1.7 The Calabi–Yau case [04N4]

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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.

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).

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 $X$ .

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.

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. $\dim\Sk(X)=n$ .

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.

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.

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.