ScalingStacks

We now let $\tilde{\omega}_{y}$ be the restriction $\tilde{\omega}_{t}|_{X_{y}}$. We have the following estimate for the volume form of $\tilde{\omega}_{y}$ on $X_{y}$:

(3.7)

$$\begin{split}\frac{\tilde{\omega}_{y}^{n-m}}{\omega_{y}^{n-m}}&=\frac{\tilde{\omega}_{t}^{n-m}\wedge\omega_{0}^{m}}{\omega_{X}^{n-m}\wedge\omega_{0}^{m}}=\frac{\tilde{\omega}_{t}^{n-m}\wedge\omega_{0}^{m}}{\tilde{\omega}_{t}^{n}}\cdot\frac{\tilde{\omega}_{t}^{n}}{H\omega_{X}^{n}}\\ &\leq\left(\frac{\tilde{\omega}_{t}^{n-1}\wedge\omega_{0}}{\tilde{\omega}_{t}^{n}}\right)^{m}\frac{c_{t}t^{n-m}e^{E}}{H}\\ &=(\textrm{tr}_{\tilde{\omega}_{t}}\omega_{0})^{m}\frac{c_{t}t^{n-m}e^{E}}{H}\leq\frac{Ct^{n-m}}{\sigma^{\lambda}}.\end{split}$$

Notice that when we restrict to $X_{y}$ we have

$$\tilde{\omega}_{y}=(\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})|_{X_{y}}=t\omega_{y}+(\sqrt{-1}\partial\overline{\partial}\varphi_{t})|_{X_{y}}.$$

It is convenient to define a function $\underline{\varphi_{t}}$ on $Y\backslash f(S)$ by

$$\underline{\varphi_{t}}(y)=\int_{X_{y}}\varphi_{t}\omega_{y}^{n-m}.$$

This is just the “integration along the fibers” of $\varphi_{t}$, and we will also denote by $\underline{\varphi_{t}}$ its pullback to $X\backslash S$ via $f$. We also define a function on $X\backslash S$ by

$$\psi=\frac{1}{t}\left(\varphi_{t}-\underline{\varphi_{t}}\right),$$

so that we have $\int_{X_{y}}\psi\omega_{y}^{n-m}=0$ and on $X_{y}$ we have

(3.8)

$$(\omega_{y}+\sqrt{-1}\partial\overline{\partial}\psi)^{n-m}=\frac{\tilde{\omega}_{y}^{n-m}}{t^{n-m}}\leq\frac{C}{\sigma^{\lambda}}\omega_{y}^{n-m}.$$

We can then apply Yau’s $L^{\infty}$ estimate for complex Monge-Ampère equations [Y1] to the inequality (3.8). Since the volume of $X_{y}$ is constant equal to $1$, the Sobolev constant of $\omega_{y}$ is uniformly bounded (Lemma 3.2) and the Poincaré constant is controlled by Lemma 3.4, Yau’s $L^{\infty}$ estimate gives

(3.9)

$$\sup_{X_{y}}\left|\varphi_{t}-\underline{\varphi_{t}}\right|=t\sup_{X_{y}}|\psi|\leq tCe^{B\sigma(y)^{-\lambda}},$$

where we increased the constant $B$ to absorb the term $\sigma^{-\lambda}$ in (3.8). Recall that from (2.8) we have a uniform bound for the oscillation of $\varphi_{t}$.