We set $J=\{0,1,\ldots,n-r\}$ such that $Z=\cap_{j\in J}D_{j}$. Since for every irreducible component $D$ of $\mathscr{X}_{k}$, the intersection $D\cap Z$ is connected by assumption, this allows us to denote by $D_{l}$ with $l\in L$ the components of $\mathscr{X}_{k}$ intersecting $Z$ transversally along $Z_{l}\coloneqq Z\cap D_{l}$, so that the toric boundary of $Z$ is given by $\Delta_{Z}=\sum_{l\in L}Z_{l}$.

We denote by $\Sigma\subset N_{\mathbb{R}}$ the fan of $Z$. Its rays are given by $\mathbb{R}_{\geqslant 0}u_{l}$ for $l\in L$, with primitive generators $u_{l}$; the maximal cones of $\Sigma$ are in bijection with the set of unordered $r$-tuples $\{i_{1},\ldots,i_{r}\}\in L^{r}$ such that $\cap_{\beta=1}^{r}D_{i_{\beta}}\cap Z\neq\varnothing$. For a maximal cone $\sigma$ of $\Sigma$, we write $L_{\sigma}\coloneqq\{l\in L\,|\,u_{l}\in\sigma\}$.

Let $\mathcal{N}\coloneqq\nu_{Z/\mathscr{X}}\xrightarrow{p}Z$ be the normal bundle of $Z$ in $\mathscr{X}$, and denote by $Z\subset\mathcal{N}$ the zero section. We write $\mathcal{O}_{\mathscr{X}}(D_{j})_{|Z}=\mathcal{O}_{Z}(F_{j})$ so that $\mathcal{N}=\oplus_{j\in J}\mathcal{O}_{Z}(F_{j})$. Since any Cartier divisor on $Z$ is linearly equivalent to a toric one, for any $j=1,\ldots,n-r$, there exist integers $\lambda_{j,l}$ such that

(2.1.4)

$$\mathcal{O}_{Z}(F_{j})=\mathcal{O}_{Z}\big(-\sum_{l\in L}\lambda_{j,l}Z_{l}\big).$$

For $j=0$ we set $\lambda_{0,l}\coloneqq 1-\sum_{j\in J\setminus\{0\}}\lambda_{j,l}$ and verify that

$\displaystyle\mathcal{O}_{Z}(F_{0})=\mathcal{O}_{\mathscr{X}}(D_{0}-\mathscr{X}_{k})_{|Z}=\mathcal{O}_{Z}\Big(-\sum_{j\in J\setminus\{0\}}F_{j}-\sum_{l\in L}Z_{l}\Big)=\mathcal{O}_{Z}(-\sum_{l\in L}\lambda_{0,l}Z_{l}).$

We obtain that $\mathcal{N}=\oplus_{j\in J}\mathcal{O}_{Z}\big(-\sum_{l\in L}\lambda_{j,l}Z_{l}\big)$ and for all $l$ in $L$

(2.1.5)

$$\sum_{j\in J}\lambda_{j,l}=1.$$

The normal bundle $\mathcal{N}$ is a toric variety of dimension $n+1$. The corresponding fan $\hat{\Sigma}$ lies in $N_{\mathbb{R}}\times\mathbb{R}^{J}$ and consists of the following cones and their faces (see [CLS11, §7.3] for a reference). Let $e_{0},\ldots,e_{n-r}$ be the standard basis of $\mathbb{R}^{J}$; given a cone $\sigma\in\Sigma$, we have

$$\hat{\sigma}=\textrm{Cone}((0,e_{0}),\ldots,(0,e_{n-r}),(u_{l},(\lambda_{j,l}))\,|\,u_{l}\in\sigma)\in\hat{\Sigma}.$$

In particular, we denote the rays of $\hat{\Sigma}$ by

$$v_{j}=(0,e_{j})\textrm{ for }j\in J,\quad v_{l}=(u_{l},(\lambda_{j,l}))\textrm{ for }l\in L.$$

The map

$$\ord(t):N_{\mathbb{R}}\times\mathbb{R}^{J}\rightarrow\mathbb{R}_{\geqslant 0}\quad(u,w)\mapsto\sum_{j=0}^{n-r}w_{j}$$

is $\mathbb{Z}$-linear, sends all the primitive generators of the rays of $\hat{\Sigma}$ to $1$ by Eq. 2.1.5, and is compatible with $\hat{\Sigma}$ and the fan of $\mathbb{A}^{1}_{k}$. Thus, it induces a toric morphism $t:\mathcal{N}\rightarrow\mathbb{A}^{1}_{k}$ whose fiber over $0$ is the toric boundary of $\mathcal{N}$. The base change $\mathscr{N}\coloneqq\mathcal{N}\times_{\mathbb{A}^{1}}R$ to $R$ is a toric $R$-scheme, whose generic fiber is isomorphic to $\mathbb{G}_{m,K}^{n}$. The special fiber $\mathscr{N}_{k}$ can be written as $\mathscr{N}_{k}=\sum_{i\in J\cup L}E_{i}$, where the combinatoric of intersections between components is exactly the same as in $\mathscr{X}_{k}$.

We prove Theorem B by constructing a formal isomorphism

$$f:\widehat{\mathscr{X}_{/Z}}\xrightarrow{\simeq}\widehat{\mathscr{N}_{/Z}}.$$

More specifically, we proceed as follows. We set the notations $\mathfrak{X}=\widehat{\mathscr{X}_{/Z}}$ and $\mathfrak{N}=\widehat{\mathscr{N}_{/Z}}$.

• •

  (Sections 2.2 and 2.3) Let $\sigma\in\Sigma$ be a maximal cone. Denote by $Z_{\sigma}$ and $\mathcal{N}_{\sigma}\coloneqq\mathcal{N}_{Z_{\sigma}/\mathscr{X}}$ the corresponding toric affine charts in $Z$ and $\mathcal{N}$ respectively. This induces an open formal subscheme of $\mathfrak{N}$, which we denote by $\mathfrak{N}_{\sigma}$. We construct a morphism

  $$f_{\sigma}:\mathfrak{X}\setminus\big(\cup_{l\in L\setminus L_{\sigma}}D_{l}\big)\eqqcolon\mathfrak{X}_{\sigma}\,\rightarrow\mathfrak{N}_{\sigma},$$

  in a similar manner to [NXY19]: we construct $n+1$ divisors $W^{\sigma}_{j}$ and $W^{\sigma}_{i}$ on $\mathscr{X}$, whose defining equations on the chart $\mathfrak{X}_{\sigma}$ yields the morphism $f_{\sigma}$. The equations are induced by sections of $\mathcal{O}_{Z}(W^{\sigma}_{j})$ and $\mathcal{O}_{Z}(W^{\sigma}_{i})$: these are first constructed on $Z$, then extended to $\mathfrak{X}$ by the nef condition on the conormal bundle $\nu_{Z/\mathscr{X}}^{*}$ assumed in Theorem B, which ensures the vanishing of higher cohomology groups for the tensor powers of $\nu_{Z/\mathscr{X}}^{*}$.

• •

  (Sections 2.4 and 2.5) Let $\sigma$ and $\sigma^{\prime}$ be two maximal cones of $\Sigma$ intersecting along a face of codimension one. We establish relations among the respective divisors and construct sections on $\mathfrak{X}_{\sigma^{\prime}}$ from those on $\mathfrak{X}_{\sigma}$. This allows us to prove that the morphisms $f_{\sigma}$ on the charts $\mathfrak{X}_{\sigma}$’s can be chosen so that they are compatible on the overlaps $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$. This yields a well defined morphism $f$ which extends the identity on $Z$ and preserves the ideal $\mathscr{I}_{Z}$, so that it turns out to be an isomorphism.

We set

$$\Delta:=\sum_{l\in L}u_{l}\otimes D_{l}\,\in\,N\otimes\Div_{0}(\mathscr{X})\simeq(\Div_{0}(\mathscr{X}))^{r};$$

this is an $r$-tuple of divisors on $\mathscr{X}$. Moreover, the restriction of any of these to $Z$ is a principal divisor by Corollary 1.2.2. Given a maximal cone $\sigma$ of $\Sigma$, for any $i\in L_{\sigma}$, we define

$$W^{\sigma}_{i}\coloneqq-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma}\setminus\{i\}})}{\det(u_{i},(u_{l})_{l\in L_{\sigma}\setminus\{i\}})}\in\Div_{0}(\mathscr{X})$$

where the column vectors $u_{l}$ are in the same order in the numerator and in the denominator, and the denominator has value $\pm 1$ by Lemma 2.1.3.

For $j\in J$, we define the divisor on $\mathscr{X}$

(2.2.2)

$\displaystyle\begin{split}W^{\sigma}_{j}&\coloneqq-D_{j}-\sum_{l\in L}\lambda_{j,l}D_{l}-\sum_{i\in L_{\sigma}}\lambda_{j,i}W^{\sigma}_{i}\\ &=-D_{j}-\sum_{l\in L\setminus L_{\sigma}}\lambda_{j,l}D_{l}-\cancel{\sum_{l\in L_{\sigma}}\lambda_{j,l}D_{l}}-\sum_{i\in L_{\sigma}}\lambda_{j,i}\big(\cancel{-D_{i}}+\sum_{l\in L\setminus L_{\sigma}}c_{i,l}D_{l}\big)\\ &=-D_{j}+\sum_{l\in L\setminus L_{\sigma}}d_{j,l}D_{l}\quad\textrm{ with }\quad d_{j,l}=-\lambda_{j,l}-\sum_{i\in L_{\sigma}}\lambda_{j,i}c_{i,l}.\end{split}$

The restriction of $W^{\sigma}_{j}$ to $Z$ is a principal divisor, as the ${W^{\sigma}_{i}}_{|Z}$ are principal and ${-D_{j}}_{|Z}$ is linearly equivalent to $\sum_{l\in L}\lambda_{j,l}Z_{l}$ by Eq. 2.1.4.

Let $\sigma$ be a maximal cone of $\Sigma$. We denote by $\mathscr{L}^{\sigma}_{j}$ and $\mathscr{L}^{\sigma}_{i}$ the line bundles on $\mathfrak{X}$ induced respectively by $\mathcal{O}_{\mathscr{X}}(W^{\sigma}_{j})$ for $j\in J$, and by $\mathcal{O}_{\mathscr{X}}(W^{\sigma}_{i})$ for $i\in L_{\sigma}$. Since $W^{\sigma}_{j}$ and $W^{\sigma}_{i}$ are principal on $Z$, the restrictions ${\mathscr{L}^{\sigma}_{j}}_{|Z}$ and ${\mathscr{L}^{\sigma}_{i}}_{|Z}$ are trivial line bundles on $Z$, thus we may choose non-zero global sections $s^{\sigma}_{j}$ and $s^{\sigma}_{i}$ on $Z$.

We now lift the sections $s^{\sigma}_{j}$ and $s^{\sigma}_{i}$ to global sections of $\mathscr{L}^{\sigma}_{j}$ and $\mathscr{L}^{\sigma}_{i}$, which we still denote by $s^{\sigma}_{j}$ and $s^{\sigma}_{i}$. Indeed, for any $n\geqslant 1$, write $(\mathscr{X}/Z)_{n}$ for the (non-reduced) subscheme of $\mathscr{X}$ defined by the ideal $\mathscr{I}^{n}_{Z}$. In the exact sequence

$$H^{0}((\mathscr{X}/Z)_{n},\mathscr{L}^{\sigma}_{j})\longrightarrow H^{0}((\mathscr{X}/Z)_{n-1},\mathscr{L}^{\sigma}_{j})\longrightarrow H^{1}(Z,(\nu^{*}_{Z/\mathscr{X}})^{\otimes n}),$$

and in the analogous one for $\mathscr{L}^{\sigma}_{i}$, the right-hand vanishes: the conormal bundle is a direct sum of line bundles on $Z$ which are nef by the hypothesis in Theorem B and so are its positive tensor powers, thus their first cohomology group vanishes by Proposition 1.2.5. We thus extend the sections constructed above to all of the $(\mathscr{X}/Z)_{n}$ by induction, which yields an extension to $\mathfrak{X}=\varprojlim_{n}(\mathscr{X}/Z)_{n}$.

Let $\sigma$ and $\sigma^{\prime}$ be two maximal cones of $\Sigma$ intersecting along a face of codimension one. Setting $L_{\sigma\sigma^{\prime}}=L_{\sigma}\cap L_{\sigma^{\prime}}$, we may write $L_{\sigma}=L_{\sigma\sigma^{\prime}}\cup\{i_{0}\}$ and $L_{\sigma^{\prime}}=L_{\sigma\sigma^{\prime}}\cup\{i_{\infty}\}$. The sets $\mathcal{B}=((v_{i})_{i\in L_{\sigma\sigma^{\prime}}},v_{i_{0}},(v_{j})_{j\in J})$ and $\mathcal{B}^{\prime}=((v_{i})_{i\in L_{\sigma\sigma^{\prime}}},v_{i_{\infty}},(v_{j})_{j\in J})$ are bases of $\hat{N}=N\oplus\mathbb{Z}^{J}$. They induce isomorphisms $\beta,\beta^{\prime}:\mathbb{Z}^{r}\oplus\mathbb{Z}^{J}\rightarrow\hat{N}$ such that the change of basis from $\mathcal{B}$ to $\mathcal{B}^{\prime}$ is

$$M_{\mathcal{B}^{\prime}\mathcal{B}}=\beta^{\prime}\circ\beta^{-1}=\begin{array}[]{cccc}L_{\sigma\sigma^{\prime}}&i_{0}&J\\ \Id&(-C\cdot D_{i})_{i\in L_{\sigma\sigma^{\prime}}}&0&L_{\sigma\sigma^{\prime}}\\ 0&-1&0&i_{\infty}\\ 0&(-C\cdot D_{j})_{j\in J}&\Id&J\\ \end{array}\quad\textrm{ and }\left(\begin{matrix}v_{i}\\ v_{i_{\infty}}\\ v_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}^{T}\left(\begin{matrix}v_{i}\\ v_{i_{0}}\\ v_{j}\end{matrix}\right).$$

Denote by $((\varepsilon_{i})_{i\in L_{\sigma\sigma^{\prime}}},\varepsilon_{i_{0}},(\varepsilon_{j})_{j\in J})$ the basis of $\hat{M}\coloneqq\Hom(\hat{N},\mathbb{Z})$ dual to $\mathcal{B}$, and $((\varepsilon^{\prime}_{i})_{i\in L_{\sigma\sigma^{\prime}}},\varepsilon^{\prime}_{i_{\infty}},(\varepsilon^{\prime}_{j})_{j\in J})$ the basis dual to $\mathcal{B}^{\prime}$. It follows that

(2.4.1)

$$\left(\begin{matrix}\varepsilon^{\prime}_{i}\\ \varepsilon^{\prime}_{i_{\infty}}\\ \varepsilon^{\prime}_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}\left(\begin{matrix}\varepsilon_{i}\\ \varepsilon_{i_{0}}\\ \varepsilon_{j}\end{matrix}\right).$$

The isomorphisms $\beta$ and $\beta^{\prime}$ allow us to view

$$W^{\sigma}\coloneqq((W^{\sigma}_{i})_{i\in L_{\sigma\sigma^{\prime}}},W^{\sigma}_{i_{0}},(W^{\sigma}_{j})_{j\in J})\,\in\,(\mathbb{Z}^{r}\oplus\mathbb{Z}^{J})\otimes\Div_{0}(\mathscr{X})\simeq(\Div_{0}(\mathscr{X}))^{n+1}$$

and $W^{\sigma^{\prime}}$ as elements of $\widehat{N}\otimes\Div_{0}(\mathscr{X})$, that we will still denote by $W^{\sigma}$ and $W^{\sigma^{\prime}}$ .

The inverse $(s^{\sigma}_{i_{0}})^{-1}$ is a section on $\mathfrak{X}$ of $\mathscr{L}^{\sigma^{\prime}}_{i_{\infty}}$, so by Lemma 2.4.2 the sections

$$\begin{cases}s^{\sigma^{\prime}}_{i}\coloneqq s^{\sigma}_{i}\cdot(s^{\sigma}_{i_{0}})^{-(C\cdot D_{i})}&\textrm{ \quad for }i\in L_{\sigma\sigma^{\prime}}\\ s^{\sigma^{\prime}}_{i_{\infty}}\coloneqq(s^{\sigma}_{i_{0}})^{-1}&\\ s^{\sigma^{\prime}}_{j}\coloneqq s^{\sigma}_{j}\cdot(s^{\sigma}_{i_{0}})^{-(C\cdot D_{j})}&\textrm{ \quad for }j\in J\end{cases}$$

are sections on $\mathfrak{X}$ of the line bundles $\mathscr{L}^{\sigma^{\prime}}_{i}$ and $\mathscr{L}^{\sigma^{\prime}}_{j}$. By Lemma 2.3.1 these give equations for $D_{i}$ and $D_{j}$ on the open subscheme $\mathfrak{X}_{\sigma^{\prime}},$ and on $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$ we have

(2.4.3)

$$\left(\begin{matrix}s^{\sigma^{\prime}}_{i}\\ s^{\sigma^{\prime}}_{i_{\infty}}\\ s^{\sigma^{\prime}}_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}\left(\begin{matrix}s^{\sigma}_{i}\\ s^{\sigma^{\prime}}_{i_{0}}\\ s^{\sigma}_{j}\end{matrix}\right)$$

where the additive notation on the matrix corresponds to the multiplicative notation on the sections. Moreover, on $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$ we have

(2.4.4)

$\displaystyle\begin{split}w_{\sigma^{\prime}}&\coloneqq t\cdot\prod_{j\in J}(s_{j}^{\sigma^{\prime}})^{-1}\cdot\prod_{i\in L_{\sigma^{\prime}}}(s_{i}^{\sigma^{\prime}})^{-1}=t\cdot\prod_{j\in J}(s^{\sigma}_{j})^{-1}(s^{\sigma}_{i_{0}})^{C\cdot D_{j}}\cdot\prod_{i\in L_{\sigma\sigma^{\prime}}}(s^{\sigma}_{i})^{-1}(s^{\sigma}_{i_{0}})^{C\cdot D_{i}}\cdot(s^{\sigma}_{i_{0}})\\ &=t\cdot\prod_{j\in J}(s^{\sigma}_{j})^{-1}\cdot\prod_{i\in L_{\sigma\sigma^{\prime}}}(s^{\sigma}_{i})^{-1}\cdot(s^{\sigma}_{i_{0}})^{\sum_{j\in J}C\cdot D_{j}+\sum_{i\in L_{\sigma\sigma^{\prime}}}C\cdot D_{i}+1}=w_{\sigma},\end{split}$

hence the invertible function $w_{\sigma}$ on $\mathfrak{X}_{\sigma}$ extends to $\mathfrak{X}_{\sigma}\cup\mathfrak{X}_{\sigma^{\prime}}$ by $w_{\sigma^{\prime}}$.

Let $\Gamma$ be the graph with vertices the maximal cones of $\Sigma$ (hence the maximal cones of $\widehat{\Sigma}$) and with an edge between $\sigma$ and $\sigma^{\prime}$ if and only if $\sigma\cap\sigma^{\prime}$ is a common face of codimension one. Note that since $Z$ is proper, if $\mathbb{S}\subset N_{\mathbb{R}}$ is a sphere with center the origin, then $\Sigma\cap\mathbb{S}$ is a triangulation of $\mathbb{S}$. In particular, $\Gamma$ is the 1-skeleton of the dual complex of a triangulation of the sphere, and is thus connected.

Let $\sigma_{0}\in\Sigma$ be a maximal cone, and $p_{0}\in\Gamma$ the corresponding vertex, that we will use as a reference point. We fix a tuple of sections $s^{\sigma_{0}}$ of $W^{\sigma_{0}}$ as in Section 2.3.
Let $\sigma\in\Sigma$ be a maximal cone, and $p\in\Gamma$ the corresponding vertex. By connectedness of $\Gamma$, there exists a path $\gamma$ from $p_{0}$ to $p$, hence a sequence of maximal cones $\sigma_{0},\ldots,\sigma_{q}=\sigma$ such that $\sigma_{h}\cap\sigma_{h+1}$ is a codimension one face of both $\sigma_{h}$ and $\sigma_{h+1}$, for $h=0,\ldots,q-1$. The construction of Section 2.4 allows us to construct inductively along $\gamma$ a tuple of sections $s^{\sigma_{h}}$ of $W^{\sigma_{h}}$.

This provides us with a tuple of sections $s^{\sigma}$ of $W^{\sigma}$ for each maximal cone $\sigma\in\Sigma$, and the function

$$w_{\sigma}=t\cdot\prod_{j\in J}(s_{j}^{\sigma})^{-1}\cdot\prod_{i\in L_{\sigma}}(s_{i}^{\sigma})^{-1}\,\in\mathcal{O}(\mathfrak{X}_{\sigma})^{\times}.$$

By Eq. 2.4.4 the $w_{\sigma}$ glue to an invertible function $w$ on $\mathfrak{X}$; $w$ admits a $(n+1)$-th root on $Z$, since it is constant, and by Hensel’s lemma we obtain an invertible function $w^{\prime}$ on $\mathfrak{X}$ such that $(w^{\prime})^{n+1}=w$. We use the sections $s^{\sigma}$ and the function $w^{\prime}$ to define a morphism

$$f_{\sigma}:\mathfrak{X}_{\sigma}\longrightarrow\mathfrak{N}_{\sigma}$$

as follows. Denoting by $((\varepsilon_{i})_{i\in L_{\sigma}},(\varepsilon_{j})_{j\in J})$ the dual basis to $((v_{i})_{i\in L_{\sigma}},(v_{j})_{j\in J})$, the toric chart $\mathfrak{N}_{\sigma}$ has the following explicit description:

$$\mathfrak{N}_{\sigma}=\Spf R[\chi^{\varepsilon_{i}},i\in L_{\sigma}][[\chi^{\varepsilon_{j}},j\in J]]/\{t-\chi^{\sum_{i\in L_{\sigma}}\varepsilon_{i}+\sum_{j\in J}\varepsilon_{j}}\}.$$

Indeed, $\mathfrak{N}_{\sigma}$ is the formal completion along $Z$ of $\mathcal{N}_{\hat{\sigma}}\times_{\mathbb{A}^{1}}R$, where $\mathcal{N}_{\hat{\sigma}}=\Spec k[(\hat{\sigma})^{\vee}\cap\hat{N}]$; since $\ord(t)=\sum_{i\in L_{\sigma}}\varepsilon_{i}+\sum_{j\in J}\varepsilon_{j}$ on $\widehat{\sigma}$, the relation $t=\chi^{\sum_{i\in L_{\sigma}}\varepsilon_{i}+\sum_{j\in J}\varepsilon_{j}}$ holds.
The map $f_{\sigma}$ is now defined at the level of function rings by

	$\displaystyle f^{\#}_{\sigma}:\mathcal{O}(\mathfrak{N}_{\sigma})$	$\displaystyle\longrightarrow\mathcal{O}(\mathfrak{X}_{\sigma})$
	$\displaystyle\chi^{\varepsilon_{i}}$	$\displaystyle\mapsto w^{\prime}s^{\sigma}_{i}\;\textrm{ \quad for }i\in L_{\sigma}$
	$\displaystyle\chi^{\varepsilon_{j}}$	$\displaystyle\mapsto w^{\prime}s_{j}^{\sigma}\;\textrm{ \quad for }j\in J$

where the sections $s^{\sigma}$ are viewed as functions on $\mathfrak{X}_{\sigma}$ thanks to the proof of Lemma 2.3.1.