Proposition 3.14 (00QL) · ScalingStacks

00QL

Proposition 3.14. As $s\to+\infty$ , the pushforward measure $(\text{Log}_{s})_{*}d\mu_{s}$ converges to the Lebesgue measure $d\mu_{\infty}$ supported on $\partial\Delta_{\lambda}^{\vee}$ . In particular

\int_{X_{s}}d\mu_{s}\to\text{Vol}(\partial\Delta_{\lambda}^{\vee})=\int_{\partial\Delta_{\lambda}^{\vee}}d\mu_{\infty}.

(15)

Morever, there is a uniform exponential measure decay estimate

d\mu_{s}(\{z\in X_{s}:\text{dist}_{\mathbb{R}^{n+1}}(\text{Log}_{s}(z),\partial\Delta_{\lambda}^{\vee})>s^{-1}\Lambda\})\leq C^{\prime}e^{-C\Lambda},\quad\forall\Lambda>0.

(16)

00QM

Proof. (Sketch) Using Lemma 3.5 and the holomorphic volume form formula (13), the neighbourhood of the toric boundary near $\mathcal{A}_{\lambda,\sigma}^{\infty}$ only contributes $O(s^{-l})$ to the normalised measure, where $l=\dim NC_{\Delta}(\sigma)$ . The same lemmas imply (16) by summing over contributions from boundary type regions. In the toric region corresponding to the neighbourhood of $\partial\Delta_{\lambda}^{\vee}$ , the convergence of the normalised volume measure follows from Prop. 3.2 and formula (12). ∎