(Taub-NUT) We take $N=2,\mathfrak{n}=1$, and $V=W=\frac{1}{2\sqrt{\mu^{2}+|\eta|^{2}}}+A$, where $A$ is a positive constant. This defines a Calabi-Yau metric whose asymptotic geometry at infinity approaches the constant solution (cf. Example 1.6), with asymptotic circles of length $\frac{2\pi}{\sqrt{A}}$ fibred over a flat 3-dimensional base. Different choices of $A$ define the same metric up to scaling. The first Chern class $c_{1}$ of the $S^{1}$-bundle over $(\mathbb{R}_{\mu}\times\mathbb{C}_{\eta})\setminus\{0\}$ evaluates to $-1$ on any sphere around the origin in $\mathbb{R}_{\mu}\times\mathbb{C}_{\eta}$; equivalently, the 3-current $-\frac{1}{2\pi}dF$ is represented by the origin $0\in\mathbb{R}_{\mu}\times\mathbb{C}_{\eta}$ viewed as a codimension 3 cycle. Written in terms of the delta function,

$$\frac{1}{2\pi}dF=\frac{1}{2\pi}(\frac{\partial^{2}}{\partial\mu^{2}}+4\frac{\partial^{2}}{\partial\eta\partial\bar{\eta}})Vd\mu\wedge d\text{Re}\eta\wedge d\text{Im}{\eta}=-\delta(\mu,\eta,\bar{\eta})d\mu\wedge d\text{Re}\eta\wedge d\text{Im}{\eta}.$$

From the holomorphic perspective, LeBrun [17] observes that the Taub-NUT space is biholomorphic to $\mathbb{C}^{2}$. To see this, recall $\zeta=Vd\mu+\sqrt{-1}\vartheta$ and notice the (1,0) form $\zeta-\frac{\mu}{2\eta\sqrt{\mu^{2}+|\eta|^{2}}}d\eta$ is closed, so locally is the differential of a holomorphic function. The line integrals

$$\log z_{1}=\int\zeta+(\frac{1}{2\eta}-\frac{\mu}{2\eta\sqrt{\mu^{2}+|\eta|^{2}}})d\eta,\quad\log z_{0}=\int-\zeta+(\frac{1}{2\eta}+\frac{\mu}{2\eta\sqrt{\mu^{2}+|\eta|^{2}}})d\eta$$

define holomorphic functions up to $2\pi\sqrt{-1}\mathbb{Z}$ over the regions $\mathbb{R}\times\mathbb{C}\setminus\{\eta=0,\mu\leq 0\}$ and $\mathbb{R}\times\mathbb{C}\setminus\{\eta=0,\mu\geq 0\}$ respectively, so $z_{1}$ and $z_{0}$ are well defined over the respective regions. Since $d\log z_{1}+d\log z_{0}=d\log\eta$ we can normalise $z_{0},z_{1}$ to satisfy the functional equation $z_{1}z_{0}=\eta$, whence $z_{1}$ and $z_{0}$ extend as global holomorphic functions. These coordinates exhibit the biholomorphism to $\mathbb{C}^{2}$. By considering the Hamiltonian vector field acting on $\log z_{1},\log z_{0}$, we idenitfy the $S^{1}$ action as

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

The holomorphic volume form is

$$\Omega=-\sqrt{-1}\zeta\wedge d\eta=-\sqrt{-1}d\log z_{1}\wedge d(z_{1}z_{0})=\sqrt{-1}dz_{0}\wedge dz_{1}.$$

Thus $\eta=z_{0}z_{1}$ defines holomorphic fibration of $\mathbb{C}^{2}$ by affine quadrics. The generic quadric fibre is topologically a cylinder, and metrically is also approaching the flat cylindrical metric near spatial infinity. When $\eta=0$, the quadric fibre degenerates into a union of two complex lines with simple normal crossing, where each line looks metrically like a cylinder with one capped end.