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

$$\Log(v_{1},\ldots,v_{n-1})=(\log|v_{1}|,\ldots,\log|v_{n-1}|).$$

(31)

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

$$\Phi:\mathbb{C}^{n-1}\rightarrow\mathbb{C}^{n-1}$$

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

$$X_{t}=\Gamma_{t}^{-1}\circ\Phi^{-1}\left((\mathbb{C}^{\ast})^{n-1}\right).$$

Denote, with slight abuse of notation,

$$\Sigma:=\{u_{1}=0\}\cap X_{0}.$$

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

$$G_{t}=\Log\circ\Phi\circ\Gamma_{t}.$$

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

$$\Delta=\{0\}\times\left(\Log\circ\Phi\circ\Gamma_{0}(\Sigma)\right).$$

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).