7. The Calabi–Yau case [017U]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

7. The Calabi–Yau case

As in §6, we assume that $X$ is a smooth projective variety over ${\mathbb{C}}(\!({t})\!)$ . Now we further assume that $K_{X}$ is trivial. Pick a trivializing section $\eta\in H^{0}(X,K_{X})$ , and denote by $\log|\eta|$ the associated model metric on $K_{X}^{\mathrm{an}}$ , determined on any model ${\mathcal{X}}$ by ${\mathcal{L}}={\mathcal{O}}_{\mathcal{X}}$ , with $\eta$ providing the identification ${\mathcal{L}}|_{X}\simeq K_{X}$ . Denote also by $\log|\eta|^{\#}$ the residually metrized model metric induced by the trivial Hermitian metric $\psi_{0}=0$ on ${\mathcal{O}}_{{\mathcal{X}}_{0}}$ .

The function $\kappa:=A_{X}-\log|\eta|=-\log|\eta|_{A_{X}}$ coincides with the weight function of [MN15, NX13]. By definition, the Kontsevich–Soibelman skeleton of $X$ is $\operatorname{Sk}(X):=\operatorname{Sk}(\log|\eta|^{\#})$ . It is indeed independent of the choice of $\eta$ , since any other trivializing section of $K_{X}$ is of the form $\eta^{\prime}=f\eta$ with $f\in{\mathbb{C}}(\!({t})\!)^{*}$ , and hence $\kappa^{\prime}=\kappa+\operatorname{ord}_{0}(f)$ .

7.1. Topology of the skeleton

By [NX13, Theorem 4.2.4], the ${\mathbb{Z}}$ -PA-space $\operatorname{Sk}(X)$ is connected, of pure dimension $d$ , and is a deformation retract of $X^{\mathrm{an}}$ . Further, $\operatorname{Sk}(X)$ is a pseudomanifold with boundary, i.e. for some (or, equivalently, any) triangulation $\Delta$ of $\operatorname{Sk}(X)$ , we have:

(a)

Non-branching property: every $(d-1)$ -simplex of $\Delta$ is contained in at most two $d$ -simplices
(b)

Strong connectedness: every pair of $n$ -simplices $\sigma$ , $\sigma^{\prime}$ is joined by a chain of $n$ -simplices $\sigma=\sigma_{1},\dots,\sigma_{N}=\sigma^{\prime}$ with $\sigma_{i}$ and $\sigma_{i+1}$ sharing a common $(n-1)$ -face.

In the maximally degenerate case $d=n$ , $\operatorname{Sk}(X)$ is even a pseudomanifold, i.e. (a) is replaced by

(a’)

every $(n-1)$ -simplex of $\Delta$ is contained in exactly two $n$ -simplices.

See also [KX15] for even more precise results on the structure of $\operatorname{Sk}(X)$ . For example, $\operatorname{Sk}(X)$ is homeomorphic to a sphere if $n\leq 3$ .

7.2. The skeletal measure

Consider the skeletal measure $\mu_{\log|\eta|^{\#}}$ on $\operatorname{Sk}(X)$ . Choose an snc model ${\mathcal{X}}$ , and write as usual ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ . The form $\eta$ defines an identification $K^{\mathrm{log}}_{{\mathcal{X}}/S}=\sum_{i\in I}a_{i}E_{i}$ , and Proposition 5.10 yields

\kappa_{\min}=\min_{i}\frac{a_{i}}{b_{i}}.

(7.1)

If $\kappa_{\min}\in{\mathbb{Z}}$ , then

\omega:=\frac{d{t}}{{t}^{\kappa_{\min}+1}}\wedge\eta

is a logarithmic form on ${\mathcal{X}}$ . For each face $\sigma$ of $\Delta({\mathcal{L}})$ , ordering the set $J\subset I$ of components cutting out the stratum $Y=Y_{\sigma}$ yields a well-defined Poincaré residue $\operatorname{Res}_{Y}(\omega)$ . By Lemma 5.12, $\operatorname{Res}_{Y}(\omega)$ is a rational section of $K_{Y}$ , with divisor

-B^{\mathcal{L}}_{Y}=\sum_{i\notin J}(a_{i}-\kappa_{\min}b_{i}-1)E_{i}|_{Y}

When $\sigma$ is a maximal face, $\operatorname{Res}_{Y}(\omega)$ is thus a holomorphic form on $Y$ ; using the formulas in §6.2, it is easy to see that the residual measure on $Y$ is given by

\operatorname{Res}_{Y}(\log|\eta|^{\#})=|\operatorname{Res}_{Y}(\omega)|^{2}.

The following result corresponds to Theorem C in the introduction.

Theorem 7.1.

Assume that $X$ is maximally degenerate, i.e. $\dim\operatorname{Sk}(X)=n$ . If $X$ has semistable reduction, then the skeletal measure $\mu_{\log|\eta|^{\#}}$ is a multiple of the integral Lebesgue measure of $\operatorname{Sk}(X)$ .

Proof.

Let ${\mathcal{X}}$ be a semistable model, i.e. ${\mathcal{X}}$ is snc with ${\mathcal{X}}_{0}$ reduced. By (7.1), we have $\kappa_{\min}\in{\mathbb{Z}}$ . Since some non-empty $E_{J}$ might have several components, the dual complex $\Delta({\mathcal{X}})$ is possibly not a triangulation of $\operatorname{Sk}(X)$ . However, the barycentric subdivision $\Delta^{\prime}$ of $\Delta({\mathcal{X}})$ is a triangulation; the corresponding toroidal modification ${\mathcal{X}}^{\prime}$ is snc, with ${\mathcal{X}}^{\prime}_{0}$ is possibly non-reduced, but $b_{\sigma}=1$ for each $n$ -simplex $\sigma$ of $\Delta^{\prime}$ . Applying the above discussion to ${\mathcal{X}}^{\prime}$ , we infer

\mu_{\log|\eta|^{\#}}=\sum_{\sigma}|\operatorname{Res}_{y_{\sigma}}(\omega)|^{2}\lambda_{\sigma},

with $\sigma$ ranging over the $n$ -dimensional faces of $\Delta^{\prime}$ , with corresponding strata $y_{\sigma}\in{\mathcal{X}}^{\prime}_{0}$ reduced to single points. It will thus be enough to show that $|\operatorname{Res}_{y_{\sigma}}(\omega)|$ is independent of $\sigma$ .

By the strong connectedness property, any two $n$ -simplices $\sigma$ , $\sigma^{\prime}$ of $\Delta^{\prime}$ can be joined by a chain of $n$ -simplices $\sigma=\sigma_{1},\dots,\sigma_{N}=\sigma^{\prime}$ with $\sigma_{i}$ and $\sigma_{i+1}$ sharing a common $(n-1)$ -face $\tau_{i}$ . Denoting by $y_{i}=y_{\sigma_{i}}$ and $Y_{i}=Y_{\tau_{i}}$ the corresponding strata in ${\mathcal{X}}^{\prime}$ , we thus have $y_{i},y_{i+1}\in Y_{i}$ . Further, the Poincaré residue $\operatorname{Res}_{Y_{i}}(\omega)$ has poles precisely at $y_{i},y_{i+1}$ , since any other pole would correspond to an $n$ -simplex of $\Delta^{\prime}$ containing $\tau_{i}$ , contradicting the non-branching property. Since $\operatorname{Res}_{y_{i}}\operatorname{Res}_{Y}(\omega)=\operatorname{Res}_{y_{i}}(\omega)$ , the residue theorem applied to the Riemann surface $Y_{i}$ yields $\operatorname{Res}_{y_{i}}(\omega)+\operatorname{Res}_{y_{i+1}}(\omega)=0$ , and hence $|\operatorname{Res}_{y_{1}}(\omega)|=\dots=|\operatorname{Res}_{y_{N}}(\omega)|$ . ∎

Remark 7.2.

Theorem 7.1 fails in general when $X$ does not have semistable reduction. Indeed, the semistable reduction theorem [KKMS] shows that the base change $p\colon X^{\prime}\to X$ to ${\mathbb{C}}(\!({t}^{1/m})\!)$ has semistable reduction for some $m$ divisible enough. By Lemma 5.14, $\dim\operatorname{Sk}(X^{\prime})=n$ , and $\mu_{\log|\eta^{\prime}|^{\#}}$ is thus a multiple of the integral Lebesgue measure $\lambda^{\prime}$ of $\operatorname{Sk}(X^{\prime})$ , by Theorem 7.1. By Theorem 6.7, $\mu_{\log|\eta|^{\#}}=m^{-n}p_{*}\lambda^{\prime}$ . However, $p_{*}\lambda^{\prime}$ is not proportional to the integral Lebesgue measure $\lambda$ of $\operatorname{Sk}(X)$ in general. Indeed, for each $n$ -simplex $\sigma$ of $\Delta({\mathcal{L}})$ , Lemma 5.13 shows that $(p_{*}\lambda^{\prime})_{\sigma}=m^{n}b_{\sigma}\lambda_{\sigma}$ , and $b_{\sigma}$ is in general not independent of $\sigma$ .