Let $X$ be a separated scheme of finite type over $K$. A separated flat $R$-scheme $\mathscr{X}$ of finite type together with an isomorphism of $K$-schemes $\mathscr{X}\times_{R}K\simeq X$ is called an $R$-model of $X$. We denote by $\mathscr{X}_{k}=\mathscr{X}\times_{R}k$ the special fiber of $\mathscr{X}$, and by $\Div_{0}(\mathscr{X})$ the group of Weil divisors on $\mathscr{X}$ supported on the special fiber.

If $Y$ is a normal variety and $D$ a Weil divisor on $Y$, whose irreducible decomposition is $D=\sum_{i\in I}a_{i}D_{i}$, a stratum of $D$ is a connected component of an intersection $D_{J}=\cap_{j\in J}D_{j}$ for some $J\subset I$. An open stratum of $D$ is a stratum $Z$ minus the irreducible components of $D$ not containing $Z$; this is denoted by $\mathring{Z}$.

We will not give a precise definition of log canonical centers here, and refer the reader to [KM98]. However, if $(\mathscr{X},\mathscr{X}_{k,\red})$ is dlt and defined over an algebraic curve - which is the most relevant case for applications - then the log canonical centers of $(\mathscr{X},\mathscr{X}_{k,\red})$ are precisely the strata of $\mathscr{X}_{k}$ by [Kol13, 4.16], so that a dlt model $\mathscr{X}$ is simple normal crossing at the generic points of the strata of $\mathscr{X}_{k}$. If $\mathscr{X}_{k}$ is reduced, it follows from the approximation arguments of [NXY19, Corollary 4.4] that this holds in the general case as well.

A dlt model $\mathscr{X}$ is good if each irreducible component of $\mathscr{X}_{k,\red}$ is $\mathbb{Q}$-Cartier. See [NXY19, §1.12-1.14] for an overview on existence results of such models.

Throughout this section, let $Z$ be an $r$-dimensional (normal) proper toric variety over $k$, in the sense of [KKMSD73]. This means that $Z$ is a normal $k$-variety, containing the torus $\mathbb{T}=\mathbb{G}^{r}_{m,k}$ as an open subset, and such that the torus action onto itself extends to an action on $Z$. The complement $\Delta_{Z}=Z\setminus\mathbb{T}$ is a reduced anticanonical Weil divisor in $Z$, called the toric boundary of $Z$; we write it as the sum of its irreducible components $\Delta_{Z}=\sum_{l\in L}Z_{l}$. We write $N=\Hom(\mathbb{G}_{m,k},\mathbb{T})$ for the free abelian group of 1-parameter subgroups of $\mathbb{T}$, and $N_{\mathbb{R}}=N\otimes\mathbb{R}$.

The variety $Z$ can be described by a combinatorial object, called its fan $\Sigma$. The fan lives inside the finite-dimensional vector space $N_{\mathbb{R}}$; $\Sigma$ is a collection of strictly convex rational polyhedral cones $\Sigma=\{\sigma\}_{\sigma\in\Sigma}$ inside $N_{\mathbb{R}}$, stable under intersection and such that each face of a cone in $\Sigma$ is itself in $\Sigma$. The cones of $\Sigma$ are in inclusion-reversing bijection with the strata of $\Delta_{Z}$; this induces in particular a bijection between the set of rays (i.e. 1-dimensional cones) of $\Sigma$ and the irreducible components of $\Delta_{Z}$.
The fan $\Sigma$ encodes various types of algebro-geometric information about $Z$. For instance, the variety $Z$ is smooth if and only if each top-dimensional cone of $\Sigma$ is $\GL(N)$-isomorphic to the standard octant $\mathbb{R}^{r}_{\geqslant 0}\subset\mathbb{R}^{r}$.

Furthermore, in the case where $Z$ is smooth (which implies it has simple normal crossing boundary), the fan allows us to give a rather explicit description of the Picard group of $Z$. Indeed, each Cartier divisor can be moved via the torus action to a $\mathbb{T}$-invariant Cartier divisor, which is thus supported on the boundary. This provides a canonical set of generators of $\Pic(Z)$, and the kernel can be described as follows.
Write $\Div^{\mathbb{T}}(Z)=\oplus_{l\in L}\mathbb{Z}Z_{l}\simeq\mathbb{Z}^{L}$ the abelian group of Weil divisors supported on the boundary. The canonical map $q:\Div^{\mathbb{T}}(Z)\longrightarrow\Pic(Z)$ sends a divisor to its class; the map $p:M\longrightarrow\Div^{\mathbb{T}}(Z)$ sends a monomial $z^{m}$ to the principal divisor $\textrm{div}(z^{m})$.

We now want to describe how the fan $\Sigma$ encodes the intersection theory on $Z$. Each $1$-cycle in $Z$ being numerically equivalent to a sum of toric strata, it is enough to study the intersection numbers $(C\cdot Z_{l})$, where $Z_{l}$ is a boundary component of $Z$ and $C$ is a 1-dimensional toric stratum, which is isomorphic to $\mathbb{P}^{1}$ by properness. The stratum $C$ is thus a rational curve with two marked points $p$ and $q$, which are the intersection points of $C$ with two components of $\Delta_{Z}$, denoted here by $Z_{p}$ and $Z_{q}$, with corresponding rays $\rho_{p}$ and $\rho_{q}$. The curve $C$ corresponds to a $(r-1)$-dimensional cone $\sigma_{C}$ of $\Sigma$, while the points $p$ and $q$ correspond to the maximal cones generated by $<\sigma_{C},\rho_{p}>$ and $<\sigma_{C},\rho_{q}>$.

Note that this lemma holds even if $Z$ is not proper.
Finally, we have the following vanishing theorem for nef divisors on a proper toric variety.

Following [NXY19], we will say that an $R$-scheme of finite type $\mathscr{Z}$ is toric if there exists a toric $k$-scheme of finite type $\mathcal{Z}$, together with a toric morphism $t:\mathcal{Z}\longrightarrow\mathbb{A}^{1}_{k}$, such that $\mathscr{Z}\simeq\mathcal{Z}\times_{\mathbb{A}^{1}}R$. Writing $\widehat{N}$ for the lattice of 1-parameter subgroups of the torus of $\mathcal{Z}$, such a scheme is described by a fan $\widehat{\Sigma}$ in $\widehat{N}_{\mathbb{R}}$, together with a linear map $\ord(t):\lvert\widehat{\Sigma}\rvert\longrightarrow\mathbb{R}_{\geq 0}$, defined by $\ord(t)(n)=\ord_{0}(t\circ n)$ for a 1-parameter subgroup $n:\mathbb{G}_{m}\rightarrow\mathcal{Z}$. Note that the map $\ord(t)$ recovers the function $t$ uniquely, since it is a monomial.

Let $X$ be a normal variety over $K$. We denote by $X^{\an}$ the Berkovich analytification of $X$. Set-theoretically, it consists of pairs $x=(\xi_{x},v_{x})$ where $\xi_{x}\in X$ and $v_{x}$ is a real-valued valuation on the residue field at $\xi_{x}$ extending the valuation $\ord_{t}$ on $K$. We denote by $\mathscr{H}(x)$ the completion of the residue field at $\xi_{x}$ with respect to $v_{x}$. We endow $X^{\an}$ with the coarsest topology such that

• -

  the forgetful map $\iota:X^{\an}\rightarrow X$, which maps $x=(\xi_{x},v_{x})$ to $\xi_{x}$, is continuous;

• -

  for any Zariski open $U\subseteq X$ and any function $f\in\mathcal{O}_{X}(U)$, the map:

  $$\lvert f\rvert:U^{\an}\coloneqq\iota^{-1}(U)\rightarrow\mathbb{R},$$

  which evaluates $f$ at $x$ associating the value $\lvert f\rvert(x)\coloneqq\exp(-v_{x}(f(\xi_{x})))$, is continuous.

This makes $X^{\an}$ a Hausdorff topological space, which is compact if and only if $X$ is proper over $K$.

Assume that $X/K$ is proper, and let $\mathscr{X}/R$ be a proper model of $X$. By the valuative criterion of properness, for any $x=(\xi_{x},v_{x})\in X^{\an}$ there is a unique lift of the point $\xi_{x}$ to the valuation ring $\mathscr{H}(x)^{\circ}$ of $\mathscr{H}(x)$:

The image of the closed point of $\Spec\mathscr{H}(x)^{\circ}$ under the extended morphism $\Spec\mathscr{H}(x)^{\circ}\rightarrow\mathscr{X}$ is called the center (or specialization) of $x$ and denoted by $c_{\mathscr{X}}(x)$. The map $c_{\mathscr{X}}:X^{\an}\longrightarrow\mathscr{X}_{k}$ turns out to be anticontinuous, i.e. the preimage of an open subset of $X^{\an}$ by $c_{\mathscr{X}}$ is closed in $\mathscr{X}_{k}$.

Let $X$ be a smooth proper variety over $K$. To every dlt model $\mathscr{X}$ of $X$, with special fiber $\mathscr{X}_{k}=\sum_{i\in I}a_{i}D_{i}$, we can associate a cell complex encoding the combinatorics of the intersections of the components $D_{i}$, whose faces are in one-to-one correspondence with strata of $\mathscr{X}_{k}$.

Given any dlt model $\mathscr{X}$ of $X$ over $R$, there is a natural embedding $i_{\mathscr{X}}$ of the dual complex $\mathcal{D}(\mathscr{X}_{k})$ into $X^{\text{an}}$, given as follows. The vertices $v_{i}$ of $\mathcal{D}(\mathscr{X}_{k})$ are in one-to-one correspondence with irreducible components $D_{i}$ of the special fiber $\mathscr{X}_{k}=\lx@nobreakspace\sum_{i\in I}a_{i}D_{i}$, so that we set

where the valuation $\ord_{D_{i}}$ associates to a meromorphic function $f\in K(X)\simeq K(\mathscr{X})$ its vanishing order along $D_{i}$ - the normalisation by $a^{-1}_{i}$ ensuring that $v_{D_{i}}(t)=1$. A valuation given in this way, for some dlt model $\mathscr{X}$ of $X$, is called divisorial. One can now somehow interpolate between those divisorial valuations using quasi-monomial valuations, in order to embed $\mathcal{D}(\mathscr{X}_{k})$ into $X^{\text{an}}$:

The above valuation is called the quasi-monomial valuation associated with the data $(Y,w)$. Then

gives a well-defined continuous injective map from $\mathcal{D}(\mathscr{X}_{k})$ to $X^{\text{an}}$.

Let $\mathscr{X}$ be a good dlt model of a smooth proper $K$-variety $X$. We can now define a retraction for the inclusion $\Sk(\mathscr{X})\subset X^{\text{an}}$ as follows: for any $v\in X^{\an}$, there exists a minimal stratum $Y\subseteq\cap_{j\in J}D_{j}$ of $\mathscr{X}_{k}$ such that the center $c_{\mathscr{X}}(v)$ of $v$ is contained in $Y$. We then associate to $v$ the quasi-monomial valuation $\rho_{\mathscr{X}}(v)$ corresponding to the data $(Y,w)$ with $w_{j}=\frac{1}{q}v(z_{j})$, where $z_{j}$ is a local equation of $qD_{j}$ at the generic point of $Y$, for some $q\in\mathbb{N}_{>0}$. This should be seen as a monomial approximation of the valuation $v$ at the generic point of $Y$, with respect to the model $\mathscr{X}$ (which is snc there).

Let $Y$ be a stratum of $\mathscr{X}_{k}$. The formal scheme $\widehat{\mathscr{X}_{/Y}}$ admits a generic fiber $\mathfrak{X}_{Y}$ in the sense of Berkovich, which is a Berkovich space and can be explicitly described as the open subset of $X^{\an}$:

It furthermore coincides with $\rho^{-1}_{\mathscr{X}}(\Star(\tau_{Y}))\subset X^{\an}$. This Berkovich space comes with a retraction:

which coincides with the restriction of the retraction $\rho_{\mathscr{X}}$. Thus, the restriction of $\rho_{\mathscr{X}}$ over $\Star(\tau_{Y})$ only depends on the formal completion $\widehat{\mathscr{X}_{/Y}}$.

An explicit example of Berkovich retractions, which we will use as a local model in the rest of the paper, is as follows. Let $\mathbb{T}=\mathbb{G}^{n}_{m,K}$ be a torus, with character lattice $M$ and cocharacter lattice $N$. We view the elements $m$ of $M$ as rational functions on $\mathbb{T}$, so that its analytification $\mathbb{T}^{\text{an}}$ comes with a continuous map:

under the identification $N_{\mathbb{R}}=\Hom(M,\mathbb{R})$. The notation $\val$ can be understood as follows: fix an isomorphism $N\simeq\mathbb{Z}^{n}$, so that $\mathbb{T}=\Spec K[M]\simeq\Spec K[X_{1}^{\pm},\ldots,X_{n}^{\pm}]$, and $v_{x}(m)=v_{x}(X^{m})=\sum_{i=1}^{n}m_{i}v_{x}(X_{i})$, so that the map $\val$ reads:

Since $v_{x}(X_{i})=-\log\lvert X_{i}(x)\rvert$, this is the non-archimedean analog of the map $(\mathbb{C}^{*})^{n}\longrightarrow\mathbb{R}^{n}$ sending $(z_{1},\ldots,z_{n})$ to $-(\log\lvert z_{1}\rvert,\ldots,\log\lvert z_{n}\rvert)$.

Now let $\mathscr{X}/R$ be a regular toric model of $\mathbb{T}$, i.e. a regular toric $R$-scheme such that $\mathscr{X}\times_{R}\;\Spec K=\mathbb{T}$, which we assume to have reduced special fiber. Such a model is described by a regular fan $\hat{\Sigma}\subset\hat{N}_{\mathbb{R}}=N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$, whose cones intersect $N_{\mathbb{R}}\times\{0\}$ only at the origin.
We consider the following open subset of $\mathbb{T}^{\an}$:

which admits a Berkovich retraction:

defined as above. In this case, the map $\rho_{\mathscr{X}}$ can be described explicitly as follows: let $\Sigma_{1}$ be the polyhedral complex obtained by intersecting the fan $\hat{\Sigma}$ with $N_{\mathbb{R}}\times\{1\}$. There is a natural identification between $\Sigma_{1}$ and $\mathcal{D}(\mathscr{X}_{k})$, sending a vertex of $\mathcal{D}(\mathscr{X}_{k})$ to the primitive generator of the corresponding ray of $\hat{\Sigma}$, and then extending on each face by linearity. Moreover, it follows from [GJKM19, Theorem A.4] that $\zeta(\lvert\Sigma_{1}\rvert)=\Sk(\mathscr{X})\subset\mathbb{T}^{\an}$.

Let $X$ be a smooth proper variety over $K$.

Note that the above definition implies that $B$ is a topological manifold at $b$; in the case of a Berkovich retraction $\rho_{\mathscr{X}}$, this does not necessarily hold at every point of $\Sk(\mathscr{X})$.

Given a continuous map $\rho:X^{\text{an}}\longrightarrow B$, we denote by $B^{\textrm{sm}}$ the locus of points in $B$ where $\rho$ is an affinoid torus fibration at; we call $B\setminus B^{\textrm{sm}}$ the discriminant or singular locus of $B$. $B^{\textrm{sm}}$ is endowed with an integral affine structure; we recall the definition and describe such structure.

If $\rho$ is an affinoid torus fibration over $B^{\textrm{sm}}\subseteq B$, the integral affine structure on $B^{\textrm{sm}}$ is the pull-back of $\textrm{Aff}_{\mathbb{R}^{n}}$ via the charts in Definition 1.6.1. An alternative description of this structure is given in [KS06, 4.1, Theorem 1]: let $U\subset B^{\textrm{sm}}$ be a connected open subset. Then if $h$ is an invertible analytic function on $\rho^{-1}(U)$, its modulus $\lvert h\rvert$ is constant on the fibers of $\rho$ by the maximum principle, so that it defines a continuous function on the base. We now have:

Let $X/K$ be a smooth $n$-dimensional Calabi–Yau variety: here, this means that $K_{X}=\mathcal{O}_{X}$ (note that this includes for instance the case of abelian varieties). This is the main case we are interested in for applications.
We consider a distinguished class of models of $X$, which we call minimal models. Note that other references may define minimal models in a slightly different way.

The existence of such models is known when $X$ is defined over an algebraic curve (and is expected to hold in the general case).

Such models are not unique, but they turn out to have the same skeleton inside $X^{\an}$ by [NX16] (even though the triangulation may differ), which is thus a canonical piecewise-linear space associated with $X$.

The essential skeleton can also be defined intrinsically as the locus where the weight function associated with a non-vanishing section $\omega\in H^{0}(X,K_{X})$, $\text{wt}_{\omega}:X^{\an}\longrightarrow\mathbb{R}$ defined in [MN15] reaches its minimum; we refer the reader to [MN15] and [NX16] for details.

The maximally degenerate case is of great interest to mirror symmetry. In such case, Kontsevich and Soibelman conjecture that the Gromov-Hausdorff limit of the (rescaled) Kähler Ricci-flat metrics on the fibers $X_{t}$ is the essential skeleton of $X$, endowed with a metric which is given in local affine coordinates by the Hessian $\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}$ of a convex function $\phi$. This is known in the case of abelian varieties by the work of [Oda18], and for Fermat hypersurfaces in $\mathbb{P}_{\mathbb{C}}^{n+1}$ by [Li19]. See also the results in [Li20] for recent progress on this conjecture in the general case.
More precisely, the metric spaces $(X_{t},\omega_{t})$ are conjectured to “look like” the total space of a Lagrangian torus fibration over $\Sk(X)$, submersive away from a singular locus of real codimension 2; the affine structure on the base being induced by action-angle coordinates. Building on these considerations, it is reasonable to expect this affine structure to be non-singular in (real) codimension one.

One of the main motivations in non-archimedean mirror symmetry is to reconstruct this affine structure through Berkovich spaces, interpreting Berkovich retractions as non-archimedean avatars of Lagrangian torus fibrations. The main theorem in [NXY19] establishes the following.