Fibrations by Lagrangian tori]Fibrations by Lagrangian tori
for maximal Calabi–Yau degenerations and beyond

Javier Fernández de Bobadilla
Javier Fernández de Bobadilla:
(1) IKERBASQUE, Basque Foundation for Science, Euskadi Plaza, 5, 48009 Bilbao, Basque Country, Spain;
(2) BCAM,  Basque Center for Applied Mathematics, Mazarredo 14, 48009 Bilbao, Basque Country, Spain;
(3) Academic Colaborator at UPV/EHU.

J.F.B. was supported by the Basque Government through the BERC 2022-2025 program, by the Ministry of Science and Innovation: BCAM Severo Ochoa accreditation CEX2021-001142-S/MICIN/AEI/10.13039/501100011033, and by the Spanish Ministry of Science, Innovation and Universities, project reference PID2020-114750GB-C33. J.F.B. and T.P. were supported by the
semester Singularity Theory and Low Dimensional Topology at Rényi Institute, Budapest.

[2020]Primary: 14D06; Secondary: 14J32, 32Q25, 53C23, 53D12

Tomasz Pełka
Tomasz Pełka: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

We introduce a general technique to construct Lagrangian torus fibrations in degenerations of Kähler manifolds. We show that such torus fibrations naturally occur at the boundary of the A'Campo space.
This space extends a degeneration over a punctured disc to a hybrid space involving its real oriented blowup and the dual complex of the special fiber; equipped with an exact modification of a given Kähler form which becomes fiberwise symplectic at the boundary. Therefore, it allows to move Lagrangian tori from the boundary  using the symplectic connection. In the setting of maximal Calabi–Yau degenerations, we prove that our tori have some asymptotic properties expected from the (conjectural) Strominger–Yau–Zaslow fibration.

§ INTRODUCTION

The Kontsevich–Soibelman conjecture [22] predicts existence of certain special Lagrangian torus fibrations in maximal Calabi–Yau degenerations. In this article, we prove an asymptotic version of this conjecture. In order to state our main results, we briefly recall the relevant definitions; for more details see Section <ref> or e.g.  [1, 12, 25].

A maximal Calabi–Yau degeneration is a complex projective morphism $f^\circ:X^\circ\to\D^*$, where $\D^{*}$ is a punctured disc, such that the fibers $X_z\de (f^\circ)^{-1}(z)$ are Calabi–Yau $n$-folds, and the monodromy has Jordan blocks of maximal size $n+1$. Applying the semistable reduction theorem [20] after a finite base change we can extend $f^{\circ}$ to a model $f\colon X\to\D$ such that $X$ is smooth, the central fiber $D=\sum_{i} D_i$ is a reduced snc divisor, and $f|_{X\setminus D}=f^{\circ}$. Since each fiber of $f^{\circ}$ is Calabi–Yau, there exists a holomorphic section $\Omega\in\Gamma(X,K_{X/\D})$ with zeros only at the central fiber. Multiplying $\Omega$ by a holomorphic function on the base $\D^{*}$, we can assume that the divisor of $\Omega$ is $\sum_{i}a_iD_i$, with $\mathrm{min}_{i} a_{i}=0$. We say that a component $D_i$ is essential if $a_i=0$. The essential skeleton $\Delta_{\cS}$ is a subcomplex of the dual complex of $D$ spanned by vertices corresponding to essential divisors. It is a closed $n$-dimensional  pseudomanifold [28], so the complement $\Delta_\cS^\circ$ of the union of its codimension $2$ faces is a topological manifold. This implies that any $(n-1)$-dimensional face $F^{n-1}$ of $\Delta_{\cS}^\circ$ is the intersection of exactly two $n$-dimensional faces $F^n_0$, $F^n_1$ of $\Delta_{\cS}^\circ$. In Section <ref> we construct from $\Delta_{\cS}^{\circ}$ the expanded essential skeleton $E_{\cS}$  by replacing any $(n-1)$-dimensional face $F^{n-1}$ by $F^{n-1}\times [0,1]$ and identifying $F^{n-1}\times\{i\}$ with the subset $F^{n-1}\subseteq F^n_i$ for each $i\in \{0,1\}$; see Figure <ref> for an example. Such $E_{\cS}$ is a smooth manifold with boundary and corners, whose interior is homeomorphic to $\Delta_{\cS}^{\circ}$.

Let $t\de -1/\log|f|$. It is known, cf. <cit.> or Section <ref>, that $\mathrm{vol}^\Omega_{\new}\de t^n(\frac{\imath}{2})^n(-1)^{\frac{n(n-1)}{2}}\Omega\wedge\overline{\Omega}$ restricts to a volume form on each $X_z$ such that the limit of the volume of $X_z$ is finite as $z\rightarrow 0$. Let $\omega_X$ be any Kähler form in $X$. Yau's proof of the Calabi conjecture [34] implies that there exists a family of potentials $\phi_z^{\CY}\colon X_z\to\RR$ such that the Calabi–Yau metric $\omega_{X_z}^{\CY}\de \omega_X|_{X_z}+dd^c\phi_z^{\CY}$ is Ricci flat for all $z$. Ricci flatness is equivalent to the existence of a function $c\colon \D^*\to\RR$ such that

\begin{equation}
\label{eq:ricciflatintro}
\tfrac{1}{n!}(\omega_{X_z}^{\CY})^n=c(z)\cdot \vol^\Omega_{\new}|_{X_z}.
\end{equation}

The rescaling in the above definition of $\vol^{\Omega}_{\new}$ ensures that $\lim_{z\rightarrow 0} c(z)$ exists and is  nonzero.

Motivated by Mirror Symmetry, Strominger, Yau and Zaslow [33] predicted that Calabi–Yau $3$-folds should admit $\omega_{X_z}^{\CY}$-Lagrangian fibrations by tori which are special with respect to $\Omega$, cf. [12]. A Lagrangian submanifold $L$ is special of phase $\varpi\in\S^1$ if the imaginary part $\Im(\varpi \Omega)|_{L}$ vanishes. It was realized by several authors, see e.g. [15, 22], that such tori fibrations are an emergent feature of a maximal Calabi–Yau degeneration, in the sense that they should appear on $X_z$ for $|z|$ sufficiently small. More precise versions of the problem (e.g. taking into account inevitable singularities of some tori) were formulated e.g. in [13, 19, 23], with strong evidence provided by various constructions in particular contexts, such as [30, 7, 14, 31, 9, 27]. We do not attempt to give a complete list here, but rather refer the reader to surveys [2, 8, 25] or <cit.>.

This article is mostly inspired by the formulation due to Kontsevich and Soibelman <cit.>. In short, the main aspects of their prediction are as follows; for details see Conjectures 1 and 2 loc. cit.

Gromov–Hausdorff limit On each fiber $X_z$, consider the metric induced by the Calabi–Yau form $\omega_{X_z}^{\CY}$, rescaled so that the diameter of $X_{z}$ is independent of $z$. Then the Gromov–Hausdorff limit of $X_{z}$  as $z\rightarrow 0$ is a metric space $\Delta$ containing a smooth affine $n$-manifold $\Delta^{\circ}$ whose complement in $\Delta$ has Hausdorff codimension at most $2$. Moreover, the Riemannian metric on $\Delta^{\circ}$ has a potential satisfying the real Monge-Ampère equation, i.e. it is locally given by a matrix $\operatorname{Hess}(K)$ for some function $K$ of the affine coordinates; such that $\det\operatorname{Hess}(K)$ is constant.
Special Lagrangian fibration There is $\delta>0$ such that for any $z\in\D_{\delta}^*$, there is a subset  $X^{\sm}_z\subseteq X_z$ and a fibration $X^{\sm}_z\to \Delta^\circ$ by special Lagrangian tori. Moreover, the pairs  $(X_z,X_z\setminus X^{\sm}_z)$ equipped with the rescaled Calabi–Yau metric as in <ref> converge in the Gromov–Hausdorff sense to $(\Delta,\Delta\setminus\Delta^\circ)$.

It is expected that as $z\rightarrow 0$, the volume of the region $X_z^{\sm}$  asymptotically fills the whole volume of $X_z$ <cit.>. Here, the volume is computed with respect to $\mathrm{vol}^\Omega_{\new}|_{X_z}$ or equivalently, by Ricci-flatness condition (<ref>), with respect to $\frac{1}{n!}(\omega^{\CY}_{X_z})^n$.

In the non-archimedean setting, Nicaise, Xu and Yu [29] constructed fibrations by tori over $\Delta_{\cS}^{\circ}$, i.e. over the complement of the codimension $2$ faces of the essential skeleton $\Delta_{\cS}$. In turn, for the Fermat family, Li [24] constructed special Lagrangian fibrations over codimension $0$ faces of $\Delta_{\cS}$, cf. [16]. Given these insights it is possible to expect that the pair $(\Delta,\Delta^\circ)$ predicted by Kontsevich and Soibelman may coincide with $(\Delta_{\cS},\Delta_{\cS}^\circ)$. This is the second source of inspiration for us.

In this article we introduce a general technique to produce fibrations by Lagrangian tori. We summarize it in Theorem <ref>. In the setting of maximal Calabi–Yau degenerations, this theorem specifies to Theorem <ref> below, and  with little additional arguments given in Section <ref>, it implies the subsequent Theorem <ref>.  The conjunction of Theorems <ref> and <ref> provides an asymptotic version of the Kontsevich and Soibelman prediction in the general case of maximal Calabi-Yau degenerations.

We denote by $\C_{\log}\de [0,\infty)\times \S^{1}$ the real oriented blowup of $\C$ at the origin, and by $\D_{\delta,\log}\subseteq \C_{\log}$ the preimage of $\D_{\delta}$ in $\C_{\log}$.

Let $f^{\circ}\colon X^{\circ}\to \C^{*}$ be a maximal Calabi–Yau degeneration admitting a semi-stable model. There is a model $f\colon X\to \D_{\delta}$ of $f^{\circ}$, with a family of holomorphic volume forms $\Omega$ as above, such that the following holds. Let $\omega_X$ be any Kähler form on $X$. There exists a family of potentials $\phi_q\colon X^{\circ}\to\RR$, smoothly depending on a parameter $q\in [0,1]$, such that the family of $2$-forms $\omega_{q}\de \omega_X+dd^c\phi_q$ satisfies the following properties.

Kähler potential For every $z\in \D_{\delta}^{*}$ and every $q\in [0,1]$, the restriction $\omega_{q}|_{X_{z}}$ is Kähler.
Lagrangian fibration There is a smooth family of maps $\ll_{z,q}\colon X_{z,q}^{\sm}\to E_{\cS}$, parametrized by $q\in (0,1]$, where each $X_{z,q}^{\sm}$ is a submanifold of $X_{z}$ of codimension zero, with boundary and corners, each $\ll_{z,q}$ is a Lagrangian torus fibration with respect to the form $\omega_{q}$, and $E_{\cS}$ is the expanded version of the essential skeleton $\Delta_{\cS}$, constructed in Section <ref>.
Gromov–Hausdorff limit Choose a smooth path $(z,q)\colon [0,\delta)\to \D_{\delta,\log} \times [0,1]$ such that $|z(0)|=q(0)=0$ and $|z(h)|,q(h)> 0$ for $h>0$, see Figure <ref>. For each $h>0$ let $X_{h}$ be the fiber $X_{z(h)}$ equipped with a Kähler metric given by $\omega_{q(h)}$, rescaled so that its diameter is independent of $h$. Let $X_{h}^{\sm}\de X^{\sm}_{z(h),q(h)}$ be the region introduced in <ref>. Then the Gromov-Hausdorff limit $\lim_{h\rightarrow 0}(X_{h},X_{h}\setminus X^{\sm}_{h})$ is the essential skeleton $(\Delta_{\cS},\Delta_\cS\setminus\Delta_\cS^\circ)$, equipped with a positive multiple of the euclidean metric.

The euclidean metric on $\Delta_{\cS}$ is restricted from the dual complex of $D$, viewed as a subset of some euclidean space $\R^{N}$. Clearly, it has a potential satisfying the real Monge-Ampère equation for the natural affine structure of the maximal faces, as required by <cit.>, see <ref> above.

The base $E_{\cS}$ in Theorem <ref><ref> is the expanded essential skeleton, described in the beginning of the introduction. It contains, as an open subset, the union of relative interiors of $n$-dimensional faces $\Delta_{\cS}$. We denote this open subset by $\Delta_{\cS}^{\gen}$, let $X^{\gen}_{z,q}\subseteq X_{z,q}^{\sm}$ be the preimage of $\Delta_{\cS}^{\gen}$ through the Lagrangian torus fibration $\ll_{z,q}$ from Theorem <ref><ref>, and refer to $X^{\gen}_{z,q}$ as the generic region of the fiber $X_z$. The next result asserts that the generic region is large, and the family $\omega_{q}$ approximates there the conjectural behavior of the Ricci-flat forms. For a path as in Theorem <ref><ref> we put $X^{\gen}_{h}\de X^{\gen}_{z(h),q(h)}\subseteq X_{h}$.

We keep the notation and assumptions from Theorem <ref>. Fix a smooth path as in Theorem <ref><ref>, and for $h>0$ let $X_{h}^{\gen}\subseteq X_{h}$ be the generic region introduced above. Then as $h\rightarrow 0$, we have the following.

Volume filling The volume of $X_{h}^{\gen}$ with respect to $\mathrm{vol}^\Omega_{\new}$ converges to the total volume of $X_h$.
Asymptotic Ricci-flatness  Let $c_{h}\colon X_h^{\gen}\to \R_{>0}$ be a smooth family of functions satisfying $\frac{1}{n!} (\omega_{q(h)})^{n}=c_{h}\cdot \vol^{\Omega}_{\new}$. Then $c_{h}$ converges to a positive constant.
Lagrangian tori are asymptotically special Assume that the path $\gamma$ is special, see Definition <ref>: this condition holds e.g. if $|z(h)|=h$ for all $h\geq 0$, see Figure <ref>. Then there is a phase $\varpi\in \S^1$ such that for every $b\in \Delta_{\cS}^{\gen}$, the fibers $L_{h}\de \ll_{h}^{-1}(b)\subseteq X_{h}^{\gen}$ of the Lagrangian torus fibration $\ll_{h}$ from Theorem <ref><ref> satisfy

\begin{equation*}
\lim_{h\rightarrow 0}\  \frac{\Im (\varpi \Omega)|_{L_{h}}}{\Re (\varpi \Omega)|_{L_{h}}}=0.
\end{equation*}

In particular, the Lagrangian tori $L_h\subseteq X_{h}^{\gen}$ asymptotically minimize volume in their homology classes, see Corollary <ref> for a precise statement.
Maslov class  All the above Lagrangian tori, i.e. all fibers of $\ll_{h}$, have trivial Maslov class.

The Maslov class of the Lagrangian torus $L$ is the element of $H^{1}(L,\Z)$ induced by the phase map $\varpi\colon L\to \S^1$, defined so that $\varpi\cdot \Omega|_{L}$ is a volume form on $L$, see <cit.>. Triviality of Maslov class is important to define grading in Lagrangian Floer homology, see <cit.>.

The specialty assumption in Theorem <ref><ref> is a mild technical condition implying that the path is not very tangent to the $q$-axis, see Figure <ref>.

[use as bounding box] (-1.6,0) rectangle (8,3);
[->] (0,0) – (0,3);
[below left] at (0,3) $q(h)$;
[->] (0,0) – (7,0);
[above] at (7,0) $|z(h)|$;
[thick] (0,0) to[out=90,in=180] (2,1.5) to[out=0,in=180] (3.5,1.2) to[out=0,in=-120] (5,2.5);

Path in Theorems <ref><ref> and <ref><ref>,<ref>.

[use as bounding box] (-1.6,0) rectangle (8,3);
[->] (0,0) – (0,3);
[below left] at (0,3) $q(h)$;
[->] (0,0) – (7,0);
[above] at (7,0) $|z(h)|$;
[thick] (0,0) to[out=60,in=180] (2,1) to[out=0,in=-120] (5,2.5);;

Special path in Theorem <ref><ref>.

Paths in Theorems <ref><ref> and <ref>.

We remark that our $2$-forms $\omega_{q(h)}$ are not expected to approximate the Ricci flat form $\omega^{\CY}_{X_h}$. In fact, the volume of the non-generic region $X_{h}\setminus X_{h}^{\gen}$ computed with respect to $\frac{1}{n!}(\omega_{q(h)})^n$ does not converge to zero as $h\rightarrow 0$, see Remark <ref>.
As a consequence, the constant $\lim_{h\rightarrow 0}c_{h}$ in Theorem <ref><ref> is smaller than the one in formula (<ref>). This explains the factor $\epsilon^{n}$ in Proposition <ref>.

Therefore, our results demonstrate that some properties distinguished by the Kontsevich–Soibelman conjecture hold for families of metrics which are not necessarily asymptotic to the Ricci-flat one.

The main novelty of this article is a general technique that allows to construct fibrations by Lagrangian tori in Kähler degenerations, where fibers do not need to be Calabi–Yau nor compact. Theorems <ref> and <ref> above are obtained by combining this technique with special geometric properties of maximal Calabi–Yau degenerations. Now, we briefly describe the construction.

Let $f\colon X\to \C$ be a holomorphic map whose fibers have dimension $n$, and the unique singular fiber $D\de f^{-1}(0)$ is simple normal crossings, with irreducible components $D_1,\dots, D_{N}$. Let $\omega_{X}$ be a Kähler form on $X$. Choose a set of indices $\cS\in \{1,...,N\}$ corresponding to components of $D$ which we call essential. In the maximal Calabi–Yau degeneration case the set $\cS$ corresponds, as above, to components of minimal discrepancy, but the construction works for any choice of $\cS$. Let $\Delta_D$ be the dual complex of $D$ and let $\Delta_{\cS}$ be the essential skeleton, defined as the subcomplex of $\Delta_{D}$ spanned by vertices corresponding to essential components.

In [10], we introduced the A'Campo space $A$ of $f$, which fits into the diagram of smooth manifolds with boundary and smooth maps

\begin{equation}\label{eq:acampodiagram}
\begin{tikzcd}
\ar[r]
\ar[d, "f_{A}"]
\ar[rd, "\pi"]
& X
\ar[d, "f"]
\\
\C_{\log}
\ar[r] & \C
\end{tikzcd}
\end{equation}

where $f_A$ is a locally trivial fibration, extending the family of smooth fibers over $\C^*$ over the radius-zero circle $\partial\C_{\log}$. The space $A$ is constructed by a hybrid construction: it compactifies $f\colon X^\circ\to \C^*$ with the radius-zero fibration combining the Kato–Nakayama space of the log structure $(X,D)$ and the dual complex $\Delta_D$ of the snc divisor $D$. By construction, any radius-zero fiber $A_{\theta}$ for $\theta\in\partial \C_{\log}$ splits naturally as $A_{\theta}=\bigsqcup_{I}A_{I,\theta}^\circ$, where the disjoint union is indexed by all faces $\Delta_{I}$ of $\Delta_{D}$. Each of the pieces $A_{I,\theta}^\circ$ admits a concrete geometric description (see Lemma <ref>), and in particular:

*  If $\dim \Delta_I=n$ (so  $\Delta_{I}$ is a maximal face) then we have  $A_{I,\theta}^\circ=\Delta_I\times (\S^1)^n$,

*  If $\dim \Delta_I=n-1$ then we have $A_{I,\theta}^\circ=\Delta_I\times (X_{I}^{\circ})_{\log}$, where $(X_{I}^{\circ})_{\log}\to X_{I}^{\circ}$ is an $(\S^1)^{n-1}$-bundle over a compact Riemann surface with as many punctures as maximal faces contain $\Delta_I$.

We remark that the idea of combining natural coordinates of $X$ with tropical ones of $\Delta_{D}$ goes back to [4] and has been applied to Calabi–Yau degenerations e.g. in [5], cf. [25]. The new feature of our approach is to further combine them with polar coordinates of the Kato–Nakayama space, which parametrize tori at radius zero. This allows to endow $A$ with a smooth structure, and to define on it a fiberwise symplectic form $\omega_A$, as an exact modification of $\pi^*\omega_X$.

In [10], the form $\omega_{A}$ was used to to compare the usual symplectic monodromy with the radius-zero model via the symplectic connection. However, compatibility of $\omega_A$ with the complex structure was not proved. In Section <ref> we introduce a family of closed $2$-forms

\begin{equation}\label{eq:omega-s}
\omega_{q}^{\epsilon}=\pi^{*}\omega_{X}+\epsilon \cdot ( q\cdot \omega^{\sharp}+(1-q)\cdot \omega^{\flat}),\quad q\in [0,1],
\end{equation}

which at positive radius are fiberwise equal to $\omega_{q}^{\epsilon}=\pi^{*}\omega_{X}+dd^{c}\phi_{q}^{\epsilon}$ for some potential $\phi_{q}^{\epsilon}$, so they are compatible with the complex structure. Moreover, we prove that for $q>0$ they are fiberwise symplectic; and fiberwise Kähler at positive radius. In the setting of maximal Calabi–Yau degenerations, this will imply Theorem <ref><ref>. The proof of the remaining statements follows the general philosophy of [10]: we perform the main constructions at radius zero, where the structure of $A$ and the symplectic form is very well known and special, and then pull them to positive radius by the symplectic connection. In the remaining part of the introduction, we highlight the main ideas and point to the sections of the paper where they are implemented.

For any maximal face $\Delta_I$ of $\Delta_{\cS}$, close to the interior of $A_{I,\theta}^\circ\subseteq \d A$ the form $\omega_{0}^{\epsilon}$ is non-degenerate, see Proposition <ref><ref>, and as can be seen from the definition (<ref>) of $\omega^{\flat}$, the induced metric  $\omega_{0}^{\epsilon}$ resembles asymptotically the semi-flat one described in [33], cf. <cit.>. This is the reason for asymptotic Ricci-flatness claimed in Theorem <ref><ref>. We prove it in Proposition <ref>.

Moreover, tori arising in the product structure <ref> become Lagrangian with respect to  $\omega_q^\epsilon$ for any $q\in [0,1]$, see Proposition <ref>. Thus they can be pulled to positive radius by the symplectic connection of $\omega_q^\epsilon$, which results in Lagrangian torus fibration claimed in Theorem <ref><ref>, see Proposition <ref>. In the maximal Calabi–Yau degeneration case, as $q$ approaches $0$ faster than the radius (see Figure <ref>) we prove in Proposition <ref> that the Lagrangian tori become asymptotically special, as claimed in Theorem <ref><ref>. As a consequence, they have trivial Maslov class, as claimed in Theorem <ref><ref>.

For any face $\Delta_I$ of dimension $n-1$, see case <ref> above, we take a proper Morse function $h_I\colon X_{I}^{\circ}\to (0,\infty)$ and prove that for each $c\in (0,\infty)$, the preimage of $h^{-1}(c)$ in $(X_{I}^{\circ})_{\log}$ is isotropic for $\omega_{q}^{\epsilon}$ for any $q\in [0,1]$. Thus again we get a Lagrangian torus fibration, away from the preimage of $\Crit(h_{I})$.

Define $A_\theta^{\sm}=\bigsqcup_{I}A_{I,\theta}^\circ$, where the union is indexed by all faces $\Delta_{I}$ of $\Delta_{\cS}$ of dimension $n$ and $n-1$. An appropriate choice of Morse functions $h_I$ guarantees that the Lagrangian fibrations defined above in each of the pieces
$A_{I,\theta}^\circ$ glue to a Lagrangian fibration

$\ll\colon A_\theta^{\sm}\to E_{\cS}$

admitting finitely many singular fibers. Here $E_{\cS}$ is the expanded skeleton, endowed with a structure of an ivy-like manifold, see Section <ref>: it is a smooth manifold with some singularities corresponding to critical points of $h_{I}$.

In the maximal Calabi–Yau degeneration case we can take a model with  $X_{I}^{\circ}\cong \C^{*}$, see Proposition <ref><ref>, and we have a canonical choice of Morse functions $h_{I}$ with no critical points, see Remark <ref>. With this choice, the map $\ll$ becomes an honest Lagrangian fibration over a smooth manifold $E_{\cS}$, which agrees with the one described at the beginning of the introduction. Pulling back the Lagrangian fibration $\ll$ by the symplectic connection associated with $\omega_{q}^{\epsilon}$ for every $q\in (0,1]$, one gets Theorem <ref><ref>, see Proposition <ref>. Moreover, just like in the semi-flat model, after a suitable rescaling we see that the above Lagrangian tori collapse as $z\rightarrow 0$, both with respect to the metric, and to the Calabi–Yau volume. This implies the Gromov–Hausdorff convergence and volume filling, claimed in Theorem <ref><ref> and <ref><ref>. We prove these results in Propositions <ref> and <ref>, respectively.

Acknowledgments This work grew during visits of the first author to IMPAN, Warsaw, the second author to BCAM, Bilbao, and both authors to Rényi Institute, Budapest and CIRM, Marseille. We are grateful to each of these institutions for hospitality and excellent working conditions. We would also like to thank Duco van Straten and Valentino Tosatti  for inspiring conversations.

§ PRELIMINARIES ON THE ESSENTIAL SKELETON

In this section, we set up the notation for the remaining part of the article, and, in case of maximal Calabi–Yau degenerations, choose an appropriate model for which our Lagrangian tori will have no singularities. The latter is achieved in Proposition <ref> by applying the Minimal Model Program.

We work in the following setting.
Let $(X,\omega_{X})$ be a Kähler manifold of dimension $n+1$, and let

\begin{equation*}
f\colon X\to \C
\end{equation*}

be a holomorphic function with unique critical value $0\in \C$, such that the fiber $D\de f^{-1}(0)$ is snc. Let $D_{1},\dots, D_{N}$ be the irreducible components of $D$, and write

\begin{equation}\label{eq:multiplicites}
\end{equation}

Moreover, we choose a subset $\cS\subseteq \{1,\dots, N\}$. In practice, we will either take $\cS$ to be $\{1,\dots, N\}$, or, in the setting of Theorem <ref>, the set of indices of essential components of $D$, see formula  (<ref>).

For $z\in \C^{*}$ we write  $X_z=f^{-1}(z)$. Since we will be interested only in fibers $X_{z}$ for $|z|$ small enough, we will often replace $X$ by the preimage $f^{-1}(\D_{\delta})$ for some $\delta>0$ such that $f|_{X\setminus D}\colon X\setminus D\to \D_{\delta}^{*}$ is a submersion; and keep shrinking $\delta>0$ whenever needed.

Throughout the article, the adjective fiberwise will always refer to the fibers of $f|_{X\setminus D}$, or its extension $f_{A}$ to the A'Campo space, introduced in Section <ref>.  We write $\imath=\sqrt{-1}$.

§.§ Dual complex of an snc divisor

Let $D=\sum_{i=1}^{N}m_i D_i$ be an snc divisor as above. For $I\subseteq \{1,\dots, N\}$ we put $X_{I}=\bigcap_{i\in I}D_{i}$, $X_{I}^{\circ}=X_{I}\setminus \bigcup_{j\not\in I}D_{j}$. This way, $X_{I}=\bar{X}_{I}^{\circ}$ and $X=\bigsqcup_{I} X_{I}^{\circ}$ is a stratification of $X$.

Let $\Delta=\{(w_{1},\dots,w_{N})\in \R^{N}: \sum_{i}w_{i}=1\}$ be the standard $(N-1)$-dimensional simplex. For a subset $I\subseteq \{1,\dots,N\}$ we put $\Delta_{I}=\{(w_1,\dots,w_{N})\in \Delta: w_{i}=0 \mbox{ for } i\neq I\}$. The dual complex of $D$ is $\Delta_{D}\de\bigcup_{\{I: X_{I}\neq \emptyset\}} \Delta_{I}\subseteq \Delta$. We equip it with a metric induced by the standard Riemannian metric  $\sum_{i}(dw_{i})^{2}$ on $\R^{N}$. For a given subset $\cS\subseteq \{1,\dots, N\}$, we put $\Delta_{\cS}=\Delta_{D}\cap (\bigcup_{I\subseteq \cS}\Delta_{I})$.

Since the divisor $D$ is snc, for every face $\Delta_{I}$ of $\Delta_{D}$ we have $\#I=\codim_{X} X_{I}=n+1-\dim X_{I}\leq n+1$. We say that the face $\Delta_{I}$ is maximal if $\#I=n+1$ (so $X_{I}$ is a point), and submaximal if $\#I=n$ (so $X_{I}$ is a curve). We will be mostly interested in the case when $\Delta_{D}$ has at least one maximal face.

Each face $\Delta_{I}$, viewed as a subset of the plane $\{\sum_{i}w_{i}=1, w_{i}=0 \mbox{ for } i\not\in I\}\subseteq \R^{N}$ is a manifold with boundary and corners, see <cit.> for precise definitions. By Proposition 3.1 loc. cit, we can and do think of a manifold with boundary and corners as a subset of a usual smooth manifold, mapped by each local chart to an open subset of $\R^{p}\times \R^{m-p}_{\geq 0}$ for some $0\leq p \leq m$. We use the notions of smooth functions, tangent space etc. inherited from the ambient smooth manifold.

§.§ Maximal Calabi–Yau degenerations and their models

A smooth projective complex manifold $X$ is Calabi–Yau if its canonical divisor $K_{X}$ is trivial, i.e. there is a nowhere vanishing holomorphic $n$-form $\Omega\in \Omega^{n,0}(X)$.  Yau's solution to the Calabi conjecture [34] asserts that any Kähler class in $H^{1,1}(X)\cap H^{2}(X,\R)$ admits a unique representative whose associated metric is Ricci flat, see <cit.>. We refer to this representative as the Calabi–Yau form, and denote it by $\omega_{\CY}$. Ricci-flatness of the metric $g_{\CY}\de \omega_{\CY}(\sdot,J\sdot)$ is equivalent to the fact that there exists a holomorphic volume form $\Omega$  satisfying the complex Monge-Ampère equation

\begin{equation}\label{eq:complex_MA}
\tfrac{1}{n!}\cdot \omega_{\CY}^{n}=\vol^{\Omega},\quad\mbox{where}\quad \vol^{\Omega}\de \left(\tfrac{\imath}{2}\right)^{n}\cdot (-1)^{\frac{1}{2}n(n-1)}\cdot  \Omega \wedge \bar{\Omega},%,\quad\mbox{where}\quad c=(-1),
\end{equation}

cf. <cit.> or <cit.>.

Let $f^{\circ}\colon X^{\circ}\to \C^{*}$ be a projective morphism whose fibers $X_{z}\de (f^{\circ})^{-1}(z)$ are Calabi–Yau manifolds of dimension $n$. We say that $f$ is a Calabi–Yau degeneration if $f^{\circ}$ is meromorphic at $0$, i.e. it admits an extension to a model $f\colon X\to \C$. It is maximal if the monodromy transformation $T\colon H^{n}(X_{z};\Q)\to H^{n}(X_{z},\Q)$ satisfies $(T-\id)^{n}\neq 0$, $(T-\id)^{n+1}=0$, cf. <cit.>.

Resolving singularities, we see that every $f^{\circ}$ admits a model $f\colon X\to \C$ which is snc, i.e. such that $X$ is a smooth Kähler manifold, and the special fiber $f^{-1}(0)$ is an snc divisor. Moreover, every $f^{\circ}$ admits, after a finite base change, an snc model which is semistable, i.e. its special fiber is reduced <cit.>. We note that after a finite base change, a maximal Calabi–Yau degeneration is still maximal. Indeed, the general fiber remains the same, while the monodromy transformation is replaced by its iterate, so its Jordan blocks remain of the same size.

We now introduce the essential skeleton $\Delta_{\cS}$ of $f^{\circ}$, which agrees with the one used in [26, 28], see also <cit.>. Let $f\colon X\to \C$ be an snc model of $f^{\circ}$. We write its special fiber as $D=\sum_{i=1}^{N} m_{i}D_{i}$, see formula (<ref>), and use notation from Section <ref>. By adjunction, for every $z\neq 0$ the smooth fiber $X_{z}$ satisfies $K_{X}|_{X_{z}}=K_{X_{z}}=0$, so the canonical bundle of $X$ admits a meromorphic section $\Theta$ with no zeros or poles off $D$. We choose one such $\Theta$, and let $\nu_{i}\in \Z$ be an integer such that $\Theta$ has a zero of order $\nu_{i}-1$ along $D_{i}$. Thus we have a linear equivalence

\begin{equation}\label{eq:discrepancies}
\end{equation}

Note that the coefficients $\nu_{i}$ depend on the choice of the section $\Theta$. If we choose another section, say $\Theta'$, then $\Theta'=uf^{k}\cdot \Theta$ for some holomorphic unit $u\in \cO_{X}^{*}$ and integer $k\in \Z$, so the corresponding coefficients are $\nu_{i}'=\nu_{i}+km_{i}$. It follows that the set

\begin{equation}\label{eq:essential}
\cS=\left\{ i:\frac{\nu_i}{m_i}=\min_{j}\frac{\nu_j}{m_j} \right\}
\end{equation}

does not depend on $\Theta$. We say that a component $D_{i}$ of $D$ is essential if $i\in \cS$; a face $\Delta_{I}$ of the dual complex $\Delta_{D}$ is essential if $I\subseteq \cS$. The essential skeleton $\Delta_{\cS}\subseteq \Delta_{D}$ is the union of all essential faces.

In <cit.>, the essential skeleton $\Delta_{\cS}$ (or rather its homeomorphic image in the Berkovich space) is called the Kontsevich–Soibelman skeleton $\Sk(X^{\circ},\Omega)$, where

\begin{equation}\label{eq:Omega}
\Omega\de \frac{\Theta}{d \log f}=\frac{f\cdot\Theta}{df}
\end{equation}

is a chosen section of the vertical canonical bundle. As explained in 4.7.3 loc. cit, the quotient $\frac{\nu_i}{m_i}$ equals $\textnormal{wt}_{\Omega}(x)+1$, where $x$ is the divisorial valuation of $\C(X^{\circ})$ corresponding to $D_i$, and $\textnormal{wt}_{\Omega}$ is the weight function introduced in 4.3.5 loc. cit. The essential skeleton is defined in 4.10 loc. cit. as $\Sk(X^{\circ})\de \bigcup_{\Omega} \Sk(X^{\circ},\Omega)$, so since $K_{X}$ is Calabi–Yau, this definition agrees with $\Sk(X^{\circ},\Omega)$, see 4.10.2 loc. cit. Hence up to a homeomorphism, our essential skeleton $\Delta_{\cS}$ agrees with the one in loc. cit.

Theorem 4.7.5 and Corollary 3.2.4 loc. cit. show that the homeomorphism type of the essential skeleton $\Delta_{\cS}$, as well as its piecewise affine structure inherited from $\Delta_{D}$ (see 3.2 loc. cit), do not depend on the snc model $X$.  By <cit.>, the Calabi–Yau degeneration $f^{\circ}$ is maximal if and only if $\dim \Delta_{\cS}=n$. In this case, Theorem 4.2.4(3) loc. cit. asserts that $\Delta_{\cS}$ is a closed pseudomanifold.

We remark that by <cit.> the property of being essential depends only on the corresponding valuation of $\C(X^{\circ})$, and not on a particular snc model. More precisely, if a valuation of $\C(X^{\circ})$ corresponds to a component $E$ (resp. $E'$) of a special fiber of some snc model $f$ (resp. $f'$), then $E$ is essential if and only if $E'$ is,.

The following proposition is known to experts, cf. e.g. <cit.>. For the reader's convenience, we include its proof.

Let $f^{\circ}\colon X^{\circ}\to \C^{*}$ be a Calabi–Yau degeneration admitting a semistable snc model. Then $f^{\circ}$ admits a (not necessarily semistable) snc model $f\colon X\to \C$ with the following properties.

*  Every essential component $D_{i}$ of the special fiber $D=f^{-1}(0)$ has multiplicity one in $D$.

*  The canonical bundle $K_{X}$ admits a section $\Theta$ such that, denoting as above by $\nu_{i}-1$ the order of $\Theta$ along a component $D_i$ of $D$, we have $\nu_{i}\geq 0$, with equality if and only if $D_{i}$ is essential.

*  Let $\Delta_{I}$ be a submaximal, essential face of the dual complex $\Delta_{D}$. Then every maximal face of $\Delta_{D}$ containing $\Delta_{I}$ is essential, too, and the following hold.

*  If $f^{\circ}$ is a maximal degeneration then $(X_{I},X_{I}^{\circ})\cong (\P^1,\C^{*})$.

*  If $f^{\circ}$ is not a maximal degeneration then $X_{I}=X_{I}^{\circ}$ is an elliptic curve.

Let $\tilde{f}\colon \tilde{X}\to \C$ be a semi-stable snc model of $f^{\circ}$, with special fiber $\tilde{D}$. By <cit.>, applying $\tilde{f}$-MMP as in  [18] to the pair  $(\tilde{X},\tilde{D})$ we get a good dlt model $\bar{f}\colon \bar{X}\to \C$ of $f^{\circ}$, with special fiber $\bar{D}$ being the direct image of $\tilde{D}$; in particular connected and reduced. The fact that $\bar{f}$ is a good dlt model implies that $\bar{X}$ is normal and $\Q$-factorial, the pair $(\bar{X},\bar{D})$ is dlt, and the Weil divisor $K_{\bar{X}}+\bar{D}$ is nef, see <cit.>. Since $\bar{D}$ is numerically trivial, it follows that $K_{\bar{X}}$ is nef, too. Let $\bar{D}_1,\dots$ be the irreducible components of $\bar{D}$, so $\bar{D}=\sum_{i}\bar{D}_i$.

For $z\neq 0$ the fiber $\bar{f}^{-1}(z)$ is Calabi–Yau, so $K_{\bar{X}}|_{\bar{X}\setminus \bar{D}}$ is trivial. Hence we have a linear equivalence of Weil divisors $K_{\bar{X}}=\sum_{i}\bar{a}_{i}\bar{D}_i$ for some integers $\bar{a}_{i}$.

We claim that all integers $\bar{a}_i$ are equal. Suppose the contrary, and say that $\bar{a}_{1}=\max_{i}\bar{a}_i>\bar{a}_2$. Because $\bar{D}$ is connected, we can order its components in such a way that $\bar{D}_1$ meets $\bar{D}_2$.  Choose a curve $C\subseteq \bar{D}_1$ such that $C\cap \bar{D}_2\neq \emptyset$ and $C\not\subseteq \bar{D}_i$ for all $i\neq 1$. Since $K_{\bar{X}}$ is nef, we have $0\leq K_{\bar{X}}\cdot C=\sum_{i}\bar{a}_i \bar{D}_i\cdot C$. Subtracting $\sum_{i} \bar{a}_{1}\bar{D}_i\cdot C=\bar{a}_{1}\bar{D}\cdot C=0$, we get $0\leq \sum_{i>1} (\bar{a}_{i}-\bar{a}_1)\bar{D}_i\cdot C$. Since $\bar{a}_i\leq \bar{a}_1$ and $C\not \subseteq \bar{D}_i$, each summand is non-positive, hence they are all zero. By assumption $C\cdot\bar{D}_2>0$, so $\bar{a}_2=\bar{a}_1$; a contradiction.

Therefore, we have a linear equivalence $K_{\bar{X}}=\bar{a}_1\bar{D}=0$. The fiber $\bar{D}$ is a trivial Cartier divisor, so the same holds for $K_{\bar{X}}$. In particular, the Weil divisor $K_{\bar{X}}+\bar{D}$ is Cartier. Recall also that $\bar{X}$ is $\Q$-factorial, so all components of $\bar{D}$ are $\Q$-Cartier. Now <cit.> implies that

\begin{equation}\label{eq:NXY}
\mbox{each one-dimensional stratum of }\bar{D}\mbox{ is contained in the snc locus of }\bar{X},
\end{equation}

where we consider the standard stratification of $\bar{D}$ by intersections $\bigcap_{i\in I} \bar{D}_{i}$, see 1.10 loc. cit.

Let $\pi\colon (X,D)\to (\bar{X},\bar{D})$ be a log resolution, i.e. a proper morphism which is an isomorphism over the locus where $\bar{X}$ is smooth and $\bar{D}$ is snc; such that $X$ is smooth and $D\de \pi^{*}\bar{D}$ is snc. Such a resolution exists, see <cit.> and references there. Write $D=\sum_{i=1}^{N}m_{i}D_{i}$, and split $\{1,\dots,N\}=\cE\sqcup \cP$, where $\Exc\pi=\sum_{i\in \cE} D_{i}$ and $\pi^{-1}_{*}\bar{D}=\sum_{i\in \cP} D_{i}$. Note that $m_i=1$ for all $i\in \cP$.

Recall that $K_{\bar{X}}+\bar{D}$ is a Cartier divisor. Hence $K_{X}+D\redd-\pi^{*}(K_{\bar{X}}+\bar{D})$ is a Cartier divisor, too. Its push-forward on $\bar{X}$ is trivial since $\pi_{*}D\redd=\bar{D}$. Thus for some integers $\nu_{i}$ we have a linear equivalence

\begin{equation*}
K_{X}+D\redd=\pi^{*}(K_{\bar{X}}+\bar{D})+\sum_{i\in \cE} \nu_i D_{i}=\sum_{i=1}^{N} \nu_{i} D_{i},
\end{equation*}

where for the second equality we put $\nu_{i}=0$ for $i\in \cP$, and use the linear equivalences $K_{\bar{X}}=\bar{D}=0$. We now choose a section $\Theta$ of $K_{X}$ with zeros of order $\nu_{i}-1$ along each $D_{i}$, see formula (<ref>).

Since the pair $(\bar{X},\bar{D})$ is dlt and the resolution $\pi$ is an isomorphism over its snc locus, we have $\nu_{i}>0$ for all $i\in \cE$, see <cit.>. Thus  $\cP=\{i:\nu_{i}=0\}=\{i:\frac{\nu_i}{m_i}=\min_{j} \frac{\nu_{j}}{m_j}\}$. We conclude that $\cP$ is the set (<ref>), i.e. $i\in \cP$ if and only if a component $D_{i}$ of $D$ is essential, cf. <cit.>. Now conditions <ref>, <ref> follow from the fact that $(m_i,\nu_i)=(1,0)$ for all $i\in \cP$.

Let now $\Delta_{I}$ be a submaximal face of the dual complex of $D$, so $\dim X_{I}=1$. Assume that $\Delta_{I}$ is essential, so $I\subseteq \cP$. By definition of $\cP$, the morphism $\pi$ is an isomorphism at a general point of $X_{I}$, so $\dim\pi(X_{I})=\dim X_{I}=1$. Now condition (<ref>) implies that $\pi(X_{I})$ is contained in the snc locus of $\bar{D}$, so $\pi$ is an isomorphism near $X_{I}$. It follows that for every component $D_{j}$ of $D$ meeting $X_{I}$, we have $j\in \cP$, hence $D_j$ is essential. This proves the first part of <ref> and shows that $(K_{X}+D)|_{X_{I}}=0$.

Write $B_{I}=(\sum_{i\not \in I} m_{i}D_{i})|_{X_{I}}$, so $B_{I}$ is a divisor on $X_{I}$ supported on $X_{I}\setminus X_{I}^{\circ}$. Since each component $D_{i}$ meeting $X_{I}$ has $i\in \cP$, so $m_i=1$, we see that $B_{I}$ is reduced. Applying Lemma <ref> below to $J=\emptyset$, we get $K_{X_{I}}+B_{I}=(K_{X}+D)|_{X_{I}}=0$. It follows that either $B_{I}=0$ and $X_{I}=X_{I}^{\circ}$ is an elliptic curve; or $B_{I}\neq 0$ and $(X_{I},X_{I}^{\circ})\cong (\P^1,\C^{*})$. Since by <cit.>, the essential skeleton $\Delta_{\cS}$ is a pseudomanifold (with boundary) we have $\dim\Delta_{\cS}=n-1$ in the first case and $\dim\Delta_{\cS}=n$ in the second case, see condition (1) of the definition of pseudomanifold given in 4.1.2 loc. cit. By Theorem 4.1.10 loc. cit., the second case happens if and only if $f^{\circ}$ is a maximal degeneration.

In the above proof, we used the following elementary lemma.

Let $D$ be an snc divisor on a smooth projective variety $X$. Write $D=\sum_{i=1}^{N} m_{i}D_{i}$, and for $I\subseteq \{1,\dots, N\}$ put $X_{I}=\bigcap_{i\in I} D_{i}$, $X_{\emptyset}=X$,  $B_{I}=\sum_{i\not\in I}m_{i}D_{i}|_{X_{I}}$. Fix two subsets $J\subseteq I\subseteq \{1,\dots, N\}$, and assume that $m_{i}=1$ for all $i\in I$. Then $K_{X_I}+B_I=(K_{X_{J}}+B_{J})|_{X_{I}}$.

By induction on $\#I\setminus J$, we can assume that $I=J\sqcup\{i\}$. By adjunction, we have $K_{X_{I}}=(K_{X_{J}}+D_{i}|_{X_{J}})|_{X_{I}}=K_{X_{J}}|_{X_{I}}+D_{i}|_{X_{I}}$. Moreover, $B_{I}=\sum_{j\not\in I} m_jD_j|_{X_{I}}=\sum_{j\not\in J} m_{j}D_{j}|_{X_{I}}-m_{i}D_{i}|_{X_{I}}=B_{J}|_{X_{I}}-D_{i}|_{X_{I}}$ since $m_{i}=1$ by assumption. Adding the two equalities we get $K_{X_{I}}+B_{I}=(K_{X_{J}}+B_{J})|_{X_{I}}$.

[Hesse pencil]
Consider a family of elliptic curves $X=\{(x_1^3+x_2^3+x_3^3)\cdot z=x_1x_2x_3\}\subseteq \P^{2}\times \C$, and let $f\colon X\to \C$ be the projection to the $z$-coordinate. The restriction $f^{\circ}$ of $f$ over $\C^{*}$ is a maximal Calabi–Yau degeneration, and $f$ is its model satisfying all conditions of Proposition <ref>. Indeed, the special fiber $D=f^{-1}(0)$ is just a union of three non-concurrent lines $D_1,D_2,D_3\subseteq \P^2$; and $K_{X}=0$, so each $D_i$ is essential. We conclude that the essential skeleton is a triangle.

Fix $p\in D$ and let $\pi\colon X'\to X$ be a blowup at $p$, so $f'\de f\circ \pi$ is another model of $f^{\circ}$. Put $D'=(f')^{-1}(0)$, $D_i'=\pi^{-1}_{*}D_i$ and $D_{0}'=\Exc\pi$, so $K_{X'}=D_{0}'$. Let $m_{i}'$, $\nu_{i}'$ be as in formulas (<ref>), (<ref>), so $\nu_{i}'/m_{i}'=1$ for $i\in \{1,2,3\}$ and $\nu_{0}'=2$.

Assume first that $p$ is a smooth point of $D$. Then $m_{0}'=1$, so $\nu_{0}'/m_{0}'=2>1$, i.e. $D_{0}'$ is not essential. Hence the model $f'$ satisfies conditions <ref><ref> and <ref>, but does not satisfy <ref><ref>. Note that the component $D_{i}'$ meeting $D_0'$ satisfies  $(D_{i}',(D_{i}'\setminus (D'-D_{i}'))\cong (\P^{1},\C^{**})$, so <ref><ref> fails, too. The essential skeleton $\Delta_{\cS}'$ computed using this model is still a triangle, in agreement with <cit.>.

Assume now that $p$ is a singular point of $D$. Then $m_{0}'=2$, so $\nu_{0}'/m_{0}'=1$, i.e. all components of $D'$ are essential. This model satisfies  conditions <ref><ref> and <ref>, but does not satisfy <ref><ref>. Now $\Delta_{\cS}'$ is a square obtained by subdividing the triangle $\Delta_{\cS}$, again in agreement with <cit.>.

§ THE A'CAMPO SPACE AND ITS SMOOTH STRUCTURES

Let $f\colon X\to \C$ be as in the beginning of Section <ref>, i.e.  $f$ is a holomorphic function whose unique singular fiber $D\de f^{-1}(0)$ is snc; and  $f|_{X\setminus D}\colon X\setminus D\to \D_{\delta}^{*}$ is a submersion. In <cit.>, we introduced the A'Campo space $A$ of $f$, which fits into the diagram (<ref>). In particular, it allows to extend the monodromy of $f$ to radius zero, i.e. to the monodromy of the restriction $f|_{\d A}\colon \d A \to \d \C_{\log}$. The above radius-zero monodromy has simpler dynamical properties than the natural one at positive radius, see Proposition 7.2 loc. cit. Most importantly, it is induced by a symplectic connection with respect to a certain fiberwise symplectic form $\omega_{A}^{\epsilon}$. Our aim is to improve this form  so that it becomes fiberwise Kähler at positive radius.

§.§ Definition of the A'Campo space

In this section, we review the construction of the A'Campo space $A$. Recall that $D=\sum_{i=1}^{N}D_i$ is the irreducible decomposition of the special fiber $D$.

We begin with the notion of an adapted chart <cit.>. For a holomorphic chart $(z_{i_1},\dots,z_{i_n})\colon U_{X}\to \C^{n+1}$ of $X$, we define its associated index set as $S=\{i\in \{1,\dots, N\}:D_i\cap U_{X}\neq \emptyset\}$. We say that this chart is adapted to $f$ if for every $i\in \{1,\dots, N\}$ we have $U_X\cap D_i=\{z_i=0\}$ and $f|_{U_X}=\prod_{i\in S}z_{i}^{m_i}$ in case $S\neq \emptyset$. We also require $U_{X}$ to be small enough, so that $\log |z_{i}|<1$ for all $i\in S$, and each coordinate $z_i$ extends to a continuous function on the compact closure $\bar{U}_X$.

For any $i\in S$ we have the following smooth functions on $U_{X}\setminus D$, see <cit.>

\begin{equation}\label{eq:basic functions_r}
r_{i}=|z_i|,\quad \theta_i=\frac{z_i}{r_i},\quad s_i=\log r_i,\quad t_i=\frac{-1}{m_i s_i}.
\end{equation}

We call $r_i$ and $\theta_i$ the radial and angular coordinates of $U_{X}$. We also define global functions on $X\setminus D$

\begin{equation}\label{eq:basic functions_t}
t=\frac{-1}{\log|f|},\quad g=\eta(t),\quad \theta=\frac{f}{|f|},
\end{equation}

where $\eta\colon [0,1]\to[0,1]$ is given by $\eta(\tau)=(1-\log \tau)^{-1}$. The key property of this auxiliary function is that its inverse is a smooth function whose all derivatives vanish at the origin.

Eventually, for each $i\in S$ we put

\begin{equation}\label{eq:basic_functions_v}
w_{i}=\frac{t}{t_i},\quad u_{i}=\eta(w_i),\quad v_{i}=t_i-u_i, \quad\mbox{and}\quad \sigma_{i}= t_{i}^{2}+t_{i}u_{i}^{2}.
\end{equation}

The functions $r_i$, $w_i$ and $v_i$ should be thought of as natural, tropical and hybrid coordinates, respectively. For more intuition, we refer the reader to <cit.>. The last function $\sigma_{i}$ plays an auxiliary role.

In order to define the A'Campo space as a $\cC^{1}$-manifold, it is convenient to choose an atlas $\{U_{X}^{p}\}_p$ consisting of adapted charts, a subordinate partition of unity $\{\tau^{p}\}_p$, and define

\begin{equation*}
\bar{u}_{i}=\sum_{p}\tau^{p}u_{i}^{p},\quad \bar{v}_i=\sum_{p}\tau^{p}v_{i}^{p},
\end{equation*}

where for every $p$ such that $U_{X}^{p}\cap D_{i}\neq \emptyset$, the functions $u_{i}^{p},v_{i}^{p}$ are the ones defined for $U_{X}^{p}$ by formulas (<ref>); and for $p$ such that $U_{X}^{p}\cap D_{i}=\emptyset$, we put $u_{i}^{p}=v_{i}^{p}=0$.

We define a topological space $\Gamma$ as the closure of the graph of $(\bar{u}_{1},\dots, \bar{u}_{N})\colon X\setminus D\to \R^{N}$.  The A'Campo space $A$ is a fibered product over $X$ of $\Gamma$ and the Kato–Nakayama space $X_{\log}$ of $(X,D)$, see diagram (<ref>). This way, $A$ is a topological space equipped with a continuous map $\pi\colon A\to X$ which is a homeomorphism over $X\setminus D$; such that for any adapted chart $U_{X}\subseteq X$, all basic functions (<ref>)–(<ref>) extend to continuous functions on the preimage $\pi^{-1}(U_X)$ <cit.>. Crucially for us, the value of each $w_{i}, u_i$ on $\pi^{-1}(U_X\cap D)$ does not depend on the choice of the adapted chart $U_X$, see Lemma 3.6(d),(e) loc. cit.

\begin{equation}\label{eq:AX-diagram}
\begin{tikzcd}
%\de (X,D)\AC
\ar[r]
\ar[d, %"\pi_{\log}"
\ar[dd, bend right=90, looseness=1, "{(g,\theta)\sim f_A}"']
\ar[dr, phantom, "\ulcorner", very near start]
\ar[dr, "\pi", start anchor = south east, shorten={5mm}]
& \Gamma\de \overline{\mbox{\small{graph}}(\mu)}
\ar[d, %"\pi_{\Gamma}"
\ar[r, hook]
%			\ar[dr, %"\mu_{\Gamma}",
%			start anchor = south east]
& X\times  \R^{N}
\ar[d]
\\
%\de (X,D\redd)_{\log}
\ar[r]%, "\tau"]
\ar[d, "f_{\log}"]
& X
\ar[r, dashed, "\mu", "{(\bar{u}_1,\ldots,\bar{u}_{N})}"']
\ar[d, "f"]
& \R^{N}
\\
\C_{\log}
%\de (\C,0)_{\log}
\ar[r] & \C &
\end{tikzcd}
\end{equation}

The stratification $X=\bigsqcup_{I}X_{I}^{\circ}$ lifts to a decomposition $A=\bigsqcup A_{I}^{\circ}$, where $A_{I}^{\circ}\de \pi^{-1}(X_{I}^{\circ})$.

The $\cC^{1}$-atlas on $A$ is introduced as follows. For an adapted chart $U_X$ with associated index set $S$, we cover its preimage by open sets $U_{i}=\{w_{i}>\frac{1}{n+2}\}$, $i\in S$, which will be used as domains of smooth charts on $A$. To make the notation more compact, we put $\rest\de ((\theta_{i})_{i\in S},(z_{j})_{j\in \{i_1,\dots,i_n\}\setminus S})$. For $k\in \{1,\dots, n+1\}$ we write $Q_{k,n+1}=[0,\infty)\times \R^{k-1}\times (\S^1)^{k}\times \C^{n+1-k}$. Now, we define the  $\cC^{1}$-charts as

\begin{equation}\label{eq:AC1-chart}
\psi_{i}=(g,({v}_{j})_{j\in S\setminus \{i\}};\rest)\colon
U_{i}\to Q_{k,n+1}.
\end{equation}

Results of <cit.> show that these charts induce a $\cC^{1}$-structure on the A'Campo space, which is independent of the choice of $\{(U_{X}^{p},\tau^{p})\}_p$. The $\cC^{\infty}$-charts are defined in 3.4 loc. cit. as

\begin{equation}\label{eq:AC-chart}
\bar{\psi}_{i}=(g,({\bar{v}}_{j})_{j\in S\setminus \{i\}};\rest)\colon
U_{i}\to Q_{k,n+1}.
\end{equation}

These charts upgrade the above $\cC^{1}$-structure to a $\cC^{\infty}$ one, which does depend on the choice of the adapted atlas $\{U_X^{p}\}_p$ and partition of unity $\{\tau^{p}\}_p$, see Remark 3.32 loc. cit.

§.§ Distance functions

Our next goal is to introduce another $\cC^{\infty}$-atlas on $A$, compatible with the $\cC^{1}$-structure given by charts (<ref>), which will allow us to introduce our fiberwise Kähler form.

We work in a fixed open set $W_{X}\subseteq X$ such that the restriction $f|_{\bar{W}_{X}}$ is proper, has connected fibers; and $f|_{\bar{W}_X\setminus D}\colon \bar{W}_{X}\setminus D\to \D_{\delta}^{*}$ is a submersion. If $f$ is an snc model of a Calabi–Yau degeneration, which is the case of most interest for us, we simply take $W_{X}=X$. For another example, let $X$ be an embedded resolution of an isolated hypersurface singularity, and take for $W_{X}$ the preimage of some neighborhood of the origin. We can and do assume that $\delta<\exp(-\max_{i}m_{i})$.

Now, we introduce global distance functions $\hat{r}_i$. To this end, we fix an atlas of adapted charts $\{U_{X}^{p}\}_{p}$, and denote by $S^{p}=\{i:U_X^p\cap D_i\neq \emptyset\}$ the associated index set of $U_{X}^{p}$. We define open sets

\begin{equation}\label{eq:Ri}
R_{i}=\bigcup_{\{p\ :\ i\in S^{p}\}} U_{X}^{p},\quad\mbox{and}\quad R_{i}^{\circ}=R_i\setminus \bigcup_{j\neq i} \bar{R}_{j},
\end{equation}

so $R_{i}$ is a neighborhood of $D_i$, and $R_{i}^{\circ}$ consists of points far away from the remaining part of $D$.

Refining the atlas $\{U_{X}^{p}\}_p$ if needed, we can and do assume the following properties.

*  For every $p$ and every $i\in S^{p}$ the radial coordinate $r_{i}^{p}$ of $U_{X}^{p}$ satisfies $r_{i}^{p}<\frac{1}{e}$.

*  For every maximal face $\Delta_{I}$ of $\Delta_{D}$, the point $X_{I}$ lies in exactly one chart $U_{X}^{p}$.

*  For every $i\in \{1,\dots, N\}$ the open set $R_{i}^{\circ}$ is non-empty.

Since the set $\bar{W}_{X}\cap D$ is compact, shrinking $\delta>0$ if needed we can further assume that

*  Every fiber $f^{-1}(z)\cap \bar{W}_X$, $z\in \D_{\delta}$ is contained in $\bigcup_{i=1}^{N}R_{i}$ and meets $R_{i}^{\circ}$ for each $i\in \{1,\dots,N\}$.

Eventually, we choose a subordinate partition of unity $\{\tau^{p}\}_{p}$, put $r_{i}^{p}=\frac{1}{e}$ for $i\not\in S^{p}$, and define

\begin{equation*}
\hat{r}_{i}=\sum_{p}\tau^{p}r_{i}^{p}\colon X\to [0,\tfrac{1}{e}].
\end{equation*}

[Distance functions]
The functions $\hat{r}_i$ defined above have the following properties.

*  For every point $x\in D_{i}$ and every adapted chart $U_{X}$ containing $x$, there is a neighborhood $V_X$ of $x$ in $U_X$ and a  smooth function $\lambda\colon \bar{V}_X\to (0,\infty)$ such that the radial coordinate $r_{i}$ of $U_X$ satisfies $\hat{r}_{i}|_{V_X}=\lambda r_{i}|_{V_X}$.

*  Let $\Delta_{I}$ be a maximal face of $\Delta_{D}$. There is an adapted chart $U_X$ around the point $X_{I}$ with coordinates $r_i$ such that on $U_X$ we have $r_i=\hat{r}_i$ if $i\in I$ and $\hat{r}_i=\frac{1}{e}$ otherwise.

*  Let $U_X\subseteq R_{i}^{\circ}$ be an adapted chart meeting $D_i$. Then its radial coordinate $r_{i}$ is equal to $\hat{r}_i$.

*  We have $\hat{r}_{i}<\frac{1}{e}$ on $R_{i}$ and $\hat{r}_{i}=\frac{1}{e}$ on $X\setminus R_{i}$.

*  We have $\hat{r}_i^{m_i}\geq |f|$ and $\hat{r}_i^{m_i}=|f|$ on $R_{i}^{\circ}\cap f^{-1}(\D_{\delta})$.

<ref> Put $P_{x}=\{p:U_{X}^{p}\cap \bar{U}_{X}\neq \emptyset\}$, so $\hat{r}_{i}|_{\bar{U}_{X}}=\sum_{p\in P_{x}}\tau^{p}r_{i}^{p}$. Since the point $x$ lies on $D_i$, shrinking the chart $U_{X}$ around $x$ if necessary, we can assume that every chart $U_{X}^{p}$ meeting $\bar{U}_X$ meets $D_i$, too, i.e. $P_{x}\subseteq \{p:i\in S^{p}\}$. Fix $p\in P_{x}$ and put $V^{p}=U_{X}^{p}\cap U_{X}$. Recall that, by definition of the adapted charts, both $r_i$ and $r_i^{p}$ extend continuously to the compact closure $\bar{V}^{p}$. Put $\lambda^{p}\de \frac{r_{i}^{p}}{r_i}\colon \bar{V}^{p} \to (0,\infty)$. Now the function $\lambda\de \sum_{p\in P_x}\tau^{p}\lambda^{p}=\frac{\hat{r}_i}{r_i}$ is defined on the whole compact closure $\bar{U}_{X}$ and satisfies the required equality $\hat{r}_i=\lambda r_i$. Moreover, each summand $\tau^{p}\lambda^{p}$ is non-negative, and at each point of $\bar{U}_X$, at least one of them does not vanish; so $\lambda$ is a positive function on $\bar{U}_X$, as needed.

<ref> By property <ref> the point $X_{I}$ lies in $U_{X}^{p}$ for a unique $p$. Now, take $U_X=U_{X}^{p}\setminus \bigcup_{q\neq p}\bar{U}_{X}^{q}$.

<ref> Because $U_X$ is an adapted chart with associated index $\{i\}$, we have $r_{i}=|f|^{1/m_i}$. Since $U_{X}\subseteq R_{i}^{\circ}$, the same holds for all charts $U_{X}^{p}$ meeting $U_{X}$, so $\hat{r}_{i}=|f|^{1/m_i}$, too. Thus $\hat{r}_i=r_i$, as claimed.

<ref> By property <ref>, on $R_i$ we have $\tau^{p}r_{i}^{p}<\tau^{p}e^{-1}\leq e^{-1}$ if $U_{X}^{p}$ meets $D_i$ and $\tau^{p}r_{i}^{p}=\tau^{p}e^{-1}\leq e^{-1}$ otherwise. Since there is at least one chart $U_{X}^{p}$ meeting $D_i$, we get $\hat{r}_i|_{R_i}<e^{-1}$. On the other hand, on $X\setminus R_i$ we have $\tau^{p}r_{i}^{p}=\tau^{p}e^{-1}$ for all $p$, so $\hat{r}_i|_{X\setminus R_i}=e^{-1}$, as claimed.

<ref> On each adapted chart $U_{X}^{p}$ we have $|f|=\prod_{j\in S^{p}} (r_{j}^{p})^{m_i}\leq (r_{i}^{p})^{m_i}$ for any $i\in S^{p}$, since  $r_i^{p}<e^{-1}<1$ by property <ref>. In turn, if $i\not\in S^{p}$ then on $U_{X}^{p}$ we have $(r_{i}^{p})^{m_i}=e^{-m_i}>\delta>|f|$ by our assumption about $\delta$. Thus for every $i$, we have $\tau^{p}r_{i}^{p}\geq \tau^{p}|f|^{1/m_{i}}$ on $U_{X}^{p}$. Away from $U_{X}^{p}$ both sides of this inequality are zero, so it holds on the entire $X$. Now $\hat{r}_{i}=\sum_{p}\tau^{p}r_{i}^{p}\geq \sum_{p}\tau^{p}|f|^{1/m_i}=|f|^{1/m_i}$, as claimed. The last assertion follows from part <ref>.

The distance functions $\hat{r}_i$ can also be constructed as follows. Fix a hermitian metric on the normal bundle $N_{D_i}$ to $D_{i}$ in $X$; and identify a tubular neighborhood $R_{i}$ of $D_{i}$ with the total space of the disc bundle in $N_{D_{i}}$. Next, define $\hat{r}_{i}|_{R_i}$ as the distance function in each fiber, composed with a smooth bijection $[0,1]\to [0,\frac{1}{e}]$ whose all derivatives at $1$ vanish. Then extend $\hat{r}_{i}$ to $X$ by $\hat{r}_i|_{X\setminus R_i}=\frac{1}{e}$.

Nonetheless, this alternative construction should not be regarded as more canonical than the one in Lemma <ref>. Indeed, choosing a hermitian metric on $N_{D_i}$ boils down to choosing a partition of unity.

§.§ A new smooth structure

Now, we repeat the definitions  of basic functions (<ref>) and (<ref>) using global distance functions $\hat{r}_i$ instead of the local $r_{i}$. That is, we put

\begin{equation}\label{eq:basic_functions_hat}
\hat{s}_i=\log \hat{r}_i,\quad \hat{t}_i=\frac{-1}{m_i \hat{s}_i},\quad	\hat{w}_{i}=\frac{t}{\hat{t}_i},\quad \hat{u}_{i}=\eta(\hat{w}_i),\quad \hat{v}_{i}=\hat{t}_i-\hat{u}_i, \quad \mbox{and }
\hat{\alpha}_i=-d^{c}\hat{s}_i.
\end{equation}

The $1$-form $\hat{\alpha}_i$, which will be used in the next section, is the analogue of the angular form $\alpha_i$ from  <cit.>. We use the convention $d^{c}=\imath (\d - \bar{\d})$. Note that Lemma <ref><ref> implies that $\hat{w}_{i}\in [0,1]$.

To write the new smooth charts, we replace $\bar{v}_i$ in formula (<ref>) by $\hat{v}_i$. That is, we define

\begin{equation}\label{eq:AC-chart_hat}
\hat{\psi}_{i}=(g,(\hat{v}_{j})_{j\in S\setminus \{i\}};\rest)\colon
U_{i}\to Q_{k,n+1}.
\end{equation}

With these definitions, we have an analogue of <cit.>.

The collection $(\hat{\psi}_{i})$ for all adapted charts $U_{X}$ and all $i$ in the associated index set of $U_X$, is a $\cC^{\infty}$ atlas on the A'Campo space $A$. This atlas is compatible with the $\cC^{1}$-atlas given by charts (<ref>). The space $A$ endowed with this $\cC^{\infty}$ structure has the following properties.

*  The map $\pi\colon A\to X$ is smooth. Its restriction $\pi|_{A\setminus \d A}\colon A\setminus \d A\to X\setminus D$ is a diffeomorphism.

*  The map $(g,\theta)\colon A\to [0,1)\times \S^{1}$ is a smooth submersion. In particular, $f_A\colon A\to \C_{\log}$ is smooth.

* For every $i\in \{1,\dots, N\}$ the functions $\hat{v}_i,\hat{w}_{i}$ and the $1$-form $\hat{\alpha}_i$ are smooth on $A$.

The proof of Proposition <ref>, which we outline below, follows the same steps as the proof of <cit.>. The key point is that we can compare the functions $\hat{t}_i,\hat{w}_i,\hat{v}_i$ with local functions $t_i,w_i,v_i$ in exactly the same way as we do in loc. cit. More precisely, we have the following formulas.

Let $U_{X}$ be an adapted chart meeting the divisor $D_i$. For a subset $I$ of the associated index set of $U_X$ put $U_{X,I}^{\circ}=U_{X}\cap X_{I}^{\circ}$. Moreover, put $a\de -m_i \log\lambda\in \cC^{\infty}(U_X)$, where $\lambda$ is the function from Lemma <ref><ref>. Then the functions introduced in (<ref>), (<ref>) and (<ref>) satisfy the following identities.

*  On $U_{X}$, we have $t_i-\hat{t}_{i}=\frac{at_i^2}{1-at_{i}}$, see <cit.>

*  On $U_{X}\setminus D$, we have $\hat{w}_{i}-w_{i}=at$, see <cit.>.

*  On $U_{X}$, we have $\hat{u}_{i}=u_i+\frac{u_{i}^{2}\log(1+a t_{i})}{1-u_{i}\log(1+a t_i)}$, see <cit.>

*  On $U_{X}\setminus D$, we have $v_{i}-\hat{v}_{i}=\frac{a t_{i}^{2}}{1+a t_{i}}+\frac{u_{i}^{2}\log(1+a t_{i})}{1-u_{i}\log(1+a t_i)}$, see <cit.>.

*  There is a bounded function $b\in \cC^{\infty}(U_{X,I}^{\circ})$ such that on $U_{X,I}^{\circ}$ we have $v_i-\hat{v}_i=\sigma_{i}b$, where $\sigma_{i}=t_{i}^{2}+t_iu_i^2$, see <cit.>.

*  Fix $\epsilon\in (0,1)$. There are bounded functions $c,q\in \cC^{\infty}(U_{X,I}^{\circ})$, and a bounded
$1$-form $\gamma\in \Omega^{1}(U_{X,I}^{\circ})$ such that

= t_{i}\cdot c\, dv_{i}+t_i^{\epsilon}\cdot q\, dg+\sigma_{i}\cdot \gamma$ on $U_{X,I}^{\circ}$,

see <cit.>.

*  On $U_{X,I}^{\circ}$ we have a fiberwise equality

$d\hat{v}_i=\sigma_{i}m_{i}(1+ct_i)\, ds_i+\sigma_i\gamma$ for some bounded $c\in \cC^{\infty}(U_{X,I}^{\circ})$, $\gamma\in \Omega^{1}(U_{X,I}^{\circ})$.

*  There is a smooth $1$-form $\beta\in \Omega^{1}(U_{X})$ such that on $U_X\setminus D$ we have $\hat{\alpha}_i=d\theta_{i}+\beta$. In particular, the $2$-form $d\hat{\alpha}_i$ extends to a smooth form on $X$, see <cit.>.

To get <ref>–<ref> and <ref>, we repeat the computation in the quoted part of [10], with the following modification. In loc. cit, we took another adapted chart $U_{X}'$, and on the intersection $U_{X}\cap U_{X}'$, we compared $t_i$, $w_i$ etc. with the corresponding functions $t_{i}'$, $w_i'$ etc. defined for $U_{X}'$. Each computation was based on the existence of smooth function $\lambda\colon \bar{U}_X\cap \bar{U}_X'\to (0,\infty)$ satisfying an equality $r_i'=\lambda r_i$. Now, we work on $U_{X}$ instead of $U_{X}\cap U_{X}'$, and use the analogous equality $\hat{r}_i=\lambda r_i$ given by Lemma <ref><ref>. Note that <ref> and <ref> are also analogues of <cit.>, with $\bar{v}_i$ replaced by $\hat{v}_i$.

For <ref>, we use a fiberwise inequality $dg=0$. As in <cit.>, from definitions (<ref>) we obtain a fiberwise equality $dv_i=\sigma_{i}m_i\, ds_i$. Substituting those two equalities to part <ref> proves <ref>.

We describe the minor adjustment needed to adapt the proof of <cit.> to the current setting.

Fix an adapted chart $U_{X}$ meeting a divisor $D_i$. The pair $(U_X,i)$ comes equipped with a transition function $a\in \cC^{\infty}(U_X)$, introduced in Lemma <ref>. We define an algebra $\cA_{i}$ in the same way as in formula (40) loc. cit, with the set $\cT_{i}$ replace by singleton $\{a\}$. This algebra keeps all the properties listed in loc. cit. Indeed, in the proof one only needs $a\in \cT_i$ to be a smooth function on $U_X$, small enough so that the denominators in Lemma <ref><ref> are bounded away from zero and infinity. Lemma <ref><ref> implies that this algebra $\cA_i$  contains functions $\hat{v}_i-v_i$.

The modified definition of $\cA_{i}$ leads to the new definition of algebras $\cI_{i}$, $\cJ$, $\cP$ and the matrix ring $\cM$. We introduce a matrix $\Psi$ in the same way as in formula (41) loc. cit, but with $\bar{v}_i$ replaced by $\hat{v}_i$. The fact that the functions $\hat{v}_i-v_i$ belong to $\cA_i$ implies that this new matrix $\Psi$ belongs to the new ring $\cM$: to see this, we repeat the proof of Lemma 3.47 loc. cit. Now, parts <ref>, <ref> and smoothness of $\hat{v}_i$ follow by repeating the proofs on p. 35 loc. cit. with this modified matrix $\Psi$.

Smoothness of the $1$-form $\hat{\alpha}_i$ follows from Lemma <ref><ref>: indeed, the function $\theta_i$ is is smooth since it is one of the coordinates of the chart (<ref>); and the $1$-form $\beta$ is smooth as a pullback of a smooth $1$-form on $U_X$ by $\pi$, which is smooth by <ref>. Similarly, to see that the function $\hat{w}_i$ is smooth, by Lemma <ref><ref> it is enough to prove smoothness of each local function $w_{i}$: indeed, the function $t=\exp(1-g^{-1})$ is smooth by <ref>, and the function $a$ is smooth by <ref>, as it is a pullback of a smooth function from $X$. Now, we note that $w_i$ belongs to the algebra $\cW_{i}'$ introduced in p. 32 loc. cit., hence to the algebra $\cA_{i}\cdot \cW_{i}'\cdot \cP$, see 3.4.5 loc. cit. The proof of Lemma 3.49 loc. cit, which is valid in the new setting, too, shows that all elements of this algebra are smooth, so $w_i$ is smooth, as needed.

§.§ The rounded dual complex

The rounded dual complex $\tilde{\Delta}_{D}$ is the (homeomorphic) image of the dual complex $\Delta_{D}$ by the map $(-\eta,\dots,-\eta)$, see <cit.>. For a face $\Delta_{I}$ of the dual complex $\Delta_{D}$, we define its rounded version $\tilde{\Delta}_{I}$ as the image of $\Delta_{I}$ by the same map. The set $\tilde{\Delta}_{I}$ is a manifold with boundary and corners, diffeomorphic to a standard simplex $\Delta_{I}$. Since the spaces $\tilde{\Delta}_{D}$, $\tilde{\Delta}_{I}$ are diffeomorphic to $\Delta_{D}$, $\Delta_{I}$ one can interchangeably think about each of them. Due to the nature of our smooth coordinate systems it will be more convenient for us to formulate statements using the rounded versions.

Recall that the stratification $X=\bigsqcup_{I}X_{I}^{\circ}$ lifts to a decomposition $A=\bigsqcup_{I} A_{I}^{\circ}$ given by $A_{I}^{\circ}=\pi^{-1}(X_{I}^{\circ})$. Lemma <ref> below gives a handy description of each piece $A_{I}^{\circ}$, cf. Lemma 3.10(c) or Proposition 3.33(d) of [10]. To state this result, we introduce some notation.

Let $\tau\colon X_{\log}\to X$ be the natural map from the Kato–Nakayama space, see diagram (<ref>). For $I\subseteq \{1,\dots, N\}$ put $(X_{I}^{\circ})_{\log}=\tau^{-1}(X_{I}^{\circ})$. The restriction of $\tau$ to $(X_{\log})_{I}^{\circ}$ is a smooth  $(\S^{1})^{\#I}$-bundle over $X_{I}^{\circ}$. Denoting by $\pi_{\log}$ the projection $A\to X_{\log}$, we have $\pi_{\log}^{-1}(X_{I}^{\circ})_{\log}=A_{I}^{\circ}$.

For every nonempty subset $I\subseteq \{1,\dots,N\}$, the map

\begin{equation}\label{eq:product}
(\pi_{\log},(\hat{v}_{i})_{i\in I})\colon A_{I}^{\circ}\to (X_{I}^{\circ})_{\log}\times \tilde{\Delta}_{I}
\end{equation}

is a diffeomorphism  of manifolds with boundary and corners.

In $A_{I}^{\circ}$ we have $\hat{t}_{i}=0$ for $i\in I$, so $\hat{v}_{i}=-\hat{u}_{i}=u_{i}=-\eta(w_i)$ for $i\in I$, where $u_i$, $w_{i}$ are functions introduced in (<ref>), for any adapted chart with $i$ in its associated index set. Hence the map (<ref>) indeed takes values in $(X_{I}^{\circ})_{\log}\times \tilde{\Delta}_{I}$. It is surjective by  <cit.>. It is injective because a point in the A'Campo space is uniquely determined by its image in $X_{\log}$ and the values of all functions $u_{i}$; and since on $A_{I}^{\circ}$ we have $u_{i}=0$ for $i\not\in I$, these data are determined by the image of the map (<ref>), as needed. To see that (<ref>) is a local diffeomorphism, fix a point $x\in A_{I}^{\circ}$ and a chart (<ref>) around $x$, with coordinates $(\hat{v}_{j})_{j\in I\setminus \{i\}}$. At the image of $x$ in $\tilde{\Delta}_{I}$, we can choose a smooth chart with coordinates $-\eta(w_j)$ for $j\in I\setminus \{i\}$. In these coordinates, the map (<ref>) is the identity, as needed.

The space $\Gamma$ in diagram (<ref>) can also be  defined as the closure of the graph of the map $(\hat{u}_1,\dots,\hat{u}_N)\colon X\setminus D\to \R^{N}$. Let $\hat{\pi}\colon \hat{A}\to X$ be the A'Campo space obtained this way. Since the limit of $w_i$ as we approach $D$ does not depend on the chosen adapted chart <cit.>, the new boundary $\d\hat{A}$ is equal to the original $\d A$, as a subset of $X_{\log}\times \R^{N}$. Thus we can define a map $\hat{\Phi}\colon \hat{A}\to A$ as the identity on $\d \hat{A}$, and $\pi^{-1}\circ \hat{\pi}$ on $\hat{A}\setminus \d\hat{A}=X\setminus D$. Exactly as in the proof of <cit.>, we see that $\hat{\Phi}$ is a homeomorphism. Moreover, if we define $\cC^{\infty}$-charts $\tilde{\psi}_i$ on $\hat{A}$ by the same formula as (<ref>), we have $\tilde{\psi}_i=\hat{\psi}_{i}\circ \hat{\Phi}$, so they endow $\hat{A}$ with a $\cC^{\infty}$-structure.

This way, we see that the A'Campo space $A$, as a smooth manifold, depends only on the initial function $f\colon X\to \C$ and the choice of the distance functions $\hat{r}_i$  as in Lemma <ref>. Formula (<ref>) for the $\cC^{1}$-charts show that the $\cC^{1}$-structure on $A$ is independent of this choice.

§ FIBERWISE KÄHLER FORMS

Let $f\colon X\to \C$ be as in Section <ref>, and let $A$ be the A'Campo space of $f$, equipped with the smooth structure given by charts (<ref>). In this section, we introduce exact forms $\omega^{\sharp}$ and $\omega^{\flat}$ appearing in formula (<ref>), and prove that the resulting forms $\omega_{q}^{\epsilon}$ are indeed fiberwise symplectic, see Proposition <ref>.

The form $\omega^{\sharp}$ is an analogue of the form $\omega_{E}$ introduced in <cit.>. It is designed so that for any sufficiently small $\epsilon>0$, the form  $\pi^{*}\omega_{X}+\epsilon \omega^{\sharp}$ extends to a fiberwise symplectic form on $A$, and in particular yields a well-behaved monodromy at radius zero. The additional feature of $\omega^{\sharp}$ with respect to $\omega_{E}$ from loc. cit. is its fiberwise $J$-compatibility, which we prove in Lemma <ref>. In turn, the form $\omega^{\flat}$ is designed so that the induced metric degenerates at radius zero to the euclidean one on $\Delta_{\cS}$,  as required by Theorem <ref><ref>. It is inspired by the semi-flat models surveyed in <cit.>.

We identify $A\setminus \d A$ with $X\setminus D$ via diffeomorphism $\pi|_{A\setminus \d A}\colon A\setminus \d A\to X\setminus D$, and denote by $J$ the induced complex structure. We fix an open set $W_{X}$ as in Section <ref>, i.e. such that $f|_{\bar{W}_{X}}\colon \bar{W}_X\to \D_{\delta}$ is proper, has connected fibers, and restricts to a submersion over $\D_{\delta}^{*}$. We put $W=\pi^{-1}(W_X)$ and write  $W_{z}$ for the fiber $f_{A}^{-1}(z)\cap W$. We also identify $\D_{\delta}^{*}$ with its preimage in $\D_{\delta,\log}$.

§.§ The A'Campo form $\omega^{\sharp}$

Recall that by Proposition <ref>, for each $i\in \{1,\dots, N\}$ we have a smooth function $\hat{v}_i\in \cC^{\infty}(A)$ and a smooth angular $1$-form $\hat{\alpha}_i\in \Omega^{1}(A)$, introduced in formula (<ref>). We now define smooth forms on $A$:

\begin{equation}\label{eq:omega_sharp}
\lambda^{\sharp}=\sum_{i=1}^{N}\hat{v}_i\cdot \hat{\alpha}_i,\quad
\omega^{\sharp}=d\lambda^{\sharp}.
\end{equation}

There is a smooth function $\phi\colon W\setminus \d A\to \R$ satisfying a fiberwise equality $\lambda^{\sharp}=d^{c}\phi$. In particular, $\omega^{\sharp}$ is fiberwise $J$-compatible, i.e. we have an equality $\omega^{\sharp}(J\sdot,J\sdot)=\omega^{\sharp}(\sdot,\sdot)$ on the tangent space to the fiber $W_{z}$ for all $z\in \D_{\delta}^{*}$.

Clearly, it is enough to prove the first assertion. If $D$ has only one component $D_1$ then $\hat{s}_1=-m_{1}t^{-1}$ is fiberwise constant, hence $\lambda^{\sharp}=-\hat{v}_1\cdot d^{c}\hat{s}_1$ is fiberwise zero. Therefore, we can assume that $D$ has at least two components. Fix $i\in \{1,\dots, N\}$.

By definition (<ref>), the function $\hat{v}_i\colon W_{X}\setminus D\to \R$ is equal to the composition $\check{v}_i\circ (\hat{s}_i,t)$, where

\begin{equation*}
\check{v}_i\colon T\to \R\quad \mbox{is defined as}\quad
\check{v}_i(\check{s},\check{t})=-(m_i\check{s})^{-1}-\eta(-m_{i}\cdot \check{s}\check{t}),
\end{equation*}

and $T\subseteq \R^2$ is the image of the map $(\hat{s}_i,t)\colon W_{X}\setminus D\to \R^2$.

We claim that $T$ is a union of some horizontal segments (or half-lines), ending on the vertical line $\check{s}=-1$. By Lemma <ref><ref> we have $\hat{s}_{i}\leq -1$, and the equality $\hat{s}_i=-1$ holds on $X\setminus R_{i}$. Since $D$ has at least two components, the set $X\setminus D_i$ contains a nonempty set $R_{j}^{\circ}$ for some $j\neq i$, which meets each fiber by property <ref> of the atlas used to define the functions $\hat{r}_i$. Thus the equality $\hat{s}_i=-1$ is attained somewhere on each fiber, and the claim follows since each fiber is connected.

Therefore, we can define a smooth  function $\check{\phi}_i\colon T\to \R$ by the formula $\check{\phi}_{i}(\check{s},\check{t})=\int_{\check{s}}^{-1} \check{v}_i(s,\check{t})\, ds$. Fundamental theorem of calculus gives $\frac{\d}{\d \check{s}}\check{\phi}_{i}=-\check{v}_i$. We define a smooth function  $\phi_{i}\colon W_X\setminus D\to \R$ as $\phi_{i}\de \check{\phi}_{i}\circ (\hat{s}_i,t)$. Since $\frac{\d}{\d \check{s}}\check{\phi}_{i}=-\check{v}_i$, we have a fiberwise equality $d\phi_{i}=-\check{v}_i\circ (\hat{s}_i,t)\, d\hat{s}_i=-\hat{v}_i\, d\hat{s}_i$, so $d^{c}\phi_{i}=-\hat{v}_i\, d^{c}\hat{s}_i=\hat{v}_i\hat{\alpha}_i$ by definition (<ref>) of $\hat{\alpha}_i$. Eventually, we put $\phi=\sum_{i=1}^{N}\phi_{i}$, so we have a fiberwise equality $d^{c}\phi=\sum_{i=1}^{N}\hat{v}_i\hat{\alpha}_i=\lambda^{\sharp}$, as needed.

§.§ The semi-flat form $\omega^{\flat}$

As in Section <ref>, we fix any subset $\cS\subseteq \{1,\dots, N\}$: in the setting of Calabi–Yau degenerations, it will correspond to essential divisors. Now, we define smooth forms on $A$:

\begin{equation}\label{eq:omega_flat}
\lambda^{\flat}=-\sum_{i\in \cS} m_{i}\, \hat{w}_i\cdot \hat{\alpha}_i,
\quad
\omega^{\flat}=d\lambda^{\flat}.
\end{equation}

Put $\phi^{\flat} = -\frac{1}{2t}\sum_{i\in \cS} \hat{w}_i^2$. Then the form $d^{c}\phi^{\flat}$ extends to a smooth form on $A$, and we have an equality $\lambda^{\flat}=d^{c}\phi^{\flat}+\frac{1}{2}\sum_{i\in \cS} \hat{w}_{i}^{2}\cdot d\theta$. In particular, we have a fiberwise equality $\lambda^{\flat}=d^{c}\phi^{\flat}$, so the form $\omega^{\flat}$ is fiberwise $J$-compatible.

Using identities  $\hat{w}_i=-m_{i}\hat{s}_{i}t$ and $-d^{c}\hat{s}_i=\hat{\alpha}_i$ from (<ref>), we get

\begin{equation*}
d^{c}\hat{w}_i=-m_{i}\, d^{c}(\hat{s}_it)=m_{i}t\, \hat{\alpha}_i - m_{i}\hat{s}_i d^{c}t=
m_{i}t\, \hat{\alpha}_i+\tfrac{1}{t}\hat{w}_i\, d^{c}t.
\end{equation*}

Put $\phi_{i}^{\flat}=-\tfrac{1}{2t}\hat{w}_i^2$, so $\phi^{\flat}=\sum_{i\in \cS} \phi_{i}^{\flat}$. The above equality gives

\begin{equation*}
d^{c} \phi_{i}^{\flat}=
-\tfrac{1}{t}\hat{w}_i\, d^{c}\hat{w}_i
+\tfrac{1}{2t^2} \hat{w}_i^2\, d^{c}t=
-m_{i} \hat{w}_i\hat{\alpha}_i
-\tfrac{1}{2t^2}\hat{w}_i^2\, d^{c}t.
\end{equation*}

Moreover, we have an equality

\begin{equation*}
\end{equation*}
Indeed, putting $s=\log |f|$ we see that $d^{c}s=-d\theta$ and $t=-s^{-1}$, so $d^{c}t=-d^{c}s^{-1}=s^{-2}d^{c}s=-t^{2}d\theta$, as claimed. Substituting this to the previous equality, we conclude that

\begin{equation*}
d^{c} \phi_{i}^{\flat}=-m_{i}\, \hat{w}_i\alpha_i+\tfrac{1}{2} \hat{w}_i^2\, d\theta.
\end{equation*}

By Proposition <ref><ref>,<ref>, the functions $\theta$, $\hat{w}_i$ and the form $\hat{\alpha}_i$ are smooth in $A$, hence so is $d^{c}\phi_{i}^{\flat}$.
The result follows after taking a sum over all $i\in \cS$.

§.§ The form omega-q is fiberwise Kähler

Recall that we have fixed a Kähler form $\omega_{X}$ on $X$, and introduced $2$-forms $\omega^{\sharp}$ and $\omega^{\flat}$ by formulas (<ref>) and (<ref>). We now  introduce a family of smooth $2$-forms on $A$ by formula (<ref>), which reads as

\begin{equation*}
\omega_{q}^{\epsilon}=\pi^{*}\omega_{X}+\epsilon \cdot ( q\cdot \omega^{\sharp}+(1-q) \cdot \omega^{\flat}), \quad q\in [0,1].
\end{equation*}

The following proposition summarizes key properties of this family.

There is an $\epsilon_0>0$ and $\delta'>0$ such that the following holds.

*  For every $q\in [0,1]$ and $\epsilon\in [0,\epsilon_0]$, the form $\omega_{q}^{\epsilon}$ is Kähler on the fiber $W_{z}$ for every $z\in \D_{\delta'}^{*}$

*  For every $q\in (0,1]$ and $\epsilon\in (0,\epsilon_0]$, the form $\omega_{q}^{\epsilon}$ is symplectic on the fiber $W_{z}$ for every $z\in \D_{\delta',\log}$.

*  Let $A^{\textnormal{int}}_{\cS}$ be the union of $\Int_{\d A} A_{I}^{\circ}$ for all faces $\Delta_{I}$ of $\Delta_{\cS}$. Then for every $\epsilon\in (0,\epsilon_0]$ there is a neighborhood $V_{\epsilon}$ of $W\cap A^{\textnormal{int}}_{\cS}$ in $A$ such that the restriction $\omega_{0}^{\epsilon}|_{V_{\epsilon}}$ is fiberwise symplectic.

The proof of Proposition <ref><ref> follows the steps of <cit.>. The starting point is the following analogue of formula (53) loc. cit.

Let $U_{X}$ be an adapted chart with associated index set $I$, and let $U=\pi^{-1}(U_X)$. Then on every fiber $f_{A}^{-1}(r,\theta)\cap U$ for $r>0$ we have equalities

*  $\omega^{\sharp}=\sum_{i\in I} \sigma_i \cdot (m_i(1+c_it_i)\, d s_i\wedge d\theta_{i}+\gamma_i\wedge d\theta_{i}+m_i(1+c_it_i)\, d s_i\wedge \beta_{i})+\check{\omega}$,

$\omega^{\flat}=\sum_{i\in I\cap \cS} t\cdot (m_i^2\, ds_i\wedge d\theta_{i}+da_i\wedge d\theta_{i}+ d s_i\wedge \beta_{i})+\tilde{\omega}$,

for some bounded functions $a_i$, $c_i$ and bounded forms $\beta_{i}$, $\gamma_{i}$, $\check{\omega}$, $\tilde{\omega}$ on $U_{X}\setminus D$.

<ref> By definition (<ref>) of $\omega^{\sharp}$ we have

\begin{equation*}
\omega^{\sharp}=\sum_{i\in I}d\hat{v}_i\wedge \hat{\alpha}_i+\check{\omega}',
\end{equation*}

where $\check{\omega}'=\sum_{i\not\in I}d\hat{v}_i\wedge \hat{\alpha}_i+\sum_{i=1}^{N}\hat{v}_i d\alpha_{i}$. Lemma <ref><ref>,<ref> implies that the forms $d\hat{v}_i$, $\hat{\alpha}_{i}$ for $i\not\in I$ and $d\hat{\alpha}_i$ for all $i$ are bounded on $U_X\setminus D$, cf. <cit.>. Thus the form $\check{\omega}$ is bounded on $U_{X}\setminus D$. Now part <ref> follows by substituting the fiberwise formulas for $d\hat{v}_i$ and $\hat{\alpha}_i$ from  Lemma <ref><ref>,<ref>.

<ref>  By definition (<ref>) of $\omega^{\flat}$ we have

\begin{equation*}
\omega^{\flat}=-\sum_{i\in I\cap \cS}m_i\, d\hat{w}_{i}\wedge \hat{\alpha}_i+\tilde{\omega}',
\end{equation*}

where $\tilde{\omega}'=-\sum_{i\in \cS\setminus I}m_i\, d\hat{w}_i\wedge \hat{\alpha}_{i}-\sum_{i\in \cS} m_i\, \hat{w}_id\hat{\alpha}_i$, so as before, the form $\tilde{\omega}\in \Omega^{2}(U_X\setminus D)$ is bounded. By Lemma <ref><ref> we have $\hat{w}_i=w_{i}+ta_{i}=-m_{i}s_{i}t+ta_{i}$ for some smooth function $a_{i}\in \cC^{\infty}(U_X)$. Combining it with the formula for $\hat{\alpha}_i$ in Lemma <ref><ref>, we get part <ref>.

Lemmas <ref> and <ref> imply that $\omega_{q}^{\epsilon}|_{A\setminus \d A}$ is fiberwise $J$-compatible. Thus it remains to prove that $\omega_{q}^{\epsilon}$ tames $J$ on each fiber $W_{z}$, $z\in \D_{\delta'}^{*}$. This result is analogous to <cit.>. We now explain how to adapt its proof.

Covering $\bar{W}_X$ with finitely many adapted charts, we see that it is enough to work locally, in an adapted chart $U_X$. Say that its associated index set is $\{1,\dots,k\}$. Reordering the components of $D$ if needed, we can assume that $\cS\cap \{1,\dots, k\}=\{1,\dots,l\}$ for some $1\leq l \leq k$. We group terms from Lemma <ref> as follows:

\begin{equation}\label{eq:omega-parts}\begin{split}
\omega_{1}&\de \sum_{i=1}^{k} \sigma_i \cdot (m_i(1+ct_i)\, d s_i\wedge d\theta_{i}+\gamma_i\wedge d\theta_{i}+m_i(1+ct_i)\, d s_i\wedge \beta_{i}),\\
\omega_{2}&\de \sum_{i=1}^{l} t\cdot (m_i^2\,ds_i\wedge d\theta_{i}+da_i\wedge d\theta_{i}+d s_i\wedge \beta_{i}), \\
\omega_{3,q}&\de q\cdot \check{\omega}+(1-q)\cdot \tilde{\omega}\quad \mbox{for } q\in [0,1].
\end{split}\end{equation}

For a $2$-form $\omega$, we denote by $\omega^{J}$ the symmetric part of the form $\omega(\sdot,J\sdot)$. With this notation, we can split the formula (<ref>) defining $\omega_{q}^{\epsilon}$ as follows.

\begin{equation*}
(\omega_{q}^{\epsilon})^{J}=q\cdot (\tfrac{1}{3}\pi^{*}\omega_{X}^{J}+\epsilon \omega_{1}^{J})+(1-q)\cdot (\tfrac{1}{3}\pi^{*}\omega_{X}^{J}+\epsilon\omega_{2}^{J})+(\tfrac{1}{3}\pi^{*}\omega_{X}^{J}+\epsilon \omega_{3,q}^{J}).
\end{equation*}

Therefore, to prove Proposition <ref><ref> it is enough to prove the following three assertions.

* There exists $\epsilon_1>0$ such that for every $\epsilon\in [0,\epsilon_1]$ the form $\frac{1}{3}\pi^{*}\omega_{X}^{J}+\epsilon \omega_{1}^{J}$ is positive definite.

*  There exists $\epsilon_2>0$ such that for every $\epsilon\in [0,\epsilon_2]$ the form $\frac{1}{3}\pi^{*}\omega_{X}^{J}+\epsilon\omega_{2}^{J}$ is positive definite.

*  There exists $\epsilon_3>0$ such that for every $\epsilon\in [0,\epsilon_3]$ and every $q\in [0,1]$ the form $\frac{1}{3}\pi^{*}\omega_{J}+\epsilon \omega_{3,q}^{J}$ is positive definite.

To see assertion <ref>, recall that $\omega_{3,q}\in \Omega^{2}(U_X\setminus D)$ is a family of bounded forms, smoothly parametrized by $q\in [0,1]$. Since the interval $[0,1]$ is compact, this family is uniformly bounded, so for sufficiently small $\epsilon>0$, positive definiteness of $\pi^{*}\omega_{X}^{J}$ implies that the sum  $\frac{1}{3}\pi^{*}\omega_{J}+\epsilon \omega_{3,q}^{J}$ is positive definite, too.

To see assertion <ref>, note that the definition of $\omega_{1}$ is entirely analogous to formula (53) in [10]. Repeating the proof of Proposition 4.3 loc. cit. from this formula onward, we get assertion <ref>.

It remains to prove <ref>. The definition of $\omega_{2}$ is still similar to the aforementioned  formula (53). We now explain how to adapt the proof of Proposition 4.3 loc. cit. to this setting.

Recall that $d^{c}s_i=-d\theta_{i}$ and $d^{c}\theta_{i}=d^{c}s_i$. Applying these rules to the definition (<ref>) of $\omega_{2}$, we get

\begin{equation}\label{eq:omega2J}
\omega_{2}^{J}=t\cdot \sum_{i=1}^{l} m_{i}^2\, ds_{i}^{2}+m_i^2\, d\theta_{i}^{2}+ds_{i} \cdot \beta_{i}'+d\theta_{i}\cdot \beta_{l+i}'
\end{equation}

for some $1$-forms $\beta_{i}'$ bounded with respect to the natural coordinates on $U_X\setminus D$.

Consider a basis $\{ds_1,\dots,  ds_l;d\theta_1,\dots,d\theta_l,dx_{1},\dots,dx_{2(n+1-l)}\}$ of the cotangent space $T^{*}(U\setminus \d A)$, where $x_{j}$ stand for real and imaginary parts of the remaining coordinates of $U_{X}$. Let $\{\nu_1,\dots,\nu_{2l},\allowbreak \xi_{1},\dots,\xi_{2(n+1-l)}\}$ be the corresponding dual basis of $T(U\setminus \d A)$. Let $||\sdot ||$ be the maximum norm in $T(U \setminus \d A)$ with respect to this basis, and let $||\cdot ||_{X}$ be a maximum norm in $TU_X$ with respect to the natural coordinates. As in <cit.>, we note the following two properties.

*  For $i\in \{1,\dots, 2l\}$, we have $||\pi_{*}\nu_{i}||_{X}\rightarrow 0$ as $r_1,\dots,r_k\rightarrow 0$.

*  Let $\Xi$ be the subspace spanned by $\{\xi_1,\dots,\xi_{2(n+1-l)}\}$. Then there is a constant $K_0>0$ such that for every $\xi\in \Xi$ we have $||\pi_{*}\xi||_{X}\geq K_{0}||\xi||$.

The matrix of $\omega_{2}^{J}$ in this basis has a block form

\begin{equation*}
t\cdot \left[\begin{matrix}
A & Q \\ Q & P
\end{matrix}\right].
\end{equation*}

We claim that, after possibly shrinking the chart $U_{X}$, the entries of each block have the following properties, cf. <cit.>.

*  The diagonal terms of $A$ are bounded from below by $\frac{1}{4}$.

*  The off-diagonal terms of $A$ converge to $0$ as $r_1,\dots,r_k\rightarrow 0$.

*  All entries of the blocks $Q$ and $P$ are bounded functions on $U\setminus \d A$.

These properties follow easily from formula (<ref>). Indeed, the $(i,j)$-term of $A$ for $i,j\in \{1,\dots, l\}$ is the function $\delta_{i}^{j}\bar{m}^2_{i}+\beta_{i}'(\nu_{j})+\beta_{j}'(\nu_{i})$, where $\delta_{i}^{j}$ is the Kronecker delta, and $\bar{m}_{i}=m_i$ if $i\leq l$ and $\bar{m}_{i}=m_{i-l}$ if $i>l$. Property <ref> and boundedness of the $1$-forms $\beta_{i}'$ imply that $\beta_{i}'(\nu_{j})\rightarrow 0$ as we approach the origin of the chart, so after possibly shrinking $U_{X}$, we get properties <ref> and <ref>. To see property <ref>, we note that $P=0$, and the $(i,j)$-term of $Q$ for $i\in \{1,\dots,2l\}$, $j\in \{1,\dots,2(n+1-k)\}$ is $\beta_{i}'(\xi_{j})$. Since the $1$-form $\beta_{i}'$ and the vector field $\xi_{j}$ are bounded with respect to natural coordinates on $U_X$, so is the function $\beta_{i}'(\xi_{j})$, as needed.
It is immediate to see that all properties <ref>–<ref> are preserved by the first $2l$ steps of the Gaussian diagonalization. Indeed, at each step we add to $\nu_{i}$ or $\xi_{j}$ a linear combination of vectors $\nu_{1},\dots,\nu_{2l}$, with bounded coefficients, so properties <ref>, <ref> are kept. In the matrix, to each entry of $A$ and $Q$ we add a function converging to $0$ as $r_1,\dots,r_k\rightarrow 0$; and to each entry of $P$ we add a bounded function. Hence after possibly shrinking $U_X$ at each step, we see that properties <ref>–<ref> are preserved, too.

This way, we get a basis of $T(U\setminus \d A)$ satisfying <ref>, <ref>, such that the matrix of $\omega_{2}^{J}$ in this basis is

\begin{equation*}
t\cdot \left[\begin{matrix}
A' & 0 \\ 0 & P'
\end{matrix}\right],
\end{equation*}

where $A'$ is a diagonal matrix with entries bounded from below by $\frac{1}{4}$; and all entries of $P'$ are bounded.

Fix a nonzero vector $v\in T(U\setminus \d A)$, and decompose it as $v=\nu+\xi$, according to the above splitting. The above block-diagonal form of $\omega_{2}^{J}$ shows that

\begin{equation*}
\omega_{2}^{J}(\nu,\nu)\geq t\cdot \tfrac{1}{4}||\nu||^2,\quad |\omega_{2}^{J}(\xi,\xi)|\leq t\cdot K||\xi||^2
\end{equation*}

for some constant $K>0$, uniform on $U\setminus \d A$.

Now, we repeat the proof of <cit.>, starting on p. 44 loc. cit, with inequalities (58) and (59) replaced by the above ones; and the forms $\pi^{*}\omega_{X}^{J}$, $\omega_{E}$ replaced by $\frac{1}{3}\pi^{*}\omega_{X}^{J}$, $\omega_2$. The proof goes through without any further changes, except for the second-to-last line. There, in the inequality bounding $C$, the number $\epsilon$ is now multiplied by $t$: i.e. the modified inequality reads as $C\geq (\frac{1}{3}K_1-\epsilon tK)||\xi||^2$. Shrinking $\delta>0$ we can make $t$ as small as needed, which proves <ref>.

We now move on to the proof of Proposition <ref><ref>,<ref>, that is, fiberwise non-degeneracy of the forms $\omega_{q}^{\epsilon}$ for $q>0$ on $W\cap \d A$; and of $\omega_{0}^{\epsilon}$ on $W\cap A_{\cS}^{\textnormal{int}}$. We follow the proof of  <cit.>, with $\tilde{\omega}^{\epsilon}_{q}$ playing the role of $\omega_{A}^{\epsilon}$. First, we introduce some notation as in Lemma 4.13 loc. cit.

Fix a smooth chart (<ref>), call it $U$. Reordering the components of $D$ if needed, we can assume that the associated index set of this chart is $S=\{1,\dots, k\}$, and the index $i$ excluded in the formula (<ref>) is $1$. Fix a subset $I\subseteq S$ containing $1$, say $I=\{1,\dots, l\}$. Then as in <cit.>, we see that at each point of $U_{I}^{\circ}\de U\cap A_{I}^{\circ}$ we have smooth coordinates

\begin{equation*}
\end{equation*}

These coordinates provide a splitting $TU_{I}^{\circ}=\cV_{I}\oplus \Theta_{I} \oplus \cZ_{I}$, where $\cV_{I}=\sspan\{\frac{\d}{\d\hat{v}_2},\dots,\frac{\d}{\d\hat{v}_{l}}\}$, $\Theta_{I}=\sspan \{\frac{\d}{\d \theta_{1}},\dots,\frac{\d}{\d\theta_{l}}\}$, and $\pi_{*}$ maps $\cZ_{I}$ isomorphically to the tangent bundle to the stratum $\pi(U)\cap X_{I}^{\circ}$.

We have the following analogue of <cit.>.

On $U_{I}^{\circ}$, the following holds.

*  For every $\beta\in \Omega^{*}(U_X \cap X_{I}^{\circ})$ we have $\pi^{*}\beta|_{\cV_{I}\oplus \Theta_{I}}=0$.

*  For every $j>l$ we have $d\hat{w}_j=0$ and $d\hat{v}_j|_{\cV_I\oplus \Theta_I}=0$.

*  For every $i\in \{1,\dots, k\}$, $j\in \{1,\dots, l\}$ we have $\hat{\alpha}_{i}|_{\cV_{I}}=0$ and $\hat{\alpha}_{i}(\tfrac{\d}{\d \theta_{j}})=\delta_{i}^{j}$.

*  We have $d\hat{v}_{1}=-\sum_{i=2}^{l}\frac{\zeta(u_i)}{\zeta(u_1)}\, d\hat{v}_i$, where $\zeta=(\eta^{-1})'\colon [0,1]\ni s\mapsto s^{-2}e^{1-s^{-1}}\in [0,\infty)$.

*  For every $i\in \{1,\dots, l\}$ we have  $d\hat{w}_{i}|_{\Theta_{I}\oplus \cZ_{I}}=0$ and $d\hat{v}_{i}|_{\Theta_{I}\oplus \cZ_{I}}=0$.

*  For every $i\in \{1,\dots, l\}$ we have $\omega_{q}^{\epsilon}(\sdot,\tfrac{\d}{d\theta_{i}})=\epsilon c_{i}\, d\hat{v}_i$, where  $c_{i}=q+(1-q)\cdot m_i \zeta(u_i)$ if $i\in \cS$ and $c_{i}=q$ if $i\not\in \cS$.

The computations are exactly the same as in <cit.>. The additional assertions about the function $\hat{w}_i$ follow from the fact that on each piece $A_{J}^{\circ}$ of $\d A$, we have $\hat{w}_i=0$ if $i\not\in J$ and $\hat{v}_i=-\eta(\hat{w}_i)$ otherwise; so $\hat{w}_i$ is either zero or a re-parametrization of $\hat{v}_i$.

First, as in <cit.> we prove that, after possibly shrinking $\epsilon>0$ and the chart $U_X$, the restriction $\omega^{\epsilon}_{q}|_{\cZ_{I}}$ is nondegenerate. To see this, we note that by Lemma <ref><ref>, <ref> we have $d\hat{w}_i|_{\cZ_I}=0$ for all $i\in \{1,\dots, N\}$, so $\omega^{\flat}|_{\cZ_I}=-\sum_{i=1}^{N} m_i\, \hat{w}_{i}d\hat{\alpha}_{i}$, which is bounded in the natural coordinates of $\pi(U_{X,I}^{\circ})$. This way, we can repeat the proof of Lemma 4.15 loc. cit., treating the contribution from $\omega^{\flat}$ as a part of the form $\check{\omega}_1$.

With non-degeneracy of $\omega^{\epsilon}_{q}|_{\cZ_{I}}$ at hand, we now repeat the proof of Proposition 4.12 loc. cit, replacing each $\bar{v}_i$ for $i\in \{1,\dots, l\}$ by $c_{i}\, d\hat{v}_i$. We now outline each step of the proof.

Fix a point $x\in U_{I}$, and assume that $q>0$ or $x\in \Int_{\d A} A_{I}^{\circ}$ and $I\subseteq \cS$ (so we are in the setting of Proposition <ref><ref> or <ref>, respectively). Fix a nonzero vector $\nu\in T_{x}(\d A)$ which is $\omega^{\epsilon}_{q}$-orthogonal to the radius-zero fiber $F\de f_{A}^{-1}(0,\theta)$. We need to prove that $\nu\not\in T_{x}F$. In our coordinates, we have $T_{x}F=\ker[\sum_{i=1}^{l}m_{i}d\theta_{i}]$ and $\nu=\sum_{i=2}^{l}a_{i}\frac{\d}{\d\hat{v}_i}+\sum_{i=1}^{l}b_{i}\frac{\d}{\d \theta_i}+\xi$ for some $a_i,b_i\in \R$ and a vector $\xi\in \cZ_{I}$, see formulas (65) and (66) loc. cit.

We claim that at the point $x$, the coefficients $c_{1},\dots,c_{l}$ from Lemma <ref><ref> are positive. This is clear if $q>0$. If $q=0$ then by assumption $I\subseteq \cS$, so $c_{i}=m_{i}\zeta(u_i)$ for all $i$, and $x\in \Int_{\d A} A_{I}^{\circ}$, so $u_{i}>0$ for all $i$, too, which proves the claim.

Next, we claim that $a_i=0$ for all $i\in \{2,\dots, l\}$. Let $\nu_{i}^{\theta}\de \frac{1}{m_{i}}\frac{\d}{\d \theta_{i}}- \frac{1}{m_{1}}\frac{\d}{\d \theta_{1}}\in TF$. Lemma <ref><ref> gives $
\epsilon \cdot( \tfrac{c_{i}}{m_{i}} a_{i}-\tfrac{c_1}{m_1}d\hat{v}_{1}(\nu))$,
so $\frac{c_{i}}{m_{i}}a_{i}=\frac{c_1}{m_{1}}d\hat{v}_{1}(\nu)$. If $a_{i}\neq 0$ then $d\hat{v}_i(\nu)\neq 0$ and by Lemma <ref><ref> we have
-1=\frac{1}{d\hat{v}_1(\nu)}\cdot \sum_{i=2}^{l}\frac{\zeta(u_i)}{\zeta(u_1)}a_{i}=
\sum_{i=2}^{l}\frac{\zeta(u_i)m_{i}c_1}{\zeta(u_1)m_1c_i}>0,
a contradiction.

Now, using non-degeneracy of $\omega^{\epsilon}_{q}|_{\cZ_{I}}$, we prove that $\xi=0$. Fix $\xi'\in \cZ_{I}$. By Lemma <ref><ref>,<ref> we have $\omega_{q}^{\epsilon}(\xi',\frac{\d}{\d \theta_{i}})=\epsilon c_{i}d\hat{v}_{i}(\xi')=0$, so  $\omega_{q}^{\epsilon}(\xi',\nu)=\omega_{q}^{\epsilon}(\xi',\xi)$ since all the coefficients $a_{i}$ vanish. Thus non-degeneracy of $\omega^{\epsilon}_{q}|_{\cZ_{I}}$ indeed implies $\xi=0$.

Eventually, we compute the coefficients $b_{i}$. Lemma <ref><ref> gives $0=\omega_{q}^{\epsilon}(\frac{\d}{\d \hat{v}_i},\nu)=\sum_{j}b_{j}\cdot \epsilon c_j d\hat{v}_{j}(\frac{\d}{\d \hat{v}_{i}})=\epsilon(b_{i}c_{i}-b_{1}c_1\frac{\zeta(u_i)}{\zeta(u_1)})$ by Lemma <ref><ref>. Hence there is a number $b$ such that for all $i$ we have $b_i=b\cdot \frac{\zeta(u_i)}{c_i}$, and $b\neq 0$ since $\nu\neq 0$. Now $\sum_{i}m_{i}d\theta_{i}(\nu)=b\sum_{i}m_{i}\frac{\zeta(u_i)}{c_i}\neq 0$, so $\nu\not\in T_{x}F$, as claimed.

For $q=1$, computing the number $b$ in the above proof shows that the $\omega_{1}^{\epsilon}$-symplectic lift to $\d A$ of the angular vector field $\frac{\d}{\d \theta}$ on $\d\D_{\delta,\log}$ is given by $\frac{1}{\sum_{i}m_{i}\zeta(u_i)}\sum_{i}\zeta(u_i)\frac{\d}{\d \theta_{i}}$, which is precisely formula (48) from [10]. Thus the radius-zero monodromy for $\omega_{1}^{\epsilon}$ is exactly the same as in loc. cit, which means that one prove the results of loc. cit. using the form $\omega_{1}^{\epsilon}$ instead of $\omega_{A}^{\epsilon}$.

Replacing $\omega_{q}^{\epsilon}$ by $\omega_{q}^{\epsilon}+C\cdot dg\wedge d\theta$ for $C\gg 0$, we can guarantee that the form $\omega_{q}^{\epsilon}$ is symplectic (and Kähler away from $\d A$), not just   fiberwise symplectic (or Kähler).

Fix $\delta_{0}\in (0,\delta')$ and a smooth function $\rho\colon [0,\infty)\to [0,1]$ such that $\rho(0)=1$ and $\rho(r)=0$ for $r\geq \delta_{0}$. Following an analogy with formula (46) in [10], we can define $\lambda^{\sharp}_{\delta_{0}}=\rho(|f|)\lambda^{\sharp}$,
and eventually $\omega_{q}^{\epsilon,\delta_{0}}=\pi^{*}\omega_{X}+\epsilon\cdot (q\cdot d\lambda^{\sharp}_{\delta_{0}}+(1-q)\cdot d\lambda^{\flat}_{\delta_{0}})$. This way, we have fiberwise equality $\omega_{q}^{\epsilon,\delta_0}=\pi^{*}\omega_{X}$ over $\C\setminus \D_{\delta_0}$; and in general $\omega_{q}^{\epsilon,\delta_{0}}=\omega_{q}^{\epsilon'}$ for some $\epsilon'\in [0,\epsilon]$, with $\epsilon'=\epsilon>0$ on $\d A$. Thus by Proposition <ref>, the form  $\omega_{q}^{\epsilon,\delta_0}$ is fiberwise symplectic for all $\epsilon\in (0,\epsilon_0]$.

§ LAGRANGIAN FIBRATION AT RADIUS ZERO

As in the previous sections, let $f\colon X\to \D_{\delta}$ be a holomorphic function such that the unique singular fiber $f^{-1}(0)=\sum_{i=1}^{N}m_{i}D_{i}$ is snc. Choose a subset $\cS\subseteq \{1,\dots, N\}$, and recall that $\Delta_{\cS}$ denotes the subcomplex of the dual complex $\Delta_{D}$ spanned by vertices with indices in $\cS$. Let $A$ be the A'Campo space of $f$, equipped with the family of forms  $\omega_{q}^{\epsilon}$ introduced in formula (<ref>).

In this section, we introduce the expanded skeleton $E_{\cS}$, together with a map

\begin{equation}\label{eq:Lagrangian}
\ll_{\cS}\colon A_{\cS}^{\sm}\to E_{\cS}, \quad \mbox{where}\quad  A_{\cS}^{\sm}= \bigsqcup\nolimits_{\#I\geq n,\ I\subseteq \cS} A_{I}^{\circ},
\end{equation}

i.e. $A_{\cS}^{\sm}$ is the union of boundary pieces of the A'Campo space which correspond to maximal and submaximal faces of the skeleton $\Delta_{\cS}$. Moreover, we introduce the generic region $A_{\cS}^{\gen}\subseteq A_{\cS}^{\sm}$ as

\begin{equation*}
A_{\cS}^{\gen}= \bigsqcup\nolimits_{\#I=n+1,\ I\subseteq \cS} \Int_{\d A} A_{I}^{\circ}.
\end{equation*}

and write $A_{I,\theta}^{\circ}$, $A_{\cS,\theta}^{\sm}$ and $A_{\cS,\theta}^{\gen}$ for the intersections of the above regions with a radius-zero fiber $f_{A}^{-1}(0,\theta)$.

Proposition <ref>, which is the main result of this section, asserts that the restriction of $\ll_{\cS}$ to $A_{\cS,\theta}^{\sm}$ is a Lagrangian torus fibration with respect to each form $\omega_{q}^{\epsilon}$, with some mild singularities.

As a set, $E_{\cS}$ is a disjoint union of maximal faces of $\Delta_{\cS}$ and products $\Delta_{I}\times C_{I}$, where $\Delta_{I}$ is a submaximal face, and $C_{I}$ is a certain graph, namely a Reeb space of a fixed Morse function on the Riemann surface $X_{I}^{\circ}$, defined in Example <ref>. These pieces, glued together in a natural way, give rise to a manifold with boundary, corners and some mild branching coming from branching vertices of $C_{I}$. To describe this structure in precise terms, we introduce a notion of an ivy-like manifold.

In the setting of maximal Calabi–Yau degenerations, Proposition <ref><ref> will give a canonical choice of $\ll_{\cS}$, which is a Lagrangian torus fibration in the usual sense.

§.§ Ivy-like manifolds

An open graph is a topological space $C$ obtained as a quotient of a finite disjoint union of closed and semi-closed intervals, by some equivalence relation identifying only some endpoints of different intervals.  The images of those endpoints are called vertices of $C$, a degree of such vertex is the number of intervals  it lies in. The set of vertices of degree at least $3$ is called a ramification locus and denoted $\Ram(C)$; similarly, the set of vertices of degree $1$ is called the boundary of $C$, and denoted by $\d C$. Clearly, $C\setminus \Ram(C)$ is a $1$-dimensional manifold with boundary $\d C$. Since we are only interested in $C$ as a topological space, we can and do assume that no vertex has degree $2$.

An ivy is an open graph $C$ together with a proper continuous map $p\colon C\to (0,\infty)$ whose restriction to each interval constituting $C$ is a homeomorphism onto its image. Note that the restriction of $p$ to $C\setminus (\Ram(C)\sqcup \d C)$ is a local homeomorphism. We choose a smooth structure on $C\setminus \Ram(C)$ such that the restriction $p|_{C\setminus \Ram(C)}$ becomes a diffeomorphism onto its image.

[Stein factorization of a Morse function, see Figure <ref>]
The above definition is motivated by the following, classical construction, explained in more detail in <cit.> and references therein. Let $M$ be a Riemann surface, and let $h\colon M\to \R$ be a proper Morse function. The Reeb space of $h$ is the quotient space $C\de M/_{\sim}$, where for two points $x,y\in M$ we put $x\sim y$ if $x$ and $y$ lie in the same connected component of a fiber of $h$. The quotient map $\bar{h}\colon M\to C$ is called a Stein factorization of $h$. Writing $h=p\circ \bar{h}$, it is easy to see that $(C,p)$ is an ivy, and that the restriction

\begin{equation*}
\bar{h}\colon M\setminus \bar{h}^{-1}(\Ram(C))\to C\setminus \Ram(C)
\end{equation*}

is smooth.

For $c\not\in \d C\cup \Ram(C)$, the fiber $M_{c}\de \bar{h}^{-1}(c)$ is a connected submanifold of dimension $1$, i.e. an embedded circle. For $c\in \d C$ the fiber $M_{c}$ is a single point, either a maximum or a minimum of $h$. Eventually, for $c\in \Ram(C)$ the fiber $M_{c}$ is a connected union of circles, meeting at saddle points of $h$ where, since $h$ is Morse, $M_{c}$ is locally of the form $\{xy=0\}$. Choosing top and bottom branch of each such singularity, we conclude that $M_{c}$ is an immersed circle (but the immersion is not canonical).

[densely dashed] (-0.5,0) to[out=10,in=170] (0.5,0);
[densely dashed] (-0.5,0) to[out=-10,in=-170] (0.5,0);
(-0.5,0) to[out=90,in=-90] (-1.5,2.5) to[out=90,in=180] (-1,3.5) to[out=0,in=180] (0,2.5) to[out=0,in=-90] (0.5,4);
[densely dashed] (0.5,4) to[out=10,in=170] (1.5,4);
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(1.5,4) – (1.5,2.5) to[out=-90,in=90] (0.5,0);
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(-1,3.5) circle (0.06);
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[densely dotted] (-1.5,2.5) to[out=10,in=170] (0,2.5);
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[densely dotted] (-1.4,2) to[out=10,in=170] (0,2);
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[densely dotted] (0,1) to[out=10,in=170] (0.8,1);
[densely dotted] (0,1) to[out=-10,in=-170] (0.8,1);

at (0.8,0.3) $M$;
[->] (2.5,2) – (4,2);
at (3.25,2.25) $\bar{h}$;

(0,0) – (0,1);
(0,1) circle (0.06);
(0,1) to[out=150,in=-150] (0,2);
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(0,2) – (0,2.5);
(0,2.5) circle (0.06);
(0,2.5) – (-0.5,3.5);
(-0.5,3.5) circle (0.06);
[above] at (-0.5,3.5) $\d C$;
(0,2.5) – (0.5,4);
at (0.3,0.3) $C$;
[gray, ->] (-1.7,0.3) to[out=0,in=180] (-0.2,2.5);
[gray, ->] (-1.7,0.3) to[out=0,in=180] (-0.2,2);
[gray, ->] (-1.7,0.3) to[out=0,in=180] (-0.2,1);
[left] at (-1.6,0.3) $\Ram(C)$;
[->] (1,2) – (2.5,2);
at (1.75,2.2) $p$;

(0,0) – (0,4);
(0,1) circle (0.06);
(0,2) circle (0.06);
(0,2.5) circle (0.06);
(0,3.5) circle (0.06);
at (0.3,0.3) $\R$;

[gray, ->] (1.8,0.3) to[out=180,in=0] (0.2,2.5);
[gray, ->] (1.8,0.3) to[out=180,in=0] (0.2,2);
[gray, ->] (1.8,0.3) to[out=180,in=0] (0.2,1);
[gray, ->] (1.8,0.3) to[out=180,in=0] (0.2,3.5);
[right] at (1.85,0.3) $\Crit(h)$;

Stein factorization of a Morse function through its Reeb space

We now return to abstract definitions. An ivy-like chart (with boundary and corners) is a homeomorphism $\varphi\colon V\to U\times C$ from a topological space $V$ to a product $U\times C$ of a smooth manifold $U$ (with boundary and corners) and an ivy $p\colon C\to (0,\infty)$. The ramification locus of such chart is $\Ram(V)\de \varphi^{-1}(U\times \Ram(C))$, and outer boundary is $\dout V\de \varphi^{-1}(U\times \d C)$. Note that $V\setminus \Ram(V)$ inherits a structure of a smooth manifold with boundary and corners from $U\times (C\setminus\Ram(C))$.

We say that a continuous, surjective map $\psi\colon N\to V$ from a smooth manifold (with boundary and corners) to an ivy-like chart $V$ as above is an ivy-like submersion if it locally comes from a Stein factorization, i.e. the following holds. There is an open covering $N=\bigcup_{i} N_{i}$ and for each $i$, there is a smooth manifold $N_{i}'$ (with boundary and corners), a Riemann surface $M_{i}$, and a diffeomorphism $\tau_{i}\colon N_{i}'\times M_i\to N_{i}$ so that the composition $\varphi|_{\psi(N_i)}\circ \psi|_{N_{i}}\circ \tau_{i}$ is of the form $(\varphi',\bar{h}_{i})$, where $\varphi'\colon N_{i}'\to U$ is a submersion onto its image, and $\bar{h}_{i}$ is the Stein factorization of some Morse function $h_{i}\colon M_i\to \R$, see Example <ref>. Clearly, an ivy-like submersion restricts to a usual submersion over $V\setminus \Ram(V)$.

It will be convenient to call a usual chart a homeomorphism $\varphi \colon V\to U$ for some smooth manifold $U$ (with boundary and corners). In this case we put $\Ram(V)=\dout V=\emptyset$.

An ivy-like manifold (with boundary and corners) is a Hausdorff topological space $E$ together with an open covering $E=\bigcup_{i}V_{i}$, such that each $V_{i}$ is a domain of a usual or ivy-like chart $\varphi_{i}$; for every $i\neq j$ we have $\Ram(V_i)\cap \Ram(V_j)=\emptyset$; and the transition maps $\varphi_{j}|_{V_{i}\cap V_{j}}\circ \varphi_{i}^{-1}|_{\varphi_{i}(V_i\cap V_j)}$ are smooth, as maps between usual manifolds (with boundary and corners). We define its ramification locus $\Ram(E)$ and outer boundary $\dout E$ as the union of, respectively, all ramification loci and outer boundaries of the charts. A collection $\{(V_i,\varphi_{i})\}$ is called an ivy-like atlas. Note that the subspace $E\setminus \Ram (E)$ is a manifold (with boundary and corners) in the usual sense; and $\dout E\subseteq \d E$.

A continuous, surjective map $\psi\colon N\to E$ from a smooth manifold $N$ with boundary and corners to an ivy-like manifold $E$ as above is an ivy-like submersion if for every $i$, the composition $\varphi_{i}\circ \psi|_{\psi^{-1}(V_i)}$ is a (usual, or ivy-like) submersion. In particular, $\psi$ restricts to a usual submersion over $E\setminus \Ram(E)$.

Let $(Y,\omega)$ be a symplectic manifold of dimension $2n$, and let $\iota\colon L\to Y$ be an embedding (or an immersion). We say that its image $\iota(L)$ is an (immersed) isotropic submanifold if $\iota^{*}\omega=0$. It is (immersed) Lagrangian if additionally $\dim L=n$.

Let $E$ be an ivy-like manifold (with boundary and corners). We say that a map $\ll\colon Y\to E$ is an ivy-like Lagrangian torus fibration if it is an ivy-like submersion and for every $p\in E$, and every connected component $T$ of the fiber $\ll^{-1}(p)$ the following hold.

* If $p\in E\setminus (\Ram E\cup \dout E)$ then $T$ is Lagrangian torus $(\S^1)^{n}$.

* If $p\in \Ram E$ then $T$ is an immersed Lagrangian torus $(\S^1)^{n}$.

* if $p\in \dout E$ then $T$ is an isotropic torus $(\S^{1})^{n-1}$.

We say that $\ll$ is a Lagrangian torus fibration if $\Ram E\cup \dout E=\emptyset$. In this case, $E$ is a manifold (with boundary and corners), the map $\ll$ is a usual submersion and all fibers of $\ll$ are Lagrangian tori.

§.§ Construction of the expanded skeleton

We return to the setting of Section <ref>, that is, we let $f\colon X\to \C$ be a holomorphic function whose fibers over $z\neq 0$ are Kähler manifolds of dimension $n$;  $f^{-1}(0)=\sum_{i=1}^{N}m_{i}D_{i}$ is snc; and we let $A$ be the associated A'Campo space with fiberwise symplectic forms $\omega_{q}^{\epsilon}$ introduced in formula (<ref>).

Our goal is to define the expanded skeleton $E_{\cS}$ together with a map (<ref>), and prove that the latter is an ivy-like Lagrangian torus fibration. We first do it piecewise, that is, we define a space $E_{I}^{\circ}$ and a map $\ll_{I}^{\circ}\colon A_{I}^{\circ}\to E_{I}^{\circ}$, for every maximal or submaximal face $\Delta_{I}$ of $\Delta_{\cS}$.

Let $\Delta_{J}$ be a maximal face of $\Delta_{\cS}$, so the stratum $X_{J}$ is a point. Fix an adapted chart around $X_{J}$ such that its preimage $U_{J}$ is covered by smooth charts (<ref>). On $U_{J}$ we have a free, smooth $(\S^{1})^{n+1}$-action, given in each of those charts by a translation in $\theta_{i}$-directions. Denote the quotient map by

\begin{equation}\label{eq:thickquotient}
\qq_{J}\colon U_{J}\to Q_{J},\quad\mbox{and define}\quad \ll_{J}^{\circ}\de \qq_{J}|_{A_{J}^{\circ}}\colon A_{J}^{\circ}\to E_{J}^{\circ},
\end{equation}

where $E_{J}^{\circ}=\qq_{J}(A_{J}^{\circ})$. Clearly, $Q_{J}$ is a smooth manifold, with global coordinates $(\hat{v}_{i})_{i\in J}$. By Lemma <ref>, $E_{J}^{\circ}$ is a codimension zero submanifold with boundary and corners of $Q_{J}$, which in the above coordinates gets identified with rounded simplex $\tilde{\Delta}_{J}$. In particular, $E_{J}^{\circ}$ is diffeomorphic to the maximal face $\Delta_{J}$.

Let now $\Delta_{I}$ be a submaximal face of $\Delta_{\cS}$, so the corresponding stratum $X_{I}$ is a Riemann surface. We choose a proper Morse function $h_{I}\colon X_{I}^{\circ}\to (0,\infty)$ satisfying the following condition: for every point $p\in X_{I}\setminus X_{I}^{\circ}$ there is a neighborhood $U_{p}$ of $p$ in $X_{I}$, and a holomorphic coordinate $z\in \cO_{X_{I}}(U_p)$ at $p$ such that on $U_{p}\setminus \{p\}\subseteq X_{I}^{\circ}$ we have $h_{I}=|z|$ or $|z|^{-1}$. Clearly, such a Morse function $h_{I}$ exists.

Let $h_{I}=p_{I}\circ \bar{h}_{I}$ be the Stein factorization of $h_{I}$, so $p_{I}\colon C_{I}\to (0,\infty)$ is its Reeb space, see Example <ref>. We define an ivy-like chart $E_{I}^{\circ}\de \tilde{\Delta}_{I}\times C_{I}$, and an ivy-like submersion

\begin{equation}\label{eq:submaximalquotient}
\ll_{I}^{\circ}\de ((\hat{v}_{i})_{i\in I},\bar{h}_{I}\circ \pi)\colon A_{I}^{\circ}\to E_{I}^{\circ}.
\end{equation}

The above maps $\ll_{I}^{\circ}$ glue to an ivy-like submersion $\ll_{\cS}\colon A_{\cS}^{\sm}\to E_{\cS}$.

Fix a maximal face $\Delta_{J}$ of $\Delta_{\cS}$ and a submaximal face $\Delta_{I}$ contained in $\Delta_{J}$. We explain how to glue $\ll_{J}^{\circ}$ with $\ll_{I}^{\circ}$. The result will follow by repeating this construction for all such inclusions $\Delta_{I}\subsetneq \Delta_{J}$.

Say that $J=\{1,\dots, n+1\}$ and $I=J\setminus \{1\}$. By Lemma <ref><ref>, there is an adapted chart $U_{X}$ around the point $X_{J}$ whose coordinates $(z_1,\dots, z_{n+1})$ satisfy $|z_i|=\hat{r}_i$, so for each $i\in J$ we have $\hat{v}_i=v_i$. Moreover, shrinking the chart $U_{X}$ if needed we can assume that the Morse function $h_{I}$ equals $|\tilde{z}|^{\pm 1}$ for some holomorphic coordinate $\tilde{z}$ of $X_{I}$ at the point  $X_{J}$. Write $\tilde{z}=\lambda_{0}\cdot \lambda(z_1)\cdot z_1$, where $\lambda_0\in \C^{*}$, and $\lambda$ is a holomorphic function on a neighborhood of $0\in \C$ such that $\lambda(0)=1$. Put $z=\lambda_{0}^{-1}\cdot \tilde{z}$, so $z$ is another holomorphic coordinate of $X_{I}$ at $X_{J}$, and we have $z=\lambda(z_1)\cdot z_1$.

Put $U=\pi^{-1}(U_X)$ and $U_{I}=A_{I}\cap U\subseteq \d A$, where $A_{I}=\pi^{-1}(X_{I})$. The holomorphic coordinate $z$ pulls back to a smooth function on $U_{I}$, which we denote by the same letter. Since $|z|=|\lambda_{0}^{-1}|\cdot h_{I}^{\pm 1}$, the fibers of $|z|$ and $h_{I}$ give the same foliations of $U\cap A_{I}^{\circ}$.

Recall from formula (<ref>) that $U\cap \d A$ is covered by charts $V_{i}\de \{w_i>\frac{1}{n+2}\}\cap \d A$, with coordinates $((v_{j})_{j\neq i},\vartheta)$. Since on $A_{I}^{\circ}$ we have $w_{1}=0$, the set $\bigcup_{i\neq 1} V_i$ is an open neighborhood of $\bar{A}_{I}^{\circ}\cap U$ in $U_{I}$.

Fix $i\in I$, say $i=n+1$, and put $V\de V_{n+1}$. The smooth coordinates on $V$ are $(v_1,\dots, v_n;\vartheta)$. We define a function $p\colon V\to \R$ piecewise, as $p=v_1$ on $A_{J}^{\circ}$ and $p=\frac{-1}{m_1\log |z|}$ on $A_{I}^{\circ}$. 	We claim that, after shrinking $U_{X}$ around $X_{J}$ if necessary,

\begin{equation}\label{eq:claim_gluing}
\mbox{the map } (p,v_2,\dots,v_{n},\vartheta) \mbox{ is a smooth coordinate system on }V.
\end{equation}

Once (<ref>) is shown, we conclude as follows. The foliation of $V$ given by the projection to the first $n$ coordinates agrees with the foliations of $V\cap A_{J}^{\circ}$ and $V\cap A_{I}^{\circ}$ given by $\ll_{J}^{\circ}$ and $\ll_{I}^{\circ}$, respectively. Hence these maps glue to a submersion $V\to E_{V}$, onto some smooth manifold $E_{V}$. Now, we define a smooth manifold $E_{I,J}$ as $\bigsqcup_{i\neq 1} E_{V_i}/_{\sim}$, where $\sim$ identifies two points whenever they are images of the same point in $\bigcup_{i\neq 1} V_{i}$. This way, we get a submersion $\bigcup_{i\neq 1} V_{i}\to E_{I,J}$, which restricts to $\ll_{I}^{\circ}$ and $\ll_{J}^{\circ}$. Eventually, we glue to $E_{I,J}$ a usual chart $\ll_{J}^{\circ}(\Int_{\d A}A_{J}^{\circ})\subseteq E_{J}^{\circ}$ and an ivy-like chart $E_{I}^{\circ}$.
It remains to prove claim (<ref>). Clearly, the function $p$ is smooth on the interior of $A_{J}^{\circ}$, where it equals the coordinate $v_1$; and on $A_{I}^{\circ}$, where it is a pullback of a smooth function from $X_{I}^{\circ}$. It remains to prove that all derivatives of $v_{1}-p$ approach $0$ as we approach $A_{J}^{\circ}$ from $A_{I}^{\circ}$, i.e. as $v_1\searrow 0$.

Recall that $z=\lambda(z_1)\cdot z_1$ for some non-vanishing, holomorphic function $\lambda$ such that $\lambda(0)=1$. Moreover, on $A_{I}^{\circ}$ we have $v_1=t_1=\frac{-1}{m_1\log|z_1|}$, so  $p=\frac{-1}{m_1\log|z|}=\frac{-1}{m_1\log|\lambda(z_1)|+m_1\log|z_1|}=\frac{v_{1}}{1-v_{1}m_1\log|\lambda(z_1)|}$. Thus $p-v_1=\frac{v_{1}^2m_1\log |\lambda(z_1)|}{1-v_1m_1 \log|\lambda(z_1)|}$. By Leibniz rule, it is enough to show that  $\log|\lambda(z_1)|$ and all its derivatives approach $0$ as $v_{1}\searrow 0$. By assumption we have  $\log|\lambda(0)|=\log 1=0$, so by chain rule it remains to show that all derivatives of $z_1$ vanish as $v_1\searrow 0$. Since $v_{1}=\frac{-1}{m_1\log|z_1|}$ on $A_{I}^{\circ}$, we have $z_1=\exp(-m_{1}^{-1}v_{1}^{-1}+2\pi \imath\cdot \theta_{1})$, where $\theta_{1}$ is a smooth coordinate on $U$. Thus $dz_{1}=\exp(-m_{1}^{-1}v_{1}^{-1})\cdot \exp(2\pi\imath \cdot \theta_1)\cdot (m_{1}^{-1}v_{1}^{-2}\, dv_{1}+2\pi\imath\, d\theta_{1})$. The result follows since all derivatives of the one-variable function $v_1\mapsto \exp(-m_{1}^{-1}v_{1}^{-1})v_{1}^{-2}$ vanish at $0$.

If for some submaximal face $\Delta_{I}$ of $\Delta_{\cS}$ we have $(X_{I},X_{I}^{\circ})\cong (\P^1,\C^{*})$, then we have a canonical choice of the Morse function $h_{I}$: namely, we let $h_{I}\de |z_{I}|$, where $z_{I}$ is a meromorphic function on $X_{I}$ with a simple zero at one point of $X_{I}\setminus X_{I}^{\circ}$ and a simple pole at the other one. Since any two such functions $z_{I}$ differ by a multiplication by a nonzero complex number or taking the inverse, for any such $z_{I}$ we get the same foliation of $X_{I}^{\circ}$, hence the same map $\ll_{I}^{\circ}$. Note that this function $h_{I}$ has no critical points, so the ivy-like submersion (<ref>) is a submersion in the usual sense.

As a consequence, if for all submaximal faces $\Delta_{I}$ of $\Delta_{\cS}$ we have $(X_{I},X_{I}^{\circ})\cong (\P^1,\C^{*})$, then the above canonical choice of Morse functions yields a canonical submersion $\ll_{\cS}\colon A_{\cS}^{\sm}\to E_{\cS}$ (in the usual, not just ivy-like sense). This happens e.g. in the setting of Theorem <ref>, when we take for $f$ an snc model of a maximal Calabi–Yau degeneration chosen in Proposition <ref>.

§.§ Lagrangian fibration at radius zero

We now prove that the map $\ll_{\cS}\colon A_{\cS}^{\sm}\to E_{\cS}$ constructed in Section <ref> restricts to an ivy-like Lagrangian torus fibration on  $A_{\cS,\theta}^{\sm}$ for each $\theta\in \S^1$. We begin by describing the fibers of $\ll_{\cS}$.

Fix a point $b\in E_{\cS}$ and let $L_{b}=\ll_{\cS}^{-1}(b)$. Then the following hold.

*  The fiber $L_{b}$ is the image of an immersion $\iota_{b}\colon (\S^{1})^{k}\to A$,  where $k=n+1$ if $b\not\in \dout E$ and $k=n$ otherwise.	If $b\not\in \Ram(E)$ then the immersion $\iota_{b}$ is an embedding.

*  For every $q\in [0,1]$ and every $\epsilon\geq 0$, we have $\iota_{b}^{*}\omega_{q}^{\epsilon}=0$.

*  Fix $\theta\in \S^{1}$ and let $F_{\theta}=f_{A}^{-1}(0,\theta)$ be the radius zero fiber. Then the restriction of $\iota_{b}$ to the preimage of $L_{b}\cap F_{\theta}$ is an immersion $\iota_{b}^{\theta}\colon T\to F_{\theta}$, where $T\subseteq (\S^{1})^{k}$ is a disjoint union of $\gcd\{m_{i}: i\in I\}$ tori $(\S^{1})^{k-1}$. If $b\not\in \Ram(E)$ then $\iota_{b}^{\theta}$ is an embedding.

We have $b\in E_{I}^{\circ}$ for some $I$ such that $\Delta_{I}$ is a maximal or submaximal face of $\Delta_{\cS}$.

Assume first that the face $\Delta_{I}$ is maximal. Then by definition (<ref>) of $\ll_{I}^{\circ}$, its fiber $L_{b}$ is contained in some chart (<ref>) with associated index set $I$, and it is parametrized by the coordinates $(\theta_{i})_{i\in I}$. Thus $L_{b}$ is an embedded torus $(\S^{1})^{n+1}$, which proves  <ref>. The tangent space $T_{p}L_{b}$ at every point $p\in L_{b}$ is spanned by coordinate vectors $(\frac{\d}{\d \theta_{i}})_{i\in I}$; so by Lemma <ref><ref> we have $d\hat{v}_{i}|_{L_{b}}=0$ for every $i\in I$. Thus Lemma <ref><ref> gives $\omega_{q}^{\epsilon}|_{L_{b}}=0$, which proves <ref>. Part <ref> follows from the fact that in coordinates (<ref>), the fiber $F_{\theta}$ is given by equation $\sum_{i\in I} m_{i}\theta_{i}=\theta$.

Assume now that $\Delta_{I}$ is submaximal. By definition (<ref>) of $\ll_{I}^{\circ}$, its fiber $L_{b}$ is contained in $A_{I}^{\circ}$, and its image through the diffeomorphism (<ref>) is the preimage in $(X_{I}^{\circ})_{\log}$ of a fiber $C_{b}$ of $\bar{h}_{I}^{-1}\colon X_{I}^{\circ}\to C_{I}$. The latter is either an embedded circle if $b\not\in \Ram E_{I}^{\circ}\cup \dout E_{I}^{\circ}$, an immersed circle if $b\in \Ram E_{I}^{\circ}$, or a point if $b\in \dout E_{I}^{\circ}$. The $(\S^{1})^{n}$-bundle $(X_{I}^{\circ})_{\log}\to X_{I}^{\circ}$ splits as a direct sum of $\S^{1}$-bundles, where each summand can be identified with the circle bundle in $\cO_{X}(D_{i})|_{X_{I}^{\circ}}$ for $i\in I$. The restriction of this summand to $C_{b}$ is an oriented circle bundle over an immersed circle (or a point), so it is trivial. It follows that $L_{b}=(\S^{1})^{n}\times C_{b}$, so $L_{b}$ is an embedded torus $(\S^{1})^{n+1}$ if $b\not\in \Ram E_{I}^{\circ}\cup \dout E_{I}^{\circ}$; an immersed torus $(\S^{1})^{n+1}$ if $b\in \Ram E_{I}^{\circ}$; or an embedded torus $(\S^{1})^{n}$ if $b\in \dout E_{I}$. This proves <ref>.

Fix a point $p\in L_{b}$, and let $V\subseteq T_{p}A_{I}^{\circ}$ be the tangent space to a branch of $L_{b}$ at $p$.
The point $p$ is contained in some chart (<ref>) with associated index set $I$. Its coordinate vectors $\frac{\d}{\d \theta_{i}}\in T_{p}A_{I}^{\circ}$ satisfy $d\hat{v}_{j}(\frac{\d}{\d \theta_{i}})=0$ for $j\in I$ and $\pi_{*}(\frac{\d}{\d \theta_i})=0$, see Lemma <ref><ref> and <ref>, so by definition (<ref>) of $\ll_{I}^{\circ}$ we have $\frac{\d}{\d \theta_{i}}\in V$ for $i\in I$. Since $d\hat{v}_{i}|_{V}=0$, Lemma <ref><ref> gives  $\omega_{q}^{\epsilon}(\sdot,\tfrac{\d}{d\theta_{i}})|_{V}=0$. However, the vectors $(\frac{\d}{\d \theta_{i}})_{i\in I}$ span a subspace of $V$ of codimension $\dim V-\#I\leq (n+1)-n=1$, so the $2$-form $\omega_{q}^{\epsilon}$ vanishes on $V$, as claimed in <ref>. Part <ref> follows as before from the fact that $F_{\theta}=\{\sum_{i\in I} m_{i}\theta_{i}=\theta\}$.

The above construction  is now summarized in the following result. Recall for a radius-zero fiber $F_{\theta}\de f_{A}^{-1}(0,\theta)\subseteq \d A$, the region $A^{\sm}_{\cS, \theta}\subseteq F_{\theta}$ is the union pieces $A_{I}^{\circ}\cap F_{\theta}$ such that $\#I\geq n$ and $I\subseteq \cS$, i.e.  $\Delta_{I}$ is a maximal or submaximal face of $\Delta_{\cS}$.

Let $\ll^{\theta}_{\cS}\colon A^{\sm}_{\cS,\theta}\to E_{\cS}$ be the restriction to $F_{\theta}$ of the map (<ref>) constructed in Lemma <ref>, for some collection of Morse functions $(h_{I})$. Fix $\epsilon\in (0,\epsilon_0]$ as in Proposition <ref><ref>. Then the following hold.

*  For every $q\in (0,1]$, the map $\ll^{\theta}_{\cS}$ is an ivy-like Lagrangian torus fibration with respect to the symplectic form $\omega_{q}^{\epsilon}$ introduced in formula (<ref>), see Definition <ref>.

*  If $\Delta_{I}$ is a maximal face of $\Delta_{\cS}$ then the image $\ll^{\theta}_{\cS}(A_{I}^{\circ}\cap F_{\theta})$ is diffeomorphic to $\Delta_{I}$. Moreover, the restriction of $\ll^{\theta}_{\cS}$ to $\Int_{\d A} A_{I}^{\circ}\cap F_{\theta}$ is a Lagrangian torus fibration with respect to $\omega_{0}^{\epsilon}$.

*  If $\Delta_{I}$ is a submaximal face of $\Delta_{\cS}$ then the image $\ll_{\cS}^{\theta}(A_{I}^{\circ}\cap F_{\theta})$ is diffeomorphic to $\Delta_{I}\times C_{I}$, where $C_{I}$ is the Reeb space of $h_{I}$, see Example <ref>.

*  Assume that for every submaximal face $\Delta_{I}$ of $\Delta_{\cS}$ we have $(X_{I},X_{I}^{\circ})\cong (\P^1,\C^{*})$. Then the canonical map $\ll_{\cS}^{\theta}$, chosen in Remark <ref>, is a Lagrangian torus fibration.

By Lemma <ref>, the map $\ll^{\theta}_{\cS}$ is an ivy-like Lagrangian torus fibration with respect to $\omega_{q}^{\epsilon}$ whenever the latter is a symplectic form. Together with Proposition <ref>, this proves <ref> and the second statement of <ref>. Lemma <ref> implies also that each fiber of $\ll_{\cS}$ meets $F_{\theta}$, so  $\ll^{\theta}_{\cS}(A_{I}^{\circ}\cap F_{\theta})=\ll_{\cS}(A_{I}^{\circ})=E_{I}^{\circ}$, which by construction is diffeomorphic to $\Delta_{I}$ if the latter is a maximal face; and to $\Delta_{I}\times C_{I}$ otherwise. This proves <ref> and <ref>. Part <ref> follows from Remark <ref>.

[Hesse pencil, see Figure <ref>]
Consider the Hesse pencil $\{(x_1^3+x_2^3+x_3^3)\cdot z=x_1x_2x_3\}$ from Example <ref>. Recall that the special fiber is a triangle $D_1+D_2+D_3$, where each $D_i\cong  \P^1$ is line in $\P^2$. In the A'Campo space, each radius-zero fiber $F$ is a torus, divided into six cylinders $F\cap A_{I}^{\circ}$.

Choose $\cS=\{1,2,3\}$, so $\Delta_{\cS}=\Delta_{D}$ is a triangle in $\R^3$. As in Remark <ref>, choose the Morse functions $h_{i}\colon D_{i}\setminus(D-D_i)\to \R$ without critical points. Now, the expanded skeleton $E_{\cS}$ is a circle, divided in six parts: three disjoint closed segments corresponding to the intersection points $D_{i}\cap D_{j}$, and three open ones, corresponding to each line $D_{i}$.

special fiber $D\subseteq X$[.21]

[use as bounding box] (-0.8,-1.5) rectangle (1.8,1.5);
(A) at (0,-1);
(B) at (0,1);
(C) at (1.732,0);
(A) to[out=110,in=-110] (B) to[out=-70,in=70] (A);
at (-0.5,0) $D_1$;
(C) to[out=130,in=-10] (B) to[out=-50,in=170] (C);
at (1.2,0.9) $D_2$;
(C) to[out=-130,in=10] (A) to[out=50,in=-170] (C);
at (1.2,-0.9) $D_3$;

[densely dotted] ($0.5*(B)+0.5*(C)+(0.09,0.18)$) to[out=-90,in=40] ($0.5*(B)+0.5*(C)-(0.09,0.18)$);
[densely dotted] ($0.5*(A)+0.5*(C)+(0.09,-0.18)$) to[out=140,in=-90] ($0.5*(A)+0.5*(C)-(0.09,-0.18)$);
[densely dotted] (-0.2,0) to[out=20,in=160] (0.2,0);

radius-zero fiber $F\subseteq A$[.28]

[use as bounding box] (-1,-0.5) rectangle (3,2.5);

[fill=black!10] (1.626,1.313) to[out=0,in=-120] (2.073,1.537) to[out=-60,in=60] (2.073,0.463) to[out=180,in=-60] (1.626,0.678) to[out=60,in=-60] (1.626,1.313);

[fill=black!10] (1,-0.2) to[out=120,in=-120] (1,0.3) to[out=180,in=-60] (0.374,0.678) to[out=180,in=60] (-0.073,0.463) to[out=-60,in=180] (1,-0.2);

[fill=black!10] (0.374,1.313) to[out=120,in=0] (-0.073,1.537) to[out=60,in=180] (1,2.2) to[out=-60,in=60] (1,1.7) to[out=180,in=60] (0.374,1.313);

(1,1) circle (1.2);
(1,1) circle (0.7);
(1,2.2) to[out=-60,in=60] (1,1.7);
(1.626,1.313) to[out=0,in=-120] (2.073,1.537);
(1.626,0.678) to[out=-60,in=180] (2.073,0.463);
(1,-0.2) to[out=120,in=-120] (1,0.3);
(0.374,0.678) to[out=180,in=60] (-0.073,0.463);
(0.374,1.313) to[out=120,in=0] (-0.073,1.537);

at (-0.5,1) $A_{1}^{\circ}$;
at (2.6,1) $A_{2,3}^{\circ}$;
at (0,2.2) $A_{1,2}^{\circ}$;
at (0,-0.2) $A_{1,3}^{\circ}$;
at (1.9,-0.2) $A_{3}^{\circ}$;
at (1.9,2.2) $A_{2}^{\circ}$;

expanded skeleton $E_{\cS}$ [.24]

[use as bounding box] (-0.8,-0.5) rectangle (2.8,2.5);
(1,1) circle (1);
(1,2.15) – (1,1.85);
(1.74,1.38) – (2.028,1.514);
(1.74,0.62) – (2.028,0.486);
(1,0.15) – (1,-0.15);
(0.26,0.62) – (-0.028,0.486);
(0.26,1.38) – (-0.028,1.514);
at (-0.3,1) $E_{1}^{\circ}$;
at (2.4,1) $E_{2,3}^{\circ}$;
at (0.2,2.1) $E_{1,2}^{\circ}$;
at (0.2,-0.1) $E_{1,3}^{\circ}$;
at (1.7,-0.1) $E_{3}^{\circ}$;
at (1.7,2.1) $E_{2}^{\circ}$;
[very thick] (1.894,1.447) to[out=-60,in=60] (1.894,0.553);
[very thick] (1,0) to[out=180,in=-60] (0.106,0.553);
[very thick] (0.106,1.447) to[out=60,in=180] (1,2);

essential skeleton $\Delta_{\cS}$[.24]

[use as bounding box] (-2.4,-1.5) rectangle (0.8,1.5);
(A) at (0,-1);
(B) at (0,1);
(C) at (-1.732,0);
[very thick] (A) – (B) – (C) – (A);
(A) circle (0.08);
(B) circle (0.08);
(C) circle (0.08);
at ($(A)-(1,-0.2)$) $\Delta_{1,3}$;
at ($(B)-(1,0.1)$) $\Delta_{1,2}$;
at ($(C)+(2.2,0)$) $\Delta_{2,3}$;
at ($(A)+(0.3,-0.1)$) $\Delta_{3}$;
at ($(B)+(0.3,0.1)$) $\Delta_{2}$;
at ($(C)-(0.3,0)$) $\Delta_{1}$;

Example <ref>: Hesse pencil $\{(x_1^3+x_2^3+x_3^3)\cdot z=x_1x_2x_3\}$.

Our results are most powerful in case $\dim\Delta_{\cS}=n+1$, e.g. for maximal Calabi–Yau degenerations. In fact, if $\dim \Delta_{\cS}<n$ then our results are vacuous. Consider the case $\dim \Delta_{\cS}=n$, so $\Delta_{\cS}$ has submaximal faces, but no maximal one. Then the expanded skeleton $E_{\cS}$ is a disjoint union of ivy-like charts $\Delta_{I}\times C_{I}$, where $\Delta_{I}$ is a submaximal face of $\Delta_{\cS}$, and $C_I$ is a Reeb space of the chosen Morse function $h_{I}\colon X_{I}^{\circ}\to [0,1]$. Since the Riemann surface $X_{I}^{\circ}$ is closed, $h_{I}$ has a maximum, so $\dout E_{\cS}\neq\emptyset$ and therefore $\ll_{\cS}^{\theta}$ has fibers of lower dimension.

Nonetheless, in the setting of Calabi–Yau degenerations this can be easily fixed. Indeed, assume that $f$ is an snc model chosen in Proposition <ref>. Then $X_{I}^{\circ}$ is an elliptic curve, diffeomorphic to $\S^{1}\times \S^{1}$. Replacing $\bar{h}_{I}$ by one of the projections, we get $E_{\cS}$ to be a disjoint union of products $\S^{1}\times \Delta_{I}$; and the same argument as in Lemma <ref><ref> shows that $\ll_{\cS}^{\theta}$ is a Lagrangian fibration.

§ LAGRANGIAN FIBRATIONS AT POSITIVE RADIUS

We keep the notation and assumptions from Section <ref>. In Proposition <ref>, we have constructed an ivy-like Lagrangian torus fibration $\ll_{\cS}$ of the part of the radius-zero fiber  corresponding to maximal and submaximal faces of $\Delta_{\cS}$. In this section, we use the symplectic connection to transport those Lagrangian tori to positive radius, and study their properties.

In order to apply  Proposition <ref>, we need to choose an open subset $W_{X}\subseteq X$ such that $f|_{\bar{W}_{X}}$ is proper; and the flows of the vector fields lifted by the symplectic connection do not escape $W_X$. If $f$ was proper to begin with, we simply take $W_X=X$. In general,  we take any open subset $V_{X}\subseteq X$ with compact closure, and obtain $W_{X}$ by slightly modifying $V_X$ near its  boundary.
This modification was introduced in <cit.>, we summarize it in Setting <ref> below.  The reader only interested in the proper case may skip this part and substitute $W_X=X$ in what follows.

[see Figure <ref> or <cit.>]
Let $V_{X}$ be a domain with smooth boundary, such that $f|_{\bar{V}_{X}}$ is proper, $f^{-1}(0)$ is transverse to $\d V_{X}$, and the intersection $f^{-1}(0)\cap \d \bar{V}_{X}$ is contained in the union of the open sets $R_{i}^{\circ}$ introduced in formula (<ref>). Shrinking $\delta>0$ if needed we get that $\bigcup_{i}R_{i}^{\circ}$ contains some neighborhood $C_{X}$ of $\d \bar{V}_{X}$. Put $C\de \pi^{-1}(C_X)$. By Lemma <ref><ref>,<ref>, on $C$ each distance function $\hat{r}_{i}$ is either equal to $\frac{1}{e}$ or $|f|^{1/m_{i}}$, hence is fiberwise constant. Thus each form $\hat{\alpha}_{i}$ is fiberwise zero there, so formulas (<ref>) and (<ref>) imply that $\omega^{\sharp}$ and $\omega^{\flat}$ are fiberwise zero. Eventually, definition (<ref>) of $\omega^{\epsilon}_{q}$ gives a fiberwise equality $\omega_{q}^{\epsilon}|_{C}=\pi^{*}\omega_{X}|_{C}$ for every $q$ and $\epsilon$.

[fill=black!5] (0.2,2.3) to[out=0,in=180] (1.8,2.7) – (1.8,-0.8) – (0.2,-0.8) – (0.2,2.3);
[decorate, decoration = calligraphic brace, very thick] (1.9,2.7) –  (1.9,-0.8);
[right] at (2,0.95) $W_{X}$;

[fill=black!10] (0.2,2.3) to[out=0,in=180] (1.8,2.7) – (1.8,1.7) to[out=180,in=0] (0.2,1.3) – (0.2,2.3);
(0.2,2.3) to[out=0,in=180] (1.8,2.7);
(0.2,1.3) to[out=0,in=180] (1.8,1.7);
[decorate, decoration = calligraphic brace, very thick] (0.1,1.3) –  (0.1,2.3);
[left] at (0.15,1.8) $C_{X}'$;

[-stealth] (1,2.25)  to[out=20,in=-175]  (1.5,2.4);
[-stealth] (1,2)  to[out=20,in=-175]  (1.5,2.15);
[-stealth] (1,1.75)  to[out=20,in=-175]  (1.5,1.9);

(-0.5,3) – (1.8,3);
(0.2,2) – (3.2,2);
at (3,1.75) $\d \bar{V}_{X}$;
(-0.5,1) – (1.8,1);
[decorate, decoration = calligraphic brace, very thick] (-0.6,1) –  (-0.6,3);
[left] at (-0.6,2) $C_{X}$;

[thick] (1,3.1) – (1,1) to[out=-90,in=60] (0.9,0.4);
[thick] (0.9,0.6) – (1.1,-0.2);
[thick] (1.1,0) – (0.9,-0.8);
[->] (1,-0.9) – (1,-1.6);
[left] at (1,-1.3) $f$;
(1,-2) ellipse (0.7 and 0.25);
[-stealth] (1,-2) – (1.5,-2);
(1,-2) circle (0.04);
(1.7,-2) circle (0.03);
at  (1.9,-1.8) $\delta$;

Setting <ref>: modifying the domain in case $f$ is not proper.

Now, for $r<\delta$ let $\Phi_{r}\colon C_{X}\cap f^{-1}(0)\to X$ be the time $r$ flow of the symplectic lift of the radial vector field $\frac{\d}{\d r}$ on a fixed ray of $\C$, with respect to $\omega_{X}$: note that it is well defined since the restriction of $f^{-1}(0)$ to the collar $C_{X}$ is smooth. Shrinking $\delta>0$ if needed, we get that for some neighborhood $C'_{0}$ of $f^{-1}(0)\cap \d \bar{V}_{X}$ in $f^{-1}(0)\cap C_{X}$, the image $\Phi_{r}(C'_{0})$ is contained in $C_{X}$ for all $r<\delta$. Now, let $C'_{X}$ be the union of the images $\Phi_{r}(C'_{0})$, for all $r<\delta$ and all rays in $\C$. Put $W_{0}=f^{-1}(0)\cap (V_{X}\cup C'_{0})$ and define $W_{X}$ as the union of $C'_{X}$ and all connected components of $X\setminus C'_{X}$ which do not meet the union of images $\Phi_{r}(\d W_{0})$, see <cit.> for details.

Having fixed $W_{X}$ and $\delta>0$ as above, we choose $\epsilon_0>0$ and shrink $\delta>0$ further so that they satisfy Proposition <ref>. That is, for every $q\in (0,1]$ and $\epsilon\in (0,\epsilon_{0}]$, the forms $\omega_{q}^{\epsilon}$ are fiberwise symplectic on $f_{A}^{-1}(\D_{\delta,\log})\cap W$, where $W=\pi^{-1}(W_X)$. The above choice of $W_{X}$ guarantees that the the symplectic lift of $\frac{\d}{\d r}$ with respect to each $\omega_{q}^{\epsilon}$ defines a flow $W\cap \d A\to W$ for all times $r<\delta$. Indeed, if $W\neq A$ then near $\d W$ we have fiberwise equality $\omega_{q}^{\epsilon}=\pi^{*}\omega_{X}$, so the above flow is equal to the one of $\pi^{*}\omega_{X}$, so it does not escape $W=\pi^{-1}(W_X)$ by construction of $W_X$.

In the following, we replace $X$ and its A'Campo space $A$ by $W_{X}$ and its preimage in $A$. As usual, we identify $X\setminus D$ with its preimage $A\setminus \d A$. We fix one $\epsilon\in (0,\epsilon_0]$, and suppress it in the notation. This way, for every $q\in (0,1]$ we have a  flow

\begin{equation}\label{eq:flow}
\Phi_{\bullet}^{q}\colon \d A\to A
\end{equation}

of the $\omega_{q}^{\epsilon}$-symplectic lift of the radial vector field on $\C_{\log}$, with respect to the submersion $(g,\theta)$.

§.§ Moving Lagrangian tori by the symplectic connection

Fix $z\in \D_{\delta}^{*}$, and put $\theta=\frac{z}{|z|}$, $\tau=\eta(-\frac{1}{\log|z|})$, so the flow $\Phi_{\tau}^{q}$ maps the radius-zero fiber $F_{\theta}=f_{A}^{-1}(0,\theta)$ to the smooth fiber $X_{z}=f^{-1}(z)$. We push the decomposition $F_{\theta}=\bigsqcup_{I}A_{I,\theta}$ to $X_{z}=\bigsqcup_{I}(X_{z}^{q})_{I}^{\circ}$, where $(X_{z}^{q})^{\circ}_{I}=\Phi_{\tau}^{q}(A_{I,\theta}^{\circ})$; and we define

\begin{equation*}
\end{equation*}

This way, $(X_{z}^{q})_{\cS}^{\sm}$ is the union of pieces $(X_{z}^{q})^{\circ}_{I}$ for all maximal and submaximal faces $\Delta_{I}$ of the skeleton $\Delta_{\cS}$; and $(X_{z}^{q})_{\cS}^{\gen}$ is the union of $\Int_{X_z} (X_{z}^{q})_{I}^{\circ}$ for all maximal faces of $\Delta_{\cS}$. Note that $(X_{z}^{q})^{\sm}_{\cS}$ is a submanifold of $X_{z}^{q}$ of codimension $0$ with boundary and corners; and $(X_{z}^{q})^{\gen}_{\cS}$ is its open subset.

We remark that the sets $\Int_{X_{z}}(X_{z}^{q})^{\circ}_{I}$ play a similar role as the sets $E_{I}$ introduced in <cit.>.

We have the following immediate consequence of Proposition <ref><ref>.

For every $q\in (0,1]$, the composition

\begin{equation}\label{eq:Lagranian-at-positive-radius}
\ll_{\cS}^{\theta}\circ (\Phi_{\tau}^{q})^{-1}\colon (X_{z}^{q})_{\cS}^{\sm}\to E_{\cS}
\end{equation}

is an ivy-like Lagrangian torus fibration with respect to the Kähler form $\omega_{q}^{\epsilon}$ on $X_{z}$. Its restriction to $(X_{z}^{q})_{\cS}^{\gen}$ is a Lagrangian torus fibration in the usual sense.

Moreover, if for every submaximal face $\Delta_{I}$ of $\Delta_{\cS}$ we have $(X_{I},X_{I}^{\circ})\cong (\P^1,\C^{*})$
then for the canonical map $\ll_{\cS}^{\theta}$ defined in Remark <ref>, the composition (<ref>) is a Lagrangian torus fibration in the usual sense.

By Proposition <ref><ref>,<ref>, the form $\omega_{0}^{\epsilon}$ is fiberwise symplectic both at positive radius, and on $A_{\cS}^{\gen}$. Therefore, we have a flow

\begin{equation*}
\Phi_{\tau}^{0}\colon A_{\cS,\theta}^{\gen}\to (X_{z}^{0})_{\cS}^{\gen}\subseteq X_{z}.
\end{equation*}

Moreover, by Proposition <ref><ref> we have a Lagrangian torus fibration

\begin{equation*}
\ll_{\cS}^{\theta} \circ (\Phi_{\tau}^{0})^{-1} \colon (X_{z}^{0})_{\cS}^{\gen} \to \Delta_{\cS}^{\gen},
\end{equation*}

where $\Delta_{\cS}^{\gen}$ is the union of relative interiors of maximal faces of $\Delta_{\cS}$.

§.§ The associated metrics

Motivated by <cit.>, we study the behavior of Kähler metrics

\begin{equation*}%\label{eq:Kahler-metrics}
\end{equation*}

on $X_{z}$ as $z\rightarrow 0$. For maximal Calabi–Yau degenerations, <cit.> expects that the fiberwise Ricci-flat metric should converge, after appropriate re-scaling, to one satisfying a real Monge–Ampére equation. In our setting, an example of such metric is given by $\omega^{\flat}$, so looking at formula (<ref>), it is reasonable to expect such limiting behavior as $q\rightarrow 0$. This motivates considering a family of Kähler metrics parametrized both by $z\in \D_{\delta}^{*}$ and $q\in (0,1]$; and studying its limit as $q,|z|\rightarrow 0$.

Before we state our result in Proposition <ref>, we need some preparations. We denote by $T_{\vert} A$ the vertical tangent bundle to the A'Campo space, i.e. the kernel of the map $df_{A}\colon TA\to T\C_{\log}$. As usual, we denote its dual by $T^{*}_{\vert}  A$, its symmetric powers by $S^{k}T^{*}_{\vert} A$, etc.

Recall that we identify $A\setminus \d A$ with its image $X\setminus D$; in particular we have there the standard complex structure $J$ induced from $X$. We study the following sections of $\otimes^{2} T^{*}(A\setminus \d A)$:

\begin{equation*}\begin{split}