Let $(X,\omega)$ be a symplectic $2n$-manifold and let $\mu:(X,\omega)\rightarrow\mathfrak{t}^{\ast}$ be the moment map of a Hamiltonian $T^{k}$-action. Let $t\in\mu(X)$ and let $\pi_{t}:\mu^{-1}(t)\rightarrow X_{t}$ be the projection modulo the $T^{k}$ action. When $t$ is a regular value of $\mu$, $X_{t}$ is a smooth manifold and the symplectic form $\omega$ descends to a symplectic form $\omega_{t}$ on $X_{t}$. When $t$ is a critical value of $\mu$, $X_{t}$ may be a singular space and $\omega_{t}$ will be only defined on the smooth part of $X_{t}$. The space $(X_{t},\omega_{t})$ is the Marsden-Weinstein reduced space at $t$.

Goldstein [5] and Gross [6] used reduced spaces to construct $T^{k}$-invariant (special) Lagrangian fibrations. The following is a particular case of [6]Thm. 1.2:

When the $T^{k}$-action has fixed points, the construction of Proposition 5.2 will produce fibrations with interesting singular fibres. We will give some explicit examples shortly.

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

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:

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

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

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:

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

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

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

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.

We now illustrate a general method to construct piecewise smooth Lagrangian fibrations using Proposition 5.2 and the observations about the reduced geometry with respect to the $S^{1}$ action as in (24).

Let $\Log:(\mathbb{C}^{\ast})^{n-1}\rightarrow\mathbb{R}^{n-1}$ be the map defined by

Clearly, the above map is a Lagrangian fibration with respect to the restriction of $\omega_{\mathbb{C}^{n-1}}$ to $(\mathbb{C}^{\ast})^{n-1}$. Moreover, it defines a trivial $T^{n-1}$-bundle over $\mathbb{R}^{n-1}$. Let the map

be a smooth symplectomorphism of the standard $\mathbb{C}^{n-1}$. Let $X_{t}$ be the open and dense subsets of $(\mathbb{C}^{n-1},\omega_{t})$ defined by

Denote, with slight abuse of notation,

Then examples of maps $G_{t}:X_{t}\rightarrow\mathbb{R}^{n-1}$ as in Proposition 5.2 can be defined by

This clearly makes sense also when $t=0$. It is also clear that, for all fixed $t\in\mathbb{R}$, $G_{t}$ is a Lagrangian fibration with respect to the reduced symplectic form (28). We summarize this in the following:

It is clear that all the singular fibres of $f$ must lie in $\mu^{-1}(0)\cap X$. In fact, the singular fibres are all the lifts of fibres of $G_{0}$ in $X_{0}$ which intersect $\Sigma$. The topology of the singularity depends on the topology of this intersection. The discriminant locus of the fibration is therefore the set $\Delta\subset\mathbb{R}^{n}$ given by

Given a point $b=(0,b_{1},\ldots,b_{n-1})\in\Delta$, the fibre $f^{-1}(b)$ looks like $S^{1}\times G_{0}^{-1}(b_{1},\ldots,b_{n-1})$ after the circles over all points in $G_{0}^{-1}(b_{1},\ldots,b_{n-1})\cap\Sigma$ have been collapsed to points (cf. Figure 6).

In the following examples we use the above construction with $n=2$ or $3$. Define the piecewise smooth map $\gamma:\mathbb{C}^{2}\rightarrow\mathbb{C}$ by

If $\pi_{t}$ is the restriction of the map (27) to $\mu^{-1}(t)$, then one can easily see that for all $(z_{1},z_{2},z_{3})\in\mu^{-1}(t)$, the map $\Gamma_{t}\circ\pi_{t}$ is given by

From Proposition 5.4, we see that $\Gamma_{t}\circ\pi_{t}$ can be twisted by a symplectomorphism $\Phi$. The topology of the resulting fibration depends on how we choose $\Phi$.

In dimension $n=2$ we have the following:

Notice that in this example we are in the extremal case of Proposition 5.2, i.e. the reduced spaces are 2-dimensional and Remark 5.3 applies. In particular, the second component of $f$ in (36) could be replaced by any $T^{2}$ invariant function $G$, i.e. depending on $t=\frac{1}{2}\left(|z_{1}|^{2}-|z_{2}|^{2}\right)$ and $u_{1}=z_{1}z_{2}$, subject to the condition that all maps $G_{t}$ have $1$-dimensional level sets. Using this idea it is easy to construct everywhere smooth fibrations, such as the one in Example 3.20 where $G(t,u_{1})=\log|u_{1}+1|$. Of course the topology of the resulting fibration depends on the topology of the maps $G_{t}$.

We have an analogous model in dimension $n=3$:

The next result states existence of Lagrangian sections of the fibrations in the previous examples.

We notice that “smooth section” in the above statement means a section whose image is a smooth, manifold. In fact there is no obvious notion of what a smooth map from the base is, since there is no notion of smooth coordinates.

In view of Proposition 5.4, the fibrations of Examples 5.5 and 5.8 are all given by piecewise $C^{\infty}$ maps. More precisely, away from $\Sigma$, they are the union of two honest $C^{\infty}$ fibrations meeting and coinciding along $\mu^{-1}(0)$. A similar phenomenon occurs in special Lagrangian geometry [21]. This kind of piecewise smoothness deserves careful attention and we study it in the next Section.