4.5. Convergence of measures [016C]

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4.5. Convergence of measures

For any locally compact Hausdorff space $Z$ , let ${\mathcal{M}}(Z)$ denote the space of signed Radon measures on $Z$ . By definition we have $X^{\mathrm{hyb}}=\varprojlim_{\mathcal{X}}{\mathcal{X}}^{\mathrm{hyb}}$ , and this induces a homeomorphism

{\mathcal{M}}(X^{\mathrm{hyb}})\overset{\sim}{\to}\varprojlim_{\mathcal{X}}{\mathcal{M}}({\mathcal{X}}^{\mathrm{hyb}}).

Theorem 3.4 now implies the following result, which is equivalent to Corollary B in the introduction.

Corollary 4.11.

Let $\pi\colon X\to{\mathbb{D}}^{*}$ be a proper submersion that is meromorphic at $0\in{\mathbb{D}}$ , and let $\psi$ be a continuous metric on $K_{X/{\mathbb{D}}^{*}}$ with analytic singularities. Then there exists a positive measure $\mu_{0}$ on $X^{\mathrm{hyb}}_{0}$ such that if $\mu_{t}:=\frac{\lambda(t)^{d}e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi)^{d}}$ , then $\lim_{t\to 0}\mu_{t}=\mu_{0}$ in the sense of weak convergence of measures on $X^{\mathrm{hyb}}$ . Further, there exists a snc model ${\mathcal{X}}\to{\mathbb{D}}$ and a ${\mathbb{Q}}$ -line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$ such that $\psi$ extends to a smooth metric on ${\mathcal{L}}$ , and

\mu_{0}:=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}(\psi)\right)b_{\sigma}^{-1}\lambda_{\sigma},

where $\sigma$ ranges over the $d$ -dimensional faces of $\Delta({\mathcal{L}})$ . Here $\lambda_{\sigma}$ denotes normalized Lebesgue measure on $\sigma$ and $b_{\sigma}=\gcd_{i\in J}b_{i}$ , where ${\mathcal{X}}_{0}=\sum_{i}b_{i}E_{i}$ and $E_{i}$ , $i\in J$ are the divisors defining $\sigma$ .