The proof of Theorem 1.1 consists of roughly three main pieces.

The first piece involves algebraic modification of the family $\mathcal{X}$. Our initial naive strategy is to start with the Tian-Yau metrics on $X_{0}\setminus D=(Y_{1}\setminus D)\cup(Y_{2}\setminus D)$, and graft them to nearby fibers $X_{t}$ for $|t|$ small to get Kähler metrics which are approximately Calabi-Yau. However, the existence of singularities of the total space $\mathcal{X}$ along $H$ imposes difficulties in performing a reasonable construction. So our first step is to modify the family $\mathcal{X}\rightarrow\Delta$ to another family $\widehat{\mathcal{X}}\rightarrow\Delta$ using base change and birational modifications (c.f. Figure 7.1). The new family $\widehat{\mathcal{X}}$ agrees with $\mathcal{X}$ away from $X_{0}$, and the new fiber $\widehat{X}_{0}$ consists of a chain of three components, with the two end components isomorphic to $Y_{1},Y_{2}$ respectively, and the middle component $\mathcal{N}$ is given by a conic bundle over $D$, as a natural hypersurface in the projective bundle $\mathbb{P}(L_{1}\oplus L_{2}\oplus\mathbb{C})$ cut out by the equation $s_{1}s_{2}+s_{3}f(x)=0$. The family of conics degenerate precisely along the divisor $H$ in $D$. The component $\mathcal{N}$ intersects transversally with $Y_{1},Y_{2}$ along $D_{1},D_{2}$, which are naturally isomorphic to $D$. Notice $\widehat{\mathcal{X}}$ is not necessarily smooth. Indeed it has singularities along $D_{1}\cup D_{2}$ which is of codimension two. However it turns out that working with $\widehat{\mathcal{X}}$ is the correct thing to do. This is done in Section 7.1.

The second piece involves the construction of the neck region. We want Calabi-Yau metrics on the smooth locus of the central fiber of $\widehat{\mathcal{X}}$. For the two end components these are provided by the complete Tian-Yau metrics. For the middle component, with a moments’ thought one realizes that it is difficult to construct a complete Calabi-Yau metric on $\mathcal{N}^{0}=\mathcal{N}\setminus(D_{1}\cup D_{2})$. The reason is that if such metric existed, it would have two ends, and Ricci-flatness would imply it must split a line, and this is not quite compatible with the complex geometry of $\mathcal{N}^{0}$. Instead we shall look for a family of incomplete Calabi-Yau metrics defined on larger and larger open subsets in $\mathcal{N}^{0}$. The fact that $\mathcal{N}$ has a natural holomorphic $\mathbb{C}^{*}$ action suggests us to look for Calabi-Yau metrics with $S^{1}$ symmetry.

In complex dimension 2, this is essentially achieved in [HSVZ18] using the classical Gibbons-Hawking ansatz (except we did not identify the underlying complex manifold). In higher dimensions the technical details are more complicated. In Section 2 we discuss a higher dimensional generalization of the Gibbons-Hawking ansatz. The corresponding reduced equation is still non-linear, and by linearization we are lead to study certain solutions to a linear elliptic PDE with singularities along a submanifold. The existence and local regularity of such solutions, which we call Green’s currents, is studied in detail in Section 3. In Section 4 we use these Green’s currents to construct a family of incomplete Kähler metrics on open subsets of $\mathcal{N}^{0}$. The fact that the singularities of the Green’s currents are non-isolated causes difficulties in understanding the regularity of the Kähler metrics. In reality we only prove the metrics are $C^{2,\alpha}$ and this suffices for our purpose. Another difference in higher dimensions is that these metrics are only approximately Calabi-Yau. In Section 4 we study the various rescaled limit geometries for this family of metrics. We also give a formula for the Kähler potential of these Kähler metrics, which is crucial for our gluing construction since we work on the fixed complex family $\widehat{\mathcal{X}}$. In Section 7.3 we graft the incomplete Calabi-Yau metrics constructed in Section 4 and the complete Tian-Yau metrics on $Y_{i}\setminus D$ to $C^{1,\alpha}$ Kähler metrics on $X_{t}$ for $|t|$ sufficiently small, which are approximately Calabi-Yau.

The third piece then involves weighted analysis. This is roughly along the same lines as in [HSVZ18]. Again a new difficult point is the proof of a Liouville theorem on the Tian-Yau spaces. This will be done in Section 5 using elementary analysis of special functions. For readers’ convenience, we also summarize the relevant formulae regarding these special functions in Appendix A. In Section 6 we use the implicit function theorem and weighted estimates to show the family of approximately Calabi-Yau metrics on the neck can be perturbed to genuine Calabi-Yau metrics. Here a subtle point is that we use Neumann boundary condition instead of Dirichlet boundary condition. One can then see directly from this the Gromov-Hausdorff collapsing behavior of the Calabi-Yau metrics. We also discuss the renormalized limit measures.

Notice for the proof of Theorem 1.1 we do not need to use these incomplete Calabi-Yau metrics, but the proof will involve similar arguments. This is explained in Section 7.4.