A family of Calabi-Yau metrics $(X_{t},\omega_{t})$ living on a flat family of compact Calabi-Yau manifolds is said to be collapsing if there is no uniform estimate

$$\text{Vol}_{\omega_{t}}(B(x_{t},r))\geq\kappa r^{\dim_{\mathbb{R}}X_{t}},\quad\forall x_{t}\in X_{t},\forall 0<r<\text{diam}(X_{t}),\quad\kappa>0.$$

Two well-studied basic mechanisms for collapsing are:

• •

  Fix the complex structure of $X_{t}=X$ and a reference Kähler class $[\omega_{X}]$ on $X$. Assume there is a holomorphic fibration $f:X\to Y$ to a lower dimensional Kähler manifold $Y$ with Kähler class $[\omega_{Y}]$. Then we take $\omega_{t}$ to be the Calabi-Yau metric in the class $[t\omega_{X}+f^{*}\omega_{Y}]$, where $t\ll 1$. Crucially the fibre volume is cohomologically determined, and the fibre length scale is much smaller compared to the diameter of the base (cf. [29]).

• •

  Fix a polarisation on a 1-parameter flat family $X_{t}$, which prescribes the Kähler class, and assume there is a holomorphic volume form $\Omega_{\mathcal{X}}$ on the total space, so there are induced holomorphic volume forms $\Omega_{t}$ on $X_{t}$ depending on $t$ in a holomorphic way. Then we study the Calabi-Yau metrics $\omega_{t}$ as we allow the complex structure to degenerate, in such a way that the central fibre $X_{0}$ has worse than klt singularities.

Kontsevich and Soibelman observe that in the polarised collapsing situation, the resolution of singularity implies

$$\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}=C(\log|t|)^{m}|t|^{k}(1+o(1)),$$

where $C$ is some constant, $k$ is an integer which can be taken as zero by adjusting $\Omega_{t}$, and $0<m\leq\dim_{\mathbb{C}}X_{t}$ if the central fibre has worse than klt singularities . The integer $m$ is determined by Hodge theory for the degeneration. The curious presence of the transcendental factor $(\log|t|)^{m}$ is interpreted by Kontsevich and Soilbelman as indicating the presence of an $m$-dimensional torus fibration; in the special case of the large complex structure limit $\dim_{\mathbb{C}}X_{t}=m$ they predict a $T^{m}$-fibration, which is compatible with the SYZ proposal (cf. Section 3.1 [16]). Transcendental phenomenon is captured by non-archimdean analysis. They also suggest that collapsing phenomenon in general involves an iterative fibration structure, based on motivations from conformal field theory (cf. Section 2.3 in [16]).

There is a simple conceptual relation between collapsing families of Calabi-Yau metrics $(X_{t},\omega_{t})$ on compact manifolds, and non-compact complete Calabi-Yau metrics. If we scale the metrics such that $\sup|\text{Rm}|=1$ inside a region of interest, then there is a dichotomy:

• •

  If the injectivity radius is bounded below, then the pointed Gromov-Hausdorff limit is a smooth complete Calabi-Yau manifold (a ‘complete bubble’).

• •

  If the injectivity is not bounded below, then we are in the situation of collapsing with bounded curvature, and we should instead look at the covering geometry.

It often happens that the original $X_{t}$ has a natural fibration structure, which would strongly motivate a complete Calabi-Yau manifold with the same kind of fibration structure.

To explain the role of the Euclidean volume growth condition for the complete Calabi-Yau manifolds, we recall a basic fact in Riemannian geometry called Bishop-Gromov monotonicity, which implies that for Ricci-flat manifolds of real dimension $N$, the normalised volume

$$\frac{\text{Vol}(B(x,r))}{\text{Vol}(B^{N}_{\text{ Euclid }}(0,r))}$$

is a decreasing function of the radius $r$. Thus if one has a geometric reason for the non-collapsing bound $\text{Vol}(B(x,R))\geq\kappa R^{N}$ at a particular distance scale $R$, then in all smaller scales $r$ we have also $\text{Vol}(B(x,r))\geq\kappa r^{N}$. In particular, even though a family of Calabi-Yau metric is collapsing globally, it can happen that in a local region of interest the non-collapsing bound holds, so the complete bubble inherits the Euclidean volume growth condition. The reader is referred to the author’s papers [19][20] for concrete examples where this phenomenon happens.

Finally, focusing on complex dimension 3, recall from subsection 2.11.2 that the Taub-NUT type metrics on $\mathbb{C}^{3}$ are expected to be primary objects, while the conjectural multi-Taub-NUT type metrics are composite objects which naturally arise in high dimensional families. We suggest that this means the Taub-NUT type metric on $\mathbb{C}^{3}$ typically occurs as a complete bubble in a suitably generic 1-parameter collapsing family of compact Calabi-Yau metrics when the Euclidean volume growth condition fails, while most other complete bubbles are relevant for multi-parameter degenerations.