We define

(4.93)

$$\begin{cases}\mathcal{M}_{-}\equiv{\mathcal{M}^{*}}\setminus\pi^{-1}(H\times[0,\infty))\\ \mathcal{M}_{+}\equiv{\mathcal{M}^{*}}\setminus\pi^{-1}(H\times(\infty,0]).\end{cases}$$

On $\mathcal{M}_{-}$ we can trivialize the $U(1)$ connection $-\sqrt{-1}\Theta$ along the $z$ direction so that the $z$ component $\Theta_{z}$ vanishes identically. Denote by $\Theta|_{z}$ the restriction of $\Theta$ to the slice $D\times\{z\}$ for $z<0$ and to $(D\setminus H)\times\{z\}$ for $z\geq 0$. From (4.33) we see that that curvature form of $-\sqrt{-1}\Theta|_{z}$ is given by $-\sqrt{-1}\partial_{z}\tilde{\omega}$.

By Section 3.4, we have

(4.94)

$$\partial_{z}\tilde{\omega}|_{z=T_{-}}=k_{-}\omega_{D}+\epsilon_{T}$$

and

(4.95)

$$[\partial_{z}\tilde{\omega}]|_{z=T_{-}}=k_{-}[\omega_{D}]\in H^{2}(D;\mathbb{R}).$$

Since $b_{1}(D)=0$, we may assume $\mathcal{M}|_{z=T_{-}}$ embeds into $L_{-}$, as the unit circle bundle defined by another hermitian metric $\|\cdot\|_{\sim}^{2}$ which differs from the fixed metric by $\epsilon_{T}$, and the connection 1-form $-\sqrt{-1}\Theta|_{T_{-}}$ agrees with the restriction of the Chern connection form. Denote by $\tilde{r}_{-}$ the norm function on $L_{-}$ corresponding to the new hermitian metric, then we have

(4.96)

$$\log\tilde{r}_{-}=\log r_{-}+\epsilon_{T}$$

Furthermore, we may extend $-\sqrt{-1}\Theta|_{T_{-}}$ naturally to the complement of the zero section ${\bf 0}_{L_{-}}$ in $L_{-}$, via the fiberwise projection, and the resulting 1-form coincides with $\sqrt{-1}\tilde{r}_{-}^{-1}J_{-}d\tilde{r}_{-}$, where $J_{-}$ denotes the complex structure on $L_{-}$.

Now we define a map $\Phi_{-}:{\mathcal{M}^{*}}_{-}\rightarrow L_{-}\setminus{\bf 0}_{L_{-}}$ where ${\bf 0}_{L_{-}}$ denotes the zero section in $L_{-}$. First at $z=T_{-}$ we define $\Phi_{-}$ to be the natural inclusion map as above, multiplied by $e^{A_{-}}$ for some constant $A_{-}$ to be determined later. Then using the trivialization of the $U(1)$ bundle ${\mathcal{M}^{*}}_{-}$ along the $z$ direction and the natural scaling map on $L_{-}$, we extend the map to the whole ${\mathcal{M}^{*}}_{-}$ by setting

(4.97)

$$\tilde{r}_{-}=e^{A_{-}-\int_{T_{-}}^{z}h(u)du}$$

Then $\Phi_{-}$ clearly commutes with the projection maps to $D$, so $\Phi_{-}^{*}\alpha=\alpha$ for any $1$-form $\alpha$ which is a pull-back from $D$. Since

(4.98)

$$\partial_{z}\Theta|_{z}=d_{D}^{c}h=-J_{D}d_{D}h$$

we have

(4.99)

$$\tilde{r}_{-}^{-1}\Phi_{-}^{*}d\tilde{r}_{-}=-hdz-\int_{T_{-}}^{z}du\wedge d_{D}h=-hdz-J_{D}(\Theta|_{z}-\Theta|_{T_{-}}),$$

noticing that $\Theta|_{z}-\Theta|_{T_{-}}$ is a 1-form pulled-back from $D$. So

(4.100)

$$\tilde{r}_{-}^{-1}\Phi_{-}^{*}(d\tilde{r}_{-}+\sqrt{-1}J_{-}d\tilde{r}_{-})=-hdz-\sqrt{-1}\Theta|_{z}-\sqrt{-1}(\Theta|_{T_{-}}-\Theta|_{z})-J_{D}(\Theta|_{z}-\Theta_{T_{-}})$$

is a $(1,0)$ form on ${\mathcal{M}^{*}}_{-}$.

Notice by definition locally

(4.101)

$$\tilde{r}_{-}^{2}=|\zeta_{-}|^{2}\cdot\|\sigma\|_{\sim}^{2}$$

so

(4.102)

$$\frac{d\zeta_{-}}{\zeta_{-}}=\frac{d\tilde{r}_{-}}{\tilde{r}_{-}}+\sqrt{-1}J_{-}\frac{d\tilde{r}_{-}}{\tilde{r}_{-}}+\partial_{D}\log|\sigma|^{2}$$

Therefore we obtain

(4.103)

$$\Phi_{-}^{*}\Omega_{L_{-}}=\Omega$$

where

(4.104)

$$\Omega_{L_{-}}\equiv-\sqrt{-1}\frac{d\zeta_{-}}{\zeta_{-}}\wedge\Omega_{D}$$

is a natural holomorphic volume form on $L_{-}\setminus{\bf 0}_{L_{-}}$. In particular $\Phi_{-}$ is a holomorphic embedding. Also, we have

(4.105)

$$d\Phi_{-}(\xi^{1,0})=\sqrt{-1}\zeta_{-}\partial_{\zeta_{-}}$$

is the natural holomorphic vector field on $L_{-}$.

Since $h$ is positive we see that the image of $\Phi_{-}$ is bounded in $L_{-}$. Since $\mathcal{M}\setminus\mathcal{M}_{-}$ is of complex codimension one, by the removable singularity theorem for bounded holomorphic functions, $\Phi_{-}$ extends to a holomorphic map on the entire $\mathcal{M}$.

Similarly we get a holomorphic embedding

(4.106)

$$\Phi_{+}:\mathcal{M}_{+}\rightarrow L_{+}$$

with

(4.107)

$$\tilde{r}_{+}=e^{A_{+}-\int_{z}^{T_{+}}h(u)du}$$

for a constant $A_{+}$ to be determined. Again $\Phi_{+}$ extends to a holomorphic map on $\mathcal{M}$.

Together we obtain

(4.108)

$$\Phi\equiv(\Phi_{+},\Phi_{-}):\mathcal{M}\rightarrow L_{+}\oplus L_{-}$$

which is an embedding on $\mathcal{M}\setminus\mathcal{P}$. It commutes with projections maps to $D$ and satisfies

(4.109)

$$d\Phi(\xi^{1,0})=\sqrt{-1}(\zeta_{-}\partial_{\zeta_{-}}-\zeta_{+}\partial_{\zeta_{+}}).$$

Now we show that with appropriate choice of $A_{\pm}$, $\Phi$ maps $\mathcal{M}$ into $\mathcal{N}^{0}$. First we notice that by (4.109),

(4.110)

$$\det\Phi\equiv\Phi_{+}\otimes\Phi_{-}:\mathcal{M}\setminus(H\times(-\infty,\infty))\rightarrow L_{+}\otimes L_{-}$$

has image lying on a non-zero holomorphic section, say $\tilde{S}$, of $L^{\otimes k}=L_{+}\otimes L_{-}$ over $D\setminus H$. By definition since $h$ is positive we know the the image of $\Phi$ is bounded in $L_{+}\oplus L_{-}$, with respect to the norm $\tilde{r}_{\pm}$, so $\tilde{S}$ is a bounded section of $L^{\otimes k}$ with respect to the norm $\tilde{r}\equiv\tilde{r}_{+}\otimes\tilde{r}_{-}$, hence again by removable singularity theorem for bounded holomorphic functions it extends to a holomorphic section on the entire $D$. By our assumption that $[H]$ is isomorphic to $L$, we see $H$ is exactly the zero locus of $\tilde{S}$, so there is a constant $C$ such that

(4.111)

$$\tilde{S}=C\cdot S_{H}.$$

Multiplying $\Phi_{-}$ by an element in $S^{1}$ we may assume $C$ is a positive real number. Now

(4.112)

$$\log C=\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|{\tilde{S}}\|\omega_{D}^{n-1}-\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|\omega_{D}^{n-1}$$

The second term is a constant independent of $T$. For the first term, by definition we have

(4.113)

$$-\log\|\tilde{S}\|=\int_{T_{-}}^{T_{+}}hdz-(A_{-}+A_{+})+\epsilon_{T}.$$

By (4.14)

(4.114)		$\displaystyle\int_{T_{-}}^{T_{+}}\int h\omega_{D}^{n-1}$	$\displaystyle=$	$\displaystyle\int_{T_{-}}^{T_{+}}T^{2-n}\int_{D}(T\omega_{D}+\psi)^{n-1}dz+T^{-1}\underline{B}_{T}$
(4.115)			$\displaystyle=$	$\displaystyle\frac{1}{n}\int_{D}\omega_{D}^{n-1}(\frac{1}{k_{-}}-\frac{1}{k_{+}})(T^{2}-1)+T^{-1}\underline{B}_{T}$

So we get that

(4.116)

$$-\log C=\frac{1}{n}(\frac{1}{k_{-}}-\frac{1}{k_{+}})(T^{2}-1)-(A_{-}+A_{+})+\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|+T^{-1}\underline{B}_{T}$$

Setting $C=1$ gives one condition on $A_{-}$ and $A_{+}$. For our later purposes we shall need additionally that

(4.117)

$$k_{-}A_{-}=k_{+}A_{+}$$

Together these determine $A_{-}$ and $A_{+}$ as

(4.118)

$$A_{-}\equiv\frac{1}{nk_{-}}(T^{2}-1)-\frac{-k_{+}}{2(k_{-}-k_{+})}\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|+T^{-1}\underline{B}_{T}$$

(4.119)

$$A_{+}\equiv\frac{1}{-nk_{+}}(T^{2}-1)-\frac{k_{-}}{2(k_{-}-k_{+})}\frac{1}{\int_{D}\omega_{D}^{n-1}}\int\log\|S_{H}\|+T^{-1}\underline{B}_{T}$$

Then we can make $\Phi$ maps $\mathcal{M}$ into $\mathcal{N}^{0}$.

It is easy to check that $\Phi$ satisfies (1), (2), (3) in the statement of the Proposition. It then follows from (2) that $\Phi$ is a holomorphic embedding also across $\mathcal{P}$. This finishes the proof of Proposition.

∎

Consider the case $z\leq-1$. Since

(4.126)

$$\frac{d\zeta_{-}}{\zeta_{-}}=\frac{dr_{-}}{r_{-}}+\sqrt{-1}J\frac{dr_{-}}{r_{-}}=-hdz+\epsilon_{T}-\sqrt{-1}Jhdz$$

The result then easily follows from the asymptotics of $h$ (4.16) and $\tilde{\omega}$ (3.349). ∎

We denote

(4.129)

$$\hat{h}_{-}=A_{-}-\int_{T_{-}}^{0}h(u)du-\frac{1}{2}\log\|S_{H}\|.$$

By the Poincaré-Lelong equation we have

(4.130)

$$d_{D}d_{D}^{c}\log{\|S_{H}\|}^{2}=4\pi\delta_{H}-(k_{-}-k_{+})\omega_{D},$$

where $\delta_{H}$ denotes the current of integration along $H$. By directly taking derivatives and use (2.13) we obtain that outside $H$,

(4.131)

$$d_{D}d_{D}^{c}(\int_{0}^{T_{-}}h(z)dz)=\int_{0}^{T_{-}}d_{D}d_{D}^{c}h(z)dz=\int_{0}^{T_{-}}-\partial_{z}^{2}\tilde{\omega}(z)dz=-\partial_{z}\tilde{\omega}|_{z=T_{-}}+\partial_{z}\tilde{\omega}|_{z=0}$$

By (3.349) and (3.381), the right hand side is given by $-\frac{1}{2}(k_{-}-k_{+})\omega_{D}+\epsilon_{T}$. Now using the asymptotics of $h$ near $P$ in (4.17), one sees that $\hat{h}_{-}$ is bounded near $H$. So the following current equation holds globally on $D$

(4.132)

$$d_{D}d_{D}^{c}\hat{h}_{-}=\epsilon_{T}$$

Now

(4.133)

$$\int_{D}\hat{h}_{-}\omega_{D}^{n-1}=A_{-}\int_{D}\omega_{D}^{n-1}+\int_{D}\int_{0}^{T_{-}}h\omega_{D}^{n-1}dz=B_{T}$$

So by standard elliptic regularity we get the conclusion for $\hat{h}_{-}$. The proof for the other equation is similar. ∎

By the previous Lemma,

(4.136)

$$A_{-}-\int_{T_{-}}^{z}h(u)=\frac{1}{2}\log{\|S_{H}\|}+B_{T}+\int_{z}^{0}h(u)du$$

When $|z|\leq 1$, if we are in the above chart, then

(4.137)

$$\int_{0}^{z}h(u)du=B_{T}+Tz+\frac{1}{2}(\log(r+z)-\log|y|)$$

Since $\log{\|S_{H}\|}=\log|y|+B_{T}$, it follows that

(4.138)

$$\log r_{-}=B_{T}-Tz+\frac{1}{2}\log(r-z)$$

Similarly we get the estimate for $\log r_{+}$.

∎

The first two items are easy consequences of the previous Lemma. For the last item we simply notice that for $C\geq 1$,

(4.139)

$$\int_{-C}^{-1}h(u)du=\int_{-C}^{-1}(T^{2-n}(T+k_{-}u)^{n-1}+\epsilon(u))du\leq CT.$$

∎

By definition $\phi$ is smooth on $Q_{T}\setminus H\times(-\infty,0]$. Using (4.17) it is easy to see that $\phi$ extends to a continuous function on $Q_{T}$. Hence for all fixed $z$, the following equation holds in the sense of currents on $D$

(4.150)

$$T\omega_{D}+d_{D}d_{D}^{c}\phi(z)=\tilde{\omega}(z)-z\partial_{z}\tilde{\omega}(z).$$

Elliptic regularity then implies that $\phi$ is smooth on each slice $\{z\}\times D$ for $z\neq 0$. Now for $z\leq 0$ we can write

(4.151)

$$\phi(z)=\int_{T_{-}}^{z}uhdu+\phi(T_{-}).$$

We then see that $\phi$ is indeed smooth on $Q_{T}\setminus P$. Over the $S^{1}$ fibration $\mathcal{M}$, we know $\phi$ is globally continuous, and it is smooth and satisfies the equation (4.140) on ${\mathcal{M}^{*}}$. Now again by standard theory on pluri-subharmonic functions we conclude the current equation holds on $\mathcal{M}$. Since we know $\omega$ is $C^{2,\alpha}$ in local holomorphic coordinates on $\mathcal{M}$, elliptic regularity gives that $\phi$ is in $C^{4,\alpha}$ in local holomorphic coordinates. This implies that $\phi$ is $C^{3,\alpha}$ in the smooth topology we defined, since we know the holomorphic coordinate functions are $C^{3,\alpha}$. ∎

We have $\phi(T_{-})=\phi(T_{+})-\Psi,$ where

(4.169)

$$\Psi=\int_{T_{-}}^{T_{+}}zhdz.$$

Away from $H$ we have

(4.170)

$$d_{D}d_{D}^{c}\Psi=\int_{T_{-}}^{T_{+}}zd_{D}d_{D}^{c}hdz=-\int_{T_{-}}^{T_{+}}z\partial_{z}^{2}\tilde{\omega}dz$$

Integration by parts we get

(4.171)

$$d_{D}d_{D}^{c}\Psi=(-z\partial_{z}\tilde{\omega}+\tilde{\omega})|^{T_{+}}_{T_{-}}=\epsilon_{T}$$

Notice since there is a factor $z$ in the integrand we do not get residue term at $z=0$. Notice $\Psi$ is continuous on $D$, and the right hand side is smooth on $D$, so elliptic regularity implies that $\Psi$ is indeed smooth on $D$, and the equation holds globally on $D$.