We will divide a neighborhood of $\widehat{X}_{0}$ into various regions (c.f. Figure 7.2)

• •

  Region $\bf{I}$ is given by $2|s_{3}|\geq\max(|s_{1}|,|s_{2}|)$;

• •

  Region $\bf{II}_{-}$ is given by $s_{1}\neq 0$, and $|s_{3}|\leq 2|s_{1}|,-\log|f_{2}|\geq-\frac{d_{1}}{2}\log|t|$;

• •

  Region $\bf{III}_{-}$ is given by $s_{1}\neq 0$, and $|f_{2}|\leq 1/2$, $-\log|f_{2}|\leq-\frac{d_{1}}{2}\log|t|+1$;

• •

  Region $\bf{IV}_{-}$ is given by $s_{1}\neq 0$ and $|f_{2}|\geq 1/4$;

• •

  Region $\bf{II}_{+}$ is given by $s_{2}\neq 0$, and $|s_{3}|\leq 2|s_{2}|,-\log|f_{1}|\geq-\frac{d_{2}}{2}\log|t|$;

• •

  Region $\bf{III}_{+}$ is given by $s_{2}\neq 0$ and $|f_{1}|\leq 1/2$, $-\log|f_{1}|\geq-\frac{d_{2}}{2}\log|t|+1$;

• •

  Region $\bf{IV}_{+}$ is given by $s_{2}\neq 0$ and $|f_{1}|\geq 1/4$

For all $|t|$ sufficiently small, then we also get a division of $\widehat{X}_{t}$ into 7 regions. Notice we have non-empty intersections between these regions and we shall need a cut-off (gluing) on the overlap.

For the convenience of later analysis, we now fix a finite cover $\mathcal{U}=\{U_{\beta}^{1},U_{\gamma}^{2},U_{\mathcal{N}},U_{-},U_{+}\}$ of a neighborhood of $\widehat{X}_{0}$ in $\widehat{\mathcal{X}}$ obtained as follows.

We first cover a neighborhood of $D_{1}$. Given any point in $(x,t,[s_{1}:s_{2}:s_{3}])\in D_{1}$, we have $t=f_{1}(x)=f_{2}(x)=s_{2}=s_{3}=0,s_{1}\neq 0$. On the open subset $\{x_{j}\neq 0\}$ in $\mathbb{C}\mathbb{P}^{n+1}$, we can view $\sigma=x_{j}$ as a trivialization of $\mathcal{O}(1)$. Without loss of generality we may assume $j=0$. Then we get affine coordinates $\{u_{i}=x_{i}/x_{0}(i=1,\cdots,n+1)\}$, and we can $v_{1}=f_{1}(u)$ and $v_{2}=f_{2}(u)$ as local holomorphic functions on $\mathbb{C}\mathbb{P}^{n+1}$. Further without loss of generality we can assume $\{v_{1},v_{2},u_{i}=x_{i}/x_{0}(i=3,\cdots)\}$ yield local holomorphic coordinates in a neighborhood of $x$ in $\mathbb{C}\mathbb{P}^{n+1}$. Correspondingly we can pull-back these to local holomorphic functions on the projective bundle $\mathbb{P}(\mathcal{O}(d_{2})\oplus\mathcal{O}(d_{1})\oplus\mathbb{C})$. As before we also introduce local holomorphic functions $\zeta_{3}=s_{3}/s_{1},\zeta_{2}=s_{2}/s_{1}$ on the projective bundle, and the space $\widehat{\mathcal{X}}$ is then defined by the equations as in (7.21), which essentially reduces to one relation $v_{2}\zeta_{3}=t^{d_{1}}$ in the three variables $v_{2},\zeta_{3},t$. We denote by $U_{\beta}^{1}$ an open subset in $\widehat{\mathcal{X}}$ defined by the inequalities $|\zeta_{3}|<3|\sigma|^{d_{2}}$, $|v_{2}|<3|\sigma|^{d_{2}}$, and $|u_{i}|<C(i=3,\cdots)$ for some fixed $C>0$. Call such an open set $U_{\beta}^{1}$, and denote the trivializing section $\sigma$ by $\sigma_{\beta}^{1}$. For $|t|$ small, $U^{1}_{\beta,t}\equiv U^{1}_{\beta}\cap\widehat{X}_{t}$ is then defined by the equation $v_{2}\zeta_{3}=t^{d_{1}}$.

We have the natural projection maps

Then the union of images $\pi^{1,D}_{\beta}(U^{1}_{\beta,t})$ form an open cover of $D$. By compactness we can choose and then fix finitely many of them which also cover $D$, and we put these $U^{1}_{\beta}$’s in $\mathcal{U}$. Then we obtain also a cover of a neighborhood of $D_{1}$ in $Y_{1}$ by $\{U_{\beta,0}\cap Y_{1}\}$ and a cover of a neighborhood of $D_{1}$ in $\mathcal{N}$ by $\{U_{\beta,0}\cap\mathcal{N}\}$ so that on each element in the cover we have holomorphic coordinates. Without loss of generality we may assume these cover the neighborhood defined by $|f_{2}|\leq 3$ and $|s_{3}/s_{1}|\leq 3$. So in particular they contain Regions $\bf{II}_{-}$ and $\bf{III}_{-}$.

We can do the same with $D_{2}$, and add the corresponding elements $U_{\gamma}^{2}$ to $\mathcal{U}$. Now away from $D_{1}\cup D_{2}$ we may find a trivialization of the fibration $\widehat{\mathcal{X}}\rightarrow\Delta$. So we can obtain three open subsets of $\widehat{\mathcal{X}}$, each of which has a differentiable trivialization over $\Delta$. Call these $U_{\mathcal{N}}$, $U_{-}$, $U_{+}$. Adding these to $\mathcal{U}$ we then obtain an open cover of a neighborhood of $\widehat{X}_{0}$. Over each of the three subsets we also have the projection map $\pi_{-},\pi_{+}$, and $\pi_{{\mathcal{N}}}$ from them into $\widehat{X}_{0}\setminus(D_{1}\cup D_{2})$. We may assume that Region $\bf{I}$ is contained in $U_{\mathcal{N}}$, Region $\bf{IV}_{\pm}$ is contained in $U_{\pm}$.

We shall fix a partition of unity $\chi_{\beta}^{1}$ of $D$ subordinate to the cover $\pi_{\beta}^{1,D}(U^{1}_{\beta,t})$, and ${\chi_{\gamma}^{2}}$ of $D$ subordinate to the cover $\pi_{\gamma}^{1,D}(U^{2}_{\gamma,t})$. We view these naturally as functions on the corresponding $U^{1}_{\beta,t}$ and $U^{2}_{\gamma,t}$, though not compactly supported (along the fiber direction).

Below we define the approximately Calabi-Yau metric $\omega(t)$ on $(\widehat{X}_{t},\Gamma_{t})$ for each region above, and we also define the weight function $\rho_{t}(\bm{x})$ simultaneously and measure the error of the Calabi-Yau equation in the weighted sense.

Obviously $\mathrm{Err}_{t}=0$ if and only if $\omega(t)$ is Calabi-Yau. Also in the meantime we discuss the gluing in the intersection of neighboring regions.

Region $\bf{I}$. In this region we define

where

Using the fixed diffeomorphism $\pi_{\mathcal{N}}$ we may view the Kähler structures $(\omega(t),\Omega(t)=\Gamma_{t})$ on $\widehat{X}_{t}$ as a perturbation of the Kähler structure $(\omega_{T},\Omega_{0})$ on $\mathcal{N}^{0}$.

Notice by Corollary 4.11.1 it is not difficult to see that $\bf{I}\cap\mathcal{N}$ is contained in the union $\bf{I}_{1}\cup\bf{I}_{2}$ (as defined in Section 4.4). So we can define

and then use (4.273) to define the weight function $\rho_{t}(\bm{x})$. We can then apply Proposition 4.24 to conclude that

Then by Proposition 4.23 we get an error estimate

At the two ends of ${\bf{I}}\cap{\widehat{X}_{t}}$, we can write down the metric $\omega(t)$ in potential form. In the negative end we have $f_{2}\neq 0$, so we can write

and

where

Similarly at the positive end we have

where

Region $\bf{IV}_{\pm}$. We only consider the region $\bf{IV}_{-}$, and the other region is similar. We define

Then for $|t|$ small we can view $(\widehat{X}_{t}\cap{\bf{IV}_{-}},\omega(t))$ as a perturbation of the Tian-Yau metric $\tilde{\omega}_{TY,1}$. It is easy to see that in the intersection $\widehat{X}_{t}\cap\bf{IV}_{-}$, for all $k\geq 0$ we have

To define the weight we let

and then define $\rho_{t}(\bm{x})$ as in (4.273).Then we obtain that

We also have by assumption the asymptotics at the end

Region $\bf{II}_{\pm}$. Again we only consider the region $\bf{II}_{-}$. We define

where

We need the following Lemma.

Now we define the weight function $\rho_{t}$. We first define

Then we define the weight function $\rho_{t}(\bm{x})$ as (4.273). Notice we have that on ${\bf{II}_{-}}\cap\widehat{X}_{t}\cap U_{\beta}^{1}$,

From this we get that

Now we understand the holomorphic volume form. Using (7.35) we get that

where $H$ is a holomorphic function in $\zeta_{3},v_{2},w_{2},\cdots,w_{n-1}$, and its derivatives is of order $\underline{\epsilon}_{T^{2}}$ in these coordinates. Then we again apply weighted Schauder estimates to get that

So by Proposition 4.23 we obtain

Notice ${\bf{II}_{-}}\cap\widehat{X}_{t}$ has two ends. Along one end it is close to the negative end of Region $\bf{I}$.

By Proposition 7.8, we can easily glue the the potentials in Region $\bf{I}_{-}$ and $\bf{II}_{-}$, using a simple cut-off function of the form

where $\chi$ is a cut-off function in $s$ satisfying

Along the other end, Region $\bf{II}_{-}$ is close to the region $\bf{III}_{-}$.

Region $\bf{III}_{\pm}$. Again we only consider the Region $\bf{III}_{-}$. The discussion here is very similar to the case of Region $\bf{II}_{-}$ so we will be sketchy. We define

where

The proof is very similar to the proof of Proposition 7.7, except one compares with the cylindrical metric and uses Lemma 7.5. We omit the details.

To define the weight, we also define the function $L_{t}$ by setting

and correspondingly the weight $\rho_{t}$ using (4.273).

Similar to the case of Region $\bf{II}_{-}$ we have the holomorphic volume form

where $H$ is a holomorphic function in $v_{2},\zeta_{3},w_{2},\cdots,w_{n-1}$ and is of order $\underline{\epsilon}_{T^{2}}$ in these coordinates. We get

Region $\bf{III}_{-}$ has two ends. One end intersects Region $\bf{IV}-$.

This is fairly easy to see, by working in a fixed $U^{1}_{\beta}$.

The other end is close to the Region $\bf{II}_{-}$.

To see this we only need to work in a fixed $U_{\beta}^{1}$ and use the asymptotics of the Tian-Yau metric $\tilde{\omega}_{TY,1}$.

Now by Proposition 7.9 and 7.12, we can choose a cut-off function to glue together $\phi_{t,\bf{II}_{-}}$ and $\phi_{t,\bf{III}_{-}}$. Similarly we may also glue the corresponding weight function $\rho_{t}(\bm{x})$. Here we need to use (7.126), the fact that

and the relation between $|t|$ and $T$ (7.72).

We also choose a cut-off function to glue together $\phi_{t,\bf{III}_{-}}$ and $\phi_{t,\bf{IV}_{-}}$ and also the corresponding weight function $\rho_{t}(\bm{x})$.

Similarly we can define the metrics $\omega(t)$ on $\bf{II}_{+},\bf{III}_{+},\bf{IV}_{+}$ and glue together in the intersections and also glue the weight functions.

To sum up, we have constructed a family of $C^{1,\alpha}$ Kähler metrics $\omega(t)$ on $\widehat{X}_{t}$ for $|t|$ small such that in the above defined weighted norm