In this subsection we shall use (2.19) to construct a family of $C^{2,\alpha}$ Kähler structures on certain $S^{1}$ fibrations over increasing domains in $Q$. So we need to construct a family of pairs $(\tilde{\omega},h)$ parametrized by $T\gg 1$. Most of the quantities defined in this subsection will depend on the parameter $T$, but for simplicity of notation we will not always keep track of this if it is clear from the context.

For $T\gg 1$, we define

(4.5)

$$\tilde{\omega}=T\omega_{D}+\psi.$$

It can be viewed as a family of closed $(1,1)$-forms $\tilde{\omega}(z)$ on $D$ parametrized by $z$. Using the Kähler identity, we obtain

(4.6)

$$\partial_{z}^{2}\tilde{\omega}=\Delta_{D}\tilde{\omega}=-d_{D}d_{D}^{c}\Tr_{\omega_{D}}\tilde{\omega},$$

So if we define

(4.7)

$$h\equiv\Tr_{\omega_{D}}\tilde{\omega}+q(z)$$

for any smooth function $q(z)$, then the pair $(\tilde{\omega},h)$ satisfies the first equation in (2.19):

(4.8)

$$\partial_{z}^{2}\tilde{\omega}+d_{D}d_{D}^{c}h=0.$$

For our purpose we need to make a special choice of the function $q(z)$. First we define $q_{0}(z)$ by the following co-homological condition

(4.9)

$$q_{0}(z)\int_{D}\omega_{D}^{n-1}+(n-1)\int_{D}\tilde{\omega}(z)\wedge\omega_{D}^{n-2}=T^{2-n}\int_{D}\tilde{\omega}(z)^{n-1},\ \ \forall z\in\mathbb{R},$$

By Lemma 3.33, we know that the cohomology class $[\psi(z)]\in H^{2}(D;\mathbb{R})$ is piecewise linear in $z\in\mathbb{R}$, so

(4.10)

$\displaystyle q_{0}(z)=\begin{cases}T^{2-n}(T+k_{+}z)^{n-1}-(n-1)(T+k_{+}z),&z>0,\\ T^{2-n}(T+k_{-}z)^{n-1}-(n-1)(T+k_{-}z),&z<0.\end{cases}$

It follows that $q_{0}(z)$ is identically zero if $n=2$, which corresponds to the case of the classical Gibbons-Hawking anstaz used in [HSVZ18]. But if $n>2$ then $q_{0}(z)$ is only $C^{1,1}$ at $z=0$ and we need to smooth it. We shall fix throughout this section a smooth function $L_{0}:\mathbb{R}\rightarrow\mathbb{R}$ satisfying

(4.11)

$\displaystyle L_{0}(z)\equiv\begin{cases}k_{+}z,&z>1,\\ 0,&z=0,\\ k_{-}z,&z<-1.\end{cases}$

and let

(4.12)

$$L_{T}(z)\equiv T+L_{0}(z).$$

Then we define

(4.13)

$$q(z)\equiv T^{2-n}L_{T}(z)^{n-1}-(n-1)L_{T}(z).$$

It follows that $q(z)$ is smooth and agrees with $q_{0}(z)$ when $|z|\geq 1$. It is also easy to see that correspondingly we have

(4.14)

$\displaystyle\int_{D}h\omega_{D}^{n-1}=\begin{cases}T^{2-n}\int_{D}\tilde{\omega}(z)^{n-1},&|z|\geq 1,\\ T^{2-n}\int_{D}\tilde{\omega}(z)^{n-1}+T^{-1}B(z),&z\in[-1,1].\end{cases}$

We refer to Remark 4.3.1 for an explanation of this choice of $q(z)$.

To apply the construction in Section 2, we need to restrict to the region in $Q$ where $\tilde{\omega}(z)$ is a positive form and $h$ is a positive function. For $T$ large we define $T_{+}>0$ and $T_{-}<0$ by

(4.15)

$$\begin{cases}T+k_{+}T_{+}=T^{\frac{n-2}{n}}\\ T+k_{-}T_{-}=T^{\frac{n-2}{n}}.\end{cases}$$

and denote by $Q_{T}\subset Q$ the region where $z\in[T_{-},T_{+}]$.

Now we define the 2-form

(4.21)

$$\Upsilon\equiv\partial_{z}\tilde{\omega}-dz\wedge d_{D}^{c}h.$$

Then (4.6) implies that $\Upsilon$ is closed on $Q\setminus P$ and hence $[\Upsilon]\in H^{2}(Q\setminus P,\mathbb{R})$. Moreover, we have

By Lemma 4.2, standard theory yields a $U(1)$ connection $1$-form $-\sqrt{-1}\Theta$ on a principal $S^{1}$-bundle

(4.32)

$$\pi:\mathcal{M}^{*}\rightarrow Q_{T}\setminus P$$

with curvature form $-\sqrt{-1}\Upsilon$. Moreover, $\mathcal{M}^{*}$ restricts to the standard Hopf bundle on each normal $S^{2}$ to $P$ (it has degree $-1$ if we use the natural orientation). Then we have the second equation in (2.19) satisfied:

(4.33)

$$d\Theta=\partial_{z}\tilde{\omega}-dz\wedge d_{D}^{c}h.$$

On ${\mathcal{M}^{*}}$ we define a real-valued 2-form

(4.34)

$$\omega\equiv T^{\frac{2-n}{n}}(\pi^{*}\tilde{\omega}+dz\wedge\Theta)$$

and a complex-valued $n$-form

(4.35)

$$\Omega\equiv\sqrt{-1}(hdz+\sqrt{-1}\Theta)\wedge\pi^{*}\Omega_{D}.$$

One can directly check that both $\omega$ and $\Omega$ are closed. By the discussion in Section 2, we know $(\omega,\Omega)$ defines a smooth Kähler metric on ${\mathcal{M}^{*}}$, so that $\Omega$ is the holomorphic volume form and $\omega$ is the Kähler form. Also $h^{-1}$ has an intrinsic geometric meaning as the norm squared of the Killing field generating the $S^{1}$ action.

By (4.1) and straightforward calculations, we have

(4.36)

$$\frac{(\sqrt{-1})^{n^{2}}2^{-n}\Omega\wedge\bar{\Omega}}{\omega^{n}/n!}=\frac{T^{-1}h\omega_{D}^{n-1}}{(\omega_{D}+T^{-1}\psi)^{n-1}}.$$

In particular, $\omega$ is a Calabi-Yau metric if $\mathrm{Err}_{CY}=0$.

Next we move on to the study the compactified geometry of ${\mathcal{M}^{*}}$ near $P$. We shall first construct a smooth model for the compactification and then study the regularity of the Kähler metric on this model.

As before we will always identify a neighborhood $\mathcal{U}$ of $P$ in $Q$ with a tubular neighborhood of the zero section in $N_{0}\oplus\mathbb{R}$ over $H$. Denote by $\mathbb{L}_{1}$ and $\mathbb{L}_{2}$ the complex line bundles over $H$ given by the restriction

(4.39)

$$\mathbb{L}_{1}\equiv L^{\otimes-k_{+}}|_{H};\ \ \mathbb{L}_{2}\equiv L^{\otimes k_{-}}|_{H}.$$

Then as complex line bundles $N_{0}$ is isomorphic to $L^{\otimes k}|_{H}\simeq\mathbb{L}_{1}\otimes\mathbb{L}_{2}$, and we fix such an isomorphism now. Notice $N_{0}$ is equipped with a natural hermitian metric induced from the Kähler metric $\omega_{D}$ on $D$ (c.f. Section 3.3). This then determines a hermitian metric on $L$ hence on $\mathbb{L}_{1}$ and $\mathbb{L}_{2}$. Define

(4.40)

$$\mathbb{L}\equiv\mathbb{L}_{1}\oplus\mathbb{L}_{2},$$

and consider the map

(4.41)

$$\tau:\mathbb{L}\rightarrow N_{0}\oplus\mathbb{R};(s_{1},s_{2})\mapsto(s_{1}\otimes s_{2},\frac{|s_{1}|^{2}-|s_{2}|^{2}}{2}).$$

Away from the zero section in $\mathbb{L}$, $\tau$ is a principal $S^{1}$ bundle, with the $S^{1}$ action given by

(4.42)

$$e^{\sqrt{-1}\mathfrak{t}}\cdot(s_{1},s_{2})=(e^{-\sqrt{-1}t}s_{1},e^{\sqrt{-1}t}s_{2}).$$

As Section 3.3, locally choosing holomorphic coordinates $\{w_{1},\cdots,w_{n-1}\}$ on $D$ centered at $p\in H$. These give rise to local coordinates $\{y,\bar{y},w_{2}^{\prime},\bar{w}_{2}^{\prime},\cdots,w_{n-1}^{\prime},\bar{w}_{n-1}^{\prime}\}$ on $N_{0}$, and also a local unitary section of $N_{0}$ in the form $\bm{e}=|\sigma|^{-1}\cdot\sigma$. Then we choose a local section $\bm{e}_{L}$ of $L|_{H}$ with $\bm{e}_{L}^{\otimes k}=\bm{e}$. Correspondingly we get local unitary sections $\bm{e}_{1}\equiv\bm{e}_{L}^{\otimes-k_{+}},\bm{e}_{2}\equiv\bm{e}_{L}^{\otimes k_{-}}$ of $\mathbb{L}_{1},\mathbb{L}_{2}$ respectively. Then we obtain local fiber coordinates $u_{1},u_{2},y$ on $\mathbb{L}_{2},\mathbb{L}_{1},N_{0}$ respectively by writing

(4.43)

$$s_{1}=u_{1}\bm{e}_{1},s_{2}=u_{2}\bm{e}_{2},s=y\bm{e}.$$

Then the map $\tau$ can be represented in coordinates as

(4.44)

$$\begin{cases}y=u_{1}u_{2}\\ z=\frac{1}{2}(|u_{1}|^{2}-|u_{2}|^{2})\end{cases}$$

Hence $\tau$ is the standard Hopf fibration $\mathbb{C}^{2}\rightarrow\mathbb{R}^{3}$ over each fiber.

By Lemma 4.4 we may glue ${\mathcal{M}^{*}}$ and $\mathbb{L}$ together to obtain a differentiable compactfication $\mathcal{M}$ of ${\mathcal{M}^{*}}$. The projection map $\pi$ naturally extends to a map

(4.49)

$$\pi:\mathcal{M}\rightarrow Q_{T}$$

which is a singular $S^{1}$ fibration, with discriminant locus given by $P$. We shall identify

(4.50)

$$\mathcal{P}\equiv\pi^{-1}(P)$$

with the zero section in $\mathbb{L}$, and identify a neighborhood of $\mathcal{P}$ with a neighborhood of the zero section in $\mathbb{L}$ and the projection map $\pi$ with the above $\tau$.

To study the regularity of the Kähler metric $(\omega,\Omega)$ on the compactification $\mathcal{M}$, we shall make a special choice of the connection 1-form $-\sqrt{-1}\Theta$ on a neighborhood $\mathcal{V}$ of $\mathcal{P}$ in $\mathbb{L}$, with curvature form $\Upsilon$, which has explicit regularity behavior across $\mathcal{P}$. To do this, we need a few steps. First, we notice that $\{u_{1},\bar{u}_{1},u_{2},\bar{u}_{2},w_{2}^{\prime},\bar{w}_{2}^{\prime},\cdots,w_{n-1}^{\prime},\bar{w}_{n-1}^{\prime}\}$ provides local coordinates on $\mathbb{L}$, and we can define a local model connection 1-form on $\mathbb{L}$ by simply taking the model formula (2.47):

(4.51)

$$\Theta_{0}=-\sqrt{-1}\frac{\bar{u}_{1}du_{1}-u_{1}d\bar{u}_{1}-\bar{u}_{2}du_{2}+u_{2}d\bar{u}_{2}}{2(|u_{1}|^{2}+|u_{2}|^{2})}.$$

Just as in the discussion in Section 2, we see $\Theta_{0}(\partial_{t})=-1$, where $\partial_{t}$ is the vector field generating the $S^{1}$ action. It is clear that the definition of $\Theta_{0}$ only depends on the choice of $\sigma$ and does not depend on the choice of $\bm{e}_{1}$ and $\bm{e}_{2}$ (which has the freedom of multiplying by a constant root of unity).

To make a globally defined connection 1-form, we need to add a correction term, and define

(4.52)

$$\Theta_{1}=\Theta_{0}+\frac{z}{r}\Gamma-\frac{k_{-}+k_{+}}{k_{-}-k_{+}}\Gamma,$$

where $\Gamma$ is the local 1-form given in Section 3.3, and we have implicitly viewed forms on $H$ as forms on $\mathbb{L}$ using the pull-back $\pi^{*}$.

The next Lemma allows us to correct $O^{\prime}(s)$ term on the right hand side. We fix any $S^{1}$ invariant Riemannian metric on $\mathbb{L}$.

Now we define a fixed connection 1-form on $\mathbb{L}$.

(4.66)

$$\Theta_{m}\equiv\Theta_{1}+\theta,$$

Therefore, in a neighborhood of $\mathcal{P}\subset\mathbb{L}$ minus $\mathcal{P}$, the original choice of $\Theta$ can be written as

(4.67)

$$\Theta=\Theta_{m}+\theta_{f},$$

where $\theta_{f}$ is a flat connection, which is gauge equivalent to the pull-back of a flat connection on $D$. Without loss of generality, we can then assume $\theta_{f}$ is smooth.