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.

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  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.

A gravitational instanton is a complete non-compact hyperKähler 4-manifold with $\int|\text{Rm}|^{2}d\text{Vol}<\infty$. The theory of gravitational instantons is very rich, with important contributions from Kronheimer, Atiyah, Hitchin, Hein, and many others. Recent breakthrough made by Chen and Chen [1] is a decisive step towards a complete classification. A conspicuous feature of this classification program is the crucial role played by the volume growth rate. In the Euclidean volume growth rate case, these are the ALE metrics (‘asymptotically locally Euclidean’) classified by Kronheimer. In the sub-Euclidean volume growth case, in all known situations the asymptotic geometry near infinity is approximately a flat torus fibration over a flat base.

We will not attempt to review this extensive literature, but limit ourselves to examine a simple class of examples known as multi-Taub-NUT metrics. In the Gibbons-Hawking coordinates (cf. Section 1.2), this is given by the potential

$$V=A+\sum_{i=1}^{k}\frac{1}{2\sqrt{|\mu-\mu_{i}|^{2}+|\eta-\eta_{i}|^{2}}},$$

where $(\mu_{i},\eta_{i})$ are disjoint given points on the base $\mathbb{R}^{3}=\mathbb{R}_{\mu}\oplus\mathbb{C}_{\eta}$, and $k\geq 1$. The asymptotic geometry is given by a degree $k$ circle bundle over the complement of a compact region in $\mathbb{R}^{3}$, whose circle fibres have approximate length $2\pi A^{-1/2}$. This behaviour is known as asymptotically locally flat, or ALF for short. The $k=1$ case is the usual Taub-NUT metric.

Let’s assume for convenience that the $\eta_{i}$ are all distinct, which is the generic situation. From the holomorphic perspective, the multi-Taub-NUT metrics lives on the smooth algebraic varieties

$$\{z_{0}z_{1}=F(\eta)=(\eta-\eta_{1})(\eta-\eta_{2})\ldots(\eta-\eta_{k})\}\subset\mathbb{C}^{3}_{z_{0},z_{1},\eta},$$

with nowhere vanishing holomorphic volume form $\Omega=\frac{1}{F^{\prime}(\eta)}\sqrt{-1}dz_{0}\wedge dz_{1}$, and the circle action is

$$e^{i\theta}\cdot(z_{0},z_{1})=(e^{-i\theta}z_{0},e^{i\theta}z_{1}),$$

which ensures $\iota_{\frac{\partial}{\partial\theta}}\Omega=d\eta$.

A crucial aspect of multi-Taub-NUT metrics is that they come in a moduli space, determined by the positions of $(\mu_{i},\eta_{i})$. In general, the fact that a family of geometric objects has natural moduli indicates the possibility that in some degenerate limit they decompose into more primary objects, and the parameters in the moduli comes from the parameters in these building blocks and the combinatorics of the gluing construction. This is the case when the spatial separation distance of the monopole points $(\mu_{i},\eta_{i})\in\mathbb{R}^{3}$ is far larger than the circle length parameter $A^{-1/2}$. Then we can view the multi-Taub-NUT metric as obtained from gluing $k$ copies of the Taub-NUT metrics, whose curvature centres are far separated and therefore whose mutual interaction is weak.

Now we can try to push this story to higher dimensions. The natural generalisation of complete hyperKähler 4-folds is complete Calabi-Yau manifolds. Since in our Taub-NUT type $\mathbb{C}^{3}$ example the Riemannian curvature does not decay at infinity along $\mathfrak{D}_{i}$, the total $L^{2}$-curvature integral is infinite. Finding the correct generalised notion of finite curvature condition is clearly fundamental to any classification program. We do not fully understand what this notion is. A tentative idea compatible with Chen and Chen’s work [1] and our Taub-NUT type $\mathbb{C}^{3}$ example is to require that $|\text{Rm}|\leq C$ globally and $|\text{Rm}|(x)\leq C\text{dist}(x,0)^{-2-\epsilon}$ for some $\epsilon>0$ in the generic region.

In the direction of constructing more examples, we comment that Hein’s existence package is by no means limited to the case of $\mathbb{C}^{n}$. Focusing on complex dimension 3, the distinguished role of our Taub-NUT type metrics on $\mathbb{C}^{3}$ is instead that they are more primary objects, and in particular ought to have a more rigid moduli space, than most of the other 3-dimensional complete Calabi-Yau metrics with similar behaviours. It is perhaps best to illustrate this by a conjectural example which generalises the multi-Taub-NUT metrics.

Let $\eta_{i}$ be all distinct and take the smooth algebraic varieties

$$\{z_{0}z_{1}z_{2}=F(\eta)=(\eta-\eta_{1})(\eta-\eta_{2})\ldots(\eta-\eta_{k})\}\subset\mathbb{C}^{4}_{z_{0},z_{1},z_{2},\eta},$$

with nowhere vanishing holomorphic volume form $\Omega=\frac{-1}{F^{\prime}(\eta)}dz_{0}\wedge dz_{1}\wedge dz_{2}$. These admit a $T^{2}$-action

$$e^{i\theta_{1}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{1}}z_{0},e^{i\theta_{1}}z_{1},z_{2}),\quad e^{i\theta_{2}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{2}}z_{0},z_{1},e^{i\theta_{2}}z_{2})$$

which ensures $\Omega(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}})=d\eta$. It seems likely that Hein’s package can be made to provide a multi-parameter family of complete Calabi-Yau metrics on these varieties. Morever, when the $T^{2}$-fibres have much smaller lengths compared to the spatial separation of $\eta_{i}$, then the author expects such metrics to have a gluing description in terms of our Taub-NUT type metric on $\mathbb{C}^{3}$. On the other extreme, if we allow $\eta_{i}$ to collide, then we may see new metric behaviours not yet understood in the literature.

We now discuss the problems of generalising the Taub-NUT type $\mathbb{C}^{3}$ to higher dimensional exotic metrics on $\mathbb{C}^{n}$. The key issue seems to be an extra layer of combinatorial complexity of recursive nature. This calls for a theory which deals with linear analysis on quasi-ALF geometry. Roughly put, a quasi-ALF geometry of complexity 1 asymptotically looks like a flat torus fibration over a flat base. A quasi-ALF geometry of complexity $k$ is a singular torus fibration, whose asymptotic behaviour away from the neighbourhood of a lower dimensional stratified singular set looks ALF, and whose behaviour transverse to the singular locus is modelled on a quasi-ALF geometry of complexity $k-1$. We shall not attempt to make a formal definition, but merely point out that theories of a very similar flavour are much studied, such as QALE spaces by Joyce [15], and QAC spaces by Degeratu and Mazzeo [4].

A conjectural example which illustrates the main ideas is the direct generalisation of our Taub-NUT type metric to $\mathbb{C}^{n}$ with $n\geq 4$. We take the holomorphic fibration

$$\mathbb{C}^{n}\xrightarrow{\eta=z_{0}z_{1}\ldots z_{n-1}}\mathbb{C}_{\eta},\quad\Omega=\sqrt{-1}^{n-1}dz_{0}\wedge dz_{1}\ldots\wedge dz_{n-1}.$$

which admits the action by the diagonal torus $T^{n-1}\subset\text{SL}(n,\mathbb{C})$. The asymptotic geometry is as follows:

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  Far away from $\cup\{z_{i}=z_{j}=0\}$, the metric looks like a flat $T^{n-1}$-fibration over a flat base. In the holomorphic persepcitive, the fibres of $\eta=z_{0}\ldots z_{n-1}$ have a almost flat cylindrical metric on $(\mathbb{C}^{*})^{n-1}$, and the horizontal part of the metric looks like the pullback of a Euclidean metric on $\mathbb{C}_{\eta}$.

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  Near $\cup\{z_{i}=z_{j}=0\}$ but far from $\cup\{z_{i}=z_{j}=z_{k}\}$, we see the Taub-NUT metric appearing in the transverse direction to $\cup\{z_{i}=z_{j}=0\}$.

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  Near $\cup\{z_{i}=z_{j}=z_{k}\}$ but far from the intersection of 4 coordinate hyperplanes, we see the Taub-NUT type $\mathbb{C}^{3}$ appearing in the transverse direction to $\cup\{z_{i}=z_{j}=z_{k}\}$.

  …

• •

  Near $\{z_{1}=\ldots z_{n-1}=0\}$ but far from $\{z_{0}=0\}$, we see the conjectural metric on $\mathbb{C}^{n-1}$ appearing in the transverse direction.

The point is that if one has a sufficiently powerful linear theory which could correct the initial volume form errors to have faster than quadratic decay near infinity, then one can invoke Hein’s package to produce a global Calabi-Yau metric. The whole construction follows a clearly inductive pattern.

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:

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  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]).

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  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:

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

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  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.