ScalingStacks

In particular, we can represent the Calabi-Yau metric $\omega_{CY,s}$ on $X_{s}$ by a potential $\varphi_{CY,s}$. The Calabi-Yau condition is

$$\omega_{CY,s}^{n}=a_{s}s^{-n}d\mu_{s},$$

(20)

where the normalising constant

$$a_{s}=\frac{\int_{X_{s}}[\Delta]^{n}}{\int_{X_{s}}d\mu_{s}}\to a_{\infty}=\frac{\int_{X_{s}}[\Delta]^{n}}{\text{Vol}(\partial\Delta_{\lambda}^{\vee})}$$

(21)

as $s\to+\infty$ (cf. Prop. 3.14).