From the proof of Theorem 1.1, it follows that similar results hold in the following more general situation. We leave it for the readers to check the details.

• –

  $p:\mathcal{X}\rightarrow\Delta$ is a proper holomorphic map from an $n+1$ dimensional normal complex analytic variety onto a disc $\Delta$ in $\mathbb{C}$.

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  For $t\neq 0$, $X_{t}\equiv p^{-1}(t)$ is a smooth $n$ dimensional compact complex manifold.

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  $X_{0}\equiv p^{-1}(0)$ is a union of two smooth $n$ dimensional Fano manifolds $Y_{1}$ and $Y_{2}$, and $Y_{1}\cap Y_{2}$ is a smooth $n-1$ dimensional Calabi-Yau manifold $D$.

• –

  $\mathcal{L}$ is a relatively ample holomorphic line bundle on $\mathcal{X}$.

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  Two positive integers $d_{1},d_{2}$, and we denote $k=d_{1}+d_{2}$.

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  Holomorphic sections $f_{,}f_{2},f$ of $\mathcal{L}^{d_{1}},\mathcal{L}^{d_{2}},\mathcal{L}^{d_{1}+d_{2}}$ respectively satisfying

  (8.37)

  $$f_{1}f_{2}+tf=0.$$

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  $Y_{1}\cap Y_{2}$ is a smooth $n-1$ dimensional Calabi-Yau manifold $D$.

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  $\mathcal{X}$ is singular along a smooth divisor $H\subset D$ given by $\{f_{1}=f_{2}=f=t=0\}$. and transverse to $H$ the singularity is modeled on $\{x_{1}x_{2}+tx_{3}=0\}$.

• –

  There is a holomorphic volume form on the smooth locus of $\mathcal{X}$.

Theorem 1.1 can be viewed as understanding the first order expansion of the family of Calabi-Yau metrics on $\widehat{\mathcal{X}}$ near $t=0$. One may ask whether it is possible to obtain a refined asymptotic expansion. In spirit, it is similar to the case of family of hyperbolic metrics on nodal degeneration of Riemann surfaces (See the recent work [MZ18] by Melrose-Zhu), and it is very likely similar techniques will be useful here. We thank Dominic Joyce and Xuwen Zhu for conversations on this.

As is mentioned in the Introduction, it remains an interesting question to directly glue together two Tian-Yau metrics with the same divisor $D$, without a priori assuming the existence of the complex family $\mathcal{X}$. As mentioned in the Introduction, in the case $n=2$ this was done in [HSVZ18] using $SU(2)$-structures, and in the case $n=3$ it is possible to use deformations of $SU(3)$-structures. This would require certain analysis (in particular Liouville theorem) on forms instead of functions. We leave this for future study.

In Section 5, we proved a Liouville Theorem on the Tian-Yau spaces using elementary analysis on special functions. Although not needed in this paper, it is interesting to see if there is a general Fredholm theory for the analysis of the Laplace operator on such spaces. We asked similar questions in the two dimensional case in [HSVZ18].

There is a different class of Tian-Yau spaces, constructed on the complement of a smooth anti-canonical divisor in a projective manifold with trivial normal bundle. In particular the ambient manifold can not be Fano. These spaces have different asymptotics at infinity from the ones we considered in this paper. Namely, they are asymptotically cylindrical. Given a smooth Fano manifold $Y$ and a pencil of anti-canonical divisors with smooth base locus $B$, let $Y^{\prime}$ be the blown-up of $Y$ along $B$, and let $D^{\prime}$ be the proper transform of a smooth element $D$ in the pencil. Then there is such an asymptotically cylindrical Calabi-Yau metric on $Y\setminus B$ (in every Kähler class). Asymptotically cylindrical Tian-Yau spaces have been important ingredients in the twisted connected sum construction of examples of compact $G_{2}$ holonomy manifolds. It is interesting to see whether the ideas of this paper can be used to construct new examples of $G_{2}$ holonomy manifolds by gluing together a suitably twisted circle fibration over various pieces.

As pointed out in Remark 1.1.1, our main result approximately reduces the understanding on the geometry of part of the Calabi-Yau manifolds $(X_{t},\omega_{CY,t})$ (the neck region) for $|t|\ll 1$ to the geometry of the Calabi-Yau metric on the one lower dimensional space $D$. One expects this can possibly lead to an inductive way to study geometry of Calabi-Yau metrics in higher dimensions through iterated degenerations. Correspondingly, it is also interesting to relate the submanifold geometry of the neck region to that of $D$. For example, suppose we have a special Lagrangian fibration on a region in $D$, can we construct special Lagrangian fibrations on the neck which are invariant under the $S^{1}$ action? At the two ends of the neck it is easy to see the pre-image of a special Lagrangian fibration under the projection map is approximately special Lagrangian. Near the singular fibers of the $S^{1}$ fibration the situation is more complicated and one expects certain singular perturbation techniques are needed. There are also similar discussions in [Li19] in the setting of Section 8.1.

In connection with algebro-geometric study of degenerations of Calabi-Yau manifolds, Theorem 1.1 shows that the normalized Gromov-Hausdorff limit in our setting is topologically the same as the essential skeleton of the degeneration $\mathcal{X}$. In the other extreme case, namely, the case of large complex structure limit of Calabi-Yau manifolds, it is a folklore conjecture (by Gross-Wilson and Kontsevich-Soibelman) that the normalized Gromov-Hausdorff limit is topologically the same as the essential skeleton of the degeneration. It is then natural to expect this conjecture may extend to general degenerations. Also it is also an interesting question to understand the algebro-geometric meaning of the normalized limit measure in Theorem 1.1, see [BJ17] for related algebro-geometric work. In the case $n=2$, there is also a plausible connection with the compactification of moduli space of hyperkähler metrics on K3 manifolds , see [OO18]. We leave all these for future exploration.