We begin with some historical remarks about the fundamental problem:

The initial guess was that the only solution is the flat metric. The rationale is that the moduli of compact Calabi-Yau manifolds depends on the cohomology class of the Kähler form and the holomorphic volume form, and since $\mathbb{C}^{n}$ has trivial topology, it seemed that there was no room to admit nontrivial Calabi-Yau metrics. The situation changed when LeBrun first observed that the Taub-NUT metric gives a counterexample on $\mathbb{C}^{2}$ (cf. Section 1.8). Hindsight shows that the necessary amount of nontrivial topology comes from an additional fibration structure. In fact the Taub-NUT metric admits two kinds of fibration structures: a holomorphic fibration $\mathbb{C}^{2}_{z_{0},z_{1}}\xrightarrow{z_{0}z_{1}}\mathbb{C}$ which gives an algebraic perspective, and a circle fibration coming from the Gibbons-Hawking ansatz which gives a transcendental perspective.

In [18] the author realised that if we take the holomorphic fibration one step further, namely if we start from the standard Lefschetz fibration $\mathbb{C}^{3}\xrightarrow{f=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}}\mathbb{C}$ on $\mathbb{C}^{3}$, then we can construct a nontrivial complete Calabi-Yau metric on $\mathbb{C}^{3}$, such that near spatial infinity, the restricted metric on the affine quadric fibres are approximately the Eguchi-Hanson metrics on the fibres, and the horizontal part of the metric is approximately the pullback of the Euclidean metric on the base. This work was soon generalised independently by Conlon-Rochon [3] and Székelyhidi [27], who developed more substantial linear analysis to treat more complicated holomorphic fibrations. In the most general known version, we start from a weighted homogeneous polynomial $f:\mathbb{C}^{n}\to\mathbb{C}$ where $n\geq 3$, such that the only singularities in the fibration $f$ are isolated singularities on the central fibre $f^{-1}(0)$, and we require the weighted cone $f^{-1}(0)$ to admit a conical Calabi-Yau metric whose Reeb vector field action is compatible with the weights. Algebro-geometrically, the singular fibre must have klt singularity, and the requirement for the existence of a conical Calabi-Yau metric imposes a stability condition on the singular fibre. Then by standard results the smoothing fibres $f^{-1}(c)$ are equipped with asymptotically conical Calabi-Yau metrics, which now play the same role as the Eguchi-Hanson metrics played in the $\mathbb{C}^{3}$ example setting. The final output of their theory is a complete Calabi-Yau metric on $\mathbb{C}^{n}$ associated to the fibration $f:\mathbb{C}^{n}\to\mathbb{C}$, equipped with the standard holomorphic volume form.

The most important Riemannian geometric aspect of this infinite class of complete Calabi-Yau metrics is that the volume of metric balls have Euclidean volume growth rate

$$C^{-1}\leq\frac{\text{Vol}(B(r))}{\text{Vol}(B_{\text{Euclid}}^{2n}(r))}\leq 1,\quad r>0.$$

Since these manifolds are Ricci-flat, it makes sense to take the tangent cone at infinity, which is identified as the singular variety $f^{-1}(0)\times\mathbb{C}$ with the product metric, and in particular has the same dimension as $\mathbb{C}^{n}$. This aspect is contrasted with the Taub-NUT metric in complex dimension 2, whose volume growth rate is $\text{Vol}(B(r)\sim r^{3}$ which is not Euclidean. This failure can be traced back to the fact that the singular fibre $z_{0}z_{1}=0$ for the Taub-NUT $\mathbb{C}^{2}$ is not even irreducible, let alone having a Calabi-Yau cone metric.

Furthermore, the metric distance to the origin for these examples on $\mathbb{C}^{n}$ are bi-Hölder equivalent to the standard Euclidean distance, but not uniformly equivalent. This has the consequence that the ring of algebraic functions on these exotic $\mathbb{C}^{n}$ coincides with the ring of holomorphic functions with polynomial growth, but the filtration structure on these functions induced by the growth rate is not the standard filtration.

Now we turn to the new Taub-NUT type Calabi-Yau metric on $\mathbb{C}^{3}$. Like the Taub-NUT $\mathbb{C}^{2}$, it is associated to both a holomorphic fibration structure and a torus fibration structure. The holomorphic fibration is given by $\mathbb{C}^{3}\xrightarrow{z_{0}z_{1}z_{2}}\mathbb{C}$, which may be viewed as a degenerate case where the fibration is allowed to have more severe singularities: here $z_{0}z_{1}z_{2}=0$ is reducible into 3 pieces, and morever its singularity is non-isolated, stretching all the way into spatial infinity. This explains why the Riemannian curvature does not decay at infinity along the locus $\{z_{i}=z_{j}=0\}$, a phenomenon similar to Joyce’s examples of quasi-ALE Calabi-Yau metrics [15]. Another viewpoint is that the generic fibre is stable while the central singular fibre is unstable. Their delicate balance produces a global metric on $\mathbb{C}^{3}$, but the instability near the singular fibre produces large quantum fluctuation effects.

However, the principal novalty of our Taub-NUT type $\mathbb{C}^{3}$ metric comes from the $T^{2}$-fibration structure. An immediate consequence of the fact that 2 spatial dimensions are ‘compactified’, is that the volume growth rate is sub-Euclidean: in fact $\text{Vol}(B(r)\sim r^{4}$ and the tangent cone at infinity is the flat $\mathbb{R}^{4}$. This sub-Euclidean growth is otherwise known as collapsing in Riemannian geometry.

An important conceptual feature of real tori is that they are inherently transcendental objects, tied up intimately with the fundamental functions $\log$ and $\exp$; we saw the pervasive presence of such transcendental functions in Section 2.4 in the metric asymptote. Another manifestation of this is that the ring of algebraic functions on the Taub-NUT type $\mathbb{C}^{3}$ is defined by holomorphic functions with an exponential type growth condition, rather than the more familiar polynomial growth which is the expected feature in the Euclidean volume growth situation.

The evidence suggests that the full mystery of complete Calabi-Yau metrics on $\mathbb{C}^{n}$ involves at least 3 fundamental phenomena:

• •

  holomorphic fibrations with a suitable notion of stability, which is associated with Euclidean volume growth rate and polynomial growth rate on holomorphic functions.

• •

  torus fibrations, which is associated with collapsing phenomenon and exponential growth rate on holomorphic functions.

• •

  an additional layer of combinatorial complexity involving iterative fibrations (cf. subsection 2.11.3 for the flavour).

This picture seems to fit well with Kontsevich and Soibelman’s conjectural picture for collapsing compact Calabi-Yau manifolds (cf. Chapter 2, 3 in [16]). The relation between the two situations will be further explained in subsection 2.11.4.