Proof. By the conical structure theorem for isolated complex surface singularities, for \(\epsilon>0\) sufficiently small, there exists a homeomorphism of germs \((X,0)\cong (\operatorname{cone}(K_X),0)\). More precisely, we have a homeomorphism \(\Phi:X\smallsetminus\{0\}\longrightarrow (0,\epsilon]\times K_X\) such that every point \(x\in X\smallsetminus\{0\}\) can be written as \(\Phi(x)=(\|x\|,u(x))\) and \(u(x)\in K_X\).

The curve germ \(\gamma:(\mathbb{C},0)\to (X,0)\) intersects the link in the circle \(C=\gamma\cap K_X\). Under the above identification, the curve germ is given by \(\Phi(\gamma\smallsetminus\{0\})=(0,\epsilon]\times C\).

Now consider the equivalence relation \(\mathrm{curl}(\gamma,d)\) on \(X\). Two points \(x,y\in X\) are equivalent if and only if either \(x=y\), or \(x,y\in\gamma\) and there exists \(m\in\mathbb{Z}\) such that \(\gamma^{-1}(x)=e^{\frac{2\pi i m}{d}}\gamma^{-1}(y)\).

Since the above identification writes points of \(\gamma\) as \((r,c)\) with \(r\in(0,\epsilon]\) and \(c\in C\), this equivalence relation does not change the value of \(r\). It only identifies the points in the second factor. Hence \((r,c_1)\sim_{\mathrm{curl}(\gamma,d)}(r,c_2)\) if and only if \(c_1\sim_{\mathrm{curl}(C,d)}c_2\).

Therefore, after applying the homeomorphism \(\Phi\), the quotient \(X/\mathrm{curl}(\gamma,d)\) is obtained from \((0,\epsilon]\times K_X\) by identifying exactly the points which are identified in the quotient \(K_X/\mathrm{curl}(C,d)\).

Consequently, we obtain a natural homeomorphism \(\frac{X}{\mathrm{curl}(\gamma,d)} \longrightarrow \operatorname{cone} \left( \frac{K_X}{\mathrm{curl}(C,d)} \right)\) given by class of the image by \(\Phi\). ◻

Proof. It is enough to prove that \(\bar h\) is well-defined. Suppose that \(\pi_{\gamma,d}(x)=\pi_{\gamma,d}(y)\). Then \(x\sim_{\mathrm{curl}(\gamma,d)}y\). If \(x=y\), then clearly \(\pi_{h(\gamma),d}(h(x)) = \pi_{h(\gamma),d}(h(y))\). Otherwise, \(x,y\in\gamma\) and there exists \(m \in \mathbb{Z}\) such that, \[\gamma^{-1}(x)=e^{\frac{2\pi i m}{d}}\gamma^{-1}(y).\] We have \((h\circ\gamma)^{-1}(h(x))=\gamma^{-1}(x)\) and \((h\circ\gamma)^{-1}(h(y))=\gamma^{-1}(y)\). Therefore, \[(h\circ\gamma)^{-1}(h(x))=e^{\frac{2\pi i m}{d}}(h\circ\gamma)^{-1}(h(y)),\] and \(h(x),h(y)\in h(\gamma)\). In other words, \(h(x)\sim_{\mathrm{curl}(h(\gamma),d)} h(y).\) So \(h\) sends equivalent points to equivalent points. Therefore the map \(\bar h([x])=[h(x)]\) is well-defined and is continuous because it is induced by the continuous map \(h\). Applying the same logic to \(h^{-1}\) gives the continuous inverse map. ◻

Proof. It follows from the fact that a generating system \(G=(\ell_1,\ldots,\ell_k)\) determines a holomorphic embedding \[\begin{array}{rcl} \ell:(X,0) &\longrightarrow& (\mathbb{C}^k,0), \\ p &\longmapsto& (\ell_1(p),\dots,\ell_k(p)). \end{array}\] ◻

Proof. The proof of [lemm2468] is identical to that of [2], and the proof of [lemma24611] is identical to that of [2]. ◻

Proof of Lemma 37. Let \(\pi:(X_{\pi},E)\longrightarrow(X,0)\) be a good resolution and \(E_v\) be an irreducible component of \(E\). Let \(p\) be a smooth point of \(E_v\) and \((x,y)\) be a local system of coordinates of \(X_{\pi}\) centered at \(p\) such that \(E_v\) has local equation \(x=0\). For every \(i,j \in \{1,\dots k\}\) we denote \[\pi^*(df_i \wedge df_j)(x,y)=x^{m_v(i,j)}\phi_{ij}(x,y)\mathrm{d}x \wedge \mathrm{d}y,\] where \(\phi_{ij}\) is an element of \(\mathbb{C}\{x,y\}\) such that \(\phi_{ij}(0,y) \neq 0\) and \(m_v(i,j)\) is a positive integer. Let \(\alpha\) and \(\beta\) be two elements of \(\mathbb{P}(\mathbb{C}^k)\). We have \[\pi^*(\mathrm{d}F_{\alpha} \wedge \mathrm{d}F_{\beta}) =\sum_{i\neq j} \alpha_i \beta_j x^{m_v(i,j)}\phi_{ij}(x,y)\mathrm{d}x\wedge \mathrm{d}y.\] By Point [l39ouvertdezariski] Lemma 34 there exists a Zariski open set \(W_{F}^{\pi}\) such that \[\pi^{*}(\mathrm{d}F_{\alpha} \wedge \mathrm{d}F_{\beta})=x^{\nu_v(F) }S_{\alpha \beta}(x,y)\mathrm{d}x \wedge \mathrm{d}y, \;\text{and} \;S_{\alpha \beta}(0,y) \neq 0 \;\forall (\alpha,\beta) \in W_{F}^{\pi},\] where \(S_{\alpha \beta}(x,y)=\sum_{i\neq j} \alpha_i \beta_j x^{m_v(i,j)-\nu_v(F)}\phi_{ij}(x,y)\). By definition \(S_{\alpha \beta}(x,y)=0\) is the local equation of the strict transform of the curve \(\Pi_{\alpha \beta}\) by \(\pi\).

Let us assume that \(p\) is a basepoint for the family of curves \(\Pi_F\). If there exists \((\alpha_0,\beta_0)\) in \(W_{F}^{\pi}\) such that \(\Pi_{\alpha_0\beta_0}^*\) does not pass through \(p\) then \(S_{\alpha_0\beta_0}\) is a unit of \(\mathbb{C}\{x,y\}\). It implies that for every \((\alpha,\beta)\) in \(W_{F}^{\pi}\) there exists \(h_{\alpha \beta}\) in \(\mathbb{C}\{x,y\}\) such that

\[\pi^{*}(\mathrm{d}F_{\alpha} \wedge \mathrm{d}F_{\beta})=h_{\alpha \beta} \pi^{*}(\mathrm{d}F_{\alpha_0} \wedge \mathrm{d}F_{\beta_0}).\] For any \((\alpha,\beta)\) in \(\mathbb{P}(\mathbb{C}^k) \times \mathbb{P}(\mathbb{C}^k)\) denote by \(\omega_{\alpha\beta}\) the holomorphic \(2\)-form \(\pi^{*}(\mathrm{d}F_{\alpha} \wedge \mathrm{d}F_{\beta})\). A direct computation shows that for any elements \(t_1,t_2\) of \(\mathbb{C}\) we have the equality \[\omega_{\alpha_0+t_1 \alpha,\beta_0+t_2 \beta}=\omega_{\alpha_{0},\beta_{0}}(1+t_1h_{\alpha,\beta_0}+t_2h_{\alpha_0,\beta}+t_1t_2h_{\alpha,\beta}).\] This last equality implies that for a generic choice of \(t_1, t_2\) the strict transform of the polar curve \(\Pi_{\alpha_0+t_1 \alpha,\beta_0+t_2 \beta}^*\) does not pass through the point \(p\). It follows that \(p\) is not a basepoint for the family \(\Pi_F\).

Assume that \(\Pi_{\alpha \beta}^*\) passes through \(p\) for every \((\alpha,\beta)\in W_F^\pi\), or equivalently that \[S_{\alpha\beta}(0,0)=0,\qquad \forall (\alpha,\beta)\in W_F^\pi.\] Denote by \(I_p\) the ideal of \(\mathbb{C}\{x,y\}\) generated by the finite set \(\{h_{ij}(x,y):=x^{m_v(i,j)-\nu_v(F)}\phi_{ij}(x,y)\}_{i\neq j}.\) Since \(S_{\alpha\beta}(0,0)=0\) for all \((\alpha,\beta)\in W_F^\pi\), it follows that \(h_{ij}(0,0)=0\) for all \(i,j\in\{1,\dots,k\}\). Hence \(I_p\) either contains a power of the maximal ideal of \(\mathbb{C}\{x,y\}\), or is prime i.e, generated by a single element. However, if the ideal \(I_p\) was prime, then for every \((\alpha,\beta)\) in a Zariski open subset, the \(2\)-form \(\pi^*(\mathrm{d}F_\alpha\wedge \mathrm{d}F_\beta)\) would vanish along the zero locus of \(I_p\). It would follow that \(\mathrm{d}f_i\wedge \mathrm{d}f_j\) vanishes along the image of that curve for all \(i,j\in\{1,\dots,k\}\), contradicting the immersion property of the map \(p\longmapsto (f_1(p),\dots,f_k(p)).\) Thus the ideal \(I_p\) is primary, i.e. it contains a power of the maximal ideal.

We now consider the blow-up \(\mathrm{BL}_{I_p}\) of the ideal \(I_p\). By Proposition 33, the modification \(\mathrm{BL}_{I_p}\) has no base points for the family of curves \[\left\{\sum_{i\neq j}\gamma_{ij}h_{ij}=0\right\}_{[\gamma_{ij}]\in\mathbb{P}(\mathbb{C}^{k^2-k})}.\] Let \(O_p\) be the Zariski open subset of \(\mathbb{P}(\mathbb{C}^{k^2-k})\) such that the strict transforms by \(\mathrm{BL}_{I_p}\) of the curves \[\left\{\sum_{i\neq j}\gamma_{ij}h_{ij}=0\right\}_{[\gamma_{ij}]\in O_p}\] do not intersect.

Consider the regular map \[P:\mathbb{P}(\mathbb{C}^k)\times \mathbb{P}(\mathbb{C}^k) \longrightarrow \mathbb{P}(\mathbb{C}^{k^2-k}), \qquad P(\alpha,\beta)=[\alpha_i\beta_j]_{i\neq j},\] and define the Zariski open subset \[V_p=P^{-1}(O_p).\] It follows that the strict transforms of the curves \[\left\{ S_{\alpha\beta} = \sum_{i\neq j}\alpha_i\beta_j h_{ij} = 0 \right\}_{(\alpha,\beta)\in V_p}\] do not intersect. Consequently, \(\mathrm{BL}_{I_p}\) has no base points for the family \[\left\{ \sum_{i\neq j}\alpha_i\beta_j h_{ij}=0 \right\}_{(\alpha,\beta)\in \mathbb{P}(\mathbb{C}^k)\times\mathbb{P}(\mathbb{C}^k)}.\]

We perform the same operation at every smooth base point of the family \(\Pi_F\) in order to obtain the desired good resolution. A similar argument applies when \(p\) is a double point of \(E\). ◻

Proof of Proposition 40 . Let \[V_F^{\pi}:=(U_F^{\pi}\times U_F^{\pi}) \cap W_{F}^{\pi} \subset \mathbb{P}(\mathbb{C}^k)\times \mathbb{P}(\mathbb{C}^k),\] where \(U_F^{\pi},W_{F}^{\pi}\) are Zariski open sets as in Lemma 34. Let \(E_v\) be an irreducible component of \(E\), and let \(p\) be a smooth point of \(E_v\). By Proposition 33 and Definition 39, there exists \((\alpha,\beta)\in V_{F}^{\pi}\) such that \(p \notin F_{\alpha}^* \cup F_{\beta}^* \cup \Pi_{{\alpha \beta}}^*.\) By Proposition 23, we have \[q_{F_{\alpha},v}^{F_{\beta}} = \frac{\mathrm{Ord}_{E_v}\big(\pi^{*} (\mathrm{d} F_{\alpha} \wedge \mathrm{d}F_{\beta})\big) - m_v(F_{\beta}) + 1}{m_v(F_{\beta})}.\]Since \((\alpha,\beta)\in V_{I}^{\pi}\), Lemma 34 implies \[m_v(F_{\alpha}) = m_v(F_{\beta}) = m_v(I) \quad \text{and} \quad \mathrm{Ord}_{E_v}\big(\pi^{*} (\mathrm{d} F_{\alpha} \wedge \mathrm{d}F_{\beta})\big) = \nu_v(F).\]

It follows that the number \(q_{F_{\alpha},v}^{F_{\beta}}\) does not depend on the choice of \((\alpha,\beta)\in V_{I}^{\pi}\), and we obtain \[q_{F_{\alpha},v}^{F_{\beta}} = q_v^{F} := \frac{\nu_v(F)-m_v(F)+1}{m_v(F)}.\] ◻

Proof. We obtain Proposition 56 by repeating the proof of [3], using Lemma 43 in place of [3]. ◻

Proof of Point [compatibilité] to [point333] of Theorem 61. Consider the \(\mathcal{O}_{X,0}\)-module \[Q := \frac{\Omega^1_{X,0}}{(\mathrm{d}f_1,\dots,\mathrm{d}f_k)},\] where \(\Omega^1_{X,0}\) denotes the module of germs of holomorphic \(1\)-forms, and \((\mathrm{d}f_1,\dots,\mathrm{d}f_k)\) is the submodule generated by the \(1\)-forms \(\mathrm{d}f_i\) for \(i \in \{1,\dots,k\}\). By the semi-completeness hypothesis on the generating system \(F\), the module \(Q\) is supported only at the maximal ideal \(\mathfrak{m}\) of \(\mathcal{O}_{X,0}\). Hence, \(Q\) has finite length and the ideal \(I\) is \(\mathfrak{m}\)-primary. Therefore, there exists an integer \(N \in \mathbb{N}\) sufficiently large such that \[\label{allerlesenfoires1223} \mathfrak{m}^N \Omega^1_{X,0} \subset (\mathrm{d}f_1,\dots,\mathrm{d}f_k) \qquad \text{and} \qquad \mathfrak{m}^N \subset I.\tag{2}\]

Let \((\ell_1,\dots,\ell_r)\) be a generating system of the maximal ideal \(\mathfrak{m}\). Consider the following generating system of \(I\): \[G := \left( f_1,\dots,f_k, \ell_1^{N+1},\dots,\ell_r^{N+1}, \ell_1^{N+2},\dots,\ell_r^{N+2} \right).\]

The generating system \(G\) is semi-complete. Indeed, consider the map \[\begin{array}{rcl} G:(X,0) &\longrightarrow& (\mathbb{C}^{k+2r},0) \\ p &\longmapsto& \left( f_1(p),\dots,f_k(p), \ell_1^{N+1}(p),\dots,\ell_r^{N+1}(p), \ell_1^{N+2}(p),\dots,\ell_r^{N+2}(p) \right). \end{array}\] This map is an immersion at every point of \(X \smallsetminus\{0\}\) because the \(1\)-forms \(\{\ell_i^N\,\mathrm{d}\ell_i\}_{1 \leq i \leq r}\) cannot vanish simultaneously along a curve. Furthermore, it follows directly from 2 that the generating system \(G\) satisfies Point [compatibilité]. Let us now prove Points [point111] and [point222].

We first check that \(G\) is injective. Since \((\ell_1,\dots,\ell_r)\) generates the maximal ideal \(\mathfrak{m}\), we have \[(\ell_1(p_1),\dots,\ell_r(p_1)) \neq (\ell_1(p_2),\dots,\ell_r(p_2))\] for all distinct points \(p_1,p_2 \in X\). Hence, \(G\) is injective.

We now prove that \(G\) is a proper modification. Since \((\ell_1,\dots,\ell_r)\) is a generating system of the maximal ideal \(\mathfrak{m}\) of \(\mathcal{O}_{X,0}\), the morphism \[\begin{array}{rcl} \ell:(X,0) &\longrightarrow& (\ell(X),0) \subset (\mathbb{C}^r,0) \\ p &\longmapsto& (\ell_1(p),\ell_2(p),\dots,\ell_r(p)) \end{array}\] is a holomorphic embedding.

Consider now the holomorphic map \[h:\mathbb{C}^{k+2r} \longrightarrow \mathbb{C}^r,\] defined by \[h(z_1,\dots,z_k,z_{1,1},\dots,z_{1,r},z_{2,1},\dots,z_{2,r}) = \left( \frac{z_{2,1}}{z_{1,1}}, \dots, \frac{z_{2,r}}{z_{1,r}} \right).\] Then \[(h \circ G)(p) = (\ell_1(p),\dots,\ell_r(p)) = \ell(p).\] Since \(\ell\) is a biholomorphism onto its image, the map \(\ell^{-1}\circ h\) defines a bimeromorphic inverse of \(G\). Moreover, we already proved that \(G\) is an injective local biholomorphism on \(X \smallsetminus\{0\}\). Hence, \(G\) induces a homeomorphism onto its image, and is therefore proper. By Remmert’s proper mapping theorem, the image \((G(X),0)\) is a complex surface germ. Finally, since \(G\) restricts to a biholomorphism \(X \smallsetminus\{0\} \longrightarrow G(X)\smallsetminus\{0\},\) the germ \((G(X),0)\) has an isolated singularity. With this, we proved that \(G\) is a homeomorphism and a modification of the complex surface germ with isolated singularity \((G(X),0)\).

\(\bullet\) Now let us study the induced map \(\widetilde{G}:\mathrm{NL}(X,0)\longrightarrow \mathrm{NL}(G(X),0)\) on the non-archimedean links. Since \(G\) is a modification, the comorphism \[G^*:\mathrm{Frac}(\mathcal{O}_{G(X),G(p)})\longrightarrow \mathrm{Frac}(\mathcal{O}_{X,p})\] is an isomorphism. Therefore, by Proposition 54, the map \(\widetilde{G}\) is one-to-one and continuous.

Let us prove that \[\mathcal{I}_{\mathfrak{m}_{G}}\circ \widetilde{G}=\mathcal{I}_{F}.\] We first show that \(\mathcal{I}_{\mathfrak{m}_{G}}\circ \widetilde{G}=\mathcal{I}_{G}.\) Let \(val\) be a divisorial valuation of \(\mathrm{NL}(X,0)\), and let \(\pi:(X_{\pi},E)\longrightarrow (X,0)\) be a good resolution such that there exists an irreducible component \(E_v\subset E\) whose associated divisorial valuation is \(val\). The image of \(val\) under \(\widetilde{G}\) is the valuation \[\widetilde{G}(val)(h) = \frac{m_v(h\circ G\circ \pi)}{m_v(\mathfrak m)} \cdot \frac{1}{val(G^*\mathfrak m_G)} = \frac{m_v(h\circ G\circ \pi)}{m_v(I)}, \qquad h\in \mathcal{O}_{G(X),0},\] because \[val(G^*\mathfrak m_G) = val(I) = \frac{m_v(I)}{m_v(\mathfrak m)}.\]On the other hand, since \(G\) is a modification, the map \[\pi_G:=G\circ \pi:(X_{\pi},E)\longrightarrow (G(X),0)\] is a good resolution of \((G(X),0)\). Denote by \(val_G\) the divisorial valuation of \(\mathrm{NL}(G(X),0)\) associated with the irreducible component \(E_v\). For every \(h\in \mathcal{O}_{G(X),0}\), we have \[\label{13003} val_G(h) = \frac{m_v(h\circ G\circ \pi)}{m_v(\mathfrak m_G)} = \widetilde{G}(val)(h),\tag{3}\] because \[\label{13000} m_v(\mathfrak m_G) = \inf \left\{ \operatorname{Ord}_{E_v}(h\circ G\circ \pi) \;\middle|\; h\in \mathfrak m_G \right\} = m_v(I).\tag{4}\]

Furthermore, \[\label{13002} \nu_v(\mathfrak m_G) = \inf \left\{ \operatorname{Ord}_{E_v}\bigl(\pi_G^*(dg\wedge df)\bigr) \;\middle|\; g,f\in \mathfrak m_G \right\} = \nu_v(G),\tag{5}\] because \(G^*\mathfrak m_G=I\).

It follows from 3 , 4 , and 5 that \[\mathcal{I}_{\mathfrak m_G}\circ \widetilde{G}(val) = \mathcal{I}_{\mathfrak m_G}(val_G) = q_v^{\mathfrak m_G} = \frac{\nu_v(\mathfrak m_G)-m_v(\mathfrak m_G)+1}{m_v(\mathfrak m_G)} = \mathcal{I}_{G}(val),\] since \[\nu_v(\mathfrak m_G)=\nu_v(G) \qquad\text{and}\qquad m_v(\mathfrak m_G)=m_v(G).\]We now prove that \[\mathcal{I}_{G}=\mathcal{I}_{F}.\] Indeed, for every divisorial valuation \(v\) of \(\mathrm{NL}(X,0)\), \[m_v(G)=m_v(F)=m_v(I),\] because \(F\) and \(G\) generate the same ideal \(I\). Moreover, \[\nu_v(G)=\nu_v(F),\] and this follows directly from Point [compatibilité]. Indeed, for all \(i,j\in \{1,\dots,r\}\), we have \[df_i\wedge df_j = \left(\sum_{s=1}^k \lambda_{s,i}\,df_s\right) \wedge \left(\sum_{t=1}^k \lambda_{t,j}\,df_t\right) = \sum_{1\le s<t\le k} \left( \lambda_{s,i}\lambda_{t,j} - \lambda_{t,i}\lambda_{s,j} \right) df_s\wedge df_t,\] which implies the desired equality.

Finally, we conclude that \[\mathcal{I}_{\mathfrak m_G}\circ \widetilde{G} = \mathcal{I}_{F}.\]

It remains to prove that \(\widetilde{G}:(\mathrm{NL}(X,0),\mathrm{d}_{I}) \longrightarrow (\mathrm{NL}(F(X),0),\mathrm{d}_{\mathfrak{m}_G})\) is an isometry. Let \(v\) and \(v'\) be two adjacent vertices of \(\Gamma_{\pi}\). From what we done before, the image of the edge \(e_{v,v'}\) by \(\widetilde{G}\) is again the edge \(e_{v,v'}\) and we have:

\[\mathrm{d}_{I}(v,v')=\frac{1}{m_v(I)m_{v'}(I)}=\frac{1}{m_v(\mathfrak{m}_G)m_{v'}(\mathfrak{m}_G)}=\mathrm{d}_{\mathfrak{m}_G}(v,v').\] It follows that \(\widetilde{G}\) is an isometry on the divisorial valuations and it follows by their density that it is an isometry on \(\mathrm{NL}(X,0)\) for the specified distance.It remains to prove Point [point444]. We will do so after introducing a few additional tools. ◻

Proof of Corollary 68. First, let us prove that the inner rate function \(\mathcal{I}_{\mathfrak{m}}\) determines the value \(m_v(\mathfrak{m})\) at every divisorial point \(v\) of \(\mathrm{NL}(X,0)\). Let \(v\) be a divisorial valuation and \(\sigma:(X_{\sigma},E_{\sigma}) \longrightarrow (X,0)\) be a good resolution which factors through the blowup of \(\mathfrak{m}\) and the minimal good resolution of \(\Pi_{\mathfrak{m}}\) (See Definition 39 to recall the notation) such that \(E_{\sigma}\) contains the irreducible component \(E_v\) associated to the valuation \(v\). Let \(p\) be a smooth point of \(E_v\) and \(E_w\) be the irreducible component that arises from the blowup of center \(p\). It follows directly from Lemma 43 that \[m_v(\mathfrak{m})=\frac{1}{q_w^{\mathfrak{m}} - q_v^{\mathfrak{m}}}.\]

Now, consider the minimal good resolution \(\pi:(X_{\pi},E) \longrightarrow (X,0)\). By Theorem 9 the topological type of \((X,0)\) determines the dual graph \(\Gamma_{\pi}\). Hence it determines all the \(\mathcal{T}\)-nodes. Now, we need to determine the \(\mathcal{L}\)-nodes and the special \(\mathcal{P}\)-nodes. For this, we need to prove that we can determine the dual graph of the minimal good resolution of \((X,0)\) which factors through the blowup of the maximal ideal of \(\mathcal{O}_{X,0}\) and the Nash transform of \((X,0)\). Indeed, let \(\Gamma\) be the smallest connected subset of \(\mathrm{NL}(X,0)\) such that:

1. The set \(\Gamma\) contains the vertices of \(\Gamma_{\pi}\).

2. Let \(v\) be a divisorial valuation of \(\Gamma\) and denote by \(E_v\) the associated irreducible component obtained by a good resolution of \((X,0)\). Let \(E_w\) be the irreducible component obtained from blowing up a point of \(E_v\). One of the following conditions must be verified:

   1. The semivaluation \(w\) associated to \(E_w\) is in \(\Gamma\).

   2. \(q_w^\mathfrak{m}=q_v^\mathfrak{m}+\frac{1}{m_v(\mathfrak{m})}\).

   3. \(q_w^\mathfrak{m}=\frac{m_v(\mathfrak{m})q_v^\mathfrak{m}+m_{v'}(\mathfrak{m})q_{v'}^\mathfrak{m}}{m_v(\mathfrak{m})+m_{v'}(\mathfrak{m})}\) such that \(v'\) is an element of \(\Gamma\).

Let \(\pi_0\) be the minimal good resolution of \((X,0)\) such that \(V(\Gamma_{\pi_0})=\Gamma\). It follows from Lemma 26 that \(\pi_{0}\) has no basepoints for the family of the hyperplane sections and the family of the polar curves associated to the linear projections on \(\mathbb{C}^2\). By Proposition 33 and Theorem 64 we get that \(\pi_{0}\) factors through the blowup of the maximal ideal of \(\mathcal{O}_{X,0}\) and the Nash transform of \((X,0)\). We can then determine the \(\mathcal{L}\)-nodes directly from this dual graph and the given inner rate function, whereas the special \(\mathcal{P}\)-nodes are obtained by applying the inner rates formula of Theorem 24, which allows us to locate the generic polar curve of the maximal ideal. The locus of this curve coincides with the exceptional locus of the Nash transform thanks to Remark 70. ◻

Proof. By definition, the modification \(\pi\) has no base points for the families of curves \(L_F\) and \(\Pi_F\) associated with \(F\). On the other hand, the family of linear forms \(P=\{p_i(z_1,\dots,z_n)=z_i\}_{1 \leq i \leq k}\) generates the ideal \(\mathfrak{m}_{F(X)}\). Obviously, the family \(\Pi_P\) coincides with the polar curves associated with linear projections onto \(\mathbb{C}^2\), and we have \(P^*:=(p_1 \circ F = f_1, \dots, p_k \circ F = f_k)=F\).

By definition of the blow-up of \(I\) and the minimal good resolution of of \(\Pi_F\), the good resolution \(\pi: (X_{\pi},E)\longrightarrow (X,0)\) has no base points for the families \(L_{P^*}=L_F\) and \(\Pi_{P^*}=\Pi_F\). It follows that the composition \(F \circ \pi\) has no base points for the families \(L_P\) and \(\Pi_P\).

Thus, it follows from Theorem 64 and Proposition 33 that \(F \circ \pi\) factors through the Nash transform of \((F(X),0)\) and the blow-up of the maximal ideal \(\mathfrak{m}_{F(X),0}\) of \(\mathcal{O}_{F(X),0}\). This proves Point [merde1] of Proposition 72.

Points [merde2]–[merde4] are direct consequences of Point [merde1] of the current statement and Point [point333] of Theorem 61. ◻

Proof of Point [point444] of Theorem 61. It is a direct consequence of Proposition 72 and Proposition [classificationBNP]. ◻

Proof. We have the following commutative diagram:

\[\xymatrix@C=6em@R=5em{ \bigl((\overline{X},0), g_G\bigr) \ar[r]^{\raisebox{0.5ex}{G}} \ar[d]_{\mathrm{Id}} & (X_G,0) \ar@{^{(}->}[r] \ar[d]^{P_k} & (\mathbb{C}^{k+r},0) \\ \bigl((\overline{X},0), g_F\bigr) \ar[r]^{\raisebox{0.5ex}{n}} & (X,0) \ar@{^{(}->}[r] & (\mathbb{C}^k,0). }\]

Recall that the map \(G\) is a homeomorphism and is the normalization of the complex surface germ \((X_G,0)\) (see Definition 63) and here \(P_k\) denotes the projection onto the first \(k\) factors.

Since the map \(G\) is a homeomorphic isometry we just have to show that the identity map \(\mathrm{Id}\) is a bilipschitz homeomorphism for the distances induced by \(g_F\) and \(g_G\). Given any point \(p \in \overline{X} \smallsetminus\{0\}\) and any tangent vector \(u \in T_p \overline{X}\), we have \[\begin{align} \|u\|_{g_G} &= \bigl\| \mathrm{d}_p G(u)\bigr\|_{\mathbb{C}^{k+r}} \\ &= \Bigl\|\bigl(\mathrm{d}_p f_1(u), \dots, \mathrm{d}_p f_k(u), \mathrm{d}_p g_1(u), \dots, \mathrm{d}_p g_r(u)\bigr)\Bigr\|_{\mathbb{C}^{k+r}} \\ &\ge \Bigl\|\bigl(\mathrm{d}_p f_1(u), \dots, \mathrm{d}_p f_k(u)\bigr)\Bigr\|_{\mathbb{C}^{k}} \\ &= \bigl\|\mathrm{d}_p (P_k \circ G)(u)\bigr\|_{\mathbb{C}^{k}} \\ &\ge \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} \\ &= \|u\|_{g_F}. \end{align}\]

On the other hand, since \(G=(f_1,\dots,f_k,g_1,\dots,g_r)\) is a completion of \(F\), there exist holomorphic functions \(\{\lambda_{i,j} \in \mathcal{O}_{\overline{X},0} \}_{1 \le i \le r, 1 \le j \le k}\) such that for all \(i=1,\dots,r\), we have \(\mathrm{d}_p g_i = \sum_{j=1}^k \lambda_{i,j}(p) \, \mathrm{d}_p f_j.\)

Let \(K \in \mathbb{R}_+\) denote the maximum of \(\{|\lambda_{i,j}|\}_{1 \le i \le r, 1 \le j \le k}\) in a small neighborhood of \(0\) in \(\overline{X}\). Then we have \[\begin{align} \|u\|_{g_G} &= \bigl\|\mathrm{d}_p G(u)\bigr\|_{\mathbb{C}^{k+r}} \\ &= \Bigl\|\bigl(\mathrm{d}_p f_1(u), \dots, \mathrm{d}_p f_k(u), 0, \dots, 0\bigr) + \bigl(0, \dots, 0, \mathrm{d}_p g_1(u), \dots, \mathrm{d}_p g_r(u)\bigr)\Bigr\|_{\mathbb{C}^{k+r}} \\ &\le \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} + \Bigl\| \bigl(\mathrm{d}_p g_1(u), \dots, \mathrm{d}_p g_r(u)\bigr)\Bigr\|_{\mathbb{C}^r} \\ &= \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} + \Bigl\| \Bigl(\sum_{j=1}^k \lambda_{1,j}(p) \mathrm{d}_p f_j(u), \dots, \sum_{j=1}^k \lambda_{r,j}(p) \mathrm{d}_p f_j(u)\Bigr)\Bigr\|_{\mathbb{C}^r} \\ &= \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} +\left(\sum_{i=1}^r \left|\sum_{j=1}^k \lambda_{i,j}(p)\,\mathrm{d}_p f_j(u)\right|^2 \right)^{1/2} \\ &\le \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} +\left(\sum_{i=1}^r \left(\sum_{j=1}^k |\lambda_{i,j(p)}|\,|\mathrm{d}_p f_j(u)|\right)^2 \right)^{1/2} \\ &\le \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} +\left(\sum_{i=1}^r \left(K \sum_{j=1}^k |\mathrm{d}_p f_j(u)|\right)^2 \right)^{1/2} \\ &= \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} + \sqrt{r}\,K \sum_{j=1}^k |\mathrm{d}_p f_j(u)| \\ &\le \bigl\|\mathrm{d}_p n(u)\bigr\|_{\mathbb{C}^{k}} +\sqrt{rk}\,K\, \bigl\|(\mathrm{d}_p f_1(u),\dots,\mathrm{d}_p f_k(u))\bigr\|_{\mathbb{C}^k}.\\ &\le \left( \sqrt{rk}K+1\right) \|u\|_{g_F} \end{align}\]

Combining these two inequalities shows that the identity map \(\mathrm{Id} : (\overline{X}, g_G) \longrightarrow (\overline{X}, g_F)\) is bilipschitz. ◻

Proof. Let us first prove that the map \[\begin{array}{ccc} \widetilde{n} : (\overline{X},0) & \longrightarrow & (\widetilde{X},0)\subset(\mathbb{C}^{2k},0) \\ p & \longmapsto & (f_1(p),\ldots,f_k(p), \psi(p)f_1(p), \dots, \psi(p)f_k(p)) \end{array}\] is an immersion. We have \[\mathrm{d}_p\widetilde{n}(v) = \bigl( \mathrm{d}_pf_1(v),\ldots,\mathrm{d}_pf_k(v), \psi(p)\mathrm{d}_pf_1(v)+f_1(p)\mathrm{d}_p\psi(v),\ldots, \psi(p)\mathrm{d}_pf_k(v)+f_k(p)\mathrm{d}_p\psi(v) \bigr).\]

The only curve along which \(\mathrm{d}\widetilde{n}\) might fail to be injective is \(\overline{\mathrm{Sing}(X)}\), because the kernel of \(\mathrm{d}n=(\mathrm{d}f_1,\ldots,\mathrm{d}f_k)\) is trivial along any other curve of \((\overline{X},0)\). Also, \(\mathrm{d}n\) does not vanish identically along an irreducible component \((C,0)\) of \(\overline{\mathrm{Sing}(X)}\), that is, for all irreducible components \((C,0)\) of \(\overline{\mathrm{Sing}(X)}\) we have \(\mathrm{d}_pn(v)\neq 0\) for \(p\in C\) close enough to \(0\) and \(v\) tangent to \(C\) at \(p\). Otherwise, \(n\) would contract such a curve to a point, contradicting the finiteness property of the normalization.

Now let us go back to \(\widetilde{n}\). Let \((C,0)\) be an irreducible component of \(\overline{\mathrm{Sing}(X)}\) along which \(n\) fails to be an immersion. That is, there exists a vector field \(V\) along the curve \(C\) such that \[\mathrm{d}_p n(V(p))=0 \quad \text{for all } p\in C.\] As we saw earlier, \(V\) cannot be tangent to \(C\), which implies that \(\mathrm{d}_p\psi(V(p))\neq 0\) for \(p\) in a small enough neighbourhood of \(0\) in \(\overline{X}\). Moreover, since \(\psi\) vanishes identically on \(C\) by definition, it follows that \[\mathrm{d}_p\widetilde{n}(V(p)) = (0,\ldots,0,f_1(p)\mathrm{d}_p\psi(V(p)),\ldots,f_k(p)\mathrm{d}_p\psi(V(p))).\]

Since the functions \(f_1,\ldots,f_k\) cannot vanish simultaneously along \(C\), it follows that \(\widetilde{n}\) is an immersion outside the singular locus of \(\overline{X}\).

Furthermore, the map \(\widetilde{n}\) is finite and proper, since \(n\) is a normalization and hence finite and proper. It follows that \(\widetilde{n}\) is the normalization of the complex surface germ \((\widetilde{X},0)\). This proves Points [pointA] and [pointB] of Proposition 79.

It remains to prove Point [pointC], namely that the map \[\nu : (\widetilde{X},0) \subset (\mathbb{C}^{2k},0) \longrightarrow (X,0) \subset (\mathbb{C}^{k},0)\] is a bi-Lipschitz homeomorphism for the inner metric. The map \(\nu\) is clearly a homeomorphism: its inverse is given by \[\begin{array}{ccc} \nu^{-1} : (X,0) \subset (\mathbb{C}^k,0) & \longrightarrow & (\widetilde{X},0) \subset (\mathbb{C}^{2k},0) \\ z & \longmapsto & \big(z,\, \psi \circ n^{-1}(z) \cdot z \big) \end{array}\] This is well-defined and continuous because \(\psi\) vanishes on \(\overline{\mathrm{Sing}(X)}\). Indeed, \(\psi \circ n^{-1}\) vanishes along \(\mathrm{Sing}(X)\) and is holomorphic on \(\widetilde{X} \smallsetminus\mathrm{Sing}(X)\). It remains to prove that \(\nu\) is a bi-Lipschitz homeomorphism.

Before proving that it is a bilipschitz homeomorphism, let us remark that if \((X,0)\) is non-pinched, then the map \(n\) is a local biholomorphism on \(\overline{X} \smallsetminus\{0\}\). It follows that the meromorphic map \(\psi \circ n^{-1}\) has no poles on \(\widetilde{X} \smallsetminus\{0\}\). Since the set of poles of a meromorphic function on a complex surface has dimension \(1\), this implies that \(\psi \circ n^{-1}\) is, in fact, holomorphic. Consequently, the map \(\nu^{-1}\) is holomorphic, and therefore \(\nu\) is a biholomorphism.

In order to prove that the map \(\nu\) is bilipschitz we consider the following diagram:

\[\xymatrix@C=6em@R=5em{ \bigl((\overline{X},0), g_F\bigr) \ar[r]^{\widetilde{n}} \ar[d]_{\mathrm{Id}} & (\widetilde{X},0) \ar@{^{(}->}[r] \ar[d]^{\nu} & (\mathbb{C}^{2k},0) \\ \bigl((\overline{X},0), g\bigr) \ar[r]^{n} & (X,0) \ar@{^{(}->}[r] & (\mathbb{C}^k,0) }\]

where \(g\) and \(g_F\) are respectively the pulled-back Euclidean metrics by \(n\) and \(\widetilde{n}\). Let us prove that the identity map between \(\bigl((\overline{X},0), g_F\bigr)\) and \(\bigl((\overline{X},0), g\bigr)\) is bilipschitz (for the arc-length distances induced by \(g_F\) and \(g\) respectively). Let \(\gamma:\left((-\epsilon,\epsilon),0\right)\longrightarrow (\overline{X},0)\) be a germ of rectifiable curve. It is obvious that

\[\|\gamma'(t)\|_{g_F} = \|\mathrm{d}_{\gamma(t)}\widetilde{n}(\gamma'(t))\|_{\mathbb{C}^{2k}} \ge \|\mathrm{d}_{\gamma(t)}n(\gamma'(t))\|_{\mathbb{C}^{k}} = \|\gamma'(t)\|_{g}.\]

which proves that the map \(\mathrm{Id}:\left( (\overline{X},0), g\right) \longrightarrow \left( (\overline{X},0), g_F \right)\) is Lipschitz. Now, let us prove that \(\mathrm{Id}:\left( (\overline{X},0), g_F\right) \longrightarrow \left( (\overline{X},0), g \right)\) is also Lipschitz:

\[\begin{align} \|\gamma'(t)\|_{g_F} &= \bigl\|\mathrm{d}_{\gamma(t)} \widetilde{n}\left(\gamma'(t)\right)\bigr\|_{\mathbb{C}^{2k}} \\ &= \Bigl\| \left(\mathrm{d}_{\gamma(t)} n(\gamma'(t)),0\right) + \psi(\gamma(t))\left(0,\mathrm{d}_{\gamma(t)} n(\gamma'(t))\right) + \mathrm{d}_{\gamma(t)}\psi(\gamma'(t)) \left( 0,n(\gamma(t)) \right) \Bigr\|_{\mathbb{C}^{2k}} \\ &\leq \|\mathrm{d}_{\gamma(t)} n(\gamma'(t))\|_{\mathbb{C}^{k}}+ |\psi(\gamma(t))|\cdot \|\mathrm{d}_{\gamma(t)} n(\gamma'(t))\|_{\mathbb{C}^{k}} + |\mathrm{d}_{\gamma(t)}\psi(\gamma'(t))|\cdot \|n(\gamma(t))\|_{\mathbb{C}^{k}} \\ &\leq \|\gamma'(t)\|_{g} + |\psi(\gamma(t))|\cdot \|\gamma'(t)\|_{g} + |\mathrm{d}_{\gamma(t)}\psi(\gamma'(t))|\cdot \|n(\gamma(t))\|_{\mathbb{C}^{k}}. \end{align}\] Without loss of generality we can assume that \((\overline{X},0)\) is embedded in some \(\mathbb{C}^N\) and we denote by \(||\cdot||\) the norm of any tangent vector to \(\overline{X}\). Denote by \(K\) the maximum of \(|\psi(p)|\) and by \(C\) the maximum of \(|\mathrm{d}_p\psi(v)|\) when \(p\) lies in a small enough neighborhood of \(0\) in \(\overline{X}\) and \(v\) lies in the unit tangent bundle of \(\overline{X}\). We then have

\[\begin{align} \|\gamma'(t)\|_{g_F} &\leq \|\gamma'(t)\|_{g} + K\|\gamma'(t)\|_{g} + ||\gamma'(t)||\cdot \left|\mathrm{d}_{\gamma(t)} \psi\left(\frac{\gamma'(t)}{||\gamma'(t)||}\right) \right|\cdot \|n(\gamma(t))\|_{\mathbb{C}^{k}}.\\ &\leq \|\gamma'(t)\|_{g} ( 1+K ) + C||\gamma'(t)|| \,\|n(\gamma(t))\|_{\mathbb{C}^k} \\ &= \|\gamma'(t)\|_{g} \left( 1+K + C\,\frac{\|n(\gamma(t))\|_{\mathbb{C}^k}}{\|\mathrm{d}_{\gamma(t)} n \left(\frac{\gamma'(t)}{||\gamma'(t)||}\right)\|_{\mathbb{C}^{k}}} \right). \end{align}\] And when \(t\) is small enough the quantity \[\frac{\|n(\gamma(t))\|_{\mathbb{C}^k}}{\|\mathrm{d}_{\gamma(t)} n \left(\frac{\gamma'(t)}{||\gamma'(t)||}\right)\|_{\mathbb{C}^{k}}}\] is smaller then \(1\) thus \[\|\gamma'(t)\|_{g_F} \leq \|\gamma'(t)\|_{g}(1+K+C).\] We proved that \(\mathrm{Id}:\left( (\overline{X},0), g\right) \longrightarrow \left( (\overline{X},0), g_F \right)\) is a bilipschitz map and that concludes the proof of this Proposition. ◻

Proof. Let \(G=(f_1,\dots,f_k)\) be a completion of the generating system \(F\), and consider the normalization map \[\begin{array}{ccc} G : (\overline{X},0) & \longrightarrow & (X_G,0) \subset (\mathbb{C}^{k},0) \\ p & \longmapsto & (f_1(p), \dots, f_k(p)). \end{array}\]

It follows from Point [merde1] of Proposition 72 that the map \(G \circ \pi\) factors through the blowup of the maximal ideal of \(\mathcal{O}_{X_G,0}\) and the Nash transform of \((X_G,0)\). Furthermore, when \((\overline{X},0)\) is equipped with the metric \(g_F\), the map \(G\) is a bilipschitz homeomorphism by Proposition 77. Hence, the inverse images of the \(B\)-pieces and \(A\)-pieces of \(X_G\) are respectively \(B\)-pieces and \(A\)-pieces with the same topology and the same rates.

Finally, thanks to Points [merde2]–[merde4] of Proposition 72 and Proposition 85, we obtain that the rational numbers defining the \(B\)-pieces and \(A\)-pieces are the inner rates associated with \(G\), which coincide, by Point [point444] of Theorem 61, with the inner rates associated with \(F\). ◻

Proof. Let us first consider the following commutative diagram following Notation 91: \[\xymatrix@C=6em@R=5em{ \bigl((\overline{X},0), d_F\bigr) \ar[r]^{\raisebox{0.5ex}{n}} \ar[d]_{\pi_n} & \bigl((X,0), d_i \bigr) \subset (\mathbb{C}^k,0) \\ \bigl((\overline{X}/\!\sim_n, 0), \overline{d_F} \bigr) \ar[ru]^{\raisebox{0.5ex}{\overline{n}}} & }\]

where \(d_i\) is the arclength distance induced by \(\mathbb{C}^k\), \(d_F\) is the distance induced by the Riemannian metric \(g_n\), defined as the pullback of the Euclidean metric of \(\mathbb{C}^k\) via \(n\) (which is well-defined since \(n\) is an immersion), and \(\overline{d_n}\) is the metric completion of the distance induced by the push-forward of \(g_n\) via \(\pi_n\).

Moreover, we define \[\begin{array}{ccc} \overline{n}:\left( \bigl(\overline{X}/\!\sim_n,\,0\bigr), \overline{d_n} \right)& \longrightarrow & \bigl((X,0), d_i \bigr) \\ p & \longmapsto & n(p), \end{array}\]

Let us prove that \(\overline{n}\) is an isometry. First, notice that by construction it is a homeomorphism, sending the singular locus of \(\bigl(\overline{X}/\!\sim_n,\,0\bigr)\) homeomorphically onto the singular locus of \((X,0)\). It remains to prove that it preserves distances.

We call a singular arc any arc passing through the singular locus of \(\bigl(\overline{X}/\!\sim_n,\,0\bigr)\) or \((X,0)\). By construction, the length of any non-singular arc \(\gamma\) is equal to the length of its image \(\overline{n}(\gamma)\). The same holds for singular arcs since both \(\mathrm{Sing}(\overline{X}/\!\sim_n)\) and \(\mathrm{Sing}(X)\) have real codimension greater than \(2\), so any path in these metric spaces can be approximated arbitrarily closely (in length) by a path lying entirely in the smooth locus. Hence, \(\overline{n}\) is an isometry.

It remains to prove that \(\overline{n}\) is a bilipschitz homeomorphism when \(\bigl(\overline{X}/\!\sim_n,\,0\bigr)\) is endowed with the distance \(\overline{d_F}\) as described in Notation 91. Consider the following commutative diagram: \[\xymatrix@C=6em@R=5em{ \bigl((\overline{X},0), d_F\bigr) \ar[r]^{\raisebox{0.5ex}{\mathrm{Id}_{\overline{X}}}} \ar[d]_{\pi_n} & \bigl((\overline{X},0), d_n\bigr) \ar[r]^{\raisebox{0.5ex}{n}} \ar[d]_{\pi_n} & \bigl((X,0), d_i \bigr) \subset (\mathbb{C}^k,0) \\ \bigl( (\overline{X}/\!\sim_n, 0), \overline{d_F}\bigr) \ar[r]^{\raisebox{0.5ex}{\mathrm{Id}_{\overline{X}/\!\sim_n}}} & \bigl( (\overline{X}/\!\sim_n, 0), \overline{d_n}\bigr) \ar[ru]^{\raisebox{0.5ex}{\overline{n}}} & }\]

All that remains is to show that \(\mathrm{Id}_{\overline{X}/\!\sim_n}\) is a bilipschitz homeomorphism. This follows from the fact that \(\mathrm{Id}_{\overline{X}} : \bigl((\overline{X},0), d_F\bigr) \longrightarrow \bigl((\overline{X},0), d_n\bigr)\) is a bilipschitz homeomorphism, as proved in Proposition 77. ◻

Proof. Thanks to Proposition 79, we may assume, without loss of generality, that all complex surface germs considered throughout this proof are non-pinched.

We first prove the invariance of the data. Let \((X,0)\subset (\mathbb{C}^k,0)\) and \((Y,0)\subset (\mathbb{C}^m,0)\) be two complex surface germs at the origin. Assume that there exists an inner bilipschitz homeomorphism \(h:(X,0)\longrightarrow (Y,0).\)

Consider their normalizations \[\begin{array}{ccc} n_X:(\overline{X},0) & \longrightarrow & (X,0), \subset (\mathbb{C}^k,0) \\ p & \longmapsto & (f_1(p),f_2(p),\dots,f_k(p)) \end{array} \;\;\begin{array}{ccc} n_Y:(\overline{Y},0) & \longrightarrow & (Y,0), \subset (\mathbb{C}^m,0) \\ p & \longmapsto & (g_1(p),g_2(p),\dots,g_m(p)) \end{array}\]

Set \(F=(f_1,\ldots,f_k)\) and \(G=(g_1,\ldots,g_m)\). We use Adkins result in [24] for the existence of a homeomorphism \(\overline{h}:(\overline{X},0) \longrightarrow (\overline{Y},0)\) such that the following diagram commute

\[\xymatrix@C=6em@R=5em{ (\overline{X},0) \ar[r]^{n_X} \ar[d]_{\overline{h}} & ({X},0) \ar@{^{(}->}[r] \ar[d]^{h} & (\mathbb{C}^{k},0) \\ (\overline{Y},0) \ar[r]^{n_Y} & (Y,0) \ar@{^{(}->}[r] & (\mathbb{C}^m,0). }\]

Since \((X,0)\) and \((Y,0)\) are non-pinched, the normalization maps \(n_X\) and \(n_Y\) are local biholomorphisms along every irreducible curve germ. Consequently, the image of a curve germ under \(n_X\) (respectively \(n_Y\)) is contained in the singular locus of \((X,0)\) (respectively \((Y,0)\)) if and only if the restriction of \(n_X\) (respectively \(n_Y\)) to that curve has degree strictly greater than \(1\). It follows that \(\overline{h}\) sends each irreducible component \(S_X\) of \(\overline{\mathrm{Sing}(X)}\) onto an irreducible component \(S_Y\) of \(\overline{\mathrm{Sing}(Y)}\), and that the degree of \({n_X}_{|S_X}\) is equal to the degree of \({n_Y}_{|S_Y}\). Therefore, the data described in Point [letypetopologiquedulieusingulier] coincide for \((X,0)\) and \((Y,0)\). Furthermore, since \(h\) is a bilipschitz homeomorphism, the map \[\overline{h}:((\overline{X},0),g_F)\longrightarrow ((\overline{Y},0),g_G)\] is also a bilipschitz homeomorphism. Therefore, by Theorem 90, the data appearing in Points [lepoint2dudernier], [lepoint3dudernier], [lepoint4dudernier] and [lepoint5dudernier] also coincide for \((X,0)\) and \((Y,0)\).
Conversely, assume that the data in Points [letypetopologiquedulieusingulier]–[lepoint5dudernier] coincide for \((X,0)\) and \((Y,0)\). By Theorem 90, there exists a germ of topological space \((T,0)\), which is the cone over a smooth manifold, together with a Riemannian metric \(g\) on \((T,0)\) such that both metric spaces \(((\overline{X},0),g_F)\) and \(((\overline{Y},0),g_G)\) are bilipschitz homeomorphic to \(((T,0),g)\). We denote by \(h_X:((\overline{X},0),g_F)\to((T,0),g)\) and \(h_Y:((\overline{Y},0),g_G)\to((T,0),g)\) the corresponding bilipschitz homeomorphisms. Let \(S_1^X,\dots,S_\ell^X\) (respectively \(S_1^Y,\dots,S_\ell^Y\)) be the irreducible components of \(\overline{\mathrm{Sing}(X)}\) (respectively \(\overline{\mathrm{Sing}(Y)}\)). Define \(d_X=(d_{X,1},\dots,d_{X,\ell})\) and \(d_Y=(d_{Y,1},\dots,d_{Y,\ell})\), where \(d_{X,i}\) (respectively \(d_{Y,i}\)) is the degree of the restriction of \(n_X\) to \(S_i^X\) (respectively of \(n_Y\) to \(S_i^Y\)).

We order the components so that \(S_i^X\) corresponds to \(S_i^Y\) up to topological equivalence. This is possible since, by Point [letypetopologiquedulieusingulier], the curves \(\overline{\mathrm{Sing}(X)}\) and \(\overline{\mathrm{Sing}(Y)}\) are topologically equivalent. Under that same assumption, we also have \(d_X=d_Y=:d\).

Finally, by Theorem 19, we obtain that \(\left((\overline{X}/\sim_n,0),\overline{g_F}\right)\) is bilipschitz homeomorphic to \[\left(\frac{T}{\mathrm{curl}\left(h_X\left(\overline{\mathrm{Sing}(X)}\right),d\right)},\overline{g}\right),\] where \(\overline{g}\) is the metric induced from \(g\) by the \(d\)-curling of \((T,0)\) along \(h_X(\overline{\mathrm{Sing}(X)})\).

Similarly, \(\left((\overline{Y}/\sim_n,0),\overline{g_G}\right)\) is bilipschitz homeomorphic to \[\left(\frac{T}{\mathrm{curl}\left(h_Y\left(\overline{\mathrm{Sing}(Y)}\right),d\right)},\overline{g}\right).\]

Since \(h_X(\overline{\mathrm{Sing}(X)})\) and \(h_Y(\overline{\mathrm{Sing}(Y)})\) are topologically equivalent, the resulting spaces are bilipschitz homeomorphic. Hence, \(\left((\overline{X}/\sim_n,0),\overline{g_F}\right)\) and \(\left((\overline{Y}/\sim_n,0),\overline{g_G}\right)\) are bilipschitz homeomorphic. Finally, by Proposition 92, we conclude that \((X,0)\) and \((Y,0)\) are bilipschitz homeomorphic. ◻