In [3] we proposed to extend the classical theory of action-angle coordinates to a particular type of piecewise smooth fibrations, which we called stitched fibrations. Here we review how this theory was further developed in [2] and extend some of those techniques to fibrations which are not proper. For details and complete proofs we refer the reader to [2]. The material in this section is primarily technical but necessary to understand the lack of regularity of the fibrations in §5. The techniques here are useful, in particular, for the construction of Lagrangian fibrations of negative type §7.

Notice that we do not require $f$ to be onto $\mathbb{R}\times M$, so we denote $B=f(X)$ and $B^{\pm}=f(X^{\pm})$. In general, a stitched fibration will only be piecewise $C^{\infty}$, however all its fibres are smooth Lagrangian tori. Observe also that $f^{\pm}$ is the restriction of a $C^{\infty}$ map, it is not a priori required to extend to a smooth Lagrangian fibration beyond $X^{\pm}$. Throughout this section we will always assume (unless otherwise stated) that the pair $(B,\Gamma)$ is diffeomorphic to the pair $(D^{n},D^{n-1})$, where $D^{k}\subset\mathbb{R}^{k}$ is an open unit ball centered at the origin and $\mathbb{R}^{n-1}$ is embedded in $\mathbb{R}^{n}$. Later on we will consider more general bases –e.g. non-simply-connected– when we speak about monodromy.

We now review some the examples given in §5:

Analogously, the piecewise smooth fibration in Example 5.7 gives rise to a stitched fibration when restricted to the complement of the union of the singular fibres. There is another important example in dimension three:

To understand the geometry of stitched fibrations in a neighborhood of a point on the wall, it is convenient to allow a more general set of coordinates than just the smooth ones.

Essentially, admissible coordinates are those such that $\phi\circ f$ is again stitched. Let $f:X\rightarrow B$ be a stitched Lagrangian fibration and let $\phi$ be a set of admissible coordinates. For $j=2,\ldots,n$, $f_{j}^{\pm}=\phi_{j}\circ f|_{X^{\pm}}$ is the restriction of a $C^{\infty}$ function on $X$ to $X^{\pm}$ and we can write $f=(\mu,f_{2}^{\pm},\ldots,f_{n}^{\pm})$. Let $\eta_{1}$ and $\eta_{j}^{\pm}$ be the Hamiltonian vector fields of $\mu$ and $f^{\pm}_{j}$ respectively. In order to measure how far $f$ is from being smooth, it makes sense to compare $\eta_{j}^{+}$ and $\eta_{j}^{-}$ in the only place where they exist simultaneously, i.e. along $Z$. In fact it is not difficult to show that there are $S^{1}$ invariant functions $a_{j}$ on $Z$ such that

Clearly, when $\phi\circ f$ is smooth $a_{2}=\cdots=a_{n}=0$.

It is convenient to interpret the $S^{1}$ invariant functions $(a_{2},\ldots,a_{n})$ in (48) as follows. First observe that the seam of a stitched fibration is an $S^{1}$-bundle $p:Z\rightarrow\bar{Z}:=Z/\penalty S^{1}$ such that:

where $\bar{Z}$ has the reduced symplectic form and $\bar{f}$ is the reduced Lagrangian fibration over the wall $\Gamma$. We also have the vertical $(n-1)$-plane distribution:

tangent to the fibres of $\bar{f}$. Clearly, a choice of coordinates around $b\in\Gamma$ induces a frame $\bar{\eta}=(\bar{\eta}_{2},\ldots,\bar{\eta}_{n})$ of $\mathfrak{L}$, where $\bar{\eta}_{j}=p_{\ast}\eta_{j}^{+}=p_{\ast}\eta_{j}^{-}$. Define $\ell_{1}$ to be the section of $\mathfrak{L}^{\ast}$ such that:

It is not difficult to see (and we prove it in [2]) that $\ell_{1}$ is fibrewise closed, i.e. when restricted to the fibres of $\bar{f}$, $\ell_{1}$ is a closed 1-form. One can prove that a different choice of coordinates around $b\in\Gamma$ induces a frame $\bar{\eta}^{\prime}$ and a section $\ell_{1}^{\prime}$ such that $\ell_{1}^{\prime}-\ell_{1}=\delta$, where $\delta$ is fibrewise constant, i.e. the Lie derivative $\mathcal{L}_{\bar{\eta}_{j}}\delta=0$ for all $j=2,\ldots,n$ (cf. [2]Proposition 4.2). As a corollary, if there is a change of coordinates in the base which makes a stitched fibration smooth, then $\ell_{1}$ is fibrewise constant. The invariant $\ell_{1}$ is a first order measure of how much $f$ fails to be smooth along $Z$. Of course one also needs to consider “higher order terms” to fully understand the behavior of a stitched fibration near the seam.

In the smooth case, action-angle coordinates defined over $B$ depend on a choice of a basis of $H_{1}(X,\mathbb{Z})$. In the case of stitched fibrations it is convenient to generalize this idea as follows. We choose a pair of bases $\gamma^{\pm}=(\gamma_{1},\gamma_{2}^{\pm},\ldots,\gamma_{n}^{\pm})$ of $H_{1}(X,\mathbb{Z})$ such that

• (a)

  $\gamma_{1}$ is represented by an orbit of the $S^{1}$ action,

• (b)

  $\gamma_{j}^{+}=\gamma_{j}^{-}+m_{j}\gamma_{1}$, for some $m_{2},\ldots,m_{n}\in\mathbb{Z}$.

Condition (b) simply means that $p_{\ast}\gamma^{+}=p_{\ast}\gamma^{-}$ under the map $p_{\ast}:H_{1}(X,\mathbb{Z})\rightarrow H_{1}(X/S^{1},\mathbb{Z})$. Such a choice of bases will be useful to understand fibrations over non simply connected bases where monodromy may occur. The following proposition generalizes the notion of action angle coordinates on the base.

Recall that to establish the existence of action-angle coordinates, in the classical case, one chooses a smooth Lagrangian section. In the stitched case we choose a continuous section $\sigma:B\rightarrow X$ such that $\sigma|_{B^{\pm}}$ are the restrictions of smooth maps and $\sigma(B)$ is a smooth Lagrangian submanifold. Such sections always exist locally, for example the one constructed in Proposition 5.9 is a section of this type. We denote a stitched fibration $f:X\rightarrow B$ together with a choice of basis $\gamma$ of $H_{1}(X,\mathbb{Z})$ and a section $\sigma$ as above by $\mathcal{F}=(X,B,f,\gamma,\sigma)$.

Notice that in the above definition we are allowed to shrink to a smaller neighborhood of $\Gamma$ but not to a smaller $\Gamma$. So germs are meant to be defined around $\Gamma$ and not around a point. In [2] we classified stitched Lagrangian fibrations up to symplectic conjugation in terms of certain invariants. We review this classification here.

First we illustrate a basic construction of stitched fibrations.

Now suppose $\mathcal{F}_{u}=(Y,B_{u},f_{u},\sigma_{0},\gamma_{0})$ is as above and let $(b,y)=(b_{1},\ldots,b_{n},y_{1},\ldots,y_{n})$ be canonical coordinates on $T^{\ast}B_{u}$ so that $y$ gives coordinates on the fibre $T_{b}^{\ast}B_{u}$. Let $W$ be a neighborhood of $\Gamma$ inside $u(V)$. If $r\in\mathbb{R}$ is a parameter, for any $b=(0,b_{2},\ldots,b_{n})\in\Gamma$, let $(r,b)$ denote the point $(r,b_{2},\ldots,b_{n})\in\mathbb{R}^{n}$. Given $(r,b)\in W$, denote by $L_{r,b}$ the fibre $u^{-1}((r,b))$. For every fibre $F_{b}\subset Z$ of $\pi$, consider the symplectomorphism

between a neighborhood of the zero section of $T^{\ast}F_{b}$ and a neighborhood of $F_{b}$ in $V$. If $W$ is sufficiently small, for every $(r,b)\in W$, the Lagrangian submanifold $L_{r,b}$ will be the image of the graph of a closed $1$-form on $F_{b}$. Due to the $S^{1}$ invariance of $u$ and the fact that $u_{1}=b_{1}$, this 1-form has to be of the type

where $\ell(r,b)$ is the pull back to $F_{b}$ of a closed one form on $\bar{F}_{b}$. Denote by $\ell(r)$ the smooth one parameter family of sections of $\mathfrak{L}^{\ast}$ such that $\ell(r)|_{\bar{F}_{b}}=\ell(r,b)$. The condition $u|_{Z}=\pi$ implies that $\ell(0,b)=0$. Furthermore, the $N$-th order Taylor series expansion of $\ell(r)$ in the parameter $r$ can be written as

where the $\ell_{k}$’s are fibrewise closed sections of $\mathfrak{L}^{\ast}$.

As above, to a given $(V,u)\in\mathscr{U}_{Z}$ we can associate a unique sequence $\ell\in\mathscr{L}_{Z}$. Conversely, in [2]§5 we showed that for any given sequence $\ell\in\mathscr{L}_{Z}$ there is some $(V,u)\in\mathscr{U}_{Z}$, therefore a normal form, associated to it. Clearly, this $(V,u)$ is not unique.

In [2] we proved that stitched fibrations are normalized according to the following:

When $\mathcal{F}$ is smooth, its normal form is $\mathcal{F}_{\pi}$. This is Arnold-Liouville theorem (cf. Corollary 3.5). Given a stitched Lagrangian fibration $\mathcal{F}=(X,B,f,\sigma,\gamma)$ with normal form $\mathcal{F}_{u}=(Y,B_{u},f_{u},\sigma_{0},\gamma_{0})$, we respectively denote by $Z_{\text{nor}}$ and $\Gamma_{\text{nor}}$ the seam and the wall of $\mathcal{F}_{u}$ and by $\bar{Z}_{\text{nor}}$ the $S^{1}$ reduction of $Z_{\text{nor}}$.

We prove in [2]Corollary 6.9 that $\inv(\mathcal{F})$ is independent on the choice of normal form.

We will now see that every specified data $(\bar{Z}_{\text{nor}},\ell)$, with $\ell_{1}$ satisfying an integrality condition can be realized as the invariants of a stitched fibration. Notice that $\bar{Z}_{\text{nor}}$ is uniquely determined by $\Gamma$ as $\bar{Z}_{\text{nor}}=T^{\ast}\Gamma/\bar{\Lambda}$, where $\bar{\Lambda}=\spn\langle{db_{2},\ldots},{db_{n}}\rangle_{\mathbb{Z}}$. We have

We also have the following (cf. [2]Theorem 6.11):

In other words, the set of germs of stitched fibrations is classified by the pairs $(\bar{Z}_{\text{nor}},\ell)$. We say that a fibration is fake stitched if it becomes smooth after an admissible change of coordinates on the base. One interesting consequence of Theorem 6.11, which we will exploit later on, is that from a given set of invariants we can form another one for example by summing to the sequence $\ell$ another sequence or by multiplying elements $\ell_{k}$ by pull backs of smooth functions on the base. The new invariants give rise to new stitched fibrations.