Consider the following $S^{1}$ action on $\mathbb{C}^{n}$, with $n\geq 2$:

$$e^{i\theta}(z_{1},z_{2},z_{3},\ldots,z_{n})=(e^{i\theta}z_{1},e^{-i\theta}z_{2},z_{3},\ldots,z_{n}).$$

(24)

This action is Hamiltonian with respect to $\omega_{\mathbb{C}^{n}}$. Clearly it is singular along the $2(n-2)$ dimensional symplectic submanifold $\Crit(\mu)=\{z_{1}=z_{2}=0\}$. The moment map is:

$$\mu(z_{1},\ldots,z_{n})=\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2}.$$

(25)

The only critical value of $\mu$ is $t=0$ and $\Crit(\mu)\subset\mu^{-1}(0)$.

Now consider the map $\bar{\pi}$ as in Remark 2.5. Recall that $\bar{\pi}$ is given by

$$\begin{array}[]{rll}\bar{\pi}:\mathbb{C}^{n}&\rightarrow&\mathbb{R}\times\mathbb{C}^{n-1}\\ (z_{1},\ldots,z_{n})&\mapsto&(\mu,z_{1}z_{2},z_{3},\ldots,z_{n}).\end{array}$$

(26)

When restricted to $\mathbb{C}^{n}-\Crit(\mu)$, the above is an $S^{1}$-bundle onto $(\mathbb{R}\times\mathbb{C}^{n-1})-\bar{\pi}(\Crit(\mu))$ with Chern class $c_{1}=1$. Let $\pi_{t}$ be the restriction to $\mu^{-1}(t)$ of the map

$$(z_{1},\ldots,z_{n})\mapsto(z_{1}z_{2},z_{3},\ldots,z_{n}).$$

(27)

Then $\pi_{t}$ can be used to identify the reduced space $\mu^{-1}(t)/S^{1}$ with $\mathbb{C}^{n-1}$. Under this identification, i.e. letting the coordinates $u_{1}=z_{1}z_{2}$ and $u_{j}=z_{j+1}$ when $2\leq j\leq n-1$, the reduced symplectic form $\omega_{t}$ can be written as:

$$\omega_{t}=\frac{i}{2}\left(\frac{1}{2\sqrt{t^{2}+|u_{1}|^{2}}}\,du_{1}\wedge d\overline{u}_{1}+\sum_{j=2}^{n-1}\,du_{j}\wedge d\overline{u}_{j}\right).$$

(28)

Clearly, away from $t=0$, the reduced spaces are smooth manifolds.

On the other hand, at $t=0$ the reduced form $\omega_{0}$ blows up along the hyperplane

$$\Sigma:=\pi_{0}(\Crit(\mu))=\{u_{1}=0\},$$

so the reduced space $(\mathbb{C}^{n-1},\omega_{0})$ is singular. However, it was observed by Guillemin and Sternberg in [15], that it can be smoothed out, i.e. it can be identified with $(\mathbb{C}^{n-1},\omega_{\mathbb{C}^{n-1}})$. Indeed, the identification is given by the following

$$\Gamma_{0}:(u_{1},u_{2},\ldots,u_{n-1})\mapsto\left(\frac{u_{1}}{\sqrt{|u_{1}|}},u_{2},\ldots,u_{n-1}\right).$$

(29)

The map $\Gamma_{0}$ is continuous, smooth away from $u_{1}=0$ and such that $\Gamma_{0}^{\ast}\omega_{\mathbb{C}^{n-1}}=\omega_{0}$. One can do more: one can identify all the reduced spaces with $(\mathbb{C}^{n-1},\omega_{\mathbb{C}^{n-1}})$ at once. Consider the map

$$\Gamma_{t}:(u_{1},u_{2},\ldots,u_{n-1})\mapsto\left(\frac{u_{1}}{\sqrt{|t|+\sqrt{t^{2}+|u_{1}|^{2}}}},u_{2},\ldots,u_{n-1}\right).$$

(30)

One can verify that $\Gamma_{t}$ is a symplectomorphism between $(\mathbb{C}^{n-1},\omega_{t})$ and the standard symplectic space $\mathbb{C}^{n-1}$. However, this identification has the problem that, although continuous and smooth for fixed $t\in\mathbb{R}$, it is not smooth in $t$ when $t=0$. In fact one can show that it cannot be otherwise.