# Asymptotics of action variables near semi-toric singularities

Christophe Wacheux111 Collaborateur scientifique at EPFL,
<EMAIL_ADDRESS>
###### Abstract

The presence of focus-focus singularities in semi-toric integrables
Hamiltonian systems is one of the reasons why there cannot exist global
Action-Angle coordinates on such systems. At focus-focus critical points, the
Liouville-Arnold-Mineur theorem does not apply. In particular, the affine
structure of the image of the moment map around has non-trivial monodromy. In
this article, we establish that the singular behaviour and the multi-vauedness
of the Action integrals is given by a complex logarithm. This extends a
previous result by San Vũ Ngọc to any dimension. We also calculate the
monodromy matrix for these systems.

###### keywords:

$37J30$ , $37J05$ , $53D20$ , Semi-Toric systems , Moment maps

##  1 Introduction, definitions and notations

Given a symplectic manifold $(M^{2n},\omega)$, an integrable Hamiltonian
system (or IHS) can be defined as a function
$F=(f_{1},\ldots,f_{n}):M\to{\mathbb{R}}^{n}$ such that its components are
Poisson-commuting and whose differential is of maximal rank almost everywhere.
From now, $F$ will always designate for us an IHS. A point $p\in M$ such that
$dF(p)$ is of rank $n$ is called a regular point for $F$; it is called
critical if otherwise, and in particular it is called fixed if $dF(p)=0$. We
shall note the rank $k_{\textup{x}}(p)$, or just $k_{\textup{x}}$ if the
context is obvious.

For IHS, the famous Liouville-Arnold-Mineur theorem provides a particularly
appropriate set of local coordinates near regular points, the Action-Angle
coordinates. It can be formulated by considering the foliation $\mathcal{F}$
given by the connected components of the fibers of $F$. The theorem states
that for regular leaves of $\mathcal{F}$ (i.e. leaves without critical
points), the germ of foliation is locally a fibration by Lagrangian tori.

The problem is that a generic IHS does have critical points on which one
cannot apply Liouville-Arnold-Mineur theorem. One question then, is to examine
what can be preserved of the inital result for critical points. Another
question is to find the largest open subset of the set of regular points of
$F$ on which the period bundle can be trivialized, that is, what are the
obstructions to having _global_ Action-Angle coordinates.

Over the last decades, a lot of work has been produced for both questions. The
study of non-degenerate critical points of IHS goes back from the works of
Birkhoff and Williamson [34], to the works of Bolsinov and Fomenko [2], Rüss,
Colin de Verdière, Vey [12], Eliasson, Zung, Miranda, San Vũ Ngọc [28],
Chaperon [4][3] and many other, including the author [30]. Concerning the
existence of global Action-Angle coordinates, we cite the works of Duistermaat
who established among other obstructions to global Action-Angle coordinates
the monodromy phenomenon which occurs in particular in our case, and
explicited the matrix associated to it [11],[8]. Dazord and Delzant [5]
extended the study to coisotropic foliations, while Dufour, Molino and Toulet
began to study the case with critical points [9],[10].

This article deals mostly with the first question, yet the two questions are
deeply linked. As we shall see, once we have the proper local model to
describe what occurs near focus-focus singularities, the monodromy matrix
becomes very easy to calculate. We consider mainly singularity of maximal
corank (see precise definition in section 2.2 for precise definition). Our
main result is the existence of suitable coordinates, in which the action
integrals near the focus-focus singulaity have a simple expression where the
singular behaviour and the multi-valuedness is expressed by a simple complex
logarithm (Theorem 2.10). With these coordinates, it is possible to compute
the monodromy matrix. The two results we establish here extend the result
proved in [28] for the case $2n=4$ to higher dimension and with possible
elliptic components.

The article is organised as following: first, we set the necessary notions
nedded for a precise formulation of the result. Next, we present in details
the counterpart of Morse theory for integrable Hamiltonian systems at the
local and semi-global scale. Then, we prove our main result, compute the
topological monodromy matrix and give a comment to the case with elliptic
components.

##  2 Statement of the result

In order to give a precise formulation of our result, we need to recall the
notion of non-degeneracy for integrable Hamiltonian systems. We shall discuss
the counterpart of the Morse theory that we obtain in that framework.

###  2.1 Critical points in integrable Hamiltonian systems

Remember first that ${\mathcal{C}^{\infty}}((M,\omega)\to{\mathbb{R}})$ is
naturally equipped with a Poisson bracket $\\{.,.\\}$ such that
$({\mathcal{C}^{\infty}}((M,\omega)\to{\mathbb{R}}),\\{.,.\\})$ is a Lie
algebra. At a fixed point $p$ of a function
$u\in{\mathcal{C}^{\infty}}(M\to{\mathbb{R}})$, one can associate a quadratic
form $\mathcal{H}[u]_{p}\in S^{2}(T_{p}M)^{*}$ by taking the Hessian of $u$ in
a local set of coordinates. It is well defined (it does not depends of the
local coordinates) because $p$ is a fixed point.

The symplectic form $\omega$ induces a Poisson bracket $\\{.,.\\}_{p}$ on
$S^{2}(T_{p}M)^{*}$ in the following way:

$\\{\mathcal{H}[f]_{p},\mathcal{H}[g]_{p}\\}_{p}:=\mathcal{H}[\\{f,g\\}]_{p},$

and $(S^{2}(T_{p}M)^{*},\\{.,.\\}_{p})$ is a Lie algebra isomorphic to
$sp(2n,{\mathbb{R}})$, the Lie algebra of the symplectic group. Now, reminding
that a Cartan subalgebra of a Lie algebra $A$ is a subalgebra of $A$ which is
abelian and self-centralizing, we can define the following

######  Definition 2.1.

A fixed point $p$ for $F$ is said to be _non-degenerate_ if
$\langle\mathcal{H}[f_{1}]_{p},\ldots,\mathcal{H}[f_{n}]_{p}\rangle$, the
subalgebra spanned by the Hessians of the components of $F$ at $p$, is a
Cartan subalgebra of $(S^{2}(T_{p}M)^{*},\\{.,.\\}_{p})$.

Now, to define a non-degeneracy condition for critical points of arbitrary
rank, we remark that $S^{2}(T_{p}M)^{*}$ is isomorphic as a Lie algebra to a
subalgebra of $\mathcal{Q}({\mathbb{R}}^{2n}\to{\mathbb{R}})$, the algebra of
quadratic forms of ${\mathbb{R}}^{2n}$. We consider a critical point $m$ of
$M$ for $F$ of rank $k_{\textup{x}}$, and we may assume without loss of
generality that $df_{n-k_{\textup{x}}+1}\wedge\ldots\wedge df_{n}\neq 0$. We
can thus apply Darboux-Caratheodory theorem to the system $\langle
f_{n-k_{\textup{x}}+1},\ldots,f_{n}\rangle$: there exists a symplectomorphism
$\varphi:(\mathcal{U},\omega)\to({\mathbb{R}}^{2n},\sum_{i=1}^{n}d\tilde{f_{i}}\wedge
du_{i})$ with $\varphi(m)=0$ and such that
$\tilde{f_{j}}=f_{j}\circ\varphi^{-1}-f_{j}(m)$ are canonical coordinates
$\xi_{j}$ for $j\geqslant n-k_{\textup{x}}+1$. In these local coordinates,
since the $f_{j}$ are Poisson commuting, $f_{1},\ldots,f_{n-k_{\textup{x}}}$
do not depend of $x_{n-k_{\textup{x}}+1},\ldots,x_{n}$. We define the function
$g_{j}:{\mathbb{R}}^{2(n-k_{\textup{x}})}\to{\mathbb{R}}$, $1\leqslant
j\leqslant n-k_{\textup{x}}$ as
$g_{j}(\bar{x},\bar{\xi})=f_{j}^{\prime}(\bar{x},0,\bar{\xi},0)$.

######  Definition 2.2.

A critical point of rank $k_{\textup{x}}$ is called _non-degenerate_ if for
the $g_{i}$’s defined above, the Hessians $\mathcal{H}[g_{i}]_{0}$ span a
Cartan subalgebra of
$(\mathcal{Q}({\mathbb{R}}^{2(n-k_{\textup{x}})}\to{\mathbb{R}}),\varphi_{*}(\\{.,.\\}_{p}))$.
A Hamiltonian system is called non-degenerate if all its critical points are
non-degenerate.

###  2.2 Toric and semi-toric systems

Relying on the decomposition of Cartan subalgebras of $sp(2n)$, one can give
the following classification result of non-degenerate critical points due to
Williamson [34]:

######  Theorem 2.3.

Let $p\in M$ a nondegenerate critical point for $F$ an IHS. Then there exists
a symplectomorphism
$\varphi:(M,\omega,p)\to({\mathbb{R}}^{2n},\omega_{0}=\sum_{i=1}^{n}d\xi_{i}\wedge
dx_{i},0)$ such that

$\varphi^{*}F=(\textup{e}_{1},\ldots,\textup{e}_{k_{\textup{e}}},\textup{h}_{1},\ldots,\textup{h}_{k_{\textup{h}}},\textup{f}^{1}_{1},\textup{f}^{2}_{1},\ldots,\textup{f}^{1}_{k_{\textup{f}}},\textup{f}^{2}_{k_{\textup{f}}},\xi_{n-k_{\textup{x}}+1},\ldots,\xi_{n})+o(2)\text{
with:}$

  * 1.

$\textup{e}_{i}=x^{2}_{i}+\xi^{2}_{i}$

  * 2.

$\textup{h}_{i}=x_{i}y_{i}$

  * 3.

$\begin{cases}\textup{f}^{1}_{i}=x^{1}_{i}\xi^{1}_{i}+x^{2}_{i}\xi^{2}_{i}\\\
\textup{f}^{2}_{i}=x^{1}_{i}\xi^{2}_{i}-x^{2}_{i}\xi^{1}_{i}\end{cases}$

The theorem introduces three classes of possible “components” at a critical
point (apart from the regular components $\xi_{i}$) : elliptic (the
$\textup{e}_{i}$’s), hyperbolic (the $\textup{h}_{i}$’s), and focus-focus (the
couples $\textup{f}_{i}=(\textup{f}^{1}_{i},\textup{f}^{2}_{i})$). We can thus
define the following notations

######  Definition 2.4.

Given $(M,\omega,F)$ an IHS, and we can associate to $m\in M$ the following
_Williamson type_ (with respect to $F$), or _Williamson index_
$\Bbbk=(k_{\textup{e}},k_{\textup{f}},k_{\textup{h}},k_{\textup{x}})\in{\mathbb{N}}^{4}$
with

  * 1.

$k_{\textup{e}}-$ number of _elliptic_ (or $E$) components,

  * 2.

$k_{\textup{f}}-$ number of _focus-focus_ (or $FF$) components,

  * 3.

$k_{\textup{h}}-$ number of _hyperbolic_ (or $H$) components,

  * 4.

$k_{\textup{x}}-$ number of _transverse_ (or $X$) components, that is the
regular components.

We may also use the notation
$FF^{k_{\textup{f}}}-E^{k_{\textup{e}}}-H^{k_{\textup{h}}}-X^{k_{\textup{x}}}$
instead of $\Bbbk$. We also define

$Q_{\Bbbk}:=(\textup{e}_{1},\ldots,\textup{e}_{k_{\textup{e}}},\textup{f}^{1}_{1},\textup{f}^{2}_{1},\ldots,\textup{f}^{1}_{k_{\textup{f}}},\textup{f}^{2}_{k_{\textup{f}}},\textup{h}_{1},\ldots,\textup{h}_{k_{\textup{h}}},\xi_{1},\ldots,\xi_{k_{\textup{x}}})$

Note that the four coefficients are linked by the following equation

$k_{\textup{e}}+2k_{\textup{f}}+k_{\textup{h}}+k_{\textup{x}}=n$ (1)

######  Definition 2.5.

The set $\mathcal{W}(F)$ is defined as the set of different Williamson types
that occurs for a given IHS $F$. When equipped with the following relation

$\Bbbk\preccurlyeq\Bbbk^{\prime}\text{ if : }k_{\textup{e}}\geqslant
k_{\textup{e}}^{\prime},k_{\textup{f}}\geqslant k_{\textup{f}}^{\prime}\text{
and }k_{\textup{h}}\geqslant k_{\textup{h}}^{\prime},$

it is a (partially) ordered set (the term _poset_ also appears in the
litterature).

Let us show that $(\mathcal{W}(F),\preccurlyeq)$ is an ordered set. Let
$\Bbbk,\Bbbk^{\prime},\Bbbk^{\prime\prime}\in\mathcal{W}(F)$.

  * 1.

reflexivity: we always have $k_{\textup{e}}\geqslant
k_{\textup{e}},k_{\textup{f}}\geqslant k_{\textup{f}}$, and
$k_{\textup{h}}\geqslant k_{\textup{h}}$, thus $\Bbbk\preccurlyeq\Bbbk$,

  * 2.

antisymetry: if $\Bbbk\preccurlyeq\Bbbk^{\prime}$ and
$\Bbbk^{\prime}\preccurlyeq\Bbbk$, then $k_{\textup{e}}\geqslant
k_{\textup{e}}^{\prime}$, $k_{\textup{f}}\geqslant k_{\textup{f}}^{\prime}$,
$k_{\textup{h}}\geqslant k_{\textup{h}}^{\prime}$ and $k_{\textup{e}}\leqslant
k_{\textup{e}}^{\prime}$, $k_{\textup{f}}\leqslant k_{\textup{f}}^{\prime}$,
$k_{\textup{h}}\leqslant k_{\textup{h}}^{\prime}$, so
$k_{\textup{e}}=k_{\textup{e}}^{\prime}$,
$k_{\textup{f}}=k_{\textup{f}}^{\prime}$,
$k_{\textup{h}}=k_{\textup{h}}^{\prime}$ and hence, by equation 1 we have
$k_{\textup{x}}=k_{\textup{x}}^{\prime}$, so $\Bbbk=\Bbbk^{\prime}$,

  * 3.

transitivity: if $\Bbbk\preccurlyeq\Bbbk^{\prime}$ and
$\Bbbk^{\prime}\preccurlyeq\Bbbk^{\prime\prime}$ then $k_{\textup{e}}\geqslant
k_{\textup{e}}^{\prime}\geqslant
k_{\textup{e}}^{\prime\prime},k_{\textup{f}}\geqslant
k_{\textup{f}}^{\prime}\geqslant
k_{\textup{f}}^{\prime\prime},k_{\textup{h}}\geqslant
k_{\textup{h}}^{\prime}\geqslant k_{\textup{h}}^{\prime\prime}$, hence
$\Bbbk\preccurlyeq\Bbbk^{\prime\prime}$.

This is the ordered set involved in the stratification mentioned in section
4.3. We can also define consistently the Williamson type of a leaf (see 3.2)
as follows

######  Definition 2.6.

Given a leaf $\Lambda\in\mathcal{F}$, the Williamson type $\Bbbk(\Lambda)$, or
$\Bbbk$ if it is unambiguous, is the Williamson type of the point of smallest
rank.

Lastly, we introduce the useful following sets

######  Definition 2.7.

Let $\mathcal{U}\in M$ be an open set. We define
$P_{\Bbbk}(\mathcal{U})\subseteq\mathcal{U}$ (resp.
$L_{\Bbbk}(\mathcal{U})\subseteq M$) is the set of critical points (resp.
leaves) of type $\Bbbk$ in $\mathcal{U}$. Finally, we define
$V_{\Bbbk}(\mathcal{U}):=F(L_{\Bbbk}(\mathcal{U}))$, the set of critical
values in $F(\mathcal{U})$ of Williamson type $\Bbbk$.

######  Remark 2.8.

In the above definition, $P_{\Bbbk}$ (resp. $L_{\Bbbk}$, resp. $V_{\Bbbk}$)
are covariant functors from the category of open sets with inclusions as
morphisms, to the categories of subsets of $M$( resp. union of leaves of
$\mathcal{F}$, resp. subsets of $F(M)$), with inclusions as morphisms. Since
the fibers of the moment map are not a priori connected, a critical value may
belong to $V_{\Bbbk}(\mathcal{U})$ and $V_{\Bbbk^{\prime}}(\mathcal{V})$ with
$\Bbbk$ and $\Bbbk^{\prime}$ possibly different, and $\mathcal{U}$ and
$\mathcal{V}$ possibly disjoint.

The Williamson type is a symplectic invariant. The aim of this article is to
examine the asymptotic behavior of Action-Angle coordinates when getting close
to a singularity with one focus-focus component and no hyperbolic component.

This question is actually motivated by a long-term program of classification
of IHS based on their dynamical behaviour. While such classification for
general systems is out of reach for now, partial results exists for subclasses
of IHS. To formulate some of these results, we introduce here a criterium
called “complexity”. The notion of complexity find its origins in the works of
Karshon an Tolman [17][18][19], Margaret Symington and Leung [25], [20], and
of San Vũ Ngọc in [29].

######  Definition 2.9.

Let $F=(f_{1},\ldots,f_{n}):(M^{2n},\omega)\to{\mathbb{R}}^{n}$ be an
integrable Hamiltonian system. It is said to be _almost-toric_ of complexity
$c\leqslant n$ if (up to a global permutation of the components of $F$), every
critical point verifies these conditions:

  * 1.

all critical points are non-degenerate.

  * 2.

there are no singularities of hyperbolic type: $k_{\textup{h}}=0$,

  * 3.

the function $\check{F}^{c}=(f_{c+1},\ldots,f_{n})$ generates a _global_
${\mathbb{T}}^{n-c}$-action.

An almost-toric system is called _semi-toric_ if $c=1$, and _toric_ if $c=0$.
For a semi-toric system, $\check{F}:=(f_{2},\ldots,f_{n})$ is the map that
generates the ${\mathbb{T}}^{n-1}$-action.

The classification of toric IHS begins with Liouville Arnold Mineur theorem :
the Action-Angle coordinates provide an (integral) affine structure on the
base space. It was shown in 1982, simultaneously by Atiyah [1], and Guillemin
& Sternberg [1], [15], [16], that for an Hamiltonian
${\mathbb{T}}^{d}$-action, the image of the moment map is a rational convex
polytope in ${\mathbb{R}}^{d}$. Delzant, in 1988, gave a constructive proof
that, in the completely integrable case $d=n$, if the action is effective, the
polytope completely determines the system, thus completing the classification
for toric IHS [6] [7].

For the semi-toric case though, the image of moment map is not a polytope, and
it is not enough to classify such IHS. However, San Vũ Ngọc and Pelayo use it
to get a classification “à la Delzant” for semi-toric systems in dimension
$2n=4$ [28],[23],[24]. Among the other classifying invariants they introduce,
there is a formal series that is explicited as the Taylor expansion of
regularized Action coordinate, thus describing, in a way, how the Lagrangian
fibration pinches at a semi-toric singularity. The principal goal of this
article is to give the general formula of this invariant, for any semi-toric
critical point in any dimension.

###  2.3 The main result

A semi-toric IHS only has elliptic critical points, which are well understood
from the study of toric system, and critical points of Williamson type
$FF-X^{k_{\textup{x}}}-E^{k_{\textup{e}}}$. From [32] we know the set
$\Gamma:=V_{FF-X^{k_{\textup{x}}}-E^{k_{\textup{e}}}}$ is a
$k_{\textup{x}}$-dimensional submanifold. We note $D^{r}Int$ the disk of
dimension $r$. In this article, we prove the following result.

######  Theorem 2.10.

Let $(M^{2n},\omega,F=(f_{1},\ldots,f_{n}))$ be a semi-toric integrable system
with $n$ degrees of freedom. Let $m$ be a $FF-X^{n-2}$ critical point,
$\Lambda_{0}$ the leaf containing $m$ and $\mathcal{U}(\Lambda_{0})$ (a germ
of) a tubular neighborhood of $\Lambda_{0}$ such that
$V_{X^{n}}(\mathcal{U}(\Lambda_{0}))$ is simply connected.

There exists (a germ of) a tubular neighborhood $W\subseteq{\mathbb{R}}^{n}$
of $W_{0}:=\\{0\\}\times\\{0\\}\times D^{n-2}$, a local diffeomorphism

$G:(W,W_{0})\to(F(\mathcal{U}(\Lambda_{0})),\Gamma),$

and a symplectomorphism

$\varphi:\left(L_{X^{n}}(\mathcal{U}(\Lambda_{0})),\omega\right)\to\left((W\setminus
W_{0})\times{\mathbb{T}}^{n},\tilde{\omega}\right)$

such that if we write
$\varphi=(I_{1},\ldots,I_{n},\theta_{1},\ldots,\theta_{n})$ we have

  1. 1.

The coordinates $(\theta_{3},\ldots,\theta_{n},I_{3},\ldots,I_{n})$ can be
extended to partial Action-Angle coordinates on $\mathcal{U}(\Lambda_{0})$.

  2. 2.

We have that

$\displaystyle I_{1}(\mathbf{v})$
$\displaystyle=S(\mathbf{v})-\mathop{\mathfrak{R}e}\nolimits(w\ln(w)-w)$
$\displaystyle I_{2}(\mathbf{v})$ $\displaystyle=v_{2}$ $\displaystyle I_{3}$
$\displaystyle=v_{3}$ $\displaystyle\ \vdots$ $\displaystyle I_{n}$
$\displaystyle=v_{n}$

where $\mathbf{v}=(v_{1},\ldots,v_{n})=G^{-1}\circ F$, $w=v_{1}+iv_{2}$, $\ln$
is a determination of the complex logarithm on
$W_{v_{1},v_{2}}=W\cap\\{v_{3}=\ldots=v_{n}=0\\}\subseteq{\mathbb{R}}^{2}\simeq{\mathbb{C}}$,
and $S$ a smooth function of $\mathbf{v}$.

From this therorem in the case of maximal codimension, we treat the general
case with possible elliptic components. While it is quite intuitive, we shall
define precisely the meaning of “partial Action-Angle coordinates” in section
3.2. To prove this result, some genericity assumptions are necessary, which we
shall indicate during the proof.

This result was obtained during my Ph.D. thesis. We present it here in the
simplest manner, with the addition of the case wwith elliptic components

##  3 A symplectic Morse theory for integrable Hamiltonian systems

The non-degeneracy condition we defined calls for an equivalent of the Morse
theory in the symplectic framework and integrable Hamiltonian systems. We
shall see that there are different level at which we can establish
counterparts of classical results in Morse theory.

###  3.1  Local and orbital theory

The first difference with classical Morse theory is that instead of a single
function, we have a family of $n$ real-functions. We exploit the relations
between the functions to get a normal form theorem, due to Eliasson. The
version of the theorem given here is an extension of the original theorem
which incorporates partial Action-Angle coordinates for the transverse
components. This version is due to Miranda and Zung [21]. Also, we give the
version only for semi-toric singularites

######  Theorem 3.1 (Eliasson-Miranda-Zung Normal Form - Semi-toric case).

Let $(M^{2n},\omega,F)$ a semi-toric integrable system with $n\geqslant 2$,
$F$ an IHS. Let $m$ be a critical point of Williamson type
$\Bbbk=FF-E^{k_{\textup{e}}}-X^{k_{\textup{x}}}$ and
$F_{X}:=(f_{n-k_{\textup{x}}+1},\ldots,f_{n})$ the transverse components in
Williamson decomposition of $F$ at $m$ that provide a global
${\mathbb{T}}^{n-2}$-action on $M$.

Then there exists a triplet $(\mathcal{U}_{m},\varphi_{\Bbbk},G_{\Bbbk})$ with
$\mathcal{U}_{m}$ an open neighborhood of $m$ saturated with respect to the
orbit $\mathcal{T}$ of $F_{X}$, $\varphi_{\Bbbk}$ a symplectomorphism of
$\mathcal{U}_{m}$ to a neighborhood of $(0\in{\mathbb{R}}^{2n},\omega_{0})$
that sends the orbit $\mathcal{T}\simeq{\mathbb{T}}^{k_{\textup{x}}}$ of $m$
to $O$, and $G_{\Bbbk}$ a local diffeomorphism of $0\in{\mathbb{R}}^{n}$ such
that:

$\varphi^{*}_{\Bbbk}F=G_{\Bbbk}\circ Q_{\Bbbk}$

This is a counterpart of the Morse lemma, here adapted to the symplectic
context: in a neighborhood of a non-degenerate critical point, an integrable
system is, up to a regular change of coordinates, a function of of the
quadratic parts determined by the Williamson type of the critical point. As we
said before, we have here $k_{\textup{x}}$ Action-Angle coordinates, thus
allowing us to get a normal form on a “wider” open set. This is what one may
call a semi-local, or an orbital result.

It was the contribution of many people that allowed eventually the statement
and proof of the original theorem of Eliasson for fixed points. The first
works to be cited here are those of Birkhoff, Vey [26], Colin de Verdière and
Vey [12], and of course Eliasson in [13] and [14]. More recently, Chaperon in
[4] and [3], Zung in [36] and [37], and San Vũ Ngọc & the author in [30]
provided new proofs and filled the technical gaps that remained in the
original proof. Miranda and Zung in [21], relying on Eliasson local normal
form provided the orbital version.

Eliasson normal form allows us to visualize the image of a neighborhoud of
non-degenerate critical points. Here is a picture of the different sets of
critical values that can occur in dimension $2n=4$.

Figure 1: Local models of critical points in dimension 2, [22]

####  3.1.1  Dynamic of the local models

A description of the Lagrangian torus foliation near a semi-toric leaf starts
with the study of the dynamic near the critical point. For each type of
critical component, it is always possible to introduce a local model using
complex coordinates. For elliptic and hyperbolic components, we introduce
natural complex coordinates established after Darboux coordinates $z=x+i\xi$,
along with their respective elliptic $\textup{e}=z\bar{z})$ and hyperbolic
$\textup{h}=\mathop{\mathfrak{I}m}\nolimits(-\frac{1}{2}z^{2})$ Hamiltonian.
However, for focus-focus components we take the following 4-dimensional model:

$\displaystyle({\mathbb{R}}^{4},0)$
$\displaystyle\xrightarrow{\;\cong\;}({\mathbb{C}}^{2},0)$
$\displaystyle(x_{1},x_{2},\xi_{1},\xi 2)$
$\displaystyle\longmapsto(\underbrace{x_{1}+ix_{2}}_{=:z_{1}},\underbrace{\xi_{1}+i\xi_{2}}_{=:z_{2}})$
$\displaystyle\omega=dx_{1}\wedge d\xi_{1}+dx_{2}\wedge d\xi_{2}$
$\displaystyle\longmapsto\omega=\mathop{\mathfrak{R}e}\nolimits(d\bar{z_{1}}\wedge
dz_{2})$ $\displaystyle\textup{f}=(\textup{f}^{1},\textup{f}^{2})$
$\displaystyle\longmapsto\textup{f}^{1}+i\textup{f}^{2}=\bar{z_{1}}z_{2}.$

We summarize in the array below some of the dynamical properties of each
component.

Component |  Critical leaf |  Expression of the flow in local coordinates
---|---|---
Elliptic |  $\\{\textup{e}=0\\}$ is a point in ${\mathbb{R}}^{2}$ |  $\phi_{\textup{e}}^{t}:z^{\textup{e}}\mapsto e^{it}z^{\textup{e}}$
Focus-focus |  $\\{f^{1}=f^{2}=0\\}$ is the union of the planes $\\{x_{1}=x_{2}=0\\}$ and $\\{\xi_{1}=\xi_{2}=0\\}$ in ${\mathbb{R}}^{4}$ |  $\phi_{f^{1}}^{t}:(z^{f}_{1},z^{f}_{2})\mapsto(e^{-t}z^{f}_{1},e^{t}z^{f}_{2})$ and $\phi_{f^{2}}^{t}:(z^{f}_{1},z^{f}_{2})\mapsto(e^{it}z^{f}_{1},e^{it}z^{f}_{2})$
Table 1: Properties of elementary blocks

Note also that f, which is here the complex counterpart to the Hamiltonian,
reminds of a hyperbolic singularity. This is why focus-focus critical points
are also called “complex hyperbolic” or also “loxodromic” in the literrature.

We give below a representation of the dynamic near an elliptic and a focus-
focus critical point. For the elliptic case, the vector field
$\chi_{\textup{e}}$ is just the rotation around the critical point. For the
focus-focus case, each vector field acts simultaneously on the two complex
planes. The “pseudo-hyperbolic” field $\chi_{\textup{f}^{1}}$ is along radial
trajectories on each plane, that is, half-lines starting at the focus-focus
critical point, while the “pseudo-elliptic” field $\chi_{\textup{f}^{2}}$ is
just the rotation around the focus-focus critical point. Here is a picture of
these two fields in dimension $2n=4$, with $\chi_{1}=\chi_{f^{1}}$ and
$\chi_{2}=\chi_{f^{2}}$.

Figure 2: Linear model of focus-focus critical point, [27]

###  3.2 Semi-global (leaf-wise) theory

Our goal here is to formulate a version of the normal form for a critical
leaf, and the foliation near it, that is, a Liouville-Arnold-Mineur theorem
for critical points. There are several problems we need to examine in order to
be able to formulate The first problem is the following. In symplectic
geometry, the orbit of a point $m$ by the ${\mathbb{R}}^{n}$ action induced by
the moment map is contained in the leaf that contains $m$, and it is easy to
see that each point of an orbit has the same Williamson type. However, while
for elliptic critical points the orbit is exactly the leaf, for hyperbolic and
focus-focus critical points this is not the case. In the semi-toric case in
dimension $2n=4$, for instance, the leaf containing the focus-focus point
contains also points for which the action is of rank 2 (in the local model,
they are the points for which $z_{1}$ or $z_{2}$ is not zero).

Fortunately, in [35], Zung describes the stratification of a given leaf
$\Lambda\in\mathcal{F}$ by the orbits of its points, and proves that all the
orbits with lowest rank have the same Williamson type, and it is thus an
invariant of the leaf. This justifies Definition 2.6

One important point also when we consider semi-toric leaves, is that a semi-
toric leaf (a leaf that contains a critical point with $k_{\textup{f}}=1$) may
carry more than one semi-toric orbit. We will not consider this case for the
present time

######  Assumption 3.2.

From now on, for $F$ a semi-toric IHS, the leaves of $F$ will only carry at
most one semi-toric orbit.

In order to expose Arnol’d-Liouville theorem for critical points, we need some
definitions.

######  Definition 3.3.

A (non-degenerate) _singularity_ shall be defined as (a germ of) a tubular
neighborhood of a (non-degenerate) leaf. We adopt here the following notation:
for $c$ in $B$ the base space of $\mathcal{F}$ and $\pi_{\mathcal{F}}:M\to B$
the projection map, $\Lambda_{c}:=\pi_{\mathcal{F}}^{-1}(c)$ is the leaf of
all points in $M$ over $c$ in the fibration, and $\mathcal{U}(\Lambda_{c})$ a
tubular neighborhood of $\Lambda_{c}$.

Two singularities are isomorphic if they are leaf-wise isomorphic.

There is a mild assumption concerning critical leaves that is required to
formulate the central result of the theorem.

######  Definition 3.4.

A non-degenerate critical leaf $\Lambda$ is called _topologically stable_ if
there exists a tubular neighborhood $\mathcal{V}$ of $\Lambda$ and a
$\mathcal{U}\subset\mathcal{V}(\Lambda)$ a small neighborhood of a point $m$
of minimal rank, such that

$\forall\Bbbk\in\mathcal{W}(F),\
V_{\Bbbk}(\mathcal{V}(\Lambda))=V_{\Bbbk}(\mathcal{U}))\ .$

An integrable system will be called _topologically stable_ if all its critical
points are non-degenerate and topologically stable.

######  Assumption 3.5.

From now on, all of our systems will be topologically stable.

The assumption of topological stability rules out some pathological behaviours
that can occur for general foliations. Note however that for all known
examples, the non-degenerate critical leaves are all topologically stable, and
it is conjectured that this is also the case for all analytic systems.

######  Definition 3.6.

  * 1.

A singularity is called of _(simple) elliptic type_ if it is isomorphic to
$\mathcal{L}^{e}$: a plane ${\mathbb{R}}^{2}$ foliated by $e=x^{2}+\xi^{2}$.

  * 2.

A singularity is called of _(simple) focus-focus type_ if it is isomorphic to
$\mathcal{L}^{f}$, where $\mathcal{L}^{f}$ is given by ${\mathbb{R}}^{4}$
locally foliated by $f^{1}=x_{1}x_{2}+\xi_{1}\xi_{2}$ and
$f^{2}=x_{1}\xi_{2}-x_{2}\xi_{1}$.

Topological properties of simple elliptic, hyperbolic and focus-focus
singularities are discussed in details in [35],[36],[37],[27]. In particular,
one can show that the focus-focus critical leaf must be homeomorphic to a
pinched torus that we note $\check{{\mathbb{T}}}^{2}$; it is topologically
equivalent to a 2-sphere with two points identified. The regular leaves around
are regular tori.

####  3.2.1 Singular Arnold-Liouville theorem

Now we can formulate an extension of Liouville-Arnold-Mineur theorem to
singular leaves. We call the next theorem a “leaf-wise” result, as the results
given hold for a _leaf_ of the system. However assertion $2.$ of the theorem
does not extend Eliasson normal form, since here the normal form of the leaf
doesn’t preserve the symplectic structure. Again, we only give the semi-toric
version.

######  Theorem 3.7 (Arnold-Liouville with semi-toric singularities, [35]).

Let $F$ be a proper semi-toric system, $\Lambda$ be a non-degenerate critical
leaf of Williamson type $\Bbbk$ and $\mathcal{V}(\Lambda)$ a tubular
neighborhood of $\Lambda$.

Then the following statements are true:

  1. 1.

There exists an effective Hamiltonian action of
${\mathbb{T}}^{k_{\textup{e}}+k_{\textup{f}}+k_{\textup{x}}}$ on
$\mathcal{V}(\Lambda)$. There is a locally free
${\mathbb{T}}^{k_{\textup{x}}}$-subaction. The number
$k_{\textup{e}}+k_{\textup{f}}+k_{\textup{x}}$ is the maximal possible for an
effective Hamiltonian action.

  2. 2.

If $\Lambda$ is topologically stable, on $(\mathcal{V}(\Lambda)$ the foliation
$\mathcal{F}$ is leaf-wise homeomorphic (and even diffeomorphic) to a product
of elliptic and focus-focus simple singularities:

$(\mathcal{V}(\Lambda),\mathcal{F})\simeq(\mathcal{U}({\mathbb{T}}^{k_{\textup{x}}}),\mathcal{F}_{\textup{x}})\times\mathcal{L}^{\textup{e}}_{1}\times\ldots\times\mathcal{L}^{\textup{e}}_{k_{\textup{e}}}\times\mathcal{L}^{\textup{f}}_{1}\times\ldots\times\mathcal{L}^{\textup{f}}_{k_{\textup{f}}}$

where $(\mathcal{U}({\mathbb{T}}^{k_{\textup{x}}}),\mathcal{F}_{\textup{x}})$
is a foliation of the full torus
$(\mathcal{U}({\mathbb{T}}^{k_{\textup{x}}}),\mathcal{F}_{\textup{x}})\subseteq{\mathbb{R}}^{2k_{\textup{x}}}$
by tori ${\mathbb{T}}^{k_{\textup{x}}}$.

  3. 3.

There exists partial Action-Angle coordinates on $\mathcal{V}(\Lambda)$ :
there exists a diffeomorphism $\varphi$ on $\mathcal{V}(\Lambda)$ such that

$\varphi^{*}\omega=\sum_{i=1}^{k_{\textup{x}}}d\theta_{i}\wedge
dI_{i}+P^{*}\omega_{1}$

where
$(\theta_{1},\ldots,\theta_{k_{\textup{x}}},I_{1},\ldots,I_{k_{\textup{x}}})$
are the action-angle coordinates on $T^{*}\mathcal{T}$ ($\mathcal{T}$ being
$F_{X}$-orbit in Eliasson-Miranda-Zung Theorem 3.1), and $\omega_{1}$ is a
symplectic form on
${\mathbb{R}}^{2(n-k_{\textup{x}})}\simeq{\mathbb{R}}^{2(k_{\textup{e}}+2k_{\textup{f}})}$.

In Definition 3.6, the description of the simple focus-focus leaf is actually
a consequence of the existence of a Hamiltonian $S^{1}$-action on a tubular
neighborhood of a simple focus-focus singularity. It is important to note that
in statement 2. of Theorem 3.7, the decomposition is at best leaf-wise
diffeomorphic, but not symplectomorphic.

##  4 Proof of the main result

We solve the case in Theorem 2.10, and complete it with a comment on the case
where some components are elliptic.

###  4.1 The transversally focus-focus case

Let us consider a $FF-X^{n-2}$ singularity $\mathcal{U}(\Lambda_{0})$ of our
foliation $\mathcal{F}$. We already know with item 3. of Theorem 3.7 that
there exists partial Action-Angle on $\mathcal{U}(\Lambda)$ that are well
defined. They are Action-Angle coordinates associated to the
${\mathbb{T}}^{n-2}$-action induced by the transverse components of $F$. For
the two other coordinates, we have that

  1. 1.

one cannot define Action-Angle coordinates on the of $\mathcal{F}$,

  2. 2.

one can only define Action-Angle coordinates canonically on the set of regular
leaves $L_{X^{n}}(\mathcal{U}(\Lambda))$, if $V_{X^{n}}(\mathcal{U}(\Lambda))$
is simply connected: this is the monodromy phenomenon.

To show these two points, we define a complete set of Action-Angle coordinates
on the regular tori foliation near a focus-focus singularity, and give the
asymptotics of the Action coordinate near the critical point. To prove the
theorem, we follow and generalize each step of the proof of Section 3 in [28].
During the proof will arise what causes the monodromy phenomenon.

###### Proof.

of Theorem 2.10

With Zung’s theorem, we can always take $\mathcal{U}(\Lambda_{0})$ small
enough so that for all $\mathbf{v}\in F(\mathcal{U}(\Lambda_{0}))$, there is a
unique leaf in $\mathcal{U}(\Lambda_{0})$, that is, for the restricted system
$(M,\omega,\mathcal{F}|_{\mathcal{U}(\Lambda_{0})})$ the fibers are connected.
We then define, for $\mathbf{v}\in F(\mathcal{U}(\Lambda_{0}))$,
$\Lambda_{\mathbf{v}}=F^{-1}(\mathbf{v})$.

Item $2.$ of Theorem 3.7 gives us the topological (and differential)
description of the foliation on the singularity $\mathcal{U}(\Lambda_{0})$,
while Eliasson-Miranda-Zung normal form (Theorem 3.1) describes it
symplectically, but only on a neighborhood $\mathcal{U}_{MZ}$ of $m$,
$\mathcal{U}_{MZ}$ stable by the flow of $\check{F}$. On $\mathcal{U}_{MZ}$
there is a local symplectomorphism $\varphi:\mathcal{U}_{MZ}\to
T^{*}{\mathbb{R}}^{2}\times T^{*}{\mathbb{T}}^{n-2}$ and a local
diffeomorphism $G:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ such that, for
$Q_{FF-X^{n-2}}:=(f^{1},f^{2},\xi_{1},\ldots,\xi_{n-2})$,

$\varphi^{*}F=G\circ Q_{FF-X^{n-2}}$

From this form, we can see first that locally $\Gamma$ is indeed a
$k_{\textup{x}}$-dimensional submanifold.

Now let us have a point $A_{0}\in\mathcal{U}_{MZ}\cap\Lambda_{0}$ different
than $m$: $A_{0}$ is on the same critical fiber as $m$, and near enough so
that Eliasson-Miranda-Zung normal form can be used. We then set $\Sigma^{n}$ a
(small enough) $n$-dimensional submanifold of $M$ which intersects
transversally the foliation $\mathcal{F}$ at $A_{0}$, with
$V_{FF-X^{n-2}}(\mathcal{U}(\Lambda_{0}))\cap\Sigma^{n}=\emptyset$. We set
$\Omega:=\\{\Lambda_{c}\in\mathcal{F}\ |\
\Lambda_{c}\cap\Sigma^{n}\neq\emptyset\\}$. The open set $\Omega$ is in
$\mathcal{U}(\Lambda_{0})$.

We have that $G^{-1}\circ F$ is a global moment map for the foliation
$\mathcal{F}$ on $\Omega$. On $\mathcal{U}_{MZ}$, $G^{-1}\circ
F=(q_{1},q_{2},\xi_{1},\ldots,\xi_{n-2})$, so $G^{-1}\circ F$ is an extention
of $Q_{FF-X^{n-2}}$ to $\Omega$. We can now simply forget $G^{-1}$ and just
take $F$ for a global momentum map that extends $Q_{FF-X^{n-2}}$, and restrict
the system to $\Omega\simeq W$, with $\Gamma\simeq W_{0}$.

For all critical leaves in $L_{FF-X^{n-2}}(\Omega)$, all the $f_{1}$-orbits
are homoclinic orbits. Since $q_{2},\xi_{1},\ldots,\xi_{n-2}$ have
$2\pi$-periodic flows on $\mathcal{U}_{MZ}$ and since for their extension
$f_{2},\ldots,f_{n}$, all the $f_{i}$’s Poisson-commute, $\check{F}$ yield a
${\mathbb{T}}^{n-1}$-action on $\Omega$ that commutes with the flow of
$f_{1}$. The ${\mathbb{T}}^{n-2}$-action induced by $\check{F}^{2}$ is free
everywhere on $\Omega$ as told by item.1 of Theorem 3.7, while the
$S^{1}$-action of $f_{2}$ is free on $P_{X^{n}}(\Omega)$.

Since the leaves of $\mathcal{F}$ are compact, for $A\in\Sigma^{n}\setminus
L_{FF-X^{n-2}}(\Omega)$ we can define $A^{\prime}$ the point of first
intersection of the $f_{1}$-orbit of $A$ with the orbit by $\check{F}$, and
$\tau_{1}$ the time of intersection. Let $(\tau_{2},\ldots,\tau_{n})$ be the
multi-time needed to come back from $A^{\prime}$ to $A$ with the flows of
$f_{2},f_{3},\ldots,f_{n}$, hence closing the trajectory

$\phi_{f_{2}}^{\tau_{2}}\circ\cdots\circ\phi_{f_{n}}^{\tau_{n}}(A^{\prime})=A.$

Since the joint flow of $F$ is transitive these times depend only of the
Lagrangian torus and not of $A$, and thus, only of the values $\mathbf{v}$ of
$F$. For any regular value $\mathbf{v}$, the set of all the
$(\alpha_{1},\ldots,\alpha_{n})\in{\mathbb{R}}^{n}$ such that
$\alpha_{1}\chi_{f_{1}}(\mathbf{v})+\ldots+\alpha_{n}\chi_{f_{n}}(\mathbf{v})$
has a $1$-periodic flow is a sublattice of ${\mathbb{R}}^{n}$ called the
period lattice. The following matrix

$\mathcal{R}=\begin{pmatrix}\tau\\\ r_{2}\\\ \vdots\\\ \vdots\\\
r_{n}\end{pmatrix}=\begin{pmatrix}\tau_{1}(v)&\cdots&\cdots&\cdots&\tau_{n}(v)\\\
0&2\pi&0&\cdots&0\\\ \vdots&0&\ddots&\ddots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&0\\\ 0&0&\cdots&0&2\pi\end{pmatrix}$

forms a ${\mathbb{Z}}$-basis of the period lattice ($\tau_{1}>0$ by
definition). These vectors can also be seen as a basis of cycles of the
Lagrangian tori foliation $\mathcal{F}$ on $\Omega$. The next proposition
proves the second item of Theorem 2.10: it gives the singular behavior of the
basis as $\mathbf{v}$ tends to $\Gamma$.

######  Proposition 4.1.

Let us fix a determination of the complex logarithm: $\ln(w)$, where
$w=v_{1}+iv_{2}$. Then the following quantities

  * 1.

$\sigma_{1}(\mathbf{v})=\tau_{1}(\mathbf{v})+\Re e(\ln(w))\in{\mathbb{R}}$,

  * 2.

$\sigma_{2}(\mathbf{v})=\tau_{2}(\mathbf{v})-\Im
m(\ln(w))\in{{\mathbb{R}}}/{2\pi{\mathbb{Z}}}$,

  * 3.

$\sigma_{3}(\mathbf{v})=\tau_{3}(\mathbf{v})\in{\mathbb{R}}/2\pi{\mathbb{Z}}\
,\ldots,\
\sigma_{n}(\mathbf{v})=\tau_{n}(\mathbf{v})\in{{\mathbb{R}}}/{2\pi{\mathbb{Z}}}$,

are defined on $W\setminus W_{0}$ and can be extended to smooth and single-
valued functions of $\mathbf{v}$ in $W$. Moreover, the differential form

$\sigma=\sum_{i=1}^{n}\sigma_{i}dv_{i}$

is closed in $W$.

###### Proof.

We fix some small $\varepsilon>0$ and we set

$\Sigma_{u}^{\boldsymbol{\alpha}}:=\\{z_{1}=\varepsilon,z_{2}\text{ small
},{\boldsymbol{\theta}}=(\theta_{1},\ldots,\theta_{k_{\textup{x}}})={\boldsymbol{\alpha}},{\boldsymbol{\xi}}\in[-\varepsilon,\varepsilon]^{k_{\textup{x}}}\\}\subseteq\mathcal{U}_{MZ}$
$\Sigma_{s}^{\boldsymbol{\alpha}}:=\\{z_{1}\text{ small
},z_{2}=\varepsilon,{\boldsymbol{\theta}}=(\theta_{1},\ldots,\theta_{k_{\textup{x}}})={\boldsymbol{\alpha}},{\boldsymbol{\xi}}\in[-\varepsilon,\varepsilon]^{k_{\textup{x}}}\\}\subseteq\mathcal{U}_{MZ}.$

These are stable and unstable local submanifolds for the dynamic of the
$f_{1}$-flot near a critical point
$m^{{\boldsymbol{\alpha}}}_{{\boldsymbol{\xi}}}=(z_{1}=0,z_{2}=0,{\boldsymbol{\theta}}=(\theta_{1},\ldots,\theta_{n-2})={\boldsymbol{\alpha}},{\boldsymbol{\xi}})\in
CrP_{FF-X^{n-2}}(\Omega)$. They are $n$-dimensional submanifolds intersecting
transversally the foliation $\mathcal{F}$ on $\Omega$. Thus, the intersections
$A(\mathbf{v},{\boldsymbol{\alpha}}):=\Lambda_{\mathbf{v}}\cap\Sigma_{u}^{\boldsymbol{\alpha}}$
and
$B(\mathbf{v},{\boldsymbol{\alpha}}):=\Lambda_{\mathbf{v}}\cap\Sigma_{s}^{\boldsymbol{\alpha}}$
are points of $M$ in the same $\chi_{f_{1}}$-orbit for $\mathbf{v}\in
V_{X^{n}}(\Omega)$, at least for $A(\mathbf{v},{\boldsymbol{\alpha}})$ and
$B(\mathbf{v},{\boldsymbol{\alpha}})$ in $L_{X^{n}}(\Omega)$. They are well-
defined smooth functions of $\mathbf{v}$ and ${\boldsymbol{\alpha}}$.

The ${\mathbb{T}}^{n-1}$-orbits of $A(\mathbf{v},{\boldsymbol{\alpha}})$ and
$B(\mathbf{v},{\boldsymbol{\alpha}})$ are transversal to the Hamiltonian flow
of $f_{1}$, thus one can define
$\tau^{A,B}_{1}(\mathbf{v},{\boldsymbol{\alpha}})$ as the time necessary for
the Hamiltonian flow of $f_{1}$ starting at
$A(\mathbf{v},{\boldsymbol{\alpha}})$ (which flows outside of
$\mathcal{U}_{MZ}$), to make first hit to the ${\mathbb{T}}^{n-1}$-orbit of
$B(\mathbf{v},{\boldsymbol{\alpha}})$. Let us call this first hit
$B^{\prime}=(b_{1}^{\prime},b_{2}^{\prime},{\boldsymbol{\alpha}}+{\boldsymbol{\theta}},{\boldsymbol{\xi}})$.
Since the ${\mathbb{T}}^{n-1}$-orbit of $B$ is in $\mathcal{U}_{MZ}$, we know
that in it $f_{2}=q_{2}$, $f_{3}=\xi_{3}$, $\ldots$,$f_{n}=\xi_{n}$, so we
have the explicit expression for the time needed to get back to $B$, which we
call $\tau^{A,B}_{2}(\mathbf{v},\alpha)$, $\ldots$,
$\tau^{A,B}_{n}(\mathbf{v},\alpha)$:

$\tau^{A,B}_{2}(\mathbf{v},{\boldsymbol{\alpha}})=\arg(b_{1}^{\prime})\text{
and for }j=3\ldots n\ ,\
\tau^{A,B}_{j}(\mathbf{v},{\boldsymbol{\alpha}})=2\pi-\theta_{j}^{\prime}$

Since the $f_{i}$’s commute, the $\tau^{A,B}_{j}$ are smooth, single-valued
functions of $\mathbf{v}$ only. We can now interchange the roles of $A$ and
$B$, and thus, of $\Sigma_{u}^{\boldsymbol{\alpha}}$ and
$\Sigma_{s}^{\boldsymbol{\alpha}}$, to define the times $\tau^{B,A}_{j}(v)$
for $j=1,\ldots,n$. The joint flow of $F$ now takes place inside
$\mathcal{U}_{MZ}$ where $L_{FF-X^{n-2}}(\Omega)\cap\mathcal{U}_{MZ}$ is a
codimension-$2$ manifold, so for $\mathbf{v}\in\Gamma$, the quantities
$\tau^{B,A}_{1}(v),\tau^{B,A}_{2}(v),\ldots,\tau^{B,A}_{n}(v)$ cannot be
defined a priori.

However, one can see that in the definition, $\tau^{B,A}_{1}(v)$ and
$\tau^{B,A}_{2}(v)$ do not depend of the value of $(f_{3},\ldots,f_{n})$: in
Eliasson-Miranda-Zung theorem, the local model is a _direct product_ of the
Eliasson normal form for the focus-focus and the Action-Angle coordinate for
the transversal component. Moreover, since everywhere it is defined, for
$j=3,\ldots,n,\tau^{B,A}_{j}(v)=0$, its limit when $\mathbf{v}\to\Gamma$ must
be $0$ also.

With the explicit formulaes of the Hamiltonian flow of $q_{1}$ and $q_{2}$
given in Table 1, we know that $\tau^{B,A}_{1}(v)$ and $\tau^{B,A}_{2}(v)$
satisfy the following equation:

$(e^{\tau^{B,A}_{1}+i\tau^{B,A}_{2}}b_{1},e^{-\tau^{B,A}_{1}+i\tau^{B,A}_{2}}b_{2},{\boldsymbol{\theta}},{\boldsymbol{\xi}})=(a_{1},a_{2},0,\ldots,0,0,\ldots,0).$
(2)

We also have the equations: $a_{1}=\varepsilon\ ,\ b_{2}=\varepsilon\text{ and
}\bar{a_{1}}a_{2}=\bar{b_{1}}b_{2}=w$. Here we introduce our determination of
the complex logarithm to give the solution of 2:

$\tau^{B,A}_{1}+i\tau^{B,A}_{2}=\ln(\frac{a_{1}}{b_{1}})=\ln(\varepsilon.\frac{\varepsilon}{\bar{w}})=\ln(\varepsilon^{2})-\ln(\bar{w}).$

Writing now
$\tau_{1}+i\tau_{2}=(\tau^{A,B}_{1}+\tau^{B,A}_{1})+i(\tau^{A,B}_{2}+\tau^{B,A}_{2})$,
we can refer to the statement announced on Proposition 4.1 concerning
$\sigma_{1}$ and $\sigma_{2}$:

$\displaystyle\sigma_{1}+i\sigma_{2}$ $\displaystyle=\tau_{1}(\mathbf{v})+\Re
e(\ln(w))+i(\tau_{2}(\mathbf{v})-\Im m(\ln(w)))$
$\displaystyle=\tau^{A,B}_{1}+i\tau^{A,B}_{2}+\tau^{B,A}_{1}+i\tau^{B,A}_{2}+\ln(\bar{w})$
$\displaystyle=\tau^{A,B}_{1}+i\tau^{A,B}_{2}+\ln(\varepsilon^{2})-\ln(\bar{w})+\ln(\bar{w})$
$\displaystyle=\tau^{A,B}_{1}+i\tau^{A,B}_{2}+\ln(\varepsilon^{2}).$

This last quantity is smooth with respect to $\mathbf{v}$. Since for
$j=3,\ldots,n$,
$\sigma_{j}(\mathbf{v})=\tau_{j}(\mathbf{v})=\tau^{A,B}_{j}(\mathbf{v})$ is
also smooth, this shows the first statement of Proposition 4.1.

Let us now show that for regular values, the 1-form
$\tau=\sum_{i=1}^{n}\tau_{i}dv_{i}$ is closed. For this we fix a regular value
$\mathbf{c}\in V_{X^{n}}(\Omega)$ and introduce the following action integral

$\mathcal{A}(\mathbf{v})=\int_{\gamma_{\mathbf{v}}}\alpha$

with $\alpha$ a Liouville 1-form defined on a tubular neighborhood
$\mathcal{V}(\Lambda_{\mathbf{c}})\subseteq\Omega$ of $\Lambda_{\mathbf{c}}$
($\omega=d\alpha$), while, for $\mathbf{v}\in U:=F(\mathcal{V}(\mathbf{c}))$,
$\mathbf{v}\mapsto\gamma_{\mathbf{v}}\subseteq\Lambda_{\mathbf{v}}$ is a
smooth family of loops with the same homotopy class in $\Lambda_{\mathbf{v}}$
as the joint flow of $F$ at the times
$(\tau_{1}(\mathbf{v}),\tau_{2}(\mathbf{v}),\ldots,\tau_{n}(\mathbf{v}))$. The
integral $\mathcal{A}$ only depends of $\mathbf{v}$ as
$\gamma_{\mathbf{v}}\subseteq\Lambda_{\mathbf{v}}$, which is Lagrangian (this
is another statement of Mineur formula).

A consequence of Darboux-Weinstein that one can found in [33], is the
following general lemma

######  Lemma 4.2.

Each Lagrangian submanifold in a tubular neighborhood
$\mathcal{V}(\Lambda_{\mathbf{c}})$ can be canonically associated with a
closed 1-form on $\Lambda_{\mathbf{c}}$.

###### Proof.

The exponential map provides a diffeomorphism between
$\mathcal{V}(\Lambda_{\mathbf{c}})$ and the normal bundle

$N\Lambda_{\mathbf{c}}=\bigsqcup_{p\in\Lambda_{\mathbf{c}}}\\!\frac{T_{p}M}{T_{p}\Lambda_{\mathbf{c}}}.$

The latter can be identified with $T^{*}\Lambda_{\mathbf{c}}$ using the
symplectic form: for $m\in\Lambda_{\mathbf{c}}$ and $X_{m}\in T_{m}M$, we
define

$\tilde{\omega}[X_{m}]_{m}:=\left.(\imath_{X_{m}}\omega_{m})\right|_{T\Lambda_{\mathbf{c}}}\in
T^{*}_{m}\Lambda_{\mathbf{c}}.$

Since $T_{m}M=T_{m}\Lambda_{\mathbf{c}}\oplus N_{m}\Lambda_{\mathbf{c}}$ with
$\Lambda_{\mathbf{c}}$ Lagrangian, the map

$\displaystyle TM$ $\displaystyle\to T^{*}\Lambda_{\mathbf{c}}$ $\displaystyle
X$ $\displaystyle\mapsto\tilde{\omega}[X]$

is linear, and $\tilde{\omega}[X]$ is non-zero if and only if the projection
of $X$ on $N\Lambda_{\mathbf{c}}$ is non-zero as a vector field.

Now, an infinitesimal deformation of the submanifold $\Lambda_{\mathbf{c}}$ is
a vector field of $\mathcal{V}(\Lambda_{\mathbf{c}})$ transversal to
$\Lambda_{\mathbf{c}}$, that is, a section $X\in N\Lambda_{\mathbf{c}}$. This
infinitesimal deformation is to be performed in the space of Lagrangian
submanifolds and only if

$\displaystyle\frac{d}{dt}\left.\left.\left[{\phi_{X}^{t}}^{*}\omega\right]\right|_{t=0}\right|_{T\Lambda_{\mathbf{c}}\times
T\Lambda_{\mathbf{c}}}$ $\displaystyle=0$
$\displaystyle\left[\mathcal{L}_{X}\omega\right]_{T\Lambda_{\mathbf{c}}\times
T\Lambda_{\mathbf{c}}}$ $\displaystyle=0$
$\displaystyle\left.\left[d\imath_{X}\omega+\imath_{X}\underbrace{d\omega}_{=0}\right]\right|_{T\Lambda_{\mathbf{c}}\times
T\Lambda_{\mathbf{c}}}$ $\displaystyle=0$ $\displaystyle
d\left[(\imath_{X}\omega)|_{T\Lambda_{\mathbf{c}}}\right]$
$\displaystyle=d\tilde{\omega}[X]=0,$

that is, if and ony if the associated 1-form $\tilde{\omega}[X]$ is closed, if
and only if $X$ is locally Hamiltonian in our case.

∎

In our case the foliation $\mathcal{F}=(\Lambda_{\mathbf{v}})_{\mathbf{v}\in
U}$ is given by the fibers of the moment map $F|_{U}$. Its deformation map is
$\mathbf{v}\mapsto\Lambda_{\mathbf{v}}$ and the associated 1-form to the
infinitesimal deformation verifies

$\left.\tilde{\omega}\left[\frac{\partial}{\partial
v_{i}}\Lambda_{\mathbf{v}}\right]\right|_{\mathbf{v}=\mathbf{c}}=\kappa_{i}(\mathbf{c})$

where $\kappa_{i}$ is the closed 1-form on $\mathcal{V}(\Lambda_{\mathbf{c}})$
defined by: $\imath_{\chi_{f_{j}}}\kappa_{i}=\delta_{i,j}$. In other words,
the integral of $\kappa_{i}$ along a trajectory of the flow of $F$ measures
the increasing of time $t_{j}$ along this trajectory. We now show the
following formula linking the variation 1-form to the infinitesimal variation
of the action $\mathcal{A}$

$\frac{\partial\mathcal{A}}{\partial
v_{i}}(\mathbf{c})=\int_{\gamma_{\mathbf{c}}}\kappa_{i}$ (3)

by proving that

$\left.\frac{\partial}{\partial
v_{i}}(\gamma_{\mathbf{v}}^{*}\alpha)\right|_{\mathbf{v}=\mathbf{c}}\text{ and
}\gamma_{\mathbf{c}}^{*}\kappa_{i}\;\text{are cohomologous on}\;S^{1}.$

We have

$\frac{\partial}{\partial
v_{i}}(\gamma_{\mathbf{v}}^{*}\alpha)=\gamma_{\mathbf{v}}^{*}\left[\mathcal{L}_{\frac{\partial\gamma_{\mathbf{v}}}{\partial
v_{i}}}\alpha\right]=\gamma_{\mathbf{v}}^{*}\left[\imath_{\frac{\partial\gamma_{\mathbf{v}}}{\partial
v_{i}}}d\alpha+d\imath_{\frac{\partial\gamma_{\mathbf{v}}}{\partial
v_{i}}}\alpha\right]$

To each $\gamma_{\mathbf{v}}$ corresponds a unique Lagrangian submanifold
$\Lambda_{\mathbf{v}}$, hence for all $p\in\gamma_{\mathbf{c}}$,
$\frac{\partial\gamma_{\mathbf{v}}}{\partial
v_{i}}|_{\mathbf{v}=\mathbf{c}}(p)=\frac{\partial}{\partial
v_{i}}\Lambda_{\mathbf{v}}|_{\mathbf{v}=\mathbf{c}}(p)$. The vector splits
into two components $X^{t}_{\mathbf{c}}(p)$ and $X^{n}_{\mathbf{c}}(p)$, with
$X^{t}_{\mathbf{c}}(p)\in T_{p}\Lambda_{\mathbf{c}}$ and
$X^{n}_{\mathbf{c}}(p)\in N_{p}\Lambda_{\mathbf{c}}$. The normal vector
$X^{n}_{\mathbf{c}}$ is by definition the infinitesimal deformation of
$\mathcal{F}$ at $\mathbf{c}$ along the direction $\frac{\partial}{\partial
v_{i}}$, that is, $\kappa_{i}$. We then have

$\left.\left(\gamma_{\mathbf{v}}^{*}\left[\imath_{\frac{\partial\gamma_{\mathbf{v}}}{\partial
v_{i}}}d\alpha\right]\right)\right|_{\mathbf{v}=\mathbf{c}}=\gamma_{\mathbf{c}}^{*}\left(\underbrace{\imath_{X^{t}_{\mathbf{c}}}d\alpha}_{=0\text{
on
}T_{p}\Lambda_{\mathbf{c}}}+\imath_{X^{n}_{\mathbf{c}}}\underbrace{d\alpha}_{=\omega}\right)=\gamma_{\mathbf{c}}^{*}\tilde{\omega}[\frac{\partial}{\partial
v_{i}}\Lambda_{\mathbf{v}}]|_{\mathbf{v}=\mathbf{c}}=\gamma_{\mathbf{c}}^{*}\kappa_{i}.$

Thus $\left.\frac{\partial}{\partial
v_{i}}(\gamma_{\mathbf{v}}^{*}\alpha)\right|_{\mathbf{v}=c}-\gamma_{\mathbf{c}}^{*}\kappa_{i}=d\gamma_{\mathbf{c}}^{*}\alpha(\frac{\partial\gamma_{\mathbf{v}}}{\partial
v_{i}})$ is exact: the two 1-forms are cohomologous. ∎

Since $\gamma_{\mathbf{c}}$ has the same homotopy class as the joint flow of
$F$ at the times
$(\tau_{1}(\mathbf{c}),\tau_{2}(\mathbf{c}),\ldots,\tau_{n}(\mathbf{c}))$, we
have

$d\mathcal{A}(\mathbf{c})=\sum_{i=1}^{n}\frac{\partial\mathcal{A}}{\partial
v_{i}}(\mathbf{c})dv_{i}=\sum_{i=1}^{n}\tau_{i}(\mathbf{c})dv_{i}=\tau(\mathbf{c})$

Thus, we can now forget about $\mathbf{c}$: $\tau$ is a closed 1-form of the
variable $\mathbf{v}$, $\mathcal{A}$ is the action integral, defined for all
$\mathbf{v}\in F(L_{X^{n}}(\mathcal{U}(\Lambda_{0})))$. The function $\ln(w)$
is also closed as a holomorphic 1-form of the variable $w$ and thus,
$\sigma=\tau+\ln(w)$ is closed for all regular values, and by continuation,
for all $\mathbf{v}\in F(\mathcal{U}(\Lambda_{0}))$. Taking the primitive of
$ln(w)$, we define the

######  Definition 4.3.

Let $S$ be the unique smooth function of $\mathbf{v}$ defined on
$F(\mathcal{U}(\Lambda_{0}))$ such that $dS=\sigma$ and $S(0)=0$. The Taylor
serie of $S$ in $(v_{1},v_{2})$ at $(0,0,v_{3},\ldots,v_{n})$ can be written
as

$(S)^{\infty}=\sum_{j_{1}=0}^{\infty}\sum_{j_{2}=0}^{\infty}\frac{\partial
S}{\partial v_{1}^{j_{1}}\partial
v_{2}^{j_{2}}}(0,0,v_{3},\ldots,v_{n})v_{1}^{j_{1}}v_{2}^{j_{2}}$

In accordance with [28] we call this double sum the symplectic invariant of
the nodal locus $\Gamma$ and we have:

$\mathcal{A}(\mathbf{v})=S(\mathbf{v})-\Re e(w\ln w-w)$

This concludes the proof of Theorem 2.10.

∎

###  4.2 Computation of the monodromy

In order to exhibit the monodromy phenomenon, we first mention that in [32],
we show that $V_{X^{n}}(\Omega)$ is not simply connected. We take a circle

$C:=\\{C(t)=(v_{1}\cos(t),v_{2}\sin(t),v_{3},\ldots,v_{n})|t\in[0,2\pi]\\}\subseteq
F(M)$

with $\mathbf{v}$ small enough so that it is in in $F(\Omega)$. Each point of
$C$ is a regular value, so to each $t$ corresponds a $n$-torus. We can take
any basis of cycle for each torus, that is, a ${\mathbb{Z}}$-basis for the
period lattice. For $C(0)$, we take the basis $\mathcal{R}$ introduced in the
proof of Theorem 2.10, and we follow it by homotopy as $t$ goes from $0$ to
$2\pi$. From Proposition 4.1, it turns out that as $t$ varies, $\tau$ is
deformed as $\tau+(0,t,0,\ldots,0)$, the other vectors staying unchanged.
Thus, at $t=2\pi$, we are back to $C(0)$, but our basis of covectors is now

$\mathcal{R}^{\prime}=\begin{pmatrix}\tau+r_{2}\\\ r_{2}\\\ \vdots\\\
r_{n}\end{pmatrix}$

Hence, we can write the following corollary of Theorem 2.10

######  Corollary 4.4.

With the same hypothesis as Theorem 2.10, the topological monodromy matrix is:

$\begin{pmatrix}1&1&0&\ldots&0\\\ 0&1&0&\ldots&0\\\
0&0&\ulcorner&\;&\urcorner\\\ \vdots&\vdots&\;&I_{n-2}&\;\\\
0&0&\llcorner&\;&\lrcorner\par\end{pmatrix}$

###  4.3 The general case

Now for the general case , we consider a critical point $m$ of Williamson type
$FF-X^{k_{\textup{x}}}-E^{k_{\textup{e}}}$ with $k_{\textup{e}}\neq 0$. We can
always assume the last $k_{\textup{e}}$ components to be the elliptic ones.
Using again Theorem 3.7, we have that for $q_{1}=q_{2}=0$ and $e_{i}\neq
0,i=1..k_{\textup{e}}$, the singularity is a $FF-X^{n-2}$ singularity.

On the other hand, although we only have proved it for $2n=6$, we conjecture
that

$\left(\mathcal{U}_{\Bbbk}(\Lambda_{0}):=\cup_{k_{\textup{x}}^{\prime}\leq
k_{\textup{x}}}L_{\Bbbk^{\prime}}(\mathcal{U}(\Lambda_{0})),\omega_{\Bbbk}:=\omega|_{T^{*}\mathcal{U}_{\Bbbk}(\Lambda_{0})},F_{\Bbbk}=F|_{\mathcal{U}_{\Bbbk}}\right)$

is a semi-toric IHS. It is the stratification of the semi-toric system by the
Williamson type, which shall be the subject of another article. If we admit
the conjecture, we have that in this system, $m$ is a $FF-X^{k_{\textup{x}}}$
critical point. We can hence apply the result above, and for
$S_{\Bbbk}(v_{1},v_{2},v_{3},\ldots,v_{k_{\textup{x}}+2})=S_{\Bbbk}(\mathbf{v}_{\Bbbk})$
the regularized action coordinate associated to it, we have

$S_{\Bbbk}(\mathbf{v}_{\Bbbk})=\lim_{\mathbf{v}\to(\mathbf{v}_{\Bbbk},0)}S(\mathbf{v}).$

Note that, while we can still describe the foliation for an elliptic
singularity, we do not have Action-Angle coordinates: polar coordinates are
not defined at the origin. Concerning the monodromy, we have for the
restricted IHS the following matrix

$\begin{pmatrix}1&1&0&\ldots&0\\\ 0&1&0&\ldots&0\\\
0&0&\ulcorner&\;&\urcorner\\\ \vdots&\vdots&\;&I_{k_{\textup{x}}}&\;\\\
0&0&\llcorner&\;&\lrcorner\par\end{pmatrix}$

##  5 Conclusion and perspectives

In this article, we have established the general formula for the Taylor
invariant introduced by San Vũ Ngọc in [28] and the monodromy matrix. Some of
the results needed to prove that it is indeed an invariant are proved given in
Taylor invariant we have provided several results using local and leaf-wise
model of singular integrable systems. These techniques are helpful in less
friendly settings. In almost-toric systems of higher complexity, for instance,
we can apply the same tools. However, it is not clear what could be a general
formulation of this result, to what extent we can simply mimetize the proof we
did here.

There are several global results in our research program (see [24] and [22]
for a description ) that necessitate more conceptual tools than the local
models like we used here. In that sense, the aim of articles [31] and [38] are
to establish these result in a fit framework.

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