While for the gluing in Proposition 4.17 it is sufficient to consider the zero order term of $H_{\Delta}$, to glue two singular Lagrangian fibrations $\mathcal{F}$ and $\mathcal{F}^{\prime}$ along their legs one should take into account all terms. This is essentially due to the fact that, gluing legs also involves gluing them along their singular fibres. We will see that Theorem 4.13 also takes care of this.

Suppose we are given a simple affine $3$-manifold with singularities $(B,\Delta,\mathscr{A})$ and two points $p$ and $p^{\prime}$ of $\Delta$ connected by an edge $J$ ($p$ and $p^{\prime}$ may be generic, positive or negative points). Let us assume that we have glued to $X(B_{0},\mathscr{A})$ the germs of singular Lagrangian fibrations $\mathcal{F}$ and $\mathcal{F}^{\prime}$ fibering over disjoint neighborhoods $V$ and $V^{\prime}$ of $p$ and $p^{\prime}$ respectively (e.g. using Proposition 4.17, if $p$ and $p^{\prime}$ are positive or generic). We do not consider only the case when $p$ and $p^{\prime}$ are either positive of generic, since we want the arguments here to hold also for negative points onto which we can glue fibrations like the ones in §7. We only assume here that $\mathcal{F}$ and $\mathcal{F}^{\prime}$ have legs with generic-singular fibres on their ends and these ends are connected by $J$. We now explain how to glue to $X(B_{0},\mathscr{A})$ a generic singular fibration along $J$ in such a way that this gluing is made compatible with the gluing of $\mathcal{F}$ and $\mathcal{F}^{\prime}$.

We can assume that there are disjoint neighborhoods $U$ and $U^{\prime}$ of the ends of $J$, as in Figure 9, and generic-singular fibrations $\mathcal{L}=\mathcal{F}|_{U}$ and $\mathcal{L}^{\prime}=\mathcal{F}^{\prime}|_{U^{\prime}}$ over $U$ and $U^{\prime}$. Let $H_{\Delta}$ and $H^{\prime}_{\Delta}$ be, respectively, the invariants of $\mathcal{L}$ and $\mathcal{L}^{\prime}$ as in Theorem 4.13.

Since $J$ is an edge of $\Delta$, there is a neighborhood $W$ of $J$, with $W\cap\Delta=J$, such that $(W,J)$ is (locally) affine isomorphic to $(D^{2}\times I,\Delta_{\tau})$ as in Example 3.9. Without loss of generality, we can assume $I=(-1,1)$ and that there exists $\delta\in(0,1)$ such that $U\cong D^{2}\times(-1,-\delta)$ and $U^{\prime}\cong D^{2}\times(\delta,1)$. Denote $I_{-\delta}=(-1,-\delta)$ and $I_{\delta}=(\delta,1)$. Clearly, we can interpret $H_{\Delta}$ and $H^{\prime}_{\Delta}$ as formal power series along $I_{-\delta}$ and $I_{\delta}$ respectively. By the arguments of the previous section, we must have that the zero order terms of $H_{\Delta}$ and $H^{\prime}_{\Delta}$ coincide with $\tau|_{I_{-\delta}}$ and $\tau|_{I_{\delta}}$ respectively.

It is now clear that we can choose a formal power series $\tilde{H}_{\Delta}$ along $I$ such that

• (a)

  the zero order term of $\tilde{H}_{\Delta}$ is $\tau$;

• (b)

  $\tilde{H}_{\Delta}$ coincides with $H_{\Delta}$ and $H^{\prime}_{\Delta}$ along $I_{-\delta}$ and $I_{\delta}$ respectively.

This can be done using cut-off functions. For this purpose it may be necessary to shrink $I_{-\delta}$ and $I_{\delta}$ by taking a slightly bigger $\delta$.

We can now apply Remark 4.12 and the first part of Theorem 4.13 to find the germ of a generic-singular Lagrangian fibration $\tilde{\mathcal{L}}$ fibering over $W$ whose invariant is $\tilde{H}_{\Delta}$. The second part of Theorem 4.13 and condition $(b)$ above imply that $\tilde{\mathcal{L}}|_{U}\cong\mathcal{L}$ and $\tilde{\mathcal{L}}|_{U^{\prime}}\cong\mathcal{L}$, moreover condition $(a)$ implies that $\tilde{\mathcal{L}}$ can be glued to $X(B_{0},\mathscr{A})$ along $J$. It is clear that the symplectic conjugations $\tilde{\mathcal{L}}|_{U}\cong\mathcal{L}$ and $\tilde{\mathcal{L}}|_{U^{\prime}}\cong\mathcal{L}^{\prime}$ coincide with the map gluing $\tilde{\mathcal{L}}$ to $X(B_{0},\mathscr{A})$.

We have proved:

The upshot of the results of this Section is that: 1) we can construct local models of generic and positive singular fibres; 2) we know how to glue them onto any given simple affine manifold with generic and positive singularities; 3) these gluings can be made compatible over common intersections. In fact, we can show:

The proof is an application of the above preparation results. Using Proposition 4.17 we can first glue in the positive vertices, then using Proposition 4.18 we glue in the generic-singular fibres over the edges. Theorem 4.19 is a particular case of our more general result we shall prove in §8, where we also include negative fibrations. We emphasize that the fibration obtained in Theorem 4.19 is smooth. This will not happen if $\Delta$ includes negative vertices. In that case, the resulting fibration will be piecewise smooth only.

As a further remark we point out that Theorem 4.19 can be generalized to dimension $n\geq 3$, since there are natural generalizations of generic and positive singularities and the analysis of their affine structures carries through as in the $n=3$ case. Our notion of simplicity can also be generalized to higher dimensions, though for $n>3$ it may no longer coincide with the notion of simplicity in the sense of Gross and Siebert [13].