From the way $f$ is defined in Example 5.8, we can assume

$$\Sigma_{h,M}=\{t=0,\ u_{1}=0,\ |u_{2}|^{2}<\epsilon/4\},$$

where $\epsilon$ is as in (46) and $M=\log(\sqrt{\epsilon}/2)$. For any $\tau>0$ denote open sets

$$N^{\tau}=\{(t,u_{1},u_{2})\ |\ \max(|u_{1}|,|u_{2}|^{2})<\tau\}.$$

From now on we assume $f$ is restricted to $N^{\epsilon/2}$. As one can easily see from the construction, the map $G_{t}$ defining $f$, restricted to $N^{\epsilon/2}$ is

$$G_{t}(u_{1},u_{2})=\left(\log|u_{2}|,\log\left|\frac{u_{1}}{\sqrt{|t|+\sqrt{t^{2}+|u_{1}|^{2}}}}-1\right|\right).$$

(70)

This is the map that we want to perturb, but just on a smaller neighborhood. We do it applying the idea already anticipated at the end of Example 5.7. In fact we notice that $G_{t}$ is invariant with respect to the $S^{1}$ action

$$e^{i\theta}(u_{1},u_{2})=(u_{1},e^{2i\theta}u_{2}),$$

which is also Hamiltonian with respect to the reduced symplectic form $\omega_{t}$ given in (28). The moment map is

$$(u_{1},u_{2})\mapsto|u_{2}|^{2}.$$

So, if $g$ is a real function depending only on $u_{1},t$ and $s=|u_{2}|^{2}$, then

$$(u_{1},u_{2})\mapsto(\log|u_{2}|,\ g(u_{1},t,|u_{2}|^{2}))$$

is a Lagrangian fibration with respect to $\omega_{t}$, provided the level sets of $u_{1}\mapsto g(u_{1},t,s)$ are one dimensional submanifolds for every $s$ and $t$. For example, consider a real non-negative function $\rho$ defined on $\mathbb{R}^{3}$ such that, for every fixed $(t,s)\in\mathbb{R}^{2}$, the map

$$u\mapsto\frac{u}{\rho(|u|^{2},t,s)}$$

(71)

is a local homeomorphism of a neighborhood of $u=0$, then $g=\log|\frac{u}{\rho}-1|$ defines a Lagrangian fibration (at least in a neighborhood of $0$). In particular

$$\rho_{0}(r,t)=\sqrt{|t|+\sqrt{t^{2}+r}},$$

with $(r,t)\in\mathbb{R}^{2}$ gives the map $G_{t}$ in (70), but it is not smooth. It is easy to see that if $\rho$ is smooth on $\mathbb{R}^{3}$ and satisfies

$$\rho>\rho_{0}$$

(72)

then the map (71) is an orientation preserving diffeomorphism (at least near $u=0$). So let us choose a smooth $\rho_{1}$, defined on $\mathbb{R}^{2}$ and satisfying $\rho_{1}>\rho_{0}$, and let

$$g_{j}=\log\left|\frac{u_{1}}{\rho_{j}(|u_{1}|^{2},t)}-1\right|,$$

for $j=0,1$. We wish to find a $g$ which interpolates between $g_{0}$ and $g_{1}$. More precisely, we want $g$ to be equal to $g_{0}$ outside $N^{3\epsilon/8}$ and to $g_{1}$ on some smaller open neighborhood of $\Sigma_{h,M}$. Clearly $(u_{1},u_{2})\in N^{3\epsilon/8}$ if and only if $(|u_{1}|^{2},|u_{2}|^{2})$ is in the rectangle

$$S_{0}=[-9\epsilon^{2}/64,9\epsilon^{2}/64]\times[-3\epsilon/8,3\epsilon/8].$$

Now let $S_{1}$ be a closed neighborhood of $0$ in $\mathbb{R}^{2}$ which is contained in the interior of $S_{0}$, e.g. a smaller rectangle. Taking a $\sigma\in C^{\infty}(\mathbb{R}^{2})$, which is $0$ outside $S_{0}$ and $1$ on $S_{1}$, let us define

$$\rho(r,t,s)=(1-\sigma(r,s))\rho_{0}(r,t)+\sigma(r,s)\rho_{1}(r,t),$$

so that $\rho$ is equal to $\rho_{0}$ outside $S_{0}$ and it is equal to $\rho_{1}$ on $S_{1}$. Clearly $\rho>\rho_{0}$. We leave it to the reader to check that choices can be made so that with this $\rho$, (71) is indeed a homeomorphism. Now define

$$g=\log\left|\frac{u_{1}}{\rho(|u_{1}|^{2},t,|u_{2}|^{2})}-1\right|.$$

Clearly $g$ is equal to $g_{0}$ outside $N^{3\epsilon/8}$ and to $g_{1}$ on

$$N_{h,M}=\{(|u_{1}|^{2},|u_{2}|^{2})\in S_{1}\}$$

which, with a suitable choice of $S_{1}$, is a neighborhood of $\Sigma_{h,M}$. Moreover $u\mapsto g(u,t,s)$ has $1$-dimensional level sets. We can therefore replace the second component of $G_{t}$ in (70) with $g$ and redefine

$$G_{t}(u_{1},u_{2})=(\log|u_{2}|,g),$$

which is smooth on $N_{h,M}$. This proves the lemma for $\Sigma_{h,M}$. A schematic picture of this smoothing is described in Figure 12. The vertical lines represent fibres of $f$ over the horizontal leg. The base of the fibration is represented by the horizontal line on the bottom of the picture; the bold segment on the right represents the region where the codimension one part of $\Delta$ begins. The shaded region represents the locus where $f$ is not smooth. The dashed region is $N_{h,M}$.

The case of the vertical leg is done in the same way. At first sight it is not so obvious that also the diagonal leg can be treated in the same way. So let us give some explanation. When $|u_{2}|^{2}\geq M$, the map $G_{t}$ becomes

$$G_{t}(u_{1},u_{2})=\left(\log\left|\frac{u_{1}}{\rho_{0}(|u_{1}|^{2},t)}-u_{2}\right|,\,\log\left|\frac{u_{1}}{\rho_{0}(|u_{1}|^{2},t)}+u_{2}\right|\right).$$

(73)

The first observation is that this map is invariant under the $S^{1}$-action

$$e^{i\theta}(u_{1},u_{2})=(e^{i\theta}u_{1},e^{i\theta}u_{2}).$$

(74)

After the following change of coordinates on the base

$$(x_{1},x_{2})\mapsto\left(\frac{e^{2x_{1}}+e^{2x_{2}}}{2},x_{1}-x_{2}\right)$$

this becomes

$$G_{t}(u_{1},u_{2})=\left(\frac{\sqrt{t^{2}+|u_{1}|^{2}}+|u_{2}|^{2}}{2}-\frac{|t|}{2},\,\log\frac{\left|u_{1}/\rho_{0}-u_{2}\right|}{\left|u_{1}/\rho_{0}+u_{2}\right|}\right).$$

(75)

One can check that for every fixed $t\in\mathbb{R}$ the map

$$(u_{1},u_{2})\mapsto\frac{\sqrt{t^{2}+|u_{1}|^{2}}+|u_{2}|^{2}}{2},$$

is the moment map of the $S^{1}$-action (74), with respect to the reduced symplectic form $\omega_{t}$. Moreover, if one replaces $u_{1}=z_{1}z_{2}$, $u_{2}=z_{3}$ and $t=\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2}$, then the above map becomes

$$\nu:(z_{1},z_{2},z_{3})\mapsto\frac{|z_{1}|^{2}+|z_{2}|^{2}}{4}+\frac{|z_{3}|^{2}}{2}$$

which is a smooth map on the total space. Let us denote

$$s=\frac{\sqrt{t^{2}+|u_{1}|^{2}}+|u_{2}|^{2}}{2}.$$

The second component of (75) can be rewritten as

$$g_{0}(u_{1},u_{2})=\log\left|\frac{2u_{1}/\rho_{0}}{u_{1}/\rho_{0}+u_{2}}-1\right|.$$

We can now apply the same strategy we used in the case of the horizontal leg. We observe that we could replace this $g_{0}$ with any other $S^{1}$-invariant function $g$. In particular we could replace $\rho_{0}$, which is $S^{1}$-invariant, with another smooth $S^{1}$-invariant $\rho_{1}$. As before, we then interpolate $\rho_{0}$ and $\rho_{1}$ with a cut off function $\sigma$ depending on $|u_{1}|^{2}$ and $s$. We avoid writing the details here, as they just follow the same argument as before.

In the end we obtain that, in a small neighborhood of $\Sigma_{d,M}$, $G_{t}$ can be written as:

$$G_{t}=\left(s-\frac{|t|}{2},\log\left|\frac{2u_{1}/\rho_{1}}{u_{1}/\rho_{1}+u_{2}}-1\right|\right),$$

where now the second component is smooth. The first component is not quite smooth yet. We saw that $s$ is smooth when lifted to the total space, but $|t|$ isn’t. The total fibration becomes of the type

$$f(z_{1},z_{2},z_{3})=\left(\mu,\nu-\frac{|\mu|}{2},g(z_{1}z_{2},z_{3},\mu,\nu)\right),$$

where $g$ is smooth. We see that after a change of coordinates on the base of the type

$$(b_{1},b_{2},b_{3})\mapsto\left(b_{1},b_{2}+\frac{|b_{1}|}{2},b_{3}\right)$$

(76)

this fibration becomes

$$f(z_{1},z_{2},z_{3})=\left(\mu,\nu,g(z_{1}z_{2},z_{3},\mu,\nu)\right),$$

which is smooth. One can find a global change of coordinates on the base which acts like (76) only in a neighborhood of the end of the diagonal leg and is the identity elsewhere. This ends the proof of the Lemma. ∎

Consider one of the fibrations $\mathcal{F}_{h,M}$, $\mathcal{F}_{v,M}$ or $\mathcal{F}_{d,M}$ as above (whenever necessary, we allow ourselves to restrict to smaller neighborhoods of $\Delta_{h,M}$, $\Delta_{v,M}$ or $\Delta_{d,M}$). To keep the notation simple we temporarily drop the subindices and denote it by $\mathcal{F}$.

Since $\mathcal{F}$ satisfies Assumption 6.22, it follows from Proposition 6.28 that we can associate to $\mathcal{F}$ a normal form of cylindrical type $\mathcal{F}_{u,H}$ together with its invariants given by a triple $(Z^{\#}_{H},\ell,H_{\Delta})$ which, in view of Theorem 6.29, uniquely determine $\mathcal{F}$ as a germ around $\Gamma=B\cap\{b_{1}=0\}$. By slight abuse of notation we will denote by the same letter $\Gamma$ both $B\cap\{b_{1}=0\}$ and $B_{u}\cap\{b_{1}=0\}$, where $B_{u}$ is the base of $\mathcal{F}_{u,H}$. For the duration of this proof $H$ will remain unchanged, so we drop the subindex $H$ and denote $\mathcal{F}_{u}:=\mathcal{F}_{u,H}$ for short.

The proof consists in suitably deforming the sequence $\ell$. Let $\bar{A}\subset\Gamma$ and $\bar{A}^{\prime}\subset\bar{A}$ be (planar) regions as depicted in Figure 14. Given a cut-off function $\rho\in C^{\infty}(\Gamma)$ such that $\rho$ is 1 on $\Gamma-\bar{A}$ and $0$ on $\bar{A}^{\prime}$, define a new (fibrewise closed) sequence $\tilde{\ell}$ whose elements are $\tilde{\ell}_{k}=(\rho\circ\bar{\pi}^{\#})\,\ell_{k}$ for each $k\in\mathbb{N}$. We obtain a triple $(Z^{\#}_{H},\tilde{\ell},H_{\Delta})$, such that $\ell|_{(\bar{\pi}^{\#})^{-1}(\Gamma-\bar{A})}=\tilde{\ell}|_{(\bar{\pi}^{\#})^{-1}(\Gamma-\bar{A})}$ and $\tilde{\ell}|_{(\bar{\pi}^{\#})^{-1}(\bar{A}^{\prime})}=0$.

In view of Proposition 6.30, $(Z^{\#}_{H},\tilde{\ell},H_{\Delta})$ gives rise to a normal form of cylindrical type $\mathcal{F}_{\tilde{u}}$ defined over a neighborhood of $\Gamma$. By construction and by Theorem 6.29, $\mathcal{F}_{u}$ and $\mathcal{F}_{\tilde{u}}$ define the same germ around $\Gamma-\bar{A}$, i.e. there are open neighborhoods $U$ and $\tilde{U}$ of $\Gamma-\bar{A}$ (satisfying $U\cap\{b_{1}=0\}=\tilde{U}\cap\{b_{1}=0\}=\Gamma-\bar{A}$) such that $\mathcal{F}_{u}|_{U}$ and $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ are symplectically conjugate. Moreover $\mathcal{F}_{\tilde{u}}$ is smooth when restricted to any open neighborhood $A^{\prime}$ of $\bar{A}^{\prime}$ such that $A^{\prime}\cap\{b_{1}=0\}=\bar{A}^{\prime}$. Now recall that $\mathcal{F}_{u}$ is symplectically conjugate to $\mathcal{F}$, so we have that $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ is symplectically conjugate $\mathcal{F}|_{U}$.

Let us summarize the result using our original notation for the horizontal leg. For $\Gamma_{h,M}=B_{h,M}\cap\{b_{1}=0\}$, we have found sets $\bar{A}^{\prime}\subset\bar{A}\subset\Gamma_{h,M}$ (as in Figure 14) and a normal form of cylindrical type $\mathcal{F}_{\tilde{u}}$, defined over a neighborhood of $\Gamma_{h,M}$, smooth over $\bar{A}^{\prime}$ and such that $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ is symplectically conjugate to $\mathcal{F}_{h,M}|_{U}$, where $U$ and $\tilde{U}$ are neighborhoods of $\Gamma_{h,M}-\bar{A}$ (satisfying $U\cap\{b_{1}=0\}=\tilde{U}\cap\{b_{1}=0\}=\Gamma_{h,M}-\bar{A}$).

If we go back denoting by $\mathcal{F}$ the fibration of Lemma 7.4, we can form a new fibration $\tilde{\mathcal{F}}$ in the following way. Let $\mathcal{F}^{\prime}=\mathcal{F}|_{\mathbb{R}^{3}-(\mathbb{R}\times\bar{A})}$ and symplectically glue $\mathcal{F}_{\tilde{u}}$ to $\mathcal{F}^{\prime}$ using the conjugation between $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ and $\mathcal{F}^{\prime}|_{U}=\mathcal{F}_{h,M}|_{U}$. The fibration $\tilde{\mathcal{F}}$ is the result of this gluing. Notice that $\tilde{\mathcal{F}}$, due to the properties of $\mathcal{F}_{\tilde{u}}$, is such that for some $m>M$ (depending on $\bar{A}^{\prime}$) and a suitable neighborhood of $B_{h,m}$ of $\Delta_{h,m}$, the restriction $\tilde{\mathcal{F}}_{h,m}$ is smooth. Notice that $\bar{A}^{\prime}$ can be chosen so that the latter holds for any $m>M$.

The above method applied to all legs, produces the required result. ∎

Recall that the fibration produced in Lemma 7.6 is a perturbation of the one in Example 5.8, whose total space is an open set of $\mathbb{C}^{3}$ with standard symplectic form, which is exact. One can see that the successive perturbations of this fibration have not modified the cohomology class of $\omega$. ∎

We apply a similar argument to the one used in the case of the positive fibre (see Proposition 4.11). Clearly, since $\gamma_{1}$ is represented by the orbits of the $S^{1}$ action

$$-\int_{\gamma_{1}(b)}\eta=b_{1},$$

which is continuous. We now prove that, for $j=2,3$

$$\alpha_{j}(b)=-\int_{\gamma_{j}(b)}\eta$$

(77)

extends continuously to points in $\Gamma_{d}$ or in $\Gamma_{e}$. As we did in Proposition 4.11, we can think of $\alpha_{j}(b)$ as

$$\alpha_{j}(b)=\int_{S}\omega,$$

where $S$ is a surface spanned by the cycles $\gamma_{j}(b^{\prime})$ as $b^{\prime}$ moves along a curve joining $\bar{b}$ and $b$. Suppose $b\in\Gamma_{e}$ (or $\Gamma_{d}$), then we need to show that $\alpha_{j}(b)$ is independent of the curve from $\bar{b}$ to $b$, or equivalently that

$$\int_{S_{1}-S_{2}}\omega=0,$$

where $S_{1}$ and $S_{2}$ are the surfaces corresponding to two different paths from $\bar{b}$ to $b$. The boundary $\partial(S_{1}-S_{2})$ is determined by monodromy. It is easy to see that $\partial(S_{1}-S_{2})$ is a multiple of $\gamma_{1}(b)$, therefore for some integer $k$ we have

$$\int_{S_{1}-S_{2}}\omega=-\int_{\partial(S_{1}-S_{2})}\eta=k\int_{\gamma_{1}(b)}\eta=0,$$

where the last equality follows from the fact that $b\in\Gamma_{d}$ or $\Gamma_{e}$. To show that $\alpha$ extends continuously also to points of $\Delta$ we can argue that (77) makes sense also over singular fibres, since both $\eta$ and $\gamma_{j}(b)$ are well defined when $b\in\Delta$. ∎

Since $\alpha_{1}(b)=b_{1}$, it is enough to show that, if for fixed $t\in\mathbb{R}$ we let $U_{t}=\{b_{1}=t\}$, then $\alpha_{t}=\alpha|_{U_{t}}$ is a bijection onto its image. If $\lambda_{2}$ and $\lambda_{3}$ are the periods of the fibration corresponding to $\gamma_{2}$ and $\gamma_{3}$, then $\alpha_{t}$ is computed by taking primitives of $\lambda_{2}|_{U_{t}}$ and $\lambda_{3}|_{U_{t}}$. If we let $X_{t}$ denote the symplectic reduction of $X$ at $t$ and $G_{t}:X_{t}\rightarrow\mathbb{R}^{2}$ the reduced fibration, then it is not difficult to see that $\lambda_{2}|_{U_{t}}$ and $\lambda_{3}|_{U_{t}}$ are in fact periods of $G_{t}$ (cf. [3]Lemma 5.9). Now the conclusion follows by simply observing that $G_{t}$ is a proper Lagrangian submersion, i.e. an integrable system. The argument works also when $t=0$.

An explicit computation of the periods was done in [3]Proposition 5.10 for the fibration in Example 5.8. There we found that

	$\displaystyle\lambda_{2}$	$\displaystyle=$	$\displaystyle\beta_{1}\,db_{1}-e^{2b_{2}}db_{2},$
	$\displaystyle\lambda_{3}$	$\displaystyle=$	$\displaystyle\beta_{2}\,db_{1}-e^{2b_{3}}db_{3},$		(78)

where $\beta_{1}$ and $\beta_{2}$ are functions depending only on $b_{1}$. The periods of the perturbed fibration obtained in Lemma 7.6 will have this same expression away from where the perturbation took place (i.e. away from the white region in Figure 15), for example in a neighborhood of the codimension 1 part of $\Delta$. It is easy to see from this expression of the periods that $\alpha$ extends continuously to $\Delta$ and that it is a bijection. ∎

The fibrations constructed in Theorem 6.19 have smooth Lagrangian sections and the action coordinates extend continuously to the whole base. Since $\mathcal{F}|_{U}$ also has a Lagrangian section (cf. Remarks 7.7) and the action coordinates extend continuously to the whole base, the statement easily follows from the results on stitched fibrations such as the existence of a normal form. The latter is found extending the maps $f^{+}$ and $f^{-}$ beyond all connected components of the seam and then using the Lagrangian section to normalize with the period map.

∎

The proof follows the same lines of Lemma 7.6. Assume that $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$ has been constructed with Theorem 6.19. In particular the wall $\Gamma$ consists of the union of three disjoint sets, denoted $\Gamma_{c}$, $\Gamma_{d}$ and $\Gamma_{e}$. The corresponding components of the seam are $Z_{c}=f^{-1}(\Gamma_{c})$, $Z_{d}=f^{-1}(\Gamma_{c})$ and $Z_{e}=f^{-1}(\Gamma_{e})$ with corresponding quotients denoted by $\bar{Z}_{c}$, $\bar{Z}_{d}$ and $\bar{Z}_{e}$. The invariants of $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$ are given by sequences $\ell^{c}$, $\ell^{d}$ and $\ell^{e}$. In particular the first order invariants satisfy the integral conditions (60) with $m_{1}=-1$ and $m_{2}=1$.

Over the same wall $\Gamma$ and seam $Z$, we could define another triple of invariants as follows. Define $(\ell^{c})^{\prime}$ to be the zero sequence, while $(\ell^{d})^{\prime}$ and $(\ell^{e})^{\prime}$ to be sequences whose only non-zero terms are the first order ones, which we define to be

$$(\ell_{1}^{d})^{\prime}=-dy_{2}\ \ \text{and}\ \ (\ell_{1}^{e})^{\prime}=dy_{3}.$$

As we saw in Example 6.21, these choices of invariants give rise to a fake stitched fibration $\mathcal{F}^{\prime}$ which is topologically conjugate to $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$.

Using Theorem 6.19 we now construct a new stitched fibration with the same wall $\Gamma$ and seam $Z$ as $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$, but whose invariants interpolate between those of $\mathcal{F}^{\prime}$ and those of $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$. Let $A^{\prime}$ be a small tubular neighborhood of $\Delta$ and denote $\bar{A}^{\prime}=A^{\prime}\cap\{b_{1}=0\}$. Assume that $\bar{A}^{\prime}$ is entirely contained in the region in Figure 15 (a) delimited by the dotted lines. In particular we want the ends of $\bar{A}^{\prime}$ to be contained in the white region where $\mathcal{F}$ is smooth. Let $A\subset A^{\prime}$ be a smaller open neighborhood of $\Delta$ and denote $\bar{A}=A\cap\{b_{1}=0\}$. Let $\rho\in C^{\infty}(\Gamma)$ be a cut-off function which is 1 on $\bar{A}$ and $0$ on $\Gamma-\bar{A}^{\prime}$. Define $\tilde{\ell}_{k}^{c}=(1-\rho)(\ell_{k}^{c})^{\prime}+\rho\,\ell_{k}^{c}$ and similarly define $\tilde{\ell}_{k}^{d}$ and $\tilde{\ell}_{k}^{e}$. It follows from Theorem 6.19 that the sequences $\tilde{\ell}_{c}$, $\tilde{\ell}_{d}$ and $\tilde{\ell}_{e}$ give rise to a stitched Lagrangian fibration $\tilde{\mathcal{F}}^{o}$ which is topologically conjugate to $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$. Moreover $\tilde{\mathcal{F}}^{o}|_{A-\Delta}$ and $\mathcal{F}|_{A-\Delta}$ are symplectically conjugate so we can glue $\mathcal{F}|_{A}$ to $\tilde{\mathcal{F}}^{o}|_{A-\Delta}$ along $\mathcal{F}|_{A-\Delta}$. This produces a piecewise smooth Lagrangian fibration $\tilde{\mathcal{F}}$ which is topologically conjugate to $\mathcal{F}$, moreover the chosen invariants guarantee that after a change of coordinates on the base $\tilde{\mathcal{F}}$ satisfies the smoothness condition $(ii)$. ∎

It only remains to show that $\mathcal{\tilde{F}}$ satisfies property (iii) of Definition 7.1, but this immediately follows from the construction. In fact, $\tilde{\mathcal{F}}|_{\mathbb{R}^{3}-A^{\prime}}$ coincides with the fibration described in Example 6.21 restricted to a suitable neighborhood of the vertex. We observed that the latter fibration induces an affine structure on the base which is affine isomorphic to a negative vertex of Example 3.12 (or of Example 3.13). This concludes the proof. ∎