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 $\cup\{z_{i}=z_{j}=z_{k}\}$.

  …

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  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.