The relationship between the parameters $t$ and $T$ can be determined by studying the matching between the Tian-Yau ends and the neck.

In our setting we need to first normalize the Tian-Yau metrics $\omega_{TY,i}$ on $Z_{i}$ (as defined in Section 7.2). We define

Then we have

By definition we can write

where

with

and

for all $k\geq 0$, where the derivatives and norms are taken with respect to the Tian-Yau metric itself (which is equivalent to taking with respect to the metric $\omega_{Z_{j}}$).

Now on the neck $\mathcal{M}_{T}$ we have the asymptotics of the Kähler potential given in Section 4.2. By the discussion there we identify $\mathcal{M}_{T}$ with an open set in $\mathcal{N}^{0}$, and we can write

with

where

and for $|z|\geq 1$ we have

Now on $\mathcal{M}_{T}$ for $|t|$ small

which gives

So if we want to graft the metrics on the three components of $\widehat{X}_{0}$ to nearby $\widehat{X}_{t}$, then we need

Similarly at the positive end we need

This suggests that we should choose

Given $|t|$ small we can find $T$ big so that (7.72) holds. It is not necessary that $T$ is uniquely determined by $t$, but we shall always fix a particular choice for each $t$ throughout this section so that (7.72) holds. With this choice it is easy to see that

From now on, we will fix weight parameters in the definition of weight spaces, which allows us to prove the uniform injectivity estimate in Proposition 7.15 and apply the implicit function theorem to complete the proof the main theorem in Section 7.4. The parameters $\delta$, $\mu$, $\nu$ are fixed as follows (similar to the specification of the parameters in Section 6.1):

1. (GP1)

   (Fix $\nu$) The parameter $\nu\in\mathbb{R}$ is chosen such that

   (7.74)

   $\displaystyle\nu\in(-1,0).$

2. (GP2)

   (Fix $\alpha$) The Hölder constant $\alpha\in(0,1)$ is chosen such that

   (7.75)

   $\displaystyle\nu+\alpha<0.$

3. (GP3)

   (Fix $\delta$) The constant $\delta>0$ is chosen such that

   (7.76)

   $$0<\delta<\delta_{G}\equiv\frac{1}{n\cdot(|k_{-}|+|k_{+}|)}\cdot\min\{\delta_{e},\delta_{Z_{1}},\delta_{Z_{2}},\epsilon_{Z_{1}},\epsilon_{Z_{2}},\sqrt{\lambda_{D}}\},$$

   where $\sqrt{\lambda_{D}}$ is in Lemma 6.7 (Liouville theorem on $Q$), $\delta_{e}>0$ is in Proposition 4.23, $\delta_{Z_{1}},\delta_{Z_{2}}$ are the constants in Proposition 7.4 applied to $Z_{1},Z_{2}$, and $\epsilon_{Z_{1}},\epsilon_{Z_{2}}$ are the constants in Theorem 5.2 applied to $Z_{1},Z_{2}$.

4. (GP4)

   (Fix $\mu$) The parameter $\mu>0$ is chosen as

   (7.77)

   $$\mu=(1-\frac{1}{n})(\nu+2+\alpha).$$

As a comparison, on the neck region $\mathcal{M}_{T}$, the corresponding choice of parameters are given in (6.10), (6.11), (6.12) and (6.13).

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.