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). ∎