ScalingStacks

Theorem 2.3. Given any $y\in Y\backslash f(S)$ denote by $X_{y}$ the fiber $f^{-1}(y)$, by $\omega_{y}$ the Kähler form $\omega_{X}|_{X_{y}}$ and by $\tilde{\omega}_{y}$ the restriction of the Ricci-flat metric $\tilde{\omega}_{t}|_{X_{y}}$. Then there are constants $A,B,C$ that only depend on the fixed data, so that on the fiber $X_{y}$ and any $0<t\leq 1$ we have

(2.10)

$$\frac{t}{Ce^{Ae^{B\sigma(y)^{-\lambda}}}}\omega_{y}\leq\tilde{\omega}_{y}\leq tCe^{Ae^{B\sigma(y)^{-\lambda}}}\omega_{y},$$

(2.11)

$$|\nabla\tilde{\omega}_{y}|^{2}_{\omega_{y}}\leq t^{1/2}Ce^{Ae^{B\sigma(y)^{-\lambda}}},$$

where $\nabla$ is the covariant derivative of $\omega_{y}$. In particular the metrics $\tilde{\omega}_{y}$ converge to zero in $C^{1}(\omega_{y})$ as $t$ approaches zero, uniformly as $y$ varies in a compact set of $Y\backslash f(S)$.