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: