ScalingStacks

Theorem 1.2. There is a smooth Kähler metric $\omega$ on $Y\backslash f(S)$ such that the Ricci-flat metrics $\tilde{\omega}_{t}$ when $t$ approaches zero converge to $f^{*}\omega$ weakly as currents and also in the $C^{1,\beta}_{loc}$ topology of potentials on compact sets of $X\backslash S$, for any $0<\beta<1$. The metric $\omega$ satisfies

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

on $Y\backslash f(S)$, where $\omega_{WP}$ is a Weil-Petersson metric measuring the change of complex structures of the fibers. Moreover for any $y\in Y\backslash f(S)$ if we restrict to $X_{y}$, the metrics $\tilde{\omega}_{t}$ converge to zero in the $C^{1}$ topology of metrics, uniformly as $y$ varies in a compact set of $Y\backslash f(S)$.