This example is the topological model for the fibration over a point of $\Delta$ in the case $n=2$. Let $D$ be the unit disc in $\mathbb{C}$ and $D^{\ast}=D-\{0\}$. Let $f_{0}:X_{0}\rightarrow D^{\ast}$ be a $T^{2}$-bundle with monodromy generated by $\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right)$. We can use Proposition 2.4 to compactify $X_{0}$ as follows. The monodromy invariant cycle, $L\in H_{1}(f_{0}^{-1}(b),\mathbb{Z})$, induces a fibre preserving $T(L)$ action, with $T(L)=L\otimes\mathbb{R}/\penalty L$. The quotient modulo this action yields an $S^{1}$-bundle $\pi_{0}:X_{0}\rightarrow Y_{0}=D^{\ast}\times S^{1}$. One can verify that $\pi_{0}$ extends to an $S^{1}$-bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}=D\times S^{1}-\{(0,p)\}$, where $p\in S^{1}$. Furthermore $c_{1}(\pi^{\prime})=\pm 1$. Then Proposition 2.4 ensures that $X^{\prime}$ compactifies to a manifold $X=X^{\prime}\cup\{pt\}$ and that there is a proper map $\pi:X\rightarrow Y=D\times S^{1}$ extending $\pi^{\prime}$. Defining $P:Y\rightarrow D$ as the projection map, we obtain a fibration $f=P\circ\pi:X\rightarrow D$ extending $f_{0}$. The only singular fibre, $f^{-1}(0)$, is homeomorphic to $T^{2}=S^{1}\times S^{1}$ after $S^{1}\times\{x\}\subset T^{2}$ is collapsed to $x$. We denote this fibre by $I_{1}$, following Kodaira’s notation for singular fibres of elliptic fibrations. In Hamiltonian mechanics, a Lagrangian fibration with this topology is known as a focus-focus fibration.

This example is the model for the fibration over a neighborhood of an edge point of $\Delta$ –in [7] this is called $(2,2)$ fibration. Let $B=D\times(0,1)$, where $D\subset\mathbb{C}$ is the unit disc, and let $Y=T^{2}\times B$. Define $\Sigma\subset Y$ to be the cylinder sitting above $\{0\}\times(0,1)\subset B$ defined as follows. Let $e_{1},e_{3}$ be a basis of $H_{1}(T^{2},\mathbb{Z})$. Let $S^{1}\subset T^{2}$ be a circle representing the homology class $e_{3}$. Define $\Sigma=S^{1}\times\{0\}\times(0,1)$. Now let $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}:=Y-\Sigma$ be an $S^{1}$-bundle with Chern class $c_{1}=1$. Then $X^{\prime}$ compactifies to a manifold $X=X^{\prime}\cup\Sigma$ and there is a proper map $\pi:X\rightarrow Y$ extending $\pi^{\prime}$. We can now define $f=P\circ\pi:X\rightarrow B$ where $P:Y\rightarrow B$ is the projection. Then it is clear $f$ is a $T^{3}$ fibration with singular fibres homeomorphic to $I_{1}\times S^{1}$ lying over $\Delta:=\{0\}\times(0,1)$. If $e_{2}$ is an orbit of $\pi$, one can take $e_{1},e_{2},e_{3}$ as a basis of $H_{1}(X_{b},\mathbb{Z})$, where $X_{b}$ is a regular fibre. In this basis, $e_{2}$ and $e_{3}$ are monodromy invariant and a generator of the monodromy group of $f$ about $\Delta$ is represented in this basis by

This example is one of the two models over a neighborhood of a point in $\Delta_{d}$ –in [7] this is called $(2,1)$ fibration. Let $Y=T^{2}\times B$ with $B$ homeomorphic to a 3-ball. Let $\Delta\subset B$ be a cone over three distinct, non-collinear points. We write $\Delta=\{b_{0}\}\cup\Delta_{1}\cup\Delta_{2}\cup\Delta_{3}$ where $b_{0}$ is the vertex of $\Delta$ and the $\Delta_{i}$ are the legs of $\Delta$. Fix a basis $e_{2}$, $e_{3}$ for $H_{1}(T^{2},\mathbb{Z})$. Define $\Sigma\subset T^{2}\times B$ to be a pair of pants lying over $\Delta$ such that for $i=1,2,3$, $\Sigma\cap(T^{2}\times\Delta_{i})$ is a leg of $\Sigma$ which is the cylinder generated by $-e_{3}$, $-e_{2}$ and $e_{2}+e_{3}$ respectively. These legs are glued together along a nodal curve or ‘figure eight’ lying over $b_{0}$. Now consider an $S^{1}$-bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}=Y-\Sigma$ with Chern class $c_{1}=1$. This bundle compactifies to $\pi:X\rightarrow Y$. Now consider the projection map $P:Y\rightarrow B$. The composition $f=P\circ\pi$ is a proper map. The generic fibre of $f$ is a 3-torus. For $b\in\Delta$ the fibre $f^{-1}(b)$ is singular along $P^{-1}(b)\cap\Sigma$, which is a circle when $b\in\Delta_{i}$, or the aforementioned figure eight when $b=b_{0}$. Thus the fibres over $\Delta_{i}$ are homeomorphic to $I_{1}\times S^{1}$, whereas the central fibre, $X_{b_{0}}$, is singular along a nodal curve. A regular fibre can be regarded as the total space of an $S^{1}$-bundle over $P^{-1}(b)$. We can take as a basis of $H_{1}(X_{b},\mathbb{Z})$, $e_{1}(b),e_{2}(b),e_{3}(b)$, where $e_{2}$ and $e_{3}$ are the 1-cycles in $P^{-1}(b)=T^{2}$ as before and $e_{1}$ is a fibre of the $S^{1}$-bundle. The cycle $e_{1}(b)$ vanishes as $b\rightarrow\Delta$. In this basis, the matrices generating the monodromy group corresponding to loops $g_{i}$ about $\Delta_{i}$ with $g_{1}g_{2}g_{3}=1$, (cf. Figure 3) are

This is the local model for a fibration over a neighborhood of a component of $\Delta_{a}$. Consider $Y$ and $\Sigma$ as in Example 2.8. Now think of making a small perturbation of $\Sigma$ just in a neighborhood of the “figure eight” –i.e. where the three cylinders forming $\Sigma$ are joined together– and leaving the rest unchanged. A generic perturbation will be such that, near the fibre over $b_{0}$, $\Sigma$ will intersect the fibres of $P:Y\rightarrow B$ in isolated points. Then $P(\Sigma)$ will have the shape of a $3$-legged amoeba. One then constructs the bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}=Y-\Sigma$ with Chern class $c_{1}=1$ and compactifies it to $\pi:X\rightarrow Y$. The total fibration is $f=P\circ\pi$.

We can give an explicit construction of a fibration of this type, following ideas in [6]§4. Consider $(\mathbb{C}^{\ast})^{2}$ with the $T^{2}$ fibration $\Log:(v_{1},v_{2})\mapsto(\log|v_{1}|,\log|v_{2}|)$. Let $Y=\mathbb{R}\times(\mathbb{C}^{\ast})^{2}$ and $P$ be the fibration

where $t\in\mathbb{R}$ and $v=(v_{1},v_{2})\in(\mathbb{C}^{\ast})^{2}$. Define a surface $\Sigma^{\prime}$ in $(\mathbb{C}^{\ast})^{2}$ to be

and view it as a surface in $\{0\}\times(\mathbb{C}^{\ast})^{2}\subset Y$. Clearly $P(\Sigma^{\prime})$ is $\{0\}\times\Log(\Sigma^{\prime})$ and one can compute that it has the shape depicted in Figure 4. Images by $\Log$ of algebraic curves in $(\mathbb{C}^{\ast})^{2}$ are known in the literature as amoebas, and this explains the name we gave to the components of $\Delta_{a}$.

As a surface in $\mathbb{C}^{2}$, $\Sigma^{\prime}$ intersects $\{v_{1}=0\}$ in $q_{1}=(0,0,-1)$ and $\{v_{2}=0\}$ in $q_{2}=(0,-1,0)$. One can see that in a small neighborhood of $q_{1}$ one can twist $\Sigma^{\prime}$ slightly, so to make it coincide, in a smaller neighborhood, with $\{v_{2}=-1\}$. Similarly one can twist $\Sigma^{\prime}$ near $q_{2}$, so to make it coincide with $\{v_{1}=-1\}$. Finally, when $|v_{1}|$ and $|v_{2}|$ are both big, we can twist $\Sigma^{\prime}$ so to coincide with $\{v_{1}+v_{2}=0\}$. Let $\Sigma$ be this new twisted version of $\Sigma^{\prime}$. A schematic description of these twistings is described in Figure 5, where $\Sigma^{\prime}$ is the light-colored diagonal line and $\Sigma$ is the over-imposed twisted dark line. It is clear that $P(\Sigma)=\{0\}\times\Log(\Sigma)$ will have the shape of a $3$-legged amoeba whose legs have been pinched to $1$-dimensional segments toward the ends, as depicted in the right-hand side of Figure 5 (Mikhalkin [25] also defines a similar construction and calls this shape a localized amoeba). The bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}=Y-\Sigma$ with Chern class $c_{1}=1$ and its compactification $\pi:X\rightarrow Y$ can again be constructed. The fibration is $f=P\circ\pi$ and $\Delta=P(\Sigma)$.

We give a description of the fibration over the codimension $1$ part of $\Delta$. One can see that the fibres of $\Log$ over a point in the interior of the amoeba intersect $\Sigma$ in two distinct points. These two points come together to a double point as the base point approaches the boundary of the amoeba. If $p_{1}$ and $p_{2}$ are two points on $T^{2}$ –which may coincide– then the singular fibres of $f$ look like $S^{1}\times T^{2}$ after $S^{1}\times\{p_{j}\}$ is collapsed to a point. This behavior is topologically the same as the one conjectured by Joyce [21] for special Lagrangian $T^{3}$ fibrations. Moreover, the singularities of the fibres are modeled on those of an explicit example of a special Lagrangian fibration with non-compact fibres (cf. Joyce [21]§5).

In view of Proposition 2.4 and Remark 2.5, the total space $X$ in this example is diffeomorphic to the one in Example 2.8, although the fibrations differ. In both cases the singularities of the fibres occur along the intersection of the critical surface $\Sigma$ with the fibres of $P$. But the intersections happen in a different way. In Example 2.8 they occur either along circles, or along a figure eight. Here they occur along circles when the fibre is over a point in the codimension $2$ part of $\Delta$ and as isolated points when the fibre is over a point in the codimension $1$ part. As argued by Joyce, the isolated singularities are more generic in certain sense (cf. [21]§3). A schematic description of the fibration over the codimension $1$ part of $\Delta$ is depicted in Figure 6. It can be compared with Figure 2. We remark that over the codimension $2$ part of $\Delta$, the fibration has the same topology of the generic singular fibration of Example 2.7. It follows that the monodromy around the legs is same as the monodromy of Example 2.8, i.e. it is represented by the matrices (3).

This model is the other possible fibration over a neighborhood of a point in $\Delta_{d}$ –in [7] this is called $(1,2)$ fibration. Let $Y=S^{1}\times B$ with $B$ and $\Delta\subset B$ as in Example 2.8. Let $Y^{\prime}=Y\setminus(\{p\}\times\Delta)$, where $p\in S^{1}$. Let $L\cong\mathbb{Z}^{2}$ and define $T(L)=L\otimes_{\mathbb{Z}}\mathbb{R}/\penalty L$. Now consider a principal $T(L)$-bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}$. Under some mild assumptions on $\pi^{\prime}$ (cf. [7] Prop. 2.9), there is a unique manifold $X$ with $X^{\prime}\subset X$ extending the topology of $X^{\prime}$ and a proper extension $\pi:X\rightarrow Y$ of $\pi^{\prime}$. The composition of $\pi$ with the projection $Y\rightarrow B$ defines a topological $T^{3}$-fibration, $f:X\rightarrow B$. The fibre of $f$ over $b\in B\setminus\Delta$ is $T^{3}$. The fibre over $b\in\Delta_{i}$ is homeomorphic to $S^{1}\times I_{1}$, whereas the fibre over the vertex $b_{0}\in\Delta$ is homeomorphic to $S^{1}\times T^{2}/\penalty(\{point\}\times T^{2})$. It is proved in [7] that the monodromy group of this model is generated, in some basis, by the inverse transpose of the matrices (3). The reader should not worry, at this point, for the lack of details in this description as we will give explicit Lagrangian models for this example later on.