For simplicity of notation call $\omega_{y}=\omega_{M}|_{M_{y}}$ and $\tilde{\omega}_{y}=\tilde{\omega}_{t}|_{M_{y}}$. On each fiber $M_{y}$ we have that $\Ric(\omega_{y})=\sqrt{-1}\partial\overline{\partial}F_{y}$ for some smooth function $F_{y}$ normalized by $\int_{M_{y}}(e^{F_{y}}-1)\omega_{y}^{n-m}=0$. The functions $F_{y}$ vary smoothly in $y\in N\backslash f(S)$, because so do the Kähler metrics $\omega_{y}$. The unique Ricci–flat metric on $M_{y}$ cohomologous to $\omega_{y}$ is given by $\omega_{SF,y}=\omega_{y}+\sqrt{-1}\partial\overline{\partial}\zeta_{y}$ and solves the complex Monge-Ampère equation on $M_{y}$

$$\omega_{SF,y}^{n-m}=(\omega_{y}+\sqrt{-1}\partial\overline{\partial}\zeta_{y})^{n-m}=e^{F_{y}}\omega_{y}^{n-m}.$$

Recall from [38, Section 2] that we have

$$\omega_{0}^{m}\wedge\omega_{M}^{n-m}=H\omega_{M}^{n},$$

where $H\geqslant 0$ is a smooth function on $M$ that vanishes precisely on $S$. A simple calculation [38, (3.5)] shows that on $M_{y}$ we have

$$\Ric(\omega_{y})=-\sqrt{-1}\partial\overline{\partial}\log H+(\Ric(\omega_{M}))|_{M_{y}}=-\sqrt{-1}\partial\overline{\partial}\log H,$$

since we picked $\omega_{M}$ to be Ricci–flat. It follows that on $M_{y}$ the functions $F_{y}$ and $-\log H$ differ by a constant, which we can identify as follows: thanks to Yau’s estimates, the functions $\zeta_{y}$ vary smoothly in $y$ and so they define a smooth function $\zeta$ on $M\backslash S$. We then defined $\omega_{SF}=\omega_{M}+\sqrt{-1}\partial\overline{\partial}\zeta$, which is a semi-flat form on $M\backslash S$ (here semi-flat means that its restriction to each fiber $M_{y}$ is Ricci–flat). This semi-flat form is in general different from the one constructed locally in section 3, although they are equal when restricted to each fiber $M_{y}$. Even though $\omega_{SF}$ is not necessarily nonnegative, on $M\backslash S$ the $(n,n)$-form $\omega_{SF}^{n-m}\wedge\omega_{0}^{m}$ is strictly positive, and so we can define a smooth positive function $G$ on $M\backslash S$ by

(4.24)

$$G=\frac{\omega_{M}^{n}}{\omega_{0}^{m}\wedge\omega_{SF}^{n-m}}.$$

It is shown in [35, Lemma 3.3], [38, p.445] that $G$ is a positive constant on each fiber $M_{y}$, and we claim we have

(4.25)

$$e^{F_{y}}=\frac{1}{GH}.$$

This is because on $M_{y}$ we have

$$\frac{1}{H}=\frac{\omega_{M}^{n}}{\omega_{0}^{m}\wedge\omega_{M}^{n-m}}=\frac{\omega_{M}^{n}}{\omega_{0}^{m}\wedge\omega_{SF}^{n-m}}\cdot\frac{\omega_{SF,y}^{n-m}}{\omega_{y}^{n-m}}=Ge^{F_{y}}.$$

On $M_{y}$ we can then write, using (1.1), (4.25)

(4.26)

$$\begin{split}\left(\frac{\tilde{\omega}_{y}}{t}\right)^{n-m}&=t^{m-n}\frac{\tilde{\omega}_{y}^{n-m}}{\omega_{y}^{n-m}}\omega_{y}^{n-m}=t^{m-n}\frac{\tilde{\omega}_{y}^{n-m}\wedge\omega_{0}^{m}}{\omega_{M}^{n-m}\wedge\omega_{0}^{m}}\omega_{y}^{n-m}\\ &=\frac{\tilde{\omega}_{y}^{n-m}\wedge\omega_{0}^{m}}{\tilde{\omega}_{t}^{n}}\cdot\frac{c_{t}}{H}\omega_{y}^{n-m}\\ &=\frac{\tilde{\omega}_{y}^{n-m}\wedge\omega_{0}^{m}}{\tilde{\omega}_{t}^{n}}(c_{t}G)e^{F_{y}}\omega_{y}^{n-m}.\end{split}$$

We also have a pointwise identity on $M_{y}$

$$\frac{\tilde{\omega}_{y}^{n-m}\wedge\omega_{0}^{m}}{\tilde{\omega}_{t}^{n}}=\frac{\tilde{\omega}_{y}^{n-m}\wedge\omega_{0}^{m}}{\binom{n}{m}\tilde{\omega}_{y}^{n-m}\wedge\tilde{\omega}_{t}^{m}}=\frac{\omega_{y}^{n-m}\wedge\omega_{0}^{m}}{\binom{n}{m}\omega_{y}^{n-m}\wedge\tilde{\omega}_{t}^{m}},$$

and we will write

$$f_{t}=c_{t}G\frac{\omega_{y}^{n-m}\wedge\omega_{0}^{m}}{\binom{n}{m}\omega_{y}^{n-m}\wedge\tilde{\omega}_{t}^{m}},$$

so that we can recast (4.26) as

(4.27)

$$\left(\frac{\tilde{\omega}_{y}}{t}\right)^{n-m}=f_{t}\omega_{SF,y}^{n-m}.$$

Notice that the functions $f_{t}$ are the restriction to $M_{y}$ of smooth functions on $M\backslash S$. We claim that as $t$ approaches zero the functions $f_{t}$ converge to $1$ in $C^{\infty}_{loc}(M\backslash S,\omega_{M})$. To see this, first of all note that by definition we have

(4.28)

$$\lim_{t\to 0}c_{t}=\binom{n}{m}\frac{\int_{M}\omega_{0}^{m}\wedge\omega_{M}^{n-m}}{\int_{M}\omega_{M}^{n}}>0,$$

see also [10], [38, (2.6)]. We now use the assumption (4.22), and so the functions $f_{t}$ converge smoothly to

(4.29)

$$G\binom{n}{m}\frac{\int_{M}\omega_{0}^{m}\wedge\omega_{M}^{n-m}}{\int_{M}\omega_{M}^{n}}\cdot\frac{\omega_{M}^{n-m}\wedge\omega_{0}^{m}}{\binom{n}{m}\omega_{M}^{n-m}\wedge(f^{*}\omega)^{m}}.$$

To see why this equals one, recall from [38, (4.3)] that the limit metric $\omega$ on $N\backslash f(S)$ satisfies

(4.30)

$$\omega^{m}=G\frac{\int_{M}\omega_{0}^{m}\wedge\omega_{M}^{n-m}}{\int_{M}\omega_{M}^{n}}\omega_{N}^{m},$$

where our function $G$ is defined so that it differs from the function $F$ in [38, (4.3)] by the constant factor $\int_{M}(\omega_{0}+\omega_{M})^{n}/\int_{M}\omega_{M}^{n}$. Substituting (4.30) into (4.29) we see that the limit of $f_{t}$ equals

$$\frac{\omega_{M}^{n-m}\wedge\omega_{0}^{m}}{\omega_{M}^{n-m}\wedge(f^{*}\omega_{N})^{m}}=1.$$

Note now that from the main result of [38] we have that on each fiber $M_{y}$

(4.31)

$$C^{-1}\omega_{y}\leqslant\frac{\tilde{\omega}_{y}}{t}\leqslant C\omega_{y},$$

where $C$ is uniform as $y$ varies in a compact set of $N\backslash f(S)$. From the definition on $M_{y}$ we have

$$\frac{\tilde{\omega}_{y}}{t}=\omega_{y}+\sqrt{-1}\partial\overline{\partial}\left(\frac{\varphi_{t}}{t}\right),$$

where $\frac{\varphi_{t}}{t}$ satisfies the $C^{0}$ estimate (4.21). The metrics $\frac{\tilde{\omega}_{y}}{t}$ satisfy the complex Monge-Ampère equations on $M_{y}$

(4.32)

$$\left(\frac{\tilde{\omega}_{y}}{t}\right)^{n-m}=\left(\omega_{y}+\sqrt{-1}\partial\overline{\partial}\left(\frac{\varphi_{t}}{t}\right)\right)^{n-m}=f_{t}e^{F_{y}}\omega_{y}^{n-m},$$

and we have just shown that the functions $f_{t}e^{F_{y}}$ are bounded in $C^{\infty}(M_{y},\omega_{y})$ and away from zero, so we can apply the theory of Evans-Krylov and Schauder estimates on $M_{y}$ to (4.32) (using (4.21) and (4.31)) to get bounds

$$\left\|\frac{\tilde{\omega}_{y}}{t}\right\|_{C^{k}(M_{y},\omega_{y})}\leqslant C(k),$$

independent of $t$. It follows that given any sequence $t_{i}\to 0$ we can find a subsequence (still denoted by $t_{i}$) and a smooth Kähler metric $\alpha_{y}$ on $M_{y}$ so that $\frac{\tilde{\omega}_{y}}{t_{i}}\to\alpha_{y}$ in $C^{\infty}(\omega_{y})$. Equation (4.27) in the limit becomes

$$\alpha_{y}^{n-m}=\omega_{SF,y}^{n-m},$$

and so by the uniqueness of Ricci–flat metrics in a given cohomology class we must have $\alpha_{y}=\omega_{SF,y}$. Therefore the whole sequence $\frac{\tilde{\omega}_{y}}{t}$ converges smoothly to $\omega_{SF,y}$ as desired, and the convergence is uniform as $y$ varies on compact sets of $N\backslash f(S)$. ∎