(2.1.1) Let $k$ be a field of characteristic zero. We set $R=k[\negthinspace[t]\negthinspace]$ and $K=k(\negthinspace(t)\negthinspace)$, and we fix a $t$-adic absolute value $|\cdot|_{K}$ on $K$ by setting $|t|_{K}=1/e$. For every $K$-scheme of finite type $Y$, we denote by $Y^{\mathrm{an}}$ the associated $K$-analytic space. For every separated $R$-scheme of finite type $\mathscr{Y}$ we set $\mathscr{Y}_{k}=\mathscr{Y}\times_{R}k$ and $\mathscr{Y}_{K}=\mathscr{Y}\times_{R}K$. Moreover, we will denote by $\widehat{\mathscr{Y}}$ the $t$-adic completion of $\mathscr{Y}$, by $\widehat{\mathscr{Y}}_{\eta}$ the generic fiber of $\widehat{\mathscr{Y}}$ in the category of $K$-analytic spaces and by

the canonical reduction map. The generic fiber $\widehat{\mathscr{Y}}_{\eta}$ is an analytic domain in $\mathscr{Y}_{K}^{\mathrm{an}}$, and it is equal to $\mathscr{Y}_{K}^{\mathrm{an}}$ if and only if $\mathscr{Y}$ is proper over $R$.

(2.1.2) Let $\mathscr{C}$ be a connected smooth algebraic curve over $k$. Let $s$ be a $k$-rational point on $\mathscr{C}$ and set $C=\mathscr{C}\setminus\{s\}$. We fix a uniformizer $t$ in $\mathcal{O}_{\mathscr{C},s}$. This choice determines an isomorphism of $k$-algebras $R\to\widehat{\mathcal{O}}_{\mathscr{C},s}$ and thus a morphism of $k$-schemes $\mathrm{Spec}\,R\to\mathscr{C}$.

(2.1.3) Let $X$ be a smooth and proper scheme over $C$ with geometrically connected fibers. A model of $X$ over $\mathscr{C}$ is a flat separated $\mathscr{C}$-scheme of finite type $\mathscr{X}$ endowed with an isomorphism of $C$-schemes $\mathscr{X}\times_{\mathscr{C}}C\to X$. Note that we do not require $\mathscr{X}$ to be proper over $\mathscr{C}$. Morphisms of models are defined in the usual way. We denote by $\mathscr{X}_{s}$ the fiber of $\mathscr{X}$ over $s$, by $\mathscr{X}_{R}$ the base change of $\mathscr{X}$ to $\mathrm{Spec}\,R$ and by $X_{K}$ the base change of $X$ to $\mathrm{Spec}\,K$. We denote by $K_{X}$ a relative canonical divisor for $X$ over $C$, and for every normal model $\mathscr{X}$ of $X$, we denote by $K_{\mathscr{X}}$ a relative canonical divisor for $\mathscr{X}$ over $\mathscr{C}$.

(2.1.4) For every $\mathscr{C}$-model $\mathscr{X}$ of $X$, we denote by $\mathscr{X}^{\mathrm{snc}}$ the subset of $\mathscr{X}$ consisting of the points where $\mathscr{X}$ is regular and $\mathscr{X}_{s}$ is a divisor with strict normal crossings (some authors use the terminology “simple normal crossings” instead). Thus $\mathscr{X}^{\mathrm{snc}}$ is the union of $X$ with the set of points $x$ of $\mathscr{X}_{s}$ such that $\mathcal{O}_{\mathscr{X},x}$ is regular and there exist a unit $u$ and a regular system of local parameters $(z_{1},\ldots,z_{n})$ in $\mathcal{O}_{\mathscr{X},x}$ and non-negative integers $N_{1},\ldots,N_{n}$ such that

The subset $\mathscr{X}^{\mathrm{snc}}$ is an open subscheme of $\mathscr{X}$ and it is again a $\mathscr{C}$-model of $X$. Moreover, if $\mathscr{X}$ is normal, then $\mathscr{X}^{\mathrm{snc}}_{s}$ is dense in $\mathscr{X}_{s}$. We say that $\mathscr{X}$ is an $snc$-model of $X$ if $\mathscr{X}=\mathscr{X}^{\mathrm{snc}}$, that is, if $\mathscr{X}$ is regular and $\mathscr{X}_{s}$ is a divisor with strict normal crossings. If $\mathscr{X}$ is a model of $X$ over $\mathscr{C}$, then a log resolution of $(\mathscr{X},\mathscr{X}_{s})$ is a proper morphism of $\mathscr{C}$-models $h:\mathscr{Y}\to\mathscr{X}$ such that $\mathscr{Y}$ is an $snc$-model of $X$.

(2.1.5) Let $h:\mathscr{Y}\to\mathscr{X}$ be a proper morphism of normal $\mathscr{C}$-models of $X$. Assume that $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is $\mathbb{Q}$-Cartier. Then the log pullback of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$ is the unique $\mathbb{Q}$-Weil divisor $\Delta$ on $\mathscr{Y}$ such that $K_{\mathscr{Y}}+\Delta$ is $\mathbb{Q}$-linearly equivalent to

and $f_{*}\Delta=(\mathscr{X}_{s})_{\mathrm{red}}$.

(2.1.6) We will use the following notations from [MN13]. If $\mathscr{X}$ is a normal model of $X$ over $\mathscr{C}$, $x$ is a point of $\widehat{\mathscr{X}}_{\eta}$ and $D$ is a divisor on $\mathscr{X}$ that is supported on $\mathscr{X}_{s}$ and Cartier at $\mathrm{red}_{\mathscr{X}}(x)$, then we set

where $f$ is any element of the local ring of $\mathscr{X}$ at $\mathrm{red}_{\mathscr{X}}(x)$ such that $D=\mathrm{div}(f)$ locally at $\mathrm{red}_{\mathscr{X}}(x)$. It is clear that $v_{x}(D)$ is linear in $D$. If $\mathscr{X}$ is regular and $\omega$ is a non-zero rational section of $\omega_{\mathscr{X}_{R}/R}^{\otimes m}$, for some $m>0$ (for instance, an $m$-pluricanonical form on $X_{K}$) then we denote by $\mathrm{div}_{\mathscr{X}}(\omega)$ the corresponding divisor on $\mathscr{X}_{R}$.

(2.2.1) A $dlt$-model of $X$ is a normal proper $\mathscr{C}$-model $\mathscr{X}$ of $X$ such that $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ is a $dlt$-pair. This means that $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ is log canonical and that each log canonical center of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ has non-empty intersection with $\mathscr{X}^{\mathrm{snc}}$. In particular, every proper $snc$-model of $X$ is a $dlt$-model. An equivalent formulation of the definition is the following: $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is $\mathbb{Q}$-Cartier, and for every log resolution $h:\mathscr{Y}\to\mathscr{X}$ of $(\mathscr{X},\mathscr{X}_{s})$ and every irreducible component $E$ of $\mathscr{Y}_{s}$, the multiplicity of $E$ in the log pullback $\Delta$ of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$ is at most $1$. Moreover, if it is equal to $1$, then $h(E)$ must have non-empty intersection with $\mathscr{X}^{\mathrm{snc}}$. In practice, we will apply the $dlt$ property via Lemma 3.2.3 below.

(2.2.2) We say that a $dlt$-model $\mathscr{X}$ of $X$ is a good minimal model if $\mathscr{X}$ is $\mathbb{Q}$-factorial and $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is semi-ample over $\mathscr{C}$.

(2.2.3) For every $dlt$-model $\mathscr{X}$ of $X$, we can define the dual complex $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ for the $dlt$-pair $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ by gluing cells corresponding to irreducible components of intersections of irreducible components of $\mathscr{X}_{s}$, as in Definition 8 in [dFKX12]. When $k$ is not algebraically closed, we note that we only glue cells corresponding to irreducible components (instead of geometrically irreducible components). In other words, $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ is the quotient of the $\mathrm{Gal}(\bar{k}/k)$-equivariant dual complex constructed in [dFKX12, §31].

(2.2.4) For every $dlt$-model $\mathscr{X}$ of $X$, the log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ are the irreducible components of intersections of irreducible components of $(\mathscr{X}_{s})_{\mathrm{red}}$, by [Ko13, 4.16]. These are also precisely the closures in $\mathscr{X}_{s}$ of the connected components of intersections of irreducible components of $\mathscr{X}^{\mathrm{snc}}_{s}$ (since these connected components are the log canonical centers of $(\mathscr{X}^{\mathrm{snc}},(\mathscr{X}_{s}^{\mathrm{snc}})_{\mathrm{red}})$). Thus, the dual intersection complex $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ is the same as the dual intersection complex of the strict normal crossings divisor $\mathscr{X}^{\mathrm{snc}}_{s}$, and the cells of this complex correspond bijectively to the log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$. See Section 2 in [dFKX12] for more background.

(2.2.5) Let $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ be two $dlt$-models of $X$ over $\mathscr{C}$. We say that $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ are crepant birational if there exist a normal proper $\mathscr{C}$-model $\mathscr{Y}$ of $X$ and morphisms of $\mathscr{C}$-models $f_{i}:\mathscr{Y}\to\mathscr{X}_{i}$ for $i=1,2$ such that the log pullbacks of $(\mathscr{X}_{1,s})_{\mathrm{red}}$ and $(\mathscr{X}_{2,s})_{\mathrm{red}}$ coincide (see [Ko13, 2.23]). Note that we can always assume that $\mathscr{Y}$ is an $snc$-model, by taking a log resolution of $(\mathscr{Y},\mathscr{Y}_{s})$. The following theorem collects two fundamental results from the Minimal Model Program.

(3.1.1) Let $\mathscr{Y}$ be a connected regular flat separated $R$-scheme of finite type such that the special fiber $\mathscr{Y}_{k}$ is a divisor with strict normal crossings. Then, as explained in [MN13, §3.1], one can associate to $\mathscr{Y}$ its skeleton $\mathrm{Sk}(\mathscr{Y})$, which is a topological subspace of the generic fiber $\widehat{\mathscr{Y}}_{\eta}$ of the formal $t$-adic completion of $\mathscr{Y}$. It is the set of points of $\widehat{\mathscr{Y}}_{\eta}$ that correspond to a real valuation on the function field of $\mathscr{Y}_{K}$ that is monomial with respect to the strict normal crossings divisor $\mathscr{Y}_{k}$. The skeleton $\mathrm{Sk}(\mathscr{Y})$ is canonically homeomorphic to the dual intersection complex $\mathcal{D}((\mathscr{Y}_{k})_{\mathrm{red}})$ of $\mathscr{Y}_{k}$, and there exists a canonical continuous retraction

Moreover, $\mathrm{Sk}(\mathscr{Y})$ carries a canonical piecewise $\mathbb{Z}$-affine structure [MN13, §3.2].

(3.1.2) We keep the notations from (2.1). For each $\mathscr{C}$-model $\mathscr{X}$ of $X$, we define the skeleton of $\mathscr{X}$ by

and we write $\rho_{\mathscr{X}}$ for $\rho_{\mathscr{X}_{R}^{\mathrm{snc}}}$. If $\mathscr{X}$ is a proper $snc$-model of $X$, one has the following crucial property.

(3.1.4) Theorem 3.1.3 can be extended to the case where $X$ is defined over $K$ instead of $C$ and $\mathscr{X}$ is a proper $snc$-model of $X$ over $R$. The general proof technique is the same as in [Th07], but one replaces the formalism of toroidal embeddings by the more flexible language of logarithmic geometry. Details will appear in [Ni13]. We will only use this generalization in the proof of Theorem 4.2.4.

(3.2.1) In the following subsections, we will make use of the weight function

associated to a non-zero $m$-pluricanonical form on $X_{K}$, for any $m>0$. Its construction and main properties are described in [MN13, 4.4.5]. For us, its most important features are the following: if $\mathscr{X}$ is an $snc$-model of $X$ over $\mathscr{C}$ and $x$ is a point of $\mathrm{Sk}(\mathscr{X})$, then

(here we use the notation recalled in (2.1)). Moreover, for every point $y$ of $\widehat{\mathscr{X}}_{\eta}$, we have

with equality if and only if $y$ lies on $\mathrm{Sk}(\mathscr{X})$. In [MN13, 4.4.5] there is no properness assumption on $X_{K}$; this allows us to deal with rational pluricanonical forms by removing the locus of poles from $X_{K}$.

(3.2.2) It will often be useful to interpret the weight function in terms of logarithmic differential forms. Let $\mathscr{Y}$ be a regular separated $R$-scheme of finite type such that $\mathscr{Y}_{k}$ is a divisor with strict normal crossings. We write $S^{+}$ for the log scheme associated to $R\setminus\{0\}\to R$ and $\mathscr{Y}^{+}$ for the log scheme obtained by endowing $\mathscr{Y}$ with the divisorial log structure associated to $\mathscr{Y}_{k}$. Then $\mathscr{Y}^{+}$ is log smooth over $S^{+}$. If we denote by $j:\mathscr{Y}_{K}\to\mathscr{Y}$ the natural open immersion, then a simple computation shows that the sub-$\mathcal{O}_{\mathscr{Y}}$-module $\omega_{\mathscr{Y}^{+}/S^{+}}$ of $j_{*}\omega_{\mathscr{Y}_{K}/K}$ is equal to $\omega_{\mathscr{Y}/R}((\mathscr{Y}_{k})_{\mathrm{red}}-\mathscr{Y}_{k})$ (it suffices to check that these line bundles coincide at the generic points of the special fiber $\mathscr{Y}_{k}$). Thus if $\omega$ is an $m$-pluricanonical form on $X_{K}$ and $\mathscr{X}$ is an $snc$-model of $X$ over $\mathscr{C}$, then

for every point $x$ of $\mathrm{Sk}(\mathscr{X})$, where we denote by $\mathrm{div}_{\mathscr{X}^{+}}(\omega)$ the divisor on $\mathscr{X}_{R}$ associated to $\omega$ viewed as a rational section of the line bundle $\omega^{\otimes m}_{\mathscr{X}^{+}_{R}/S^{+}}$.

(3.2.6) Corollary 3.2.5 implies, in particular, that the skeleta $\mathrm{Sk}(\mathscr{X}_{1})=\mathrm{Sk}(\mathscr{X}_{2})$ are isomorphic as topological spaces with piecewise affine structure, by [MN13, §3.2]. Since $\mathrm{Sk}(\mathscr{X}_{i})$ is canonically homeomorphic to the dual complex associated to the reduced special fiber of $\mathscr{X}_{i}$, for $i=1,2$, this also follows from Proposition 11 in [dFKX12], whose proof relies on Weak Factorization. The proofs of Corollary 3.2.5 and [MN13, §3.2] do not use Weak Factorization.

(3.3.1) In [MN13, §4.5], Mustaţă and the first-named author associated to every non-zero regular pluricanonical form $\omega$ on $X_{K}$ a skeleton $\mathrm{Sk}(X_{K},\omega)$ in $X_{K}^{\mathrm{an}}$, generalizing a construction of Kontsevich and Soibelman [KS06]. The skeleton $\mathrm{Sk}(X_{K},\omega)$ is precisely the locus of points of $X_{K}^{\mathrm{an}}$ where the weight function $\mathrm{wt}_{\omega}$ reaches its minimal value. If $\mathscr{X}$ is any $snc$-model of $X$ over $\mathscr{C}$, then $\mathrm{Sk}(X_{K},\omega)$ is a union of closed faces of $\mathrm{Sk}(\mathscr{X})$, which can be explicitly computed [MN13, 4.5.5]. Taking the union of the skeleta $\mathrm{Sk}(X_{K},\omega)$ over all non-zero pluricanonical forms $\omega$ on $X_{K}$, one obtains a topological subspace $\mathrm{Sk}(X_{K})$ of $X_{K}^{\mathrm{an}}$ that was called the essential skeleton of $X_{K}$ in [MN13, 4.6.2]. It is an interesting birational invariant of $X_{K}$. In this subsection, we will compare the essential skeleton to the skeleton of a good minimal $dlt$-model of $X$.

(4.1.1) The case where $X$ is a family of Calabi-Yau varieties over $C$ is of particular interest; the connections with homological mirror symmetry were the main motivation for Kontsevich and Soibelman to define the skeleton in [KS06]. If $\omega$ is a volume form on $X_{K}$ (i.e., a nowhere vanishing differential form of maximal degree) then $\mathrm{Sk}(X_{K})=\mathrm{Sk}(X_{K},\omega)$ by [MN13, 4.6.4]. We will now prove that the underlying topological space of the essential skeleton $\mathrm{Sk}(X_{K})$ is a pseudo-manifold with boundary; this result is implicitly contained in [KK10, Ko11].

(4.1.2) A topological space $T$ is called an $n$-dimensional pseudo-manifold with boundary if it admits a triangulation $\mathscr{T}$ satisfying the following conditions:

1. (1)

   (dimensional homogeneity) $T=|\mathscr{T}|$ is the union of all $n$-simplices.

2. (2)

   (non-branching) Every $(n-1)$-simplex is a face of precisely one or two $n$-simplices.

3. (3)

   (strong connectedness) For every pair of $n$-simplices $\sigma$ and $\sigma^{\prime}$ in $\mathscr{T}$, there is a sequence of $n$-simplices

   $$\sigma=\sigma_{0},\sigma_{1},\ldots,\sigma_{\ell}=\sigma^{\prime}$$

   such that the intersection $\sigma_{i}\cap\sigma_{i+1}$ is an $(n-1)$-simplex for all $i$.

We say that $T$ is a closed pseudo-manifold if we can replace condition (2) by the property that every $(n-1)$-simplex is a face of precisely two $n$-simplices. A typical example of a 2-dimensional closed pseudo-manifold which is not a manifold is the pinched torus.

(4.1.3) For the reader’s convenience, we include some basic facts about adjunction for $dlt$-pairs. We refer to Chapter 4 of [Ko13] for more background. Let $(Y,\Delta)$ be a $dlt$-pair over $k$, and let $D$ be a log canonical center of $(Y,\Delta)$. Then $D$ is normal, by [Ko13, 4.16]. There is a well defined $\mathbb{Q}$-divisor $\Delta_{D}$ on $D$, called the different of $\Delta$ on $D$ [Ko13, 4.18], which is induced by the Poincaré map and satisfies the equation

In the sequel, whenever we write such an equation it will be understood that $\Delta_{D}$ is the different of $\Delta$ on $D$. The pair $(D,\Delta_{D})$ is again a $dlt$-pair, by [Ko13, 4.19]. Write $\lfloor\Delta\rfloor=\sum_{i\in I}D_{i}$. If $J$ is a subset of $I$ and $D$ is a component of $\bigcap_{j\in J}{\Delta}_{j}$, it is not hard to see that for every non-empty subset $J^{\prime}$ of $I\setminus J$, every irreducible component of the intersection