We start giving examples of non-proper Lagrangian fibrations describing the singular behavior of (i) and (ii) near $\Crit(f)$. Let $D^{k}\subseteq\mathbb{R}^{k}$ be the standard open ball.

Now we give explicit examples of Lagrangian positive and generic-singular fibrations.

The reader should be aware that the above are just examples of Lagrangian positive and generic-singular fibrations. In fact, there are infinitely many germs of such fibrations [1].

Now we describe the integral affine structures induced by the above models by giving their period lattices explicitly. For the details we refer the reader to [1]. Fibrations with generic-singular fibres can be normalized near $\Crit(f)$ according to the following:

The above is a corollary of a result due to Miranda and Zung [26]; we refer the reader to [1]§3 for the details.

As in the 2-dimensional focus-focus fibration, one can choose suitable branches of $\lambda_{0}$ and define action coordinates on these branches. One can easily verify that this defines a simple singular affine structure on $B$. We have:

The case of Lagrangian fibrations of positive type is analogous. Positive fibrations are locally modeled on the fibration in Example 4.3 in a neighborhood of its critical locus. One can use this local description to compute the periods. We have (cf. [1]Theorem 4.19):

We now prove that the affine structure on the base of a positive fibration is simple.

Given a simple affine manifold with singularities, we show how to symplectically glue singular fibres of positive or generic type to the associated $T^{3}$ bundle. This gives us a (partial) symplectic compactification over positive and generic points of the singular locus.

Consider a cylinder $D^{2}\times I$ inside $\mathbb{R}^{2}\times\mathbb{R}$, where $I$ is an open interval, and let $\Delta=\{0\}\times I$. Let $H$ be a smooth real-valued function on $D^{2}\times I$. The germ of $H$ along $\Delta$, denoted $H_{\Delta}$, is the Taylor expansion series of $H$ along $\Delta$. This is a formal power series in two variables whose coefficients are smooth functions on $I$.

Recall that the generators of the period lattice of a generic-singular fibration may be written as $\lambda_{1}=\lambda_{0}+dH$, $\lambda_{2}=2\pi db_{2}$ and $\lambda_{3}=db_{3}$, where $(b_{1},b_{2},b_{3})$ are coordinates in in $D^{2}\times I$, $\lambda_{0}$ as (17) and $H$ a smooth function. One can prove the following (cf. [1]):

We call $H_{\Delta}$ the invariant of the fibration $\mathcal{F}_{H}$. We proved in Corollary 4.9 that the affine base of a generic-singular fibration is always simple, isomorphic to Example 3.9. Furthermore, the shape of its discriminant locus (in affine coordinates), as well as the isomorphism class of its singular affine base is determined by the function $\tau(r)=H(0,0,r)$ which is the restriction of $H$ to $\Delta$. In other words, by the zero order term of the germ $H_{\Delta}$. In the special case when the zero order term of $H_{\Delta}$ vanishes, the base is affine isomorphic to the product of an affine disc with a node times the standard affine interval, in this case we call the associated fibration $\mathcal{F}_{H}$ straight, in all other cases we call it twisted.

Analogously, positive fibrations are also classified by germs $H_{\Delta}$, where in this case $\Delta\subset D^{3}$ is a trivalent vertex and $H$ a smooth function on $D^{3}$ as in Proposition 4.10; for the details we refer to [1]. Given a positive fibration, Proposition 4.11 tells us that its base is locally isomorphic to $(\mathbb{R}^{3},\Delta_{\tau},\mathscr{A})$ as in Example 3.11. A particular case is when $\tau=0$ which gives a straight vertex. More generally we showed (cf. proof of Proposition 4.11) that $\tau=H|_{\Delta}$. In particular, we have:

We stress that the constructions described in Lemmas 4.14 and 4.15 only involve the zero order term of $H_{\Delta}$, which is enough for determining the affine structure. From [1] it follows that we have many possible choices of $H_{\Delta}$ giving the same affine structure:

Observe that the above result holds also in the case when $\tau\equiv 0$, i.e. when the discriminant is completely straight. Exploiting the flexibility given by Lemmas 4.14 and 4.15, we can show that we can always locally compactify a torus bundle given by simple affine manifolds with singularities near a positive or generic point of the discriminant locus:

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].