Take as a symplectomorphism $\Phi$ the linear map

$$\Phi(u_{1},u_{2})=\frac{1}{\sqrt{2}}\left(u_{1}-u_{2},u_{1}+u_{2}-\sqrt{2}\right).$$

(34)

Then the fibration resulting from Proposition 5.4 can be written explicitly in the coordinates of the total space. We obtain:

$$f(z_{1},z_{2},z_{3})=\left(\frac{1}{2}\left(|z_{1}|^{2}-|z_{2}|^{2}\right),\log\frac{1}{\sqrt{2}}\left|\gamma-z_{3}\right|,\log\frac{1}{\sqrt{2}}\left|\gamma+z_{3}-\sqrt{2}\right|\right),$$

(35)

where $\gamma$ is as in (33). It is not difficult to see that $\Phi\circ\Gamma_{0}$ sends $\Sigma$ to the surface in $(\mathbb{C}^{\ast})^{2}$ given by

$$\Sigma^{\prime}=\{v_{1}+v_{2}+1=0\},$$

which is, topologically, a pair of pants. Then the discriminant locus is

$$\Delta=\{0\}\times\Log(\Sigma^{\prime}),$$

which has the shape in Figure 4. This example is topologically conjugate to the one in Example 2.9, before the surface $\Sigma^{\prime}$ has been twisted. For the discussion of the topology of the fibres in this example we refer to Example 2.9.

Using Proposition 5.4 we can obtain the following piecewise smooth fibration:

$$f(z_{1},z_{2})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|\gamma(z_{1},z_{2})+1|\right).$$

(36)

It is clearly well defined on $X=\{(z_{1},z_{2})\in\mathbb{C}^{2}\ |\ \gamma(z_{1},z_{2})+1\neq 0\}.$ Observe that $f$ is topologically conjugate to a focus-focus fibration, hence to Example 2.6. The only singular fibre is $f^{-1}(0)$ and it is a pinched torus. The fibration fails to be smooth on $\mu^{-1}(0)$. This example consists of the union of two smooth Lagrangian fibrations meeting along the “stitch”, $\mu^{-1}(0)$. We study this kind of piecewise smoothness in detail in §6.

Consider the following affine symplectomorphism of $(\mathbb{C}^{2},\omega_{\mathbb{C}^{2}})$

$$\Phi:(u_{1},u_{2})\mapsto(-u_{2},u_{1}-1).$$

(37)

The surface $\Sigma$ is sent by $\Phi\circ\Gamma_{0}$ to $\Sigma^{\prime}=\{v_{2}+1=0\}$. The amoeba of $\Sigma^{\prime}$ is just a straight line. The resulting fibration $f$ is

$$f(z_{1},z_{2},z_{3})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|z_{3}|,\,\log|\gamma(z_{1},z_{2})-1|\right).$$

(38)

The discriminant locus is $\{0\}\times\mathbb{R}\times\{0\}\subset\mathbb{R}^{3}$, a horizontal line in the plane $\{0\}\times\mathbb{R}^{2}$. The fibration is a piecewise smooth version of the generic singular fibration in Example 4.5. Notice that this fibration is invariant under the Hamiltonian $T^{2}$-action

$$(e^{i\theta_{1}},e^{i\theta_{2}})\cdot(z_{1},z_{2},z_{3})=(e^{i\theta_{1}}z_{1},\,e^{-i\theta_{1}}z_{2},\,e^{2i\theta_{2}}z_{3}),$$

(39)

whose moment map is

$$(z_{1},z_{2},z_{3})\mapsto\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,|z_{3}|^{2}\right).$$

There are other choices of symplectomorphisms $\Phi$ giving piecewise smooth generic fibrations. Although not very different from the previous one, we will write other two for convenience, since we will need them in the next example. The first one is

$$\Phi:(u_{1},u_{2})\mapsto(u_{1}-1,u_{2}-\sqrt{2}).$$

(40)

It gives the fibration

$$f(z_{1},z_{2},z_{3})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|\gamma(z_{1},z_{2})-1|,\,\log\left|z_{3}-\sqrt{2}\right|\right),$$

(41)

whose discriminant locus is the vertical line $\{0\}\times\{0\}\times\mathbb{R}\subset\mathbb{R}^{3}$. Also in this case it is clearly invariant under a $T^{2}$ action. The last choice of $\Phi$ is

$$\Phi:(u_{1},u_{2})\mapsto\frac{1}{\sqrt{2}}(u_{1}-u_{2},\,u_{1}+u_{2}),$$

(42)

giving

$$f(z_{1},z_{2},z_{3})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|\gamma(z_{1},z_{2})-z_{3}|,\,\log|\gamma(z_{1},z_{2})+z_{3}|\right),$$

(43)

whose discriminant is the slope +1 diagonal through zero in $\{0\}\times\mathbb{R}^{2}$. The $T^{2}$ action in this case is given by

$$(e^{i\theta_{1}},e^{i\theta_{2}})\cdot(z_{1},z_{2},z_{3})=(e^{i(\theta_{2}+\theta_{1})}z_{1},\,e^{i(\theta_{2}-\theta_{1})}z_{2},\,e^{2i\theta_{2}}z_{3}),$$

(44)

whose moment map is

$$(z_{1},z_{2},z_{3})\mapsto\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\frac{|z_{1}|^{2}+|z_{2}|^{2}}{2}+|z_{3}|^{2}\right).$$

In the above examples, the reduced spaces are all 2-dimensional. Using Remark 5.3 we can construct variations of (38) by replacing the last component of (38) with any function depending on $t=\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2}$, $s=|z_{3}|^{2}$ and $u_{1}=z_{1}z_{2}$, subject to the condition that all the maps $G_{t}$ have $1$-dimensional level sets. A choice providing an example of a smooth fibration is given by $G=\log|u_{1}-1|$, which gives us Example 4.5. One can do more. In fact, one can take a function $G$ which gives an interpolation between the piecewise smooth fibration in (38) and the smooth one in Example 4.5. This can be done by taking $G$ depending also on $s$, such that $G$ is equal to $\log|\gamma(z_{1},z_{2})-1|$ when $s$ is big and equal to $\log|u_{1}-1|$ when $s$ is small. We will say more about this later on, as this idea is useful in an important step of the main construction of the paper.

We now construct an example which interpolates Example 5.5 and 5.7. Consider the smooth function:

$$H_{0}=\frac{\pi}{4}\im(u_{1}\overline{u}_{2})$$

and let $\eta_{H_{0}}$ be the Hamiltonian vector field associated to $H_{0}$. If $\Phi_{s}$ is the flow generated by $\eta_{H_{0}}$, then the Hamiltonian symplectomorphism associated to $H_{0}$ is defined to be $\Phi_{H_{0}}=\Phi_{1}$. One computes that in our case

$$\Phi_{H_{0}}:(u_{1},u_{2})\mapsto\frac{1}{\sqrt{2}}(u_{1}-u_{2},u_{1}+u_{2}).$$

It maps $\{u_{1}=0\}$ to $\{v_{1}+v_{2}=0\}$. We now want a symplectomorphism which acts like $\Phi_{H_{0}}$ in a small ball centered at the origin and like the identity outside a slightly bigger ball. So choose a cut-off function $k:\mathbb{R}_{\geq 0}\rightarrow[0,1]$ such that, for some $\epsilon>0$,

$$k(t)=\left\{\begin{array}[]{l}1\ \quad\text{when}\ 0<t\leq\epsilon;\\ 0\ \quad\text{when}\ t\geq 2\epsilon\end{array}\right.$$

(45)

and define the Hamiltonian

$$H=k(|u_{1}|^{2}+|u_{2}|^{2})H_{0}.$$

The Hamiltonian symplectomorphism $\Phi_{H}$ associated to $H$ satisfies

$$\Phi_{H}(u_{1},u_{2})=\begin{cases}\id_{\mathbb{C}^{2}},&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\geq 2\epsilon;\\ \\ \frac{1}{\sqrt{2}}(u_{1}-u_{2},u_{1}+u_{2}),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\leq\epsilon.\end{cases}$$

Now let $\Psi$ be the affine symplectomorphism

$$\Psi:(v_{1},v_{2})\mapsto\frac{1}{\sqrt{2}}(v_{1}-v_{2},v_{1}+v_{2}-\sqrt{2}).$$

and finally, define $\Phi=\Psi\circ\Phi_{H}$. It is clear that

$$\Phi(u_{1},u_{2})=\begin{cases}\Psi,&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\geq 2\epsilon;\\ \\ (-u_{2},u_{1}-1),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\leq\epsilon.\end{cases}$$

Notice that $\Phi$ acts like in (37) on the ball of radius $\sqrt{\epsilon}$ around the origin, i.e. in a neighborhood of the surface $\{u_{1}=0\}$, and like in (34) outside a larger ball. We use this $\Phi$ to construct a fibration $f$ using Proposition 5.4. One can then see that $\Phi\circ\Gamma_{0}$ sends $\Sigma$ to a surface $\Sigma^{\prime}$ such that $\Log(\Sigma^{\prime})\subset\mathbb{R}^{2}$ is a 3-legged amoeba with the end of the horizontal leg pinched down to a straight line. The discriminant locus of $f$ is then $\Delta=\{0\}\times\Log(\Sigma^{\prime})\subset\mathbb{R}^{3}$. Of course, $f$ fails to be smooth on the slice $\mu^{-1}(0)$. Using the same method we can twist $\Sigma$ suitably and obtain a fibrations having discriminant locus an amoeba with three thin legs (cf. Figure 5). For example, to pinch the diagonal leg to a thin line, choose a smooth function $H_{0}$ generating the Hamiltonian symplectomorphism

$$(u_{1},u_{2})\rightarrow\left(u_{1}+\frac{1}{\sqrt{2}},u_{2}+\frac{1}{\sqrt{2}}\right).$$

Cut $H_{0}$ off with a function $\rho$ which vanishes when $|u_{2}|^{2}\leq M/2$, for some big $M$, and is equal to $1$ when $|u_{2}|^{2}\geq M$. This produces a Hamiltonian $H$. Now one proceeds as before. With an almost identical procedure one pinches down the vertical leg. The final choice of symplectomorphism $\Phi$ pinching down all three legs simultaneously may look like:

$$\Phi(u_{1},u_{2})=\begin{cases}(-u_{2},u_{1}-1),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\leq\epsilon;\\ \\ (u_{1}-1,u_{2}-\sqrt{2}),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}-\sqrt{2}|^{2}\leq\epsilon;\\ \\ \frac{1}{\sqrt{2}}(u_{1}-u_{2},\,u_{1}+u_{2}),&\quad\text{when}\ |u_{2}|^{2}\geq M;\\ \\ \Psi,&\quad\text{everywhere else}.\end{cases}$$

(46)

It is clear that this piecewise smooth example is topologically conjugate to the one in Example 2.9. Here we have made explicit the twistings described there. In §7 we will show that this fibration can be modified so that it is actually smooth towards the ends of the three legs. For this we will develop further the smoothing method sketched at the end of Example 5.7. Also in §7, we will show that this fibration can be modified so that it is smooth away from a neighborhood homeomorphic to a 2-disk containing the codimension 1 part of its discriminant.