This estimate is contained in the second-named author’s work [38], although it is not explicitly stated there. To see this, start from [38, (3.24)], which gives a constant $C$ so that on $U$ we have

$$C^{-1}(t\omega_{M})\leqslant\tilde{\omega}_{t}.$$

Then use [38, Lemma 3.1] to get

$$C^{-1}\omega_{0}\leqslant\tilde{\omega}_{t},$$

and so adding these two inequalities we get

$$C^{-1}(\omega_{0}+t\omega_{M})\leqslant\tilde{\omega}_{t},$$

or in other words $\textrm{tr}_{\tilde{\omega}_{t}}\omega_{t}\leqslant C$ on $U$, where $\omega_{t}=\omega_{0}+t\omega_{M}$ as before. To get the reverse inequality, we note that on $U$ we have

$$\textrm{tr}_{\omega_{t}}\tilde{\omega}_{t}\leqslant(\textrm{tr}_{\tilde{\omega}_{t}}\omega_{t})^{n-1}\frac{\tilde{\omega}_{t}^{n}}{\omega_{t}^{n}}\leqslant C\frac{\tilde{\omega}_{t}^{n}}{\omega_{t}^{n}}\leqslant C,$$

where the last inequality follows from [38, (3.23)]. We thus get the reverse inequality

$$\tilde{\omega}_{t}\leqslant C(\omega_{0}+t\omega_{M}),$$

thus proving (4.1). ∎

First of all notice that after replacing $U$ with a slightly smaller open set, the semi-flat metric $\omega_{0}+\omega_{SF}$ is uniformly equivalent to $\omega_{M}$, which implies that

(4.4)

$$C^{-1}(\omega_{0}+t\omega_{SF})\leqslant\omega_{0}+t\omega_{M}\leqslant C(\omega_{0}+t\omega_{SF}),$$

for all small $t>0$. Thanks to Lemma 4.1 on $U$ we have that

$$C^{-1}(\omega_{0}+tT_{-\sigma}^{*}\omega_{M})\leqslant T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(\omega_{0}+tT_{-\sigma}^{*}\omega_{M}),$$

and since $T_{-\sigma}^{*}\omega_{M}$ is uniformly equivalent to $\omega_{M}$ we also have that

(4.5)

$$C^{-1}(\omega_{0}+t\omega_{M})\leqslant T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(\omega_{0}+t\omega_{M}),$$

and combining (4.4) and (4.5) we get

(4.6)

$$C^{-1}(\omega_{0}+t\omega_{SF})\leqslant T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(\omega_{0}+t\omega_{SF}),$$

on $U$. If we pull back (4.6) by $p\circ\lambda_{t}$ we get

(4.7)

$$C^{-1}(p^{*}\omega_{0}+t\lambda_{t}^{*}p^{*}\omega_{SF})\leqslant\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(p^{*}\omega_{0}+t\lambda_{t}^{*}p^{*}\omega_{SF}),$$

on all of $B\times\mathbb{C}^{n-m}$. We claim that on the whole of $B\times\mathbb{C}^{n-m}$ we have that

(4.8)

$$t\lambda_{t}^{*}p^{*}\omega_{SF}=p^{*}\omega_{SF}.$$

In fact, the construction of $\omega_{SF}$ in section 3 gives that $p^{*}\omega_{SF}=\sqrt{-1}\partial\overline{\partial}\eta,$ for a function $\eta$ on $B\times\mathbb{C}^{n-m}$ that satisfies

(4.9)

$$\eta\circ\lambda_{t}(y,z)=\eta\left(y,\frac{z}{\sqrt{t}}\right)=\frac{1}{t}\eta(y,z),$$

for all $(y,z)$ in $B\times\mathbb{C}^{n-m}$ and any $t>0$. It follows then that

(4.10)

$$t\lambda_{t}^{*}p^{*}\omega_{SF}=t\lambda_{t}^{*}\sqrt{-1}\partial\overline{\partial}\eta=t\sqrt{-1}\partial\overline{\partial}(\eta\circ\lambda_{t})=\sqrt{-1}\partial\overline{\partial}\eta=p^{*}\omega_{SF},$$

as claimed. Combining (4.7) and (4.8) we get the bound (4.3). ∎

We pull back (1.1) via $T_{-\sigma}\circ p\circ\lambda_{t}$ and get

$$\begin{split}(\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t})^{n}(y,z)&=c_{t}t^{n-m}(\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\omega_{M})^{n}(y,z)\\ &=c_{t}(p^{*}T_{-\sigma}^{*}\omega_{M})^{n}\left(y,\frac{z}{\sqrt{t}}\right),\end{split}$$

since the pullback under $\lambda_{t}$ of any volume form $f(y,z)dy^{1}\wedge\dots\wedge d\overline{z}^{n-m}$ on $B\times\mathbb{C}^{n-m}$ equals $t^{m-n}f(y,\frac{z}{\sqrt{t}})dy^{1}\wedge\dots\wedge d\overline{z}^{n-m}.$ We now claim that in fact we have

$$(p^{*}T_{-\sigma}^{*}\omega_{M})^{n}\left(y,\frac{z}{\sqrt{t}}\right)=(p^{*}T_{-\sigma}^{*}\omega_{M})^{n}(y,z).$$

To see this, consider the $(n,0)$-form

$$dy^{1}\wedge\dots\wedge dy^{m}\wedge dz^{1}\wedge\dots\wedge dz^{n-m}$$

on $B\times\mathbb{C}^{n-m}$. This form is invariant under the $\mathbb{Z}^{2n-2m}$-action described above

$$(n_{1},\dots,n_{2n-2m})\cdot(y,z)=(y,z+\sum_{i}n_{i}v_{i}(y)),$$

where $(y,z)=(y_{1},\dots,y_{m},z_{1},\dots,z_{n-m})$, and so it descends to a holomorphic $(n,0)$-form to the quotient $(B\times\mathbb{C}^{n-m})/\Lambda$ and using the biholomorphism with $U$ we get a holomorphic $(n,0)$-form $\Omega$ on $U$. We can then consider the volume form $(\sqrt{-1})^{n^{2}}\Omega\wedge\overline{\Omega}$, and we have

$$T_{-\sigma}^{*}\omega_{M}^{n}=h\cdot(\sqrt{-1})^{n^{2}}\Omega\wedge\overline{\Omega},$$

where $h$ is a smooth positive function on $U$. Taking $\sqrt{-1}\partial\overline{\partial}\log$ of both sides we get

$$\sqrt{-1}\partial\overline{\partial}\log h=\sqrt{-1}\partial\overline{\partial}\log\frac{T_{-\sigma}^{*}\omega_{M}^{n}}{(\sqrt{-1})^{n^{2}}\Omega\wedge\overline{\Omega}}=0,$$

since $T_{-\sigma}^{*}\omega_{M}$ is Ricci–flat and $\Omega$ is a holomorphic $(n,0)$-form. So $\log h$ is pluriharmonic on $U$, and this implies that its restriction to any fiber $M_{y}$ with $y\in B$ is constant. Pulling back via $p$ we get

$$(p^{*}T_{-\sigma}^{*}\omega_{M}^{n})(y,z)=(h\circ p)(y,z)(\sqrt{-1})^{n^{2}}dy^{1}\wedge\dots\wedge d\overline{z}^{n-m},$$

but since $h$ is constant along the fibers of $f$ and $p$ is compatible with the projection to $B$ we get that the function $(h\circ p)(y,z)$ on $B\times\mathbb{C}^{n-m}$ is independent of $z$. In particular we have

$$(p^{*}T_{-\sigma}^{*}\omega_{M})^{n}\left(y,\frac{z}{\sqrt{t}}\right)=(p^{*}T_{-\sigma}^{*}\omega_{M})^{n}(y,z),$$

and so the rescaled metrics $\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}$ satisfy the nondegenerate complex Monge-Ampère equation

$$(\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t})^{n}=(p^{*}\omega_{0}+t\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\omega_{M}+\sqrt{-1}\partial\overline{\partial}\tilde{\varphi}_{t})^{n}=c_{t}(p^{*}T_{-\sigma}^{*}\omega_{M})^{n}$$

on $B\times\mathbb{C}^{n-m}$, where we have set

$$\tilde{\varphi}_{t}=\varphi_{t}\circ T_{-\sigma}\circ p\circ\lambda_{t}.$$

We claim that the estimates (4.11) hold. To see this, we use (4.2) and get

(4.12)

$$p^{*}\omega_{SF}=p^{*}T_{-\sigma}^{*}\omega_{M}+p^{*}T_{-\sigma}^{*}\sqrt{-1}\partial\overline{\partial}\xi,$$

for a function $\xi$ on $U$. On $B\times\mathbb{C}^{n-m}$ we can then use (4.10) and (4.12) and write

(4.13)

$$\begin{split}\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}&=p^{*}\omega_{0}+t\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\omega_{M}+\sqrt{-1}\partial\overline{\partial}\tilde{\varphi}_{t}\\ &=p^{*}\omega_{0}+t\lambda_{t}^{*}p^{*}(\omega_{SF}-T_{-\sigma}^{*}\sqrt{-1}\partial\overline{\partial}\xi)+\sqrt{-1}\partial\overline{\partial}\tilde{\varphi}_{t}\\ &=p^{*}\omega_{0}+p^{*}\omega_{SF}-t\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\sqrt{-1}\partial\overline{\partial}\xi+\sqrt{-1}\partial\overline{\partial}\tilde{\varphi}_{t}\\ &=p^{*}(\omega_{0}+\omega_{SF})+\sqrt{-1}\partial\overline{\partial}u_{t},\end{split}$$

where for simplicity we write $u_{t}=\tilde{\varphi}_{t}-t(\xi\circ T_{-\sigma}\circ p\circ\lambda_{t}).$ The functions $u_{t}$ are uniformly bounded in $C^{0}(B\times\mathbb{C}^{n-m})$ because of the $L^{\infty}$ bound for $\varphi_{t}$ from [9, 10] and because $\xi$ is a fixed function on $U$. The functions $u_{t}$ satisfy the complex Monge-Ampère equations

(4.14)

$$(p^{*}\omega_{0}+p^{*}\omega_{SF}+\sqrt{-1}\partial\overline{\partial}u_{t})^{n}=c_{t}(p^{*}T_{-\sigma}^{*}\omega_{M})^{n}$$

on $B\times\mathbb{C}^{n-m}$, and on any compact subset $K$ of $B\times\mathbb{C}^{n-m}$ the Kähler metric $p^{*}(\omega_{0}+\omega_{SF})$ is $C^{\infty}$ equivalent to the Euclidean metric $\delta$ (with constants that depend only on $K$). The bounds (4.3) imply that

$$C^{-1}\delta\leqslant p^{*}(\omega_{0}+\omega_{SF})+\sqrt{-1}\partial\overline{\partial}u_{t}\leqslant C\delta,$$

on $K$ for all small $t>0$, where $C$ depends on $K$. The constants $c_{t}$ are bounded uniformly and away from zero. After shrinking $K$ slightly we can then apply the Evans-Krylov theory (as explained for example in [13, 32]) and Schauder estimates to get higher order estimates $\|u_{t}\|_{C^{k}(K,\delta)}\leqslant C(k)$ for all $k\geqslant 0$, thus proving (4.11). ∎

We can assume that $K$ is sufficiently small so that $f(K)\subset B$ for a ball $B$ as before, and that there is a compact set $K^{\prime}\subset B\times\mathbb{C}^{n-m}$ so that $p:K^{\prime}\to T_{\sigma}(K)$ is a biholomorphism. We then have

$$\begin{split}\sup_{K}|\mathrm{Sec}(\tilde{\omega}_{t})|&=\sup_{T_{\sigma}(K)}|\mathrm{Sec}(T_{-\sigma}^{*}\tilde{\omega}_{t})|=\sup_{K^{\prime}}|\mathrm{Sec}(p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t})|\\ &=\sup_{\lambda_{t}^{-1}(K^{\prime})}|\mathrm{Sec}(\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t})|.\end{split}$$

For $t>0$ small enough, the sets $\lambda_{t}^{-1}(K)$ are all contained in a fixed compact set $K^{\prime\prime}\subset B\times\mathbb{C}^{n-m}$. From (4.3) and (4.11) we then get a uniform bound for the sectional curvatures of $\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}$ on $K^{\prime\prime}$, and this proves (4.15). ∎

Given $K$, for all $t>0$ small enough the sets $\lambda_{t}^{-1}(K)$ are all contained in a fixed compact set $K^{\prime}\subset B\times\mathbb{C}^{n-m}$. We wish to deduce (4.16) from (4.11). To see this, write on $B\times\mathbb{C}^{n-m}$

$$\begin{split}\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}=&\sqrt{-1}\bigg(\sum_{i,j}A_{i\overline{j}}(t,y,z)dz^{i}\wedge d\overline{z}^{j}+\sum_{i,j}B_{i\overline{j}}(t,y,z)dy^{i}\wedge d\overline{y}^{j}\\ &+\sum_{i,j}C_{i\overline{j}}(t,y,z)dy^{i}\wedge d\overline{z}^{j}+\sum_{i,j}D_{i\overline{j}}(t,y,z)dz^{i}\wedge d\overline{y}^{j}\bigg).\end{split}$$

Thanks to (4.11), on $K^{\prime}$ the coefficents $A,B,C,D$ satisfy uniform $C^{k}$ estimates in the variables $(y,z)$ independent of $t$. We then pull back this equation via the map $\lambda_{1/t}$ (the inverse of $\lambda_{t}$) and get

$$\begin{split}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}&=\sqrt{-1}\bigg(t\sum_{i,j}A_{i\overline{j}}(t,y,z\sqrt{t})dz^{i}\wedge d\overline{z}^{j}+\sum_{i,j}B_{i\overline{j}}(t,y,z\sqrt{t})dy^{i}\wedge d\overline{y}^{j}\\ &+\sqrt{t}\sum_{i,j}C_{i\overline{j}}(t,y,z\sqrt{t})dy^{i}\wedge d\overline{z}^{j}+\sqrt{t}\sum_{i,j}D_{i\overline{j}}(t,y,z\sqrt{t})dz^{i}\wedge d\overline{y}^{j}\bigg),\end{split}$$

and the new coefficients are uniformly bounded in $C^{k}$ on $K$, thus proving (4.16). ∎

Recall that $\tilde{\omega}_{t}=\omega_{0}+t\omega_{M}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$, so that

$$p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}=p^{*}\omega_{0}+tp^{*}T_{-\sigma}^{*}\omega_{M}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}\circ T_{-\sigma}\circ p).$$

We now fix a compact set $K\subset M\backslash S$, which we can assume is sufficiently small so that $f(K)\subset B$ for a ball $B$ as before, and that there is a compact set $K^{\prime}\subset B\times\mathbb{C}^{n-m}$ such that $p:K^{\prime}\to T_{\sigma}(K)$ is a biholomorphism. From (4.16) (together with the $L^{\infty}$ bound for $\varphi_{t}$ from [9, 10]) we see that

$$\|\varphi_{t}\circ T_{-\sigma}\circ p\|_{C^{k}(K^{\prime},\delta)}\leqslant C(k),$$

and therefore also

(4.17)

$$\|\varphi_{t}\|_{C^{k}(K,\omega_{M})}\leqslant C(k),$$

since $T_{-\sigma}\circ p:K^{\prime}\to K$ is a fixed biholomorphism. From [38] we know that $\varphi_{t}\to f^{*}\varphi$ in $C^{1,\alpha}_{loc}(M\backslash S,\omega_{M})$, and so (4.17) implies that $\varphi_{t}\to f^{*}\varphi$ in $C^{\infty}_{loc}(M\backslash S,\omega_{M})$, and therefore that $\tilde{\omega}_{t}\to f^{*}\omega$ in $C^{\infty}_{loc}(M\backslash S,\omega_{M})$. ∎

Recall that from (4.13) we see that on $B\times\mathbb{C}^{n-m}$

$$\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}=p^{*}(\omega_{0}+\omega_{SF})+\sqrt{-1}\partial\overline{\partial}u_{t},$$

where the functions $u_{t}=\tilde{\varphi}_{t}-t\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\xi$ have uniform $C^{\infty}$ bounds on compact sets. We need to show that as $t$ goes to zero we have $u_{t}\to(f\circ p)^{*}\varphi$ in $C^{\infty}_{loc}(B\times\mathbb{C}^{n-m},\delta)$, where $f^{*}\varphi$ is the $C^{1,\alpha}$ limit of $\varphi_{t}$ from [38]. To prove this we need another estimate from the second-named author’s work [38, (3.9)], which implies that there is a constant $C$ (that depends on the initial choice of $B$) so that for all $0<t\leqslant 1$ we have

(4.21)

$$\sup_{y\in B}\mathrm{osc}_{M_{y}}\varphi_{t}\leqslant Ct.$$

We now use this together with the fact that $\varphi_{t}\to f^{*}\varphi$ in $C^{0}$ to get that for any $(y,z)$ in $B\times\mathbb{C}^{n-m}$ we have

$$\begin{split}|\tilde{\varphi}_{t}(y,z)-(f\circ p)^{*}\varphi(y,z)|&=\left|\varphi_{t}\circ T_{-\sigma}\circ p\left(y,\frac{z}{\sqrt{t}}\right)-\varphi(y)\right|\\ &\leqslant\left|\varphi_{t}\circ p\left(y,\frac{z}{\sqrt{t}}-\tilde{\sigma}(y)\right)-\varphi_{t}\circ p(y,z)\right|\\ &\ \ \ \ +|\varphi_{t}\circ p(y,z)-((f^{*}\varphi)\circ p)(y)|\\ &\leqslant Ct+\sup_{U}|\varphi_{t}-f^{*}\varphi|,\end{split}$$

where in the last line we used (4.21) because the points $p(y,\frac{z}{\sqrt{t}}-\tilde{\sigma}(y))$ and $p(y,z)$ lie in the same fiber $M_{y}$. Letting $t$ go to zero we see that $\tilde{\varphi}_{t}\to(f\circ p)^{*}\varphi$ in $C^{0}(B\times\mathbb{C}^{n-m})$. On the other hand we have that $t\lambda_{t}^{*}p^{*}\xi\to 0$ in $C^{0}(B\times\mathbb{C}^{n-m})$, and so $u_{t}\to(f\circ p)^{*}\varphi$ in $C^{0}(B\times\mathbb{C}^{n-m})$. Thanks to the higher order estimates for $u_{t}$, we also have that $u_{t}\to(f\circ p)^{*}\varphi$ in $C^{\infty}_{loc}(B\times\mathbb{C}^{n-m},\delta)$, up to shrinking $B$ slightly. ∎

Recall that thanks to Lemma 4.7, on $B\times\mathbb{C}^{n-m}$ we can write

$$\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}-p^{*}(\omega_{SF}+f^{*}\omega)=E_{t},$$

where the error term $E_{t}$ is a $(1,1)$-form that goes to zero smoothly on compact sets. From (4.8) we also have that

$$E_{t}=\lambda_{t}^{*}p^{*}(T_{-\sigma}^{*}\tilde{\omega}_{t}-f^{*}\omega-t\omega_{SF}).$$

If we restrict the form $T_{-\sigma}^{*}\tilde{\omega}_{t}-f^{*}\omega-t\omega_{SF}$ to a fiber $M_{y}$ and divide by $t$ we get

$$\frac{E_{t}}{t}\bigg|_{\{y\}\times\mathbb{C}^{n-m}}=\lambda_{t}^{*}p^{*}\left(\frac{T_{-\sigma}^{*}\tilde{\omega}_{t}|_{M_{y}}}{t}-\omega_{SF,y}\right)$$

Pulling back this via the map $\lambda_{1/t}$ (the inverse of $\lambda_{t}$) we get

$$\frac{\lambda_{1/t}^{*}E_{t}}{t}\bigg|_{\{y\}\times\mathbb{C}^{n-m}}=p^{*}\left(\frac{T_{-\sigma}^{*}\tilde{\omega}_{t}|_{M_{y}}}{t}-\omega_{SF,y}\right).$$

Explicitly we have $\lambda_{1/t}(y,z)=(y,z\sqrt{t})$, which implies that $\lambda_{1/t}^{*}dz^{i}=\sqrt{t}dz^{i}$, and so

$$\frac{\lambda_{1/t}^{*}E_{t}}{t}\bigg|_{\{y\}\times\mathbb{C}^{n-m}}(y,z)=E_{t}\bigg|_{\{y\}\times\mathbb{C}^{n-m}}(y,z\sqrt{t}),$$

which goes to zero smoothly as $t$ approaches zero, uniformly in $y$. It follows that $\frac{T_{-\sigma}^{*}\tilde{\omega}_{t}|_{M_{y}}}{t}$ converges smoothly to $\omega_{SF,y}$, and the convergence is uniform as $y$ varies on compact sets of $N\backslash f(S)$. Pulling back via $T_{\sigma}$, and using the fact that $T_{\sigma}^{*}\omega_{SF,y}=\omega_{SF,y}$, we see that also $\frac{\tilde{\omega}_{t}|_{M_{y}}}{t}$ converges smoothly to $\omega_{SF,y}$, as desired. ∎

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 $N\backslash f(S)$. ∎