ScalingStacks

Theorem 2.4. As $t\to 0$ the Ricci-flat metrics $\tilde{\omega}_{t}$ on $X\backslash S$ converge to a smooth Kähler metric $\omega$ on $Y\backslash f(S)$ weakly as currents and also in the $C^{1,\beta}_{loc}$ topology of Kähler potentials for any $0<\beta<1$. The metric $\omega$ satisfies

$$\mathrm{Ric}(\omega)=\omega_{WP},$$

on $Y\backslash f(S)$, where $\omega_{WP}$ is the pullback of the Weil-Petersson metric from the moduli space of the Calabi-Yau fibers, and it measures the change of complex structures of the fibers.