Let $f_{V}:X_{V}\rightarrow V$ be a focus-focus fibration over a small open neighborhood $V$ of its node $0\in V$. Let $V^{\ast}=V-\{0\}$ and denote by $(V^{\ast},\mathscr{A}_{V})$ the integral affine manifold induced by $f_{V}$. Let $X(V^{\ast},\mathscr{A}_{V})$ be the associated Lagrangian $T^{2}$ bundle over $V^{\ast}$. It can be shown that $f_{V}$ has a Lagrangian section $s:V\rightarrow X_{V}$ such that $s(V)\cap\Crit(f_{V})=\varnothing$. Then from Theorem 3.3 it follows that $f^{-1}_{V}(V^{\ast})\subset X_{V}$ is symplectically conjugate to $X(V^{\ast},\mathscr{A}_{V})$.

Now let $P\in\Delta$ and let $U\subset B$ be a small neighborhood of $P$. Denote by $U^{\ast}=U-P$ and by $X(U^{\ast},\mathscr{A})$ the Lagrangian $T^{2}$ bundle over $U^{\ast}$ given by the restriction of $X(B_{0},\mathscr{A})$ to $U^{\ast}$. Recall that both $U$ and $V$ are simple affine manifold with singularities. Then, after taking $U$ and $V$ small enough, there exists an integral affine isomorphism $V^{\ast}\cong U^{\ast}$. From Corollary 3.4, the latter isomorphism induces is a symplectic conjugation,

which can be used to symplectically glue $X_{V}$ to $X(B_{0})$. Define $(X,\omega)$ to be the symplectic manifold obtained after applying this gluing over all points $P\in\Delta$ and $f:X\rightarrow B$ the resulting fibration. It is clear that $(X,\omega)$ is a semi-stable compactification of $(X(B_{0},\mathscr{A}),\omega_{0})$ such that $\iota^{\ast}\omega=\omega_{0}$. It is easy to check that $(X,\omega,f,B)$ is topologically conjugate to a simply connected elliptic fibration with 24 singular fibres of type $I_{1}$. It follows that $X$ is diffeomorphic to a K3 surface. ∎