8.1. A singular version of Theorem A [0181]

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8.1. A singular version of Theorem A

Let $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ be a projective, flat holomorphic map of a normal complex space onto the disc, with $X:=\pi^{-1}({\mathbb{D}}^{*})$ smooth over ${\mathbb{D}}^{*}$ . Since $\pi$ is projective, it defines a smooth projective variety $X_{{\mathbb{C}}(\!({t})\!)}$ over ${\mathbb{C}}(\!({t})\!)$ , as well as a model ${\mathcal{X}}_{{\mathbb{C}}[\![{t}]\!]}$ .

Let ${\mathcal{L}}$ be a ${\mathbb{Q}}$ -line bundle on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$ , and $\psi$ a continuous Hermitian metric on ${\mathcal{L}}$ . This data induces a continuous Hermitian metric $\psi_{t}$ on $K_{X_{t}}$ for $t\in{\mathbb{D}}^{*}$ , as well as a residually metrized model ${\mathcal{L}}^{\#}$ of $K_{X_{{\mathbb{C}}(\!({t})\!)}}$ , the model given by ${\mathcal{L}}_{{\mathbb{C}}[\![{t}]\!]}$ and the metric by the restriction of $\psi$ to ${\mathcal{L}}_{0}={\mathcal{L}}|_{{\mathcal{X}}_{0}}$ . Thus we obtain a skeletal measure $\mu_{{\mathcal{L}}^{\#}}$ on $X^{\mathrm{an}}_{{\mathbb{C}}(\!({t})\!)}$ .

Denote by ${\mathcal{L}}^{\prime}$ (resp. $\psi^{\prime}$ ) the pull-back of ${\mathcal{L}}$ (resp. $\psi$ ) to a log resolution ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . By invariance of skeletal measures under pull-back, we have $\mu_{{\mathcal{L}}^{\prime\#}}=\mu_{{\mathcal{L}}^{\#}}$ , and Theorem 3.4 therefore implies:

Theorem 8.1.

The rescaled measures

\mu_{t}:=\frac{e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}},

viewed as measures on ${\mathcal{X}}^{\prime\mathrm{hyb}}$ , converge weakly to $\mu_{{\mathcal{L}}^{\#}}$ .

Corollary 8.2.

If ${\mathcal{X}}$ (i.e. the pair $({\mathcal{X}},{\mathcal{X}}_{0,\mathrm{red}})$ ) is dlt, then

\lim_{t\to 0}\frac{\int_{{\mathcal{X}}_{t}}e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}}=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}({\mathcal{L}}^{\#})\right)b_{\sigma}^{-1}\operatorname{Vol}(\sigma),

where $\sigma$ runs over the $d$ -dimensional faces of $\Delta({\mathcal{L}})$ .

When $d=0$ , this implies the following slight generalization of [Li13, Lemma 1].

Corollary 8.3.

Assume that ${\mathcal{X}}_{0}$ has klt singularities (and hence ${\mathcal{X}}$ is dlt by inversion of adjunction). Let $\psi$ be a continuous metric on $K_{{\mathcal{X}}/{\mathbb{D}}}$ . Then $t\mapsto\int_{{\mathcal{X}}_{t}}e^{2\psi_{t}}$ is continuous at $t=0$ .