Symplectic monodromy at radius zero]
Symplectic monodromy at radius zero
and equimultiplicity of $\mu$-constant families

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 Spanish Ministry of Science, Innovation and Universities, project reference MTM2016-76868-C2-1-P (UCM), J.F.B. and T. P. were supported by the Basque Government through the BERC 2018-2021 program, and Gobierno Vasco Grant IT1094-16,
by the Spanish Ministry of Science, Innovation and Universities BCAM Severo Ochoa accreditation SEV-2017-0718, by the Spanish Ministry of Science, Innovation and Universities, project reference PID2020-114750GB-C33, and by CIRM Jean Morlet Semester Singularity Theory from Modern Perspectives.

Tomasz Pełka
Tomasz Pełka:
(1) BCAM,  Basque Center for Applied Mathematics, Mazarredo 14, 48009 Bilbao, Basque Country, Spain;
(2) Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

[2020]14B05, 14J17, 32S25, 32S30, 32S55, 53D40

We show that every family of isolated hypersurface singularities with constant Milnor number has constant multiplicity. To achieve this, we endow the A'Campo model of radius zero monodromy with a symplectic structure. This new approach allows to generalize a spectral sequence of McLean converging to fixed point Floer homology of iterates of the monodromy to a more general setting which is well suited to study $\mu$-constant families.

§ INTRODUCTION

Let $f\in\CC\lbr z_1,...,z_n\rbr $ be a formal power series. We write $f=\sum_{\iota\in\NN^n}a_{\iota}z^\iota$ with the usual multi-index notation. The multiplicity of $f$ is the biggest number $\nu(f)$ such that $f\in \mathfrak{m}^{\nu(f)}$, where $\mathfrak{m}$ is the maximal ideal of $\CC\lbr x_1,...,x_n\rbr $. The Milnor number of $f$ is $\mu(f)=\dim_{\C}\C \lbr z_1,\dots,z_n \rbr/\langle \tfrac{\d f}{\d z_1},\dots, \tfrac{\d f}{\d z_{n}}\rangle$, see  <cit.>. A continuous family of power series parametrized by $t\in [0,1]$ assigns to each $t$ a  power series $f_t=\sum_{\iota\in\NN^n}a_{\iota}z^\iota$ in such a way that each coefficient $a_{\iota}(t)$ is continuous in $t$.

The main result of this article is the following.

Let $(f_t)$ be a continuous family of power series. If the Milnor number $\mu(f_t)$ is independent of $t$ and finite then the multiplicity $\nu(f_t)$ is also independent of $t$.

Theorem <ref> implies a positive answer to Zariski Multiplicity Question for the case of families of isolated singularities, see Corollary <ref>. A brief survey on Zariski Question is given in Section <ref>: for more extensive ones we refer to [24, 25] or [27].

§.§ Summary of our approach

The starting point of our approach towards proving Theorem <ref> is a result of McLean <cit.>, which characterizes the multiplicity of an isolated singularity as the minimal integer $m\geq 1$ such that a version $\HF^*(\phi_f^m,+)$ of the fixed point Floer cohomology of the $m$-th iterate of the symplectic monodromy $\phi_f\colon \mathbb{F}_f\to \mathbb{F}_f$ of the Milnor fibration does not vanish. Here, $\mathbb{F}_f$ denotes the symplectic Milnor fiber (the precise definition of $(\mathbb{F}_f,\phi_f)$ requires some care, but we skip this subtlety for the sake of clarity at the introduction). To prove Theorem <ref>, we compare the groups  $\HF^*(\phi_{f_t}^m,+)$, for different members $f_t$ of  a $\mu$-constant family. We will now outline what is the key difficulty of this approach, why the setting of [49] is not sufficient to cover it, and how our new method allows to overcome this obstacle.

Let $f$ be a germ of an isolated hypersurface singularity. It is known that the pair $(\mathbb{F}_{f},\phi_f)$ is a Liouville domain with a compactly supported exact symplectomorphism [49]. Any two such pairs which are symplectically isotopic (in a certain sense)  give rise to the same Floer cohomology, see <cit.> or Proposition <ref>. So, the natural approach to prove Theorem <ref> would be to prove that, for small $t>0$, the pair $(\mathbb{F}_{f_{t}},\phi_{f_t})$ is symplectically isotopic to $(\mathbb{F}_{f_{0}},\phi_{f_{0}})$. However, even proving that $\mathbb{F}_{f_{t}}$ and $\mathbb{F}_{f_{0}}$ are isotopic as Liouville domains seems hard. The main reason is that it is not known whether the Milnor radius $\epsilon_t$ for $f_t$ can be chosen independent of $t$: in principle, we could have $\epsilon_t\rightarrow 0$ as $t\rightarrow 0$. This obstructs the direct comparison of Milnor fibrations for $f_t$ and $f_0$.

In [49], McLean uses an equivalence relation which is weaker than isotopy. Namely, to each pair $(\mathbb{F},\phi)$ as above, a construction of Giroux [32, 33] associates a pair $L_{\phi}\subseteq S_{\phi}$ of contact manifolds. Now, <cit.>, roughly speaking, shows that $\HF^{*}(\phi^m,+)$ depends only on this associated contact pair. For a symplectic monodromy of the Milnor fiber, [49] shows that this pair is the link $f^{-1}(0)\cap \S^n_{\epsilon}\subseteq \S^n_{\epsilon}$, where $0<\epsilon\ll 1$ is a Milnor radius, with the standard contact structure introduced by Varchenko [75].  This provides the basis for the main theorem of [49], asserting that the multiplicity
only depends on the embedded contact type of the link.

However, the above mentioned McLean's main result cannot be easily applied in our setting, due to the radius degeneration problem explained in the paragraph above. Indeed, let $(f_{t})_{t\in [0,1]}$ be a $\mu$-constant family, and let $\epsilon_{t}$ be a Milnor radius of $f_{t}$. Fix $0<t\ll 1$ and choose $\epsilon_t<\epsilon_0$.  Then

\begin{equation}
\label{eq:cobordismintro}
(\overline{\B_{\epsilon_{0}}\setminus \B_{\epsilon_t}},\overline{\B_{\epsilon_{0}}\setminus \B_{\epsilon_t}}\cap f_{t}^{-1}(0))
\end{equation}

defines a symplectic cobordism between the contact pair of $f_t$ and a contact pair which is contactomorphic to the one for $f_{0}$. This cobordism is known to be topologically trivial if $n\geq 4$ [46], but proving its symplectic triviality seems hard.

In order to explain our approach let us briefly recall known facts about the Milnor fibration. Let $f\colon (\CC^n,0)\to(\CC,0)$ be a holomorphic function germ. A classical result  [50] asserts that  there is  $\epsilon_0>0$ such that for any $0<\epsilon<\epsilon_0$ and any $0<\delta\ll \epsilon$ the restrictions

\begin{equation}
\label{eq:milfibsphintro}
\frac{f}{|f|}\colon \S_\epsilon \setminus f^{-1}(0) \to\S^1
\end{equation}

\begin{equation}
\label{eq:milfibtbintro}
f\colon \B_\epsilon\cap f^{-1}(\D_\delta^*)\to \D_\delta^*
\end{equation}

are locally trivial fibrations. The first one (called Milnor fibration in the sphere) is smoothly equivalent to the restriction of the second one (called Milnor fibration in the tube) to $\Int\B_{\epsilon}\cap f^{-1}(\d\D_{\delta})$. The numbers $\epsilon,\delta$ are called Milnor radii for the ball and the disc, respectively.
If $f$ has an isolated singularity at the origin then the monodromy of the above fibrations can be chosen compactly supported.

The closures of the fibers of both (<ref>) and (<ref>) are Liouville domains for the standard Liouville form $-\frac{\pi}{2}d^{c}||z||^2$. For the fibration (<ref>) McLean associates a monodromy $\phi_f:\mathbb{F}_f\to\mathbb{F}_f$ which is a compactly supported exact symplectomorphism and studies the Floer cohomology of the pair $(\mathbb{F}_f,\phi_f)$.

In turn, we note that the fibration (<ref>) will not be, in general, locally trivial in a symplectic sense, since there is no reason why the volume of the fibers should be constant in $z\in\D^*_\delta$. Nonetheless, in Example <ref> we fix this issue by slightly modifying the shape of the ball $\B_{\epsilon}$.

In <cit.> McLean provides a spectral sequence converging to $\HF^{*}(\phi_f^m,+)$, whose first page allows to compute the multiplicity of $f$. This first page is closely related to the A'Campo formula [3] for the Lefschetz numbers and zeta function of $\phi_{f}^m$, which we now recall.

Let $h\colon X\to\CC^n$ be a log resolution of $f$ that only modifies $\C^n$ over the origin. Write $(f\comp h)^{-1}(0)=\sum_{i=0}^Nm_i D_i$, where $D_0$ is the strict transform of $f^{-1}(0)$.  Then the function $f\comp h$ can be expressed in local coordinates $(z_{1},...,z_n)$ as $\prod_{i=1}^kz_{i}^{m_{i}}$.
Denote by $\tau:\D_{\log}\to\D_\delta$ the real oriented blow up at the origin. Then $\d\D_{\log}:=\tau^{-1}(0)$ can be thought as a circle at radius 0. Exploiting the local structure of $f\comp h$, A'Campo constructed a topological locally trivial fibration

\begin{equation}
\label{eq:ACampofibintro}
f_A\colon A\to\D_{\log}
\end{equation}

whose restriction over $\D_{\delta}^{*}$ coincides with (<ref>) and whose restriction

\begin{equation}
\label{eq:ACampofibintrorad0}
f_A|_{f_A^{-1}(\d \D_{\log})}\colon f_A^{-1}(\d \D_{\log})\to \d\D_{\log}
\end{equation}

over the radius $0$ circle admits a monodromy $\phi_A$ such that the fixed point set of $\phi_A^m$ is equal to

\begin{equation}
\label{eq:fixedsetintro}
\bigsqcup_{m_i|m}B_i,
\end{equation}

where $B_i$ is a cyclic covering of degree $m_i$ of $D_i\setminus (\bigcup_{j\neq i} D_i)$ determined by $f$, cf. <cit.>. Therefore, we have the equality of Lefschetz numbers

\begin{equation}\label{eq:ACampo-formula}
\Lambda(\phi_f^m)=\sum_{m_i|m}\chi(B_i),
\end{equation}

where $\chi(\sdot)$ denotes the topological Euler characteristic. Notice that the summand $\chi(B_0)$ appears for every $m$, but it is equal to $0$, since $B_0$ is homeomorphic to $(f^{-1}(0)\cap\S_{\epsilon_0})\times (0,\epsilon_0]$ by the conical structure theorem <cit.>, and $\chi(f^{-1}(0)\cap\S_{\epsilon_0})=0$ since $f^{-1}(0)\cap\S_{\epsilon_0}$ is an odd-dimensional, smooth manifold. At this stage, we would also like to remark that the space $A$ constructed by A'Campo is a manifold with corners, and $f_A$ has no further structure than the topological locally trivial fibration.

As we will explain below, the main step of our proof is to endow $A$ with a smooth structure, and define a fiberwise symplectic form on $A$ such that $\phi_{A}$ is realized as the symplectic lift of the angular vector field on $\d\D_{\log}$. In particular, the flow of the symplectic lift of the radial vector field on $\D_{\log}$ provides a symplectic isotopy between $\phi_{A}$ and the monodromy of the Milnor fibration (<ref>).

In turn, McLean in  <cit.> constructs a certain Liouville domain $\mathbb{F}$ with a compactly supported exact symplectomorphism $\phi$, whose dynamics coincides with the one of $\phi_A$; and whose associated contact pair is contactomorphic to the link of $f$. Hence even though $(\F_{f},\phi_{f})$ and $(\F,\phi)$ might not be  symplectically isotopic, one still has that $\HF^{*}(\phi^m,+)=\HF^{*}(\phi_{f}^m,+)$ is the required Floer cohomology. Now, given such $(\mathbb{F},\phi)$, the first step to define $\HF^*(\phi,+)$ is to deform $\phi$ slightly by a Hamiltonian near $\d\mathbb{F}$ so that the deformed $\check\phi$ has no fixed points in a collar near $\d\mathbb{F}$. Therefore, the fixed point set of the deformed McLean monodromy $\check{\phi}^m$ is of the form

\begin{equation}
\label{eq:mcleandecomp}
\bigsqcup_{m_i|m,\ i\neq 0}B_i
\end{equation}

where $B_i$ have the same properties as above. Note that the term $B_0$ got removed by the deformation near $\d \mathbb{F}$. This allows to construct a spectral sequence converging to $\HF^*(\phi,+)$ whose $E_1$-pieces are the ordinary homologies of the components $B_i$ for $m_i|m$, $i\neq 0$, see <cit.>. Since $\chi(\HF^*(\phi^m,+))=\Lambda(\phi^m)$, this result may be regarded as a lifting  of the A'Campo formula (<ref>) to the setting of Floer cohomology.

Since the component $B_0$ is not present in (<ref>), the $E_1$-page is empty for $m$ is strictly smaller than the multiplicity $\nu(f)$. It is also easy to show that $\HF^*(\phi^m,+)\neq 0$ for $m=\nu(f)$.

To have a grasp of why the deformation $\check\phi$ can be performed so that the component $B_0$ is not present, observe that, exploiting the topological product structure $B_0\cong (f^{-1}(0)\cap\S_{\epsilon})\times (0,\epsilon]$, where $\epsilon$ is the Milnor radius of $f$, the monodromy homeomorphism $\phi_A$ constructed by A'Campo can be isotoped to a homeomorphism $\check{\phi}_A$ that eliminates the fixed point component $B_0$ and leaves the other ones. Now for a sufficiently small Milnor radius $\epsilon>0$, $B_0$ becomes symplectically trivial, that is, a piece of a symplectization $(f^{-1}(0)\cap\S_{\epsilon}) \times (0,\epsilon)$ of the contact link. This allows to realize the above isotopy in the symplectic category, see <cit.>.

The Milnor fibration in the tube (<ref>) is better suited to study families, in particular $\mu$-constant ones. Indeed, let $F:\CC^n\times\D_\xi\to\CC\times\D_\xi$ be holomorphic of the form $F(x,t)=(f_t(x),t)$ such that $f_t$ has a  $\mu$-constant isolated singularity at the origin. Then, in [46] it is proved that there exits Milnor radii $\epsilon_0,\delta_0$ for $f_0$ such that, after shrinking $\xi>0$, the restriction

\begin{equation}
\label{eq:auxfibintro}
F\colon (\B_{\epsilon_0}\times\D_\xi)\cap F^{-1}(\D_{\delta_0}^*\times\D_\xi)\to \D_{\delta_0}^*\times\D_\xi
\end{equation}

is a locally trivial fibration. So, fixed any $t\in \D^*_\xi$ and any $\delta_t<\delta_0$, the restriction

\begin{equation}
\label{eq:auxfibintro2}
f_t\colon \B_{\epsilon_0}\cap f_t^{-1}(\D_{\delta_t}^*)\to \D_{\delta_t}^*
\end{equation}

is diffeomorphic to the Milnor fibration in the tube for $f_0$. Now we pick $\epsilon_t<\epsilon_0$ and $\delta_t<\delta_0$, Milnor radii for $f_t$. Then, for this particular choice of radii, the Milnor fibration in the tube for $f_t$ is embedded in the fibration (<ref>), and according with [46], for any $z\in \D_{\delta_{t}}$ the difference between the fiber of (<ref>) over $z$ and the Milnor fiber $f_t(z)\cap\B_{\epsilon_t}$ of $f_t$ over $z$ is a cobordism diffeomorphic to (<ref>). Moreover, the monodromy of (<ref>) can be chosen with support at the interior of $f_t^{-1}(z)\cap\B_{\epsilon_t}$.  Thus in our case, the component $B_0$ of the fixed point set (<ref>) of $\phi_A$ corresponds to the  cobordism (<ref>), which is not known to be symplectically trivial. So, $B_0$ cannot be eliminated by a small Hamiltonian perturbation.

Our proof of Theorem <ref>
can be summarized in the following steps:

*  We modify the fibration (<ref>) so that it becomes symplectically locally trivial and maintains the topological properties stated above. This is a matter of modifying the shape of the domain $(\B_{\epsilon_0}\times\D_\xi)\cap F^{-1}(\D_{\delta_0}^*\times\D_\xi)$ in such a way that the fibers can be symplectomorphic, in particular have constant volume. As a consequence, for each $t\in\D_\xi$ we define a pair $(\mathbb{F}_t,\phi_t)$ formed by a Liouville domain and an exact symplectomorphism, whose topology coincides with the monodromy of the fibration (<ref>), and such that for any small $t>0$, the pairs $(\mathbb{F}_{t},\phi_{t}^m)$ and $(\mathbb{F}_{0},\phi_{0}^m)$ are symplectically isotopic. In particular, $\HF^*(\phi_{t}^m,+)\cong \HF^*(\phi_{0}^m,+)$ for every $m$.

*  For each fixed $t$, we produce a symplectic version of A'Campo construction. That is, we construct a symplectically locally trivial fibration (<ref>) satisfying the following two properties.

*  The restrictions of $f_A$ and of the fibration (<ref>) coincide symplectically over $\D_\delta\setminus\D_{\delta'}\subset\D_{\log}$ for a certain $0<\delta'<\delta$,

*  The monodromy of (<ref>) induced by the lifting of the unit vector field in $\d\D_{\log}$ has exactly the same dynamical properties as the topological monodromy constructed by A'Campo.

We call this monodromy  the symplectic monodromy at radius zero and denote it by $(\mathbb{F}^0_{t},\phi_{t}^0)$. Property <ref> gives an isotopy between $(\mathbb{F}_{t},\phi_{t})$ and $(\mathbb{F}^0_{t},\phi_{t}^0)$, so we have $\HF^*(\phi_{t}^m,+)\cong \HF^*((\phi_{t}^0)^m,+)$ for any $m$.

As we explain below, this step gets rather involved, and creates a technique that may be useful for other degeneration problems in algebraic geometry. Our construction is, in fact, performed in a much broader generality, covering e.g. the case when $f$ is a function from Stein space $Y$, and both $Y$ and $f^{-1}(0)$ can have many isolated singularities.

We slightly generalize McLean's spectral sequence in such a way that the component $B_0$, corresponding to the cobordism (<ref>), is taken into account. The new terms appearing in the first page of the spectral sequence are summands of the Borel-Moore homology $H_*^{BM}(B_0,\ZZ_2)$. They vanish because the cobordism (<ref>) is homologically trivial. Thus we get the same $E_1$ page as <cit.>, hence the same characterization of multiplicity – now for the Milnor radius $\epsilon_0$ fixed within the family. This result can also be stated in more generality indicated in Step <ref> above, see Propositions <ref>,  <ref>.

Our proof does not use Giroux construction [32, 33] or McLean result <cit.> asserting that Floer homology depends only on the associated contact link. We directly deform the natural symplectic monodromy arising from the Milnor fibration in the tube (<ref>) to a radius zero monodromy.
This way, for any fixed radius we get a model $(\mathbb{F}_A,\phi_A)$ which is actually isotopic to the fibration $(\mathbb{F}_{f},\phi_f)$ at that fixed radius, and has dynamical properties which allow to derive McLean spectral sequence. This is done without relying on most constructions in [49].

As a byproduct of our construction, in Corollary <ref> we produce, for any singular hypersurface germ, a representative of its monodromy in the symplectic mapping class group of its Milnor fiber, which has no fixed points. We also produce one whose fixed point set is precisely the boundary of the Milnor fiber. This extends McLean results, who produces models with the same properties, but instead of being symplectically isotopic to the monodromy, they just share the same contact pairs.

§.§ Outline

We will now indicate the sections of the article where each of the above steps is realized.

The technical core of the proof lies in Step <ref>, that is, in the construction of the symplectic structure on the A'Campo space $A$. It is carried out in Sections <ref> and <ref>. Definition <ref> of the space $A$ can be explained in two steps. As before, let $h\colon X\to \C^n$ be a log resolution of $f$, and let $D=\sum_{i}m_{i}D_{i}$ be the total transform of $f^{-1}(0)$. First, we perform a real oriented blowup $X_{\log}\to X$ which extends the fibration over $\D_{\delta}^{*}$ to one over $\D_{\log}$, see (<ref>). This can be phrased conveniently in terms of Kato–Nakayama construction [44] in log geometry, see [12, 14] for applications in this context. The resulting monodromy has the required behavior over each $D_{i}$, but is not continuous over the intersections, see Example <ref>. To remedy this problem, we multiply the preimage of each intersection $\bigcap_{i\in I} D_{i}$ by a corresponding face $\Delta_{I}$ of the dual complex of $D$. This face is parametrized by tropical coordinates $w_{i}$, see (<ref>), which measure the relative speed of convergence toward each $D_i$. The idea of such a hybrid construction, combining classical and tropical coordinates, goes back to [6], and has been successfully applied to various degeneration problems,  see e.g. [8, 22]. For our application, we need to push this idea further.

Indeed, the space $A$ constructed above is a manifold with corners. We need to choose its smoothing in a very careful way,  which allows to introduce a fiberwise symplectic form satisfying restrictive properties <ref>–<ref> above. To do this, we introduce smooth charts (<ref>) in such a way that both radial coordinates $r_i=|z_i|$ of $X$; and tropical coordinates $w_i$ of the simplices in $\d A$, decay exponentially
towards their zero loci. This way, the local coordinate $r_i$ corresponding to a component $D_i$ gets replaced by a global smooth function $\bar{v}_{i}\colon A\to \R$ measuring, in a sense, a hybrid distance to $D_i$. Using these functions in (<ref>) we provide an explicit formula for the symplectic form. The resulting monodromy $\phi_A|_{\d A}$ at radius zero  is then a time one flow of the vector field given by the formula (<ref>).

The guiding principle of the technical proofs in Sections <ref>–<ref> is that all choices made along the construction become irrelevant at radius zero. Hence a local computation can be used to prove that at radius zero, $\phi_{A}$ is a rotation by $\tfrac{2\pi}{m_i}$ about $D_i$, and an interpolating Dehn twist in between, see Figure <ref>. This way, we get a symplectic representative of the monodromy as required in <ref> and <ref>.

Step <ref> is addressed in Section <ref>, where we recall the necessary notions from the theory of Liouville domains and their fibrations [33, 67]. In particular, in Definition <ref> we recall McLean's notion of graded abstract contact open book [49], i.e. pair $(\mathbb{F},\phi)$ as above, for which $\HF^{*}(\phi,+)$ is defined. Then, we describe a procedure which makes a symplectic monodromy into a graded abstract contact open book, see Definition <ref>. Our setting is closely related to Liouville fibrations with singularities considered in [67], see Remark <ref>.

In Section <ref> we recall the definition of Floer cohomology groups $\HF^{*}(\phi,+)$. To be precise,  we will use Floer homology introduced by Uljarevic in [74], cf. [65]: in Section <ref> we explain how to compare it with the setting of [49]. In Proposition <ref>, we prove a slight generalization of McLean spectral sequence <cit.> to the setting which includes the fixed-point component $B_0$. In Proposition <ref>, we apply this spectral sequence to the monodromy $\phi_{A}$ constructed in Step <ref>, using its dynamical properties listed in Proposition <ref>. This way, we get a spectral sequence similar to <cit.>, but with term $B_0$ taken into account. Theorem <ref> is deduced in Section <ref>.

In order to make the article more accessible to readers who are not specialists in symplectic geometry, we try to be reasonably self-contained at various points.

§.§ A brief history of the Zariski multiplicity conjecture

In his seminal paper [81], Zariski posed eight questions which had a major impact on the current shape of singularity theory. Out of these questions, the only one which remains open is the following.

Let $f,g\colon (\C^n,0)\to (\C,0)$ be holomorphic germs of the same topological type. Is it true that $f$ and $g$ have the same multiplicity?

Here, $f,g$ are said to be of the same topological type if there exists a germ of a homeomorphism $\Phi\colon (\CC^n,0)\to (\CC^n,0)$ such that $\Phi(V(f))=V(g)$, where $V(f)$ denotes the zero set of $f$.

If the above homeomorphism $\Phi$ is assumed to be $\cC^{1}$, then the affirmative answer to Question <ref> was given by  Ephraim [23], and, independently, by Trotman [72]. The answer is also known to be positive for $n=2$, i.e. for plane curve singularities [80], and if one of the germs is smooth [3, 45]: in fact, the latter result follows from the A'Campo formula (<ref>) for the Lefschetz number. It is also true if $n=3$ and one of the germs has multiplicity two, see Navarro-Aznar [53]. If $n=3$ and the homeomorphism $\Phi$ is additionally bilipschitz, then a positive answer to Question <ref> was given in <cit.> by  Fernandes, Sampaio and the first author.

Assume that $n=3$, and both $f$ and $g$ are germs of isolated surface singularities. Then, the answer to Question <ref> is known to be positive if both $f,g$ are quasihomogeneous (Xu–Yau [76, 77]); if one has arithmetic genus at most two (Yau [78]), or geometric genus at most $3$, or  Milnor number at most 26 (Yau–Zuo [79]). It is also true for suspensions of irreducible plane curve singularities (Mendris–Némethi [51]). For $n>3$, a positive answer is known if one of the germs has Milnor number at most $2^{n}-1$ [79].

In general, if $f$ and $g$ are topologically right-left equivalent by bilipschitz homeomorphisms, Risler–Trotman [61] proved that they have the same multiplicity.

An important special case of Question <ref>, which was also widely studied since [81], concerns families of topologically equivalent germs. Theorem <ref> gives a positive answer to this question in case of isolated singularities. More precisely, we have the following.

Let $f_t\colon (\C^n,0)\to (\C,0)$ be a continuous family of holomorphic germs of isolated singularities, parametrized by $t\in [0,1]$. Assume that the topological type of $f_t$ is independent of $t$. Then the multiplicity $\nu(f_t)$ is independent of $t$, too.

Indeed, families of isolated singularities with constant topological type have constant Milnor number $\mu$, because $\mu$ is a topological invariant <cit.>, see [45, 71]. Thus Corollary <ref> follows from Theorem <ref>.  We remark that the question whether every $\mu$-constant family has constant topological type has positive answer if $n\neq 3$ [46], and is open if $n=3$, i.e. for families of surface singularities.

In Example <ref>, we show that there do exist isolated singularities of the same topological type, which are not connected by a $\mu$-constant family. Therefore, while Corollary <ref> gives a positive answer to Question <ref> for families of isolated singularities, the non-family version remains open. The non-isolated case of Question <ref>, even for families, remains open, too. Nonetheless, David Massey observed that our results imply a positive answer for a particular class of aligned singularities, see Corollary <ref>.

Up to now, Theorem <ref> and Corollary <ref> were established only in certain special cases. Gabrielov and Kushnirenko [34] proved Theorem <ref> for $f_0$ homogeneous. Greuel [38] and O'Shea [57] proved it for $\mu$-constant deformations of functions $f_0$, with the only hypothesis that $f_0$ is quasi-homogeneous. The case when $f_0$ is Newton-nondegenerate was settled by Abderrahmane [2], cf. [69]. Other particular cases were studied e.g. by Greuel–Pfister <cit.>, Massey <cit.> or Plenat–Trotman [58]. Let us also remark that classical results of Hironaka [40] imply that Whitney equisingular families are equimultiple. In [73], Trotman and van Straten proved that weak Whitney equisingularity implies equimultiplicity, too.

As a final remark, we note that Theorem <ref> is easily reduced to the case when the family $(f_t)$ is algebraic, i.e. each coefficient $a_{\iota}$ is a polynomial in $t$. This way, Theorem <ref> becomes a purely algebraic statement, which can be formulated over any field. Nonetheless, even in the curve case ($n=2$), no purely algebraic proof of this statement is known. In fact, the only proof over the complex numbers consists in proving that a $\mu$-constant family is topologically trivial, and then applying the affirmative answer to Question <ref>, known for $n=2$. For families of functions defined over an algebraically closed field of characteristic zero, an application of Lefschetz Principle gives the same conclusion, see Remark <ref>. Answering this question was one of the main motivations for development of the computer algebra program Singular [18], see [37] for a historical account.

Recently, Nero Budur, Lê Quy Thuong, Honc Duc Nguyen and the first author have conjectured that the fixed point Floer cohomology of the $m$-th iterate of the monodromy coincides with the compactly supported cohomology of the $m$-th restricted contact locus <cit.>. This might suggest an algebraic approach to the proof of Theorem <ref>, based on studying variation of contact loci in $\mu$-constant families. However, this problem does not seem easy at all.

Part of this project was realized during the semester Singularity Theory from Modern Perspectives at Centre International de Rencontres Mathématiques in Marseille. We would like to thank CIRM for their hospitality and great working conditions. We would also like to thank Norbert A'Campo, Enrique Artal, Nero Budur,  Ailsa Keating, Ignacio Luengo, David Massey, András Némethi, Adam Parusiński, María Pe Pereira, Patrick Popescu-Pampu, Edson Sampaio, and Duco van Straten for helpful discussions. We also thank the referees for valuable comments.

§ PRELIMINARIES

In this section, we review some standard notation which used throughout the article; and recall those basic definitions from symplectic geometry which will be needed to introduce the fiberwise symplectic structure on the A'Campo space in Section <ref>. Further notions, needed to define symplectic monodromy and McLean spectral sequences, will be introduced in Sections <ref> and <ref>.

§.§ Basic notation

We write $\log$ for the natural logarithm.  For $r\geq 0$ we denote by $\B_{r}^{n}$, or simply $\B_{r}$, the closed Euclidean ball of radius $r$ in $\R^n$, and by $\D_{r} \subseteq \C$ the closed disk of radius $r$ in $\C$. We put $\D_{r}^{*}=\D_{r}\setminus \{0\}$.

We denote by $\id_{X}$ the identity map $X\to X$; and by $\id_{k\times k}$ the $k\times k$ identity matrix. If $X$ or $k$ is clear from the context, we will sometimes write $\id$ for $\id_X$ or $\id_{k\times k}$. For a map $f\colon X\to X$, we denote its fixed point set by $\Fix f\de \{x\in X: f(x)=x\}$. We write $\pr_{X}$, $\pr_{Y}$ for the projections from a subset of $X\times Y$ to $X$ and $Y$, respectively.

Let $X$ be a topological space. For a subset $U\subseteq X$, we denote by $\bar{U}$ its topological closure, and by $\Int U$ (or $\Int_{X}U$) its interior. Given a sequence $(x_{\nu})_{\nu=1}^{\infty}$ of elements of $X$ (which we abbreviate by $(x_{\nu})\subseteq X$), we write $x_{\nu}\rightarrow x$ for the convergence $\limn x_{\nu}=x$. Similarly, for a subset $Z\subseteq X$ and continuous functions $h$ and $a$ on $X\setminus Z$ and $Z$, respectively, we say that $h\rightarrow a$ on $Z$ if $h$ extends to a continuous function on $X$ which restricts to $a$ on $Z$. We denote this continuous extension by $h$, too.

For a smooth manifold $X$, we denote by $\Omega^{k}$ the sheaf of smooth $k$-forms, and put $\Omega^{*}=\bigoplus_{k\geq 0}\Omega^{k}$. Thus for an open subset $U\subseteq X$, the $\R$-algebra $\Omega^{0}(U)$ is the $\R$-algebra $\cC^{\infty}(U)$ of smooth functions on $U$. Whenever no confusion is likely to occur, we will identify a function with its restriction, e.g. by writing $\cC^{\infty}(V)\subseteq \cC^{\infty}(U)$ for an open subset $V\subseteq U$.

§.§ Symplectic manifolds

A $2$-form $\omega\in \Omega^2(M)$ is symplectic if it is closed and nondegenerate, i.e. $T_{x}M\ni v\mapsto \omega(v,\sdot)\in T^{*}_x M$ is an isomorphism for all $x\in M$. A $1$-form $\lambda$ is Liouville if $d\lambda$ is symplectic. Its Liouville vector field $X_{\lambda}$ is defined by $d\lambda(X_{\lambda},\sdot)=\lambda$. We say that $(M,\lambda)$ is a Liouville domain if $M$ is a compact connected manifold with boundary and $\lambda\in \Omega^{1}(M)$ is a Liouville form such that $X_{\lambda}$ points outwards $\d M$.

Let $J\in \operatorname{End}(TM)$ be an almost complex structure, i.e. $J^2=-\id$. We say that $\omega\in \Omega^{2}(M)$ tames $J$ if $\omega(v,Jv)>0$ for all nonzero $v\in T_{x}M$, $x\in M$: in this case, $\omega$ is nondegenerate. We say that $\omega$ is $J$-compatible if it tames $J$ and $\omega(J\sdot,J\sdot)=\omega(\sdot,\sdot)$. Note that these definitions vary slightly among the literature; here we follow <cit.>.

Assume that $M$ is a complex manifold, and let $J$ be the multiplication by $\imath=\sqrt{-1}$. A $2$-form on $M$ is Kähler if it is $J$-compatible and closed, hence symplectic.

For a smooth function $\varrho\colon M\to \R$ we write $d^{c}\varrho=d\varrho\circ J$. Thus $d^{c}=\imath (\d-\bar{\d})$. Again, we warn the reader that the definition of $d^{c}$ varies among the literature, e.g. <cit.> defines $d^{c}$ as $-\imath (\d-\bar{\d})$, and <cit.> as $-\frac{\imath}{4\pi}(\d-\bar{\d})$. Our definition of $d^{c}$, adapted from <cit.>, is used e.g. in <cit.> in a context which is very similar to ours, see Example <ref>.

We say that a smooth function $\varrho\colon M\to\R$ is strictly plurisubharmonic  if $-dd^{c}\varrho$ tames $J$, see <cit.>. In this case, the form $-dd^{c}\varrho$ is Kähler. A function $\varrho$ is exhaustive if $\varrho^{-1}(-\infty,c]$ is compact for every $c\in \R$. If $\varrho$ is an exhaustive strictly plurisubharmonic function then  $\varrho^{-1}(-\infty,c]$ with $1$-form $-d^{c}\varrho$ is a Liouville domain. Indeed, the Liouville vector field $X_{-d^{c}\varrho}$ is the gradient field of $\varrho$ with respect to the metric $-dd^{c}\varrho(\sdot,J\sdot)$, see <cit.>. Every Stein manifold admits an exhaustive strictly plurisubharmonic function, see <cit.>.

Let $M=\C^{*}$, and let $J(v)=\imath \cdot v$, i.e. $J$ is a rotation by $\pi$. Then in polar coordinates $z=r\cdot \exp(2\pi \imath\cdot \theta)$ we have $J\frac{\d}{\d r}= \frac{1}{r}\frac{\d}{\d \theta}$ and $J\frac{\d}{\d \theta}=-r\tfrac{\d}{\d r}$. Hence $d^{c}r=-r\, d\theta$ and $d^{c}\theta=\frac{1}{r}\, d r$.

Now if $M=\C^n$, $\varrho(z)=\frac{\pi}{2}\cdot ||z||^2=\frac{\pi}{2}\cdot\sum_{i}r_{i}^{2}$, then

$\lambda\std\de -d^{c}\varrho =\pi\cdot\sum_{i}r_{i}^{2}d\theta_i$,
$\omega\std \de -dd^{c}\varrho=2\pi\cdot\sum_{i} r_{i}\, dr_{i}\wedge d\theta_{i}$ is the standard area form, and $X_{\lambda\std}=\pi\cdot\sum_{i}r_{i}\frac{\d}{\d r_{i}}=\nabla\varrho$ is the radial vector field.

Let $f\colon \C^n\to \C$ be a holomorphic function with critical value $0\in \C$, and $\epsilon>0$ be such that $\B_{\epsilon}\trans f^{-1}(0)$. Then for $0<\delta\ll \epsilon $, the Milnor fiber $F\de f^{-1}(\delta)\cap \B_{\epsilon}$ is smooth, and $f^{-1}(\delta)\trans \B_{\epsilon}$, so $(F,\lambda\std|_{F})$, is a Liouville domain.

For a more general example, replace $\C^{n}$ by a Stein space $Y$, smooth off $f^{-1}(0)$; $\B_{\epsilon}$ by a sub-level set of an exhaustive strictly plurisubharmonic function $\varrho$; and the standard Liouville form by $-d^{c}\varrho$.

Let $f\colon M\to B$ be a submersion. A form $\omega\in \Omega^{2}(M)$ is fiberwise symplectic if it is closed, and its restriction to each fiber of $f$ is nondegenerate (hence symplectic). A fiberwise symplectic form defines an Ehressmann connection $\{v\in TM: \forall_{w\in \ker f_{*}}\colon \omega(v,w)=0\}$, called a symplectic connection <cit.>. Given a vector field on the base $B$, its lifting to the symplectic connection is called the symplectic lift. If the flow  the symplectic lift of a vector field in $B$ exists at a given time, then it induces symplectomorphisms between fibers, see <cit.>.

§ THE A'CAMPO SPACE: MILNOR FIBRATION AT RADIUS ZERO

Let $X$ be a complex manifold. Let $f\colon X\to \C$ be a holomorphic function with only one critical value $0\in \C$, such that the central fiber $f^{-1}(0)$ is snc. In this section, we recall the A'Campo [3] model $A$ of the radius zero monodromy of $f$, see Figure <ref>, and construct a smooth atlas on it.

After some preparation in Section <ref>, we introduce the topological space $A$ in Definition <ref> by a hybrid procedure using tropical coordinates and Kato-Nakayama spaces at radius $0$, and classical geometry at positive radius. We list its basic properties in Section <ref>. In Section <ref>, we endow $A$ with a structure of a $\cC^1$-manifold with boundary, which depends only on $f$. In Section <ref> we improve this structure to a $\cC^{\infty}$ one, designed so that we can introduce a well-behaved fiberwise symplectic form (<ref>) in Section <ref>. This $\cC^{\infty}$ structure will depend on some additional data chosen along the construction.

In a topological setting, $A$ was introduced by A'Campo in [3], who used it to compute the zeta function of the monodromy. We note that loc. cit.  predates the Kato–Nakayama construction [44], so instead of $X_{\log}$ it uses a real blowup. In turn, tropical (or hybrid) constructions have recently been used, for example in [8, 5, 22], to define topological spaces related to our space $\Gamma$ in Definition <ref>, together with some additional structure inherited from $f^{-1}(\delta)$ as $\delta\rightarrow 0$: e.g. a measure in <cit.> or a Lagrangian torus fibration in [22].

§.§ Construction of the A'Campo space

Let $X$ be a complex manifold of dimension $n$. Let $f\colon X\to \C$ be a holomorphic function, and let $D$ be the (scheme-theoretic) fiber $f^{-1}(0)$. Assume that $D\redd$ is snc. Since we are only interested in $f^{-1}(0)$ and nearby fibers, we assume $\log |f|<-1$ and that $f|_{X\setminus D}$ is a submersion onto $\D_{e^{-1}}^{*}$, so $0\in \C$ is its only critical value. Whenever no confusion is likely to occur, we identify divisors with their supports, e.g. by writing $X\setminus D$ for $X\setminus \operatorname{Supp} D$, i.e. for $f^{-1}(\D_{e^{-1}}^{*})$; or writing $x\in D$ instead of $x\in \operatorname{Supp} D$.

Let $D_1,\dots, D_N$ be the irreducible components of $D\de f^{-1}(0)$. Write

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

where each $m_{i}$ is a positive integer. Put $X_{\emptyset}=X\setminus D$, and for a nonempty subset $I\subseteq \{1,\dots, N\}$ put

\begin{equation}\label{eq:stratification}
%X_{I}=\bigcap_{i\in I} D_{i},\quad
X_{I}^{\circ}=\bigcap_{i\in I} D_{i}\setminus \bigcup_{j\not \in I} D_{j}.
\end{equation}

Then $X=\bigsqcup_{I}X_{I}^{\circ}$ is a stratification of $X$.

§.§.§ Charts adapted to $f$
In order to fix the notation, we recall some basic definitions. We say that $(U_X,\psi_X)$ is a holomorphic chart for $X$ if $U_X\subseteq X$ is an open set, and $\psi_X\colon U_X\to \C^{n}$ is a biholomorphism onto its image. The map $\psi_X$ will often  be clear from the context, in which case we will abuse the language and use the term chart to refer to $U_X$. We will usually keep the subscript $X$ when referring to a chart in $X$, and drop it when taking its preimage in the A'Campo space.

A holomorphic atlas is a collection $\cU_X=\{(U_X^{p},\psi_X^{p}):p\in R\}$ of holomorphic charts which cover $X$, i.e.  $X=\bigcup_{p\in R}U^{p}_X$.
A partition of unity inscribed in $\cU_X$ is a collection $\{\tau^{p}:p\in R\}$ of smooth functions $\tau^{p}\colon X\to [0,1]$ such that $\tau^{p}|_{X\setminus U_X^p}=0$, the set $\{p\in R:\tau^{p}(x)\neq 0\}$ is finite for every $x\in X$, and $\sum_{p\in R}\tau^{p}(x)=1$.
We will use charts of the following form.

Let $(U_X,\psi_X)$ be a holomorphic chart for $X$. Its associated index set is

\begin{equation}\label{eq:index-set}
S=\{i: D_i\cap \bar{U}_X\neq \emptyset\}\subseteq \{1,\dots, N\}.
\end{equation}

Write $\psi_X=(z_{i_1},\dots,z_{i_n})\colon U\to \C^{n}$ for some positive integers $i_1,\dots,i_n$. We say that the chart  $(U_X,\psi_X)$ is adapted to $f$ if the following conditions are satisfied.

*  For every $i\in \{1,\dots, N\}$, we have $D_{i}\cap U_X=\{x\in U_X: z_{i}(x)=0\}$.

*  If $S\neq \emptyset$ then the function $f$ restricts to $f|_{U_X}=\prod_{i\in S}z_{i}^{m_i}$.

*  For every $i\in S$, we have $\log |z_{i}|<-1$.

*  The set $U_X$ has compact closure $\bar{U}_X$, and the map $\psi_X$ extends continuously to $\bar{U}_X$.

A holomorphic atlas is adapted to $f$ if all its charts are. We say that a pair $(\cU_X,\btau)$ is adapted to $f$ if $\cU_X$ is a holomorphic atlas adapted to $f$, and $\btau$ is a partition of unity inscribed in $\cU_X$.

Note that, by <ref>, the indices of $z_{j}$'s are either in $S$ – in which case they correspond to the fixed order of components of $D$ – or are integers bigger than $N$. Although a bit unusual, this convention will make further notation much more convenient. We also note that by <ref>, each component of $D$ meeting the closure $\bar{U}_X$ (i.e. whose index is in $S$) actually meets $U_X$.

Because $D\redd$ is snc, $X$ admits an atlas adapted to $f$. Indeed, the definition of snc implies that, after suitably indexing the variables, every point of $X$ lies in a chart satisfying  <ref>, <ref>. Then, conditions  <ref>, <ref> can be achieved by shrinking those charts.

§.§.§ The basic functions

We will use a continuous bijection $\eta\colon [0,1]\to [0,1]$, given by

\begin{equation}\label{eq:smooth_simplex}
\eta(0)=0,\quad \eta(s)=(1-\log s)^{-1}\mbox{ for } s>0,
\end{equation}

see Figure <ref>. Note that the restriction $\eta|_{(0,1]}$ is smooth.
Since we have assumed $\log |f|<-1$, we have smooth functions

\begin{equation}\label{eq:def-t-g}
t\de (-\log|f|)^{-1}\colon X\setminus D\to [0,1),\qquad
g\de \eta(t)\colon X\setminus D\to [0,1).
\end{equation}

We extend them to continuous functions on $X$ by putting $t|_{D}=0$, $g|_{D}=0$. We have a smooth map

\begin{equation}\label{eq:def-theta}
\theta\de \frac{f}{|f|}\colon X\setminus D\to \S^{1}.
\end{equation}

Fix a chart $(z_{i_1},\dots, z_{i_n})\colon U_X\to \C^{n}$ adapted to $f$. Fix $i$ in its index set $S=\{i:\bar{U}_X\cap D_i\neq\emptyset\}$, see (<ref>). By  Definition  <ref><ref>, $\log|z_{i}|<-1$, so in particular $|z_{i}|<1$. We have radial coordinates

\begin{equation}\label{eq:def-r_i-t_i}
r_{i}\de |z_{i}|\colon U_X \to [0,1),
\quad\mbox{and}\quad
t_{i}\de (-m_{i}\log r_{i})^{-1}\colon U_X\setminus D_i\to (0,1).
\end{equation}

We extend $t_i$ to a continuous function $U_X\to [0,\infty)$ by setting $t_{i}|_{U_X\cap D_{i}}=0$. We also have an angular coordinate:

\begin{equation}\label{eq:def-theta_i}
\theta_{i}\de \frac{z_{i}}{|z_{i}|}\colon U_X \setminus D_i\to \S^1.
\end{equation}

We will compare the speeds of convergence towards $D_i$ using tropical functions:

\begin{equation}\label{eq:def-w_i-u_i}
w_{i}\de \frac{t}{t_{i}}\colon U_X\setminus D_i\to [0,1],\qquad
u_i\de \eta(w_i)\colon U_X\setminus D_i\to [0,1].
\end{equation}

Note that $\sum_{i\in S} w_{i}=1$, see Lemma <ref><ref>. Eventually, we introduce a hybrid coordinate:

\begin{equation}\label{eq:def-v_i}
v_{i}\de t_{i}-u_{i}\colon U_X\setminus D_i\to [-1,1).
\end{equation}

To simplify the notation, we introduce a map

\begin{equation}\label{eq:def-rest}
\rest\de ((\theta_{i})_{i\in S},(z_{j})_{j\in \{i_1,\dots,i_n\}\setminus S}).
\end{equation}

It puts together those coordinates which have no tropical part: as we will see, they are easier to handle when defining a smooth structure on the A'Campo space $A$.

Throughout the article, we will identify $\S^{1}$ with the additive group $\R / \Z$. This way,  $\theta\in \S^1$ denotes the angle $2\pi \theta$, and we have  $\theta+1=\theta$.

§.§.§ The Kato–Nakayama space $X_{\log}$

As a first step of our construction, we will extend polar coordinates on $X$ to radius zero. This can be conveniently phrased in terms of the Kato–Nakayama construction [44]. In general, it  associates to a log structure (i.e. a scheme with a sheaf of monoids, subject to some additional conditions), a topological space. We will use it only for the monoid sheaf $\cM\de \cO_{X}\cap i_{*}\cO_{X\setminus D}^{*}$, where $i\colon X\setminus D\into X$ is the inclusion. We follow  <cit.>.

For $x\in X$, let $\cM_{x}$ be the stalk of $\cM=\cO_{X}\cap i_{*}\cO_{X\setminus D}^{*}$ at $x$, so $\cM_{x}$ consists of holomorphic germs which are invertible off $D$. Let $\cM_{x}\gp$ be the group associated to the monoid $\cM_{x}$, i.e. $\cM_{x}\gp=\{ab^{-1} : a,b\in \cM_{x}\}$ with a natural identification $(ab^{-1})\cdot (ba^{-1})=1$. Let

\begin{equation*}
X_{\log}=\{(x,h): x\in X,\ h\in \Hom(\cM_{X,x}\gp,\S^1),\ h(f)=\tfrac{f(x)}{|f(x)|} \mbox{ if } f\in \cO_{X,x}^{*}\},
\end{equation*}

and let $\tau\colon X_{\log}\to X$ be the first projection. To define a topology on $X_{\log}$, fix a chart $U_X$ adapted to $f$, see Definition <ref>. Reordering the components of $D$ if needed, we can assume that its associated index set is $\{1,\dots, k\}$. Put $U_{\log}=\tau^{-1}(U_{X})\subseteq X_{\log}$. A Kato–Nakayama chart over $U_X$ is a  map $U_{\log}\to [0,1)^{k}\times (\S^1)^{k}\times \D^{n-k}$, given by

\begin{equation}\label{eq:KN-chart}
(x,h)\mapsto (r_1,\dots,r_k;\theta_1,\dots,\theta_k;z_{i_{k+1}},\dots,z_{i_n}), \quad \mbox{where}\quad r_i=|z_i|,\quad  \theta_i=h(z_i).
\end{equation}

Note that the coordinates $r_i$ and $\theta_{i}$ extend to $U_{\log}$ the pullbacks of the maps $r_i$, $\theta_i$ defined in (<ref>), (<ref>). The topology on $(X,D)_{\log}$ is the coarsest one which makes these charts continuous. Clearly, $X_{\log}$ is a topological manifold with boundary, whose interior is homeomorphic with $X\setminus D$.

Applying this construction to $(X,D)=(\C,0)$, we get

\begin{equation*}
\C_{\log}\de (\C,0)_{\log}\cong [0,\infty)\times \S^1,
\end{equation*}

so $\C_{\log}$ extends the polar coordinates to a radius zero circle $\{0\}\times \S^1$, cf.  <cit.>. The holomorphic function $f\colon X\to \C$ extends to a continuous map $f_{\log}\colon X_{\log}\to \C_{\log}$. In local coordinates (<ref>) over an adapted chart meting $D$, the map $f_{\log}$ is given by $(r_{1}^{m_1}\cdot\ldots\cdot r_{k}^{m_k},m_1\theta_1+\ldots+m_k\theta_k)$: recall from  Notation <ref> that we write the angular coordinates additively. We get a commutative diagram

\begin{equation}\label{eq:KN-diagram}
\begin{tikzcd}
\ar[r]
\ar[d, "f_{\log}"]
& X
\ar[d, "f"]
\\
\C_{\log}
\ar[r] & \C
\end{tikzcd}
\end{equation}

For a subset $I\subseteq \{1,\dots, N\}$ we define $(X_{I}^{\circ})_{\log}\subseteq X_{\log}$ as the preimage of the stratum $X_{I}^{\circ}\subseteq X$, see (<ref>). In coordinates (<ref>), $(X_{I}^{\circ})_{\log}$ is defined by $r_i=0$ for $i\in I$ and $r_j\neq 0$ for $j\not\in I$.

§.§.§ Definition of the A'Campo space $A$.

Fix a holomorphic atlas $\cU_{X}=\{(U^{p}_{X},\psi^{p}_{X}):p\in R\}$ adapted to $f$, and a partition of unity $\btau=\{\tau^{p}:p\in R\}$ inscribed in $\cU_{X}$. We write $u_i^{p}$, $v_{i}^{p}$ etc. for the corresponding maps from Section <ref>. For convenience, if $U_{X}^{p}\cap D_{i}=\emptyset$ we set $u_{i}^{p}=0$, $v_{i}^{p}=0$.

Now, we

define smooth maps $\bar{u}_i,\bar{v}_{i}\colon X\setminus D\to \R$, $\mu\colon X\setminus D\to \R^{N}$ by

\begin{equation}\label{eq:def-vbar-ubar-mu}
\bar{u}_{i}\de \sum_{p\in R} \tau^{p}u_{i}^{p}, \quad
\bar{v}_{i}\de \sum_{p\in R}\tau^{p}v_{i}^{p}, \quad
\mu\de (\bar{u}_{1},\dots,\bar{u}_{N}).
\end{equation}

Let $X_{\log}$ be the Kato–Nakayama space associated to $(X,D)$, see Section <ref>. Let $\Gamma\subseteq X\times \R^{N}$ be the closure of the graph of $\mu$. The A'Campo space $A$ is the topological space $A=\Gamma \times_{X} X_{\log}$, i.e. such that the top-left square of (<ref>) is cartesian.

\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\AC}"']
\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}

Put $f\AC=(|f| \circ\pi,\theta)\colon A\to \C_{\log}=[0,\infty)\times \S^1$. It is a composition of the natural map $A\to X_{\log}$ with the map $f_{\log}$ induced by $f$ between the Kato–Nakayama spaces, see (<ref>). Fibers of $f\AC$ agree with those of $(t,\theta)$ and $(g,\theta)$. Figure <ref> in Example <ref> shows the fibers of $f_{A}$ and $f_{\log}$ over $\d \C_{\log}$.

We pull back the stratification (<ref>) of $X$ by putting

\begin{equation}\label{eq:stratification-pullback}
\end{equation}

The partition $A=\bigsqcup_{I}A_{I}^{\circ}$ obtained this way is not a stratification anymore.

We have $A\setminus \pi^{-1}(D)=A_{\emptyset}^{\circ}$ and $\pi^{-1}(D_i)=\bigsqcup_{i\in I}A_{I}^{\circ}$. Thus, for each chart $U_X$ adapted to $f$, the zero locus of the function $t_i$ introduced in (<ref>) is $\bigsqcup_{i\in I} A_{I}^{\circ}\cap U$, where $U=\pi^{-1}(U_X)$. The deepest subset of this partition, namely the one corresponding to the associated index set (<ref>), equals $A_{S}^{\circ}\cap U=\{x\in U : \forall_{i\in S}\colon\, t_{i}(x)=0\}$ and is closed in $U$.

We put $\d A\de \pi^{-1}(D)=\bigsqcup_{I\neq \emptyset}A_{I}^{\circ}$. In the manifold with boundary structure that we will define in Section <ref>, the set $\d A$ will indeed be the boundary of $A$. Note that $\pi|_{A\setminus \d A}\colon A\setminus \d A\to X\setminus D$ is a homeomorphism. Whenever no confusion is likely to occur, we will identify $A\setminus \d A$ with $X\setminus D$. In particular,  $A\setminus \d A$ has a natural smooth structure, pulled back from $X\setminus D$ via $\pi|_{A\setminus \d A}$.

The topological space $A$ introduced in Definition <ref> agrees with the one originally constructed by A'Campo, see <cit.>.

§.§ Elementary properties of the A'Campo space

In this section, we list some simple properties of the A'Campo space $A$ introduced in Definition <ref>. They are summarized in Lemma <ref>, which studies the lifts of basic functions introduced in Section <ref>; and in Proposition <ref>, which describes the topological structure of $A$.

§.§.§ Lifts of the basic functions

The following notation, which we will keep throughout Section <ref>, will help to make our formulas lighter.

For a subset $U_X\subseteq X$, we denote a map from $U_X$ and its pullback to $U\de \pi^{-1}(U_X)$ by the same letter. This way, for every $i\in \{1,\dots, N\}$ we have continuous functions $\bar{u}_{i},\bar{v}_{i},g\colon A\to \R$ given by the formulas (<ref>) and (<ref>).

Let $U_X$ be a chart adapted to $f$. Pulling back the functions (<ref>)–(<ref>), we get functions $r_i,t_{i}\colon U\to [0,1)$, $\theta_{i}\colon U\setminus \d A\to \S^1$ and  $w_{i},u_{i},v_i\colon U\setminus \d A\to \R$. In Lemma <ref><ref> we will see that they extend continuously to $U$, and we will denote these extensions by the same letters, too.

Whenever a chart $U_X$ is fixed, we will put $U=\pi^{-1}(U_X)$, denote its associated index set (<ref>) by $S$, write $k=\# S$ and use the maps $t_i,w_i$ introduced in Section <ref> without further emphasis on the dependence on $U_X$. If the chart is called $U_X'$ or $U_X^{p}$ for some index $p$, we will call the corresponding maps $t_{i}',w_{i}',\dots $ or $t_{i}^p,w_i^p,\dots$, respectively.

In Section <ref> we will prove the following result.

Fix a chart $U_X$ adapted to $f$, not necessarily one from the atlas $\cU_X$ fixed in Definition <ref>. Let $S$ be its associated index set (<ref>), and let $U=\pi^{-1}(U_X)$. Assume $S\neq \emptyset$. The following hold.

*  For every $i\in S$, the maps $t_{i},w_{i},u_{i},v_{i},\theta_{i}$ from Section <ref> extend to continuous maps on $U$.

*  Fix $i\in S$ and a nonempty subset $I\subseteq \{1,\dots, N\}$ such that $i\not \in I$. We have $w_{i}|_{A_{I}^{\circ}\cap U}\equiv 0$.

*  For every $i\in S$ we have $w_{i}\in [0,1]$, and $\sum_{i\in S} w_{i}=1$.

Let $U_{X}'$ be another chart adapted to $f$, let $U'=\pi^{-1}(U_X')$, and let $w_{i}'$ be the corresponding map from (<ref>). Then on $\d A\cap U\cap U'$ we have $w_{i}'=w_{i}$ for all $i$ such that $U_X\cap U_X'\cap D_i\neq \emptyset$.

*  For every $i\in \{1,\dots, N\}$ we have $\bar{u}_{i}|_{U\cap \d A}=u_{i}|_{U\cap \d A}$ if $i\in S$ and $\bar{u}_{i}|_{U\cap \d A}=0$ if $i\not\in S$.

Very importantly for us, part <ref> expresses the idea that at radius zero, the values of $w_{i}$ do not depend on the choice of the chart.
Before entering in the technicalities of the proof, we proceed with the topological description of the A'Campo space.

§.§.§ Rounded simplices

To state the next result, we introduce the following additional notation. By $\Delta^{N-1}\subseteq \R^N$, we denote the image of the standard ($N-1$)-simplex through $(\eta,\ldots,\eta)$, i.e.

\begin{equation*}
\Delta^{N-1}=\{(\eta(s_1),\dots,\eta(s_{N})): s_{i}\geq 0,\ \textstyle\sum_{i=1}^{N}s_{i}=1\}\subseteq \R^{N},
\end{equation*}

see Figure <ref>.
For $\emptyset\neq I\subseteq \{1,\dots, N\}$ we denote by $\Delta_{I}^{N-1}$ (or simply $\Delta_{I}$) the face $\{u_{j}= 0, j\not\in I\}\subseteq \Delta^{N-1}$ of $\Delta^{N-1}$ corresponding to $I$, and put $\Delta_{I}^{\circ}=\{u_{i}\neq 0, u_{j}=0: i\in I, j\not\in I\}\subseteq \Delta_{I}$. Hence e.g.  $\Delta_{1}=\Delta_{1}^{\circ}=\{(1,0,\dots,0)\}$. Note that we identify a subscript $\{i\}$ with $i$.

[use as bounding box] (-0.5,-0.5) rectangle (3,2);
[->] (-0.5,0) – (2,0);
[->] (0,-0.5) – (0,2);
[thick] (0,0) to[out=90,in=-135] (1.5,1.5) – (2,2);
[right] at (0,2) $\eta(t)$;
[dotted] (0,1.5) – (1.5,1.5) – (1.5,0);
(-0.1,1.5) – (0.1,1.5);
[left] at (0,1.5)  $1$;
(1.5,0.1) – (1.5,-0.1);
[below] at (1.5,0) $1$;
[below] at (2,0) $t$;

[use as bounding box] (-0.5,-0.5) rectangle (3,2);
[->, dotted] (-0.5,0) – (2,0);
[above] at (1.5,0) $1$;
[->, dotted] (0,-0.5) – (0,2);
[right] at (0,1.5) $1$;
[thick, dashed] (-0.5,1.5) – (0,1.5);
[thick] (0,1.5) – (1.5,0);
[thick, dashed] (1.5,0) – (1.5,-0.5);
[right] at (0,2) $t_2$;
[below] at (2,0) $t_1$;

[above] at (3.25,0.5) $(\eta,\ldots,\eta)$;
[->] (2.5,0.5) – (4,0.5);

[use as bounding box] (-1.5,-0.5) rectangle (2,2);
[->, dotted] (-0.5,0) – (2,0);
[above left] at (1.5,0) $1$;
[->, dotted] (0,-0.5) – (0,2);
[below right] at (0,1.5) $1$;
[thick, dashed] (-0.5,1.5) – (0,1.5);
[thick] (0,1.5) to[out=0,in=90] (1.5,0);
[thick, dashed] (1.5,0) – (1.5,-0.5);
[right] at (0,2) $t_2$;
[below] at (2,0) $t_1$;
[above right]  at (1,1) $\Delta^{1}$;
(1.5,0) circle (2pt);
[below left] at (1.5,0) $\Delta^{1}_{1}$;
(0,1.5) circle (2pt);
[below left] at (0,1.5) $\Delta^{1}_{2}$;

Function $\eta$ from (<ref>) gives a convenient embedding of an ($N-1$)-simplex to $\R^{N}$

Let $X=\bigsqcup_I X_{I}$ be the stratification (<ref>) of $X$ associated to $D$. The dual complex of $D$ is

\begin{equation}\label{eq:dual-complex}
%	\Delta_{D}\de %\Delta_{D\redd}=
\bigcup_{\{I\neq\emptyset\ :\ X_{I}\neq \emptyset\}}\Delta_{I}\subseteq \Delta^{N-1}.
\end{equation}

§.§.§ Topological structure.

The second main result of the current Section <ref> is the following.

Let $(\cU_X,\btau)$ be a pair adapted to $f$ and let $A$ be the A'Campo space corresponding to $(\cU_X,\btau)$. Then the following hold.

*  The maps $\pi\colon A\to X$, $\mu\colon A\to \R^N$ and $\bar{v}_i\colon A\to [-1,1]$ for $i\in \{1,\dots, N\}$ are continuous.

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

*  For every nonempty subset $I\subseteq \{1,\dots, N\}$, the restriction $(\pi,\mu)|_{A_{I}^{\circ}}\colon A_{I}^{\circ}\to X_{I}^{\circ}\times \Delta_{I}$ is a topological locally trivial fibration, with fiber $(\S^{1})^{\#I}$. Hence $\mu(\d A)$ is the dual complex of $D$.

*  The subset $\d A \subseteq X_{\log}\times \R^{N}$
does not depend on the choice of $(\cU_{X},\btau)$.

Let $A'$ be the A'Campo space corresponding to another pair $(\cU_X',\btau')$ adapted to $f$. Using <ref> and <ref>, we can define a map
\Phi\colon A\to A'

\begin{equation}\label{eq:canonical_homeo}
\Phi|_{\d A}=\id_{\d A} \quad\mbox{and}\quad \Phi|_{A\setminus \d A}=\pi'|_{A'\setminus \d A}\circ \pi^{-1}|_{X\setminus D}.
\end{equation}

Then $\Phi$ is a homeomorphism.

Later, we will show that $A$ is a topological manifold with boundary. We postpone this statement, since at the same time  we will endow $A$ with a $\cC^1$-structure; in such a way that the map (<ref>) becomes a $\cC^1$-diffeomorphism. In Section <ref>, we will even make $A$ a $\cC^\infty$-manifold with boundary. As we will see, the $\cC^\infty$ structure is harder to produce, and the map (<ref>) will not be a $\cC^\infty$-diffeomorphism in general.

Let $X=\D^{2}_{\epsilon}$ for some $\epsilon\in (0,\frac{1}{e})$, and let $f=z_1z_2$. Consider $\cU_X$ consisting of just one chart $(z_1,z_2)$, and $\btau=\{1\}$. Then $(\cU_X,\btau)$ is adapted to $f$. The corresponding A'Campo space $A$ is the shaded part of Figure <ref>, multiplied by $(\S^{1})^{2}$. Similarly, for $X=\D^{3}_{\epsilon}$, $f=z_1 z_2 z_3$ and trivial $(\cU_X,\btau)$, the corresponding $A$ is Figure <ref> times $(\S^1)^{3}$.

[use as bounding box] (-2.5,-2.5) rectangle (3,3);
[fill=black!1] (-1,2) – (-1,0) to [out=270,in=180] (0,-1) – (2,-1) – (2,2) – (-1,2);
[->, dotted] (-1.5,0) – (2.4,0);
[left] at (-1,0) $-1$;
[above] at (2.4,0) $\bar{v}_1$;
[->, dotted] (0,-1.5) – (0,2.4);
[below] at (0,-1) $-1$;
[left] at (0,2.4) $\bar{v}_2$;
[thick, gray] (-1,2) – (-1,0);
[thick] (-1,0) to [out=270,in=180] (0,-1);
[thick, gray] (0,-1) – (2,-1);

at (.5,1) $A_{\emptyset}^{\circ}=X\setminus D$;
[left] at (-1,1) $A_{1}^{\circ}$;
[below] at (1,-1) $A_{2}^{\circ}$;
[below left] at (-.5,-.5) $A_{1,2}^{\circ}$;

$D=\{z_1 z_2=0\}\subseteq \C^2$

[use as bounding box] (-2.5,-2.5) rectangle (3,3);
[fill=black!8] (0.4,0.4) to [out=180,in=45] (-1,0) to [out=270,in=180] (0,-1) to [out=45,in=270] (0.4,0.4);
at (-0.3,-0.3) $A_{1,2,3}^{\circ}$;
[fill=black!4] (0.4,0.4) to [out=180,in=45] (-1,0) – (-1,2.4) to [out=45,in=180] (.4,2.8) – (0.4,0.4);
at (-.3,1.8) $A_{1,3}^{\circ}$;
[fill=black!4] (-1,0) to [out=270,in=180] (0,-1) – (-1.5,-2.5) to [out=180,in=270] (-2.5,-1.5) – (-1,0);
at (-1.4,-1.4) $A_{1,2}^{\circ}$;
[fill=black!4] (0.4,0.4) to [out=270,in=45] (0,-1) – (2.4,-1) to [out=45,in=270] (2.8,0.4) – (-0.4,0.4);
at (1.8,-0.3) $A_{2,3}^{\circ}$;
[fill=black!1] (0.4,0.4) –(0.4,2.8) – (2.8,2.8) – (2.8,0.4) – (0.4,0.4);
at (1.8,1.8) $A_{3}^{\circ}$;
[fill=black!1] (-1,0) – (-1,2.4) – (-2.5,0.9) – (-2.5,-1.5) – (-1,0);
at (-1.9,0.5) $A_{1}^{\circ}$;
[fill=black!1] (0,-1) – (2.4,-1) – (0.9,-2.5) – (-1.5,-2.5) – (0,-1);
at (0.5,-1.9) $A_{2}^{\circ}$;

[->, dotted] (-2,0) – (3,0);
[left] at (-1,0) $-1$;
[below] at (3,0) $\bar{v}_1$;
[->, dotted] (0,-2) – (0,3);
[below] at (0,-1) $-1$;
[left] at (0,3) $\bar{v}_2$;
[->, dotted] (1,1) – (-2.5,-2.5);
[above right] at (.4,.4) $-1$;
[above] at (-2.5,-2.5) $\bar{v}_3$;

[thick] (-1,2.4) – (-1,0) to [out=270,in=180]	(0,-1) – (2.4,-1);
[black!20] (-1.6,1.8) – (-1.6,-0.6) to [out=270,in=180] (-0.6,-1.6) – (1.8,-1.6);
[black!20] (-2.2,1.2) – (-2.2,-1.2) to [out=270,in=180] (-1.2,-2.2) – (1.2,-2.2);

[thick] (2.8,0.4) – (.4,.4) to [out=180,in=45] (-1,0) – (-2.5,-1.5);
[black!20] (2.8,1.4) – (.5,1.4) to [out=180,in=45] (-1,1) – (-2.5,-0.5);
[black!20] (2.8,2.4) – (.5,2.4) to [out=180,in=45] (-1,2) – (-2.5,0.5);

[thick] (0.4,2.8) – (.4,.4) to [out=270,in=45] (0,-1) – (-1.5,-2.5);
[black!20] (1.4,2.8) – (1.4,.5) to [out=270,in=45] (1,-1) – (-0.5,-2.5);
[black!20] (2.4,2.8) – (2.4,.5) to [out=270,in=45] (2,-1) – (0.5,-2.5);

$D=\{z_1 z_2 z_3 =0\}\subseteq \C^3$

Radial coordinates $\bar{v}_{i}$ of the A'Campo space, cf. <cit.>.

§.§.§ Proofs of <ref>–<ref>

The remaining part of Section <ref> is devoted to the proof of Lemma <ref> and Proposition <ref>. We remind the reader to use Notation <ref>.

First, we give a precise formula relating $t_{i}$, $w_{i}$ with corresponding $t_{i}'$, $w_{i}'$ defined by another chart. We will use it both to prove the equality  $w_{i}|_{\d A}=w_{i}'|_{\d A}$ asserted in Lemma <ref><ref>, and to make further computations in Lemma <ref>.

Let $U_X,U'_X$ be two charts adapted to $f$. Put $V_X=U_X\cap U_X'$, $V=\pi^{-1}(U_X\cap U_X')$, and use Notation <ref>. For every $i\in S\cap S'$, the following hold.

*  The functions $t_{i}$, $t_{i}'$ are continuous on $V$.

*  There is a nonvanishing holomorphic function $\lambda \in \cO_{X}^{*}(V_X)$ such that $z_{i}'=\lambda \cdot z_{i}$.

*  Let $\lambda$ be as in <ref>, and let  $a =-m_i\log|\lambda|$. Then $a \in \cC^{\infty}(V_X)$, $1+at_{i}\neq 0$, and on $V$ we have:

\begin{equation*}\label{eq:t-comparison}
t_{i}'=t_{i}\cdot (1+a \cdot t_{i})^{-1}
\quad \mbox{and} \quad
w_{i}'=w_{i}\cdot (1+a \cdot t_{i}).
\end{equation*}

*  The functions $w_i$ and $w_i'$ extend to continuous functions on $V$, such that $w_{i}|_{V\cap \d A}=w_{i}'|_{V\cap \d A}$.

<ref> Recall that $t_{i}$, $t_{i}'$ denote the pullbacks by $\pi$ of continuous functions $t_i$, $t_{i}'$ introduced in (<ref>). By Definition <ref> of $A$, the map $\pi$ is continuous, so $t_{i},t_{i}'$ are continuous, too.

<ref> By Definition <ref><ref>, we have $D_{i}\cap V=\{z_i=0\}=\{z_i'=0\}$.
Hence $z_i=\lambda z_i'$ for some $\lambda\in \cO_{X}^{*}(V)$, as claimed.

<ref> Since $\lambda$ is smooth and nonvanishing, the function $a=-m_i\log|\lambda|$ is smooth. By definition, on $V\cap \pi^{-1}(D_i)$ both  $t_i$ and $t_i'$ are zero, so $1+at_i\neq 0$ and  the first equality of <ref> holds trivially there. Away from $\pi^{-1}(D_i)$, we have $t_i, t_{i}'\neq 0$ and $t_{i}'=-(m_{i}\log|z_{i}'|)^{-1}$ by formula (<ref>). Part <ref> and the definition of $a$ give

\begin{equation*}
t_{i}' \cdot (1+at_i)=-\frac{1+at_i}{m_{i}\log|\lambda z_{i}|}=
\end{equation*}

so $1+at_i\neq 0$ and $t_{i}'=t_i\cdot (1+at_i)^{-1}$, as needed.

We will now prove that the equality $w_i'=w_i\cdot (1+at_i)$ holds on $V\setminus \pi^{-1}(D_i)$. There, substituting the equality $t_{i}'=t_i\cdot (1+at_i)^{-1}$ to the definition (<ref>) of $w_{i}'$, we get

\begin{equation*}
w_{i}'=t\cdot (t_{i}')^{-1}=t\cdot t_{i}^{-1} \cdot (1+at_{i})=w_{i} \cdot (1+at_i).
\end{equation*}

Once we prove in <ref> that $w_{i}$ and $w_{i}'$ are continuous, the above equality will extend to $\pi^{-1}(D_i)$.

<ref> As in the proof of <ref>, continuity of $\pi$ implies that $w_i,w_i'$ are continuous on $V \setminus \pi^{-1}(D_i)$. Fix $x\in V\cap \pi^{-1}(D_i)$ and a sequence $(x_{\nu})\subset V\setminus \pi^{-1}(D_i)$ such that $\limn x_{\nu}=x$. Since $w_{i}(x_{\nu})\in [0,1]$, after passing to a subsequence we can assume that both $w_{i}(x_{\nu})$ and $w_{i}'(x_{\nu})$ have limits in $[0,1]$. Call these limits $w$ and $w'$, respectively. Since $t_i(x)=0$, applying the second formula in <ref> away from $\pi^{-1}(D_i)$ and passing to the limit, we get $w=w'$. Taking for $U_X'$ a chart $U_X^p$ of the atlas $\cU_X$ fixed in Definition <ref>, and writing $w_{i}^{p}$ for the corresponding map, we get $\limn w_{i}^{p}(x_{\nu})=w$, for every $p\in R$ such that $x\in U^p$. Continuity of $\eta$ implies that $\limn u_{i}^{p}(x_{\nu})=\limn \eta(w_{i}^{p}(x_{\nu}))=\eta(w)$. Since each $\tau^{p}$ is continuous and
supported in $U_{X}^{p}$, we get $\limn \tau^{p}(x_{\nu})u_{i}^{p}(x_{\nu})=\tau^{p}(x)\eta(w)$ for all $p\in R$.

By Definition <ref>, the function $\bar{u}_{i}$ is continuous. Hence
$\bar{u}_{i}(x)=\sum_{p} \limn \tau^{p}(x_{\nu})u_{i}^{p}(x_{\nu})=\sum_{p}\tau^{p}(x)\eta(w)=\eta(w)$. Bijectivity of $\eta$ gives $w=\eta^{-1}(\bar{u}_{i}(x))$.

Thus from any sequence $(x_{\nu})\subset V\setminus \pi^{-1}(D_i)$ such that $\limn x_{\nu}=x$, we can choose a subsequence $(x_{\nu'})$ such that $\lim_{\nu'\rightarrow \infty} w_{i}(x_{\nu'})=\eta^{-1}(\bar{u}_{i}(x))$. In fact, the same is true for all sequences $(x_{\nu})\subset V$, because by Definition <ref> of $A$, every point of $V$ is a limit of a sequence of points in $V\setminus \d A$. It follows that the function  $w_i$ is continuous at $x$, and the value of $w_i$ at $x$ equals $\eta^{-1}(\bar{u}_{i}(x))$, so it does not depend on the choice of the chart $U_{X}$ defining $w_i$, as claimed.

It follows that the second formula in <ref> holds on $\pi^{-1}(D_i)$, too, which completes its proof.

<ref> Applying Lemma <ref><ref>,<ref> to $U_X$, we infer that $t_{i}$, $w_{i}$ extend to continuous functions on $U$. Since $\eta$ is continuous, the functions $u_{i}=\eta(w_{i})$ and $v_{i}=t_{i}-u_{i}$ are continuous, too. Eventually, $\theta_{i}$ pulls back to a continuous coordinate on the preimage of $U$ in the Kato–Nakayama space $X_{\log}$, see (<ref>), so its further pullback to $A$ is continuous, too.

<ref> Since $I\neq \emptyset$, we have $A_{I}^{\circ}\subseteq \d A$, so $t|_{A_{I}^{\circ}}=0$. For every $i\not \in I$, definition of $A_{I}^{\circ}$ gives $U\cap A_{I}^{\circ}\subseteq U\setminus \pi^{-1}(D_{i})$, so $t_{i}|_{U\cap A_{I}^{\circ}}>0$. By definition (<ref>) of $w_i$, on $U\cap A_{I}^{\circ}$ we have $w_{i}=t\cdot t_{i}^{-1}=0$.

<ref> Because $U_X$ is adapted to $f$, by Definition <ref><ref> we have  $|f|_{U}|=\prod_{i\in S} |z_{i}|^{m_i}$. Hence $\log |f|_{U}|=\sum_{i\in S} m_{i}\log |z_{i}|$. On $U\setminus \d A$, substituting the formulas (<ref>), (<ref>) for $t$ and $t_i$, gives $-t^{-1}=-\sum_{i\in S} t_{i}^{-1}$, so $1=\sum_{i\in S} tt_{i}^{-1}=\sum_{i\in S} w_{i}$ by definition (<ref>) of $w_i$. Since by <ref> the functions $w_{i}$ are  continuous on $U$, we infer that the equality $1=\sum_{i\in S}w_{i}$ holds on $U\cap \d A$, too, as claimed.

<ref> Put $V=U\cap U'$, and fix $i$ such that $D_i\cap V\neq \emptyset$. The set $V\cap \d A$ is a disjoint union of $V\cap \pi^{-1}(D_{i})$ and $V\cap A_{I}^{\circ}$ for all nonempty $I$ not containing $i$. By Lemma <ref><ref>, $w_{i}=w_{i}'$ on $V\cap \pi^{-1}(D_{i})$. In turn, by <ref>, both $w_{i}$ and $w_{i}'$ are zero on $V\cap A_{I}^{\circ}$ whenever $i\not \in I$. Thus $w_i=w_i'$ on the whole set $V\cap \d A$.

<ref> Fix $i\in S$. By <ref>, for every chart $U_{X}^{p}$ of the atlas $\cU_{X}$ fixed in Definition <ref>, the corresponding function $w_{i}^{p}$ satisfies $w_{i}^{p}|_{U\cap U^{p}\cap \d A}=w_{i}|_{U\cap U^{p}\cap \d A}$. Hence  $\bar{u}_{i}=\sum_{p}\tau^{p}\eta(w_{i}^{p})=\sum_{p}\tau^{p}\eta(w_{i})=\eta(w_{i})=u_{i}$ on $U\cap \d A$, as claimed.

Now, fix $i\not\in S$, and $x\in U\cap \d A$. Since $i\not \in S$, we have $x\in A_{I}^{\circ}$ for some nonempty $I$ not containing $i$. Take any $p\in R$ such that $\tau^{p}(x)\neq 0$, so $x\in U^p$. Applying <ref> to $U^{p}$, we infer that $w_{i}^{p}(x)=0$, so $u_{i}^{p}(x)=0$. Therefore, $\bar{u}_{i}(x)=\sum_{p} \tau^{p}(x)u_{i}^{p}(x)=0$, as claimed.

For a nonempty subset $I\subseteq \{1,\dots, N\}$, the following hold.

*  For every $x\in X_{I}^{\circ}$, we have  $\mu(\pi^{-1}(x))=\Delta_{I}\subseteq \Delta^{N-1}$.

*  The projection $\Gamma\to X$ is proper.

*  The subset $A_{I}^{\circ}\subseteq X_{\log}\times \R^{N-1}$ equals $(X_{I}^{\circ})_{\log}\times \Delta_{I}$.

<ref> Let $U_X$ be a chart adapted to $f$ containing $x$, let $S$ be its associated index set, and let $U=\pi^{-1}(U_X)$. Fix $y\in \pi^{-1}(x)\subseteq U\cap A_{I}^{\circ}$. For any $i\not\in S$, we have $\bar{u}_{i}(y)=0$ by Lemma <ref><ref>. Fix $i\in S$. By Lemma <ref><ref>, we have  $\bar{u}_{i}(y)=u_i(y)=\eta(w_i(y))$. Thus Lemma <ref><ref> implies that $\mu(y)\in \Delta^{N-1}$. Moreover, if $i\not \in I$ then by Lemma <ref><ref> we have $w_{i}(y)=0$, so $\bar{u}_{i}(y)=0$. We conclude that $\mu(y)\in \Delta_{I}$, so  $\mu(\pi^{-1}(x))\subseteq \Delta_{I}$.

For the reverse inclusion,  note that the subset $\mu(\pi^{-1}(x))\subseteq \Delta_{I}$ is closed. Indeed, it consists of all possible limits of $\mu(x_{\nu})$, where $(x_{\nu})\subset X\setminus D$ is a sequence converging to $x$. Therefore, it is sufficient to prove that $\Delta_{I}^{\circ}\subseteq \mu(\pi^{-1}(x))$. Fix $b\in \Delta_{I}^{\circ}$. Since the function $\eta|_{[0,1]}$ is bijective, for each $i\in I$ we get a unique $b_{i}\in (0,1]$ such that $\eta(b_{i})$ is the $i$-th coordinate of $b$; and $\sum_{i\in I} b_{i}=1$. Put $b_{i}=0$ for $i\not\in I$.

Recall that $z_{i}$, for $i\in S$, are some of the coordinates of the chart $U_X$. Hence for sufficiently small $\epsilon>0$, there is a curve $\gamma\colon [0,\epsilon]\to U_X$ such that $z_{i}(\gamma(s))=s^{b_i/m_i}$ for all $s\in [0,\epsilon]$ and all $i\in S$. By Definition <ref><ref>, we have $f(\gamma(s))=\prod_{i\in S} z_{i}(\gamma(s))^{m_i}=s^{\sum_{i\in S} b_{i}}=s$, so $t(\gamma(s))=-(\log s)^{-1}$ for $s>0$. By definition (<ref>) of $t_i$, for $s>0$ we have $t_{i}(\gamma(s))=-(m_i\log(s^{b_i/m_i}))^{-1}=-(b_i\log s)^{-1}$, and by (<ref>),  $w_{i}(\gamma(s))=t(\gamma(s))\cdot t_{i}(\gamma(s))^{-1}=b_{i}$. Therefore, $\mu(\gamma(s))=b$ for all $s\in (0,\epsilon]$. Passing with $s$ to $0$ we get $y\in A$ such that $\pi(y)=\gamma(0)=x$ and $\mu(y)=b$, as needed.

<ref>, <ref> Follow immediately from <ref>.

<ref> Follows directly from Definition <ref>.

<ref> Properness follows from the properness of the maps $\Gamma\to X$ (see Lemma <ref><ref>) and $X_{\log}\to X$, and the rest is immediate by the definitions.

<ref> Let $U_{X}\subseteq X$ be a chart adapted to $f$. Let $\tau\colon X_{\log}\to X$ be the natural map from the Kato–Nakayama space. In coordinates (<ref>) on $\tau^{-1}(U_X)$, the restriction of $\tau$ to $\tau^{-1}(U \cap X_{I}^{\circ})$ maps $\theta_{i}$ to $0$ for $i\in I$, and keeps the remaining coordinates intact. It follows that $\tau|_{\tau^{-1}(X_{I}^{\circ})}\colon \tau^{-1}(X_{I}^{\circ})\to X_{I}^{\circ}$ is a topological $(\S^{1})^{\#I}$-bundle. Thus the first part of <ref> follows Lemma <ref><ref>. The second follows from the definition (<ref>) of the dual complex.

<ref> By Lemma <ref><ref>, for every nonempty $I\subseteq \{1,\dots, N\}$, the subset $A_{I}^{\circ}\subseteq X_{\log}\times \R^{N}$ does not depend on the choice of $(\cU_{X},\btau)$. Therefore, neither does their disjoint union $\bigsqcup_{I\neq \emptyset}A_{I}^{\circ}=\d A$.

<ref> Let $\pi_{\log}\colon A\to X_{\log}$ be the natural map induced by $\pi$ in Definition <ref>. Let $\pi_{\log}'$, $\mu'$, $\bar{u}_{i}'$ etc. denote the maps from Definition <ref> associated to $A'$. By <ref>, the subsets $\d A,\d A'\subseteq X_{\log}\times \R^{N}$ are equal, so the restriction $\Phi|_{\d A}$, defined as $\id|_{\d A}$, indeed maps $\d A$ to $\d A'$. Clearly, this restriction is continuous. In turn, $\Phi|_{A\setminus \d A}\colon A\setminus \d A\to A'\setminus \d A'$ is continuous by <ref>. Since $\d A$ is a closed subset of $A$, to prove continuity of $\Phi$ it is sufficient to prove that, for every $y\in \d A$, and every sequence $(y_{\nu})\subset A\setminus \d A$ converging to $y$, the values $y_{\nu}'\de \Phi(y_\nu)$ converge to $y$, too.

Since a point $y'\in A'$ is by definition a pair $(\pi'_{\log}(y'),\mu'(y'))$, we need to show that $\pi_{\log}'(y_{\nu}')\rightarrow \pi_{\log}(y)$ and $\mu'(y'_{\nu})\rightarrow \mu(y)$. By definition of $\Phi|_{A\setminus \d A}$, we have $\pi(y_{\nu})=\pi'(y_{\nu}')\in X\setminus D$, so $\pi_{\log}(y_{\nu})=\pi'_{\log}(y_{\nu}')$, because the map $X_{\log}\to X$ is a homeomorphism away from $D$. This proves the first convergence.

For the second one, choose a chart $U_X$ adapted to $f$ containing $\pi'(y_{\nu}')$ for $\nu\gg 1$, and fix $i$ in the associated index set (<ref>). By Lemma <ref><ref>, the function $u_i$ from (<ref>) extends to continuous functions $u_{i}\colon U\to [0,1]$ and $u_{i}'\colon U'\to [0,1]$, where $U=\pi^{-1}(U_X)$, $U'=(\pi')^{-1}(U_X)$. Put $x_\nu=\pi(y_{\nu})$, $x_{\nu}'=\pi'(y_{\nu}')$. Clearly, $x_{\nu}=x_{\nu}'$. Therefore, we have the equality

\begin{equation*}
\limn u_{i}'(y_{\nu}')=
\limn u_{i}(x_{\nu}')=
\limn u_{i}(x_{\nu})=
\limn u_{i}(y_{\nu})=
\bar{u}_{i}(y),
\end{equation*}

where the last equality follows from Lemma <ref><ref>. Applying it to all charts $U_{X}$ of $\cU_{X}'$, we infer that $\bar{u}_{i}'(y_{\nu}')\rightarrow \bar{u}_{i}(y)$. Thus by definition (<ref>) of $\mu$ we get $\mu'(y_{\nu}')\rightarrow \mu(y)$, as needed.

Therefore, $\Phi$ is continuous. Continuity of the inverse follows by reversing the roles of $A$ and $A'$.

§.§ The $\cC^1$-atlas on the A'Campo space

Fix a pair $(\cU_X,\btau)$ adapted to $f$, see Definition <ref>. In this section we will introduce a $\cC^1$-atlas on the A'Campo space and we will show that if $(\cU'_X,\btau')$ is another pair adapted to $f$, then the homeomorphism (<ref>) connecting the corresponding A'Campo spaces is a $\cC^1$-diffeomorphism. As the reader may expect, this $\cC^1$-structure agrees outside $\partial A$ with the natural smooth structure in $A\setminus\partial A$, i.e. the one pulled back from $X\setminus D$ by the homeomorphism $\pi|_{A\setminus \d A}\colon A\setminus \d A\to X\setminus D$.

§.§.§ Definition of the $\cC^1$-atlas

Fix a chart $U_X$ adapted to $f$, not necessarily one from the fixed atlas $\cU_X$. Put $U=\pi^{-1}(U_X)$, let $S=\{i: U_X\cap D_i\neq \emptyset\}$ be the associated index set, and let $w_i,v_i,\rest,\dots$ be the maps introduced in Section <ref>. Assume $S\neq \emptyset$. For every $i\in S$, we define

\begin{equation}\label{eq:Ui}
U_{i}=\{x\in U: w_{i}(x)>\tfrac{1}{n+1}\}.
\end{equation}

By Lemma <ref><ref>, the subset $U_{i}\subseteq U$ is open. By Lemma  <ref><ref>, the open sets $U_{i}$ for $i\in S$ cover $U$. Put $k=\#S$ and $Q_{k,n}\de [0,\infty)\times \R^{k-1}\times (\S^1)^{k}\times \C^{n-k}$. By Lemma <ref><ref>, for every $i\in S$, we have a continuous map

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

see (<ref>), (<ref>) and (<ref>) for definitions of $g$, $v_i$ and $\rest$, respectively.

The pairs $(U_{i},\psi_{i})$ will be called the $\cC^{1}$-charts corresponding to the chart $U_{X}$ adapted to $f$. The following lemmas, which we will prove later in this section, assert that these charts indeed make $A$ a $\cC^1$-manifold with boundary.

Each map (<ref>) is a homeomorphism onto an open subset of $Q_{k,n}$. Moreover the restriction $\psi_i$ to $U_i\setminus\partial A$ is $\cC^\infty$ for the natural smooth structure in $A\setminus \partial A$.

Lemma <ref> endows $A$ with a structure of topological manifold with boundary $\d A=\{g=0\}=\pi^{-1}(D)$. The $\cC^1$-structure is provided by Lemma <ref> below, which studies the transition functions between charts (<ref>), possibly associated with  different charts adapted to $f$.

Let $U_X$, $U'_X$ be charts adapted to $f$, let  $S$, $S'$ be their associated index sets (<ref>), and let $k=\#S$, $k'=\#S'$. For $i\in S$ and $j\in S'$ let $(U_i,\psi_i)$, $(U_j',\psi_{j}')$ be the $\cC^{1}$-charts defined in (<ref>) above. Then the transition map

\begin{equation*}
\psi'_{j}\circ \psi_{i}^{-1}|_{\psi_{i}(U_i\cap U'_j)}\colon \psi_{i}(U_i\cap U'_j)\to \psi'_j(U_i\cap U'_j)
\end{equation*}

is a $\cC^{1}$-diffeomorphism between open subsets of $Q_{k,n}$ and $Q_{k',n}$.

[fill=black!5] (-0.6,2) – (-0.6,-1.85)
to [out=170,in=270] (-2,0) – (-2,2) – (-0.6,2);
[fill=black!5] (2,-0.6) – (-1.85,-0.6)
to [out=-80,in=180] (0,-2) – (2,-2) – (2,-0.6);
[fill=black!10] (-1.85,-0.6)
to [out=-80,in=170] (-0.6,-1.85) – (-0.6,-0.6) – (-1.85,-0.6);
[dashed] (-0.6,2) – (-0.6,-1.85);
[dashed] (2,-0.6) – (-1.85,-0.6);
[->, dotted] (-2.5,0) – (2.4,0);

[->] (-2.2,0.1) – (-2.2,1);
at (-2.8,0.6) $r_2>0$;
(-2,0) circle (2pt);
at (-1.8,0.2) $-1$;
[->] (-2.2,-0.1) to [out=270,in=130] (-1.9,-1.1);
at (-2.8,-0.6) $w_2>0$;

[->] (0.1,-2.2) – (1,-2.2);
at (0.7,-2.5) $r_1>0$;
at (-0.2,-1.8) $-1$;
(0,-2) circle (2pt);
[->] (-0.1,-2.2) to [out=180,in=-40] (-1.1,-1.9);
at (-0.7,-2.5) $w_1>0$;

[above] at (2.4,0) ${v}_1$;
[->, dotted] (0,-2.5) – (0,2.4);

[left] at (0,2.4) ${v}_2$;
[thick, gray] (-2,2) – (-2,0);
[thick] (-2,0) to [out=270,in=180] (0,-2);
[thick, gray] (0,-2) – (2,-2);

at (-1.3,1) $U_{1}$;
at (1,-1.3) $U_{2}$;
at (-2.3,1.7) $A_{1}^{\circ}$;
at (1.8,-2.3) $A_{2}^{\circ}$;

[fill=black!5] (-1,2.5) – (-1,0) to [out=270,in=180]	(0,-1) – (2.5,-1)
to [out=70,in=-90] (2.8,0.4) – (2.8,2.8) – (0.4,2.8) to [out=180, in=20] (-1,2.5);

[->, dotted] (-2,-0.15) – (3,-0.15);
[below] at (3,-0.15) $v_1$;
[->, dotted] (-0.15,-2) – (-0.15,3);
[left] at (-0.15,3) ${v}_2$;
[->, dotted] (1.4,1.4) – (-2,-2);
[above] at (-2,-2) $v_3$;
at (-1.8,0) $-1$;
at (-0.3,-1.4) $-1$;
at (0.5,1) $-1$;

(-1.55,2.4) – (-1.55,0) to [out=270,in=180]	(0,-1.55) – (2.4,-1.55);
[dashed] (-1,2.5) – (-1,0) to [out=270,in=180]	(0,-1) – (2.5,-1) – (2.5,2.5) – (-1,2.5);
at (-0.6,2.2) $U_3$;

[thick] (2.8,0.8) – (.4,0.8) to [out=180,in=45] (-1,0.4) – (-2.4,-1);

[thick] (0.8,2.8) – (0.8,.4) to [out=270,in=45] (0.4,-1) – (-0.9,-2.4);
(-0.15,-1.55)  circle (1.5pt);
(-1.55,-0.15) circle (1.5pt);

at (1.8,1.8) $A_{3}^{\circ}$;
at (-2.1,0.7) $A_{1}^{\circ}$;
at (0.7,-2.1) $A_{2}^{\circ}$;
[draw] at (4.4,-1.2) $w_1=0$;
[->] (3.6,-1.2) to[out=180,in=0] (0.9,-0.3);
[draw] at (4.4,1.6) $w_2=0$;
[->] (3.6,1.6) to[out=180,in=90] (1.8,0.95);
[draw,align=center] at (4.4,0.2) $w_1=0$
(0.8,0.8) circle (2pt);
[->] (3.6,0.2) to[out=180,in=-45] (0.9,0.7);

Coordinate charts (<ref>) of the A'Campo space, cf. Figure <ref>.

Since for every $k\in \{1,\dots, n\}$, the space $Q_{k,n}$ is a $2n$-manifold with boundary, refining the $\cC^1$-charts corresponding to each adapted chart $U_X$, we obtain an honest $\cC^1$-atlas for a neighborhood of $\partial A$, whose charts have images in half-euclidean spaces. In order to have a $\cC^1$-atlas of the whole $A$, we notice that by Lemma <ref> this $\cC^1$-structure agrees with the natural smooth structure in $A\setminus\partial A=X\setminus D$.

Proposition <ref> below allows to speak about the A'Campo space of $f$, as a $\cC^1$-manifold with boundary. The key properties of this space are listed in Proposition <ref>.

Fix two pairs $(\cU_X,\btau)$ and $(\cU'_X,\btau')$ adapted to $f$. Let $A$ and $A'$ be the corresponding A'Campo spaces, each endowed with the $\cC^1$-structure defined above. Then the homeomorphism $\Phi:A\to A'$ introduced in Proposition <ref><ref>  is a $\cC^1$ diffeomorphism.

Let $A$ be the A'Campo space of $f$, with the $\cC^1$-structure defined above.

*  The map $\pi\colon A\to X$ is $\cC^1$. Its restriction $\pi|_{A\setminus \d A}\colon A\setminus \d A\to X\setminus D$ is a $\cC^1$-diffeomorphism.

*  Let $g,\theta$ be as in (<ref>), (<ref>). Then the map $(g,\theta)\colon A\to [0,1)\times \S^{1}$ is a $\cC^1$-submersion. In particular, the map $f\AC\colon A\to \C_{\log}$ extending $f$ in (<ref>) is $\cC^1$.

*  For every $i\in \{1,\dots, N\}$, the function $\bar{v}_i:A\to[-1,1]$ associated in (<ref>) to the pair $(\cU_{X},\btau)$ adapted to $f$, is $\cC^1$.

The $\cC^1$-structure on $A$ constructed above may not be $\cC^2$. Indeed, let $X=\D_{\epsilon}^2$ for some $\epsilon\in (0,\frac{1}{e})$, and let $f=z_1z_2$. Consider two charts $U_X$, $U_X'$ adapted to $f$: one with the standard coordinates $(z_1,z_2)$, and the other with coordinates $(z_1',z_2')$ given by

\begin{equation*}
(z_1',z_2')=(e^{-1} \cdot z_1, e \cdot z_2).
\end{equation*}

Let $(U_1,\psi_1)$ and $(U_1',\psi_1')$ be the corresponding $\cC^1$-charts. We will show that the transition map $\psi_{1}'\circ\psi_{1}^{-1}\colon (g,v_2,\theta_1,\theta_2)\mapsto (g,v_2',\theta_2,\theta_2')$ is not $\cC^2$ along  $\{t_2=0,u_2=0\}$.

Using definitions (<ref>),  (<ref>) of $t_2$ and $u_2$, we compute

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

so $t_{2}'-t_{2}=\frac{t_{2}^{2}}{1-t_2}$ and $u_2'-u_2=\frac{u_{2}^{2}\log(1-t_2)}{1-u_2\log(1-t_2)}$, cf. Lemmas <ref><ref> and <ref><ref>. Substituting these equalities to the definition (<ref>) of $v_2$, we get

\begin{equation}\label{eq:ex-not-C2}
v_{2}'=v_{2}+\frac{t_{2}^{2}}{1-t_{2}}-\frac{u_2^2 \log(1-t_2)}{1-u_2\log (1-t_2)}.
\end{equation}

Now, we will approach $\{t_2=0,u_2=0\}$ along $\{t_2=0,u_2>0\}=\Int_{\d A}A_{1,2}^{\circ}$ and $\{t_2>0, u_2=0\}=A_{1}^{\circ}$. In Figure <ref>, these are, respectively, the circle-arc and the bold vertical line. If $t_{2}=0$, $u_2>0$ then (<ref>) reads as $v_2'=v_2$, so $(\frac{\d}{\d v_{2}})^2 v_2'=0$. In turn, if $t_2> 0$, $u_2=0$ then $v_2=t_2$, so (<ref>) reads as $v_2'=v_2+\frac{v_2^2}{1-v_2}=v_2+v_2^2+v_2^3+\dots$, and therefore $(\frac{\d}{\d v_{2}})^2 v_2'=1$. This shows that the transition map $v_2\mapsto v_2'$ is not $\cC^2$, as claimed.

In the remaining part of Section <ref>, we prove Lemmas <ref>, <ref> and Propositions <ref>, <ref> stated above. First, in Section <ref> we prove Lemma <ref>, which makes $A$ a topological manifold with boundary. To prove that this manifold is $\cC^1$, we will need some preparatory results, which will be useful for the construction of a $\cC^\infty$ atlas, too. In Section <ref>, we gather some handy identities. Next, in Section <ref> we prove that, as we approach $\d A$, the $\cC^1$-charts corresponding to different charts adapted to $f$ become $\cC^1$-close to each other. To state it in a precise way, we introduce a technical notion of differential forms bounded from $X$. With these preparations at hand, we will prove Lemma <ref> and Propositions <ref>, <ref> in Section <ref>.

§.§.§ The A'Campo space is a topological manifold

In this section, we prove Lemma <ref>, which implies that the formulas (<ref>) define topological charts. We begin with two simple observations.

Let $U_X$ be a chart adapted to $f$. Let $S=\{i:U_X\cap D_i\neq \emptyset\}$ be the associated index set (<ref>), and for $i\in S$ let $U_i=\{w_i>\frac{1}{n+1}\}$ be the open subset of $U\de \pi^{-1}(U_X)$ defined in (<ref>). Then

*  The intersection $U_{i}\cap \d A$ is an open subset of $\pi^{-1}(D_i)$.

*  Fix $x\in U_i\cap \d A$. Then $x\in A_{I}^{\circ}$ for some $I\subseteq S$ containing $i$. Moreover, for every $j\in S$ we have $v_{j}(x)=-u_{j}(x)\leq 0$ if $j\in I$ and $v_j(x)=t_j(x)>0$ if $j\not \in I$.

*  Fix $i\in S$ and $I\subseteq S$ not containing $i$. Then the restrictions of $t_i,w_i,u_i$ and $v_i$ to $A_{I}^{\circ}\cap U$ are pullbacks of smooth functions on $X_{I}^{\circ}\cap U_{X}$.

<ref> By Lemma <ref><ref>, the function $w_i$ is continuous, so $U_i=\{w_i>\frac{1}{n+1}\}$ is an open subset of $A$. To see that $U_i\cap \d A\subseteq \pi^{-1}(D_i)$, fix $x\in U_i\cap\d A$. Since $w_{i}(x)>0$, the formula (<ref>) implies that $t_{i}(x)=t(x)\cdot w_{i}(x)^{-1}$. If $x\in \d A$ then $t(x)=0$, so we get $t_i(x)=0$, i.e. $x\in \pi^{-1}(D_i)$, as claimed.

<ref> By <ref>, we have $i\in I$. If $j\in I$ then by definition (<ref>) of $t_j$ we have $t_j(x)=0$, so $v_j(x)=-u_j(x)\leq 0$ by (<ref>). If $j\not\in I$ then $u_j(x)=0$ by Lemma <ref><ref>, so $v_j(x)=t_j(x)>0$.

<ref> For $I=\emptyset$ we have $A_{I}^{\circ}\cap U=U\setminus \d A=U_X\setminus D$, where the assertion is clear. Assume $I\neq \emptyset$. Since $i\not \in I$, by Lemma <ref><ref> the restrictions to $A_{I}^{\circ}\cap U$ of $w_i$, and hence of $u_i$, are zero. Thus $v_i$ restricts to $t_i$, which is a smooth function on $U_X\setminus D_i$, in particular on $X_{I}^{\circ}\cap U_X$.

The function defining the hybrid coordinate $v_i$ in (<ref>) has the following useful property.

Fix a positive number $\check{t}>0$ and consider a function $\check{v}_{\check{t}}\colon [\check{t},\infty)\ni s\mapsto s-\eta(\check{t}\cdot s^{-1})\in \R$.

*  The function $\check{v}_{\check{t}}$ is strictly increasing. Its image equals $[\check{t}-1,\infty)$.

*  Let $U_X$ be a chart adapted to $f$, and for $i$ in the associated index set (<ref>) let $v_i$ be the function defined in (<ref>). Then $v_{i}(x)=\check{v}_{t(x)}(t_i(x))$ for every $x\in U_X\setminus D$.

For $s\in [\check{t},\infty)$ we have $\check{t}s^{-1}\in (0,1]$, so the function $\check{v}_{\check{t}}$ is well defined. Its derivative is $\check{v}_{\check{t}}'(s)=1+\eta'(\check{t}s^{-1})\cdot \check{t}s^{-2}>1$, since $\eta$ is strictly increasing. Thus $\check{v}_{\check{t}}$ is strictly increasing, too. Since  $\check{v}_{\check{t}}(\check{t})=\check{t}-\eta(1)=\check{t}-1$, this proves <ref>. Part <ref> follows directly from the definition (<ref>) of $v_i$.

Recall that on $A\setminus \d A$ we have the natural smooth structure pulled back from $X\setminus D$. The following lemma asserts that the candidate charts (<ref>) are compatible with this structure.

Let $U_{X}$ be a chart adapted to $f$, and let $\psi_{i}\colon U_i\to Q_{k,n}$ be an associated chart (<ref>).  Then the restriction $\psi_{i}|_{U_i\setminus \d A}$ is a diffeomorphism onto its image.

Reordering the components of $D$ if needed, we can assume that the index set associated to $U_X$ is $\{1,\dots, k\}$, and our fixed chart is $\psi_{1}=(g,v_2,\dots,v_k,\rest)$.

Recall from Definition <ref> that $(z_{i_1},\dots,z_{i_n})$ is a holomorphic coordinate system on $U_X$, and $D\cap U_X$ is the zero locus of $f|_{U_X}=\prod_{i\in S} z_{i}^{m_i}$. Therefore, the map $U_X\setminus D\to Q_{k,n}$ given by polar coordinates $(r_1,\dots,r_k,\rest)$ is a diffeomorphism onto its image.
Since by definition (<ref>) we have $t_i=-(m_i\log r_i)^{-1}$, the same is true for the map $(t_1,\dots,t_k,\rest)$. Thus $(t_1,\dots,t_k,\rest)$ is a smooth coordinate system on $U_1\setminus \d A$, for the smooth structure inherited from $X$.

We claim that the map $\varphi\de(t,t_2,\dots,t_k,\rest)\colon U_1\setminus \d A\to Q_{k,n}$ is a diffeomorphism onto its image as well. Recall that $|f||_{U_X}=\prod_{i=1}^{k}r_{i}^{m_i}$, $t=-(\log|f|)^{-1}$ and $t_i=-(m_i\log r_i)^{-1}$, so

\begin{equation}\label{eq:t-sum}
\end{equation}

thus in our smooth coordinates $(t_1,\dots,t_k,\rest)$ we have

\begin{equation*}
\tfrac{\d t}{\d t_1}=\tfrac{\d}{\d t_{1}}\textstyle(\sum_{i=1}^{k}t_{i}^{-1})^{-1}=-t^{2}\tfrac{\d}{\d t_1} t_{1}^{-1}=t^{2}t_{1}^{-2}>0.
\end{equation*}

Therefore, $\varphi$ is a local diffeomorphism. Suppose that for some $x,y\in U \setminus \d A$ we have $\varphi(x)=\varphi(y)$. Then $t(x)=t(y)>0$ and $t_{i}(x)=t_{i}(y)>0$ for all $i\in \{2,\dots, k\}$, so $t_1(x)=t_1(y)$ by (<ref>). Since  $\rest(x)=\rest(y)$, too, we get $x=y$. Thus $\varphi$ is injective, hence a diffeomorphism onto its image, as claimed.

The above claim shows that $(t,t_2,\dots, t_k,\rest)$ is another smooth coordinate system on $U\setminus \d A$. With respect to these coordinates, we have $\frac{\d g}{\d t}=\eta'(g)>0$ by (<ref>) and $\frac{\d v_{i}}{\d t_{j}}=0$ for $i\neq j$ by (<ref>). For a fixed $t>0$, the function $t_{i}\mapsto t_{i}-\eta(t^{-1}\cdot t_i)=v_i$ is strictly increasing by Lemma <ref>, so $\frac{\d v_{i}}{\d t_{i}}>0$. Thus the Jacobian determinant of $\psi_1$ in coordinates $(t,t_2,\dots, t_k,\rest)$ is $\frac{\d g}{\d t}\cdot \prod_{i=2}^{k} \frac{\d v_i}{\d t_i}>0$, so $\psi_1$ is a local diffeomorphism.

Assume  $\psi_{1}(x)=\psi_{1}(y)$ for some $x,y\in U\setminus \d A$. Then $g(x)=g(y)>0$, so both $t(x)$ and $t(y)$ are equal to some number $\check{t}>0$. By Lemma <ref><ref>, $v_{i}(x)=\check{v}_{\check{t}}(t_{i}(x))$ for every $i\in \{2,\dots, k\}$, and similarly $v_{i}(y)=\check{v}_{\check{t}}(t_{i}(y))$. Since $v_i(x)=v_i(y)$, Lemma <ref><ref> implies that $t_i(x)=t_{i}(y)$ for all $i\in \{2,\dots, k\}$. The equality $\psi_1(x)=\psi_1(y)$ implies that $\rest(x)=\rest (y)$, too, so the values of all coordinates $(t,t_2,\dots,t_k,\rest)$ at $x$ and $y$ are equal. Thus $x=y$, so $\psi_1$ is injective, as claimed.

The image of each chart (<ref>) is an open subset of $Q_{k,n}$.

Fix a chart $U_X$ adapted to $f$. As before, we order the components of $D$ so that the index set (<ref>) associated to $U_X$ is $\{1,\dots, k\}$, and study the map $\psi_{1}\colon U_1\to Q_{k,n}$ introduced in (<ref>).

Fix a point $x\in U_1$. We need to prove that some neighborhood of $\psi_1(x)$ in $Q_{n,k}$ lies in $\psi_1(U_1)$. This is clear if $x\not\in \d A$, since by Lemma <ref> the restriction $\psi_{1}|_{U\setminus \d A}$ is a diffeomorphism onto its image. Hence we can assume $x\in \d A$. Take a sequence $(y^{\nu})\subseteq Q_{k,n}$ such that $y^{\nu}\rightarrow \psi_{1}(x)$. We need to show that $y^{\nu}\in \psi_{1}(U_1)$ for all $\nu\gg 1$. To do this, we will construct a sequence $(x^{\nu})$ such that $\psi_{1}(x^{\nu})=y^{\nu}$ and $x^{\nu}\in U_1$ for $\nu\gg 1$.

Write $y^{\nu}=(g^{\nu},v^{\nu}_1,\dots,v_{k}^{\nu},\rest^{\nu})$ for some $g^{\nu}\geq 0$, $v_i^{\nu}\in \R$ and $\rest^{\nu}\in (\S^{1})^{k}\times \C^{n-k}$. Then $g^{\nu}\rightarrow g(x)=0$, $v_{i}^{\nu}\rightarrow v_i(x)$ and $\rest^{\nu}\rightarrow \rest(x)$. Passing to a subsequence, we can assume that either $g^{\nu}>0$ for all $\nu$, or $g^{\nu}=0$ for all $\nu$: in other words, $y^{\nu}$ approaches $\psi_1(x)$ either from $\Int Q_{k,n}$, or from $\d Q_{k,n}$.
Consider the case $g^{\nu}>0$. Put $t^{\nu}:=\eta^{-1}(g^{\nu})$. Since $\eta^{-1}$ is continuous, we have $t^{\nu}\rightarrow \eta^{-1}(0)=0$. By definition (<ref>) of $U_i$, we have $w_1(x)>\tfrac{1}{n+1}$, so by Lemma <ref><ref>, for $i\in \{2,\dots, k\}$ we have $w_{i}(x)=1-w_1(x)-\sum_{j\neq 1,i} w_j(x)<1-\tfrac{1}{n+1}<1$. The formula (<ref>) for $v_i$ implies that $v_{i}(x)>-1$. Hence there is a positive number $\epsilon>0$ such that, after taking $\nu$ sufficiently large, we have $t^{\nu}<\epsilon$ and $v_{i}^{\nu}>\epsilon-1$ for $i\in \{2,\dots, k\}$. Therefore, each number $v_{i}^{\nu}$ lies in the image of the function $\check{v}_{t^{\nu}}$ from Lemma <ref><ref>. In other words, there is a number $t_{i}^{\nu}\geq t^{\nu}$ such that $\check{v}_{t^{\nu}}(t_{i}^{\nu})=v_{i}^{\nu}$.

We claim that for each $i\in \{2,\dots, k\}$, the sequence $(t_{i}^{\nu})$ is bounded. Suppose the contrary. Passing to a subsequence, we get $t_{i}^{\nu}\rightarrow \infty$. Since $t^{\nu}\rightarrow 0$, we get $t^{\nu}\cdot (t_{i}^{\nu})^{-1}\rightarrow 0$, so $\eta(t^{\nu}\cdot (t_{i}^{\nu})^{-1})\rightarrow \eta(0)=0$. Now $v_{i}^{\nu}=\check{v}_{t^{\nu}}(t_{i}^{\nu})=t_{i}^{\nu}-\eta(t^{\nu}\cdot (t_{i}^{\nu})^{-1})\rightarrow \infty-0=\infty$, a contradiction since $v_{i}^{\nu}\rightarrow v_i(x)\in \R$.

Thus passing to a subsequence, we can assume that $t_{i}^{\nu}\rightarrow \check{t}_{i}$ for some $\check{t}_{i}\geq 0$. We claim that $\check{t}_{i}=t_i(x)$. If $\check{t}_{i}>0$, then the sequence $(t_{i}^{\nu})$ is bounded from below by a positive number, so $t^{\nu}\cdot (t_{i}^{\nu})^{-1}\rightarrow 0$, and therefore $\check{v}_{t^{\nu}}(t_{i}^{\nu})=t_i^{\nu}-\eta(t^{\nu}\cdot (t_{i}^{\nu})^{-1})\rightarrow \check{t}_{i}$. Hence $\check{t}_{i}=v_{i}(x)$. In particular, $v_i(x)> 0$, so by Lemma <ref><ref> we have $v_i(x)=t_i(x)$, and therefore $\check{t}_i=t_i(x)$, as claimed. On the other hand, if $\check{t}_{i}=0$ then $v_{i}(x)=\limn \check{v}_{t^{\nu}}(t_{i}^{\nu})=\limn -\eta(t^{\nu}\cdot (t_{i}^{\nu})^{-1})\leq 0$, so Lemma <ref><ref> gives $t_{i}(x)=0$, as needed.

Thus for all $i\in \{2,\dots, k\}$ we have $t_{i}^{\nu}\rightarrow t_{i}(x)$. Put $w_{i}^{\nu}:=t^{\nu}\cdot (t^{\nu}_{i})^{-1}$. By definition of $t_{i}^{\nu}$ we have  $v_{i}^{\nu}=\check{v}_{t^{\nu}}(t_{i}^{\nu})$, so $v_{i}^{\nu}=t_{i}^{\nu}-\eta(w_{i}^{\nu})$ by definition of the function $\check{v}_{t^{\nu}}$. Hence $w_{i}^{\nu}=\eta^{-1}(t_{i}^{\nu}-v_{i}^{\nu})\rightarrow \eta^{-1}(t_{i}(x)-v_{i}(x))=\eta^{-1}(u_i(x))=w_{i}(x)$.

By definition (<ref>) of $U_1$, we have $w_{1}(x)>\frac{1}{n+1}$. Lemma <ref><ref> implies that $\sum_{i=2}^{k}w_{i}(x)=1-w_{1}(x)< \tfrac{n}{n+1}$. Hence for $\nu\gg 1$ we have $\sum_{i=2}^{k}w_{i}^{\nu}<\tfrac{n}{n+1}$, too. By definition of $w_{i}^{\nu}$ we get $\sum_{i=2}^{k}(t_{i}^{\nu})^{-1}<\tfrac{n}{n+1} (t^{\nu})^{-1}<(t^{\nu})^{-1}$, so the positive number $t^{\nu}_{1}\de ((t^{\nu})^{-1} - \sum_{i=2}^{k}(t_{i}^{\nu})^{-1})^{-1}$ satisfies $(t^{\nu}_1)^{-1}=(t^{\nu})^{-1} - \sum_{i=2}^{k}(t_{i}^{\nu})^{-1}>\tfrac{1}{n+1}(t^{\nu})^{-1}$, and therefore $t_{1}^{\nu}<(n+1)t^{\nu}$. Since $t^{\nu}\rightarrow 0$, we infer that $t_{1}^{\nu}\rightarrow 0$, too. By Lemma <ref><ref> we have $t_{1}(x)=0$, so the convergence $t_{i}^{\nu}\rightarrow t_i(x)$ holds for all $i\in \{1,\dots, k\}$.

For $i\in \{1,\dots, k\}$ put $r_{i}^{\nu}:=e^{-(m_{i}t_{i}^{\nu})^{-1}}$. By (<ref>), we have $r_{i}(x)=0$ if $t_i(x)=0$ and $r_{i}(x)=e^{-(m_i t_i(x))^{-1}}$ otherwise, so $r_i^{\nu}\rightarrow r_i(x)$ for all $i\in \{1,\dots, k\}$.

Let $x_{\log}$ be the image of $x$ in the Kato–Nakayama space $X_{\log}$. In the chart (<ref>), the coordinates of $x_{\log}$ are $(r_1(x),\dots,r_k(x),\rest(x))$. Since $r_{i}^{\nu}\rightarrow r_{i}(x)$ and $\rest^{\nu}\rightarrow \rest(x)$ by definition of $\rest^{\nu}$, for $\nu\gg 1$ the point $(r_1^{\nu},\dots, r_k^{\nu},\rest^{\nu})$ lies in the image of that chart, too. Let $x_{\log}^{\nu}$ be the corresponding point of $X_{\log}$, and let $x^{\nu}$ be its image in $X$. Since ${r}^{\nu}_{i}>0$ for all $i\in \{1,\dots,k\}$, we have $x^{\nu}\not\in D$, so we can identify $x^{\nu}$ with its preimage in $A\setminus \d A$. This way, $\psi_{1}(x^{\nu})=y^{\nu}$ and $x^{\nu}\rightarrow x$, so $x^{\nu}\in U_1$ for $\nu\gg 1$ because $U_1$ is open. This ends the proof in case $g^{\nu}>0$.
Consider now the case when $g^{\nu}=0$ for all $\nu$. Say that $x\in A_{I}^{\circ}$ for some $I\subseteq \{1,\dots, k\}$. By Lemma <ref><ref> we have $1\in I$, $v_{i}(x)=-u_{i}(x)$ if $i\in I$ and $v_i(x)=t_i(x)>0$ if $i\not \in I$. Since $v_{i}^{\nu}\rightarrow v_{i}(x)$ for all $i\in \{2,\dots, k\}$, after passing to a subsequence we get a subset $J\subseteq I\setminus \{1\}$ such that $v_{i}^{\nu}\leq 0$ if $i\in J$ and $v_{i}^{\nu}> 0$ if $i\not\in J$.

For $i\in J$ put $w_{i}^{\nu}=\eta^{-1}(-v_{i}^{\nu})$. Since $\eta^{-1}$ is continuous, we get $w_{i}^{\nu}\rightarrow \eta^{-1}(-v_{i}(x))=w_{i}(x)$. For $i\in I\setminus J$ we have $-u_{i}(x)=v_{i}(x)=0$, so $w_{i}(x)=0$. For $i\not\in I$ we have $w_{i}(x)=0$ by Lemma <ref><ref>. Thus by Lemma <ref><ref> we have $\sum_{i\in J}w_i(x)=1-w_{1}(x)<\tfrac{n}{n+1}$, where the inequality follows from definition (<ref>) of $U_i$. Since $w_{i}^{\nu}\rightarrow w_{i}(x)$ for $i\in J$, we infer that for $\nu\gg 1$, the numbers $w_{1}^{\nu}\de 1-\sum_{i\in J}w_{i}^{\nu}$ are positive. Put $v_{1}^{\nu}=-\eta(w_{1}^{\nu})$. Then $v_{1}^{\nu}\rightarrow -\eta(w_1(x))=v_{1}(x)$ by Lemma <ref><ref>. Let $u^{\nu}$ be the point of $\R^{N}$ whose $i$-th coordinate, for $i\in J\cup\{1\}$, is $-v_{i}^{\nu}$, and the remaining coordinates are zero. Then $u^{\nu}\in \Delta_{J\cup \{1\}}\subseteq \Delta^{N-1}$.

For $i \not\in J$ put $r_{i}^{\nu}=e^{-(m_{i}v_{i}^{\nu})^{-1}}$. Like before, we have $r_{i}^{\nu}\rightarrow e^{-(m_{i}v_{i}(x))^{-1}}=e^{-(m_{i}t_{i}(x))^{-1}}=r_i(x)$. For $i\in J\cup \{1\}$ put $r_{i}^{\nu}=0$, so $r_{i}^{\nu}\rightarrow r_i(x)$, too. The image of $x$ in the chart (<ref>) of $X_{\log}$ has coordinates $(r_1(x),\dots,r_k(x),\rest(x))$, so for $\nu\gg 1$ the point $(r_{1}^{\nu},\dots,r_{k}^{\nu},\rest^{\nu})$ lies in the image of this chart, too. The corresponding point $x_{\log}^{\nu}\in X_{\log}$ lies in $(X_{J\cup\{1\}}^{\circ})_{\log}$.  Since $u^{\nu}\in \Delta_{J\cup \{1\}}$, the pair $x^{\nu}\de (x_{\log}^{\nu},u^{\nu})\in X_{\log}\times \R^{N}$ lies in $A_{J\cup\{1\}}^{\circ}$ by Lemma <ref><ref>. By construction, $x^{\nu}\rightarrow x$ and $\psi_{1}(x^{\nu})=y^{\nu}$. Since $U_{1}$ is open, we conclude that $x^{\nu}\in U_1$ for $\nu\gg 1$, as required.

Each chart (<ref>) is a homeomorphism onto its image.

Fix a chart (<ref>) corresponding to a chart $U_X$ adapted to $f$, say $\psi_{1}\colon U_1\to Q_{k.n}$. Let $S$ be the index set (<ref>) associated to $U_X$. By Definition <ref><ref>, the coordinates of $U_X$ extend continuously to the compact closure $\bar{U}_X$. Together with Lemma <ref><ref>, this implies that $\psi_1$ extends to a continuous map $\bar{U}_{1}\to Q_{k,n}$, which we will denote by the same letter. Since $\bar{U}_1$ is compact, it is sufficient to prove that this extension is injective.

For a subset $I\subseteq  S$ put $V_{I}=\bar{U}_{1}\cap A_{I}^{\circ}$. We have $\psi_{1}(V_{\emptyset})\subseteq \Int Q_{k,n}$ and $\psi(V_{I})\subseteq \d Q_{k,n}$ for all $I\neq \emptyset$, so $\psi_{1}(V_{\emptyset})\cap \psi_{1}(V_{I})=\emptyset$. Lemma <ref><ref> implies that $\psi_{1}(V_{I})\cap \psi_{1}(V_{J})=\emptyset$ for all $I\neq J$. Therefore, it is sufficient to prove the injectivity of each restriction $\psi_{1}|_{V_{I}}$.

For $I=\emptyset$, this follows Lemma <ref>. Assume $I\neq \emptyset$ and fix $x\in V_I$. By  Lemma <ref><ref>, the point $x$ is a pair $(x_{\log},x_{\Delta})\in (X_{I}^{\circ})_{\log}\times \Delta_{I}$. In coordinates (<ref>) on $X_{\log}$, the point $x_{\log}$ is given by $((r_{i}(x))_{i\in S},\rest(x))$, where $r_{i}(x)=0$ if $i\in I$ and, by (<ref>), $r_{i}(x)=e^{-(m_i t_i(x))^{-1}}$ if $i\not\in I$. By Lemma <ref><ref>, for $i\not\in I$ we have $i\neq 1$ and  $t_i(x)=v_i(x)$, so the point $x_{\log}$ is determined by the value of $\psi_1$ at $x$. In turn, the $i$-th coordinate of $x_{\Delta}\in \Delta_{I}\subseteq \R^N$ is $\bar{u}_{i}(x)$ if $i\in I$ and zero otherwise. For $i\in I$, we have $\bar{u}_{i}(x)=u_{i}(x)$ by Lemma <ref><ref> and $u_i(x)=-v_{i}(x)$ by Lemma  <ref><ref>. Hence for $i\neq 1$, the $i$-th coordinate of $x_{\Delta}\in \R^{N}$ is determined by the value of $\psi_1$ at $x$. The remaining coordinate is determined by the fact that $x_{\Delta}$ lies in the rounded simplex $\Delta_{I}$.

By Lemma <ref>, each chart (<ref>) is a homeomorphism onto its image, which is an open subset of $Q_{k,n}$ by Lemma <ref>. By Lemma <ref>, the chart (<ref>) restricts to a diffeomorphism away from $\d A$, as needed.

§.§.§ Useful computations

By Lemma <ref>, our atlas makes $A$ a topological manifold. Now, our goal is to show that this manifold is $\cC^1$. Lemmas <ref>–<ref> gather some technical computations which will be useful throughout the remaining part of Section <ref>. To state them in a convenient way, we introduce the following notation.

Recall that in (<ref>) we defined $t=-(\log|f|)^{-1}$ and $g=\eta(t)$. We put $t'=\tfrac{d}{dg} t$, so $t'=\zeta(g)$, where $\zeta\colon s\mapsto s^{-2}e^{1-s^{-1}}$ is the derivative of the inverse of $\eta$, see (<ref>). More generally, for every $l\geq 0$ we put

\begin{equation}\label{eq:def-tl}
t^{(l)}=(\tfrac{d}{d g})^{l}\, t=\zeta^{(l-1)}(g).
\end{equation}

Let $U_X$ be a chart adapted to $f$. For every $i$ in its associated index set (<ref>), we define functions

\begin{equation}\label{eq:rho-sigma}
\rho_{i}=(t_{i}+u_i^2)^{-1}
\quad \mbox{and}\quad
\sigma_{i}=t_{i}^{2}+t_{i}u_{i}^{2},
\end{equation}

where $t_{i}, u_{i}$ are the functions introduced in (<ref>) and (<ref>). Note that $\sigma_{i}=t_{i}\rho_{i}^{-1}$.

Let $U_X$ be a chart adapted to $f$. Fix $i$ in its associated index set (<ref>). The functions introduced in Section <ref> and in (<ref>), (<ref>) above satisfy the following identities on $U_X\setminus D$:

*  $r_{i}=e^{-(m_{i}t_{i})^{-1}}$,

*  $t=t_{i}\cdot w_{i}$,

*  $t'=t\cdot g^{-2}=t\cdot (1-\log t)^{2}$,

*  for every integer $l\geq 0$, there is a polynomial $p_{l}\in \R[s]$ such that $t^{(l)}=t\cdot p_{l}(\log t)$,

*  $u_{i}=g\cdot(1+g\log t_{i})^{-1}$,

*  $\rho_{i}=(1+g\log t_{i})^{2}\cdot (t_{i}(1+g\log t_{i})^{2}+g^2)^{-1}$,

*  $\eta'(w_{i})=u_{i}^{2}w_{i}^{-1}$, where $\eta\colon s\mapsto (1-\log s)^{-1}$ is the function defined in (<ref>),

*  $\eta'(w_i)t'=t_{i}(1+g\log t_{i})^{-2}$,

*  for every $\epsilon\in (0,1)$, there is a bounded function $l\in \cC^{\infty}(U_X\setminus D)$ such that $\eta'(w_{i})t'=t_{i}^{\epsilon}\cdot l$.

<ref> In (<ref>), the function $t_i$ was defined as $t_{i}=-(m_{i}\log r_{i})^{-1}$, so $\log r_{i}=-(m_it_i)^{-1}$, and therefore $r_{i}=e^{-(m_{i}t_{i})^{-1}}$, as claimed.

<ref> In (<ref>), the function $w_i$ was defined as $w_{i}=tt_{i}^{-1}$, so $t=t_iw_i$.

<ref> Recall from (<ref>) and (<ref>) that $g=\eta(t)=(1-\log t)^{-1}$, so $t=e^{1-g^{-1}}$. Now $t'=\frac{d t}{d g}=e^{1-g^{-1}} g^{-2}=tg^{-2}=t(1-\log t)^{2}$, as needed.

<ref> We argue by induction on $l\geq 0$. For $l=0$ we put $p_0=1$. Assume that for some $l\geq 0$, there is $p_{l}\in \R[s]$ such that $t^{(l)}=p_{l}(\log t)t$. Since $t=e^{1-g^{-1}}$ and $\log t=1-g^{-1}$, we have

\begin{equation*}
\begin{split}
t^{(l+1)}&=\tfrac{d}{d g}t^{(l)}=\tfrac{d}{dg} [p_{l}(1-g^{-1})e^{1-g^{-1}}]=p_{l}'(1-g^{-1})g^{-2}e^{1-g^{-1}}+p_{l}(1-g^{-1})g^{-2}e^{1-g^{-1}}=\\
&= (p_{l}(1-g^{-1})+p_{l}'(1-g^{-1}))g^{-2} e^{1-g^{-1}}
=(p_{l}(\log t)+p_{l}'(\log t)) (1-\log t)^{2} t.
\end{split}
\end{equation*}

The inductive claim follows by putting $p_{l+1}(s)=(p_{l}(s)+p_{l}'(s))(1-s)^{2}\in \R[s]$.

<ref> The function $u_i$ is defined in (<ref>) as $u_{i}=\eta(w_{i})=(1-\log w_{i})^{-1}$. By (<ref>), we have $(1-\log w_{i})^{-1}=(1-\log t+\log t_{i})^{-1}$. Since by definition $g=(1-\log t)^{-1}$, we have $(1-\log t+\log t_{i})^{-1}=(g^{-1}+\log t_{i})^{-1}=g\cdot (1+g\log t_{i})^{-1}$. Thus eventually $u_{i}=g\cdot (1+g\log t_{i})^{-1}$, as claimed.

<ref> The function $\rho_{i}$ was defined in (<ref>) by $\rho_i=(t_{i}+u_{i}^{2})^{-1}$. Substituting the formula for $u_i$ from <ref>, we get $\rho_{i}=(t_{i}+g^2\cdot (1+g \log t_i)^{-2})^{-1}=(1+g\log t_{i})^{2}\cdot (t_{i}(1+g\log t_{i})^2+g^2)^{-1}$.

<ref> By (<ref>), we have $u_{i}=\eta(w_i)=(1-\log w_{i})^{-1}$, so $\eta'(w_{i})=(1-\log w_{i})^{-2}w_{i}^{-1}=u_{i}^2w_{i}^{-1}$.

<ref> 	Substituting  the formula for $u_i$ from <ref> to <ref>, we get $\eta'(w_{i})=g^{2}\cdot (1+g\log t_{i})^{-2}\cdot w_{i}^{-1}$. By (<ref>) we have $w_{i}=t\cdot t_{i}^{-1}$, so $\eta'(w_{i})=g^{2}\cdot (1+g\log t_{i})^{-2} \cdot t^{-1}t_{i}$. In turn, by <ref> we have $t'=tg^{-2}$, so $t'\eta'(w_i)=tg^{-2}\cdot g^{2}(1+g\log t_{i})^{-2}\cdot t^{-1}t_{i}=t_{i}(1+g\log t_{i})^{-2}$, as claimed.

<ref> Recall from (<ref>) that $g=\eta(t)=(1-\log t)^{-1}$. By <ref>, we have  $t=t_{i}w_{i}$, so $g=(1-\log t_{i}-\log w_{i})^{-1}$. Substituting this formula to <ref>, we get

\begin{equation*}
\eta'(w_{i})t'=t_{i}\cdot\left(1+\frac{\log t_{i}}{1-\log t_{i}-\log w_{i}}\right)^{-2}=
t_{i}\cdot \left( \frac{1-\log w_{i}}{1-\log t_{i}-\log w_{i}}\right)^{-2}=
t_{i} \cdot \left( 1-\frac{\log t_{i}}{1-\log w_{i}}\right)^{2}.
\end{equation*}

Since $w_{i}\in (0,1]$, we have $\log w_{i}\leq 0$, so the function $s\de (1-\log w_{i})^{-1}$ is bounded. Fix $\epsilon\in (0,1)$ and put $\delta=\frac{1}{2}(1-\epsilon)>0$. Then

\begin{equation*}
\eta'(w_{i})t'=t_{i}\cdot (1-s\cdot \log t_{i})^{2}=t_{i}^{\epsilon}\cdot (t_{i}^{\delta}-s\cdot t_{i}^{\delta}\log t_{i})^{2}.
\end{equation*}

Since $\delta>0$, we have $t_{i}^{\delta}\log t_{i}\rightarrow 0$ as $t_{i}\rightarrow 0$, so $l\de (t_{i}^{\delta}-s\cdot t_{i}^{\delta}\log t_{i})^{2}$ is bounded, as claimed.

We write $d$ for the exterior derivative. We will apply it to smooth functions on (subsets of) $X\setminus D$, and study the behavior of the resulting forms as we approach $D$.

In the setting of Lemma <ref>, the following identities hold on $U_X\setminus D$:

*  $dr_{i}=m_{i}^{-1}t_{i}^{-2}r_{i}\, dt_{i}$,

*  $	dv_{i}=(1+t_{i}^{-1}u_{i}^{2})\, dt_{i}-t_{i}^{-1}\eta'(w_i)t'\, dg$,

*  $d t_i = \rho_{i}(t_{i}\, dv_{i}+u_{i}^{2}w_{i}^{-1}t'\, dg)$,

*  $dt_i=t_{i}\cdot (t_i(1+g\log t_i)^{2}+g^2)^{-1}\cdot ((1+g\log t_i)^{2}\, dv_{i}+ dg)$,

*  $d w_{i}=-\rho_{i}(w_{i}\, dv_{i}-t'\, dg)$,

*  $d u_{i}=-\rho_{i}(u_{i}^{2}\, dv_{i}-\eta'(w_i)t'\, dg)$,

*  $du_{i}=-t_{i}^{-1}(u_{i}^{2}\, dt_{i}-\eta'(w_{i})t'\, dg)$.

<ref> By Lemma <ref><ref>, we have
$r_{i}=e^{-(m_{i}t_{i})^{-1}}$, so $dr_{i}=m_{i}\cdot (m_{i}t_{i})^{-2} \cdot e^{-(m_{i}t_{i})^{-1}}\, dt_{i}=m_{i}^{-1}t_{i}^{-2}r_{i}\, dt_{i}$, as claimed.

<ref> The functions $v_i$, $w_i$ were defined in (<ref>) and (<ref>) as  $v_{i}=t_{i}-\eta(w_i)$ and $w_{i}=tt_{i}^{-1}$. Hence

\begin{equation*}
dv_{i}=dt_{i}-\eta'(w_{i})\cdot d(tt_{i}^{-1})=dt_{i}-\eta'(w_{i})\cdot (-tt_{i}^{-2}\, dt_{i}+t_{i}^{-1}\, dt)=(1+\eta'(w_{i})tt_{i}^{-2})\, dt_{i}-t_{i}^{-1}\eta'(w_{i})\, dt.
\end{equation*}

By Lemma <ref><ref>, we have $\eta'(w_{i})=u_{i}^{2}w_{i}^{-1}$. Since $t=t_{i}w_{i}$ by Lemma <ref><ref>, we get  $\eta'(w_{i})tt_{i}^{-2}=u_{i}^{2}w_{i}^{-1}\cdot t_{i}w_{i}\cdot t_{i}^{-2}=u_{i}^{2}t_{i}^{-1}$.
In turn, $dt=t' dg$ by definition (<ref>) of $t'$. Thus $dv_{i}=(1+u_{i}^{2}t_{i}^{-1})dt_{i}-t_{i}^{-1}\eta'(w_{i})t'\, dg$, as claimed.

<ref> Solving <ref> for $dt_{i}$, we get

\begin{equation*}
dt_{i}=(1+t_{i}^{-1}u_{i}^{2})^{-1}\cdot (dv_{i}+t_{i}^{-1}\eta'(w_{i})t'\, dg)=(t_{i}+u_{i}^{2})^{-1}\cdot (t_{i}\, dv_{i}+\eta'(w_{i})t'\, dg).
\end{equation*}

We have $(t_{i}+u_{i}^{2})^{-1}=\rho_{i}$ by (<ref>) and $\eta'(w_{i})=u_{i}^{2}w_{i}^{-1}$ by Lemma  <ref><ref>, so  $dt_{i}=\rho_{i}(t_{i}\, dv_{i}+u_{i}^{2}w_{i}^{-1}t'\, dg)$, as claimed.

<ref>  Substituting the formula for $\eta'(w_i)$ from Lemma <ref><ref> to Lemma <ref><ref>, we get the identity $u_{i}^{2}w_{i}^{-1}t'=t_{i}(1+g\log t_{i})^{-2}$. Substituting it to <ref> gives

\begin{equation*}
dt_{i}=\rho_{i}(t_{i}\,dv_{i}+t_{i}(1+g\log t_{i})^{-2}\, dg)=\rho_{i}t_{i} (1+g\log t_{i})^{-2} \cdot((1+g\log t_{i})^{2} dv_{i}+dg).
\end{equation*}

The claim follows by substituting the formula for $\rho_{i}$ from Lemma <ref><ref>.

<ref> The function $w_i$ was defined in (<ref>) as $w_{i}=tt_{i}^{-1}$, so $dw_{i}=-tt_{i}^{-2}\, dt_{i}+t_{i}^{-1}\, dt=-w_{i}t_{i}^{-1}\, dt_{i}+t_{i}^{-1}\, dt$. We have $dt_{i}=\rho_{i}(t_{i}\, dv_{i}+u_{i}^{2}w_{i}^{-1}t'\, dg)$ by <ref> and $dt=t'\, dg$ by definition (<ref>) of $t'$, so

\begin{equation*}
dw_{i}=-w_{i}t_{i}^{-1}\rho_{i}(t_{i}\, dv_{i}+u_{i}^{2}w_{i}^{-1}t'\, dg)+t_{i}^{-1}t'\, dg=
-\rho_{i}(w_{i}\, dv_{i}+(u_{i}^{2}-\rho_{i}^{-1})t_{i}^{-1}t'\, dg).
\end{equation*}

By definition (<ref>) of $\rho_{i}$, we have $(u_{i}^{2}-\rho_{i}^{-1})t_{i}^{-1}=(u_{i}^{2}-t_{i}-u_{i}^{2})t_{i}^{-1}=-1$, which proves the claim.

<ref> By (<ref>), we have $u_i=\eta(w_i)$, so using <ref> we get   $du_{i}=\eta'(w_i)dw_{i}=-\rho_{i}(\eta'(w_i)w_{i}\, dv_{i}-\eta'(w_i)t'\, dg)$. By Lemma <ref><ref>, we have $\eta'(w_{i})w_{i}=u_{i}^{2}$, so $du_{i}=-\rho_{i}(u_{i}^{2}\, dv_{i}-\eta'(w_i)t'\, dg)$, as claimed.

<ref> Substituting <ref> to <ref>, we get

\begin{equation*}
\begin{split}
d u_{i}&=-\rho_{i}(u_{i}^{2}\cdot ((1+t_{i}^{-1}u_{i}^{2})\, dt_{i}-t_{i}^{-1}\eta'(w_i)t'\, dg)-\eta'(w_i)t'\, dg)=\\
-\rho_{i}(u_{i}^{2}(1+t_{i}^{-1}u_{i}^{2})\, dt_{i}-(u_{i}^{2}t_{i}^{-1}+1)\eta'(w_{i})t'\, dg)=\\
-\rho_{i}(1+t_{i}^{-1}u_{i}^{2}) (u_{i}^{2}\, dt_{i}-\eta'(w_{i})t'\, dg)=
-t_{i}^{-1}(u_{i}^{2}\, dt_{i}-\eta'(w_{i})t'\, dg),
\end{split}
\end{equation*}

where the last equality follows from definition (<ref>) of $\rho_{i}$.

§.§.§ Forms bounded from $X$

Once we endow $A$ with a smooth structure, it will be important to distinguish those forms on $A$ which come from bounded forms on $X$. To this end, we introduce the following definition.

Let $U_{X}$ be a chart adapted to $f$, and let $V$ be an open subset of $\pi^{-1}(U_X)$. On $V\setminus \d A$, we consider the natural smooth structure pulled back from $\pi(V)\setminus D$. We say that a form $\beta\in \Omega^{*}(V\setminus \d A)$ is bounded from $X$ if $\beta=\pi^{*}\beta'$ for some $\beta'\in \Omega^{*}(\pi(V)\setminus D)$ which is bounded in the usual sense, with respect to coordinates $(z_{i_1},\dots,z_{i_n})$ of the adapted chart $U_X$.

Let $U_i$ be a domain of a chart (<ref>). A function on $U_i\setminus \d A$ is bounded from $X$ if and only if it is bounded. Since the closure $\bar{U}_i$ is compact by Definition <ref><ref>, all continuous functions on $\bar{U}_i$, in particular those from Lemma <ref><ref>, are bounded from $X$.

If two forms $\beta$ and $\beta'$ are bounded from $X$, then so are $\beta+\beta'$ and $\beta\wedge \beta'$. Nonetheless, $d\beta$ might not be bounded from $X$. For example, functions $\theta_i$, $t_{i}$ are bounded from $X$, but the $1$-forms $d\theta_i$, $d t_i$ are not. Indeed, $d\theta_{i}=-\frac{1}{r_i}\, d^{c}r_i$ has a logarithmic pole along $D_i$. For $dt_{i}$, note that by Lemma <ref><ref> we have $d t_{i}=m_{i}t_{i}^{2}r_{i}^{-1}\, dr_{i}$, so Lemma <ref><ref> gives

\begin{equation*}
dt_{i}=m_{i}t_{i}^{2}e^{(m_{i}t_{i})^{-1}}\, dr_{i}\rightarrow \infty\quad \mbox{as } t_i\rightarrow 0.
\end{equation*}

In principle, one should think of $1$-forms bounded from $X$ as the ones decaying exponentially fast as we approach $\d A$. One way of making this precise will be given in Lemma <ref><ref>.

In the next computation, the following elementary observation will be useful.

Let $a,b,x,y\in \R$. Assume $a,b \geq 0$ and $a+b>0$. Then the following inequality holds:

\begin{equation*}
\frac{|ax+by|}{a+b}\leq |x|+|y|.
\end{equation*}

By symmetry, we can assume $a\geq b$. Then $a>0$ and $1\geq \frac{b}{a} \geq  0$, so

\begin{equation*}
\frac{|ax+by|}{a+b}=\frac{|x+\frac{b}{a}y|}{1+\frac{b}{a}}\leq \frac{|x|+\frac{b}{a}|y|}{1+\frac{b}{a}} \leq |x|+|y|. \qedhere
\end{equation*}

We will now use Lemma <ref><ref> to compare the $1$-forms $d v_{i}$ defined by two charts adapted to $f$. We warn the reader that functions $b,c,q,s$ and a $1$-form $\gamma$ introduced below depend on the fixed charts $U_X$, $U_X'$ and index $i$; even though the notation does not reflect this explicitly. The same warning applies to functions $\lambda,a$ from Lemma <ref><ref>,<ref>.

Let $U_X,U'_X$ be two charts adapted to $f$. As in Lemma <ref>, we put $V_X=U_X\cap U_X'$, $V=\pi^{-1}(U_X\cap U_X')$, and use Notation <ref>. Fix $i\in S\cap S'$ and a subset $I\subseteq S\cap S'$ not containing $i$. Put $V_{X,I}^{\circ}=V_{X}\cap X_{I}^{\circ}$. Then the following hold.

*  The map $\theta'_i-\theta_i\colon V\to \S^1$ is a pullback of a smooth map $V_{X}\to \S^1$.

*  Let $a\in \cC^{\infty}(V_X)$ be as in Lemma <ref><ref>. Then the functions $|a|$, $|1+at_{i}|^{-1}$, $|\log(1+at_{i})|$ and $|1-u_i\log(1+at_i)|^{-1}$ are continuous and bounded on $V$.

*  Let $a\in \cC^{\infty}(V_X)$ be as in <ref>. The following identity holds on $V$

\begin{equation*}\label{eq:vp-v}
v_{i}-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)}.
\end{equation*}

*  There is a bounded function $b\in \cC^{\infty}(V_{X,I}^{\circ})$ such that $v_i-v_i'=\sigma_i b$ on $V_{X,I}^{\circ}$.

*  For every $\epsilon\in (0,1)$, there are bounded functions $c,q\in \cC^{\infty}(V_{X,I}^{\circ})$, and a bounded
$1$-form $\gamma\in \Omega^{1}(V_{X,I}^{\circ})$, such that the following equality holds on $V_{X,I}^{\circ}$:

\begin{equation*}
%=\sigma_{i}(ct_{i}^{-1}\, dt_{i}+\gamma)+ t_i^{\epsilon} u_{i} q\, dg%, \\
= t_{i}\cdot c\, dv_{i}+t_i^{\epsilon}\cdot q\, dg+\sigma_{i}\cdot \gamma.
\end{equation*}

In particular, taking $I=\emptyset$, we get that the equalities from <ref>, <ref> hold on $V_{X}\setminus D=V\setminus \d A$, where functions $b,c,q$ are bounded, and the $1$-form $\gamma$ is bounded from $X$.

<ref> By Lemma <ref><ref>, there is a nonvanishing holomorphic function $\lambda \in \cO_{X}^{*}(V_X)$ such that $z_{i}'=\lambda z_{i}$. In polar coordinates, we have $\lambda=|\lambda|e^{2\pi\imath\cdot \beta}$ for some smooth function $\beta$. Using the additive notation <ref> on $\S^1$, we get $\theta'_i-\theta_i=\beta$, as claimed.

<ref> Recall that by Definition <ref><ref>, the coordinates of an adapted chart $U_X$ extend continuously to the closure $\bar{U}_X$, which is compact. Therefore, the function $a\in \cC^{\infty}(V_X)$ introduced in Lemma <ref><ref> extends to a continuous function on a compact set $\bar{V}_X$. In particular, the function $|a|$ is bounded.

Similarly, the function $1+at_{i}$ extends to a continuous function on $\bar{V}_X$, which does not vanish by Lemma <ref><ref>. Thus the functions $|1+at_{i}|^{-1}$ and $|\log(1+at_{i})|$ are continuous on $\bar{V}_X$, hence bounded because $\bar{V}_{X}$ is compact.

On $U_X\setminus D_i$, we have $u_{i}=\eta(w_i)=(1-\log w_{i})^{-1}$ by the formulas (<ref>) and (<ref>). Using the second equation from Lemma <ref><ref>, we get that on $V \setminus \pi^{-1}(D_i)$, the following identity holds:

\begin{equation*}
\frac{1}{1-\log w_{i}-\log(1+at_{i})}-u_i=
\frac{1}{u_{i}^{-1}-\log(1+at_{i})}-u_i=
\frac{u_i}{1-u_{i}\log(1+at_{i})}-u_i.
\end{equation*}

\begin{equation}\label{eq:u-u'}
\end{equation}

On $V\cap \pi^{-1}(D_i)$ we have $t_i=0$ by definition (<ref>) of $t_i$, so the right-hand side of (<ref>) is well defined there. By Lemma <ref><ref> we have $u'_i=u_i$ on $V\cap \pi^{-1}(D_i)$, so both sides of (<ref>) are zero. Thus (<ref>) holds everywhere on $V$. As before, it extends to continuously to $\bar{V}$. In particular, the function $|1-u_{i}\log(1+at_{i})|^{-1}$ is continuous on $\bar{V}$: to see this, divide both sides of (<ref>) by $u_i^2 \log(1+at_i)$ and use the right or left side of the equality when $u_i$ is close or away from $0$, respectively. Since the map $\pi$ is proper by Proposition <ref><ref>, the set $\bar{V}=\pi^{-1}(\bar{V}_{X})$ is compact. Thus $|1-u_{i}\log(1+at_{i})|^{-1}$ is bounded, as claimed.

By Lemma <ref><ref>, we have

\begin{equation}\label{eq:t-t'}
\end{equation}

By (<ref>), we have $v_i-v_i'=(t_i-u_i)-(t_i'-u_i')=(t_i-t_i')+(u_i'-u_i)$, so <ref> follows from  (<ref>) and (<ref>).

Since $i\not \in I$, by Lemma <ref><ref> the functions $t_{i}$, $u_{i}$, $v_i$, $v_i'$ are smooth on $V_{X,I}^{\circ}$. Moreover, since $t_i>0$ on $V_{X,I}^{\circ}$, it  follows that $\sigma_{i}|_{V_{X,I}^{\circ}}$ is a smooth, positive function, hence the restriction  $\sigma_{i}^{-1}(v_i-v_{i}')|_{V_{X,I}^{\circ}}$ is smooth. We need to prove that it is bounded.

By <ref>, the functions $\check{b}\de \frac{a}{1+at_{i}}$ and  $(1-u_{i}\log(1+at_{i}))^{-1}$ are bounded. Since $\frac{1}{t_i}\log(1+a t_{i})=a-\tfrac{1}{2}a^{2}t_{i}+\dots$ is bounded, too, we infer that $\check{h} \de \frac{1}{t_{i}} \cdot \frac{\log(1+a t_{i})}{1-u_{i}\log(1+a t_{i})}$ is bounded as well.
By <ref>, we have $v_{i}-v_{i}'=\check{b}t_{i}^2+\check{h} t_{i}u_{i}^{2}$. By definition (<ref>) of $\sigma_i$, we have $\sigma_{i}=t_{i}^2+t_{i}u_{i}^2$.  Lemma <ref> implies that

\begin{equation*}
\frac{|v_{i}-v_{i}'|}{\sigma_{i}}=\frac{|\check{b} t_{i}^{2}+\check{h} t_{i}u_{i}^{2}|}{t_{i}^{2}+t_{i}u_{i}^{2}} \leq |\check{b}|+|\check{h}|.
\end{equation*}

Therefore, $v_{i}-v_{i}'=\sigma_{i}b $ for some bounded function $b$, as claimed.

<ref> As before, we note that since $i\not \in I$, the functions $t_{i}$ and $u_{i}$ are smooth on $V_{X,I}^{\circ}$. Thus we can apply the exterior derivative $d$ to the formulas (<ref>) and (<ref>). We get bounded functions $\tilde{c},\tilde{q},\tilde{s} \in \cC^{\infty}(V_{X,I}^{\circ})$ and bounded $1$-forms $\tilde{\beta}$, $\tilde{\gamma} \in \Omega^{1}(V_{X,I}^{\circ})$, such that the following identities hold on $V_{X,I}^{\circ}$:

\begin{equation*}
\begin{split}
& d(t_{i}-t_{i}') =\frac{t_{i}^{2}\, da +2a t_{i}\, dt_{i}}{1+a t_{i}}-
\frac{a t_{i}^{2}(a \, dt_{i}+t_{i}\, da )}{(1+a t_{i})^{2}}=
t_{i} \tilde{c} \, dt_{i}+t_{i}^{2}\, \tilde{\beta}, \\%, \quad\mbox{and}\quad  \\
& d(u_{i}' -u_{i})=\frac{2u_{i}\log(1+a t_{i})\, du_i}{1-u_{i}\log(1+a t_{i})}
+\frac{u_{i}^{2}(a \, dt_{i}+t_{i}\ da )}{(1+a t_{i})(1-u_{i}\log(1+a t_{i}))}+\\
\frac{u_{i}^{2}\log(1+a t_{i})}{(1-u_{i}\log(1+a t_{i}))^{2}}\cdot \left(
\log(1+a t_{i})\, du_{i}+
\frac{u_{i}(a \, dt_{i}+t_{i}\, da )}{1+a t_{i}}
\right)= u_{i}t_{i}\tilde{s} \, du_{i}+
u_{i}^{2}\tilde{q} \, dt_{i}+t_{i}u_{i}^{2}\, \tilde{\gamma} .
\end{split}
\end{equation*}

By Lemma <ref><ref>, we have $du_{i}=-t_{i}^{-1}(u_{i}^{2}\, dt_{i}-\eta'(w_i)t'\, dg)$. Fix $\epsilon\in (0,1)$. By Lemma <ref><ref>, $\eta'(w_i)t'=t_{i}^{\epsilon}l$ for some bounded function $l$. Therefore, $du_{i}=-t_{i}^{-1}(u_{i}^{2}\, dt_{i}-t_{i}^{\epsilon}l\, dg)$. Substituting this to the above formula for $d(u_{i}'-u_{i})$, we get that for some bounded functions $\check{q}$, $s$, the following holds:

\begin{equation*}
d(u_{i}' -u_{i})=-u_{i}\tilde{s} (u_{i}^{2}\, dt_{i}-t_{i}^{\epsilon}l\, dg)+
u_{i}^{2} \tilde{q} \, dt_{i}+t_{i}u_{i}^{2}\, \tilde{\gamma}=
u_{i}^{2}\check{q}\, dt_{i}+t_{i}^{\epsilon}u_{i}s\, dg+t_{i}u_{i}^{2}\, \tilde{\gamma}.
\end{equation*}

Adding this equality to the above formula for $d(t_{i}-t_{i}')$ gives

\begin{equation*}
d(v_{i}'-v_{i})=(t_{i}\tilde{c}  + u_{i}^2\check{q} )\, dt_{i}+
t_i^{\epsilon}u_{i} s \, dg+
(t_{i}^{2}\, \tilde{\beta} +t_{i}u_{i}^{2}\, \tilde{\gamma} ).
\end{equation*}

By Lemma <ref>, $\frac{|t_{i}\tilde{c}+u_{i}^{2}\check{q}|}{t_{i}+u_{i}^{2}}\leq |\tilde{c}|+|\check{q}|$, so $t_{i}\tilde{c}+u_{i}^{2}\check{q}=(t_{i}+u_{i}^{2})c$ for some bounded function $c$. By definition (<ref>) of $\sigma_i$, we get $t_{i}\tilde{c}+u_{i}^{2}\check{s}=\sigma_{i}t_{i}^{-1}c$.

Recall that, by Definition <ref>, the holomorphic coordinates on $U_X$ are $(z_{i_1},\dots,z_{i_n})$. Write $x_{j}$, $y_{j}$ for the real and imaginary part of $z_{i_j}$, respectively, and put $\gamma_{2j-1}=dx_{j}$, $\gamma_{2j}=dy_{j}$. Then we can write $\tilde{\beta}=\sum_{j=1}^{2n} \tilde{b}_{j}\gamma_{j}$ and $\tilde{\gamma}=\sum_{j=1}^{2n} \tilde{q}_{j}\gamma_{j}$ for some bounded functions $\tilde{b}_{j}$, $\tilde{q}_{j}$. As before, Lemma <ref> shows that there is a bounded function $\tilde{c}_{j}$ such that   $t_{i}^{2}\tilde{b}_{j}+t_{i}u_{i}^{2}\tilde{q}_{j}=(t_{i}^{2}+t_{i}u_{i}^{2})\tilde{c}_{j}=\sigma_{i}\tilde{c}_{j}$, where the last equality follows from definition (<ref>) of $\sigma_{i}$. Therefore,  $t_{i}^{2}\tilde{\beta}+t_{i}u_{i}^{2}\tilde{\gamma}=\sigma_{i}\sum_{j=1}^{2n}\tilde{c}_{j}\gamma_{j}=\sigma_{i}\gamma$ for some $1$-form $\gamma$ bounded from $X$. Substituting the above equalities to the formula for $d(v_i'-v_i)$, we get

\begin{equation*}
d(v_{i}'-v_i)=\sigma_{i}t_{i}^{-1}c\, dt_{i}+
t_i^{\epsilon}u_{i} s \, dg+
\sigma_{i}\gamma=
\sigma_{i}(ct_{i}^{-1}\, dt_{i}+\gamma)+
t_i^{\epsilon}u_{i} s \, dg.
\end{equation*}

In case $I\neq \emptyset$, we have $u_{i}=0$, so $v_{i}=t_{i}$ and $\sigma_{i}=t_{i}^2$, and therefore, the above equality reads as $d(v_{i}-v_{i}')=t_{i}c\, dv_{i}+\sigma_{i}\gamma$, which is what we wanted to prove. In case $I=\emptyset$, we substitute the formula for $dt_{i}$ from Lemma <ref><ref>. Since by (<ref>) we have $\sigma_{i}t_{i}^{-1}\rho_{i}=1$, we get

\begin{equation*}
\begin{split}
& d(v_{i}'-v_i)=
\sigma_{i}(ct_{i}^{-1}\, dt_{i}+\gamma)+
t_i^{\epsilon}u_{i} s \, dg
=\ &
\sigma_{i}ct_{i}^{-1}\, (\rho_{i}(t_{i}\, dv_{i}+u_{i}^{2}w_{i}^{-1}t'\, dg)) +\sigma_{i}\gamma+
t_i^{\epsilon}u_{i} s \, dg
= \\
=\ &
ct_{i}\, dv_{i}+(cu_{i}^{2}w_{i}^{-1}t'+t_{i}^{\epsilon}u_{i}s)\, dg+\sigma_{i}\gamma.
\end{split}
\end{equation*}

By Lemma <ref><ref>, $u_{i}^{2}w_{i}^{-1}=\eta'(w_{i})$, and by Lemma <ref><ref>, $\eta'(w_{i})t'=t_{i}^{\epsilon}l$ for some bounded function $l$. Hence the coefficient near $dg$ above equals $t_{i}^{\epsilon}q$ for some bounded function $q$, which proves

§.§.§ The functions $r_j$, $w_j$ are $\cC^1$.
We aim at proving Lemma <ref>, which asserts that the  transition maps for our atlas are $\cC^1$. The key step is to prove that in a chart $(U_i,\psi_i)$ introduced in (<ref>), all functions $r_j$, $w_j$ for $j\neq i$ are $\cC^1$. As a consequence, in Lemma <ref> we will infer that the function $v_i$, which is not a coordinate of $\psi_i$ (see formula (<ref>)); as well as the pullbacks of all smooth functions from $U_X$, are $\cC^1$. In Lemmas <ref><ref> and <ref><ref> we will prove a stronger result, replacing  $\cC^1$ by $\cC^{\infty}$. Nonetheless, we provide an independent, direct argument here, too. This way, we give a complete construction of a $\cC^1$-atlas without introducing further technical tools, and illustrate computations behind the proof of Lemmas <ref>, <ref>, which is more complicated.

Fix a chart $U_X$ adapted to $f$ such that $U_X\cap D\neq \emptyset$. We reorder the components of $D$ so that the associated index set (<ref>) is $S=\{1,\dots, k\}$. By Lemma <ref> the map $\psi_{1}\colon U_1\to Q_{k,n}$ is a homeomorphism onto an open subset of a smooth manifold with boundary, so  it endows $U_1$ with a smooth structure.
In other words, we take $(g,v_2,\dots,v_k,\rest)$ as smooth coordinates on $U_1$.

For a subset $V\subseteq U_1$, we denote by $\cS(V)$ (respectively, $\cS^{l}(V)$ for $l\in \N$) the $\R$-algebra of functions on $V$ which are smooth (respectively, $\cC^{l}$), with respect to this smooth structure. We do not use the symbol $\cC^{\infty}$ to avoid confusion with the eventual $\cC^{\infty}$-structure on $A$, which we will introduce in Section <ref>, and whose restriction to $U_1$ will differ from the $\cC^{\infty}$-structure that $U_1$ inherits via $\psi_1$. Since by Lemma <ref> the map $\pi|_{U_1\setminus \d A}$ is a diffeomorphism onto its image,  we can – identifying functions on $X$ with their pullbacks to $A$  – write an inclusion $\cC^{\infty}(U_X\setminus D)\subseteq \cS(U_1\setminus \d A)$.

For $j\in \{1,\dots, k\}$, we introduce differential operators $\d_{j}\colon \cS^{l}(V)\to \cS^{l-1}(V)$ by

\begin{equation*}
\d_{1}=\tfrac{\d}{\d g},\quad \d_{i}=\tfrac{\d}{\d v_{i}}\quad \mbox{for } i\in \{2,\dots, k\},
\end{equation*}

and for $\alpha=(\alpha_1,\dots,\alpha_k)\in \N^k=\Z_{\geq 0}^k$ we write $\d^{\alpha}=\d_{1}^{\alpha_{1}}\dots\d^{\alpha_k}_{k}$, $|\alpha|=\alpha_1+\dots+\alpha_k$.

For $i\in \{2,\dots, k\}$, we put

\begin{equation*}
W_{i}=\{x\in U_1: w_{i}(x)>0\}\quad\mbox{and}\quad T_{i}=\{x\in U_1:t_{i}(x)>0\}.
\end{equation*}

Using the above notation, we have the following immediate corollary of Lemma <ref>.

For every $i,j\in \{2,\dots,k\}$, such that $i\neq j$, we have

\begin{equation*}
\d_j v_i=0,\quad
\d_j r_i=0,\quad
\d_j t_i=0,\quad
\d_j w_i=0,\quad
\d_j u_i=0\quad\mbox{and}\quad
\d_{j}t^{(l)}=0\mbox{ for any } l\geq 0.
\end{equation*}

Since $(g,v_2,\dots,v_k)$ are coordinates on $U_1$, we have $\d_j v_i=0$ for $j\neq i$ and $\d_{j}g=0$ for $j\neq 1$. This proves the first equality. The last one follows by definition (<ref>) of $t^{(l)}$.

For the remaining ones, note that the basis of $T^{*}_{y}U_1$ at any point $y\in U_1$ is given by $dg,dv_2,\dots,d v_k$ and $1$-forms $dz$, where $z$ ranges through the coordinates of $\rest$. For a function $h$, the coefficient of $dh$ near $dv_j$ is $\d_{j} h$. By Lemma <ref><ref>, <ref>, <ref> and <ref>, this coefficient is zero for $h=r_i,t_i,w_i,u_i$ whenever $i\neq 1,j$.

In the setting of Notation <ref>, we have

* 	The function $t$ is smooth on $U_1$, i.e. $t\in \cS(U_1)$,

*  For all $i\in \{1,\dots, k\}$, the functions $v_i,w_{i}$ are in $\cS^{1}(U_1)$.

*  For all $i\in \{1,\dots, k\}$, the function $r_{i}$ is in $\cS^{1}(U_1)$.

*  The pullback of any $\cC^{1}$-function $h$ on $U_X$ is in $\cS^{1}(U_1)$, i.e. $\cC^{1}(U_X)\subseteq \cS^{1}(U_1)$.

<ref> Since $t=\eta^{-1}(g)=e^{1-g^{-1}}$ by (<ref>), and $g$ is one of the coordinates on $U_1$, we have $t\in \cS(U_1)$.

<ref> Fix $i\in \{2,\dots, k\}$. By definition (<ref>) of the coordinate chart $\psi_{1}$, we have  $v_{i}\in \cS^{1}(U_1)$.

By Lemma <ref><ref>, we have $\d_i w_i=-\rho_{i}w_{i}$, $\d_1 w_i=\rho_{i}t'$ and $\d_j w_i=0$ for $j\neq i,1$. Recall that $\rho_i$ was defined in (<ref>) by  $\rho_{i}=(t_{i}+u_{i}^{2})^{-1}$, so since $t_i,u_i$ are continuous by Lemma <ref><ref>, the function $\rho_{i}$ extends to a continuous function away from the common zero locus of $t_i$ and $u_i$, i.e. on $T_i\cup W_i$. It follows that $w_{i}\in \cS^{1}(T_i\cup W_i)$. We claim that $w_{i}\in \cS^{1}(U_1)$.

By definitions (<ref>) and (<ref>), we have $\rho_{i}=(t_{i}+u_i^2)^{-1}$ and $u_i=\eta(w_i)=(1-\log w_{i})^{-1}$,  so

\begin{equation*}
-\d_{i}w_{i}=\rho_{i}w_{i}=\frac{w_{i}}{t_{i}+u_{i}^{2}}=\frac{w_{i}(1-\log w_{i})^{2}}{t_{i}(1-\log w_{i})^{2}+1}\rightarrow 0\quad \mbox{as } w_{i}\rightarrow 0.
\end{equation*}

By Lemma <ref><ref> and <ref>, we have $t'=t(1-\log t)^{2}$ and $t=t_{i}w_i$, so, using Lemma <ref><ref>, we get

\begin{equation*}
\d_{1}w_{i}=\rho_{i} t'=\frac{t'(1-\log w_{i})^{2}}{t_{i}(1-\log w_{i})^{2}+1}=\frac{t_{i}w_{i}(1-\log t_{i}-\log w_{i})^{2}(1-\log w_{i})^{2}}{t_{i}(1-\log w_{i})^2+1} \rightarrow 0 \quad\mbox{as } t_{i},w_i\rightarrow 0.
\end{equation*}

Therefore, $\d_{i}w_{i}$ and $\d_1 w_{i}$ extend to continuous functions on $U_1$, so $w_i\in \cS^{1}(U_1)$, as claimed.

By Lemma <ref><ref>, we have $w_{1}=-\sum_{i=2}^{k}w_{i}$, so $w_{1}\in \cS^{1}(U_1)$, too. The remaining function $v_1$ is defined in (<ref>) as $v_{1}=t_{1}-u_{1}=tw_{1}^{-1}-\eta(w_1)$. Recall that $t\in \cS(U_1)$ by <ref>, and the restriction $\eta|_{(0,1]}$ is smooth. Therefore, since $w_{1}|_{U_{1}}>0$ by definition (<ref>) of $U_{1}$, we infer that $v_{1}\in \cS^{1}(U_1)$, as claimed.

<ref> Consider first the case $i\in \{2,\dots, k\}$. Lemma <ref><ref> implies that $\d_{1}t_{i}$ and $\d_{i}t_{i}$ are continuous on $T_i$, and $\d_{j}t_{i}=0$ for $j\neq i,1$, so like before we conclude that $t_{i}\in \cS^{1}(T_i)$. By Lemma <ref><ref>, we have $r_i=e^{(-m_it_i)^{-1}}$, so $r_i\in \cS^{1}(T_i)$, too. We need to show that $r_i$ is $\cC^1$ on its zero locus $U_1\setminus T_i$.

First, we claim that for every $j\in \{1,\dots, k\}$, the function $t_{i}\cdot \d_{j} t_{i}$ is bounded. By Lemma <ref><ref>, we have $\d_{i}t_{i}=\rho_{i}t_{i}$, $\d_{1}t_{i}=\rho_{i}u_{i}^{2}w_{i}^{-1}t'$ and $\d_jt_i=0$ for $j\neq 1,i$. By definition (<ref>) of $\rho_{i}$, we have $\rho_{i}t_{i}=(1+u_{i}^{2}t_{i}^{-1})^{-1}\leq 1$, so $\d_{i}t_{i}$ is bounded, hence $t_i \cdot \d_i t_i$ is bounded, too, as needed. To deal with $\d_{1}t_{i}=\rho_{i}u_{i}^{2}w_{i}^{-1}t'$, recall that $u_{i}^{2}w_{i}^{-1}=\eta'(w_{i})$ by Lemma <ref><ref>, and by Lemma <ref><ref>, we have $\eta'(w_i)t'=t_{i}^{\epsilon}l$ for some bounded function $l$ and $\epsilon>0$. It follows that  $\d_1t_i=\rho_it_{i}^{\epsilon} l$. We have seen that $\rho_{i}t_{i}=(1+t_{i}^{-1}u_{i}^{2})^{-1}$ is bounded, so $t_{i}\d_{1}t_{i}=\rho_{i}t_{i}^{\epsilon+1}l$ is bounded, too, which proves the claim.

Now by Lemma <ref><ref>, we have $\d_{j}r_{i}=m_{i}^{-1}\cdot t_{i}^{-3}r_{i}\cdot t_{i}\d_j t_i$. By Lemma <ref><ref>, we have $r_{i}=e^{(-m_{i}t_{i})^{-1}}$, so $t_{i}^{-3}r_{i}\rightarrow 0$ as $t_{i}\rightarrow 0$. Since we have shown that $t_{i}\d_{j}t_{i}$ is bounded, we infer that $\d_j r_{i}\rightarrow 0$ as $t_{i}\rightarrow 0$, so $r_{i}\in \cS^{1}(U_1)$, as needed.

It remains to show that $r_{1}\in \cS^{1}(U_1)$. By Lemma <ref><ref> we have  $t_{1}=tw_{1}^{-1}$, and $w_{1}>0$ on $U_1$ by definition  (<ref>). By <ref> and <ref>, we have $t,w_{1}\in \cS^{1}(U_1)$, so $t_1\in \cS^{1}(U_1)$. Now Lemma <ref><ref> gives $r_{1}=e^{-(m_{1}t_{1})^{-1}}\in \cS^{1}(U_1)$, as claimed.

<ref> For $i\in \{1,\dots,k\}$ we have $r_{i}\in \cS^{1}(U_1)$ by <ref> and $\theta_{i}\in \cS^{1}(U_1)$ by definition (<ref>) of the chart. Thus the smooth coordinates $r_i\cos(2\pi\theta_i)$ and $r_i\sin(2\pi\theta_i)$ on $U_X$ are in $\cS^1(U_1)$. By (<ref>), the remaining coordinates $z_{i_{k+1}},\dots,z_{i_n}$ are in $\cS^{1}(U_1)$, too. Now, part <ref> follows from the chain rule.

§.§.§ The A'Campo space is a $\cC^1$-manifold
We are now in the position to prove the remaining results stated in Section <ref>, which endow $A$ with a $\cC^1$-structure.

Fix two $\cC^{1}$-charts $(U_i,\psi_i)$ and $(U_j',\psi_{j}')$ associated to charts $(U_X,\psi_X)$ and $(U'_{X},\psi_{X}')$ adapted to $f$. By Lemma <ref>, the transition map $\psi_{j}'\circ \psi_{i}^{-1}|_{\psi_{i}(U_i\cap U_j')}$ is a homeomorphism between open subsets of $Q_{k,n}$ and $Q_{k',n}$. To prove that it is a $\cC^1$-diffeomorphism, we need to prove that it is $\cC^1$ at $\psi_{i}(x)$ for any $x\in U_i\cap U_{j}'$, and that its Jacobian determinant does not vanish there.

If $x\not\in \d A$ then near $x$ we have the natural smooth structure pulled back from $X\setminus D$, and by Lemma <ref> both $\psi_{i}$ and $\psi_{j}$ are diffeomorphisms with respect to this structure; so the assertion follows.

Assume $x\in \d A$, so $x\in A_{I}^{\circ}$ for some nonempty $I\subseteq \{1,\dots, N\}$, say $I=\{1,\dots, k\}$. Without loss of generality, we can assume that $U_{X}\subseteq U_{X}'$,  and that the associated index set (<ref>) for $U_X$ is $\{1,\dots, k\}$.