In the classical Gibbons-Hawking ansatz, to get interesting topology one often needs to allow the $S^{1}$ action to have fixed points. This corresponds to the harmonic function $V$ having Dirac type singularities. For the convenience of later discussion we shall briefly recall the relevant formulae in this model situation, using our description with a preferred complex structure.

We start with $X=\mathbb{C}^{2}$, with standard holomorphic coordinates $(u_{1},u_{2})$, and flat Kähler metric

(2.41)

$$\begin{cases}\omega_{\mathbb{C}^{2}}=\frac{\sqrt{-1}}{2}(du_{1}\wedge d\bar{u}_{1}+du_{2}\wedge d\bar{u}_{2})\\ \Omega_{\mathbb{C}^{2}}=du_{1}\wedge du_{2}.\end{cases}$$

Consider the $S^{1}$ action on $\mathbb{C}^{2}$

(2.42)

$$e^{\sqrt{-1}t}\cdot(u_{1},u_{2})\equiv(e^{-\sqrt{-1}t}u_{1},e^{\sqrt{-1}t}u_{2}).$$

with infinitesimal generator

(2.43)

$$\partial_{t}=-\sqrt{-1}(u_{1}\partial_{u_{1}}-u_{2}\partial_{u_{2}})+\sqrt{-1}(\bar{u}_{1}\partial_{\bar{u}_{1}}-\bar{u}_{2}\partial_{\bar{u}_{2}}).$$

Then we have a moment map $z$ for the $S^{1}$ action with respect to $\omega_{\mathbb{C}^{2}}$ and a complex moment map $y$ for the complexified $\mathbb{C}^{*}$ action with respect to $\Omega_{\mathbb{C}^{2}}$. Together we obtain the standard Hopf map $\pi:\mathbb{C}^{2}\rightarrow Q_{0}\equiv\mathbb{C}\oplus\mathbb{R}$

(2.44)

$$\begin{cases}z=\frac{1}{2}(|u_{1}|^{2}-|u_{2}|^{2})\\ y=u_{1}u_{2}.\end{cases}$$

Then the holomorphic quotient is $D_{0}=\mathbb{C}$ with holomorphic coordinate $y=y_{1}+\sqrt{-1}y_{2}$, and we can calculate that

(2.45)

$$\begin{cases}\tilde{\omega}_{0}=\frac{\sqrt{-1}}{4r}dy\wedge d\bar{y}\\ \Omega_{0}=dy\\ h_{0}=\frac{1}{2r}\\ V=\frac{1}{2r}\end{cases}$$

where $r=\sqrt{y_{1}^{2}+y_{2}^{2}+z^{2}}$ is the standard radial function on $Q_{0}$, and we have the relation

(2.46)

$$r=\frac{1}{2}(|u_{1}|^{2}+|u_{2}|^{2}).$$

The connection 1-form $\Theta_{0}$ on $\mathbb{C}^{2}$ can also be written down explicitly as

(2.47)

$$\Theta_{0}=h\cdot Jdz=\sqrt{-1}\frac{u_{1}d\bar{u}_{1}-\bar{u}_{1}du_{1}+\bar{u}_{2}du_{2}-u_{2}d\bar{u}_{2}}{2(|u_{1}|^{2}+|u_{2}|^{2})}.$$

Define the curvature 2-form on $Q_{0}$

(2.48)

$$\Upsilon_{0}\equiv\partial_{z}\tilde{\omega}_{0}-dz\wedge d_{\mathbb{C}}^{c}h_{0}$$

So we have

(2.49)

$$\Upsilon_{0}=-\frac{\sqrt{-1}}{4r^{3}}(zdy\wedge d\bar{y}+yd\bar{y}\wedge dz-\bar{y}dy\wedge dz)$$

From our above discussion we have the following holds

(2.50)

$$\begin{cases}d\Theta_{0}=\Upsilon_{0}\\ \tilde{\omega}_{0}+dz\wedge\Theta_{0}=\omega_{\mathbb{C}^{2}}\end{cases}$$

where we have implicitly viewed a form on $Q_{0}$ as a form on $\mathbb{C}^{2}$ using the pull-back $\pi^{*}$. In other words, the flat metric on $\mathbb{C}^{2}$ together with the above $S^{1}$ action can be recovered via the Gibbons-Hawking ansatz applied to the function $V=\frac{1}{2r}$ on $Q_{0}=\mathbb{C}\oplus\mathbb{R}$.

Now the above flat metric admits a one-parameter non-flat perturbation, corresponding to replacing $V$ by $V+T$ for a positive constant $T$. Correspondingly we have

(2.51)

$$\begin{cases}\tilde{\omega}_{0,T}=(\frac{1}{2r}+T)\frac{\sqrt{-1}}{2}dy\wedge d\bar{y}\\ h_{0,T}=\frac{1}{2r}+T\\ \end{cases}$$

This yields a family of Taub-NUT metrics $(\omega_{TN,T},\Omega_{TN,T})$ on $\mathbb{R}^{4}$ with

(2.52)

$$\begin{cases}\omega_{TN,T}\equiv(\frac{1}{2r}+T)\frac{\sqrt{-1}}{2}dy\wedge d\bar{y}+dz\wedge\Theta_{0}\\ \Omega_{TN,T}\equiv\sqrt{-1}((\frac{1}{2r}+T)dz+\Theta_{0})\wedge dy.\end{cases}$$

We can still view these metrics as defined on $\mathbb{R}^{4}$ with coordinates $u_{1},u_{2},\bar{u}_{1},\bar{u}_{2}$ via the above Hopf map, but the coordinate functions $u_{1},u_{2}$ are no longer holomorphic. Indeed one can write down explicitly the holomorphic volume form

(2.53)

$$\Omega_{TN,T}=du_{1}\wedge du_{2}+\frac{T}{2}dz\wedge dy.$$

We also have

(2.54)

$$h_{0,T}dz+\sqrt{-1}\Theta_{0}=Tdz+\frac{1}{2}(\frac{du_{1}}{u_{1}}-\frac{du_{2}}{u_{2}}).$$

LeBrun [LeB91] showed that if we make a (non-holomorphic) coordinate change on $\mathbb{C}^{2}$

(2.55)

$$\begin{cases}\eta_{+}=u_{1}e^{\frac{T}{2}(|u_{1}|^{2}-|u_{2}|^{2})}\\ \eta_{-}=u_{2}e^{\frac{T}{2}(|u_{2}|^{2}-|u_{1}|^{2})}\end{cases}$$

then we have

(2.56)

$$\Omega_{TN,T}=d\eta_{+}\wedge d\eta_{-}.$$

So that the underlying complex manifold is still bi-holomorphic to $\mathbb{C}^{2}$ with holomorphic coordinates $\eta_{+}$ and $\eta_{-}$, and one can write down a global Kähler potential

(2.57)

$$\begin{cases}\omega_{TN,T}=\sqrt{-1}\partial\bar{\partial}\varphi_{T},\\ \varphi_{T}=\frac{1}{2}(|u_{1}|^{2}+|u_{2}|^{2})+\frac{T}{4}(|u_{1}|^{4}+|u_{2}|^{4}).\end{cases}$$

Again in Section 4.2, Remark 4.12.3 we shall see this follows from a more general fact.