Toric geometry and integral affine structures
in non-archimedean mirror symmetry

Enrica Mazzon Léonard Pille-Schneider

Abstract

We study integral dlt models of a proper $\mathbb{C}((t))$ -variety $X$ along a toric stratum of the special fiber. We prove that the associated Berkovich retraction - from the non-archimedean analytification of $X$ onto the dual complex of the model - is an affinoid torus fibration around the simplex corresponding to the toric stratum, which extends results in [NXY19]. This allows us to construct new types of non-archimedean retractions for maximally degenerate families of quartic K3 surfaces and quintic $3$ -folds, by gluing several non-archimedean SYZ fibrations, each one toric along a codimension one stratum. We then show that the new retractions induce the same singular integral affine structures that arise on the dual complex of toric degenerations in the Gross-Siebert program, as well as on the Gromov-Hausdorff limit of the family.

[04M4]

Introduction

Let $(X,L)$ be a polarized family of $n$ -dimensional Calabi–Yau varieties over the punctured disk $\mathbb{D}^{*}\subset\mathbb{C}$ ; each fiber $X_{t}$ additionally carries a unique Ricci-flat Kähler-metric $\omega_{t}\in c_{1}(L_{t})$ , according to the celebrated Yau theorem. We will be primarily interested in such families that are maximally degenerate, in the following sense: the monodromy acting on the degree $n$ cohomology of the general fiber has a Jordan block of maximal (that is, $n+1$ ) size.

In this setting, the Strominger-Yau-Zaslow conjecture predicts that the general fiber $X_{t}$ admits a fibration $\rho_{t}:X_{t}\longrightarrow S$ , called an SYZ fibration, whose base $S$ is a real $n$ -dimensional topological manifold (even a sphere if the $X_{t}$ are strict Calabi–Yau), and whose fibers are special Lagrangian tori away from a discriminant locus of codimension $2$ in $S$ .
An SYZ fibration endows $S$ with a singular integral affine structure, induced by action-angle coordinates. This means that, in the complement of the discriminant locus of the fibration, the transition functions between charts of $S$ are affine transformations in $\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ .
Moreover, the limit for $t\rightarrow 0$ of the metric spaces $(X_{t},\omega_{t})$ should correspond to the metric collapse of the torus fibers of $\rho_{t}$ . Then, the (suitably rescaled) Gromov-Hausdorff limit of $(X_{t},\omega_{t})$ should coincide with the space $S$ , endowed with a metric which in affine coordinates satisfies a real Monge–Ampère equation away from the discriminant locus.

While some examples of special Lagrangian torus fibrations can be produced, dealing with the general case seems very difficult. The insight of Kontsevich and Soibelman is to replace the above conjecture by an analogous one in the non-archimedean world, and to interpret the latter as an asymptotic limit of the complex phenomenon when $t\rightarrow 0$ . We now elaborate on this idea.

Consider the field $K=\mathbb{C}((t))$ of Laurent power series, which comes equipped with the non-archimedean valuation $\ord_{t}$ , order of vanishing at $t=0$ ; the family $X$ can be viewed as a variety over $K$ . Within this framework, we associate with $X/K$ a topological space, called the Berkovich space $X^{\an}$ of $X$ ; this is a space of real (semi)valuations on $X$ (see Section 1.3).
A way to construct and visualize points of $X^{\an}$ is to consider models of $X$ over $R=\mathbb{C}[[t]]$ . Indeed, any suitably regular (dlt) model $\mathscr{X}$ of $X$ has an associated simplicial subset $\Sk(\mathscr{X})\subset X^{\an}$ , called the skeleton of $\mathscr{X}$ and homeomorphic to the dual (intersection) complex of the degenerate fiber $\mathscr{X}_{0}$ of $\mathscr{X}$ , and a continuous retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ (see Sections 1.4 and 1.5 for more details). It follows that $X^{\an}$ encodes geometric information coming from degenerations of $X$ and about combinatorics of models of $X$ .
Among various models and associated skeletons, minimal (in the sense of MMP) models $\mathscr{X}$ of $X$ determine a canonical skeleton $\Sk(X)=\Sk(\mathscr{X})$ , called the essential skeleton of $X$ and independent of the choice of the minimal model. The essential skeleton and the retractions $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})=\Sk(X)$ , which do depend on $\mathscr{X}$ , are of particular relevance in the non-archimedean reformulation of the SYZ conjecture as the following conjectures point out.

The key idea is that Berkovich theory should allow to construct (non-unique) non-archimedean avatars of SYZ fibrations. More precisely, in [KS06] Kontsevich and Soibelman conjecture that the essential skeleton $\Sk(X)$ can be endowed with an integral affine structure outside of a codimension 2 piecewise-affine subset $\Gamma\subset\Sk(X)$ , such that the following holds. The space $\Sk(X)$ can be recovered from the Kähler geometry of $X$ , as a (suitably rescaled) Gromov-Hausdorff limit of the metric spaces $(X_{t},\omega_{t})$ . Moreover, the limiting metric on $\Sk(X)$ should satisfy the following: outside of $\Gamma$ , the metric is given locally in affine coordinates by the Hessian of a convex function, satisfying a real Monge-Ampère equation. It is furthermore expected that this limiting affine structure can be recovered by a map $\rho:X^{\an}\rightarrow\Sk(X)$ , which is a non-archimedean analog of the SYZ fibration.
The construction of the above fibration is made more rigorous in [NXY19]. The authors prove that the retraction $\rho_{\mathscr{X}}$ associated with a minimal model $\mathscr{X}$ is an affinoid torus fibration away from a codimension 2 locus of the base - the non-archimedean analog of a smooth torus fibration - and induces an integral affine structure there, as the SYZ heuristic and the conjecture by Kontsevich and Soibelman predict. Here in particular the transition functions of the integral affine structure are in $\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{Z}^{n}$ .

The local model for affinoid torus fibrations is the tropicalization map $\textrm{val}:\mathbb{T}^{\an}\rightarrow N_{\mathbb{R}}$ , where $\mathbb{T}=\mathbb{G}_{m,K}^{n}$ and $N$ is the cocharacter lattice of the torus $\mathbb{T}$ . Global examples of such retractions are given as follows: given a (non-proper) toric variety $\mathscr{Y}$ over $R=\mathbb{C}[[t]]$ which is a model of $\mathbb{T}$ , the retraction $\rho_{\mathscr{Y}}$ is a restriction of $\mathrm{val}$ . This reduces the proof of the result in [NXY19] to showing that minimal models $\mathscr{X}$ are in fact toric along one-dimensional strata of the special fiber when the latter is reduced.

At this point the base of the SYZ fibration appears to be well identified - as the essential skeleton or equivalently the dual complex of any minimal model - while the affine structure and the metric are not. In fact, the construction in [NXY19] yields integral affine structures that depend on the additional choice of a model, while the Kontsevich–Soibelman conjecture predicts uniqueness, at least of the metric space. Moreover, the location and the nature of the singularities obtained in [NXY19] differ from previous constructions in mirror symmetry.
Such discrepancy already appears in the case of quintic three-folds in $\mathbb{P}^{4}$ . On one side, the constructions in [Rua01, Gro01] - using symplectic and toric geometry - yield an affine structure on a triangulated $3$ -sphere whose singularities are located away from the vertices. On the other side, the discriminant locus of the non-archimedean SYZ fibration constructed in [NXY19] passes through the vertices of the triangulation.
Moreover, the recent work in [Li19] provides evidence that for a degeneration of Fermat hypersurfaces, the affine structure on the Gromov-Hausdorff limit of the Kähler Ricci-flat metric on the nearby fibers has its singularities located inside the cells of codimension one and away from the vertices.

In this paper we deal with the apparent incompatibility raised by the expected affine structures on the essential skeleton and the ones induced by non-archimedean SYZ fibration. To this purpose, we further develop the non-archimedean approach, and produce examples of a new type of non-archimedean retractions. This allows us to construct singular integral affine structures which are both compatible with SYZ mirror symmetry, and built by means of non-archimedean tools. In this respect, our results provide new evidence for the dictionary between the SYZ heuristic and the non-archimedean interpretation of mirror symmetry.
Inspired by the example [KS06, §4.2.5] of an integral affine structure on the sphere $\mathbb{S}^{2}$ with $24$ singular points, associated with a degeneration of $K3$ surfaces, we move to the $3$ -dimensional case and consider the quintic $3$ -fold as testing ground of our results. More precisely, let $X\rightarrow\mathbb{D}^{*}$ be a generic family of quintics:

X=\{tF_{5}(z_{1},z_{2},z_{3},z_{4},z_{5})+z_{1}z_{2}z_{3}z_{4}z_{5}=0\}\subset\mathbb{P}^{4}\times\mathbb{D}^{*}.

We endow $\Sk(X)\simeq\mathbb{S}^{3}$ with the simplicial structure induced by the identification with the dual complex of $\mathscr{X}_{0}$ , with $\mathscr{X}$ being the closure of $X$ in $\mathbb{P}^{4}\times\mathbb{D}$ , and $\mathscr{X}_{0}$ the fiber over $0$ .

[04M5]

Theorem A.

There exists a continuous retraction $\pi:X^{\an}\rightarrow\Sk(X)$ such that

•

$\pi$ can be written as a composition $\pi^{\prime}\circ\rho_{\mathscr{X}^{\prime}}$ , with $\mathscr{X}^{\prime}$ being an snc model of $X$ and $\pi^{\prime}:\Sk(\mathscr{X}^{\prime})\rightarrow\Sk(X)$ a piecewise-linear map;
•

$\pi$ is an affinoid torus fibration outside a piecewise-linear locus $\Gamma$ , that has codimension $2$ and is contained in the $2$ -skeleton of $\Sk(X)$ ;
•

$\pi$ induces an integral affine structure on $\Sk(X)\setminus\Gamma$ , which is isomorphic to the ones constructed in [Gro01], [Gro05], and [Rua01].

The theorem holds in particular for the Fermat family of quintics; in this case, the Gromov-Hausdorff limit of the family is known [Li19], and naturally induces a singular affine structure on $\mathbb{S}^{3}$ . We are able to show (see Proposition 4.8.1) that the non-archimedean retraction in Theorem A, the metric limit as determined by Li, and the works of Gross and Ruan, all induce the same integral affine structure on $\mathbb{S}^{3}$ ; this provides a new piece of evidence for the dictionary between SYZ mirror symmetry and Berkovich geometry.

The main idea behind our construction is to consider the Berkovich retractions $\rho_{\mathscr{X}_{i}}$ associated with several minimal models $\mathscr{X}_{1},\ldots,\mathscr{X}_{N}$ of $X$ , adapted to different regions of $\Sk(X)$ and glued together. In order to prove Theorem A and describe the singular locus of the retraction $\pi$ , we establish the following result, which generalizes [NXY19, Proposition 5.4]:

[04M6]

Theorem B.

Let $X/K$ be a smooth projective variety, and $\mathscr{X}/R$ be a dlt model of $X$ with reduced special fiber $\mathscr{X}_{k}$ , such that the irreducible components of $\mathscr{X}_{k}$ are all Cartier divisors.
Let $Z$ be a stratum of $\mathscr{X}_{k}$ , such that:

•

$\mathring{Z}\subset Z$ is a torus embedding, where $\mathring{Z}$ is the open stratum of $Z$ ;
•

the conormal bundle $\nu_{Z/\mathscr{X}}^{*}$ is a nef vector bundle on $Z$ ;
•

the intersection of $Z$ with any irreducible component of $\mathscr{X}_{k}$ is connected.

Then $\mathscr{X}$ is toric along $Z$ (in the sense of Definition 1.2.6).

Note that the dlt assumption, combined with the fact that $Z$ is toric, imply that $Z$ is smooth (see Remark 2.1.1). By assumption, $Z$ is (a connected component of) the intersection of the divisors $D_{j}\subset\mathscr{X}_{k}$ containing $Z$ and the $D_{j}$ ’s are Cartier, so that $\nu^{*}_{Z/\mathscr{X}}=\bigoplus_{j\in J}\mathcal{O}_{Z}(-D_{j})$ and the nef assumption simply means that each of the $\mathcal{O}_{Z}(-D_{j})$ is a nef divisor.
Using the positivity of the conormal bundle, we then prove that in a formal neighbourhood of $Z$ , $\mathscr{X}$ is isomorphic to the normal bundle of $Z$ , which is a toric variety. This is similar in spirit to the classical work of [Gra62, Satz 7, p. 363] on holomorphic tubular neighbourhoods, as well as Grothendieck’s algebraization theorem [Gro61, Theorem 5.1.4]; the key technical point being the vanishing of the higher cohomology groups of the powers of $\nu^{*}_{Z/\mathscr{X}}$ which allows us to extend combinatorial data from $Z$ to a formal neighbourhood.

[04M7]

Corollary C.

The retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ is an affinoid torus fibration over $\Star(\tau_{Z})$ .

The subset $\Star(\tau_{Z})$ is the open star of the face determined by $Z$ (see Definition 1.4.2). Theorem B and Corollary C show that the discriminant locus of the retraction $\rho_{\mathscr{X}}$ measures the defect of a stratum to being toric. Therefore, to prove Theorem A, we combine retractions $\rho_{\mathscr{X}_{i}}$ coming from different models $\mathscr{X}_{1},\ldots,\mathscr{X}_{N}$ with the following property: for each region of $\Sk(X)\setminus\Gamma$ , there exists a model $\mathscr{X}_{i}$ such that $\pi=\rho_{\mathscr{X}_{i}}$ and $\mathscr{X}_{i}$ satisfies the hypothesis of Theorem B, hence defines an affinoid torus fibration over the corresponding region.
If $\dim Z=1$ and $Z$ is a rational curve, then the positivity assumption can always be achieved via a finite number of blow-ups and Corollary C holds; this was established in [NXY19, Proposition 5.4].

The connection between toric geometry and mirror symmetry has been explored in several ways; in particular, the Gross–Siebert program considers toric degenerations of Calabi–Yau varieties. Such degenerations satisfy assumptions similar to the ones in Theorem B, as the irreducible components of the special fiber are all assumed to be toric varieties; however, note that they are not assumed to be $\mathbb{Q}$ -Cartier, so that there may not be an associated Berkovich retraction. In [GS06] the authors then glue together the fans of the various components of the special fiber to combinatorially construct an affine structure on the skeleton of the degeneration; see Section 3.1.1 for an example in dimension 2. Using tools from non-archimedean geometry, Theorem B in particular enables us to generalize this construction to degenerations that are not necessarily toric.

The retraction in Theorem A should be closely related to the tropical contractions constructed in [Yam21], from a tropical Calabi–Yau variety $V$ to an associated integral affine manifold $B$ . In particular, the setting of Yamamoto applies to toric degenerations of Calabi–Yau varieties constructed by Gross [Gro05]; in such case, the tropical contraction maps onto the dual complex of the degeneration and induces on it the singular integral affine structure defined in [Gro05]. Relations between (co)homology groups of $V$ and $B$ are also studied in [Yam21].

Finally, the non-archimedean retraction $\pi$ of Theorem A yields an integral affine structure whose discriminant locus is of codimension 2. We recall that the $2$ -codimensionality of the discriminant is expected from the SYZ heuristic at the topological level; in Theorem A, this is achieved by construction of $\pi^{\prime}$ , building on the results in [NXY19]. In the setting of the Gross–Siebert program, given a singular integral affine manifold $B$ , one can produce, using classical moment maps, a topological torus fibration over $B$ , which however has a discriminant locus of codimension 1, see [RS20, §2.1]. In the series of recent or upcoming papers [RZ21a, RZ21b, RZ], Ruddat and Zharkov develop a strategy which solves this problem - at least at the topological level - and works in arbitrary dimension.
More precisely, the authors are able to construct a torus fibration with discriminant locus of codimension 2 in $B$ , isotopic to the previous fibration, and with a symplectic structure on the complement of a codimension 2 subset in the total space. Note that the total space of this resulting fibration will not be a manifold in general, as it could have orbifold singularities.

Let us briefly describe the organization of the paper.
In Section 1, we introduce some notation and collect some basic facts about toric varieties. We also define Berkovich spaces and recall the definition of skeletons and retractions we will be using. Section 2 is devoted to the proof of Theorem B. In Section 3 we give a detailed description of the monodromy of the integral affine structures induced by Berkovich retractions, or combinations of them. Finally, in Section 4 we study in detail the example of the degeneration of quintic 3-folds and prove Theorem A by applying the results of the previous sections. We also compare our results to various constructions existing in the literature.

Acknowledgements. We would like to thank Sébastien Boucksom and Mirko Mauri for their comments on the first version of this paper. We are also grateful to Omid Amini, Johannes Nicaise, Helge Ruddat, Yuto Yamamoto for helpful conversations. Enrica Mazzon was partially supported by Max Planck Institute for Mathematics in Bonn during the preparation of this paper.

[04M8]

1 Preliminaries

Throughout this paper, $k$ is an algebraically closed field of characteristic zero, $K=k((t))$ and $R=k[[t]]$ . The field $K$ is endowed with the non-archimedean absolute value $\lvert\cdot\rvert=e^{-\ord_{t}}$ , which makes $K$ a complete non-archimedean field with valuation ring $R$ .

[04M9]

1.1 Models

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 $\mathring{Z}$ .

[04MA]

Definition 1.1.1.

Let $\mathscr{X}/R$ be a model of $X$ . We say that $\mathscr{X}$ is a dlt (divisorially log terminal) model of $X$ if the following conditions hold:

-

the pair $(\mathscr{X},\mathscr{X}_{k,\red})$ is log canonical in the sense of the Minimal Model Program (see [KM98]);
-

the pair $(\mathscr{X},\mathscr{X}_{k,\red})$ is simple normal crossing at the generic points of log canonical centers of $(\mathscr{X},\mathscr{X}_{k,\red})$ .

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.

[04MB]

1.2 Toric geometry

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 $\textrm{div}(z^{m})$ .

[04MC]

Lemma 1.2.1 ([Ful93, 3.4]).

The following sequence

0\longrightarrow M\xrightarrow{p}\Div^{\mathbb{T}}(Z)\xrightarrow{q}\Pic(Z)\longrightarrow 0

is exact.

Let us rephrase this in term of coordinates, after fixing an isomorphism $N\simeq\mathbb{Z}^{r}$ and denoting $L=\{u_{1},\ldots,u_{s}\}$ the primitive generators of the 1-dimensional cones of $\Sigma$ . Since by [Ful93, Lemma p.61], we have $\ord_{Z_{l}}(z^{m})=\langle u_{l},m\rangle$ , we obtain $\textrm{div}(z^{m})=\sum_{l\in L}\langle u_{l},m\rangle Z_{l}$ and hence $p(m)=(\langle u_{l},m\rangle)_{l\in L}$ . We deduce the following explicit description of $\Pic(Z)$ :

[04MD]

Corollary 1.2.2.

Let $u_{l}=(u_{l,1},\ldots,u_{l,r})$ for $l\in\{1,\ldots,s\}$ . Then $\Pic(Z)$ is generated by the line bundles $\mathcal{O}_{Z}(Z_{l})$ , with the $r$ relations:

\mathcal{O}_{Z}(\sum_{l=1}^{m}u_{l,1}Z_{l})=\ldots=\mathcal{O}_{Z}(\sum_{l=1}^{m}u_{l,r}Z_{l})=0.

In particular, the divisors in the $r$ -tuple

\Delta=\sum_{l\in L}u_{l}\otimes Z_{l}\,\in\,N\otimes\Div^{\mathbb{T}}(Z)\simeq(\Div^{\mathbb{T}}(Z))^{r}

are principal.

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 $<\sigma_{C},\rho_{q}>$ .

[04ME]

Lemma 1.2.3 ([Ful93, p. 99]).

The primitive generators of the rays of the fan satisfy the following relation:

u_{p}+u_{q}=-\sum_{u_{l}\in\sigma_{C}}(C\cdot Z_{l})u_{l}.

Observing that we have $(C\cdot Z_{p})=(C\cdot Z_{q})=1$ , and $(C\cdot Z_{l})=0$ for any other $l$ , this may be rewritten in a more synthetic way:

(1.2.4)

\sum_{l\in L}(C\cdot Z_{l})u_{l}=0.

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.

[04MF]

Proposition 1.2.5.

Let $D$ be a nef Cartier divisor on a proper toric variety $Z$ . Then $H^{i}(Z,\mathcal{O}_{Z}(D))=0$ for $i>0$ .

[04MG]

Proof.

The divisor $D$ being nef is equivalent to it being globally generated, by [Mus02, Theorem 3.1]. Thus, the result follows directly from [Ful93, p. 74]. ∎

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.

[04MH]

Definition 1.2.6.

Let $\mathscr{X}$ be a normal $R$ -scheme of finite type, and $Y$ be a stratum of $\mathscr{X}_{k}$ . We say that $\mathscr{X}$ is toric along Y if there exists a toric $R$ -scheme $\mathscr{Z}$ , a stratum $W$ of $\mathscr{Z}_{k}$ and a formal isomorphism over $R$

\widehat{\mathscr{X}_{/Y}}\simeq\widehat{\mathscr{Z}_{/W}}.

[04MI]

1.3 Berkovich spaces

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 $\mathscr{X}_{k}$ .

[04MJ]

1.4 Skeletons

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 $\mathscr{X}_{k}$ .

[04MK]

Definition 1.4.1.

We call simplex a topological space, endowed with a $\mathbb{Z}$ -affine structure, which is $\mathbb{Z}$ -affine isomorphic to a space of the form:

\tau=\{\sum_{i=0}^{m}a_{i}w_{i}=1\}\subset\mathbb{R}^{m+1},\quad\textrm{ for some $a_{i}\in\mathbb{N}_{\geqslant 0}$}.

[04ML]

Definition 1.4.2.

Let $\mathscr{X}$ be a dlt model of $X$ . To each stratum $Y$ of $\mathscr{X}_{k}$ which is a connected component of $D_{J}$ , we associate a simplex:

\tau_{Y}=\{w\in\mathbb{R}_{\geqslant 0}^{|J|}\,|\sum_{j\in J}a_{j}w_{j}=1\}.

We define the cell complex $\mathcal{D}(\mathscr{X}_{k})$ by the following incidence relations: $\tau_{Y}$ is a face of $\tau_{Y^{\prime}}$ if and only if $Y^{\prime}\subset Y$ .

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

i_{\mathscr{X}}(v_{i})=v_{D_{i}}\coloneqq a^{-1}_{i}\ord_{D_{i}},

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 $X^{\text{an}}$ :

[04MM]

Proposition 1.4.3 ([MN15, Proposition 2.4.4]).

Let $\mathscr{X}$ be a dlt model of $\mathscr{X}$ , with special fiber $\mathscr{X}_{k}=\sum_{i\in I}a_{i}D_{i}$ . Let $J\subset I$ such that $D_{J}=\cap_{j\in J}D_{j}$ is non-empty, and $Y$ a connected component of $D_{J}$ , with generic point $\eta$ . We furthermore fix a local equation $z_{j}\in\mathcal{O}_{\mathscr{X},\eta}$ for $D_{j}$ , for any $j\in J$ .
Then, for any $w\in\tau_{Y}=\{w\in\mathbb{R}^{|J|}_{\geqslant 0}\,|\sum_{j\in J}a_{j}w_{j}=1\}$ , there exists a unique valuation

v_{w}:\mathcal{O}_{\mathscr{X},\eta}\longrightarrow\mathbb{R}_{\geqslant 0}\cup\{+\infty\}

such that for every $f\in\mathcal{O}_{\mathscr{X},\eta}$ , with expansion $f=\sum_{\beta\in\mathbb{N}^{|J|}}c_{\beta}z^{\beta}$ (with $c_{\beta}$ either zero or unit), we have:

v_{w}(f)=\min\{(w\cdot\beta)\,|\beta\in\mathbb{N}^{|J|},c_{\beta}\neq 0\},

where $(\;\cdot\;)$ is the usual scalar product on $\mathbb{R}^{|J|}$ .

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

	$\displaystyle i_{\mathscr{X}}:$	$\displaystyle\quad\mathcal{D}(\mathscr{X}_{k})\rightarrow X^{\text{an}}$
		$\displaystyle\quad\tau_{Y}\ni w\mapsto v_{w}$

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

[04MN]

Definition 1.4.4.

We call the image of $\mathcal{D}(\mathscr{X}_{k})$ by $i_{\mathscr{X}}$ the skeleton of $\mathscr{X}$ , written as $\Sk(\mathscr{X})\subset X^{\emph{an}}$ . It is a cell complex of dimension at most $\dim X$ .

By compactness of $\mathcal{D}(\mathscr{X}_{k})$ , $i_{\mathscr{X}}$ induces a homeomorphism between $\mathcal{D}(\mathscr{X}_{k})$ and $\Sk(\mathscr{X})$ , so that we will sometimes abusively identify $\mathcal{D}(\mathscr{X}_{k})$ with $\Sk(\mathscr{X})$ .

[04MP]

Definition 1.4.5.

Let $Y$ be a stratum of $\mathscr{X}_{k}$ . We define $\Star(\tau_{Y})$ as the union of open faces in $\Sk(\mathscr{X})$ whose closure contains $\tau_{Y}$ .

[04MQ]

1.5 Berkovich retractions

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).

[04MR]

Definition 1.5.1.

The above map $\rho_{\mathscr{X}}:X^{\an}\longrightarrow\Sk(\mathscr{X})$ is the Berkovich retraction associated with the model $\mathscr{X}/R$ .

The Berkovich retraction is continuous, restricts to the identity on $\Sk(\mathscr{X})$ , and by [Thu07, Ber99] $\rho_{\mathscr{X}}$ is a strong deformation retraction, i.e. there is a homotopy between $\rho_{\mathscr{X}}$ and the identity on $X^{\an}$ that fixes the points of $\Sk(\mathscr{X})$ . It follows that $X^{\an}$ and $\Sk(\mathscr{X})$ are homotopy equivalent.

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 $X^{\an}$ :

\mathfrak{X}_{Y}=\{x\in X^{\an}\lvert\,c_{\mathscr{X}}(v_{x})\in Y\}.

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

\rho_{Y}:\mathfrak{X}_{Y}\longrightarrow\Star(\tau_{Y}),

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:

	$\displaystyle\val:$	$\displaystyle\,\mathbb{T}^{\text{an}}\longrightarrow N_{\mathbb{R}},$
		$\displaystyle v_{x}\longmapsto(m\mapsto v_{x}(m)),$

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:

	$\displaystyle\val:$	$\displaystyle\,\mathbb{T}^{\text{an}}\longrightarrow\mathbb{R}^{n},$
		$\displaystyle v_{x}\longmapsto(v_{x}(X_{i}))_{i=1,\ldots,n}.$

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 $-(\log\lvert z_{1}\rvert,\ldots,\log\lvert z_{n}\rvert)$ .

The map $\val$ admits a continuous section $\zeta:N_{\mathbb{R}}\longrightarrow\mathbb{T}^{\text{an}}$ , sending a point $n\in N_{\mathbb{R}}$ to the Gauss point of the affinoid torus $\val^{-1}(n)$ . More explicitly, for $x\in\mathbb{T}^{\an}$ , the valuation $\zeta(\val(x))$ is the valuation on the function field of $\mathbb{T}$ defined by the following formula:

\zeta(\val(x))\big(\sum_{m\in M}\alpha_{m}z^{m}\big)=\min_{\alpha_{m}\neq 0}\big(\ord_{t}(\alpha_{m})+v_{x}(z^{m})\big).

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 $\mathbb{T}^{\an}$ :

\widehat{\mathscr{X}}_{\eta}:=\{v_{x}\in\mathbb{T}^{\an}\lvert\,v_{x}\;\text{has a center on}\;\mathscr{X}\},

which admits a Berkovich retraction:

\rho_{\mathscr{X}}:\widehat{\mathscr{X}}_{\eta}\longrightarrow\Sk(\mathscr{X})

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 $\zeta(\lvert\Sigma_{1}\rvert)=\Sk(\mathscr{X})\subset\mathbb{T}^{\an}$ .

[04MS]

Proposition 1.5.2 ([NXY19, Example 3.5]).

The equality:

\widehat{\mathscr{X}}_{\eta}=\val^{-1}(\lvert\Sigma_{1}\rvert)

holds, and $\rho_{\mathscr{X}}=\val_{|\widehat{\mathscr{X}}_{\eta}}$ .

[04MT]

Proof.

We start by proving the first equality. Let $x\in\mathbb{T}^{\an}$ , we know from Lemma 1.5.3 below that $x$ has a center on $\mathscr{X}$ if and only if $\zeta(\val(x))$ has a center on $\mathscr{X}$ . Thus, it is enough to prove that for $n\in N_{\mathbb{R}}$ and $y=\zeta(n)$ , $y$ has a center on $\mathscr{X}$ if and only if $n\in\lvert\Sigma_{1}\rvert$ .
The elements $y\in\zeta(N_{\mathbb{R}})$ are precisely the valuations invariant under the torus action, hence if $y$ has a center on $\mathscr{X}$ , it must be the closure of a torus orbit $Y\subset\mathscr{X}_{k}$ . By [KKMSD73, Theorem 6], there exists a cone $\sigma\in\hat{\Sigma}$ such that the generic point of $Y$ is contained in the associated toric affine chart $\mathscr{X}_{\sigma}=\Spec R[\check{\sigma}\cap\hat{M}]$ . In particular, for any monomial $z^{m}$ that is regular on $\mathscr{X}_{\sigma}$ , we have $v_{y}(z^{m})\geq 0$ . In other words, writing $y=\zeta(n)$ , we have $\langle n,m\rangle\geq 0$ for all $m\in\check{\sigma}$ , so that $n\in\sigma$ . Since $v_{y}(t)=1$ , $y\in\zeta(\lvert\Sigma_{1}\rvert)$ .
By the same argument, if $n\in\lvert\Sigma_{1}\rvert$ , there exists a cone $\sigma$ such that $n\in\sigma$ , which means that $v_{\zeta(n)}$ has positive value on each monomial $m\in\check{\sigma}$ , and thus has a center on $\mathscr{X}_{\sigma}$ and in particular on $\mathscr{X}$ .

To prove the second equality, since $\rho_{\mathscr{X}}$ is the identity on $\zeta(\lvert\Sigma_{1}\rvert)=\Sk(\mathscr{X})$ , we merely have to prove that $\rho_{\mathscr{X}}=\rho_{\mathscr{X}}\circ\val$ . However this follows directly from the definition of $\rho_{\mathscr{X}}$ , and the fact that $c_{{\mathscr{X}}}(x)\in\overline{c_{{\mathscr{X}}}(\zeta(\val(x)))}$ for $x\in\widehat{\mathscr{X}}_{\eta}$ by Lemma 1.5.3. Indeed, $\rho_{\mathscr{X}}(x)$ only depends on the values $v_{x}(z)$ , where $z$ is a local equation for a component of $\mathscr{X}_{k}$ at $c_{\mathscr{X}}(x)$ . Since $\mathscr{X}$ is a toric model, these local equations can be taken to be monomials, so that the result follows from the fact that $x$ and $\zeta(\val(x))$ take the same values on monomials. ∎

[04MU]

Lemma 1.5.3.

Let $x\in\mathbb{T}^{\an}$ . Then $x$ has a center on $\mathscr{X}$ if and only if $\zeta(\val(x))$ has a center on $\mathscr{X}$ . Moreover, if this holds, we have $c_{{\mathscr{X}}}(x)\in\overline{c_{{\mathscr{X}}}(\zeta(\val(x)))}$ .

[04MV]

Proof.

Let $\mathscr{X}\subset\bar{\mathscr{X}}$ be a toric compactification of $\mathscr{X}$ , i.e. a proper toric $R$ -scheme containing $\mathscr{X}$ as a torus-invariant open subset. By the valuative criterion of properness, any valuation of $\mathbb{T}^{\an}$ has a center on $\bar{\mathscr{X}}$ . We write $c_{\bar{\mathscr{X}}}(x)$ for the center of $x\in\mathbb{T}^{\an}$ .

We start by proving that $c_{\bar{\mathscr{X}}}(\zeta(\val(x))$ is the generic point of the minimal closed torus orbit $Z$ in $\bar{\mathscr{X}}$ containing $c_{\bar{\mathscr{X}}}(x)$ . We may work on the toric affine chart $\mathscr{X}_{\sigma}=\Spec R[\check{\sigma}\cap\hat{M}]$ associated with $Z$ . Since the valuation $\zeta(\val(x))$ is monomial, it is enough to prove that $\zeta(\val(x))(z^{m})=v_{x}(z^{m})\geq 0$ for $m\in\check{\sigma}\cap M$ and that $\zeta(\val(x))(z)>0$ for $z$ a local equation of any torus invariant divisor containing $Z$ , to have that $c_{\bar{\mathscr{X}}}(\zeta(\val(x))$ lies in $Z$ . Since $z^{m}$ is regular on $\mathscr{X}_{\sigma}$ , the first condition holds; the local equation $z$ is monomial and $\zeta(\val(x))(z)=v_{x}(z)>0$ since $Z$ contains $c_{\bar{\mathscr{X}}}(x)$ . Moreover, we conclude that $c_{\bar{\mathscr{X}}}(x)$ must be contained in the toric interior of $Z$ by minimality of $Z$ .

Now assume that $v_{x}$ is centered on $\mathscr{X}$ , i.e. $c_{\bar{\mathscr{X}}}(x)\in\mathscr{X}$ . Since $\mathscr{X}$ is torus-invariant and $c_{\bar{\mathscr{X}}}(x)\in Z$ , we have $Z\subset\mathscr{X}$ , hence its generic point $c_{\bar{\mathscr{X}}}(\zeta(\val(x)))\in\mathscr{X}$ . This implies that $\zeta(\val(x))$ has center on $\mathscr{X}$ . Conversely, if $\zeta(\val(x))$ has a center on $\mathscr{X}$ , then $Z\subset\mathscr{X}$ and $c_{\bar{\mathscr{X}}}(x)\in Z$ as mentioned above; thus, $c_{\mathscr{X}}(x)\in\mathscr{X}$ , which concludes the proof. ∎

[04MW]

1.6 Affinoid torus fibrations and integral affine structures

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

[04MX]

Definition 1.6.1.

Let $\rho:X^{\text{an}}\longrightarrow B$ be a continuous map to a topological space $B$ . For any point $b\in B$ , we say that $\rho$ is an affinoid torus fibration at $b$ if there exists an open neighbourhood $U$ of $b$ in $B$ , such that the restriction to $\rho^{-1}(U)$ fits into a commutative diagram:

$V$ being an open subset of $\mathbb{R}^{n}$ , the upper horizontal map an isomorphism of analytic spaces, the lower horizontal map a homeomorphism, and the map $\val$ defined as in Section 1.5.

[04MY]

Example 1.6.2.

It follows from the definition of good dlt model $\mathscr{X}$ of $X$ that the Berkovich retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ is an affinoid torus fibration over the interior of the maximal faces $\tau$ of $\Sk(\mathscr{X})$ . Indeed, the retraction over $\textrm{Int}({\tau})$ only depends on the formal completion of $\mathscr{X}$ along the corresponding 0-dimensional stratum $p$ . The pair $(\mathscr{X},\mathscr{X}_{k})$ is snc at $p$ , hence the claim.

[04MZ]

Example 1.6.3.

If $\mathscr{X}/R$ is a toric model of $X=\mathbb{T}$ , it follows from Proposition 1.5.2 that the Berkovich retraction:

\rho_{\mathscr{X}}:\widehat{\mathscr{X}}_{\eta}\longrightarrow\Sk(\mathscr{X})

is an affinoid torus fibration over the interior of $\Sk(\mathscr{X})$ . This also holds when $X$ is a regular proper toric variety over $K$ , and $\mathscr{X}$ a regular proper toric model, by [GJKM19, Theorem A.4].

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.

[04N0]

Definition 1.6.4.

An integral affine structure on a topological manifold is an atlas of charts with transition functions in $\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ .

[04N1]

Definition 1.6.5.

An integral affine function on an open subset of $\mathbb{R}^{n}$ is a continuous real-valued function locally of the form $f(x_{1},\ldots,x_{n})=a_{1}x_{1}+\ldots+a_{n}x_{n}+b$ , with $a_{i}\in\mathbb{Z}$ and $b\in\mathbb{R}$ . We denote by $\textrm{Aff}_{\mathbb{R}^{n}}$ the sheaf of integral affine functions on $\mathbb{R}^{n}$ .

[04N2]

Lemma 1.6.6 ([KS06, 2.1]).

An integral affine structure on a topological manifold $M$ is equivalent to the datum of a subsheaf $\mathrm{Aff}_{M}$ of the sheaf of continuous functions on $M$ such that $(M,\mathrm{Aff}_{M})$ is locally isomorphic to $(\mathbb{R}^{n},\mathrm{Aff}_{\mathbb{R}^{n}})$ .

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:

\mathrm{Aff}_{B^{\textrm{sm}}}(U)=\{-\log\lvert h\rvert\,|\,h\in\mathcal{O}^{\times}_{X^{\an}}(\rho^{-1}(U))\}.

[04N3]

Remark 1.6.7.

Given an integral affine structure on a topological manifold $M$ , there is a monodromy representation

T:\pi_{1}(M)\rightarrow\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}

defined by covering a loop in $M$ by affine charts and composing the corresponding transition functions. See [KS06, 2.2] for more details.

[04N4]

1.7 The Calabi–Yau case

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.

[04N5]

Definition 1.7.1.

Let $X/K$ be a Calabi–Yau variety. A minimal model of $X$ is a good dlt model $\mathscr{X}/R$ , such that the logarithmic relative canonical divisor is trivial, i.e.

K^{\log}_{\mathscr{X}/R}\coloneqq K_{\mathscr{X}/R}+\mathscr{X}_{k,\red}-\mathscr{X}_{k}\sim\mathcal{O}_{\mathscr{X}}.

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

[04N6]

Theorem 1.7.2 ([NXY19, Theorem 1.13]).

Let $X/K$ be a projective Calabi–Yau variety, and assume that $X$ is defined over an algebraic curve. Then there exists a minimal model $\mathscr{X}/R$ of $X$ . Furthermore, there exists a finite extension $K^{\prime}/K$ such that the base change $X_{K^{\prime}}$ admits a minimal model with reduced special fiber.

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 $X$ .

[04N7]

Definition 1.7.3.

Let $X/K$ be a Calabi–Yau variety. The essential skeleton $\Sk(X)\subset X^{\an}$ is the skeleton of any minimal model $\mathscr{X}/R$ of $X$ .

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.

[04N8]

Definition 1.7.4.

Let $X/K$ be a Calabi–Yau variety. We will say that $X$ is maximally degenerate if the skeleton $\Sk(X)$ has maximal dimension, i.e. $\dim\Sk(X)=n$ .

[04N9]

Example 1.7.5.

In the $2$ -dimensional case, maximally degenerate Calabi–Yau surfaces coincide with $K3$ surfaces of Type III.

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.

[04NA]

Theorem 1.7.6 ([NXY19, Theorem 6.1]).

Let $X/K$ be a maximally degenerate projective Calabi–Yau variety, and let $\mathscr{X}/R$ be a minimal model of $X$ with reduced special fiber. Then the Berkovich retraction

\rho_{\mathscr{X}}:X^{\an}\longrightarrow\Sk(X)

is an affinoid torus fibration away from the codimension 2 locus of the triangulation induced by the homeomorphism $\Sk(X)\simeq\mathcal{D}(\mathscr{X}_{k})$ .

This statement is proved by showing that $\mathscr{X}$ is toric along the 1-dimensional strata of the special fiber, which yields on the way a complete description of such models along these strata. This description also provides a way to compute the singular $\mathbb{Z}$ -affine structure induced on $\Sk(X)$ , as well as its monodromy. In Section 3.1 and Section 3.3.1 we give more details and examples of this computations.

[04NB]

Example 1.7.7.

If $S$ is a K3 surface of Type III, and $\mathscr{X}/R$ a minimal model of $S$ , then the map $\rho_{\mathscr{X}}$ is an affinoid torus fibration away from the vertices of $\mathcal{D}(\mathscr{X}_{k})$ .
The induced $\mathbb{Z}$ -affine structure with isolated singularities on the 2-sphere matches the one constructed in [GHK15, 1.2] and [Eng18, Proposition 3.10], and has no singularity at a vertex $v_{D}$ if and only if the corresponding component $D$ of $\mathscr{X}_{k}$ is toric.

[04NC]

2 Toric structure along toric strata

In this section we prove Theorem B. We recall the statement and fix the notation.

[04ND]

Theorem B.

Let $X/K$ be a smooth projective variety of dimension $n$ , and $\mathscr{X}/R$ be a dlt model of $X$ with reduced special fiber $\mathscr{X}_{k}=\sum_{\alpha}D_{\alpha}$ , such that every $D_{\alpha}$ is a Cartier divisor.
Let $Z=D_{0}\cap D_{1}\cap\ldots\cap D_{n-r}$ be an $r$ -dimensional stratum of $\mathscr{X}_{k}$ , such that:

•

$\mathring{Z}\subset Z$ is a torus embedding, where $\mathring{Z}=Z\setminus\cup_{\alpha\neq 0,1,\ldots,n-r}D_{\alpha}$ ;
•

the conormal bundle $\nu_{Z/\mathscr{X}}^{*}$ is a nef vector bundle on $Z$ ;
•

for each $\alpha\notin\{0,...,n-r\}$ , the intersection $D_{\alpha}\cap Z$ is either empty or connected.

Then the formal completion $\widehat{\mathscr{X}_{/Z}}$ is isomorphic to the formal completion of the normal bundle $\mathcal{N}=\nu_{Z/\mathscr{X}}$ along the zero section. In particular, $\mathscr{X}$ is toric along $Z$ (in the sense of Definition 1.2.6).

Note that the assumptions in Theorem B imply that $Z$ is the smooth complete intersection of the irreducible components $D_{j}$ of $\mathscr{X}_{k}$ containing $Z$ , and thus has simple normal crossing boundary, see Remark 2.1.1. Since $Z$ is a complete intersection, the conormal bundle $\nu^{*}_{Z/\mathscr{X}}$ is the direct sum of the line bundles $\mathcal{O}_{Z}(-D_{j})$ . Hence, the nefness assumption simply means that the $D_{j}$ ’s containing $Z$ are anti-nef divisors on $Z$ .

As an immediate consequence of the theorem, we prove that

[04NE]

Corollary C.

The retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ is an $n$ -dimensional affinoid torus fibration over $\Star(\tau_{Z})$ . In particular, the integral affine structure induced by $\rho_{\mathscr{X}}$ on the complement of the faces of $\Sk(\mathscr{X})$ of codimension $\geqslant 2$ extends to $\Star(\tau_{Z})$ with no singularities.

[04NF]

Proof.

Although this follows from Theorem B by [NXY19, Theorem 6.1] (end of the proof) and by [NXY19, §3.4], we sketch the proof for reader’s convenience.
As mentioned in Section 1.5, the retraction $\rho_{\mathscr{X}}$ over $\Star(\tau_{Z})$ only depends on $\widehat{\mathscr{X}_{/Z}}$ , so that by Theorem B we may assume that $\mathscr{X}$ is a toric $R$ -scheme. The equality $\rho_{\mathscr{X}}=\val$ now holds over $\Star(\tau_{Z})$ by Proposition 1.5.2, so that it follows from Definition 1.6.1 that $\rho_{\mathscr{X}}$ is an affinoid fibration over $\Star(\tau_{Z})$ . ∎

[04NG]

2.1 Notation and strategy

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 $\Delta_{Z}=\sum_{l\in L}Z_{l}$ .

[04NH]

Remark 2.1.1.

The dlt assumption on $\mathscr{X}$ and the toricness of $Z$ ensure that $Z$ is smooth, and that $(Z,\Delta_{Z})$ is an snc pair. Indeed, the singular locus of $Z$ is a union of torus invariant subvarieties (see [CLS11, Proposition 11.1.2]), hence the generic point of a component of the singular locus is the generic point of a stratum of $\Delta_{Z}$ . However, $(Z,\Delta_{Z})$ is a dlt pair, thus snc at the generic point of each stratum of $\Delta_{Z}$ .

[04NI]

Remark 2.1.2.

The smoothness of $Z$ and the assumption that the components of $\mathscr{X}_{k}$ are Cartier divisors imply that $\mathscr{X}$ is regular at any point of $Z$ . Indeed, for any point $p\in Z$ and $j\in J$ , let $z_{j}\in\mathcal{O}_{\mathscr{X},p}$ be a local equation of $D_{j}$ at $p$ . As $\mathcal{O}_{Z,p}\simeq\mathcal{O}_{\mathscr{X},p}/(z_{0},\ldots,z_{n-r})$ is a regular local ring of dimension $r$ , $(z_{0},\ldots,z_{n-r})$ can be extended to form a regular system of parameters for $\mathcal{O}_{\mathscr{X},p}$ .

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 $L_{\sigma}\coloneqq\{l\in L\,|\,u_{l}\in\sigma\}$ .

[04NJ]

Lemma 2.1.3.

For any maximal cone $\sigma$ of $\Sigma$ , we have $\det((u_{l})_{l\in L_{\sigma}})=\pm 1.$

[04NK]

Proof.

The smoothness of $Z$ (see Remark 2.1.1) implies that the primitive generators of $\sigma$ form a $\mathbb{Z}$ -basis of $N$ , which is equivalent to the condition $\det((u_{l})_{l\in L_{\sigma}})=\pm 1.$ ∎

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.

[04NL]

Proposition 2.1.6.

For any 1-dimensional toric stratum $C\subseteq Z$

(2.1.7)

\sum_{j\in J}(C\cdot D_{j})v_{j}+\sum_{l\in L}(C\cdot D_{l})v_{l}=0\;\text{in}\;N_{\mathbb{R}}\times\mathbb{R}^{J}.

[04NM]

Proof.

The relation in Eq. 2.1.7 boils down to the two following:

\begin{cases}\sum_{l\in L}(C\cdot D_{l})u_{l}=0\\ (C\cdot D_{j})+\sum_{l\in L}\lambda_{j,l}(C\cdot D_{l})=0.\\ \end{cases}

The first one follows directly from Eq. 1.2.4 in the fan $\Sigma$ of $Z$ ; the second comes from the construction of $\lambda_{l}$ , and in particular from $C\cdot D_{j}=C\cdot F_{j}=-C\cdot\sum_{l\in L}\lambda_{j,l}Z_{l}$ . ∎

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.

[04NN]

2.2 Construction of the divisors

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.

[04NP]

Lemma 2.2.1.

The divisor $W^{\sigma}_{i}$ has multiplicity $-1$ along $D_{i}$ , multiplicity $0$ along $D_{l}$ for $l\in L_{\sigma}\setminus\{i\}$ , and along $D_{j}$ for $j\in J$ . In other words, we may write:

W^{\sigma}_{i}=-D_{i}+\sum_{l\in L\setminus L_{\sigma}}c_{i,l}D_{l}

for some coefficients $c_{i,l}\in\mathbb{Z}$ . Moreover, the restriction of $W^{\sigma}_{i}$ to $Z$ is principal.

[04NQ]

Proof.

The statement on the multiplicities follows from the definition of $W^{\sigma}_{i}$ , as

W^{\sigma}_{i}=-\sum_{l\in L}\frac{\det(u_{l},(u_{l^{\prime}})_{l^{\prime}\in L_{\sigma}\setminus\{i\}})}{\det(u_{i},(u_{l^{\prime}})_{l^{\prime}\in L_{\sigma}\setminus\{i\}})}D_{l}.

Moreover, $W^{\sigma}_{i}$ is a linear combination of the divisors of the $r$ -tuple $\Delta$ , hence its restriction to $Z$ is principal by Corollary 1.2.2. ∎

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.

[04NR]

Lemma 2.2.3.

The relation $\sum_{j\in J}W^{\sigma}_{j}+\sum_{i\in L_{\sigma}}W^{\sigma}_{i}=-\sum_{j\in J}D_{j}-\sum_{l\in L}D_{l}$ holds.

[04NS]

Proof.

Write $W\coloneqq\sum_{j\in J}W^{\sigma}_{j}+\sum_{i\in L_{\sigma}}W^{\sigma}_{i}\in\Div_{0}(\mathscr{X}).$ We have

	$\displaystyle\textrm{for }j\in J\quad\ord_{D_{j}}(W)$	$\displaystyle=\ord_{D_{j}}(W^{\sigma}_{j})=-1$
	$\displaystyle\textrm{for }i\in L_{\sigma}\quad\ord_{D_{i}}(W)$	$\displaystyle=\ord_{D_{i}}(W^{\sigma}_{i})=-1$
	$\displaystyle\textrm{for }l\in L\setminus L_{\sigma}\quad\ord_{D_{l}}(W)$	$\displaystyle=\sum_{j\in J}d_{j,l}+\sum_{i\in L_{\sigma}}c_{i,l}=-\sum_{j\in J}\lambda_{j,l}+\sum_{i\in L_{\sigma}}c_{i,l}(1-\sum_{j\in J}\lambda_{j,i})=-1$

by Lemma 2.2.1, Eq. 2.2.2 and Eq. 2.1.5. ∎

[04NT]

2.3 Construction of the sections for a maximal cone

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 $\mathfrak{X}=\varprojlim_{n}(\mathscr{X}/Z)_{n}$ .

[04NU]

Lemma 2.3.1.

The restrictions of $s^{\sigma}_{j}$ and $s^{\sigma}_{i}$ to $\mathfrak{X}_{\sigma}$ are equations for $D_{j}$ and $D_{i}$ , and thus

w_{\sigma}\coloneqq t\cdot\prod_{j\in J}(s_{j}^{\sigma})^{-1}\cdot\prod_{i\in L_{\sigma}}(s_{i}^{\sigma})^{-1}

is an invertible function on $\mathfrak{X}_{\sigma}$ .

[04NV]

Proof.

We show that $s^{\sigma}_{i}$ is an equation for $D_{i}$ on $\mathfrak{X}_{\sigma}$ ; the proof is analogous for $s^{\sigma}_{j}$ .

On $Z$ , ${W_{i}^{\sigma}}_{|Z}=\textrm{div}(h)$ and $s^{\sigma}_{i}$ is a non-zero global section, which means that

	$\displaystyle{\mathscr{L}^{\sigma}_{i}}_{\|Z}(Z)=\mathcal{O}_{Z}({W^{\sigma}_{i}})(Z)=\{f\in\mathcal{K}(Z)\,\|\,\textrm{div}(f)+\textrm{div}(h)\geqslant 0\}$	$\displaystyle\xrightarrow{\simeq}\mathcal{O}_{Z}(Z)=k$
	$\displaystyle f$	$\displaystyle\mapsto fh$
	$\displaystyle s^{\sigma}_{i}$	$\displaystyle\mapsto s^{\sigma}_{i}h=\lambda\in k^{\times}.$

Let $\mathcal{U}$ be an open cover of $\mathscr{X}\setminus\big(\cup_{i^{\prime}\notin J\cup L_{\sigma}}D_{i^{\prime}}\big)$ such that ${D_{i}}_{|U}=\textrm{div}(g_{U})$ for any $U\in\mathcal{U}$ ; this is possible as $D_{i}$ is a Cartier divisor. On $U$ , ${W^{\sigma}_{i}}_{|U}=-{D_{i}}_{|U}=\textrm{div}(g_{U}^{-1})$ and

	$\displaystyle\mathscr{L}^{\sigma}_{i}(\mathfrak{X}_{\sigma}\cap U)$	$\displaystyle\xrightarrow{\simeq}\mathcal{O}_{\mathfrak{X}_{\sigma}}(\mathfrak{X}_{\sigma}\cap U)$
	$\displaystyle f$	$\displaystyle\mapsto fg_{U}^{-1}$
	$\displaystyle s^{\sigma}_{i}$	$\displaystyle\mapsto s^{\sigma}_{i}g_{U}^{-1}\in\mathcal{O}_{\mathfrak{X}_{\sigma}}^{\times}(\mathfrak{X}_{\sigma}\cap U),$

where $s^{\sigma}_{i}g_{U}^{-1}$ is a regular invertible function on $\mathfrak{X}_{\sigma}\cap U$ , as its reduction to $Z$ is invertible. Finally, the section $s^{\sigma}_{i}$ is defined globally on $\widehat{\mathscr{X}_{/Z}}$ and on each open $\mathfrak{X}_{\sigma}\cap U$ gives a local equation of the divisor $D_{i}$ , hence it is a equation for $D_{i}$ on $\mathfrak{X}_{\sigma}$ . ∎

[04NW]

2.4 Construction for two adjacent maximal cones

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 $W^{\sigma^{\prime}}$ .

[04NX]

Lemma 2.4.2.

Let $C\subseteq Z$ be the curve associated with the cone $\sigma\cap\sigma^{\prime}$ . We have

\begin{cases}W^{\sigma^{\prime}}_{i}=W^{\sigma}_{i}-(C\cdot D_{i})W^{\sigma}_{i_{0}}&\textrm{ \hskip 10.22217ptfor }i\in L_{\sigma\sigma^{\prime}}\\ W^{\sigma^{\prime}}_{i_{\infty}}=-W^{\sigma}_{i_{0}}&\\ W^{\sigma^{\prime}}_{j}=W^{\sigma}_{j}-(C\cdot D_{j})W^{\sigma}_{i_{0}}&\textrm{ \hskip 10.22217ptfor }j\in J\end{cases}

In other words, the relation $W^{\sigma^{\prime}}=(M_{\mathcal{B}^{\prime}\mathcal{B}}\otimes\Id)W^{\sigma}$ holds.

[04NY]

Proof.

By Eq. 1.2.4 we have $u_{i_{\infty}}=-u_{i_{0}}-\sum_{m\in L_{\sigma\sigma^{\prime}}}(C\cdot D_{m})u_{m}$ , so

	$\displaystyle\textrm{for $i\in L_{\sigma\sigma^{\prime}}$, }\quad W^{\sigma^{\prime}}_{i}$	$\displaystyle=-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{\infty}})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{\infty}})}$
		$\displaystyle=-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}-\sum_{m\in L_{\sigma\sigma^{\prime}}}(C\cdot D_{m})\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{m})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}$
		$\displaystyle=W^{\sigma}_{i}-(C\cdot D_{i})\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}=W^{\sigma}_{i}-(C\cdot D_{i})W^{\sigma}_{i_{0}};$
	$\displaystyle\textrm{for $i=i_{\infty}$, }\quad W^{\sigma^{\prime}}_{i_{\infty}}$	$\displaystyle=\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}{\det(u_{i_{\infty}},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}=-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}{\det(u_{i_{0}},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}=-W^{\sigma}_{i_{0}}.$

For $j\in J$

	$\displaystyle\sum_{i\in L_{\sigma^{\prime}}}\lambda_{j,i}W^{\sigma^{\prime}}_{i}$	$\displaystyle=-\lambda_{j,i_{\infty}}W^{\sigma}_{i_{0}}+\sum_{i\in L_{\sigma\sigma^{\prime}}}\lambda_{j,i}\left(W^{\sigma}_{i}-(C\cdot D_{i})W^{\sigma}_{i_{0}}\right)$
		$\displaystyle=\Big(-\lambda_{j,i_{\infty}}-\sum_{i\in L_{\sigma\sigma^{\prime}}}\lambda_{j,i}(C\cdot D_{i})-\lambda_{j,i_{0}}\Big)W^{\sigma}_{i_{0}}+\sum_{i\in L_{\sigma}}\lambda_{j,i}W^{\sigma}_{i}$
		$\displaystyle=(C\cdot D_{j})W^{\sigma}_{i_{0}}+\sum_{i\in L_{\sigma}}\lambda_{i,j}W^{\sigma}_{i}\hskip 30.0pt\textrm{by \lx@cref{creftype~refnum}{equ fan}}$
	$\displaystyle W^{\sigma^{\prime}}_{j}$	$\displaystyle=-D_{j}-\sum_{l\in L}\lambda_{j,l}D_{l}-\sum_{i\in L_{\sigma^{\prime}}}\lambda_{j,i}W^{\sigma^{\prime}}_{i}$
		$\displaystyle=-D_{j}-\sum_{l\in L}\lambda_{j,l}D_{l}-\sum_{i\in L_{\sigma}}\lambda_{j,i}W^{\sigma}_{i}-(C\cdot D_{j})W^{\sigma}_{i_{0}}=W^{\sigma}_{j}-(C\cdot D_{j})W^{\sigma}_{i_{0}}.$

These relations can be summed up as $\left(\begin{matrix}W^{\sigma^{\prime}}_{i}\\ W^{\sigma^{\prime}}_{i_{\infty}}\\ W^{\sigma^{\prime}}_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}\left(\begin{matrix}W^{\sigma}_{i}\\ W^{\sigma}_{i_{0}}\\ W^{\sigma}_{j}\end{matrix}\right)$ , i.e. $W^{\sigma^{\prime}}=(M_{\mathcal{B}^{\prime}\mathcal{B}}\otimes\Id)W^{\sigma}$ . ∎

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 $w_{\sigma^{\prime}}$ .

[04NZ]

2.5 Construction of the morphism

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 $W^{\sigma_{h}}$ .

[04P0]

Lemma 2.5.1.

The tuple of sections $s^{\sigma}$ is independent on the choice of path.

[04P1]

Proof.

By Eq. 2.4.3, for any $h=0,\ldots,q-1$ , the sections $s^{\sigma_{h+1}}$ are constructed from $s^{\sigma_{h}}$ by multiplication by the matrix for the change of basis from $((v_{i})_{i\in L_{\sigma_{h}}},(v_{j})_{j\in J})$ to $((v_{i})_{i\in L_{\sigma_{h+1}}},(v_{j})_{j\in J})$ . Thus, by composition, the sections $s^{\sigma}$ only depends on $s^{\sigma_{0}}$ and the change of basis from $((v_{i})_{i\in L_{\sigma_{0}}},(v_{j})_{j\in J})$ to $((v_{i})_{i\in L_{\sigma}},(v_{j})_{j\in J})$ . ∎

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.

[04P2]

Lemma 2.5.2.

For any pair of maximal cones $\sigma,\sigma^{\prime}$ intersecting along a codimension one face, the morphisms $f_{\sigma}$ and $f_{\sigma^{\prime}}$ coincide on the overlap $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$ .

[04P3]

Proof.

The cones $\sigma$ and $\sigma^{\prime}$ correspond to adjacent vertices in $\Gamma$ . Thus, by Lemma 2.5.1 we construct $s^{\sigma}$ from any path joining $\sigma_{0}$ to $\sigma$ , and $s^{\sigma^{\prime}}$ from $s^{\sigma}$ by the relation $s^{\sigma^{\prime}}=M_{\mathcal{B^{\prime}}\mathcal{B}}\,s^{\sigma}$ in Eq. 2.4.3.

The functions $\chi^{\varepsilon}$ transform into $\chi^{\varepsilon^{\prime}}$ via the change of dual bases, which is given by $\varepsilon^{\prime}=M_{\mathcal{B^{\prime}}\mathcal{B}}\,\varepsilon$ in Eq. 2.4.1. Comparing the two formulas, it follows that $f_{\sigma}=f_{\sigma^{\prime}}$ on $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$ . ∎

[04P4]

Proposition 2.5.3.

The morphism of formal $R$ -schemes $f:\widehat{\mathscr{X}_{/Z}}\longrightarrow\widehat{\mathscr{N}_{/Z}}$ obtained by gluing the morphisms $f_{\sigma}$ is an isomorphism.

[04P5]

Proof.

We follow the argument in [NXY19, Proposition 5.4].
Since the source and the target have same dimension and are integral, it is enough to check that $f$ is a closed immersion.
If $\mathscr{J}$ is the largest ideal of definition of $\widehat{\mathscr{N}_{/Z}}$ , i.e. the defining ideal of $Z\subset\mathscr{N}$ , then $f^{*}\mathscr{J}=\mathscr{I}_{Z}$ is the largest ideal of definition of $\widehat{\mathscr{X}_{/Z}}$ . Indeed, since $Z$ is cut out inside $\mathscr{X}$ by the $D_{j}$ for $j\in J$ , the ideal $\mathscr{I}_{Z}$ is locally generated by the $s_{j}$ for $j\in J$ ; the same reasoning shows that $\mathscr{J}$ is locally generated by the $\chi^{\varepsilon_{j}}$ . The equality $f^{*}\mathscr{J}=\mathscr{I}_{Z}$ now follows directly from the local definition of $f$ .
We use [Gro61, 4.8.10] and the fact that $f$ induces an isomorphism on the reductions to infer that $f$ is a closed immersion, and thus an isomorphism by equality of dimensions. ∎

This concludes the proof of Theorem B: $\mathscr{X}$ is toric along $Z$ .

[04P6]

2.6 Integral affine structure and toric irreducible components

The case where $Z$ is an irreducible component $D$ of $\mathscr{X}_{k}$ is particularly relevant for proving Theorem A. Under the assumptions of Theorem B, we proved that $\mathscr{X}$ is toric along $D$ , and $\rho_{\mathscr{X}}$ is an affinoid torus fibration over $\Star(v_{D})$ by Corollary C. Moreover, we have the following explicit description of the $\mathbb{Z}$ -affine structure on $\Star(v_{D})$ induced by $\rho_{\mathscr{X}}$ - note that it only depends on $D$ and not on how $D$ sits inside $\mathscr{X}$ .

[04P7]

Corollary 2.6.1.

In the setting of Theorem B, let $Z=D$ be an irreducible component of $\mathscr{X}_{k}$ . Then there is a natural $\mathbb{Z}$ -linear embedding of $\Star(v_{D})$ inside the fan $\Sigma_{D}$ of $D$ which sends the polyhedral decomposition of $\Star(v_{D})$ to the cone decomposition of $\Sigma_{D}$ .

[04P8]

Proof.

By the proof of Theorem B and Proposition 1.5.2 we have the following diagram:

where the upper arrow is an isomorphism of analytic spaces, and the lower one a homeomorphism. Here $\mathfrak{X}_{D}$ and $\mathfrak{N}_{D}$ are the generic fibers (in the sense of Berkovich) of the formal completions $\widehat{\mathscr{X}_{/D}}$ and $\widehat{\mathscr{N}_{/D}}$ respectively, and $\mathrm{Int}(\Sigma_{1})$ denotes the interior of the polyhedral complex $\Sigma_{1}$ obtained by intersecting the fan $\hat{\Sigma}\subset N_{\mathbb{R}}\times\mathbb{R}$ of the normal bundle of $D$ in $\mathscr{X}$ with $N_{\mathbb{R}}\times\{1\}$ . In particular, $\mathrm{Int}(\Sigma_{1})$ is embedded in $\Sigma_{D}\simeq N_{\mathbb{R}}\simeq\mathbb{R}^{n}$ , the polyhedral decomposition of $\Star(v_{D})$ is the same of $\Sigma_{D}$ , and the vertex $v_{D}$ corresponds to the origin. By Section 1.6, the integral affine structure on $\Star(v_{D})$ is the pullback via $\varphi$ of the integral affine structure on $\Sigma_{D}$ , and this concludes the proof. ∎

[04P9]

3 Integral affine structures

Let $X/K$ be a smooth $n$ -dimensional maximally degenerate Calabi–Yau variety. In this chapter we compute the transition functions between the charts of the integral affine structure on $\Sk(X)$ associated with a minimal model of $X$ , or obtained by combining several minimal models. This relies on and generalizes the construction in [NXY19].
We then focus on certain degenerations of quartic $K3$ surfaces (Section 3.3), and later of quintic $3$ -folds (Section 4): we apply Theorem B to reconstruct integral affine structures on the essential skeleton, and provide explicit formulas for the monodromy transformations around the singularities.

[04PA]

3.1 Integral affine structure induced by a model

Let $\mathscr{X}/R$ be a minimal model of $X$ ; we assume that the special fiber $\mathscr{X}_{k}=\sum_{i\in I}D_{i}$ is reduced. We consider a one-dimensional stratum $C=D_{1}\cap\ldots\cap D_{n}$ of $\mathscr{X}_{k}$ , which is therefore a smooth rational curve, and is such that $(\mathscr{X},\mathscr{X}_{k})$ is an snc pair in a formal neighbourhood of $C$ by [NXY19, Corollary 4.6]. Since $(C,\Delta_{C})$ is log Calabi–Yau, we may write its boundary as $\Delta_{C}=p_{0}+p_{\infty}$ , where $p_{0}=C\cap D_{0}$ and $p_{\infty}=C\cap D_{\infty}$ for two irreducible components $D_{0},D_{\infty}$ of $\mathscr{X}_{k}$ meeting $C$ transversally.

Following [NXY19], we write $b_{i}=-(C\cdot D_{i})$ for $i=1,\ldots,n$ ; from $C\cdot\mathscr{X}_{k}=0$ we infer $\sum_{i=1}^{n}b_{i}=2$ . The $\Star(\tau_{C})$ consists on the union of two maximal faces corresponding to the zero-dimensional strata $p_{0},p_{\infty}$ , meeting along $\tau_{C}$ . The goal of this section is to describe the integral affine structure on $\Star(\tau_{C})$ in terms of the intersection numbers $b_{i}$ ’s, with no assumption on their positivity.

[04PB]

Proposition 3.1.1.

Let $\rho_{\mathscr{X}}$ be the retraction associated with the model $\mathscr{X}$ , and endow $\emph{Sk}(X)$ with the $\mathbb{Z}$ -affine structure induced by $\rho_{\mathscr{X}}$ away from the codimension 2 faces of $\Sk(\mathscr{X})$ . Then $\Star(\tau_{C})$ is $\mathbb{Z}$ -affine isomorphic to the union of the simplices $<v_{0},v_{1},\ldots,v_{n}>$ and $<v_{1},\ldots,v_{n},v_{\infty}>$ in $\mathbb{R}^{n}$ where

$v_{0}=(1,0,\ldots,0)$ , $v_{1}=(0,1,\ldots,0)$ ,…, $v_{n}=0$ and $v_{\infty}=(-1,b_{1},\ldots,b_{n-1})$ .

[04PC]

Proof.

We write $b=\min_{i\leqslant n}b_{i}$ ; we assume $b$ to be negative or zero by the condition $b_{1}+\ldots+b_{n}=2$ , as the case $n=2$ and $b_{1}=b_{2}=1$ is already treated in the proof of [NXY19, prop. 5.4].

The blow-up $\mathscr{X}_{1}$ of the point $p_{\infty}$ in $\mathscr{X}$ yields a new irreducible component $D_{\infty,1}$ (we denote the strict transforms by the same letters for notational simplicity) with multiplicity $N_{\infty,1}=n+1$ , the point $p_{\infty,1}=C\cap D_{\infty,1}$ and the intersection numbers $b_{i,1}\coloneqq-(C\cdot D_{i})_{\mathscr{X}_{1}}=b_{i}+1$ . If we repeat the process $s$ times, we obtain the models $\mathscr{X}_{s}$ , the exceptional divisors $D_{\infty,s}$ with multiplicity $N_{\infty,s}=ns+1$ , the points $p_{\infty,s}=C\cap D_{\infty,s}$ and the intersection numbers $b_{i,s}\coloneqq-(C\cdot D_{i})_{\mathscr{X}_{s}}=b_{i}+s$ .

For $s=1-b$ , we have $\min_{i\leqslant n}\{b_{i,1-b}\}>0$ , and by [NXY19] the integral affine structure induced by $\mathscr{X}_{1-b}$ on $\Star(\tau_{C})$ is given by $v_{0},\ldots,v_{n}$ and

(3.1.2)

v_{\infty,1-b}=\frac{1}{n(1-b)+1}(-1,b_{1}+1-b,\ldots,b_{n-1}+1-b).

The sequence of blow-ups $\mathscr{X}_{s+1}\rightarrow\mathscr{X}_{s}$ induces (weighted) barycentric subdivisions of the faces $\tau_{p_{\infty,s}}$ with vertices such that

(3.1.3)

N_{\infty,s+1}v_{\infty,s+1}=N_{\infty,s}v_{\infty,s}+\sum_{i=1}^{n}v_{i}.

Combining Eq. 3.1.2 and Eq. 3.1.3, at each step we obtain that

v_{\infty,s}=\frac{1}{ns+1}(-1,b_{1}+s,\ldots,b_{n-1}+s),

and in particular $v_{\infty}=(-1,b_{1},\ldots,b_{n-1})$ . The proposition follows from the following lemma. ∎

[04PD]

Lemma 3.1.4.

Let $B=\tau_{1}\cup\tau_{2}$ be the union of two $n$ -dimensional simplices along a face of codimension one. Assume we are given a $\mathbb{Z}$ -affine structure on $B$ , compatible with those on the $\tau_{i}$ ’s.

Suppose there exists a sequence of (weighted) star subdivisions of $\tau_{1}$ such that $B^{\prime}\coloneqq\Star(\tau_{1}\cap\tau_{2})$ (with respect to this subdivision) can be embedded in $\mathbb{R}^{n}$ compatibly with the $\mathbb{Z}$ -affine structure. Then this embedding extends to $B$ , and the $\mathbb{Z}$ -affine structure on $B$ is uniquely recovered by this embedding.

[04PE]

Proof.

The assumptions yield two charts for the $\mathbb{Z}$ -affine structure on $B$ : the $\mathbb{Z}$ -affine subsets $B^{\prime}$ and $\tau_{1}$ . These two charts are glued along $B^{\prime}\cap\tau_{1}$ which is a simplex and thus has no non-trivial $\mathbb{Z}$ -automorphisms preserving the vertices, hence the affine structure on $B$ is uniquely determined. The set $B^{\prime}\subset\mathbb{R}^{n}$ can be obtained as the result of the same star subdivisions of a subset $\tilde{B}\subset\mathbb{R}^{n}$ , and uniqueness of the affine structure ensures a $\mathbb{Z}$ -affine isomorphism $B\simeq\tilde{B}$ . ∎

[04PF]

Remark 3.1.5.

Consider an irreducible component $D_{i}$ of $\mathscr{X}_{k}$ and write $\Delta_{D_{i}}=\sum_{j\neq i}D_{j}\cap D_{i}$ . By adjunction, the pair $(D_{i},\Delta_{D_{i}})$ is log Calabi–Yau, i.e. $D_{i}$ is a smooth projective variety over $k$ and $\Delta_{D_{i}}$ is a divisor such that $K_{D_{i}}+\Delta_{D_{i}}$ is trivial. By [EM21, Theorem 6.14] there exists a Lagrangian torus fibration

\phi:\mathcal{U}\rightarrow B\subseteq\Star(v_{D_{i}})\setminus W

where $\mathcal{U}$ is a symplectic tubular neighborhood of the $1$ -dimensional strata of $\Delta_{D_{i}}$ , $B$ is a retract of $\Star(v_{D_{i}})\setminus W$ , and $W$ is the union of cells of codimension $\geqslant 2$ in $\Sk(\mathscr{X})$ . The fibration $\phi$ is constructed gluing toric moment maps defined in the neighborhood of each stratum curve of $\Delta_{D_{i}}$ . Evans and Mauri compare the monodromy $T_{\phi}$ induced by $\phi$ on $B$ to the monodromy $T_{\rho_{\mathscr{X}}}$ induced by the affinoid torus fibration

{\rho_{\mathscr{X}}}:\rho_{\mathscr{X}}^{-1}(\Star(v_{D_{i}})\setminus W)\rightarrow\Star(v_{D_{i}})\setminus W

and conclude that they are dual. This means that given a loop $\gamma\in\pi_{1}(B)\simeq\pi_{1}(\Star(v_{D_{i}})\setminus W)$ , we have $T_{\rho_{\mathscr{X}}}(\gamma)=(T_{\phi}(\gamma)^{-1})^{T}$ . Thus the affine structure constructed in [NXY19] has a symplectic topological analog. The duality is due to the fact that the image of the moment maps is $M_{\mathbb{R}}$ , while the image of the tropicalization map $\val$ is in $N_{\mathbb{R}}$ .

[04PG]

3.1.1 Case of K3 surfaces

Let $X/K$ be a maximally degenerate $K3$ surface and let $\mathscr{X}/R$ be a minimal model of $X$ with reduced special fiber $\mathscr{X}_{k}=\sum_{i\in I}D_{i}$ . The dual complex $\mathcal{D}(\mathscr{X}_{k})$ is well-known to be a triangulated sphere, whose vertices correspond to the irreducible components of $\mathscr{X}_{k}$ .
We focus our attention to such a vertex $v_{D}$ , and hence to the corresponding irreducible component $D$ of $\mathscr{X}_{k}$ , which has boundary $\Delta_{D}\coloneqq\sum_{i=1}^{r}(D_{i}\cap D)=\sum_{i=1}^{r}C_{i}$ . Since the simple normal crossing curve $\Delta_{D}\in\lvert-K_{D}\rvert$ is an anticanonical curve by adjunction, it follows from general surface theory that $\Delta_{D}$ is a cycle of rational curves $(C_{i})_{i\leq r}$ , whose geometry is encoded by the $b_{i}=-(C_{i}\cdot D)=-(C_{i}^{2})_{D}$ . We label the curves so that for $i\leq r$ , $C_{i}\cap C_{i+1}\neq\varnothing$ , with convention $C_{r+1}=C_{1}$ .

One can associate to the pair $(D,\Delta_{D})$ a pseudo-fan, which is a singular affine structure on $\mathbb{R}^{2}$ , singular at most at $0$ . The singularity at $0$ is a way to measure the defect of $(D,\Delta_{D})$ of being toric: the affine structure affine extends smoothly at $0$ if and only $(D,\Delta_{D})$ is a toric pair [Eng18, Proposition 3.9].
The construction, as explained in [GHK15, §1.2], is the following. For each node $p_{i}=C_{i}\cap C_{i+1}$ , consider a cone $\sigma_{i}\coloneqq\mathbb{R}_{\geq 0}v_{i}+\mathbb{R}_{\geq 0}v_{i+1}\subset\mathbb{R}^{2}$ , $(v_{i},v_{i+1})$ being a basis of the lattice $\mathbb{Z}^{2}$ . The cones $\sigma_{i}$ and $\sigma_{i+1}$ are then glued to each other along $\mathbb{R}_{\geq 0}v_{i+1}$ , and the affine structure is extended through the edge by pretending that the pair $(D,\Delta_{D})$ is toric. If the pair was toric, the $\sigma_{i}$ ’s would be the maximal cones of its fan, and the relation

v_{i+2}+v_{i}=b_{i+1}v_{i+1}

would hold by Eq. 1.2.4, so that the chart $\psi_{i}:\sigma_{i}\cup\sigma_{i+1}$ that defines the $\mathbb{Z}$ -affine structure satisfies $\psi_{i}(0)=0$ , $\psi_{i}(v_{i})=(1,0)$ , $\psi_{i}(v_{i+1})=(0,1)$ and $\psi_{i}(v_{i+2})=(-1,b_{i+1})$ , and is extended by dilatation. The unions of the $\sigma_{i}$ ’s glued along the successive edge is homeomorphic to $\mathbb{R}^{2}$ , and we obtain this way an $\mathbb{Z}$ -affine structure away from the origin, extending to $0$ if and only the pair is toric.

It follows from Proposition 3.1.1 that the singular $\mathbb{Z}$ -affine structure induced by the Berkovich retraction $\rho_{\mathscr{X}}$ coincides with the one described above. We now determine the monodromy around the singularities.

[04PH]

Corollary 3.1.6.

Let $D$ be a component of $\mathscr{X}_{k}$ , with boundary $\Delta_{D}=\sum_{i=1}^{r}C_{i}$ . Writing $b_{i}=-(C_{i}^{2})_{D}$ , the monodromy $T_{\rho_{\mathscr{X}}}$ of the $\mathbb{Z}$ -affine structure induced by $\rho_{\mathscr{X}}$ around $v_{D}$ is given by

T_{\rho_{\mathscr{X}}}=\left(\begin{matrix}b_{r}&1\\ -1&0\end{matrix}\right)\cdot\ldots\cdot\left(\begin{matrix}b_{2}&1\\ -1&0\end{matrix}\right)\cdot\left(\begin{matrix}b_{1}&1\\ -1&0\end{matrix}\right)

with respect to the basis $(v_{D_{r}},v_{D_{1}})$ and origin $v_{D}$ .

[04PI]

Proof.

By Proposition 3.1.1 the integral affine structure on $\Star(\tau_{C_{i}})$ identifies $(v_{D_{i-1}},v_{D_{i}},v_{D},v_{D_{i+1}})$ with

(v_{0}=(1,0),v_{1}=(0,1),v_{2}=(0,0),v_{\infty}=(-1,b_{i})),

while on $\Star(\tau_{C_{i+1}})$ identifies $(v_{D_{i}},v_{D_{i}+1},v_{D},v_{D_{i+2}})$ with

(v_{0}=(1,0),v_{1}=(0,1),v_{2}=(0,0),v_{\infty}=(-1,b_{i+1})).

It follows that the transition map from the chart $\Star(\tau_{C_{i}})$ to $\Star(\tau_{C_{i+1}})$ of the integral affine structure on $\Star(\tau_{C_{i}})\cap\Star(\tau_{C_{i+1}})$ is given by the matrix $\left(\begin{matrix}b_{i}&1\\ -1&0\end{matrix}\right)$ . Thus, the composition of such matrices gives the monodromy around $v_{D}$ , along a loop oriented as the path connecting $v_{D_{1}},v_{D_{2}},\ldots,v_{D_{r}},v_{D_{1}}$ . ∎

[04PJ]

Remark 3.1.7.

It is well-known (see for instance [GHK15]) that $T_{\rho_{\mathscr{X}}}=\Id$ if and only the pair $(D,\Delta_{D}=\sum_{i=1}^{r}C_{i})$ is toric, or if and only if the charge $Q$ vanishes, where

Q=\chi_{\text{top}}(D\setminus\Delta_{D})=12+\sum_{i=1}^{r}(b_{i}-3).

[04PK]

3.2 Integral affine structure induced by combining several models

We start with a general definition.

[04PL]

Definition 3.2.1.

Let $\tau$ be a simplex of dimension $m$ and consider the first barycentric subdivision $\tau^{\prime}$ of $\tau$ . For each vertex $v$ of $\tau$ , we denote the star of $v$ in $\tau^{\prime}$ by $\Star(v)^{\prime}$ and define $\Gamma_{m-1}$ to be the polyhedral complex of dimension $m-1$ given by

\Gamma_{m-1}\coloneqq\tau\setminus\bigcup_{v\in\tau}\Star(v)^{\prime}\subset\tau.

For instance, if $m=2$ , $\Gamma_{1}$ is the union of the three line segments joining the barycenter of the triangle to the barycenters of the edges.

We return to the setting of Section 3, that is, let $X/K$ be a smooth $n$ -dimensional maximally degenerate Calabi–Yau variety. Assume we are given two minimal models $\mathscr{X}$ , $\mathscr{X}^{\prime}$ of $X$ such that $\mathcal{D}(\mathscr{X}_{k})=\mathcal{D}(\mathscr{X}^{\prime}_{k})$ , so that $\Sk(\mathscr{X})=\Sk(\mathscr{X}^{\prime})=\Sk(X)$ , not only as sets but also with the same triangulation. We fix an ordered labelling $(1,\ldots,s)$ of the vertices of $\Sk(X)$ , equivalently of the irreducible components of the special fiber of $\mathscr{X}$ (resp. $\mathscr{X}^{\prime}$ ).
Fix a codimension 1 face $\tau$ of $\mathcal{D}(\mathscr{X}_{k})=\mathcal{D}(\mathscr{X}^{\prime}_{k})$ , with vertices $v_{i_{1}},\ldots,v_{i_{n}}$ . We write $C$ (resp. $C^{\prime}$ ) for the corresponding strata curves of $\mathscr{X}$ (resp. $\mathscr{X}^{\prime}$ ), and $D_{i_{l}}$ (resp. $D^{\prime}_{i_{l}}$ ), $l=1,\ldots,n$ the corresponding components. We then have $C=D_{i_{1}}\cap\ldots\cap D_{i_{n}}$ , and similarly for $C^{\prime}$ . We write

b_{i_{l}}=-(C\cdot D_{i_{l}})_{\mathscr{X}}

the intersection number computed inside $\mathscr{X}$ , and similarly:

b^{\prime}_{i_{l}}=-(C^{\prime}\cdot D^{\prime}_{i_{l}})_{\mathscr{X}^{\prime}}

for $l=1,\ldots,n$ . The $(n-1)$ -dimensional face $\tau$ is contained in two maximal faces $\tau_{p_{0}}$ and $\tau_{p_{\infty}}$ of $\mathcal{D}(\mathscr{X}_{k})$ , since the boundary of $C$ in $(\mathscr{X},\mathscr{X}_{k})$ consists of two strata points $p_{0}=C\cap D_{i_{0}}$ and $p_{\infty}=C\cap D_{i_{\infty}}$ ; we assume $i_{0}<i_{\infty}$ . We set $\mathcal{S}\coloneqq\tau_{p_{0}}\cup\tau_{p_{\infty}}\subset\Sk(X)$ , and $\Gamma\coloneqq\Gamma_{n-2}\subset\tau_{C}$ as in Definition 3.2.1.
Given two vertices $v_{i_{m}},v_{i_{m^{\prime}}}$ of $\tau_{C}$ , with corresponding components $D_{i_{m}}$ and $D_{i_{m^{\prime}}}$ of $\mathscr{X}_{k}$ containing $C$ , we assume $i_{m}<i_{m^{\prime}}$ and construct a loop $\gamma$ as follows:

-

$\gamma$ is contained in $\mathcal{S}\setminus\Gamma$ ;
-

$\gamma$ goes around the segment joining the barycenter of $\tau_{C}$ with the barycenter of the edge between $v_{i_{m}}$ and $v_{i_{m^{\prime}}}$ ;
-

$\gamma$ has an orientation induced by the fixed ordered labelling on the vertices of $\Sk(X)$ in the following way: in $\mathcal{S}\setminus\Gamma$ , $\gamma$ is homotopy equivalent to the closed path given by the edges which connect in order

$v_{D_{i_{0}}},v_{D_{i_{m}}},v_{D_{i_{\infty}}},v_{D_{i_{m^{\prime}}}}.$

Figure 1: *

Two examples of loops $\gamma$ in the case $n=3$ and $C=D_{2}\cap D_{3}\cap D_{4}$

Suppose we are given a retraction $\rho:X^{\an}\longrightarrow\Sk(X)$ such that

\rho=\begin{cases}\rho_{\mathscr{X}}&\textrm{ over }U_{i_{m}}\coloneqq\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup\Star(v_{i_{m}})^{\prime}\\ \rho_{\mathscr{X}^{\prime}}&\textrm{ over }U_{i_{m^{\prime}}}\coloneqq\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup\Star(v_{i_{m^{\prime}}})^{\prime}.\end{cases}

[04PM]

Proposition 3.2.2.

The monodromy along the loop $\gamma$ , of the $\mathbb{Z}$ -affine structure induced by $\rho$ on $U_{i_{m}}\cup U_{i_{m^{\prime}}}$ is

(3.2.3)

T_{\rho}(\gamma)=\left(\begin{matrix}1&0&0&\ldots&0\\ b_{i_{1}}-b^{\prime}_{i_{1}}&1&0&\ldots&0\\ b_{i_{2}}-b^{\prime}_{i_{2}}&0&1&\ldots&0\\ \vdots&\vdots&&\ddots&\\ b_{i_{n-1}}-b^{\prime}_{i_{n-1}}&0&0&\ldots&1\end{matrix}\right)

with respect to the basis $(v_{D_{i_{0}}},v_{D_{i_{1}}},\ldots,v_{D_{i_{n-1}}})$ and origin $v_{D_{i_{n}}}$ .

[04PN]

Proof.

We need to compute the parallel transport of the vectors

(u_{0},\ldots,u_{n-1})\coloneqq(v_{D_{i_{0}}}-v_{D_{i_{n}}},v_{D_{i_{1}}}-v_{D_{i_{n}}},\ldots,v_{D_{i_{n-1}}}-v_{D_{i_{n}}})

along the loop $\gamma$ . By Proposition 3.1.1 the $\mathbb{Z}$ -affine structure on $\Star(\tau_{C})$ induced by $\rho_{\mathscr{X}}$ is described by the chart which has the following vertices:

$v_{0}=(1,0,\ldots,0)$ , $v_{1}=(0,1,\ldots,0),\,\ldots,\,v_{n}=0$ and $v_{\infty}=(-1,b_{i_{1}},\ldots,b_{i_{n-1}});$

while the $\mathbb{Z}$ -affine structure induced by $\rho_{\mathscr{X}^{\prime}}$ is given by:

$v^{\prime}_{0}=(1,0,\ldots,0)$ , $v^{\prime}_{1}=(0,1,\ldots,0),\,\ldots,\,v^{\prime}_{n}=0$ and $v^{\prime}_{\infty}=(-1,b^{\prime}_{i_{1}},\ldots,b^{\prime}_{i_{n-1}}).$

Moreover, the vectors $u_{l}$ correspond to the vectors $v_{l}$ (resp. $v^{\prime}_{l}$ ) in the chart for $\rho_{\mathscr{X}}$ (resp. $\rho_{\mathscr{X}^{\prime}}$ ). We now have $v_{0}=-v_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v_{l}$ , so that the vectors we are transporting are written on $\tau_{p_{\infty}}$

(-v_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v_{l}\,,v_{1},\ldots,v_{n-1})

in the chart for $\rho_{\mathscr{X}}$ . These are thus mapped to the tuple $(-v^{\prime}_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v^{\prime}_{l},v^{\prime}_{1},\ldots,v^{\prime}_{n-1})$ by the chart for $\rho_{\mathscr{X}^{\prime}}$ . We now transport back across $\tau_{C}$ in the chart for $\rho_{\mathscr{X}^{\prime}}$ , to get the tuple of vectors

(-v^{\prime}_{0}-\sum_{l=1}^{n-1}b^{\prime}_{i_{l}}v^{\prime}_{l}+\sum_{l=1}^{n-1}b_{i_{l}}v^{\prime}_{l}\,,v^{\prime}_{1},\ldots,v^{\prime}_{n-1})

according to the relation $-v^{\prime}_{\infty}=-v^{\prime}_{0}-\sum_{l=1}^{n-1}b^{\prime}_{i_{l}}v^{\prime}_{l}$ . We now see that after parallel transport the vectors $(u_{0},\ldots,u_{n-1})$ have changed to

(u_{0}+\sum_{l=1}^{n-1}(b_{i_{l}}-b^{\prime}_{i_{l}})u_{l}\,,u_{1},\ldots,u_{n-1}),

hence the formula Eq. 3.2.3 for the monodromy matrix. ∎

[04PP]

3.2.1 Case of K3 surfaces

We focus on the case of a maximally degenerate $K3$ surface $X/K$ . We have $C=D_{i_{m}}\cap D_{i_{m^{\prime}}}$ , $\Gamma=\{a\}$ is a point in the interior of $\tau_{C}$ and $\gamma$ is a loop around $a$ , oriented as the path joining in order $v_{D_{i_{0}}},v_{D_{i_{m}}},v_{D_{i_{\infty}}},v_{D_{i_{m^{\prime}}}}$ . We assume we have a retraction $\rho:X^{\an}\longrightarrow\Sk(X)$ such that

\rho=\begin{cases}\rho_{\mathscr{X}}&\textrm{ over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m}},a)\\ \rho_{\mathscr{X}^{\prime}}&\textrm{ over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m^{\prime}}},a)\end{cases}

where $[v_{i_{\cdot}},a)$ is the part of the edge $\tau_{C}$ joining the vertex to $a$ , but not including $a$ . Then Proposition 3.2.2 may be rewritten as follows.

[04PQ]

Corollary 3.2.4.

The monodromy along the loop $\gamma$ , of the $\mathbb{Z}$ -affine structure induced by $\rho$ on $\Star(\tau_{C})\setminus\{a\}$ , is

(3.2.5)

T_{\rho}(\gamma)=\left(\begin{matrix}1&0\\ b_{i_{m}}-b^{\prime}_{i_{m}}&1\end{matrix}\right)

with respect to the basis $(v_{D_{i_{0}}},v_{D_{i_{m}}})$ and origin $v_{D_{i_{m^{\prime}}}}$ .

[04PR]

3.3 Degeneration of quartic K3 surfaces

We consider $\mathscr{X}=\{z_{1}z_{2}z_{3}z_{4}+tF_{4}(z_{1},z_{2},z_{3},z_{4})=0\}\subset\mathbb{P}^{3}_{R}$ , where $F_{4}$ is a generic homogeneous polynomial of degree 4. The degeneration $\mathscr{X}$ has the following properties:

1.

the special fiber $\mathscr{X}_{k}$ is reduced, consisting of four Weil divisors, i.e. $D_{i}=\{z_{i}=0,t=0\}$ ;
2.

$\mathscr{X}$ has $24$ singular points, given by $\{z_{i}=0,z_{j}=0,t=0,F_{4}=0\}$ , hence 4 on each $D_{i}\cap D_{j}$ ; the local model around a singular point is given by $\mathscr{U}=\{xy=zt\}\subset\mathbb{A}^{3}_{R}$ ;
3.

the pair $(\mathscr{X},\mathscr{X}_{k})$ is dlt. Indeed, it is snc away from the singularities, and around a singular point $(\mathscr{U},\mathscr{U}_{k})$ is log canonical by [CLS11, Proposition 11.4.24];
4.

the dual complex $\mathcal{D}(\mathscr{X}_{k})$ is a PL-isomorphic to a tetrahedron, and hence homeomorphic to $\mathbb{S}^{2}$ ;
5.

by adjunction, the canonical bundle $K_{\mathscr{X}}$ is trivial.

We conclude that $\mathscr{X}$ is a minimal dlt model of the K3 surface $X\coloneqq\mathscr{X}_{K}$ , but it is not good in the sense of Section 1.1 since the prime components of the special fiber are not $\mathbb{Q}$ -Cartier.

Our goal is to construct some explicit minimal models of $X$ starting from $\mathscr{X}$ , and then to study the integral affine structure on $\Sk(X)$ induced by these models or by combining several of them. To this purpose, we will apply Corollary 3.1.6 and Corollary 3.2.4.

Some good minimal models of $X$ are obtained by performing the following small resolutions of $\mathscr{X}$ . For any triple of elements $i,j,k$ in $\{1,\ldots,4\}$ and any fixed order $(i,j,k)$ on them, we blow-up in order the divisors $D_{i}$ , $D_{j}$ and $D_{k}$ , and denote the resulting model by $\mathscr{X}_{ijk}$ and the morphism by

g_{ijk}:\mathscr{X}_{ijk}\rightarrow\mathscr{X}.

The exceptional locus of $g_{ijk}$ consists of $24$ smooth rational curves whose images via $g_{ijk}$ are the singular points of $\mathscr{X}$ . In particular, the strict transform of $D_{i}$ is isomorphic to the blow-up of $D_{i}$ along the $12$ singular points in $D_{i}$ ; similarly for $D_{j}$ at $8$ points, and for $D_{k}$ at the $4$ remaining singular points. Instead, for $h\neq i,j,k$ , $D_{h}$ is isomorphic to its strict transform. These facts follow from local computations on $\mathscr{U}$ . Blowing-up $D_{x}\coloneqq\{x=t=0\}$ induces an exceptional curve inside the strict transform $\tilde{D}_{x}$ , which is isomorphic to the blow-up of $D_{x}$ along $q$ . The above claims now follow, since the singularities of $\mathscr{X}$ are isolated.
If we denote by $\tilde{D}_{m}$ for $m\in\{1,\ldots,4\}$ the irreducible components of the special fiber of $\mathscr{X}_{ijk}$ , and by $C_{mm^{\prime}}=\tilde{D}_{m}\cap\tilde{D}_{m^{\prime}}$ the strata curves, then the intersection numbers in $\mathscr{X}_{ijk}$ are

(3.3.1)

	$\tilde{D}_{i}$	$\tilde{D}_{j}$	$\tilde{D}_{k}$	$\tilde{D}_{h}$
$C_{ij}$	1	-3	1	1
$C_{ik}$	1	1	-3	1
$C_{ih}$	1	1	1	-3
$C_{jk}$	1	1	-3	1
$C_{jh}$	1	1	1	-3
$C_{kh}$	1	1	1	-3

[04PS]

3.3.1 Integral affine structure induced by the model $\mathscr{X}_{ijk}$

By [NXY19] the non-archimedean SYZ fibration $\rho_{\mathscr{X}_{ijk}}:S^{\an}\rightarrow\Sk(\mathscr{X}_{ijk})=\Sk(X)\simeq\mathbb{S}^{2}$ is an affinoid torus fibration (at least) away from the vertices of the triangulation of $\Sk(X)$ induced by the special fiber of $\mathscr{X}_{ijk}$ , i.e. away from the $v_{D_{m}}$ ’s.
By Theorem B, $\rho_{\mathscr{X}_{ijk}}$ is an affinoid torus fibration over $\Star(\tau_{D_{h}})$ for $h\neq i,j,k$ , as $D_{h}\simeq\mathbb{P}^{2}$ and $g_{ijk}$ is an isomorphism on the strict transform of $D_{h}$ . Moreover, by Remark 3.1.7 and Eq. 3.3.1 the integral affine structure induced by $\rho_{\mathscr{X}_{ijk}}$ does not extend to $v_{D_{i}}$ , $v_{D_{j}}$ and $v_{D_{k}}$ .

We conclude that the singular points of the affine structure on $\Sk(X)$ induced by $\rho_{\mathscr{X}_{ijk}}$ are precisely $v_{D_{i}}$ , $v_{D_{j}}$ and $v_{D_{k}}$ . Corollary 3.1.6 establishes that the monodromies around these vertices are

	$\displaystyle T_{\rho_{\mathscr{X}_{ijk}}}$	$\displaystyle(\gamma_{i})=\left(\begin{matrix}21&8\\ -8&-3\end{matrix}\right)$
		$\displaystyle\text{in the basis $(v_{D_{j}},v_{D_{k}})$ and origin $v_{D_{i}}$},$
	$\displaystyle T_{\rho_{\mathscr{X}_{ijk}}}$	$\displaystyle(\gamma_{j})=\left(\begin{matrix}-15&-4\\ 4&1\end{matrix}\right)$
		$\displaystyle\text{in the basis $(v_{D_{k}},v_{D_{h}})$ and origin $v_{D_{j}}$},$
	$\displaystyle T_{\rho_{\mathscr{X}_{ijk}}}$	$\displaystyle(\gamma_{k})=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)$
		$\displaystyle\text{in the basis $(v_{D_{h}},v_{D_{i}})$ and origin $v_{D_{k}}$}.$

[04PT]

3.3.2 Integral affine structure induced combining more models

We recall a construction from [KS06, §4.2.5]. Consider the resolution $h:\mathscr{Z}\rightarrow\mathscr{X}$ obtained by blowing-up the $24$ singular points of $\mathscr{X}$ , which in particular dominates any model $\mathscr{X}_{ijk}$ . Then the special fiber is $\mathscr{Z}_{k}=\sum_{i=1}^{4}D_{i}+\sum_{q=1}^{24}E_{q}$ and the associated dual complex is the boundary of a tetrahedron with four additional $2$ -cells glued along each edge of the tetrahedron; following Kontsevich–Soibelman we call such $2$ -cells wings.

We parametrize each edge $e$ of $\mathcal{D}(\mathscr{X}_{k})$ by the interval $[-1,1]$ , and each wing $W_{q}$ glued to $e$ by the $2$ -simplex in $\mathbb{R}^{2}_{(x,y)}$ bounded by $e$ and $0\leqslant y\leqslant 1-|x|$ .

[04PU]

Lemma 3.3.2.

Let $W_{q}$ be a wing over the edge $e_{lh}$ , for $l\in\{i,j,k\}$ and assume $i,j,k,h$ all distinct. Then the retraction $\rho_{\mathscr{X}_{ijk}}:W_{q}\rightarrow e_{lh}$ is the contraction of $W_{q}$ to the edge $e_{lh}$ parallel to the edge $e_{lq}$ :

	$\displaystyle\rho_{\mathscr{X}_{ijk}}:W_{q}$	$\displaystyle\rightarrow e_{lh}$
	$\displaystyle(x,y)$	$\displaystyle\mapsto(x-y,0)$

[04PV]

Proof.

The morphism $\mathscr{Z}\rightarrow\mathscr{X}_{ijk}$ is the blow-up of the 24 exceptional curves of $g_{ijk}$ . In particular, the exceptional divisor $E_{q}$ is the preimage in $\mathscr{Z}$ of a curve contained in $\tilde{D}_{l}$ ; it follows that $v_{E_{q}}(\tilde{z}_{l})=1$ and $v_{E_{q}}(\tilde{z}_{h})=0$ , where $\tilde{z}_{l},\tilde{z}_{h}$ are local equations for $\tilde{D}_{l},\tilde{D}_{h}$ on $\mathscr{X}_{ijk}$ . The Berkovich retraction $\rho_{\mathscr{X}_{ijk}}$ is linear on $W_{q}$ and hence depends only on the image of $v_{q}$ , which is determined by $v_{E_{q}}(\tilde{z}_{l})$ and $v_{E_{q}}(\tilde{z}_{h})$ . Thus we conclude that $\rho_{\mathscr{X}_{ijk}}(v_{q})=v_{l}$ and we have the result. ∎

Kontsevich and Soibelman define a retraction

\rho:X^{\textrm{an}}\xrightarrow{\rho_{\mathscr{Z}}}\Sk(\mathscr{Z})\xrightarrow{\rho^{\prime}}\mathbb{S}^{2}\simeq\Sk(X)=\Sk(\mathscr{X})

where $\rho_{\mathscr{Z}}$ is the Berkovich retraction onto the skeleton $\Sk(\mathscr{Z})$ , and $\rho^{\prime}$ is a retraction of the 24 wings of $\Sk(\mathscr{Z})$ to the sphere given as follows. For each edge $e$ of $\mathcal{D}(\mathscr{X}_{k})$ we choose a point $a_{e}=(a_{e},0)$ in the interior of $e$ , and define the retraction of $W_{q}$ onto $e$ by

	$(x+y,0)$	if $x+y\leqslant a_{e}$
$\rho^{\prime}:\,(x,y)\mapsto$	$(x-y,0)$	if $x-y\geqslant a_{e}$
	$(a_{e},0)$	otherwise.

We note that

-

over the interior of any $2$ -dimensional face $\tau\subset\Sk(\mathscr{X})$ , $\rho$ is equal to $\rho_{\mathscr{Z}}$ , thus it is an affinoid torus fibration (see Example 1.6.2).
-

Around any vertex $v_{D}$ , $\rho$ is equal to $\rho_{\mathscr{X}_{ijk}}$ for any triple such that $D\neq D_{i},D_{j},D_{k}$ , as follows from the previous lemma. Thus, from Section 3.3.1, $\rho$ is an affinoid torus fibration around $v_{D}$ , and the affine structure induced there is the fan structure induced by $D$ , by Corollary 2.6.1.

For any edge $e$ corresponding to $C_{e}=D_{i_{m}}\cap D_{i_{m^{\prime}}}$ , adopting the notation of Section 3.2,

\rho=\begin{cases}\rho_{\mathscr{X}_{ijk}}&\textrm{ for $i_{m}\neq i,j,k$, over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m}},a_{e})\\ \rho_{\mathscr{X}_{i^{\prime}j^{\prime}k^{\prime}}}&\textrm{ for $i_{m^{\prime}}\neq i^{\prime},j^{\prime},k^{\prime}$, over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m^{\prime}}},a_{e}),\end{cases}

and thus is an affinoid torus fibration over the union of these two open sets.

We conclude that $\rho$ induces an integral affine structure on $\Sk(X)$ away from the points $a_{e}$ . By Corollary 3.2.4, we can compute the monodromy around the singularities. As all these computations are analogous, we exhibit the case $C_{e}=D_{1}\cap D_{2}$ :

	$\displaystyle T_{\rho}(\gamma_{a_{e}})$	$\displaystyle=\left(\begin{matrix}1&0\\ b_{1,\mathscr{X}_{234}}-b_{1,\mathscr{X}_{134}}&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&0\\ 3-(-1)&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)$

with respect to the basis $(v_{3},v_{1})$ and origin $v_{2}$ . This formula was already stated in [KS06, §4.2.5].

[04PW]

3.3.3 Dispersion of singularities

We construct a third singular integral affine structure on $\Sk(X)$ pushing forward the techniques developed so far. This can be viewed as a dispersion of singularieties with respect to the integral affine structure studied in Section 3.3.2: on each edge we pass from one singular point around which the monodromy is $\begin{pmatrix}1&0\\ 4&1\end{pmatrix}$ , to $4$ singular points around each of which the monodromy is $\begin{pmatrix}1&0\\ 1&1\end{pmatrix}$ . in the literature Such singularities are called focus-focus and are the most standard examples of singularities for $\mathbb{Z}$ -affine structures in dimension 2. Those arise for instance when considering the hyperkähler rotation of a generic elliptic K3 surface $f:S\longrightarrow\mathbb{C}\mathbb{P}^{1}$ , with $f$ an elliptic fibration: the hyperkähler rotation $S^{\HK}$ is a complex surface with same underlying topological space as $S$ , and hence comes with a map $f^{\HK}:S^{\HK}\longrightarrow\mathbb{S}^{2}$ , induced by $f$ at the level of topological spaces. The map $f^{\HK}$ is no longer holomorphic, but is a symplectic torus fibration inducing a $\mathbb{Z}$ -affine structure with 24 focus-focus singularities on $\mathbb{S}^{2}$ and acting as an SYZ fibration for $S^{\HK}$ . We refer the reader to [GW00] for more details.

Let $e$ be an edge of $\Sk(\mathscr{X})$ , let $C_{e}=D_{e_{1}}\cap D_{e_{2}}$ be the corresponding stratum curve in $\mathscr{X}_{k}$ . We recall that as the degree four polynomial $F_{4}$ is generic, $C_{e}$ contains four singular points $p_{1},\ldots,p_{4}$ of $\mathscr{X}$ , which are ordinary double points. Around each $p_{i}$ , $\mathscr{X}$ is étale locally of the form $\{xy=wt\}\subset\mathbb{A}^{3}_{R}$ , with $x$ and $y$ being local equations for $D_{e_{1}}$ and $D_{e_{2}}$ away from $p_{i}$ . Blowing-up the singular point $p_{i}$ yields an exceptional divisor $E\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$ . Contracting one or the other ruling of $E$ , we obtain two distinct small resolutions of $\widehat{\mathscr{X}_{/C_{e}}}$ around $p_{i}$ , respectively with an exceptional curve inside $D_{e_{1}}$ or $D_{e_{2}}$ .

For $j\in\{0,\ldots,4\}$ , we denote by $\mathscr{X}_{e,j}$ the following small resolution of $\widehat{\mathscr{X}_{/C_{e}}}$ : around $p_{i}$ for $i\leqslant j$ we consider the small resolution such that the exceptional curve over $p_{i}$ lies in $D_{e_{1}}$ , while for $i>j$ the small resolution such that the exceptional curves lie in $D_{e_{2}}$ . The gluing of these local small resolutions is done in the étale topology, so that in general the obtained models are no longer schemes but only algebraic spaces. Nevertheless, $\mathscr{X}_{e,j}$ is dominated by $\mathscr{Z}$ (defined in Section 3.3.2) and still induces a Berkovich retraction $\rho_{\mathscr{X}_{e,j}}$ , as described in Section 5. In particular, by Proposition 5.0.4, $\rho_{\mathscr{X}_{e,j}}$ is an affinoid torus fibration over $\Star(\tau_{C_{e}})$ .

We construct the following continuous retraction

\overline{\rho}:X^{\an}\xrightarrow{\rho_{\mathscr{Z}}}\Sk(\mathscr{Z})\xrightarrow{\overline{\rho}^{\prime}}\Sk(X)=\Sk(\mathscr{X}),

where $\rho_{\mathscr{Z}}$ is the Berkovich retraction onto the skeleton $\Sk(\mathscr{Z})$ , and $\overline{\rho}^{\prime}$ is a retraction of the 24 wings of $\Sk(\mathscr{Z})$ to the sphere given as follows. We fix four distinct, ordered, interior points $a_{e,1},\ldots,a_{e,4}$ of each edge $e$ . Then the map $\overline{\rho}^{\prime}$ on the wing $W_{i}$ attached to $e$ is defined as the map $\rho^{\prime}$ of Section 3.3.2, setting $a_{e}=a_{e,i}$ , for each $i\in\{1,\ldots,4\}.$

[04PX]

Proposition 3.3.3.

The map $\overline{\rho}$ is an affinoid torus fibration away from the $24$ points $a_{e,i}$ . Furthermore, the monodromy of the $\mathbb{Z}$ -affine structure induced by $\overline{\rho}$ , around each singular point, is $\SL_{2}(\mathbb{Z})$ -conjugate to

T_{\overline{\rho}}=\left(\begin{matrix}1&0\\ 1&1\end{matrix}\right).

[04PY]

Proof.

Over $\textrm{Int}(\tau)$ of any $2$ -dimensional face $\tau\subset\Sk(\mathscr{X})$ , $\overline{\rho}$ is equal to $\rho_{\mathscr{Z}}$ , hence is an affinoid torus fibration. Around any vertex $v_{D}$ , $\overline{\rho}$ is equal to $\rho_{\mathscr{X}_{ijk}}$ for any triple such that $D\neq D_{i},D_{j},D_{k}$ . It follows from Section 3.3.1 that $\overline{\rho}$ is an affinoid torus fibration around $v_{D}$ . We denote by $p_{0}+p_{\infty}$ the boundary of $C_{e}$ , with $p_{0}=C_{e}\cap D_{i_{0}}$ and $p_{\infty}=C_{e}\cap D_{i_{\infty}}$ ; we write $a_{e,0}=v_{D_{e_{1}}}$ and $a_{e,5}=v_{D_{e_{2}}}$ , and denote by $(\cdot,\cdot)$ the open segment joining two points. Then, for $j\in\{0,\ldots,4\}$ , $\overline{\rho}$ is equal to $\rho_{\mathscr{X}_{e,j}}$ over $\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup(a_{e,j},a_{e,j+1})$ , thus is an affinoid torus fibration. We conclude that $\overline{\rho}$ is an affinoid torus fibration away from the points $a_{e,i}$ for $i\in\{1,\ldots,4\}$ .

For a singular point $a_{e,i}$ , we consider a loop $\gamma$ around it and contained in $\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup(a_{e,i-1},a_{e,i})\cup(a_{e,i},a_{e,i+1})$ . We apply Corollary 3.2.4 to compute the monodromy along $\gamma$ : the numbers $b_{e_{1},\mathscr{X}_{e,i-1}}$ and $b_{e_{1},\mathscr{X}_{e,i}}$ differ by $1$ , as the model $\mathscr{X}_{e,i}$ has an additional exceptional curves in $D_{e_{1}}$ with respect to $\mathscr{X}_{e,i-1}$ . Therefore, we obtain

	$\displaystyle T_{\overline{\rho}}(\gamma_{a_{e,i}})$	$\displaystyle=\left(\begin{matrix}1&0\\ b_{e_{1},\mathscr{X}_{e,i-1}}-b_{e_{1},\mathscr{X}_{e,i}}&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&0\\ 3-(i-1)-(3-i)&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&0\\ 1&1\end{matrix}\right)$

with respect to the basis $(v_{D_{i_{0}}},v_{D_{e_{1}}})$ and origin $v_{D_{e_{2}}}$ . ∎

Note that for a generic family of quartic surfaces $X/K$ , the metric aspects of the Kontsevich-Soibelman conjecture suggest that there should exist a distinguished singular affine structure on $\Sk(X)$ (coming from the Gromov-Hausdorff limit of the family), and hence a canonical choice of interior points $a_{e,i}$ for each edge. To the authors’ knowledge there does not exist a way to produce such a canonical set of $a_{e,i}$ ’s using non-archimedean techniques.

[04PZ]

3.3.4 Collision of singularities

In Section 3.3.2, the retraction $\rho^{\prime}$ depends on the choice of the points $a_{e}$ ; the same holds for the induced integral affine structure, whose singular locus consists indeed of the points $a_{e}$ . Moving a point $a_{e}$ in the interior of the edge $e$ affects the location of the singular points, but it does not change the monodromy around the point (see Section 3.3.3). We observe now, in two examples, what happens if we let a point $a_{e}$ move to a vertex of $\Sk(\mathscr{X})$ . We write $\rho=\rho_{a_{e}}$ to emphasize the dependency on the choice of singular points.

When all the points $a_{e}$ lie in the interior of the respective edges as in Section 3.3.2, $\rho_{a_{e}}=\rho_{\mathscr{X}_{ikh}}$ on $\Star(v_{j})^{\prime}$ , the induced integral affine structure is smooth at $v_{j}$ and such that

	$\displaystyle T_{\rho}(\gamma_{a_{e_{24}}})$	$\displaystyle=\left(\begin{matrix}1&0\\ b_{4,\mathscr{X}_{123}}-b_{4,\mathscr{X}_{134}}&1\end{matrix}\right)=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)$
		$\displaystyle\text{in the basis $(v_{D_{k}},v_{D_{h}})$ and origin $v_{D_{j}}$},$
	$\displaystyle T_{\rho}(\gamma_{a_{e_{23}}})$	$\displaystyle=\left(\begin{matrix}1&0\\ b_{3,\mathscr{X}_{124}}-b_{3,\mathscr{X}_{134}}&1\end{matrix}\right)^{-1}=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)^{-1}=\left(\begin{matrix}1&0\\ -4&1\end{matrix}\right)$
		$\displaystyle\text{in the basis $(v_{D_{h}},v_{D_{k}})$ and origin $v_{D_{j}}$}.$

When $a_{e_{24}}$ collides with the vertex $v_{j}$ , $\rho_{a_{e}}$ on $\Star(v_{j})^{\prime}$ is equal to $\rho_{\mathscr{X}_{ikj}}$ , the integral affine structure is singular at $v_{j}$ with

	$\displaystyle T_{\rho_{\mathscr{X}_{ikj}}}(\gamma_{j})$	$\displaystyle=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)=T_{\rho}(\gamma_{a_{e_{24}}})$
		$\displaystyle\text{in the basis $(v_{D_{k}},v_{D_{h}})$ and origin $v_{D_{j}}$}.$

When both $a_{e_{24}}$ and $a_{e_{23}}$ collide with $v_{j}$ , $\rho_{a_{e}}$ on $\Star(v_{j})^{\prime}$ is equal to $\rho_{\mathscr{X}_{ijk}}$ , the integral affine structure is singular at $v_{j}$ with

	$\displaystyle T_{\rho_{\mathscr{X}_{ijk}}}(\gamma_{j})$	$\displaystyle=\left(\begin{matrix}-15&-4\\ 4&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&-4\\ 0&1\end{matrix}\right)\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)=T_{\rho}(\gamma_{a_{e_{23}}})T_{\rho}(\gamma_{a_{e_{24}}})$
		$\displaystyle\text{in the basis $(v_{D_{k}},v_{D_{h}})$ and origin $v_{D_{j}}$}.$

The computations above suggest that the singularities and the monodromy representation induced by the non-archimedean SYZ fibration $\rho_{\mathscr{X}_{ijk}}$ can be viewed respectively as a collision of singular points and a product of monodromies induced by the $\rho_{a_{e}}$ when the $a_{e}$ ’s collide. The affine structure induced by $\rho$ turns out to be more symmetric and simpler, as all the singular points have the same monodromy and are of focus-focus type.

[04Q0]

4 Degeneration of quintic 3-folds

As discussed in the introduction, given a maximally degenerate family $X=(X_{t})_{t\in\mathbb{D}^{*}}$ of Calabi–Yau varieties, Kontsevich and Soibelman predict that the base $S$ of the conjectural SYZ fibration $\rho_{t}:X_{t}\rightarrow S$ matches the essential skeleton of $X$ . The SYZ fibration induces an integral affine structure (with singularities) on the base $S$ , which is relevant to the reconstruction of the mirror family; in the non-archimedean interpretation of mirror symmetry of [KS06] such structure is induced by Berkovich retractions $X^{\an}\rightarrow\Sk(X)$ .

The results in [NXY19] and in Section 2 can be used to construct non-archimedean retractions and integral affine structures on $\Sk(X)$ , using minimal models of $X$ . As quintic Calabi–Yau hypersurfaces, and in particular Fermat type quintics, have played a key role in the development of mirror symmetry, it is natural to apply and test the non-archimedean approach on this family. In this section we prove Theorem A; this yields an integral affine structure which is compatible with the existing literature in mirror symmetry (see Remark 4.1.1, Remark 4.7.2 and Section 4.8).

[04Q1]

4.1 Setting and plan of the proof

We consider $\mathscr{X}=\{z_{1}z_{2}z_{3}z_{4}z_{5}+tF_{5}(z_{1},z_{2},z_{3},z_{4},z_{5})=0\}\subset\mathbb{P}^{4}_{R}$ , where $F_{5}$ is a generic homogeneous polynomial of degree $5$ . The degeneration $\mathscr{X}$ has the following properties:

1.

the special fiber $\mathscr{X}_{k}$ is reduced, consisting of five Weil divisors, i.e. $D_{i}=\{z_{i}=t=0\}$ . We denote $D_{i}^{\prime}:=\{z_{i}=F_{5}=0\}$ ;
2.

the singular locus ${\mathscr{X}}^{\text{sing}}$ of the total space $\mathscr{X}$ is contained in the special fiber, and is the intersection in $\mathbb{P}_{k}^{4}$ of $\{F_{5}=0\}$ and the union of surfaces $S_{ij}=\{z_{i}=z_{j}=0\}$ for $i\neq j$ . In particular, each $D_{i}$ intersects ${\mathscr{X}}^{\text{sing}}$ along the union of four quintic curves $C_{ij}$ and by genericity of $F_{5}$ , we may assume that $C_{ij}$ does not intersect the torus fixed points of $D_{i}$ ; ${\mathscr{X}}^{\text{sing}}\cap D_{i}=\bigcup_{\begin{subarray}{c}j=1\\ j\neq i\end{subarray}}^{5}C_{ij}$ $C_{ij}\subseteq D_{i}\cap D_{j}$ $C_{ij}\cap C_{ij^{\prime}}=\{5\text{ points}\}\text{ for }j\neq j^{\prime}$

Figure 2: *

Irreducible component $D_{i}$
3.

the pair $(\mathscr{X},\mathscr{X}_{k})$ is dlt, in particular snc away from ${\mathscr{X}}^{\text{sing}}$ ; we refer to Section 4.2 for a local study of the pair at the singular points;
4.

the dual complex $\mathcal{D}(\mathscr{X}_{k})$ of the special fiber is homeomorphic to the $3$ -sphere $\mathbb{S}^{3}$ , and the triangulation of $\mathcal{D}(\mathscr{X}_{k})$ is the same as the standard one on the boundary of a $4$ -simplex;
5.

by adjunction the canonical bundle $K_{\mathscr{X}}$ is trivial.

We conclude that $\mathscr{X}$ is a minimal dlt model of the quintic $3$ -fold $X\coloneqq\mathscr{X}_{K}$ , but it is not good in the sense of Section 1.1 since the prime components of the special fiber are not $\mathbb{Q}$ -Cartier. In particular, even if the dual complex $\mathcal{D}(\mathscr{X}_{k})$ is well-defined, $\mathscr{X}$ does not induce a well-defined retraction of $X^{\text{an}}$ onto $\mathcal{D}(\mathscr{X}_{k})=\Sk(X)$ .

Similarly to Section 3.3, the aim is to explicitly construct several explicit minimal models of $X$ starting from $\mathscr{X}$ , then apply the results from Section 3.1 and Section 3.2 to study the integral affine structures on $\Sk(X)$ , induced by Berkovich retractions or their combinations. We will proceed as follows.

-

(Section 4.5) For any order $(i,j,k,l,h)$ on $\{1,\ldots,5\}$ , we construct a small resolution of $\mathscr{X}$ by blowing-up in order the four divisors $D_{i}$ , $D_{j}$ , $D_{k}$ and $D_{l}$ . The resulting resolution is denoted $\mathscr{X}_{ijkl}$ , is a minimal model of $X$ and comes equipped with the Berkovich retraction

$\rho_{\mathscr{X}_{ijkl}}:X^{\an}\rightarrow\Sk(\mathscr{X}_{ijkl})=\Sk(X)\simeq\mathbb{S}^{3}.$

The skeleton $\Sk(\mathscr{X}_{ijkl})$ coincides with $\mathcal{D}(\mathscr{X}_{k})$ as simplicial complex; thus, independently on the order, all skeletons $\Sk(\mathscr{X}_{ijkl})$ define the same simplicial structure on $\Sk(X)$ .
-

(Section 4.6) We construct a model $\mathscr{Z}$ of $X$ which dominates any model $\mathscr{X}_{ijkl}$ , so that $\rho_{\mathscr{X}_{ijkl}}$ factors through $\rho_{\mathscr{Z}}$ . We then define a combinatorial retraction $\pi^{\prime}$ which contracts the skeleton $\Sk(\mathscr{Z})$ onto $\Sk(X)$ . This allows us to consider the composition

$\pi:X^{\an}\xrightarrow{\rho_{\mathscr{Z}}}\Sk(\mathscr{Z})\xrightarrow{\pi^{\prime}}\Sk(X)$

which is at the core of the statement of Theorem A. The retraction $\pi^{\prime}$ is constructed so that the composition $\pi$ is locally equal to a $\rho_{\mathscr{X}_{ijkl}}$ , the order depending on the region of $\Sk(X)$ .
-

(Section 4.2 to Section 4.4) The constructions and properties of $\mathscr{X}_{ijkl},\mathscr{Z}$ and $\pi^{\prime}$ rely on a local study of the model $\mathscr{X}$ : étale locally around each point of ${\mathscr{X}}^{\text{sing}}\cap D_{i}\cap D_{j}\cap D_{j^{\prime}}$ , $\mathscr{X}$ is isomorphic to a toric variety and the resolutions are given by refinements of the associated fan.

We will show that

[04Q2]

Theorem A.

•

$\pi^{\prime}$ is a piecewise-linear map, thus $\pi$ pulls back any piecewise-linear function on $\Sk(X)$ to a model function on $X^{\an}$ ;
•

$\pi$ is an affinoid torus fibration away from a graph $\Gamma\subset\Sk(X)$ ; the vertices of $\Gamma$ are the barycenters of the 1 and 2-dimensional cells of $\Sk(X)$ , and the edges join the barycenter of a 2-dimensional cell with the barycenters of its 1-dimensional faces;
•

in a neighbourhood of a vertex $v_{i}\in\Sk(X)$ , the affine structure induced by $\pi$ is determined by the toric geometry of $D_{i}$ : there is a natural $\mathbb{Z}$ -linear embedding of $\Star(v_{D_{i}})$ inside the fan of $D_{i}$ , preserving the polytopal decomposition and sending $v_{D_{i}}$ to the origin;
•

$\pi$ induces an integral affine structure on $\Sk(X)\setminus\Gamma$ , which is isomorphic to the ones constructed in [Gro01] and [Rua01], and in [Li19] for Fermat families.

[04Q3]

Remark 4.1.1.

The affine structure we obtain in Theorem A coincides also with the one defined in [Gro05, §1]. Indeed, it will follow from the construction and Corollary 2.6.1 that the affine structure induced by $\pi$ yields the fan structure (in the sense of [Gro05]) coming from $D_{i}$ at each vertex $v_{D_{i}}$ , and that those are glued (after removing $\Gamma$ ) along the maximal cells viewed as standard simplices; this is the very definition of the singular affine structure in [Gro05].
In particular, the results of Section 4.7 follow from the more general [Gro05, Proposition 2.13], [HZ02, §2.3], but we include our full computations to highlight the use of Proposition 3.2.2, which holds even outside the context of toric degenerations.

[04Q4]

4.2 Local resolution

We consider a point in ${\mathscr{X}}^{\text{sing}}\cap D_{1}\cap D_{2}\cap D_{3}$ ; the singular points in the other strata curves can be treated analogously. Étale locally around such a point, $\mathscr{X}$ is isomorphic to the toric variety $\mathscr{U}\coloneqq V(xyz-wt)\subset\mathbb{A}^{5}_{k}$ , where

D_{1}|_{\mathscr{U}}=\{x=t=0\},\quad D_{2}|_{\mathscr{U}}=\{y=t=0\}\,\text{ and }\,D_{3}|_{\mathscr{U}}=\{z=t=0\}

are the components of the special fiber $\{t=0\}$ in $\mathscr{U}$ . We still denote these by $D_{1},D_{2}$ and $D_{3}$ ; they form the toric boundary of $\mathscr{U}$ together with

D_{1}^{\prime}|_{\mathscr{U}}=\{x=w=0\},\quad D_{2}^{\prime}|_{\mathscr{U}}=\{y=w=0\}\,\text{ and }\,D_{3}^{\prime}|_{\mathscr{U}}=\{z=w=0\}.

The pair $(\mathscr{X},\mathscr{X}_{k})_{|\mathscr{U}}=(\mathscr{U},\sum_{i=1,2,3}D_{1}|_{\mathscr{U}})$ is log canonical by [CLS11, Proposition 11.4.24].

We denote the strata surfaces of the special fiber by $D_{ij}\coloneqq D_{j}\cap D_{j}$ , for $i,j\in\{1,2,3\}$ and $i\neq j$ , and the stratum curve by $D_{123}\coloneqq\{x=y=z=t=0\}$ . The singular locus ${\mathscr{U}}^{\text{sing}}$ consists of the torus invariant curves

C_{12}|_{\mathscr{U}}=\{x=y=w=t=0\}=D_{1}\cap D_{2}\cap D_{1}^{\prime}\cap D_{2}^{\prime},

C_{13}|_{\mathscr{U}}=\{x=z=t=w=0\}\,\text{ and }\,C_{23}|_{\mathscr{U}}=\{y=z=w=t=0\},

which intersect each other at the torus invariant point $p\coloneqq\{x=y=z=w=t=0\}$ .

Figure 3: *

Special fiber of $\mathscr{U}$ and a slice of the fan of the toric variety $\mathscr{U}$

The toric blow-up $G_{1}:\mathscr{U}_{1}\coloneqq\Bl_{D_{1}}\mathscr{U}\rightarrow\mathscr{U}$ along $D_{1}$ resolves the singularities along $C_{12}$ and $C_{13}$ except at the point $p$ . The exceptional locus of $G_{1}$ consists of two surfaces $S_{12}$ and $S_{13}$ intersecting each other along a curve: these surfaces are mapped by $G_{1}$ to the respective singular curves, are contained in the strict transform of $D_{1}$ , and correspond to two new edges in the slice of the fan of $\mathscr{U}_{1}$ . The intersection of $S_{12}$ and $S_{13}$ corresponds to a new $2$ -dimensional face in the slice of the fan. With a slight abuse of notation we keep the same notation for the strict transforms in $\mathscr{U}_{1}$ .

A resolution of $\mathscr{U}$ is given by the composition of $G_{1}$ and the toric blow-up $G_{12}:\mathscr{U}_{12}\coloneqq\Bl_{D_{2}}\mathscr{U}_{1}\rightarrow\mathscr{U}_{1}$ along $D_{2}$ ; the latter indeed resolves the singularities along $C_{23}$ . The exceptional locus of $G_{12}$ is a surface $S_{23}$ , which is mapped by $G_{12}$ to $C_{23}$ and is contained in the strict transform of $D_{2}$ . The morphism $G_{12}$ induces a new $2$ -dimensional face in the slice of the fan of $\mathscr{U}_{12}$ , and a new edge corresponding to the surface $S_{23}$ . In particular, after the blow-up $G_{12}$ , the strict transforms of the surface $S_{12}$ and of the divisor $D_{3}$ have empty intersection.

Figure 4: *

Slices of the fan of $\mathscr{U}$ , $\mathscr{U}_{1}$ and $\mathscr{U}_{12}$

The small resolution $\mathscr{U}_{12}$ of $\mathscr{U}$ induces an isomorphism on the strict transform of $D_{13}$ and of $D_{23}$ , while its restriction to the strict transform of $D_{12}$ is the blow-up of $D_{12}$ along a general point. These facts can be checked computing the charts of the blow-ups $G_{1}$ and $G_{12}$ . Alternatively, they can be verified looking at the fans of the strata surfaces $D_{ij}$ in the slice of the fans of $\mathscr{U}$ , $\mathscr{U}_{1}$ and $\mathscr{U}_{12}$ ; indeed, the fan of $D_{ij}$ is induced by the intersection of the slice with a normal plane to the edge corresponding to $D_{ij}$ .

[04Q5]

4.3 Local dominating model

We now introduce the local model for the dominating model $\mathscr{Z}\longrightarrow\mathscr{X}$ we will construct.
We consider the blow-up of the exceptional surfaces $S_{12},S_{13}$ and $S_{23}$ one after the other

$\begin{array}[]{ccccccccc}\mathscr{V}_{123}&\xrightarrow[H_{23}]{\text{blow-up of }S_{23}}&\mathscr{V}_{13}&\xrightarrow[H_{13}]{\text{blow-up of }S_{13}}&\mathscr{V}_{12}&\xrightarrow[H_{12}]{\text{blow-up of }S_{12}}&\mathscr{U}_{12}&\xrightarrow[G_{12}\circ G_{1}]{}&\mathscr{U}\\ \cup&&\cup&&\cup&&&&\downarrow\\ E_{23}&&E_{13}&&E_{12}&&&&\mathbb{A}^{1}_{t}.\end{array}$

As these surfaces are toric strata of $\mathscr{U}_{12}$ , the blow-ups are toric as well and the corresponding fans are refinements of the fan of $\mathscr{U}_{12}$ . We note that

-

the dual complexes of the special fibers of $\mathscr{U}_{12},\mathscr{V}_{12},\mathscr{V}_{13}$ and $\mathscr{V}_{123}$ are obtained from the slices of the corresponding fans by removing the vertices corresponding to $D_{1}^{\prime}$ , $D_{2}^{\prime}$ and $D_{3}^{\prime}$ , as well as each face containing one of these.

Then $\Sk(\mathscr{V}_{123})$ consists of four $3$ -cells: $<v_{13},v_{1},v_{2},v_{3}>$ , $<v_{13},v_{1},v_{2},v_{12}>$ , $<v_{23},v_{13},v_{2},v_{3}>$ and $<v_{23},v_{13},v_{2},v_{12}>$ ; it has only one edge in the interior, which is $<v_{2},v_{13}>$ .

The remaining $3$ -dimensional simplices of the slice of the fan of $\mathscr{V}_{123}$ are

		$\displaystyle<v_{12},v_{1}^{\prime},v_{2}^{\prime},v_{3}^{\prime}>$
		$\displaystyle<v_{13},v_{1}^{\prime},v_{3}^{\prime},v_{12}>$
		$\displaystyle<v_{23},v_{12},v_{2}^{\prime},v_{3}^{\prime}>$
		$\displaystyle<v_{23},v_{13},v_{3}^{\prime},v_{12}>$

		$\displaystyle<v_{1},v_{1}^{\prime},v_{12},v_{13}>$
		$\displaystyle<v_{2},v_{2}^{\prime},v_{12},v_{23}>$
		$\displaystyle<v_{3},v_{3}^{\prime},v_{13},v_{23}>$

-

The Berkovich retractions associated with the models $\mathscr{U}_{12}$ , $\mathscr{V}_{12}$ and $\mathscr{V}_{13}$ map $v_{12}$ and $v_{13}$ to $v_{1}$ , and $v_{23}$ to $v_{2}$ .

We study more in details the retraction $\rho_{\mathscr{V}_{12}}$ near the vertex $v_{3}$ , as this will be relevant later in the construction of the local combinatorial retraction (see Eq. 4.4.1). We observe that $\rho_{\mathscr{V}_{12}}$ collapses the convex hull $P(v_{1},v_{2},v_{3},v_{13},v_{23})$ of $v_{1},v_{2},v_{3},v_{13}$ and $v_{23}$ onto the face $<v_{1},v_{2},v_{3}>$ . If we identify the skeleton $\Sk(\mathscr{V}_{123})$ with the polyhedron in $\mathbb{R}^{3}_{(x,y,z)}$ below, $\rho_{\mathscr{V}_{12}}$ on $P(v_{1},v_{2},v_{3},v_{13},v_{23})$ is written explicitly as follows:

(4.3.1)

\displaystyle\begin{split}&\text{for }(x,y,z)\in P(v_{1},v_{2},v_{3},v_{13},v_{23})\setminus\{v_{3}\},\\ &\rho_{\mathscr{V}_{12}}\big((x,y,z)\big)=\Big(x+\left(1-\frac{t}{2}\right)z,y+\frac{t}{2}z,0\Big)\\ &\text{where }t=\frac{2y}{x+y+1}.\end{split}

The function $t$ on $<v_{1},v_{2},v_{3}>$ is the slope of the line segment joining the vertex $v_{3}$ to $tv_{1}+(1-t)v_{2}$ for $t\in[0,1]$ . We give a picture of the retraction $\rho_{\mathscr{V}_{12}}$ for various values of $t$ :

For purposes which will be clear in the construction of the local combinatorial retraction in Section 4.4, we consider a further toric blow-up. Let $H_{123}:\mathscr{G}\rightarrow\mathscr{V}_{123}$ be the blow-up along the disjoint toric strata $D_{2}\cap E_{13}$ and $E_{12}\cap D_{3}^{\prime}$ ; this yields two new components in the toric boundary, denoted by $E_{123}$ and $E^{\prime}_{123}$ . It follows that the slice of the fan of $\mathscr{G}$ is obtained from the slice of $\mathscr{V}_{123}$ as star subdivision along the edges $<v_{2},v_{13}>$ and $<v_{12},v_{3}^{\prime}>$ .

In particular, the skeleton $\Sk(\mathscr{G})$ is obtained from $\Sk(\mathscr{V}_{123})$ by

1.

the star subdivision of the edge $<v_{2},v_{13}>$ , which turns the four $3$ -cells of $\Sk(\mathscr{V}_{123})$ into eight $3$ -cells;
2.

adding an additional $3$ -cell $\tau=<v_{12},v_{13},v_{23},v^{\prime}_{123}>$ , where we denote by $v^{\prime}_{123}$ the new vertex corresponding to $E^{\prime}_{123}$ .

The diagram below summarizes the resolutions of $\mathscr{U}$ we constructed and studied so far:

$\begin{array}[]{ccccccccc}\mathscr{G}&\xrightarrow[H_{123}]{\begin{subarray}{c}\text{blow-up of }\\ D_{2}\cap E_{13},E_{12}\cap D^{\prime}_{3}\end{subarray}}&\mathscr{V}_{123}&\xrightarrow[H_{23}\circ H_{13}\circ H_{12}]{\begin{subarray}{c}\text{blow-up of }\\ S_{12},S_{13},S_{23}\end{subarray}}&\mathscr{U}_{12}&\xrightarrow[G_{12}\circ G_{1}]{\begin{subarray}{c}\text{blow-up of }\\ D_{1},D_{2}\end{subarray}}&\mathscr{U}.\\ \cup&&\cup&&\cup&&\\ E_{123},E^{\prime}_{123}&&E_{12},E_{13},E_{23}&&S_{12},S_{13},S_{23}&&\end{array}$

[04Q6]

4.4 Local combinatorial retraction

Other resolutions of $\mathscr{U}$ can be obtained by blowing-up the divisors of the special fiber in a different order. Given any order $(i_{1},i_{2},i_{3})$ on $\{1,2,3\}$ , we denote

$\begin{array}[]{ccccccccc}\mathscr{V}_{i_{1}i_{2}i_{3}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ S_{i_{1}i_{3}},S_{i_{2}i_{3}}\end{subarray}}&\mathscr{V}_{i_{1}i_{2}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ S_{i_{1}i_{2}}\end{subarray}}&\mathscr{U}_{i_{1}i_{2}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ D_{i_{1}},D_{i_{2}}\end{subarray}}&\mathscr{U}.\\ \cup&&\cup&&\cup&&\\ E_{i_{1}i_{3}},E_{i_{2}i_{3}}&&E_{i_{1}i_{2}}&&S_{i_{1}i_{2}},S_{i_{1}i_{3}},S_{i_{2}i_{3}}&&\end{array}$

The refinement of the fan of $\mathscr{U}$ corresponding to $\mathscr{V}_{i_{1}i_{2}i_{3}}$ is such that the skeleton $\Sk(\mathscr{V}_{i_{1}i_{2}i_{3}})=\Sk(\mathscr{V}_{123})$ as subspaces in the Berkovich space of $\mathscr{U}_{K}$ ; it is independent on the chosen order so that we simply denote this subspace by $\Sk(\mathscr{V})$ . However, the models $\mathscr{V}_{i_{1}i_{2}i_{3}}$ and $\mathscr{V}_{123}$ induce in general different simplicial subdivisions and different retractions onto $<v_{1},v_{2},v_{3}>$ . For instance, the only edge in the interior of $\Sk(\mathscr{V}_{i_{1}i_{2}i_{3}})$ is $<v_{i_{2}},v_{i_{1}i_{3}}>$ , which indeed depends on the chosen order. Here below we illustrate the skeletons and the Berkovich retractions in a couple of examples.

The blow-up of $\mathscr{V}_{i_{1}i_{2}i_{3}}$ along the toric strata $D_{i_{2}}\cap E_{i_{1}i_{3}}$ and $E_{i_{1}i_{2}}\cap D_{i_{3}}^{\prime}$ yields a refinement of the fan which coincides with the fan of $\mathscr{G}$ , constructed at the end of Section 4.3. It follows that the model $\mathscr{G}$ dominates all resolutions $\mathscr{V}_{i_{1}i_{2}i_{3}}$ independently on the order, hence all Berkovich retractions $\rho_{\mathscr{V}_{i_{1}i_{2}i_{3}}}$ , $\rho_{\mathscr{V}_{i_{1}i_{2}}}$ and $\rho_{\mathscr{U}_{i_{1}i_{2}}}$ factors through $\rho_{\mathscr{G}}$ .

Our goal is to construct a map $\pi$ , composing the Berkovich retraction $\rho_{\mathscr{G}}$ with a collapse $\kappa$ of the additional 3-cell $\tau$ and a combinatorial retraction $\rho$

$\begin{array}[]{ccccccccc}&\pi:&\mathscr{U}_{K}^{\an}&\xrightarrow{\rho_{\mathscr{G}}}&\Sk(\mathscr{G})&\xrightarrow[\text{collapse}]{\kappa}&\Sk(\mathscr{V})&\xrightarrow[\text{retraction}]{\rho}&\Sk(\mathscr{U})\\ &\rotatebox[origin={c}]{270.0}{$=$}&\cup&&\cup&&\cup&&\cup\\ \text{over }\Star(v_{j})^{\prime}\text{\hskip 10.0pt}&\rho_{\mathscr{U}_{i_{1}i_{2}}}:&\pi^{-1}(\Star(v_{j})^{\prime})&\xrightarrow{\rho_{\mathscr{G}}}&\Sk(\mathscr{G})&\xrightarrow{\rho_{\mathscr{V}_{i_{1}i_{2}j}}}&\Sk(\mathscr{V}_{i_{1}i_{2}j})&\xrightarrow{\rho_{\mathscr{U}_{i_{1}i_{2}}}}&\Star(v_{j})^{\prime}\end{array}$

such that, given any vertex $v_{j}$ in $\Sk(\mathscr{U})$ , the restriction of $\pi$ over $\Star(v_{j})^{\prime}$ (the $\Star$ is taken with respect to the first barycentric subdivision, as in Definition 3.2.1) is $\rho_{\mathscr{U}_{i_{1}i_{2}}}$ for any order $(i_{1},i_{2},j)$ on $\{1,2,3\}$ , i.e. any order where the index $j$ is the biggest. This guarantees that around each $v_{j}$ , the map $\pi$ is the Berkovich retraction induced by a small resolution $\mathscr{U}_{i_{1}i_{2}}$ where the strict transform of $D_{j}$ is isomorphic to $D_{j}$ , so that we are in the set-up of Corollary C.

The retraction $\rho$ . We identify again the skeleton $\Sk(\mathscr{V})$ with the polyhedron in $\mathbb{R}^{3}$ described in Section 4.3. On the convex hull $P$ of $v_{23},(-1/4,1/4,1),(0,1/3,1),(0,0,0),v_{3}$ and $(0,1/3,0)$ , the retraction $\rho$ is given as follows

(4.4.1)

\displaystyle\begin{split}&(x,y,z)\in P\mapsto\begin{cases}\Big(x+\left(1-\frac{t}{2}\right)z,y+\frac{t}{2}z,0\Big)&\text{ if }x+\left(1-\frac{t}{2}\right)z\leqslant 0\\ \Big(0,\frac{y}{x+1},0\Big)&\text{ if }x+\left(1-\frac{t}{2}\right)z\geqslant 0\end{cases}\\ &\text{where }t=\frac{2y}{x+y+1}.\end{split}

Here is a pictorial description for certain values of $t$ :

We extend the definition of $\rho$ to $\Sk(\mathscr{V})$ by symmetry along the medians of the triangles $<v_{1},v_{2},v_{3}>$ and $<v_{12},v_{13},v_{23}>$ . In particular, we note that the image of $<v_{12},v_{13},v_{23}>$ is the graph in $<v_{1},v_{2},v_{3}>$ of Definition 3.2.1.

-

The combinatorial retraction $\pi^{\prime}$ . We define the collapse $\kappa$ as the projection of the additional $3$ -cell $\tau$ of $\Sk(\mathscr{G})$ onto $<v_{12},v_{13},v_{23}>$ along the $z$ -direction. We call $\pi^{\prime}\coloneqq\rho\circ\kappa$ the combinatorial retraction of the skeleton $\Sk(\mathscr{G})$ onto $\Sk(\mathscr{U})=<v_{1},v_{2},v_{3}>$ .
-

Finally, we check that $\pi^{\prime}=\rho_{\mathscr{U}_{i_{1}i_{2}}}$ over $\Star(v_{j})^{\prime}$ . As the preimage of $\Star(v_{j})^{\prime}$ is disjoint from $<v_{12},v_{13},v_{23},v^{\prime}_{123}>$ , we have to prove that $\rho=\rho_{\mathscr{U}_{i_{1}i_{2}}}$ . By symmetry of $\rho$ , it is enough to check this for $v_{3}$ . Over $\Star(v_{3})^{\prime}$ we have $\rho_{\mathscr{U}_{i_{1}i_{2}}}=\rho_{\mathscr{V}_{i_{1}i_{2}}}$ ; there, the expression of $\rho_{\mathscr{V}_{i_{1}i_{2}}}$ determined in Eq. 4.3.1 coincides with the definition of $\rho$ in Eq. 4.4.1, hence we conclude.

[04Q7]

4.5 Minimal models $\mathscr{X}_{ijkl}$

We return to the setting of Section 4.1. The purpose of this section is to compute various intersection numbers on the small resolutions of $\mathscr{X}$ used to define the retraction $\pi$ , as this will allow us to compute the monodromy of the associated $\mathbb{Z}$ -affine structure on $\Sk(X)$ .
Fix an order $(i,j,k,l,h)$ on $\{1,\ldots,5\}$ and consider the small resolution $\mathscr{X}_{ijkl}$ , obtained from $\mathscr{X}$ by blowing-up the divisors $D_{i}$ , $D_{j}$ , $D_{k}$ , $D_{l}$ in that order. It now follows from the local study of the singularities of $\mathscr{X}$ that this is indeed a small resolution of $\mathscr{X}$ .
In $\mathscr{X}_{ijkl}$ we still denote the strict transforms of the strata of $\mathscr{X}_{k}$ by $D_{m}$ , by $D_{mm^{\prime}}=D_{m}\cap D_{m^{\prime}}$ and by $D_{mm^{\prime}m^{\prime\prime}}=D_{m}\cap D_{m^{\prime}}\cap D_{m^{\prime\prime}}$ with $m,m^{\prime},m^{\prime\prime}\in\{1,\ldots,5\}$ . By the study of the local model in Section 4.2, the exceptional locus of $g_{ijkl}:\mathscr{X}_{ijkl}\rightarrow\mathscr{X}$

$\begin{array}[]{cccccccccc}g_{ijkl}:&\mathscr{X}_{ijkl}&\xrightarrow[G_{ijkl}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{l}\end{subarray}}&\mathscr{X}_{ijk}&\xrightarrow[G_{ijk}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{k}\end{subarray}}&\mathscr{X}_{ij}&\xrightarrow[G_{ij}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{j}\end{subarray}}&\mathscr{X}_{i}&\xrightarrow[G_{i}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{i}\end{subarray}}&\mathscr{X}\\ &\cup&&\cup&&\cup&&\cup&&\\ &S_{lh}&&S_{kl},S_{kh}&&S_{jk},S_{jl},S_{jh}&&S_{ij},S_{ik},S_{il},S_{ih}&&\end{array}$

consists of ten surfaces $S_{mm^{\prime}}$ , with $m,m^{\prime}\in\{1,\ldots,5\}$ and $m<m^{\prime}$ in the order $(i,j,k,l,h)$ . The surface $S_{mm^{\prime}}$ is mapped via $g_{ijkl}$ to the singular curve $C_{mm^{\prime}}$ , and is contained in the strict transform of $D_{m}$ . The component $D_{h}$ (corresponding to the biggest index in the chosen order) is the only one isomorphic to its strict transform.

Given a pair $(m,m^{\prime})$ with $m<m^{\prime}$ , the morphism $g_{ijkl}$ induces on $D_{mm^{\prime}}$ the blow-up along $5$ distinct general points on each $D_{mm^{\prime}m^{\prime\prime}}$ with $m^{\prime}<m^{\prime\prime}$ . Thus, the intersection numbers between strata curves and strata divisors in $\mathscr{X}_{ijkl}$ are:

	$D_{i}$	$D_{j}$	$D_{k}$	$D_{l}$	$D_{h}$
$C_{ijk}$	1	1	-4	1	1
$C_{ijl}$	1	1	1	-4	1
$C_{ijh}$	1	1	1	1	-4
$C_{ikl}$	1	1	1	-4	1
$C_{ikh}$	1	1	1	1	-4
$C_{ilh}$	1	1	1	1	-4
$C_{jkl}$	1	1	1	-4	1
$C_{jkh}$	1	1	1	1	-4
$C_{jlh}$	1	1	1	1	-4
$C_{klh}$	1	1	1	1	-4

[04Q8]

4.6 Dominating model and combinatorial retraction

We consider the blow-up $h_{ijkl}:\mathscr{W}_{ijkl}\rightarrow\mathscr{X}_{ijkl}$ of the surfaces $S_{mm^{\prime}}$ in lexicographical order with respect to $(i,j,k,l,h)$ ; we denote by $E_{mm^{\prime}}$ the corresponding exceptional divisors. The skeleton $\Sk(\mathscr{W}_{ijkl})$ consists of the union of the skeleton $\Sk(X)$ with four additional $3$ -cells for each 2-dimensional face $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ of $\Sk(X)$ : for each ordered triple $m<m^{\prime}<m^{\prime\prime}$ , the union of the additional cells is isomorphic to $\Sk(\mathscr{V}_{123})$ in Section 4.3, where we identify $v_{1}=v_{m}$ , $v_{2}=v_{m^{\prime}}$ and $v_{3}=v_{m^{\prime\prime}}$ . The retraction $\rho_{\mathscr{X}_{ijkl}}$ collapses the additional faces onto $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ as $\rho_{\mathscr{U}_{12}}\circ\rho_{\mathscr{V}_{12}}$ .

Additional $3$ -cells of $\Sk(\mathscr{W}_{ijkl})$ over $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$

with a pictorial description of the retraction $\rho_{\mathscr{X}_{ijkl}}$

Given another order $(i^{\prime},j^{\prime},k^{\prime},l^{\prime},h^{\prime})$ on $\{1,2,3,4,5\}$ , the skeleton $\Sk(\mathscr{W}_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}})$ coincides with $\Sk(\mathscr{W}_{ijkl})$ as subspace of $X^{\an}$ ; we denote this simply by $\Sk(\mathscr{W})$ . Instead, the triangulation and the retraction depend on the order. Our goal is therefore to construct a model $\mathscr{Z}$ which dominates all models $\mathscr{W}_{ijkl}$ regardless of the order, so that all retractions $\rho_{\mathscr{W}_{ijkl}}$ factors through $\rho_{\mathscr{Z}}$ .

Along the same lines of Section 4.3, we define $\mathscr{Z}$ as the blow-up of $\mathscr{W}_{ijkl}$ along $D_{m^{\prime}}\cap E_{mm^{\prime\prime}}$ and $E_{mm^{\prime}}\cap D_{m^{\prime\prime}}^{\prime}$ , for all ordered triples $m<m^{\prime}<m^{\prime\prime}$ in the order $(i,j,k,l,h)$ :

$\begin{array}[]{ccccccccc}\mathscr{Z}&\xrightarrow[\text{for all }m<m^{\prime}<m^{\prime\prime}]{\begin{subarray}{c}\text{blow-up of }\\ D_{m^{\prime}}\cap E_{mm^{\prime\prime}},E_{mm^{\prime}}\cap D_{m^{\prime\prime}}^{\prime}\end{subarray}}&\mathscr{W}_{ijkl}&\xrightarrow[h_{ijkl}]{\begin{subarray}{c}\text{blow-up of }S_{mm^{\prime}}\\ \text{for all }m<m^{\prime}\end{subarray}}&\mathscr{X}_{ijkl}&\xrightarrow[g_{ijkl}]{\begin{subarray}{c}\text{blow-up of }\\ D_{i},D_{j},D_{k},D_{l}\end{subarray}}&\mathscr{X}.\\ \cup&&\cup&&\cup&&\\ E_{mm^{\prime}m^{\prime\prime}},E^{\prime}_{mm^{\prime}m^{\prime\prime}}&&E_{mm^{\prime}}&&S_{mm^{\prime}}&&\end{array}$

We denote by $E_{mm^{\prime}m^{\prime\prime}}$ and $E^{\prime}_{mm^{\prime}m^{\prime\prime}}$ the corresponding exceptional divisors, and deduce from the local study of these morphisms in Section 4.3 that $\Sk(\mathscr{Z})$ is obtained from $\Sk(\mathscr{W})$ by adding a new $3$ -cell $\tau_{mm^{\prime}m^{\prime\prime}}:=<v_{mm^{\prime}},v_{mm^{\prime\prime}},v_{m^{\prime}m^{\prime\prime}},v^{\prime}_{mm^{\prime}m^{\prime\prime}}>$ for each triple $m<m^{\prime}<m^{\prime\prime}$ .

We now define the combinatorial retraction of $\Sk(\mathscr{Z})$ onto $\Sk(X)$ : given the 2-cell $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ , we identify $v_{1}=v_{m}$ , $v_{2}=v_{m^{\prime}}$ and $v_{3}=v_{m^{\prime\prime}}$ and contract onto $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ the additional cells of $\Sk(\mathscr{Z})$ over $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ , via the combinatorial retraction $\pi^{\prime}=\rho\circ\kappa$ constructed in Section 4.4. With a slight abuse of notation, we still denote this map by $\pi^{\prime}$ . By construction, the composition $\pi=\pi^{\prime}\circ\rho_{\mathscr{Z}}$

$\begin{array}[]{ccccccc}&\pi:&X^{\an}&\xrightarrow{\rho_{\mathscr{Z}}}&\Sk(\mathscr{Z})&\xrightarrow[\begin{subarray}{c}\text{combinatorial}\\ \text{retraction}\end{subarray}]{\pi^{\prime}}&\Sk(X)\\ &\rotatebox[origin={c}]{270.0}{$=$}&\cup&&\cup&&\cup\\ \text{over }\Star(v_{m})^{\prime}\text{\hskip 10.0pt}&\rho_{\mathscr{X}_{ijkl}}:&\pi^{-1}(\Star(v_{m})^{\prime})&\xrightarrow{\rho_{\mathscr{Z}}}&\Sk(\mathscr{Z})&\xrightarrow{\rho_{\mathscr{X}_{ijkl}}}&\Star(v_{m})^{\prime}\end{array}$

coincides with $\rho_{\mathscr{X}_{ijkl}}$ over $\Star(v_{m})^{\prime}$ for any order $(i,j,k,l,m)$ on $\{1,2,3,4,5\}$ . In other words, around each vertex $v_{m}$ , the map $\pi$ is the Berkovich retraction induced by a small resolution $\mathscr{X}_{ijkl}$ of $\mathscr{X}$ , where the strict transform of $D_{m}$ is isomorphic to $D_{m}$ , thus in particular $\mathring{D_{m}}\subset D_{m}$ is a torus embedding.

For each $2$ -dimensional face $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ of $\Sk(X)$ , we denote by $\Gamma_{mm^{\prime}m^{\prime\prime}}$ the graph defined in Definition 3.2.1, and its vertices by $p_{mm^{\prime}},p_{mm^{\prime\prime}},p_{m^{\prime}m^{\prime\prime}}$ and $p_{mm^{\prime}m^{\prime\prime}}$ . We set

\Gamma\coloneqq\bigcup_{m<m^{\prime}<m^{\prime\prime}}\Gamma_{mm^{\prime}m^{\prime\prime}}.

By construction, around any point of $\Sk(X)\setminus\Gamma$ , the retraction $\pi$ is equal to the Berkovich retraction induced by a suitable minimal model $\mathscr{X}_{ijkl}$ of $X$ . It follows from the results in [NXY19] that $\pi$ induces an integral affine structure with singularities on $\Sk(X)$ . By Theorem B and Corollary C, we obtain that this integral affine structure has no singularities outside $\Gamma$ . We will furthermore prove in the next subsection that this affine structure does not extend across any edge of $\Gamma$ , i.e. is indeed singular along $\Gamma$ .

[04Q9]

4.7 Monodromy representation

We study the monodromy representation of the integral affine structure induce by $\pi$ on $\Sk(X)\setminus\Gamma$ ; we exhibit the explicit computations along loops in a neighborhood of the vertices $p_{234}$ and $p_{24}$ , as all the others are analogous. We then compare the matrices we obtain with the ones obtained in various constructions existing in the mirror symmetry literature.

Monodromy near $p_{234}$

We denote by $C$ the stratum curve $D_{234}=D_{2}\cap D_{3}\cap D_{4}$ , by $\tau_{C}$ the corresponding $2$ -dimensional face, and by $q_{1}=C\cap D_{1}$ and $q_{5}=C\cap D_{5}$ the two components of the boundary of $C$ . We set $U_{i}=\textrm{Int}(\tau_{p_{1}})\cup\textrm{Int}(\tau_{p_{5}})\cup\Star(v_{i})^{\prime}$ , for $i=2,3,4$ . The integral affine structure induced by $\pi$ over $U_{i}$ identifies $\Star(\tau_{C})$ with the $3$ -dimensional subset of $\mathbb{R}^{3}$ with vertices

v_{1}=(1,0,0),\quad v_{2}=(0,1,0),\quad v_{3}=(0,0,1),\quad v_{4}=(0,0,0)

v_{5}=-v_{1}-\sum_{j=2}^{4}(C\cdot D_{j})v_{j}=\Big(-1,-(C\cdot D_{2}),-(C\cdot D_{3})\Big)=\begin{cases}(-1,4,-1)&\text{ if }i=2\\ (-1,-1,4)&\text{ if }i=3\\ (-1,-1,-1)&\text{ if }i=4.\\ \end{cases}

The intersection numbers $(C\cdot D_{j})$ are computed in Section 4.5, accordingly to the minimal model whose Berkovich retraction induces the integral affine structure on each $U_{i}$ for $i=2,3,4$ ; for instance, $\pi$ coincides with $\rho_{\mathscr{X}_{1345}}$ over $U_{2}$ , so $(C\cdot D_{j})=(C\cdot D_{j})_{\mathscr{X}_{1345}}$ on $U_{2}$ .

There are three edges in $\Gamma$ having the vertex $p_{234}$ as endpoint. We consider the monodromy along the three corresponding loops, which are oriented as described in Section 3.2.

By Proposition 3.2.2, the monodromy matrices are

T_{\pi}(\gamma_{234,34})\eqqcolon T_{{234,34}}=\left(\begin{matrix}1&0&0\\ 0&1&0\\ 5&0&1\end{matrix}\right),\hskip 15.0ptT_{{234,23}}=\left(\begin{matrix}1&0&0\\ 5&1&0\\ -5&0&1\end{matrix}\right),\hskip 15.0ptT_{{234,24}}=\left(\begin{matrix}1&0&0\\ 5&1&0\\ 0&0&1\end{matrix}\right);

we notice that $T_{{234,34}}T_{{234,23}}=T_{{234,24}}$ , a relation which also follows from the corresponding equality at the level of loops inside $\pi_{1}(\Sk(X)\setminus\Gamma)$ .

Monodromy near $p_{24}$

We consider the vertex $p_{24}$ of the graph $\Gamma$ . As $p_{24}$ is the endpoint of three edges of $\Gamma$ , respectively contained in the $2$ -dimensional faces $\tau_{D_{234}}$ , $\tau_{D_{124}}$ and $\tau_{D_{245}}$ , we compute the monodromy along the three corresponding loops, whose orientation is prescribed in Section 3.2.

For $i=2,4$ , the integral affine structure induced by $\pi$ over $\Star(v_{i})^{\prime}$ identifies $\Star(\tau_{D_{234}})$ with the $3$ -dimensional subset of $\mathbb{R}^{3}$ with vertices

	$\displaystyle v_{1}=$	$\displaystyle(1,0,0),\quad v_{2}=(0,1,0),\quad v_{3}=(0,0,1),\quad v_{4}=(0,0,0),$
	$\displaystyle v_{5}=$	$\displaystyle-v_{1}-(D_{234}\cdot D_{2})v_{2}-(D_{234}\cdot D_{3})v_{3}-(D_{234}\cdot D_{4})v_{4}=$
	$\displaystyle=$	$\displaystyle\Big(-1,-(D_{234}\cdot D_{2}),-(D_{234}\cdot D_{3})\Big)=\begin{cases}(-1,4,-1)&\text{ if }i=2\\ (-1,-1,-1)&\text{ if }i=4;\\ \end{cases}$

the intersection numbers are computed accordingly to the minimal model inducing the integral affine structure on $\Star(v_{i})^{\prime}$ . By Proposition 3.2.2, the monodromy matrix along $\gamma_{234,24}$ is given in the basis $\mathcal{B}=(v_{1},v_{2},v_{3})$ by

T_{{234,24}}=\left(\begin{matrix}1&0&0\\ 5&1&0\\ 0&0&1\end{matrix}\right).

For $i=2,4$ , the integral affine structure induced by $\pi$ over $\Star(v_{i})^{\prime}$ identifies $\Star(\tau_{D_{124}})$ with the $3$ -dimensional subset of $\mathbb{R}^{3}$ with vertices

	$\displaystyle v_{1}=$	$\displaystyle(1,0,0),\quad v_{2}=(0,1,0),\quad v_{3}=(0,0,1),\quad v_{4}=(0,0,0),$
	$\displaystyle v_{5}=$	$\displaystyle-v_{3}-(D_{124}\cdot D_{1})v_{1}-(D_{124}\cdot D_{2})v_{2}-(D_{124}\cdot D_{4})v_{4}=$
	$\displaystyle=$	$\displaystyle\Big(-(D_{124}\cdot D_{1}),-(D_{124}\cdot D_{2}),-1\Big)=\begin{cases}(-1,4,-1)&\text{ if }i=2\\ (-1,-1,-1)&\text{ if }i=4.\\ \end{cases}$

As above, the intersection numbers are computed accordingly to the minimal model inducing the integral affine structure on $\Star(v_{i})^{\prime}$ and the monodromy matrix along $\gamma_{124,24}$ with respect to the basis $\mathcal{B}$ is

T_{{124,24}}=\left(\begin{matrix}1&0&0\\ 0&1&5\\ 0&0&1\end{matrix}\right).

For $i=2,4$ , the integral affine structure induced by $\pi$ over $\Star(v_{i})^{\prime}$ identifies $\Star(\tau_{D_{245}})$ with the $3$ -dimensional subset of $\mathbb{R}^{3}$ with vertices

	$\displaystyle v_{1}=$	$\displaystyle(1,0,0),\quad v_{2}=(0,1,0),\quad v_{5}=(0,0,1),\quad v_{4}=(0,0,0),$
	$\displaystyle v_{3}=$	$\displaystyle-v_{1}-(D_{245}\cdot D_{2})v_{2}-(D_{245}\cdot D_{4})v_{4}-(D_{245}\cdot D_{5})v_{5}=$
	$\displaystyle=$	$\displaystyle\Big(-1,-(D_{245}\cdot D_{2}),-(D_{245}\cdot D_{5})\Big)=\begin{cases}(-1,4,-1)&\text{ if }i=2\\ (-1,-1,-1)&\text{ if }i=4.\\ \end{cases}$

The monodromy along $\gamma_{245,24}$ in the basis $\mathcal{B}^{\prime}=(v_{1},v_{2},v_{5})$ is

T^{\prime}_{{245,24}}=\left(\begin{matrix}1&0&0\\ 5&1&0\\ 0&0&1\end{matrix}\right).

By change of basis from $\mathcal{B}^{\prime}$ to $\mathcal{B}$ , we write monodromy along $\gamma_{245,24}$ with respect to $\mathcal{B}$ :

P=\left(\begin{matrix}1&0&-1\\ 0&1&4\\ 0&0&-1\end{matrix}\right)\quad\text{ and }\quad T_{{245,24}}=P^{-1}\left(\begin{matrix}1&0&0\\ 5&1&0\\ 0&0&1\end{matrix}\right)P=\left(\begin{matrix}1&0&0\\ 5&1&-5\\ 0&0&1\end{matrix}\right).

We observe that $T_{124,24}T_{245,24}=T_{234,24}$ , a relation which holds indeed among the corresponding loops.

[04QA]

Remark 4.7.1.

We will now show that the affine structure we constructed on $\Sk(X)\setminus\Gamma$ is semi-simple polytopal in the sense of [RZ21a, Definition 4]. Integral affine manifolds with semi-simple polytopal singularities are the tropical analog of local complete intersections in algebraic geometry, and are the relevant class of affine structures on the base of the topological SYZ fibration in the context of the Gross–Siebert program. Indeed, given such a manifold $B$ , Ruddat and Zharkov construct a topological space $Y$ and torus fibration $Y\rightarrow B$ with discriminant of codimension 2 in $B$ , inducing the given affine structure. In [RZ21a] the authors describe the strategy in the 3-dimensional case; the general results will appear in [RZ], building on the local constructions of [RZ21b].
In the case of the quintic 3-fold, let $p=p_{ijk}$ be a vertex of the discriminant contained in the interior of a 2-face $\tau$ , and $q=p_{ik}$ a vertex contained in the interior of an edge $e$ of $\tau$ . Up to relabelling, we may assume that the lattice $L_{p}$ of invariant vectors around $p$ (i.e. the sections of the sheaf of integral affine tangent vectors on a small neighbourhood of $p$ ) is freely generated by $v_{i}$ and $v_{j}$ , in which case the three monodromy matrices around $p$ are of the form $T=\Id+5v_{k}^{\vee}\otimes w$ for some primitive $w\in L_{p}$ . Hence, writing $L(p)=L_{p}$ and $L^{\vee}(p)=5L_{p}^{\bot}$ , as well as $L(q)=L_{q}$ and $L^{\vee}(q)=5L_{q}^{\bot}$ we see that we are in the setting of [RZ21a]: the singularities of the affine structure are semi-simple abelian. Moreover, the vertices $p_{ijk}$ are negative, while the vertices $p_{ik}$ are positive.
Note that $\tau$ can be canonically realized inside $L_{p}$ , sending the vertex $v_{k}$ to the origin; in addition we set $\tau^{\vee}=<0,5v_{k}^{\vee}>$ to be the convex hull of $0$ and $5v_{k}^{\vee}$ in $\subset L^{\vee}(p)$ . The three loops described above are canonically indexed by the edges of $\tau$ , and hence by the pairs $(e,f)$ , with $e$ an edge of $\tau$ and $f$ the edge of $\tau^{\vee}$ . The upshot of working with $5L_{p}^{\bot}$ instead of $L_{p}^{\bot}$ (and similarly for $q$ ) is now that the monodromy along the loop $\gamma_{e,f}$ is now simply given by the formula $T(\gamma_{e,f})=\Id+e\otimes f$ .
Similarly for $q$ , we realize the edge $e_{ik}$ inside $L_{q}$ as the unit segment, and set $e^{\vee}=<0,5v_{i}^{\vee},5v_{k}^{\vee}>\subset L^{\vee}(q)$ . Then we may once again label the three loops around $q$ by pairs $(e,f)$ with $e=e_{ik}$ and $f$ an edge of $e^{\vee}$ , so that the formula $T(\gamma_{e,f})=\Id+e\otimes f$ holds.
Since $e_{ik}$ is a face of $\tau$ , we conclude from this that our affine structure is semi-simple polytopal.

[04QB]

Remark 4.7.2.

In [Rua01], Ruan develops a symplectic method based on gradient flow and constructs a Lagrangian torus fibration for Fermat type quintic Calabi–Yau hypersurfaces

\mathscr{X}=\{z_{0}\ldots z_{4}+t(z_{0}^{4}+\ldots+z_{4}^{4})=0\}\subset\mathbb{P}^{4}_{\mathbb{C}}\times\mathbb{D},

later extended to generic quintic hypersurfaces in toric varieties. The idea is to realize $\Sk(\mathscr{X})$ very explicitely as the boundary of the standard 4-simplex $\tau^{4}=\{\sum_{i=0}^{4}w_{i}=1\}\subset\mathbb{R}_{\geqslant 0}^{5}$ , and to spread the map:

F:\mathscr{X}_{0}\longrightarrow\partial\tau^{4}

[z_{0}:\ldots:z_{4}]\longmapsto\Bigg(\frac{\lvert z_{0}\rvert^{2}}{\lVert z\rVert^{2}},\ldots,\frac{\lvert z_{4}\rvert^{2}}{\lVert z\rVert^{2}}\Bigg)

to the nearby fibers using a gradient flow. This yields a Lagrangian fibration on the $\mathscr{X}_{t}$ ’s for small enough $t$ , which Ruan expects to be deformable towards a special Lagrangian fibration.
In addition, he describes the discriminant locus and the monodromy transformations of the expected special Lagrangian fibration, assuming that the singular locus is of codimension $2$ . The predictions in [Rua01, §4.4, §4.5] match precisely our computations above.

In [Gro01] Gross defines a class of topological $3$ -dimensional torus fibrations and proves they admit dual fibration. Building on Ruan’s description of monodromy, Gross shows that generic quintic threefolds in $\mathbb{P}^{4}$ can be endowed with such a fibration. It follows that the induced integral affine structure on the sphere $\mathbb{S}^{3}$ coincides with the one in Section 4.7.

Note that both in Ruan’s and in Gross’ aforementioned works, the polyhedral decomposition on $\mathbb{S}^{3}$ is induced by the intersection complex of the central fiber $\mathscr{X}_{0}$ (i.e., vertices correspond to zero-dimensional strata of the special fiber and so on); we work instead with the dual intersection complex associated with $\mathscr{X}_{0}$ , which is isomorphic to the intersection complex in the examples we are considering.

[04QC]

4.8 Comparison to Gromov-Hausdorff limit of Fermat families

In the previous sections, we constructed a $\mathbb{Z}$ -affine structure for generic quintic hypersurfaces in $\mathbb{P}^{4}_{K}$ via minimal models. This applies in particular to the hypersurfaces in the Fermat family, i.e. to

\mathscr{X}=\{z_{0}\ldots z_{n+1}+t(z_{0}^{n+2}+\ldots+z_{n+1}^{n+2})=0\}\subset\mathbb{P}^{n+1}_{\mathbb{C}}\times\mathbb{D}

with $n=3$ . For the hypersurfaces $X_{t}\subset\mathscr{X}$ and for arbitrary $n$ , in [Li19] Li constructs special Lagrangian torus fibrations on generic regions of $X_{t}$ . More precisely, endow $X_{t}$ with the unique Calabi–Yau metric $\omega_{t}$ in the class induced by $\mathcal{O}_{\mathbb{P}}(1)$ . Then the family of rescaled metrics $(\log\lvert t\rvert^{-1})^{-1}\omega_{t}$ on $X_{t}$ has bounded diameter and converges in the Gromov-Hausdorff sense to a smooth metric on $\mathbb{S}^{n}\setminus\Gamma$ as $t\rightarrow 0$ ; here $\mathbb{S}^{n}$ is triangulated as the boundary of a standard simplex of dimension $n+1$ , and $\Gamma$ is the complement of the open stars of the vertices of $\mathbb{S}^{n}$ in the first barycentric subdivision (see Definition 3.2.1). The metric limit obtained this way is a real Monge–Ampère metric with respect to a certain affine structure on $\mathbb{S}^{n}\setminus\Gamma$ , which is described in [Li19, §3.2, §3.5].

In this final section we prove that the integral affine structure constructed by Li coincides with the one from Theorem A when $n=3$ , which further motivates the main results of this paper. Indeed, it shows that for Fermat quintics, the essential skeleton equipped with the new type of retraction we built recovers the Gromov-Hausdorff limit with the affine structure induced by an SYZ fibration, as expected from the conjecture by Kontsevich and Soibelman.

To recall the details of Li’s construction we start by fixing some notation. The toric variety $Z=\mathbb{P}_{K}^{n+1}$ has homogeneous coordinates $[z_{0}:\ldots:z_{n+1}]$ , and we write $\mathbb{T}\subset Z$ the open dense torus. We denote by $N$ the abelian group of 1-parameter subgroups of $\mathbb{T}$ , and its dual $M=\Hom(N,\mathbb{Z})$ . We identify $M_{\mathbb{R}}$ with $\{\sum_{i=0}^{n+1}m_{i}=0\}\subset\mathbb{R}^{n+2}$ , so that $(m_{0},\ldots,m_{n+1})$ defines the character $z^{m}=\prod_{i=0}^{n+1}z_{i}^{m_{i}}$ . It follows that $N_{\mathbb{R}}\simeq\mathbb{R}^{n+2}/(1,\ldots,1)$ .
The variety $X=\mathscr{X}_{K}$ is embedded in $Z$ and the toric structure of $Z$ allows us to realize $\Sk(X)$ as a simplicial subset of $N_{\mathbb{R}}$ . Indeed, recall that the analytification of the torus $\mathbb{T}^{\an}$ comes with a tropicalization map

\val:\mathbb{T}^{\an}\longrightarrow N_{\mathbb{R}},

defined in Section 1.5. The generic point of $X$ lies in $\mathbb{T}$ , thus the set of birational points of $X^{\an}$ and in particular $\Sk(X)$ is contained inside $\mathbb{T}^{\an}$ . This yields a well-defined continuous map

\val:\Sk(X)\longrightarrow N_{\mathbb{R}},

which we claim to be an embedding. Let $i\in\{0,\ldots,n+1\}$ and $\tau_{i}\subset\Sk(\mathscr{X})=\Sk(X)$ be a top-dimensional face, corresponding to a zero-dimensional stratum $p_{i}=\cap_{j\neq i}D_{j}$ of $\mathscr{X}_{k}=\sum_{i=0}^{n+1}D_{i}$ . The points of $\tau_{i}$ are quasi-monomial valuations $v_{w}$ with weights $w=(w_{j})_{j\neq i}$ such that $\sum_{j\neq i}w_{j}=1$ , where $w_{j}=v_{w}(z_{j}/z_{i})$ . By definition of the tropicalization map, we have

\langle\val(v_{w}),m\rangle=v_{w}(z^{m})=v_{w}\Big(\prod_{j=0}^{n+1}z_{j}^{m_{j}}\Big)=v_{w}\Big(\prod_{j\neq i}(z_{j}/z_{i})^{m_{j}}\Big)=\sum_{j\neq i}m_{j}w_{j}

for any $m\in M_{\mathbb{R}}$ . Hence the tropicalization map sends the face $\tau_{i}$ to the $n$ -simplex

\theta_{i}=\{x_{i}=0\}\cap\{\sum_{j\neq i}x_{j}=1\}\subset N_{\mathbb{R}}\simeq\mathbb{R}^{n+2}/(1,\ldots,1),

and the image of $\Sk(X)$ by $\val$ is the boundary $\partial\Delta^{\vee}$ of the standard $(n+1)$ -simplex $\Delta^{\vee}$ generated by the vertices $e_{i}=(0,\ldots,1,\ldots,0)$ , for $i=0,\ldots,n+1$ , inside $N_{\mathbb{R}}$ (note that this is still an $(n+1)$ -simplex when passing to the quotient). Moreover, $\Delta^{\vee}$ is the dual polytope of the convex hull $\Delta$ of the characters $(-1,\ldots,(n+1),\ldots,-1)\in M_{\mathbb{R}}$ , i.e.

\Delta^{\vee}=\{x\in N_{\mathbb{R}}\,|\,\langle x,m\rangle\leq 1,\,\forall m\in\Delta\}.

It now follows from an elementary computation that the simplex $\Delta^{\vee}_{\lambda}$ defined in [Li19] by the formula

\Delta^{\vee}_{\lambda}=\{x\in N_{\mathbb{R}}\,|\,\max_{i=0,\ldots,n+1}(n+1)x_{i}-\sum_{j\neq i}x_{j}=1\}

is such that $-\partial\Delta^{\vee}_{\lambda}=\partial\Delta^{\vee}(=\Sk(X)\simeq\mathbb{S}^{n}$ ); observe that in $\mathbb{R}^{n+2}$ , the preimage of $-\Delta^{\vee}_{\lambda}$ by the quotient map is the Minkowski sum of the standard simplex and $\mathbb{R}(1,\ldots,1)$ . The discrepancy in sign conventions is due to the fact that in Li’s work, the tropicalisation map is taken to be $\log\lvert\cdot\rvert$ , instead of $\val=-\log\lvert\cdot\rvert$ , which is the standard non-archimedean convention.

We can now describe the integral affine structure constructed by Li on $\partial\Delta^{\vee}_{\lambda}$ . Fix a vertex $v_{i}\in\Sk(X)$ , $v_{i}$ is identified via the tropicalization map with the vertex $e_{i}\in\partial\Delta^{\vee}\subset N_{\mathbb{R}}$ , and corresponds to a codimension 1 face of $\Delta$ . Then the $\mathbb{Z}$ -linear functions on $\Star(v_{i})$ are generated by

	$\displaystyle\{m$	$\displaystyle\in M\,\|\,\langle m,e_{i}\rangle=0\}=\{m\in\mathbb{Z}^{n+2}\,\|\,\sum_{i=0}^{n+1}m_{i}=1\,,\,m_{i}=0\}$
		$\displaystyle=\{(1,0,\ldots,\underset{i}{0},\ldots,-1),(0,1,\ldots,\underset{i}{0},\ldots,-1),\ldots,(0,0,\ldots,\underset{i}{0},\ldots,1,-1)\}$

This yields an atlas of $(n+1)$ charts $U_{i}=\Star(v_{i})$ on $\Sk(X)$

	$\displaystyle f_{i}:U_{i}$	$\displaystyle\simeq\Star(e_{i})\subset\partial\Delta^{\vee}\subset N_{\mathbb{R}}\rightarrow\mathbb{R}^{n}$
	$\displaystyle x$	$\displaystyle\mapsto(x_{0}-x_{n+1},x_{1}-x_{n+1},\ldots,x_{i-1}-x_{n+1},x_{i+1}-x_{n+1},\ldots,x_{n}-x_{n+1})$

whose overlaps are the $U_{ij}=\Star(e_{ij})$ , with $e_{ij}$ being the edge joining $v_{i}$ to $v_{j}$ . One can easily check that the transition functions between those charts are piecewise-linear on $U_{ij}$ , but not linear as they induce a corner precisely along the codimension 1 faces of $U_{ij}$ .
To overcome this problem, Li uses the additional $\mathfrak{S}_{n+2}$ symmetry of the Fermat hypersurface to extend the affine structure in codimension 1, as follows. Consider the first barycentric subdivision of $\Sk(X)$ , and denote by $V_{i}$ the open star of a vertex $v_{i}$ for this new simplicial structure. We now endow $\Sk(X)$ with the atlas of charts consisting of $(V_{i},f_{i}|_{V_{i}})$ and of the top-dimensional open faces of $\Sk(X)$ . This atlas covers precisely $\Sk(X)\setminus\Gamma$ , and since the overlaps between the charts are always contained in a top-dimensional face, this yields a $\mathbb{Z}$ -affine structure on $\Sk(X)\setminus\Gamma$ .

[04QD]

Proposition 4.8.1.

The singular affine structure on $\mathbb{S}^{n}$ of [Li19] matches the one induced on $\Sk(X)$ by the retraction $\pi$ constructed in Theorem A when $n=3$ , and by $\rho$ in Section 3.3.2 when $n=2$ .

[04QE]

Proof.

For notational simplicity, we do the proof for $n=3$ , the $n=2$ case being even simpler. By $\mathfrak{S}_{5}$ -symmetry, it is enough to check this on the open $U_{0}$ .

On $U_{0}$ the $\mathbb{Z}$ -affine structure induced by $\pi$ matches the one associated with a minimal model $\mathscr{X}^{\prime}$ , such the strict transform of $D_{0}$ inside $\mathscr{X}^{\prime}$ is isomorphic to $\mathbb{C}\mathbb{P}^{3}$ and the hypotheses of Theorem B hold for the stratum $D_{0}$ . For the affine structure induced by an affinoid torus fibration, $\mathbb{Z}$ -affine functions on $U_{0}$ are given by $-\log\lvert h\rvert$ , where $h$ is a non-vanishing analytic function on $\pi^{-1}(U_{0})$ (see Section 1.6), and $\pi^{-1}(U_{0})$ is the generic fiber (in the sense of Berkovich) of $\widehat{\mathscr{X}^{\prime}_{/D_{0}}}$ (see Section 1.5).

Using the results of Section 2, we may assume that we are working on the generic fiber of $\widehat{\mathscr{N}_{/D_{0}}}$ , which we denote by $\mathfrak{N}_{D_{0}}$ ; this is an open subset of the analytification of the torus $\mathbb{T}$ of $\mathscr{N}=\mathcal{N}\times_{\mathbb{A}^{1}_{k}}R$ , where $\mathcal{N}=\nu_{D_{0}/\mathscr{X}^{\prime}}$ . Thus we replace $\pi:\pi^{-1}(U_{0})\rightarrow U_{0}$ with $\val:\mathfrak{N}_{D_{0}}\subset\mathbb{T}^{\an}\rightarrow\Star(e_{0})\simeq U_{0}$ .

The torus $\mathbb{T}_{\mathcal{N}}$ of $\mathcal{N}$ is the direct product of the torus of $D_{0}$ with $\mathbb{G}_{m,k}$ , i.e. in coordinates

\mathbb{T}_{\mathcal{N}}=\mathbb{T}_{D_{0}}\times_{k}\mathbb{G}_{m,k}=\Spec\,k\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm},u^{\pm}\Big].

The normal bundle is endowed with a morphism $t:\mathcal{N}\rightarrow\mathbb{A}^{1}_{k}$ , whose restriction $\mathbb{T}_{\mathcal{N}}\rightarrow\mathbb{G}_{m,k}$ corresponds to the morphism of rings

k[t^{\pm}]\rightarrow k\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm},u^{\pm}\Big],\quad t\mapsto u\prod_{i=1}^{3}\frac{z_{i}}{z_{4}}.

We obtain that

\mathbb{T}=\mathbb{T}_{\mathcal{N}}\times_{\mathbb{G}_{m,k}}K=\Spec\frac{K\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm},u^{\pm}\Big]}{t-u\prod_{i=1}^{3}\frac{z_{i}}{z_{4}}}=\Spec\,K\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm}\Big],

so that $\mathbb{Z}$ -affine functions on $\Star(e_{0})$ are integral linear combinations of the $-\log\lvert\frac{z_{i}}{z_{4}}\rvert$ , for $i=1,2,3$ . But those functions are precisely $m_{1}-m_{4}$ , $m_{2}-m_{4}$ , $m_{3}-m_{4}$ , i.e. $m\in M$ satisfying $\langle m,e_{0}\rangle=0$ and generating the $\mathbb{Z}$ -linear functions on $U_{0}$ in [Li19]. ∎

[04QF]

5 Appendix

The purpose of this appendix is to extend some of the constructions and results in this paper to the following situation: let $X$ be a smooth family of $n$ -dimensional projective varieties over a punctured disk $\mathbb{D}^{*}\subset\mathbb{C}$ centered at $0$ , and let $\mathscr{X}$ be an snc proper regular algebraic space over $\mathbb{D}$ extending the family $X$ to $0$ . In particular, the special fiber $\mathscr{X}_{0}$ is strict normal crossing, the morphism $\mathscr{X}\rightarrow\mathbb{D}$ is a proper holomorphic submersion, and $\mathscr{X}$ is not assumed to be projective; we still call such space $\mathscr{X}$ an snc model of $X$ . Our goal is to define a skeleton $\Sk(\mathscr{X})\subset X^{\an}$ and a retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ associated with $\mathscr{X}$ .

We start by recalling a construction from [BJ17, 4.2]. Let $\mathscr{X}^{\prime}$ be another snc model of $X$ that dominates $\mathscr{X}$ ; we write $h:\mathscr{X}^{\prime}\longrightarrow\mathscr{X}$ and the special fibers $\mathscr{X}_{0}=\sum_{i\in I}a_{i}D_{i}$ and $\mathscr{X}^{\prime}_{0}=\sum_{i\in I^{\prime}}a^{\prime}_{i}D^{\prime}_{i}$ . Then there exists an integral affine retraction

r_{\mathscr{X}^{\prime}\mathscr{X}}:\mathcal{D}(\mathscr{X}^{\prime}_{0})\longrightarrow\mathcal{D}(\mathscr{X}_{0}),

as follows. Let $\tau^{\prime}$ be the simplex in $\mathcal{D}(\mathscr{X}^{\prime}_{0})$ corresponding to a stratum $Y^{\prime}\subseteq D^{\prime}_{0}\cap\ldots\cap D^{\prime}_{q}$ . Let $Y$ be the minimal stratum of $\mathscr{X}_{0}$ such that $h(Y^{\prime})\subset Y$ ; we denote by $\tau$ the simplex in $\mathcal{D}(\mathscr{X}_{0})$ corresponding to $Y$ and we write $D_{0},\ldots,D_{p}$ the irreducible components of $\mathscr{X}_{0}$ containing $Y$ . Then

h^{*}D_{i}=\sum_{j=0}^{q}a_{ij}D^{\prime}_{j}+\sum_{h\in I^{\prime}\setminus\{0,\ldots,q\}}a_{ih}D^{\prime}_{h}

and we define the map $r_{\mathscr{X}^{\prime}\mathscr{X}}$ on $\tau^{\prime}$ by the formula:

\tau^{\prime}\ni w=(w_{0},\ldots,w_{q})\mapsto r_{\mathscr{X}^{\prime}\mathscr{X}}(w^{\prime})=\Big(\sum_{j=0}^{q}a_{ij}w^{\prime}_{j}\Big)_{0\leqslant i\leqslant p}\in\tau.

This yields a continuous integral affine map. Furthermore, we have the following transitivity property: if $\mathscr{X}^{\prime\prime}\longrightarrow\mathscr{X}^{\prime}\longrightarrow\mathscr{X}$ are three snc models of $X$ , then $r_{\mathscr{X}^{\prime\prime}\mathscr{X}}=r_{\mathscr{X}^{\prime}\mathscr{X}}\circ r_{\mathscr{X}^{\prime\prime}\mathscr{X}^{\prime}}$ .

[04QG]

Definition 5.0.1.

Let $\tau^{\prime}$ be a face of $\mathcal{D}(\mathscr{X}^{\prime}_{0})$ . We say $\tau^{\prime}$ is active for $r_{\mathscr{X}^{\prime}\mathscr{X}}$ if:

-

$h:Y^{\prime}\longrightarrow Y$ is a bimeromorphic morphism,
-

the $\mathbb{Q}$ -linear map inducing $r_{\mathscr{X}^{\prime}\mathscr{X}}:\tau^{\prime}\longrightarrow\tau$ is an isomorphism.

We write $A_{\mathscr{X}^{\prime}\mathscr{X}}$ for the union of active faces in $\mathcal{D}(\mathscr{X}^{\prime}_{0})$ . It follows from [BJ17, Proposition 4.3] that $r_{\mathscr{X}^{\prime}\mathscr{X}}$ induces a homeomorphism from $A_{\mathscr{X}^{\prime}\mathscr{X}}$ onto $\mathcal{D}(\mathscr{X}_{0})$ .

[04QH]

Definition 5.0.2.

Let $\mathscr{X}$ be an snc model of $X$ , and assume there exists a projective snc model $\mathscr{X}^{\prime}$ and $h:\mathscr{X}^{\prime}\longrightarrow\mathscr{X}$ . We define the skeleton $\Sk(\mathscr{X})\subset X^{\an}$ as the image of $A_{\mathscr{X}^{\prime}\mathscr{X}}$ by the embedding $\mathcal{D}(\mathscr{X}^{\prime}_{0})\hookrightarrow X^{\an}$ , and the Berkovich retraction $\rho_{\mathscr{X}}:X^{\an}\longrightarrow\Sk(\mathscr{X})$ as the composition $r_{\mathscr{X}^{\prime}\mathscr{X}}\circ\rho_{\mathscr{X}^{\prime}}$ (after identifying $\mathcal{D}(\mathscr{X}^{\prime}_{0})$ with $\Sk(\mathscr{X}^{\prime})$ ).

It follows directly from the transitivity property that this does not depend on the choice of a projective model $\mathscr{X}^{\prime}$ .

[04QI]

Lemma 5.0.3.

Let $\mathscr{X}$ be an snc model of $X$ and assume that $\mathscr{X}$ admits a dominating projective snc model. Then for any stratum $Y$ of $\mathscr{X}_{0}$ , the retraction $\rho_{\mathscr{X}}$ over $\Star(\tau_{Y})$ only depends on the formal completion $\widehat{\mathscr{X}_{/Y}}$ .

[04QJ]

Proof.

Let $\mathscr{X}^{\prime}$ be a projective snc model dominating $\mathscr{X}$ and write $h:\mathscr{X}^{\prime}\longrightarrow\mathscr{X}$ . We denote by $D_{0},\ldots,D_{p}$ the components of $\mathscr{X}_{0}$ containing $Y$ .

Let $Z^{\prime}\subseteq D^{\prime}_{0}\cap\ldots\cap D^{\prime}_{q}$ be any stratum of $\mathscr{X}_{0}^{\prime}$ , and denote by $\tau^{\prime}$ the corresponding simplex in $\mathcal{D}(\mathscr{X}^{\prime}_{0})$ . By construction of $r_{\mathscr{X}^{\prime}\mathscr{X}}$ , we have $\textrm{Int}(\tau_{Z^{\prime}})\subseteq r_{\mathscr{X}^{\prime}\mathscr{X}}^{-1}(\Star(\tau_{Y}))$ if and only if $h(Z^{\prime})\subseteq Y$ ; in this case, $h(D^{\prime}_{j})\cap Y\neq\varnothing$ for any $j=0,\ldots,q$ . Thus, if $D_{\alpha}$ is an irreducible component of $\mathscr{X}_{0}$ not cutting $Y$ , it follows that $h^{*}D_{\alpha}$ does not have any component along the $D^{\prime}_{j}$ for $j=0,\ldots,q$ .
We deduce from this that for each $D_{i}$ component of $\mathscr{X}_{0}$ containing $Y$ and $j\in\{0,\ldots,q\}$ , the coefficient of $D^{\prime}_{j}$ in $h^{*}(D_{i})$ is determined by $h$ and a local equation of $D_{i}$ in a formal neighbourhood of $Y$ . This proves that $r_{\mathscr{X}^{\prime}\mathscr{X}}$ over $\Star(\tau_{Y})$ only depends on $h$ over $\widehat{\mathscr{X}_{/Y}}$ .

By construction of Berkovich retraction, $\rho_{\mathscr{X}^{\prime}}$ only depends on $\widehat{\mathscr{X}^{\prime}_{/Z^{\prime}}}$ above $\Star(\tau_{Z^{\prime}})$ (see Section 1.5). If moreover $h(Z^{\prime})\subseteq Y$ , then $h$ induces a morphism $\widehat{\mathscr{X}^{\prime}_{/Z^{\prime}}}\rightarrow\widehat{\mathscr{X}_{/Y}}$ , hence $\rho_{\mathscr{X}^{\prime}}$ only depends on $h$ over $\widehat{\mathscr{X}_{/Y}}$ .

By the independence of $\rho_{\mathscr{X}}$ on the choice of projective model $\mathscr{X}^{\prime}$ and morphism $h$ , we conclude that $\rho_{\mathscr{X}}$ over $\Star(\tau_{Y})$ only depends on the formal completion $\widehat{\mathscr{X}_{/Y}}$ . ∎

[04QK]

Proposition 5.0.4.

Let $\mathscr{X}$ be an snc model of $X$ and assume that $\mathscr{X}$ admits a dominating projective snc model. Let $C$ be a one-dimensional stratum of $\mathscr{X}_{0}$ and assume that $C$ is isomorphic to $\mathbb{P}^{1}$ and $C\setminus\mathring{C}$ consists of two points. Then $\rho_{\mathscr{X}}$ is an affinoid torus fibration over $\Star(\tau_{C})\subset\Sk(\mathscr{X})$ , and the induced affine structure over $\Star(\tau_{C})$ is described as in Proposition 3.1.1.

[04QL]

Proof.

By Lemma 5.0.3, the retraction $\rho_{\mathscr{X}}$ over $\Star(\tau_{C})$ only depends on the formal completion $\widehat{\mathscr{X}_{/C}}$ . By [Knu71, V, Theorem 2.5], since $C$ is a scheme, the formal algebraic space $\widehat{\mathscr{X}_{/C}}$ is a formal scheme. We are therefore in the setting of [NXY19, Proposition 5.4, Theorem 6.1] and we can conclude. ∎

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Toric geometry and integral affine structures in non-archimedean mirror symmetry

Abstract

Introduction

Theorem A.

Theorem B.

Corollary C.

1 Preliminaries

1.1 Models

Definition 1.1.1.

1.2 Toric geometry

Lemma 1.2.1 ([Ful93, 3.4]).

Corollary 1.2.2.

Lemma 1.2.3 ([Ful93, p. 99]).

Proposition 1.2.5.

Proof.

Definition 1.2.6.

1.3 Berkovich spaces

1.4 Skeletons

Definition 1.4.1.

Definition 1.4.2.

Proposition 1.4.3 ([MN15, Proposition 2.4.4]).

Definition 1.4.4.

Definition 1.4.5.

1.5 Berkovich retractions

Definition 1.5.1.

Proposition 1.5.2 ([NXY19, Example 3.5]).

Proof.

Lemma 1.5.3.

Proof.

1.6 Affinoid torus fibrations and integral affine structures

Definition 1.6.1.

Example 1.6.2.

Example 1.6.3.

Definition 1.6.4.

Definition 1.6.5.

Lemma 1.6.6 ([KS06, 2.1]).

Remark 1.6.7.

1.7 The Calabi–Yau case

Definition 1.7.1.

Theorem 1.7.2 ([NXY19, Theorem 1.13]).

Definition 1.7.3.

Definition 1.7.4.

Example 1.7.5.

Theorem 1.7.6 ([NXY19, Theorem 6.1]).

Example 1.7.7.

2 Toric structure along toric strata

Theorem B.

Corollary C.

Proof.

2.1 Notation and strategy

Remark 2.1.1.

Remark 2.1.2.

Lemma 2.1.3.

Proof.

Proposition 2.1.6.

Proof.

2.2 Construction of the divisors

Lemma 2.2.1.

Proof.

Lemma 2.2.3.

Proof.

2.3 Construction of the sections for a maximal cone

Lemma 2.3.1.

Proof.

2.4 Construction for two adjacent maximal cones

Lemma 2.4.2.

Proof.

2.5 Construction of the morphism

Lemma 2.5.1.

Proof.

Lemma 2.5.2.

Proof.

Proposition 2.5.3.

Proof.

2.6 Integral affine structure and toric irreducible components

Corollary 2.6.1.

Proof.

3 Integral affine structures

3.1 Integral affine structure induced by a model

Proposition 3.1.1.

Toric geometry and integral affine structures
in non-archimedean mirror symmetry

3.3.1 Integral affine structure induced by the model $\mathscr{X}_{ijk}$

4.5 Minimal models $\mathscr{X}_{ijkl}$