ScalingStacks

We then fix $\eta$ a smooth function with support contained in $K$, and we will also denote by $\eta$ its pullback to $X$ via $f$. Recall that we have called $\tilde{\omega}_{1}$ the Ricci-flat metric in the class $[\omega_{1}]$, and $\Omega=\tilde{\omega}_{1}^{n}$. Then from the Monge-Ampère equation (2.5) we have

(4.6)

$$\int_{X}\eta\Omega=\frac{1}{a_{t}}\int_{X}\eta(\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n},$$

where the constants $a_{t}$ are equal to

$$\frac{\int_{X}\omega_{t}^{n}}{\int_{X}\omega_{1}^{n}},$$

and behave like (2.6). We can also write

(4.7)

$$\int_{X}\eta\Omega=\int_{X}\eta F\omega_{SF}^{n-m}\wedge\omega_{0}^{m}.$$

We are now going to estimate $\frac{1}{a_{t}}\int_{X}\eta(\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}.$ We have

$$\begin{split}&\frac{1}{a_{t}}\int_{X}\eta(\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}\\ &=\frac{1}{a_{t}}\int_{X}\eta\left((\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})+(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}})\right)^{n}\\ &=\frac{1}{a_{t}}\int_{X}\eta\sum_{k=0}^{n}\binom{n}{k}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{n-k}\end{split}$$

First of all observe that the form $\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}}$ is the pullback of a form on $Y$, and it can be wedged with itself at most $m$ times, so all terms in the sum with $k>m$ are zero. Next, we claim that all the terms with $k<m$ go to zero as $t\to 0$. To see this, start by observing that on the compact set $K$ the estimate (3.29) gives a constant $C$ (that depends on $K$) such that

(4.8)

$$-C\omega_{X}\leq\sqrt{-1}\partial\overline{\partial}\varphi_{t}\leq C\omega_{X}.$$

Moreover from the equation

$$\partial\overline{\partial}\underline{\varphi_{t}}=f_{*}(\partial\overline{\partial}\varphi_{t}\wedge\omega_{X}^{n-m})$$

together with (4.8), (3.12), we see that on $f(K)$ we have

(4.9)

$$-C\omega_{Y}\leq\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}}\leq C\omega_{Y}.$$

We also need to use (3.9) which on $K$ gives

(4.10)

$$\sup_{K}|\varphi_{t}-\underline{\varphi_{t}}|\leq Ct.$$

Then any term with $k<m$ is equal to

$$\frac{\binom{n}{k}}{a_{t}}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{n-k},$$

and it can be expanded into

$$\frac{\binom{n}{k}}{a_{t}}\sum_{i=0}^{n-k}\binom{n-k}{i}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X})^{n-k-i}\wedge(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{i}.$$

On $K$ the $(1,1)$-form $\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}}$ is bounded by (4.9). Since $a_{t}=O(t^{n-m})$ from (2.6), we see that the term in this sum with $i=0$ goes to zero. Any term with $i>0$ is comparable to

(4.11)

$$\frac{1}{t^{n-m}}\int_{X}(\varphi_{t}-\underline{\varphi_{t}})\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X})^{n-k-i}\wedge(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{i-1}.$$

Notice that all the $(1,1)$-forms appearing inside the integral are bounded by (4.8), (4.9), and that the function $\varphi_{t}-\underline{\varphi_{t}}$ is $O(t)$ by (4.10). On $K$ the estimate (2.10) gives

(4.12)

$$-Ct\omega_{y}\leq(\sqrt{-1}\partial\overline{\partial}\varphi_{t})|_{X_{y}}=(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))|_{X_{y}}\leq Ct\omega_{y}.$$

The form $\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}$ is the pullback of a form from $Y$, and so we can use (4.12) to estimate

$$\left|\frac{\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X})^{n-k-i}\wedge(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{i-1}}{\omega_{X}^{n}}\right|\leq Ct^{n-m},$$

and so the term (4.11) goes to zero. This proves our claim.