In dimension 2 it is much easier to produce symplectic semi-stable compactifications. Now we will show how Example 3.16 gives rise to a symplectic semi-stable compactification diffeomorphic to a K3 surface. This will require a local model of Lagrangian $T^{2}$ fibration with a semi-stable singular fibre, such as the one in the following:

Lagrangian fibrations with semi-stable singular fibres, e.g., conjugate to the fibration in Example 2.6, are called focus-focus fibrations. They have been studied extensively in Hamiltonian Mechanics [4], [34] –where they got their name– and more recently in symplectic topology [24], [33] and Mirror Symmetry [14].

Let $\arg:\mathbb{C}^{\ast}\rightarrow\mathbb{R}$ be the multi-valued function $\rho e^{i\theta}\mapsto\theta$. Denote by $D\subseteq\mathbb{C}$ the unit open disk and let $D^{\ast}=D-\{0\}$. Let $\mathcal{F}=(X,\omega,f,D)$ be a focus-focus fibration. It has been shown [33] that there are coordinates $b=(b_{1},b_{2})$ on $\mathbb{R}^{2}$, with values in $D$, a smooth function $q:D\rightarrow\mathbb{R}$ such that $q(0)=0$ and a choice of generators of $H_{1}(f^{-1}(b),\mathbb{Z})$ with respect to which the periods $\lambda_{1}$ and $\lambda_{2}$ of $\mathcal{F}$ can be written as

	$\displaystyle\lambda_{1}$	$\displaystyle=$	$\displaystyle-\log|b|\ db_{1}+\arg b\ db_{2}+dq$
	$\displaystyle\lambda_{2}$	$\displaystyle=$	$\displaystyle 2\pi\,db_{2}.$

Clearly $\lambda_{1}$ is multi-valued and blows up as $b\rightarrow 0$. The lattice

$$\Lambda=\spn_{\mathbb{Z}}\langle{\lambda_{1}},{\lambda_{2}}\rangle$$

has monodromy given by $T$ as in (6). We now describe the affine structure induced on $D^{\ast}$. Consider the two open subsets

	$\displaystyle U_{1}$	$\displaystyle=$	$\displaystyle D-\{\im b=0\ \text{and}\re b\geq 0\},$
	$\displaystyle U_{2}$	$\displaystyle=$	$\displaystyle D-\{\im b=0\ \text{and}\re b\leq 0\}.$

On $U_{1}$ we chose the branch of $\arg$ with values in $(0,2\pi)$ and we denote it by $\arg_{1}$. On $U_{2}$ we chose the branch with values in $(-\pi,\pi)$ which we denote by $\arg_{2}$. Clearly on $U_{1}\cap U_{2}$ we have $\arg_{1}=\arg_{2}+2\pi$. A computation shows that the maps $\psi_{j}:U_{j}\rightarrow\mathbb{R}^{2}$ given by

$$\psi_{j}(b)=(-b_{1}\log|b|+b_{1}+q(b)+b_{2}\arg_{j}b,\,2\pi b_{2}),$$

with $q(0)=0$, are a choice of affine coordinates associated to $\lambda_{1}$ and $\lambda_{2}$.

It is easy to check that the map $\psi_{1}$ (or $\psi_{2}$) extends continuously to $D$. Call $\alpha:D\rightarrow\mathbb{R}^{2}$ the extended map. On a sufficiently small neighborhood $V\subseteq D$ of $0$, the map $\alpha$ is a homeomorphism of $V$ onto $\alpha(V)$. The reader may verify that $0\in V$ is a node with respect to the affine structure given by $\{U_{j},\psi_{j}\}$. In other words, the map $\alpha$ restricted to $V^{\ast}=V-\{0\}$ is an affine isomorphism between $V^{\ast}$ and the affine manifold $\alpha(V^{\ast})$ whose affine structure is the restriction of the one in Example 3.7. The affine structure with singularities on $D$ induced by a focus-focus fibration is therefore simple. In particular, the affine structure induced by Example 3.20 is simple.