As is well-known, there is a natural bijection between (smooth, positive) volume forms on a complex manifold and smooth Hermitian metrics on its canonical bundle. Consequently, the data of a smooth family $(\nu_{t})_{t\in{\mathbb{D}}^{*}}$ of volume forms on a holomorphic family $(X_{t})_{t\in{\mathbb{D}}^{*}}$ of compact complex manifolds is equivalent to that of a proper holomorphic submersion $\pi\colon X\to{\mathbb{D}}^{*}$ together with a smooth metric $\psi$ on the relative canonical bundle $K_{X/{\mathbb{D}}^{*}}$.

We say that the family $(\nu_{t})$ has analytic singularities at $t=0$ if the following conditions hold:

• (i)

  $\pi\colon X\to{\mathbb{D}}^{*}$ is meromorphic at $0\in{\mathbb{D}}$ in the sense that it extends to a proper, flat map $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$, with ${\mathcal{X}}$ normal;

• (ii)

  ${\mathcal{X}}$ can be chosen so that $K_{X/{\mathbb{D}}^{*}}$ extends to a ${\mathbb{Q}}$-line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$, and $\psi$ extends continuously to ${\mathcal{L}}$.

When (i) holds, we call ${\mathcal{X}}$ a model of $X$. Using resolution of singularities, we can always choose ${\mathcal{X}}$ as an snc model, that is, ${\mathcal{X}}$ is smooth and ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ has simple normal crossing support. To ${\mathcal{X}}$ is then associated a dual complex $\Delta({\mathcal{X}})$, with one vertex $e_{i}$ for each $E_{i}$, and a face $\sigma$ for each connected component $Y$ of a non-empty intersection $E_{J}=\bigcap_{i\in J}E_{i}$ with $J\subset I$.

In the spirit of the Morgan-Shalen topological compactification of affine varieties [MS84], we introduce a natural “hybrid” space

associated to ${\mathcal{X}}$; it is equipped with a topology defined in terms of a tropicalization map $X\to\Delta({\mathcal{X}})$, measuring the logarithmic rate of convergence of local coordinates compatible with ${\mathcal{X}}_{0}$.

Our first main result says that, after normalizing to unit mass, the volume forms $\nu_{t}$ admit a “tropical” limit inside ${\mathcal{X}}^{\mathrm{hyb}}$.

The invariant $\kappa_{\min}$ and the subcomplex $\Delta({\mathcal{L}})$ only depend on ${\mathcal{L}}$ (and not on the metric on ${\mathcal{L}}$). Consider the logarithmic relative canonical bundle

and write $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}={\mathcal{L}}+\sum_{i\in I}a_{i}E_{i}$ with $a_{i}\in{\mathbb{Q}}$. Setting $\kappa_{i}:=a_{i}/b_{i}$, we then have $\kappa_{\min}=\min_{i\in I}\kappa_{i}$, and $\Delta({\mathcal{L}})$ is the subcomplex of $\Delta({\mathcal{X}})$ whose vertices $e_{i}$ correspond to those $i\in I$ achieving the minimum.

On the other hand, the limit measure $\mu_{0}$ does depend on $\psi$; it is given by

Here, $\sigma$ ranges over the $d$-dimensional faces of $\Delta({\mathcal{L}})$, with corresponding strata $Y_{\sigma}\subset{\mathcal{X}}_{0}$, $\operatorname{Res}_{Y_{\sigma}}(\psi)$ is a naturally defined residual positive measure on $Y_{\sigma}$, $\lambda_{\sigma}$ is the Lebesgue measure of $\sigma$ normalized by its natural integral affine structure, and $b_{\sigma}\in{\mathbb{Z}}_{>0}$ is an arithmetic coefficient.

The study of the asymptotics of integrals is a very classical subject and has been pursued by many people; see for example the book [AGZV88]. The assertions in Theorem A are closely related to results by Chambert-Loir and Tschinkel (who also worked over general local fields and in an adelic setting). Specifically, the estimate for $\nu_{t}(X_{t})$, suitably averaged over $t$, is essentially equivalent to [CLT10, Theorem 1.2]. It also appears in [KS01, §3.1] and is exploited in [BHJ16].

The convergence result for the measures $\mu_{t}$ is also closely related to [CLT10, Corollary 4.8], where, however, the limit measure lives on ${\mathcal{X}}_{0}$ and not on $\Delta({\mathcal{X}})$.¹¹ 1 A. Chambert-Loir has pointed out that [CLT10, Corollary 4.8] is sufficiently precise, so that when applying it to toric blowups of ${\mathcal{X}}$ one can see the form of the limit measure $\mu_{0}$ in Theorem A. The main new feature of Theorem A is the precise and explicit convergence of the measure $\mu_{t}$ to a “tropical” limit $\mu_{0}$, living on a simplicial complex.

The following examples illustrate Theorem A. First consider the subvariety

where $0<\varepsilon\ll 1$. Write $X:=\mathrm{pr}_{1}^{-1}({\mathbb{C}}^{*})$. The fiber $X_{t}$ over $t\in{\mathbb{D}}^{*}$ is a Calabi-Yau manifold, and we can choose a nonvanishing holomorphic $n$-form $\eta_{t}$ on $X_{t}$ to define a smooth metric $\psi$ on $K_{X/{\mathbb{D}}^{*}}$ that extends continuously to ${\mathcal{L}}=K_{{\mathcal{X}}/{\mathbb{D}}}$. In the terminology of Theorem A we have $\nu_{t}:=2^{-n}i^{n^{2}}\eta_{t}\wedge\overline{\eta}_{t}$. Here ${\mathcal{X}}_{0}$ is smooth, so $\Delta({\mathcal{X}})$ is a single point. Thus $\nu_{t}(X_{t})\sim c$ for some $c>0$, and the limit measure $\mu_{0}$ is a point mass.

Now consider instead

In this case, $\Delta({\mathcal{L}})=\Delta({\mathcal{X}})$ is a union of $(n+1)$ simplices of dimension $n$, and topologically a sphere. We have $\nu_{t}(X_{t})\sim c(\log|t|^{-1})^{n}$ and the limit measure is a weighted sum of Lebesgue measures on each simplex. In fact, it is clear by symmetry that the weights are equal; this also follows from Theorem C below.

We also prove a logarithmic version of Theorem A, for a log smooth klt pair $(X,B)$, and a metric $\psi$ on $K_{(X,B)/{\mathbb{D}}^{*}}$, see Theorem 8.4.

The space ${\mathcal{X}}^{\mathrm{hyb}}$ and the measure $\mu_{0}$ depend on the choice of snc model ${\mathcal{X}}$. We obtain a more canonical situation by considering all possible snc models simultaneously. Namely, the set of snc models of $X$ is directed, and in §4 we define a locally compact (Hausdorff) topological space

fibering over ${\mathbb{D}}$, with central fiber $X^{\mathrm{hyb}}_{0}:=\varprojlim\Delta({\mathcal{X}})$. For any ${\mathcal{X}}$, the dual complex $\Delta({\mathcal{X}})$ embeds in the central fiber $X^{\mathrm{hyb}}_{0}$ of $X^{\mathrm{hyb}}$.

Now consider the case when $X\to{\mathbb{D}}^{*}$ is projective. As we now explain, the central fiber of $X^{\mathrm{hyb}}$ is then a non-Archimedean space. Namely, $X$ induces a smooth projective variety $X_{K}$ over the non-Archimedean field $K$ of complex formal Laurent series, to which we can associate a Berkovich analytification $X_{K}^{\mathrm{an}}$. Similarly, any projective snc model ${\mathcal{X}}\to{\mathbb{D}}$ of $X$ induces a projective model ${\mathcal{X}}_{R}$ over the valuation ring $R$ of $K$. The dual complex $\Delta({\mathcal{X}})$ then has a canonical realization as a compact ${\mathbb{Z}}$-PA subspace $\operatorname{Sk}({\mathcal{X}})\subset X_{K}^{\mathrm{an}}$, the skeleton of ${\mathcal{X}}$. In fact, it is well known (see e.g. [BFJ16]) that there is a homeomorphism $X_{K}^{\mathrm{an}}\overset{\sim}{\to}\varprojlim_{\mathcal{X}}\operatorname{Sk}({\mathcal{X}})$, so we can identify the central fiber of the space $X^{\mathrm{hyb}}$ with the analytification $X_{K}^{\mathrm{an}}$. In fact, as shown in Appendix A.6, using ideas from [Berk09], we can view the restriction of $X^{\mathrm{hyb}}\to{\mathbb{D}}$ to a closed subdisc $\overline{{\mathbb{D}}}_{r}$ as the analytification of the base change of $X$ to a suitable Banach ring $A_{r}$.

Assuming $X\to{\mathbb{D}}^{*}$ is projective, we can describe the limit measure $\mu_{0}$ and its support $\operatorname{Sk}({\mathcal{L}})\simeq\Delta({\mathcal{L}})$ inside $X_{K}^{\mathrm{an}}$ in more detail. The skeleton $\operatorname{Sk}({\mathcal{L}})$ is of purely non-Archimedean nature, and can be seen as a mild generalization of the Kontsevich–Soibelman skeleton introduced in [KS06] and studied in [MN15, NX13, NX16]. The skeletal measure $\mu_{0}$, on the other hand, depends on both Archimedean and non-Archimedean data. Namely, it is supported on the skeleton $\operatorname{Sk}({\mathcal{L}})$, but depends on the choice of metric on the restriction of the line bundle ${\mathcal{L}}$ to the central fiber ${\mathcal{X}}_{0}$ (viewed as a complex space) of any snc model ${\mathcal{X}}$.

We also study both the skeleton and the skeletal measure in the more general case when the model ${\mathcal{X}}$ is allowed to have mild (dlt) singularities.

One major motivation for studying the above general setting comes from degenerations of Calabi–Yau manifolds. Thus suppose $X\to{\mathbb{D}}^{*}$ is a projective holomorphic submersion, meromorphic at $0\in{\mathbb{D}}$, such that $K_{X/{\mathbb{D}}^{*}}={\mathcal{O}}_{X}$. Any trivializing section $\eta\in H^{0}(X,K_{X/{\mathbb{D}}^{*}})$ then defines a family $\eta_{t}:=\eta|_{X_{t}}$ of trivializations of $K_{X_{t}}$, and hence a smooth family of volume forms $\nu_{t}:=|\eta_{t}|^{2}$ with analytic singularities at $t=0$. Indeed, for any snc model ${\mathcal{X}}\to{\mathbb{D}}$, $\eta$ extends to a nowhere vanishing section of ${\mathcal{L}}:={\mathcal{O}}_{\mathcal{X}}$, and $\psi:=\log|\eta|$ defines a smooth metric on ${\mathcal{L}}$.

The total mass $\nu_{t}(X_{t})=\int_{X_{t}}|\eta_{t}|^{2}$ is then nothing but the $L^{2}$ (or Hodge) metric on the direct image of $K_{X/{\mathbb{D}}^{*}}$, whose asymptotic behavior at $t=0$ is described in a very precise way by Schmid’s nilpotent orbit theorem [Sch73, Theorem 4.9] (compare for instance [GTZ13b, Proposition 2.1]).

On the other hand, the skeleton $\operatorname{Sk}({\mathcal{L}})$ described above coincides in the current context with the Kontsevich–Soibelman skeleton $\operatorname{Sk}(X)$ [KS06, MN15, NX13]. Its dimension $d$, which features as the exponent of the log term in the asymptotics of the mass, measures how “bad” the degeneration is. Further, the family $X\to{\mathbb{D}}^{*}$ admits a relative minimal model ${\mathcal{X}}$, with certain mild (dlt) singularities [KNX15], and the essential skeleton can be identified with the dual complex of ${\mathcal{X}}$ [NX13]. In particular, $d=0$ if and only if $X$ can filled in with a central fiber ${\mathcal{X}}_{0}$ which is a Calabi–Yau variety with canonical singularities.

At the other end of the spectrum, $d=n=\dim X_{t}$ if and only if $X$ is maximally degenerate, i.e. a “large complex structure limit”. In that case, the essential skeleton $\operatorname{Sk}(X)$ is shown to be a pseudomanifold in [NX13]. Building on this, we prove:

This theorem also holds in the purely non-Archimedean setting of Calabi–Yau varieties defined over the field of Laurent series. The semistable reduction condition means that $X$ admits an snc model ${\mathcal{X}}$ with ${\mathcal{X}}_{0}$ reduced. This condition is always satisfied after a finite base change.

Theorem C describes measure-theoretic degenerations of Calabi–Yau varieties. Let us briefly discuss the case of metric degenerations. Consider a smooth projective family $X\to{\mathbb{D}}^{*}$ of Calabi–Yau varieties, meromorphic at $0\in{\mathbb{D}}$, and suppose the family is polarized, that is, we are given a relative ample line bundle $A$ on $X$. By Yau’s theorem [Yau78], each fiber $X_{t}$ carries a unique Ricci-flat Kähler metric $\omega_{t}$ in the cohomology class of $A_{t}$.

By [Wan03, Tos15, Taka15], the diameter $D_{t}$ of $(X_{t},\omega_{t})$ remains bounded if and only if $d=0$, that is, $X$ admits a model ${\mathcal{X}}$ such that ${\mathcal{X}}_{0}$ has klt singularities. In this case, it is shown in [RZ11, RZ13], building in part on [DS14], that $(X_{t},\omega_{t})$ converges in the Gromov-Hausdorff sense to the Calabi–Yau variety ${\mathcal{X}}_{0}$, endowed with the metric completion of its singular Ricci-flat Kähler metric in the sense of [EGZ09].

The maximally degenerate case $d=n$ is the object of the Kontsevich–Soibelman conjecture [KS06]²² 2 Essentially the same conjecture was stated independently by Gross–Wilson [GW00] and Todorov., which states that $(X_{t},D_{t}^{-2}\omega_{t})$ (which has diameter one) converges in the Gromov-Hausdorff sense to the essential skeleton $\operatorname{Sk}(X)$ endowed with a piecewise smooth metric of Monge-Ampère type, i.e. locally given as the Hessian of a convex function satisfying a real Monge-Ampère equation. This conjecture has been verified for abelian varieties see e.g. [Oda14] but is largely open in general. The “mirror” situation, when one fixes the complex structure and degenerates the cohomology class of the Ricci-flat Kähler metric (along a line segment in the Kähler cone), is better understood [GW00, Tos09, Tos10, GTZ13a, GTZ13b, HT14, TWY14]. By performing a “hyper-Kähler rotation”, this implies a version of the Kontsevich–Soibelman conjecture for special cases of Type III degenerations of K3 surfaces [GW00].

Theorems A and C indicate a possible approach to the Kontsevich–Soibelman conjecture. Indeed, recall that the metric $\omega_{t}$ for $t\in{\mathbb{D}}^{*}$ is constructed as the curvature form of a smooth metric $\phi_{t}$ on $A_{t}$, where $\phi_{t}$ in turn is obtained as a solution of the complex Monge-Ampère equation $\operatorname{MA}(\phi_{t})=\mu_{t}$.

On the central fiber $X^{\mathrm{hyb}}_{0}=X_{K}^{\mathrm{an}}$ of $X^{\mathrm{hyb}}$, it was shown in [BFJ15] that there exists a metric on the line bundle $A_{K}^{\mathrm{an}}$, unique up to scaling, solving the non-Archimedean Monge-Ampère equation $\operatorname{MA}(\phi_{0})=\mu_{0}$ (at least when $X$ is defined over an algebraic curve). It is now tempting to approach the Kontsevich–Soibelman conjecture by studying the behavior of $\phi_{t}$ as $t\to 0$. However, this seems to be a delicate issue since there is no a priori reason why the weak continuity at $t=0$ of $t\mapsto\mu_{t}$ would imply continuity of the solutions $t\mapsto\phi_{t}$.

Instead of Calabi-Yau manifolds, it would be interesting to study degenerating families $X\to{\mathbb{D}}^{*}$ of canonically polarized projective manifolds, where the metric on $K_{X_{t}}$ would be the Kähler-Einstein metric or the Bergman metric, and prove versions of Theorems A and C in this context.

The paper is organized as follows. After recalling various facts in §1 we define in §2 the hybrid space ${\mathcal{X}}^{\mathrm{hyb}}$ associated to an SNC model ${\mathcal{X}}$. The proof of Theorem A is given in §3. In §4 we define the space $X^{\mathrm{hyb}}$ associated to a degeneration as an inverse limit of the spaces ${\mathcal{X}}^{\mathrm{hyb}}$, and prove Corollary B. Various notions of skeleta are defined and studied in §5, and in §6 we formalize the notion of a residually metrized model of the canonical bundle, and associate to such an object a positive measure on the relevant Berkovich space. Degenerations of Calabi–Yau varieties are studied in §7 where we prove Theorem C. In §8 we study various extensions, and in Appendix A we recall the Berkovich analytification of a scheme over a Banach ring.