ScalingStacks

We are then left with only the term with $k=m$, which is

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

and if we expand the term $(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{n-m}$, we get

$$\begin{split}\frac{1}{a_{t}}&\int_{X}\eta\binom{n}{m}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge(t\omega_{X})^{n-m}\\ &+\frac{1}{a_{t}}\int_{X}\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge\dots,\end{split}$$

and the second term is zero because $\partial\overline{\partial}\eta$ is the pullback of a form from the base. We are then left with the term

(4.13)

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

which we need to further estimate. Using (4.8) we see that, up to taking a further subsequence, the functions $\varphi_{t_{i}}$ converge to $\hat{\varphi}$ in the $C^{1,\beta}(K)$ topology, and (4.10) implies that the functions $\underline{\varphi_{t_{i}}}$ also converge to $\hat{\varphi}$ uniformly. We can then rewrite (4.13) as

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

Using (2.6) we see that as $t$ goes to zero the coefficient $\frac{t^{n-m}\binom{n}{m}}{a_{t}}$ converges to

$$\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}.$$

On the other hand we have

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

The term with $k=0$ is independent of $t$, while any term with $k>0$ can be written as

(4.14)

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

The $(n,n)$-form $\sqrt{-1}\partial\overline{\partial}\eta\wedge\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k-1}\wedge\omega_{X}^{n-m}$ is supported in $K$ and is uniformly bounded by (4.9), and the functions $\underline{\varphi_{t_{i}}}$ converge uniformly to $\hat{\varphi}$, and so along the sequence $t_{i}$ the term (4.14) has the same limit as

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

But this is equal to

$$\int_{X}\underline{\varphi_{t}}\sqrt{-1}\partial\overline{\partial}\eta\wedge\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k-2}\wedge\sqrt{-1}\partial\overline{\partial}\hat{\varphi}\wedge\omega_{X}^{n-m},$$

and repeating the same argument $k-1$ times we see that along the sequence $t_{i}$ the term (4.14) converges to

$$\int_{X}\eta\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{k}\wedge\omega_{X}^{n-m}.$$

It follows that along the sequence $t_{i}$ the term (4.13) converges to

$$\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m}\wedge\omega_{X}^{n-m},$$

and using (4.6), (4.7) we get

$$\int_{X}\eta F\omega_{SF}^{n-m}\wedge\omega_{0}^{m}=\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m}\wedge\omega_{X}^{n-m}.$$

We then integrate first along the fibers and get

$$\begin{split}\int_{Y}&\eta F\omega_{Y}^{m}\left(\int_{X_{y}}\omega_{SF,y}^{n-m}\right)\\ &=\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{Y}\eta(\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m}\left(\int_{X_{y}}\omega_{y}^{n-m}\right),\end{split}$$

and since $\omega_{y}$ is cohomologous to $\omega_{SF,y}$, we get

$$\int_{Y}\eta F\omega_{Y}^{m}=\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{Y}\eta(\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m},$$

which is just the weak form of (4.3). This shows that any weak limit $\hat{\omega}$ of $\tilde{\omega}_{t}$ as $t\to 0$ satisfies (4.3) weakly, and we have already remarked that we can write $\hat{\omega}=\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi}$ with $\hat{\varphi}$ in $L^{\infty}$. By Kołodziej’s uniqueness of $L^{\infty}$ weak solutions of (4.3) (see [ST2, Theorem 3.2] and [EGZ1, Z]), we must have $\hat{\varphi}=\psi$, and so the whole sequence $\tilde{\omega}_{t}$ converges weakly to $\omega$ as $t\to 0$. Then the bound (2.9) implies that $\varphi_{t}$ actually converges to $\psi$ in the $C^{1,\beta}_{loc}$ topology on $X\backslash S$. ∎