerate, it is enough to prove that for every vertical vector field $ν$ we have  $ω_0^ϵ(/ṭ-ν_t^0,ν)∈$.

Since $ν_t^0$ is orthogonal to the fibers, we have $ω_0^ϵ(/ṭ-ν_t^0,ν)=ω_0^ϵ(/ṭ,ν)$. Recall from formula \eqref{eq:omega-s} that $ω_0^ϵ=π^*ω_X+ϵω^♭$. Writing $ω_X=∑_iβ_i∧β_i'$ for some smooth $1$-forms $β_i,β_i'$ on $U_X$, we see from \ref{item:Q-beta} that $π^*ω_X(/ṭ,ν)∈$. In turn, formula \eqref{eq:omega-easy} and Lemma \ref{lem:new-coordinates}\ref{item:V-dt} imply that $ω^♭(/ṭ,ν)=0$, as needed.

\ref{item:Q-flow} By \ref{item:Q-lift} we have $ν_t^0-/ṭ=t^kν$ for some smooth vertical vector field $ν$. We treat $t^kν$ as a time-dependent vector field on $Ṿ$, and look at its integral curves. A point at time $h$ has coordinates of the form $∫_0^hs^ka(s)  ds$ for some smooth, time-dependent function $a$, namely a coordinate of $ν$. The result follows since the first $k-1$ derivatives with respect to $h$ of such an integral vanish at $h=0$.
\end{proof}

Consider a product $V×[0,1]$, where $V$ is equipped with the  smooth structure from Lemma \ref{lem:new-coordinates}, and let $q$ be the projection onto $[0,1]$. The family of forms $(ω^ϵ_q)$ yields a smooth form on $V×[0,1]$, which is fiberwise symplectic with respect to the submersion $(t,θ,q)$. Let $ν_t$ be the symplectic lift of the radial vector field on $_log$, and let $ΨṾ×[0,1]×[0,δ̃)→V×[0,1]$ be the flow of $ν_t$. Here $δ̃=exp(-δ^-1)>0$ is a small positive number such that the flow $Ψ_h$ is defined for all times $h∈[0,δ̃)$. Let $Ψ^q$ be the restriction of $Ψ$ to the slice $Ạ×{q}×[0,δ̃)$. Note that for $q>0$, this flow agrees with $Ψ^q$ introduced in Notation \ref{not:admissible}, once we identify the slice  $Ạ×{q}×[0,δ̃)$ with $Ạ×[0,δ̃)$. In turn, the flow $Ψ^0$ is described in Lemma \ref{lem:Q}\ref{item:Q-flow}.
\smallskip

Now, we fix a special admissible path $γ$, see Definition \ref{def:special_path}, and let $[0,ρ)∋h↦(h,θ(h),q(h))∈_log×[0,1]$ be its normal parametrization. We put $ΓṾ×{(q(h),h): h∈[0,ρ)}$. Since $γ$ is special, $Γ$ is a $^2$ submanifold of $Ṿ×[0,1]×[0,ρ)$.

\begin{lema}\label{lem:piecewise_Lie}
Define a smooth $1$-form $\mathfrak{a}_{j}$ on $\d V\times [0,1]\times [0,\rho)$ piecewise, as
$\frac{1}{h}((\Psi_{h}^{q})^{*}dw_{j}-dw_{j})$ for $h>0$ and as the Lie derivative $\cL_{\nu_{t}}(\frac{\d}{\d w_j})$ for $h=0$; where $h$ is the projection to the last coordinate. Let $\mathfrak{a}_{j}^{\gamma}=\mathfrak{a}_{j}|_{\Gamma}$. Then the form $h^{-2}\cdot \mathfrak{a}_{j}^{\gamma}$ extends to a continuous form on $\Gamma$.%; in particular $\mathfrak{a}_{j}^{\gamma}$ vanishes at $h=0$.
\end{lema}
\begin{proof}
We need to prove that $\mathfrak{a}_{j}^{\gamma}$ and its first derivative with respect to $h$ vanishes at $h=0$. Since the path $\gamma$ is admissible, the value $h=0$ is attained precisely on the slice $\d V\times \{(0,0)\}$. Thus $\mathfrak{a}_{j}^{\gamma}|_{{h=0}}=\mathfrak{a}_{j}|_{(h,q)=(0,0)}=0$ by Lemma \ref{lem:Q}\ref{item:Q-flow}. By chain rule
\begin{equation*}
\frac{\d \mathfrak{a}_j^{\gamma}}{\d h}\Big|_{h=0} =\frac{\d\mathfrak{a}_j}{\d h}\Big|_{(h,q)=(0,0)}+\frac{\d\mathfrak{a}_j}{\d q}\Big|_{(h,q)=(0,0)} \cdot q'(0)=0
\end{equation*}
since $\frac{\d\mathfrak{a}_j}{\d h}|_{(h,q)=(0,0)}=0$ by Lemma \ref{lem:Q}\ref{item:Q-flow} and  $q'(0)=0$ by specialty of $\gamma$, see Definition \ref{def:special_path}.
\end{proof}

Recall that $L_h⊆X_h^γ$ is the fiber of the $ω_h^γ$-Lagrangian fibration \eqref{eq:Lagranian-at-positive-radius} over the fixed point $b∈E_$. Thus in our notation, $L_h×{q(h)}=Ψ_h^q(h)(L_0×{q(h)})$.

\begin{lema}\label{lem:Omega-factor}
Let $\Omega_{j}\de dw_{j}-t^2\imath d\theta_{j}$ be a factor in formula \eqref{eq:Omega_new}. There is a continuous, complex valued $1$-form $\beta_{j}$ on $\Gamma$ such that for every small $h>0$ we have the equality of restrictions
\begin{equation}\label{eq:Omega_factor}
(\Psi_{h}^{q(h)})^{*}(\Omega_{j}|_{L_{h}})= h^2 \cdot (h\cdot \beta_{j}- \imath d\theta_j)|_{L_0}.
\end{equation}
\end{lema}
\begin{proof}
By definition of the Lagrangian torus fibration \eqref{eq:Lagrangian}, see Section \ref{sec:expanded_construction}, its fiber $L_0$ is a level set of $(w_{1},\dots,w_{n+1})$, so $(dw_{j})|_{L_{0}}=0$. Thus on $L_0$ we have the equality of restrictions
\begin{equation*}
\begin{split}
h^{-2}\cdot (\Psi_{h}^{q(h)})^{*}\Omega_{j} & =
h\cdot \left( h^{-2}\cdot \frac{(\Psi_{h}^{q(h)})^{*}dw_{j}-dw_{j}}{h}
\imath \cdot \frac{(\Psi_{h}^{q(h)})^{*}d\theta_{j}-d\theta_{j}}{h}\right)
\imath d\theta_{j} = \\
& = h\left( h^{-2}\mathfrak{a}_j^{\gamma}+\imath \mathfrak{b}_j^{\gamma}\right)-\imath d\theta_{j}.
\end{split}
\end{equation*}
where $\mathfrak{a}_{j}^{\gamma}$ is the smooth form defined in Lemma \ref{lem:piecewise_Lie}; and $\mathfrak{b}_{j}^{\gamma}$ is defined analogously, with $\theta_{j}$ instead of $w_j$. Lemma \ref{lem:piecewise_Lie} shows that $\beta_{j}\de  h^{-2}\mathfrak{a}_j^{\gamma}+\imath \mathfrak{b}_j^{\gamma}$ extends to a continuous form on $\Gamma$, as needed.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:special}]
Let $\vol_{0}$ be a volume form on $L_0$ given by  $\bigwedge_{i=1}^{n}d\theta_{i}$. Taking a wedge product in formula \eqref{eq:Omega_factor} and applying Lemma \ref{lem:volume-extends}\ref{item:vol_formulas}, we get the following equality on $L_0$:
\begin{equation*}
(\Psi_{h}^{q(h)})^{*}(\Omega_{\new}|_{L_h})= \pm (c\circ \Psi_{h}^{q(h)}) \cdot h^{2n}\cdot ((-\imath)^{n}\vol_{0}+h\cdot \beta),
\end{equation*}
for some continuous, complex-valued $n$-form $\beta$ on $L_0$. Write $\beta=b\cdot  \vol_0$, for some continuous, complex-valued function $b$. Let $c_{h}=c\circ \Psi_{h}^{q(h)}$, so $c_{h}$ is a complex-valued function approaching a constant $c_0$ as $h\rightarrow 0$, see Lemma \ref{lem:volume-extends}\ref{item:c_0}. Now for small $h>0$, the phase map $\varpi$ from Definition \ref{def:phase} equals
\begin{equation*}
\varpi=\frac{\pm c_h h^{2n}((-\imath)^{n}+hb)}{|c_h h^{2n} ((-\imath)^{n}+hb)|}=\pm \imath^{n}\cdot \frac{c_{h}}{|c_h|} \cdot \tilde{\varpi},
\quad\mbox{where}\quad
\tilde{\varpi}=\frac{1-h\cdot b\imath^{n}}{|1-h\cdot b\imath^{n}|}\xrightarrow{\ \small{h\rightarrow 0}\ }1,
\end{equation*}
so $\varpi$ extends to a continuous map $L\to \S^{1}$, whose value at $L_{0}$ equals $\pm \imath^{n}\cdot  \frac{c_{0}}{|c_0|}$, as claimed.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor:volume_minimizing}]
We follow closely the argument of \cite[\sec III.1]{HL}, proving that the special Lagrangians of phase $\varpi_0$ are $\Re(\varpi_0 \Omega)$-calibrated, hence volume minimizing. The only difference in our setting is that some bounds hold only up to asymptotically small errors.

We work in $(X_{h}^{\gamma})^{\gen}_{I}$ for a fixed maximal essential face $\Delta_{I}$. Dividing the chosen holomorphic form $\Theta$ by the constant \eqref{eq:asymptotic_specialty}, we can assume that the value of the phase map $\varpi$ on $L_0$ is $1$. Lemma \ref{lem:c+} gives
\begin{equation}\label{eq:ca2}
\tfrac{1}{\sqrt{c_{+}}}\cdot \vol^{g}_{\new}|_{L_h}=\Re \Omega_{\new}|_{L_h} + \beta,\quad
\mbox{where } \beta\rightarrow 0\mbox{ as } h\rightarrow 0.
\end{equation}
Now, we argue as in \cite[Lemma 1.9]{HL}. Fix $h>0$, $x\in L_{h}$, and let $e_1,\dots, e_n$ be an orthonormal basis of $T_{x}L_{h}$. Let $H$ be another $n$-dimensional subspace of $T_{x}X_{h}^{\gamma}$, let  $\epsilon_{1},\dots, \epsilon_{n}$ be an orthonormal basis of $H$, and let $M$ be a linear automorphism of $T_{x}X_{h}^{\gamma}$ defined by $Me_i=\epsilon_i$, $MJe_i=J\epsilon_i$. Note that $M$ is $\C$-linear, so we can compute both the complex and real determinant of $M$, and they are related by $|\det_{\C}M|^2=\det_{\R}M$. Using Lemma \ref{lem:c+}, we compute
\begin{equation*}
(\Re \Omega_{\new}(\epsilon_1,\dots, \epsilon_n))^2\leq |\Omega_{\new}(\epsilon_1,\dots, \epsilon_n)|^2=|\det\nolimits_{\C}M|^2 \cdot |\Omega_{\new}(e_1,\dots, e_n)|^2
\det\nolimits_{\R} M\cdot c_{+}^{-1} \leq c_{+}^{-1}
\end{equation*}
by Hadamard inequality and the fact that the vectors $\epsilon_1,\dots,\epsilon_n$ have length one. Thus on $n$-dimensional subspaces of $T_{x}X_{h}^{\gamma}$ we have an inequality
\begin{equation}\label{eq:calibration}
\Re \Omega_{\new} \leq \tfrac{1}{\sqrt{c_{+}}}\cdot \vol^{g}_{\new}.
\end{equation}
On $T_{x}L_{h}$, the above inequality becomes the \enquote{asymptotic} equality \eqref{eq:ca2}, cf.\ \cite[Theorem 1.10]{HL}.

Now fix $\Gamma\in [L_{h}]$. The form $\Omega$, hence $\Re \Omega$ is closed. Its rescaled version $\Re\Omega_{\new}$ is therefore fiberwise closed, so Stokes theorem gives $\int_{L_h} \Re\Omega_{\new}=\int_{\Gamma} \Re \Omega_{\new}$. We conclude that
\begin{equation*}
\int_{L_h} \tfrac{1}{\sqrt{c_{+}}} \cdot \vol^{g}_{\new}
\overset{\mbox{\tiny{\eqref{eq:ca2}}}}{=}
\int_{L_h} \Re \Omega_{\new}
+ \int_{L_h} \beta
\int_{\Gamma} \Re \Omega_{\new}
+ \int_{L_h} \beta
\overset{\mbox{\tiny{\eqref{eq:calibration}}}}{\leq}
\int_{\Gamma} \tfrac{1}{\sqrt{c_{+}}} \cdot \vol^{g}_{\new}
\int_{L_h} \beta.
\end{equation*}
The required convergence  \eqref{eq:volume_minimizing} follows since $\lim_{h\rightarrow 0}\beta=0$ and $\lim_{h\rightarrow 0} c_+$ is a positive constant.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theo:CY_af}]
Parts \ref{item:CY_Lagrangian_volume} and \ref{item:assflat} are proved in Propositions \ref{prop:volume} and  \ref{prop:MA_weak}. To see the first statement of \ref{item:CY_asymptotic specialty}, take for $\varpi \in \S^1$ the inverse of the constant \eqref{eq:asymptotic_specialty} and apply Proposition \ref{prop:special}: note that the path with $|z(h)|=h$ is special by Example \ref{ex:special}. The second statement of \ref{item:CY_asymptotic specialty} is proved in Corollary \ref{cor:volume_minimizing}. Part \ref{item:CY_Maslov} follows from \ref{item:CY_asymptotic specialty} since the Maslov class is locally constant and the base $E_{\cS}$ of the Lagrangian fibration $\ll_{h}$ is connected.
\end{proof}

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\end{document}