# Connectedness of level sets for non-degenerate integrable systems that
extend complexity one torus actions

Daniele Sepe  Escuela de Matemáticas, Universidad Nacional de Colombia sede
Medellín, Colombia, and Instituto de Matemática e Estatística, Departamento de
Matemática Aplicada (GMA), Universidade Federal Fluminense, Brazil.
<EMAIL_ADDRESS>and  Susan Tolman  Department of Mathematics, University of
Illinois at Urbana-Champaign, Urbana, IL 61801<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the connectedness of the level sets of integrable
systems that have only non-degenerate singular points and extend complexity
one $T$-spaces with proper moment maps. Our main result states that if there
are no singular points with a hyperbolic block and connected $T$-stabilizer,
then each level set is connected. Moreover, we prove that the above condition
is necessary if either some reduced space is simply connected or the moment
map for the integrable system is generic in a natural sense.

_2020 Mathematics Subject Classification_. Primary 37J35, 53D20. Secondary
37J39, 53D35.

Keywords: Completely integrable Hamiltonian systems, non-degenerate singular
points, complexity one actions, connected level sets.

###### Contents

  1. 1 Introduction
  2. 2 Tall local models
  3. 3 The key proposition
  4. 4 Purely elliptic type
  5. 5 The semitoric case
  6. 6 Non-degenerate singular points
  7. 7 Proof of the main result

##  1\. Introduction

An important step in studying Hamiltonian group actions is understanding the
level sets of the moment map. In this direction, connectedness of the level
sets has been established in the following cases:

  1. (1)

Hamiltonian group actions of compact abelian Lie groups with proper moment
maps [At, GS2, LMTW]; and

  2. (2)

semi-toric systems in dimension four and their natural generalization in
dimension six [VN, Wa].

In both cases, the connectedness of the level sets of the moment map is key to
understanding the moment map image and plays an important role in
classification.

In this paper we study the connectedness of the level sets of integrable
systems that are extensions of complexity one torus actions, and have only
non-degenerate singular points. Complexity one torus actions are the simplest
after the toric case and have been partially classified in [KT1, KT2, KT3].
Non-degenerate singular points of integrable systems are generic, and can be
thought of as the symplectic analog of critical points for Morse-Bott
functions; they have attracted much interest, both for classical and quantum
integrable systems (see [BF, SV] and references therein). In particular, the
integrable systems that we study include the semi-toric systems of [VN], and
their higher dimensional generalization in [Wa].

In our main result, Theorem 1.6 we give sufficient conditions for level sets
of the moment map to be connected, generalizing the results in [VN, Wa]. Morse
theory plays a central role in the theorems (1) and (2) above, and our theorem
both uses and generalizes the following simple Morse-theoretic result: Let $N$
be a closed, connected 2-dimensional manifold and let $\overline{g}\colon
N\to{\mathbb{R}}$ be a Morse function. If $\overline{g}$ has no singular point
of index one, then each level set is connected (see Lemma 7.3). We also
explore the extent to which the hypotheses in our main result are necessary
(see Theorem 1.10).

Before stating our results precisely, we begin with few definitions from
integrable systems.

######  Definition 1.1.

An integrable system is a quadruple $\left(M,\omega,V,f\right)$, where
$\left(M,\omega\right)$ is a connected $2n$-dimensional symplectic manifold,
$V$ is an $n$-dimensional real vector space, and $f\colon M\to V^{*}$ is a
smooth map such that:

  1. (1)

the Poisson bracket $\big{\\{}\langle f,\xi_{1}\rangle,\langle
f,\xi_{2}\rangle\big{\\}}=0$ for all $\xi_{1},\xi_{2}\in V$, and

  2. (2)

the function $f$ is regular on a dense subset of $M$.

Here $\langle\cdot,\cdot\rangle\colon V^{*}\times V\to{\mathbb{R}}$ is the
natural pairing.

The standard definition of integrable systems takes $V={\mathbb{R}}^{n}$. In
this paper, we assume that every integrable system is complete, that is, $f$
is a moment map for a Hamiltonian $V$-action.

Fix an integrable system $\left(M,\omega,V,f\right)$. A point $p\in M$ is
regular if $D_{p}f$ has full rank and singular otherwise. Given a point $p\in
M$, the $V$-orbit $\mathcal{O}$ through $p$ is isotropic. Hence, the vector
space $\left(T_{p}\mathcal{O}\right)^{\omega}/T_{p}\mathcal{O}$ inherits a
natural symplectic structure. Moreover, the stabilizer $W\subseteq V$ of $p$
acts by linear symplectomorphisms on $T_{p}M$ and fixes $T_{p}\mathcal{O}.$ We
call the induced symplectic action of $W$ on
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ the linearized action at $p$.

######  Definition 1.2.

Let $p$ be a point in an integrable system $\left(M,\omega,V,f\right)$. Let
$W_{0}\subseteq V$ be the Lie algebra of the stabilizer of $p$. The point $p$
is non-degenerate if there exist identifications
$W_{0}\simeq{\mathbb{R}}^{\dim W}$ and
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}\simeq{\mathbb{C}}^{\dim W}$ such
that the homogeneous moment map for the linearized action at $p$ is a product
of the following “blocks”:

  * •

(elliptic) $z\in{\mathbb{C}}\mapsto\frac{1}{2}|z|^{2}$;

  * •

(hyperbolic) $z\in{\mathbb{C}}\mapsto{\mathfrak{I}}(z^{2})$;

  * •

(focus-focus)
$(z_{1},z_{2})\in{\mathbb{C}}^{2}\mapsto\left(\frac{1}{2}(|z_{1}|^{2}-|z_{2}|^{2}),{\mathfrak{I}}(z_{1}z_{2})\right)$.

Here ${\mathfrak{I}}(w)$ denotes the imaginary part of $w\in{\mathbb{C}}$ and
${\mathbb{C}}^{\dim W}$ has the standard symplectic structure. If all the
blocks are elliptic, we say that $p$ has purely elliptic type.

######  Remark 1.3.

The above definition of non-degeneracy is equivalent to the standard one (see,
for instance, [SV, Theorem 4.8]). In particular, the focus-focus block in the
above definition can be transformed to the one in the integrable systems
literature by applying a linear symplectomorphism.

######  Remark 1.4.

The notion of non-degeneracy can be viewed as a “symplectic Morse-Bott”
condition. In particular, in dimension two, a point is non-degenerate in the
sense of Definition 1.2 if and only if it is non-degenerate in the sense of
Morse theory. Moreover, it is hyperbolic if and only if it has index one and
elliptic otherwise.

Next we introduce complexity one torus actions. Let $T$ be an
$(n-1)$-dimensional torus with Lie algebra ${\mathfrak{t}}$.

######  Definition 1.5.

A complexity one $\boldsymbol{T}$-space is a triple $(M,\omega,\Phi)$, where
$(M,\omega)$ is a connected $2n$-dimensional symplectic manifold and
$\Phi\colon M\to{\mathfrak{t}}^{*}$ is a moment map for an effective
$T$-action.

Let $(M,\omega,\Phi)$ be a complexity one $T$-space. Given
$\alpha\in{\mathfrak{t}}^{*}$, the reduced space at $\alpha$ is the quotient
$\Phi^{-1}(\alpha)/T$; if $\alpha=0$, we write ${M/\\!/T}$ for the reduced
space. A point $p\in M$ is short if $[p]$ is an isolated point in the reduced
space $\Phi^{-1}(\Phi(p))/T$ and is tall otherwise. By [KT1, Proposition 6.1],
if $\Phi$ is proper each reduced space $\Phi^{-1}(\alpha)/T$ containing more
than one point is topologically a closed, connected oriented 2-dimensional
manifold; moreover, any two non-empty reduced spaces have the same fundamental
group [KT1, Lemma 5.7 and Corollary 9.7].

The main objects that we study in this paper are non-degenerate integrable
systems that extend complexity one torus actions, i.e., integrable systems
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=(\Phi,g)\right)$ such that
every singular point is non-degenerate and such that $(M,\omega,\Phi)$ is a
complexity one $T$-space. Our main result is as follows.

######  Theorem 1.6.

Let $\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=(\Phi,g)\right)$ be an
integrable system so that $(M,\omega,\Phi)$ is a complexity one $T$-space with
a proper moment map. Assume that each tall singular point in
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f\right)$ is non-degenerate.
If $M$ has no non-degenerate singular points with a hyperbolic block and
connected $T$-stabilizer, then the level set $f^{-1}(\alpha,c)$ is connected
for each $(\alpha,c)\in{\mathfrak{t}}^{*}\times{\mathbb{R}}$; moreover, the
reduced space $\Phi^{-1}(\alpha)/T$ is simply connected.

######  Remark 1.7.

Theorem 1.6 implies connectedness of the level sets for the higher dimensional
analogs of semi-toric systems that Wacheux studied in [Wa]: Integrable systems
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
so that $\left(M,\omega,\Phi\right)$ is a complexity one $T$-space with a
proper moment map all of whose singular points are non-degenerate without
hyperbolic blocks. In particular, our main result generalizes [Wa, Theorem
4.2.4] to arbitrary dimensions.

######  Remark 1.8.

Theorem 1.6 is comparable to [PRV, Theorems 1 – 3] in that all these results
assume that the singular points of an integrable system are non-degenerate,
disallow (certain) singular points with a hyperbolic block, and prove that the
level sets are connected. However, the results in [PRV] do not require the
first component of the moment map to generate a Hamiltonian circle action, but
only hold in dimension four and have additional hypotheses on the moment map.

Unfortunately, Theorem 1.6 is not quite sharp. If a reduced space has positive
genus, then $M$ must have non-degenerate singular points with a hyperbolic
block and connected $T$-stabilizer; see Remark 1.14. However, as we show
below, it is still possible for all the level sets of $f$ to be connected in
this case.

######  Example 1.9.

Let $\omega$ be the standard symplectic form on the torus
$N={\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$. The function $g\colon N\to{\mathbb{R}}$
defined by $g([x,y])=\cos(2\pi x)+\cos(2\pi y)$ is Morse and has connected
fibers, a unique minimum and maximum, and two points of index one. Hence,
$(N,\omega,{\mathbb{R}},g)$ is an integrable system with two hyperbolic
singular points. (Since the compact torus $T$ is trivial, all points have
connected $T$-stabilizer.) To form higher dimensional examples, take the
product of this integrable system with a compact, connected symplectic toric
manifold.

Nevertheless, as our next result shows, Theorem 1.6 is quite close to being
sharp.

######  Theorem 1.10.

Let $\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=(\Phi,g)\right)$ be an
integrable system so that $(M,\omega,\Phi)$ is a complexity one $T$-space with
a proper moment map. Assume that each tall singular point in
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f\right)$ is non-degenerate.
Moreover, assume that one of the following holds:

  * •

The reduced space $\Phi^{-1}(\beta)/T$ is simply connected for some
$\beta\in\Phi(M).$

  * •

Each level set of $f$ contains at most one orbit of non-degenerate singular
points with a hyperbolic block.

Then the level set $f^{-1}(\alpha,c)$ is connected for all
$(\alpha,c)\in{\mathfrak{t}}^{*}\times{\mathbb{R}}$ if and only $M$ has no
non-degenerate singular points with a hyperbolic block and connected
$T$-stabilizer.

######  Remark 1.11.

If in Theorems 1.6 and 1.10 we additionally assume that $M$ is compact, then
by a theorem of Hui Li [Li], the fundamental group of $M$ is equal to that of
any reduced space. Hence, in this case, the phrase “the reduced space
$\Phi^{-1}(\alpha)/T$” can be changed to “$M$”.

The fundamental idea behind the proofs of Theorems 1.6 and 1.10 is quite
simple: Since
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f=(\Phi,g)\right)$ is an
integrable system, the function $g$ is $T$-invariant, and hence induces a
function $\overline{g}\colon\Phi^{-1}(\alpha)/T\to{\mathbb{R}}$. By the key
technical result in our paper, Proposition 7.1, each non-trivial component of
the reduced space $\Phi^{-1}(\alpha)/T$ can be given the structure of a smooth
oriented surface so that
$\overline{g}\colon\Phi^{-1}(\alpha)/T\to{\mathbb{R}}$ is a Morse function.
Moreover, the critical points of index $1$ correspond exactly to orbits of
non-degenerate singular points with a hyperbolic black and connected
$T$-stabilizer. Since $\Phi$ is proper, the reduced space
$\Phi^{-1}(\alpha)/T$ is compact and connected by Theorem 5.5. Therefore, the
proofs reduce to standard results in Morse theory; see Lemmas 7.3 and 7.4.

This method can be used more generally on integrable systems that extend a
complexity one torus action but have (sufficiently nice) degenerate singular
points. For example, in a forthcoming paper, we will prove that the analogs of
Theorems 1.6 and 1.10 hold if we also allow a broader class of “ephemeral”
singular points; like focus-focus points and hyperbolic points with
disconnected $T$-stabilizers, these correspond to regular points on the
symplectic quotients.

######  Remark 1.12.

The second bullet point in Theorem 1.10 can be thought of as a genericity
condition because generic Morse functions have at most one critical point (of
index 1) in each level set.

######  Remark 1.13.

More generally, Theorems 1.6 and 1.10 both hold if $\Phi$ is proper as a map
to an open convex subset of ${\mathfrak{t}}^{*}$, because the analog of
Theorem 5.5 still holds in this case [LMTW, Theorem 4.3]. Additionally, if $f$
is proper and the fibers of $\Phi$ are connected, then the first statement in
Theorem 1.6 still holds. This is because a proper Morse function on a
connected surface with no critical points of index $1$ has connected fibers
(Proposition 5.6).

######  Remark 1.14.

Let $(M,\omega,{\mathfrak{t}}^{*}\times{\mathbb{R}},f)$ be an integrable
system satisfying the assumptions in the first two sentences of Theorem 1.6,
but assume that the reduced space $\Phi^{-1}(\alpha)/T$ has positive genus
$k$, and hence Euler characteristic $2-2k$. In this case, by calculating the
Euler characteristic, we see that $\Phi^{-1}(\alpha)/T$ has at least $2k$
orbits of non-degenerate singular points with a hyperbolic block and connected
$T$-stabilizer because because $\overline{g}$ has at least one minimum and
maximum. Note that if $\Phi^{-1}(\alpha)/T$ has more than $2k$ such orbits,
then the same argument shows that $\overline{g}$ must have additional local
minima or maxima; hence the fibers of $f$ are not connected by Lemma 7.2.

### Structure of the paper

In Section 2, we review some facts about tall complexity one local models and
then briefly explore a few additional properties. Next, in Section 3 we prove
that we can endow the symplectic quotient $\Phi^{-1}(\alpha)/T$ with a smooth
structure so that $\overline{g}$ is a Morse function – as long as most
singular points satisfy a certain technical condition that guarantees that we
can do this locally. Most of the rest of the paper is dedicated to proving
that non-degenerate singular points satisfy this condition. For example, in
Section 4 we prove that tall points that have purely elliptic type satisfy it,
and in Section 5 we show that, if $M$ is four-dimensional, focus-focus points
satisfy it. At this point, we are able to demonstrate our technique by using
it to give a new proof that that level sets of a semi-toric system are
connected. In Section 6, we prove that all the remaining types of non-
degenerate singular points also satisfy this technical condition. Therefore,
we can complete the proof of our theorems in Section 7.

### Acknowledgments

D. Sepe was partially supported by FAPERJ grant JCNE E-26/202.913/2019 and by
a CAPES/Alexander von Humboldt Fellowship for Experienced Researchers
88881.512955/2020-01. This study was financed in part by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior – Brazil (CAPES) – Finance code
001. S. Tolman was partial supported by NSF DMS Award 2204359 and Simons
Foundation Collaboration Grant 637995.

##  2\. Tall local models

To begin, we review some facts about tall complexity one local models. In
particular, we introduce (tall) complexity one local models, construct the
“defining polynomial” for tall local models, and use it to describe the
sympletic quotient ${Y/\\!/T}$. Moreover, we provide a generating set for
$T$-invariant polynomials on these models.

Fix a complexity one $T$-space $(M,\omega,\Phi)$ and let $\mathfrak{O}$ be the
$T$-orbit through a point $p\in M$. Since $\mathfrak{O}$ is isotropic, the
vector space $(T_{p}\mathfrak{O})^{\omega}/T_{p}\mathfrak{O}$ inherits a
natural symplectic structure. Moreover, the natural action of the stabilizer
$H\subseteq T$ of $p$ on $T_{p}M$ induces a symplectic representation of $H$
on $(T_{p}\mathfrak{O})^{\omega}/T_{p}\mathfrak{O}$, called the symplectic
slice representation at $p$. It is isomorphic to the representation of $H$ on
${\mathbb{C}}^{h+1}$ associated to some injective homomorphism $\rho\colon
H\to(S^{1})^{h+1}$, where $h=\dim H$. Let $\eta_{i}\in{\mathfrak{h}}^{*}$ be
the differential of the $i$’th component of $\rho$. The (not necessarily
distinct) vectors $\eta_{0},\dots,\eta_{h}$ are called the isotropy weights at
$p$; the multiset of isotropy weights at $p$ is uniquely determined.

Set $Y:=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, where
${\mathfrak{h}}^{\circ}$ is the annihilator of ${\mathfrak{h}}$ in
${\mathfrak{t}}^{*}$ and $H$ acts on $T$ by multiplication, on
${\mathfrak{h}}^{\circ}$ trivially, and on ${\mathbb{C}}^{h+1}$ by $h\cdot
z=\rho(h^{-1})z$. Fix an inner product on the Lie algebra ${\mathfrak{t}}$
once and for all, and use it to identify ${\mathfrak{t}}^{*}$ with
${\mathfrak{h}}^{\circ}\oplus{\mathfrak{h}}^{*}.$ There exists a canonical
symplectic form $\omega_{Y}$ on $Y$ so that the $T$-action
$s\cdot[t,\alpha,z]=[st,\alpha,z]$ has homogeneous moment map $\Phi_{Y}\colon
Y\to{\mathfrak{t}}^{*}$ given by

$\textstyle\Phi_{Y}([t,\alpha,z])=\alpha+\Phi_{H}(z),$

where $\Phi_{H}(z)=\frac{1}{2}\sum_{i}\eta_{i}|z_{i}|^{2}$ is the homogeneous
moment map for the $H$-action on ${\mathbb{C}}^{h+1}$; see Remark 4.2. By the
Marle-Guillemin-Sternberg local normal form theorem, there is a
$T$-equivariant symplectomorphism from an invariant neighborhood of
$\mathfrak{O}$ to an open subset of $Y$ taking $p$ to $[1,0,0]$. We call
$(Y,\omega_{Y},\Phi_{Y})$ the local model for $p$. We say that $Y$ is short if
$\Phi_{Y}^{-1}(0)$ consists of a single $T$-orbit and tall otherwise. By Lemma
2.7 below, the level set $\Phi^{-1}_{Y}(0)$ contains exactly one $T$-orbit
exactly if $[p]\in\Phi^{-1}(\Phi(p))$ is an isolated point. Hence, $p$ is
short exactly if its local model is short.

As shown in [KT1], tall local models have an important invariant, called the
defining polynomial.

######  Lemma 2.1.

Let $Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be a local
model associated to an injective homomorphism $\rho:H\to(S^{1})^{h+1}$. Then
there exists $\xi=(\xi_{0},\ldots,\xi_{h})\in{\mathbb{Z}}^{h+1}$ such that the
homomorphism $(S^{1})^{h+1}\to S^{1}$ given by
$\lambda\mapsto\prod\limits_{j=0}^{h}\lambda_{j}^{\xi_{j}}$ induces a short
exact sequence

(2.2) $1\to H\stackrel{{\scriptstyle\rho}}{{\to}}(S^{1})^{h+1}\to S^{1}\to 1;$

moreover, the vector $\xi$ is unique up to sign. Finally, $\xi_{i}\xi_{j}\geq
0$ for all $i,j$ if and only if the local model $Y$ is tall.

###### Proof.

Existence of $\xi$ and its uniqueness up to sign follow from [KT1, Lemma 5.8].
The local model $Y$ is tall if and only if $\Phi^{-1}_{Y}(0)$ consists of more
than one orbit. By [KT1, Lemma 5.4], the latter occurs exactly if $\Phi_{Y}$
is not proper, which, by [KT1, Lemma 5.8], is equivalent to
$\xi_{i}\xi_{j}\geq 0$ for all $i,j$. ∎

######  Remark 2.3.

Let $\eta_{0},\ldots,\eta_{h}\in{\mathfrak{h}}^{*}$ be the isotropy weights at
$p=[1,0,0]\in Y$. By (2.2), the homomorphism
${\mathbb{R}}^{h+1}\to{\mathbb{R}}$ given by
$a\mapsto\sum\limits_{j}a_{j}\xi_{j}$ induces dual short exact sequences

$\displaystyle
0\to{\mathfrak{h}}\stackrel{{\scriptstyle\rho_{*}}}{{\to}}{\mathbb{R}}^{h+1}\to{\mathbb{R}}\to
0;$ $\displaystyle
0\to{\mathbb{R}}^{*}\to\left({\mathbb{R}}^{h+1}\right)^{*}\stackrel{{\scriptstyle\rho^{*}}}{{\to}}{\mathfrak{h}}^{*}\to
0.$

In particular, $b=(b_{0},\ldots,b_{h})\in{\mathbb{R}}^{h+1}$ lies in
$\rho_{*}({\mathfrak{h}})$ exactly if $\sum\limits_{j}b_{j}\xi_{j}=0$, while
$\rho^{*}(a)=\sum\limits_{j}a_{j}\eta_{j}$ vanishes exactly if $a$ is a
multiple of $\xi$. Hence, if $Y$ is tall then
$(1,\dots,1)\notin\rho_{*}({\mathfrak{h}})$ because $\xi\neq 0$ can be chosen
to lie in ${\mathbb{Z}}_{\geq 0}^{h+1}.$

######  Definition 2.4.

Fix a tall local model111Here, and throughout the paper, we only consider
complexity one local models. $Y$. By Lemma 2.1, there exists a unique
$\xi=(\xi_{0},\ldots,\xi_{h})\in{\mathbb{Z}}_{\geq 0}^{h+1}$ inducing the
short exact sequence (2.2). The map $P\colon{\mathbb{C}}^{h+1}\to{\mathbb{C}}$
given by $P(z)=\prod_{j=0}^{h}z_{j}^{\xi_{j}}$ is called the defining
polynomial of $Y$. By a slight abuse of notation, we use the same name for the
map $P\colon Y\to{\mathbb{C}}$ given by
$[t,\alpha,z]\mapsto\prod_{j=0}^{h}z_{j}^{\xi_{j}}$. The sum
$N:=\sum_{j=0}^{h}\xi_{j}$ is called the degree of $P$.

######  Example 2.5.

Let ${\mathbb{Z}}_{2}=\\{\pm 1\\}$ act on ${\mathbb{C}}$ by $\alpha\cdot
z=\alpha z$. The associated local model
$S^{1}\times_{{\mathbb{Z}}_{2}}{\mathbb{R}}^{*}\times{\mathbb{C}}$ is tall.
The defining polynomial $P\colon{\mathbb{C}}\to{\mathbb{C}}$ is given by
$P(z)=z^{2}$ and has degree $2$.

######  Example 2.6.

Let $S^{1}$ act effectively on ${\mathbb{C}}^{2}$ by
$\alpha\cdot(z_{1},z_{2})=(\alpha z_{1},\alpha^{-1}z_{2})$ with moment map
$\Phi(z_{1},z_{2})=\frac{1}{2}(|z_{1}|^{2}-|z_{2}|^{2})$. This is a tall local
model. The defining polynomial $P\colon{\mathbb{C}}^{2}\to{\mathbb{C}}$ is
given by $P(z_{1},z_{2})=z_{1}z_{2}$ and also has degree $2$.

The reduced space ${Y/\\!/T}=\Phi^{-1}_{Y}(0)/T$ is endowed with the
subquotient topology and a sheaf of smooth functions defined as follows: For
any open subset $U\subseteq{Y/\\!/T}$, let $C^{\infty}(U)$ be the set of
functions $\overline{g}\colon U\to{\mathbb{R}}$ whose pullback to
$\Phi_{Y}^{-1}(0)$ is the restriction of a smooth $T$-invariant function
$g\colon Y\to{\mathbb{R}}$. A homeomorphism between topological spaces endowed
with sheaves of smooth functions is a diffeomorphism if it induces an
isomorphism between the sheaves of smooth functions.

######  Lemma 2.7.

Let $Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be a tall
local model with moment map $\Phi_{Y}\colon Y\to{\mathfrak{t}}^{*}$. The
defining polynomial $P\colon Y\to{\mathbb{C}}$ induces a homeomorphism
$\overline{P}\colon{Y/\\!/T}\to{\mathbb{C}}$ that restricts to a
diffeomorphism on the complement of $\left\\{[1,0,0]\right\\}$.

###### Proof.

Following [KT1], we say that an orbit is exceptional if every nearby orbit in
the same moment level set has strictly smaller stabilizer. Arguing as in the
proof of Lemma 2.1, since $Y$ is tall the moment map $\Phi_{Y}$ is not proper.
The first claim follows at once by [KT1, Corollary 6.3]. By [KT1, Lemmas 5.15
and 5.17], the orbit of $[1,0,0]$ is the only possible exceptional orbit in
${Y/\\!/T}$. The second claim now follows at once by [KT1, Corollary 7.2]. ∎

In most cases, the homeomorphism $\overline{P}\colon{Y/\\!/T}\to{\mathbb{C}}$
is not globally a diffeomorphism; however, as the lemma below shows, it is a
diffeomorphism when $P(z)=z_{0}$.

######  Lemma 2.8.

Let $Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be a tall
local model with defining polynomial $P(z)=\prod_{i=0}^{h}z_{i}^{\xi_{i}}$.
Assume that $\xi_{i}=0$ for all $i>k$. Let the group

$\textstyle
K:=\\{\lambda\in(S^{1})^{k+1}\mid\prod_{j=0}^{k}\lambda_{j}^{\xi_{j}}=1\\}$

act naturally on ${\mathbb{C}}^{k+1}$. The inclusion
${\mathbb{C}}^{k+1}\hookrightarrow Y$ given by

$(z_{0},\ldots,z_{k})\mapsto[1,0,(z_{0},\ldots,z_{k},0,\ldots,0)]$

induces a diffeomorphism between ${\mathbb{C}}^{k+1}/\\!/K$ and ${Y/\\!/T}$.
In particular, if $P(z)=z_{0}$ then
$\overline{P}\colon{Y/\\!/T}\to{\mathbb{C}}$ is a diffeomorphism with inverse
$z\mapsto[1,0,(z,0)].$

###### Proof.

Let $H$ act on ${\mathbb{C}}^{h+1}$ via $\rho\colon
H\hookrightarrow(S^{1})^{h+1}$. Let
$\Phi_{H}\colon{\mathbb{C}}^{h+1}\to{\mathfrak{h}}^{*}$ and
$\Phi_{K}\colon{\mathbb{C}}^{k+1}\to\mathfrak{k}^{*}$ be the homogeneous
moment maps for the $H$-action on ${\mathbb{C}}^{h+1}$ and the $K$-action on
${\mathbb{C}}^{k+1}$, respectively. Since $\xi_{i}=0$ for all $i>k$, we have
$\Phi_{H}^{-1}(0)=\Phi_{K}^{-1}(0)\times\\{0\\}$, and so
$\Phi_{Y}^{-1}(0)=T\times_{H}\\{0\\}\times(\Phi_{K}^{-1}(0)\times\\{0\\})$,
where $\Phi_{Y}:Y\to{\mathfrak{t}}^{*}$ is the moment map for the $T$-action
on $Y$. Since $\rho(H)=K\times(S^{1})^{h-k}$, the claim follows immediately. ∎

On the reduced space ${Y/\\!/T}$ there is a simple relation between the
magnitudes of the defining polynomial and of $z$.

######  Lemma 2.9.

Let $Y:=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be a tall
local model with moment map $\Phi_{Y}$. Let $N$ be the degree of the defining
polynomial $P(z)=\prod_{j}z_{j}^{\xi_{j}}$. Then for all
$[t,\alpha,z]\in{Y/\\!/T}$,

$\textstyle\big{\lvert}z\rvert^{2}=N\big{(}\prod_{j}\xi_{j}^{\xi_{j}}\big{)}^{-\frac{1}{N}}\lvert
P(z)\big{\rvert}^{\frac{2}{N}},$

where $0^{0}=1$ by convention.

###### Proof.

Let $\eta_{0},\dots,\eta_{h}\in\mathfrak{h}^{*}$ be the weights for the
$H$-action on ${\mathbb{C}}^{h+1}$. The moment map $\Phi_{Y}$ sends
$[t,\alpha,z]\in Y$ to $\alpha+\Phi_{H}(z)\in{\mathfrak{t}}^{*}$, where
$\Phi_{H}(z)=\sum_{j=0}^{h}\frac{1}{2}|z_{j}|^{2}\eta_{j}$. Thus it suffices
to check that $\lvert
z\rvert^{2}=\big{(}\prod_{j}\xi_{j}^{\xi_{j}}\big{)}^{-\frac{1}{N}}\big{\lvert}P(z)\big{\rvert}^{\frac{2}{N}}$
for all $z\in\Phi_{H}^{-1}(0)$. Fix $z\in\Phi_{H}^{-1}(0)$. Since the local
model $Y$ is tall and
$\Phi_{H}(z)=\sum_{j=0}^{h}\frac{1}{2}|z_{j}|^{2}\eta_{j}=0$, by Remark 2.3
there exists $c\geq 0$ such that $|z_{j}|^{2}=c\xi_{j}$ for all $j$.
Therefore,

$\begin{split}\lvert z\rvert^{2}&=\sum\limits_{j=0}^{h}\lvert
z_{j}\rvert^{2}=cN;\\\
|P(z)|^{2}=\Big{\lvert}\prod_{j=0}^{h}z_{j}^{\xi_{j}}\Big{\rvert}^{2}&=\prod\limits_{j=0}^{h}\lvert
z_{j}\rvert^{2\xi_{j}}=c^{N}\prod\limits_{j=0}^{h}\xi_{j}^{\xi_{j}}.\end{split}$

The result holds by combining these equations. ∎

To study integrable systems that extend complexity one $T$-actions, we need to
understand $T$-invariant functions on these spaces. We start by considering
the simplest $T$-invariant functions on local models: $T$-invariant
polynomials.

######  Definition 2.10.

Let $Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be a local
model. By a slight abuse of notation, we call a $T$-invariant function
$g\colon Y\to{\mathbb{R}}$ a $\boldsymbol{T}$-invariant (homogeneous)
polynomial if its restriction to
${\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ is a (homogeneous)
polynomial.

######  Lemma 2.11.

Let $Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be a tall
local model with moment map $\Phi_{Y}\colon Y\to{\mathfrak{t}}^{*}$. The
algebra of $T$-invariant polynomials on $Y$ is generated by the real and
imaginary parts of the defining polynomial $P\colon Y\to{\mathbb{C}}$, the map
$[t,\alpha,z]\mapsto|z|^{2}$, and the components of $\Phi_{Y}$.

###### Proof.

By definition, there is a one-to-one correspondence between $T$-invariant
polynomials on $Y$ and $H$-invariant polynomials on
${\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$. Let $H$ act on
${\mathbb{C}}^{h+1}$ via $\rho\colon H\hookrightarrow(S^{1})^{h+1}$ and let
$\Phi_{H}\colon{\mathbb{C}}^{h+1}\to{\mathfrak{h}}^{*}$ be the homogeneous
moment map. Since $\Phi_{Y}([t,\alpha,z])=\alpha+\Phi_{H}(z)$ for all
$[t,\alpha,z]\in Y$, it suffices to prove that the algebra of $H$-invariant
polynomials on ${\mathbb{C}}^{h+1}$ is generated by the real and imaginary
parts of the defining polynomial $P\colon{\mathbb{C}}^{h+1}\to{\mathbb{C}}$,
the map $z\mapsto|z|^{2}$, and the components of $\Phi_{H}$. Since the local
model $Y$ is tall, Remark 2.3 implies that
$(1,\ldots,1)\notin\rho_{*}({\mathfrak{h}})$. Hence, since
$\Phi_{H}(z)=\frac{1}{2}\rho^{*}(|z_{0}|^{2},\ldots,|z_{h}|^{2})$, the
subalgebra of $H$-invariant polynomials on ${\mathbb{C}}^{h+1}$ generated by
the map $z\mapsto|z|^{2}$ and the components of $\Phi_{H}$ is also generated
by the functions $z\mapsto|z_{i}|^{2}$ for $i=0,\ldots,h$.

Let $P\colon{\mathbb{C}}^{h+1}\to{\mathbb{C}}$ be given by
$P(z)=\prod_{i}z_{i}^{\xi_{i}}$. Consider a complex-valued polynomial of the
form $Q(z)=\prod_{i}z_{i}^{j_{i}}\,\overline{z}_{i}^{k_{i}}$. Given
$\lambda\in(S^{1})^{h+1}$, $Q(\lambda
z)=\prod_{i}\lambda_{i}^{j_{i}-k_{i}}Q(z)$. In particular, $Q$ is
$H$-invariant exactly if $\prod_{i}\lambda_{i}^{j_{i}-k_{i}}=1$ for all
$\lambda\in\rho(H)\varsubsetneq(S^{1})^{h+1}$. Hence, by Lemma 2.1, $Q$ is
$H$-invariant exactly if the vector
$(j_{0}-k_{0},j_{1}-k_{1},\dots,j_{h}-k_{h})$ is a multiple of
$(\xi_{0},\dots,\xi_{h})$, i.e., exactly if $Q$ is some product of $P$, its
conjugate, and polynomials of the form $z\mapsto|z_{i}|^{2}$. Since the the
vector space of complex-valued polynomials on ${\mathbb{C}}^{h+1}$ decomposes
as the direct sum of weight spaces that are spanned by elements of the form
$z\mapsto\prod_{i}z_{i}^{j_{i}}\overline{z}_{i}^{k_{i}}$, it follows that
$H$-invariant complex-valued polynomials are generated by $P$, its conjugate,
and the polynomials $z\mapsto|z_{i}|^{2}$. The claim follows immediately. ∎

######  Remark 2.12.

The proof of Lemma 2.11 provides an equivalent generating set for the algebra
of $T$-invariant polynomials on a tall local model $Y$, namely the real and
imaginary parts of the defining polynomial $P\colon Y\to{\mathbb{C}}$, the
maps $|z_{i}|^{2}\colon Y\to{\mathbb{R}}$ that send $[t,\alpha,z]$ to
$|z_{i}|^{2}$ for $i=0,\ldots,h$, and the components of the map
$[t,\alpha,z]\mapsto\alpha$.

##  3\. The key proposition

In this section, we prove that we can endow each non-trivial symplectic
quotient with the structure of a smooth oriented surface so that the induced
function $\overline{g}$ is Morse – as long as most singular points satisfy a
certain technical condition (see Proposition 3.4). This eventually allows us
to use Morse theory in two dimensions to obtain the desired results.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. As in the Introduction, we denote the induced function on the
reduced space $\Phi^{-1}(\Phi(p))/T$ by $\overline{g}$. We need the following
definition.

######  Definition 3.1.

A point $p\in M$ is a critical point of $\boldsymbol{g}$ modulo
$\boldsymbol{\Phi}$ if $D_{p}g$ vanishes on ${\operatorname{ker}}D_{p}\Phi$;
otherwise, $p$ is a regular point of $\boldsymbol{g}$ modulo
$\boldsymbol{\Phi}$.

Near each regular point of $g$ modulo $\Phi$ there is a natural structure of
smooth manifold on the reduced space so that $\overline{g}$ is smooth with no
critical points. More precisely, the following holds.

######  Lemma 3.2.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. Let $p\in M$ be a regular point of $g$ modulo $\Phi$. Then
there exists an open neighborhood $U_{p}$ of $[p]$ in the reduced space
$\Phi^{-1}(\Phi(p))/T$ and a map $\Psi_{p}\colon U_{p}\to{\mathbb{R}}^{2}$
satisfying:

  1. (a)

$\Psi_{p}\colon U_{p}\to\Psi(U_{p})$ is a diffeomorphism onto an open set; and

  2. (b)

$\overline{g}\circ\Psi_{p}^{-1}\colon\Psi_{p}(U_{p})\to{\mathbb{R}}$ is a
smooth function with no critical point.

Moreover, $p$ has purely elliptic type and the degree $N$ of the defining
polynomial of the local model for $p$ is $1$.

###### Proof.

The result is local. Hence, by the Marle-Guillemin-Sternberg local normal form
theorem, we may assume that $M$ is a local model
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, that
$\Phi=\Phi_{Y}$, and that $p=[1,0,0]$. First we show that $Y$ is tall. If it
is short, then $\Phi^{-1}(\Phi(p))=T\cdot p$. Since $\Phi$ is invariant under
the Hamiltonian flow of $g$, this implies that $(T\times{\mathbb{R}})\cdot
p=T\cdot p$. Hence $p$ is a critical point of $g$ modulo $\Phi$, which
contradicts our assumption.

We identify $T_{p}Y$ with
${\mathfrak{h}}^{\perp}\times{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$
via the map
${\mathfrak{h}}^{\perp}\times{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}\to
Y$ given by $(\xi,\alpha,z)\mapsto[{\operatorname{exp}}(\xi),\alpha,z]$. The
kernel of $D_{p}\Phi$ is given by
${\mathfrak{h}}^{\perp}\times\\{0\\}\times{\mathbb{C}}^{h+1}$. By Remark 2.12,
if the degree $N$ of the defining polynomial $P$ is greater than 1, then the
only $T$-invariant polynomials of degree 1 on $Y$ are the components of the
map $[t,\alpha,z]\mapsto\alpha$, while if $N=1$, the real and imaginary parts
of $P$ are also $T$-invariant polynomials of degree 1. Since the derivative of
the map $[t,\alpha,z]\mapsto\alpha$ at $p$ vanishes when restricted to
${\operatorname{ker}}D_{p}\Phi$, this implies that $N=1$. If we assume,
without loss of generality, that $P(z)=z_{0}$, it also implies that $D_{p}g$
restricts to a non-zero $H$-invariant linear function on
$\\{0\\}\times\\{0\\}\times({\mathbb{C}}\times\\{0\\})$. By Lemma 2.8,
$\overline{P}$ is a diffeomorphism and
$\overline{g}\circ\overline{P}^{-1}\colon{\mathbb{C}}\simeq{\mathbb{R}}^{2}\to{\mathbb{R}}$
is equal to the restriction of $g$ to
${\mathbb{C}}\simeq\\{1\\}\times_{H}\\{0\\}\times({\mathbb{C}}\times\\{0\\})\subseteq
Y$. Hence, since the restriction of $D_{p}g$ to
$\\{0\\}\times\\{0\\}\times({\mathbb{C}}\times\\{0\\})$ is non-zero, there
exists an open neighborhood $U_{p}$ of $[p]$ in ${Y/\\!/T}$ such that the
restriction of $\overline{g}\circ\overline{P}^{-1}$ to $\overline{P}(U_{p})$
is smooth and has no critical points. Setting $\Psi_{p}\colon
U_{p}\to{\mathbb{R}}^{2}$ to be the restriction of $\overline{P}$ to $U_{p}$,
the first result follows.

Let $\mathcal{O}$ be the $(T\times{\mathbb{R}})$-orbit through $p$. Since
$D_{p}g$ restricts to a non-zero function on ${\mathbb{C}}\times\\{0\\}$, the
vector space $\left(T_{p}\mathcal{O}\right)^{\omega}/T_{p}\mathcal{O}$ is
isomorphic to ${\mathbb{C}}^{h}$ and the Lie algebra of the
$(T\times{\mathbb{R}})$-stabilizer of $p$ is
${\mathfrak{h}}\times\\{0\\}\varsubsetneq{\mathfrak{t}}\times{\mathbb{R}}$.
Hence, the linearized action on
$\left(T_{p}\mathcal{O}\right)^{\omega}/T_{p}\mathcal{O}$ is isomorphic to the
$H$-action on ${\mathbb{C}}^{h}$. Finally, since $P(z)=z_{0}$, $H$ acts on
${\mathbb{C}}^{h+1}\simeq{\mathbb{C}}\times{\mathbb{C}}^{h}$ through an
isomorphism $H\simeq\\{1\\}\times(S^{1})^{h}$. Therefore, $p$ has purely
elliptic type. ∎

######  Remark 3.3.

Clearly, every critical point of $g$ modulo $\Phi$ is a singular point of
$(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=(\Phi,g))$. The converse holds
over the interior of $\Phi(M)$ if $\Phi$ is proper, and hence open. To see
this, note that if $p$ is a regular point of $g$ modulo $\Phi$, then by Lemma
3.2 the degree of the defining polynomial is 1. Hence, either the
$T$-stabilizer of $p$ is trivial or $\Phi(p)$ lies on the boundary of
$\Phi(M)$. Additionally, the following are all critical points of $g$ modulo
$\Phi$: non-degenerate singular points with a non-elliptic block, ephemeral
points, and short points.

By Lemma 3.2, to construct the desired Morse function on a reduced space we
need to impose local conditions on each critical point of $g$ modulo $\Phi$.
More precisely, the following key proposition holds.

######  Proposition 3.4.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. Fix $\alpha\in{\mathfrak{t}}^{*}$. Assume that for each
critical point $p\in\Phi^{-1}(\alpha)$ of $g$ modulo $\Phi$ there exists an
open neighborhood $U_{p}$ of $[p]$ in the reduced space $\Phi^{-1}(\alpha)/T$
and a map $\Psi_{p}\colon U_{p}\to{\mathbb{R}}^{2}$ satisfying:

  1. (1)

$\Psi_{p}\colon U_{p}\to\Psi_{p}(U_{p})$ is a homeomorphism onto an open set;

  2. (2)

$\Psi_{p}$ restricts to a diffeomorphism on $U_{p}\smallsetminus\\{[p]\\}$;
and

  3. (3)

$\overline{g}\circ\Psi_{p}^{-1}\colon\Psi_{p}(U_{p})\to{\mathbb{R}}$ is a
Morse function.

Then $\Phi^{-1}(\alpha)/T$ can be given the structure of a smooth oriented
surface such that $\overline{g}\colon\Phi^{-1}(\alpha)/T\to{\mathbb{R}}$ is a
Morse function. Moreover, $[p]$ is a critical point of $\overline{g}$ of index
$\lambda$ exactly if both $p$ is a critical point of $g$ modulo $\Phi$ and
$\Psi_{p}([p])$ is a critical point of $\overline{g}\circ\Psi_{p}^{-1}$ of
index $\lambda$.

###### Proof.

Assume for simplicity that $\alpha=0$. By hypothesis and Lemma 3.2, for each
$p\in\Phi^{-1}(0)$ there exists an open neighborhood $U_{p}$ of $[p]$ in the
reduced space ${M/\\!/T}:=\Phi^{-1}(0)/T$ and a map $\Psi_{p}\colon
U_{p}\to{\mathbb{R}}^{2}$ satisfying properties (1) – (3) above. Since
${M/\\!/T}$ is paracompact, the open cover $\\{U_{p}\\}_{p\in\Phi^{-1}(0)}$
has a locally finite refinement. Hence, there exists a proper subset
$\mathcal{P}\varsubsetneq\Phi^{-1}(0)$ and a locally finite open cover
$\\{U_{p}\\}_{p\in\mathcal{P}}$ such that there exists a map $\Psi_{p}\colon
U_{p}\to{\mathbb{R}}^{2}$ satisfying properties (1) – (3) above for each
$p\in\mathcal{P}$; however, we may no longer assume that $U_{p}$ contains $p$.
Since a locally finite set of points is closed, we may also assume that if
$[p]\in U_{p}$ then $[p]\notin U_{p^{\prime}}$ for all $p^{\prime}\neq p$ in
$\mathcal{P}$. By construction, for all $p,p^{\prime}\in\mathcal{P}$, the
transition function $\Psi_{p}\circ\Psi_{p^{\prime}}^{-1}$ is a diffeomorphism
from $\Psi_{p^{\prime}}(U_{p}\cap U_{p^{\prime}})$ to $\Psi_{p}(U_{p}\cap
U_{p^{\prime}})$. Hence, the coordinate charts $\Psi_{p}$ endow the reduced
space ${M/\\!/T}$ with the structure of a $2$-dimensional smooth manifold.
Moreover, with respect to this smooth structure,
$\overline{g}\colon{M/\\!/T}\to{\mathbb{R}}$ is a Morse function.
Additionally, $[p]$ is a critical point of $\overline{g}$ exactly if both $p$
is a critical point of $g$ modulo $\Phi$ and $\Psi_{p}([p])$ is a critical
point of $\overline{g}\circ\Psi_{p}^{-1}$; in this case, the index of $[p]$ is
the index of $\Psi_{p}([p])$. Finally, since every point in $\Phi^{-1}(0)$ is
tall, ${M/\\!/T}$ is oriented by [KT1, Proposition 6.1]. ∎

The aim of Sections 4 and 6 is to show that the hypotheses of Proposition 3.4
hold for non-degenerate singular points.

##  4\. Purely elliptic type

Our goal in this section is to prove that tall critical points of $g$ modulo
$\Phi$ that have purely elliptic type satisfy the hypotheses of Proposition
3.4 (see Proposition 4.5 below). An important ingredient is the local
classification of such points (see Lemma 4.4).

We begin by giving an important example and describing the symplectic form on
(complexity one) local models.

######  Example 4.1.

Fix $\ell\in\\{0,1,\dots,n\\}.$ Let
$\omega_{0}\in\Omega^{2}(T^{*}(S^{1})^{n-\ell}\times{\mathbb{C}}^{\ell})$ be
the sum of the pullbacks of the canonical symplectic form on
$T^{*}(S^{1})^{n-\ell}$ and the standard symplectic form on
${\mathbb{C}}^{\ell}$. Let $\mu\colon
T^{*}(S^{1})^{n-\ell}\times{\mathbb{C}}^{\ell}\simeq(S^{1})^{n-\ell}\times\left({\mathbb{R}}^{n-\ell}\right)^{*}\times{\mathbb{C}}^{\ell}\to({\mathbb{R}}^{n})^{*}$
be the moment map for the standard
$(S^{1})^{n}=(S^{1})^{n-\ell}\times(S^{1})^{\ell}$-action on
$T^{*}(S^{1})^{n-\ell}\times{\mathbb{C}}^{\ell}$ given by
$\mu(s,a,w)=(a,\frac{1}{2}|w|^{2})$, where here
$\frac{1}{2}|w|^{2}:=\frac{1}{2}(|w_{1}|^{2},\ldots,|w_{\ell}|^{2})$. Then
$(T^{*}(S^{1})^{n-\ell}\times{\mathbb{C}}^{\ell},\omega_{0},{\mathbb{R}}^{n},\mu)$
is an integrable system; moreover, $p=(s,a,w)$ is singular exactly if
$\prod_{i}w_{i}=0$, in which case $p$ has purely elliptic type.

######  Remark 4.2.

Let $Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be a
complexity one local model, where $H$ acts on ${\mathbb{C}}^{h+1}$ via
$\rho:H\to(S^{1})^{h+1}$ with homogeneous moment map
$\Phi_{H}\colon{\mathbb{C}}^{h+1}\to{\mathfrak{h}}^{*}$. Let
$i^{*}\colon{\mathfrak{t}}^{*}\to{\mathfrak{h}}^{*}$ be the projection induced
by the inclusion $H\hookrightarrow T$. Equip $T^{*}T\times{\mathbb{C}}^{h+1}$
with the symplectic form given by the sum of the pullbacks of the canonical
symplectic form on $T^{*}T$ and the standard symplectic form on
${\mathbb{C}}^{h+1}$. The $H$-action on $T^{*}T\times{\mathbb{C}}^{h+1}\cong
T\times{\mathfrak{t}}^{*}\times{\mathbb{C}}^{h+1}$ given by
$h\cdot(t,\beta,w):=(ht,\beta,\rho(t^{-1})w)$ has a moment map
$T\times{\mathfrak{t}}^{*}\times{\mathbb{C}}^{h+1}\to{\mathfrak{h}}^{*}$ that
takes $(t,\beta,z)$ to $i^{*}(\beta)-\Phi_{H}(z)$. The reduced space at $0$ is
smooth, and the map
$Y\to(T\times{\mathfrak{t}}^{*}\times{\mathbb{C}}^{h+1})/\\!/H$ taking
$[t,\alpha,z]$ to $[t,\alpha+\Phi_{H}(z),z]$ is a diffeomorphism. By
definition, the symplectic form $\omega_{Y}$ on $Y$ is the pullback of the
reduced symplectic form along this diffeomorphism.

Next we define “isomorphism” for the class of integrable systems that we
consider.

######  Definition 4.3.

Let
$(M_{i},\omega_{i},{\mathfrak{t}}\times{\mathbb{R}},f_{i}=(\Phi_{i},g_{i}))$
be an integrable system such that $(M_{i},\omega_{i},\Phi_{i})$ is a
complexity one $T$-space for $i=1,2$. An isomorphism from
$(M_{1},\omega_{1},{\mathfrak{t}}\times{\mathbb{R}},f_{1})$ to
$(M_{2},\omega_{2},{\mathfrak{t}}\times{\mathbb{R}},f_{2})$ is a
symplectomorphism $\Psi\colon(M_{1},\omega_{1})\to(M_{2},\omega_{2})$ and a
diffeomorphism222Let $X,Z$ be manifolds and let $A\subseteq X$. A map
$\psi\colon A\to Z$ is smooth if, for every point $a\in A$, there exists an
open set $U_{a}\subseteq X$ and a smooth map $\psi_{a}\colon U_{a}\to Z$
extending $\psi$. $\psi\colon
f_{1}(M_{1})\subseteq{\mathfrak{t}}^{*}\times{\mathbb{R}}\to
f_{2}(M_{2})\subseteq{\mathfrak{t}}^{*}\times{\mathbb{R}}$ of the form

$\psi(\beta,\lambda)=(\tau(\beta),\psi_{{\mathbb{R}}}(\beta,\lambda))\text{
for }(\beta,\lambda)\in f_{1}(M_{1}),$

where $\tau\colon{\mathfrak{t}}^{*}\to{\mathfrak{t}}^{*}$ is a translation,
such that $f_{2}\circ\Psi=\psi\circ f_{1}$.

We can now give the local classification for critical points of $g$ modulo
$\Phi$ that have purely elliptic type.

######  Lemma 4.4.

Let $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=(\Phi,g))$ be an integrable
system such that $(M,\omega,\Phi)$ is a complexity one $T$-space. Let $p\in M$
be a critical point of $g$ modulo $\Phi$ that has purely elliptic type. Let
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ be the local
model for $p$, where $H$ acts on ${\mathbb{C}}^{h+1}$ via $\rho\colon
H\to(S^{1})^{h+1}$. Given
$b\in{\mathbb{R}}^{h+1}\smallsetminus\rho_{*}({\mathfrak{h}})$, a neighborhood
of $T\cdot p$ in $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f=(\Phi,g))$ is
isomorphic to a neighborhood of $T\cdot[1,0,0]$ in
$(Y,\omega_{Y},{\mathfrak{t}}\times{\mathbb{R}},f_{b}=(\Phi_{Y},g_{b}))$,
where $g_{b}\colon Y\to{\mathbb{R}}$ is given by
$g_{b}([t,\alpha,z])=\sum_{i=0}^{h}b_{i}|z_{i}|^{2}$.

###### Proof.

Since $p$ is a critical point of $g$ modulo $\Phi$,
${\operatorname{ker}}D_{p}f={\operatorname{ker}}D_{p}\Phi$, and so the
$\left(T\times{\mathbb{R}}\right)$-orbit of $p$ is equal to the $T$-orbit of
$p$. In particular, the rank of $D_{p}f$ is $n-h-1$ and the
$\left(T\times{\mathbb{R}}\right)$-orbit of $p$ is compact, where $h$ is the
dimension of the stabilizer of $p$ for the $T$-action. By the local normal
form for compact orbits of points that have purely elliptic type [E], we may
assume that $M$ and the symplectic form are as in Example 4.1, where
$\ell=h+1$, that $p=(1,0,0)$, and that there exists a diffeomorphism
$\varphi\colon\left({\mathbb{R}}^{n}\right)^{*}\to{\mathfrak{t}}^{*}\times{\mathbb{R}}$
such that $f=\varphi\circ\mu$, where $\mu$ is also as in Example 4.1.

Since $\varphi\circ\mu=f=(\Phi,g)$, there exist an injective Lie group
homomorphism $\Lambda\colon
T\to(S^{1})^{n}=(S^{1})^{n-h-1}\times(S^{1})^{h+1}$,
$\beta\in{\mathfrak{t}}^{*}$, and a smooth map
$\varphi_{\mathbb{R}}\colon({\mathbb{R}}^{n})^{*}\to{\mathbb{R}}$ such that
$\varphi=(\Lambda^{*}+\beta,\varphi_{{\mathbb{R}}})$, where $\Lambda^{*}$ is
the dual of the Lie algebra homomorphism
$\Lambda_{*}:{\mathfrak{t}}\to{\mathbb{R}}^{n}$. By replacing $f$ by its pre-
composition with the translation by $-\beta$, we may assume that $\beta=0$.
Hence, the composite
$D_{0}\varphi\circ\varphi^{-1}\colon{\mathfrak{t}}^{*}\times{\mathbb{R}}\to{\mathfrak{t}}^{*}\times{\mathbb{R}}$
is the identity on ${\mathfrak{t}}^{*}$. Therefore, by replacing $f$ by the
composition $D_{0}\varphi\circ\varphi^{-1}\circ f=D_{0}\varphi\circ\mu$, we
may assume that $\varphi_{{\mathbb{R}}}$ is linear.

Let $\Lambda_{1}$ and $\Lambda_{2}$ denote the projections of $\Lambda$ onto
$(S^{1})^{n-h-1}$ and $(S^{1})^{h+1}$, and let $\Lambda_{1}^{*}$ and
$\Lambda^{*}_{2}$ be the dual of the Lie algebra homomorphisms induced by
$\Lambda_{1}$ and $\Lambda_{2}$ respectively. Then
${\operatorname{ker}}\,\Lambda_{1}=H$ and $\Lambda_{2}|_{H}=\rho$. Hence, if
$b\in{\mathbb{R}}^{h+1}\smallsetminus\rho_{*}({\mathfrak{h}})$, then
$(0,b)\in{\mathbb{R}}^{n}\smallsetminus\Lambda_{*}({\mathfrak{t}})$. Thus
there exists a linear isomorphism $L$ of
${\mathfrak{t}}^{*}\times{\mathbb{R}}$ that restricts to the identity on
${\mathfrak{t}}^{*}$ so that, by further replacing $f$ by the composition
$L\circ f$, we may assume that $\varphi_{{\mathbb{R}}}=(0,2b)$. In particular,
$g(s,a,w)=\sum_{i=0}^{h}b_{i}|w_{i}|^{2}$.

By construction, the map given by

$[t,\alpha,z]\mapsto[t,\alpha+\Phi_{H}(z),z]$

is a symplectomorphism between the local model $Y$ and the reduced space
$(T\times{\mathfrak{t}}^{*}\times{\mathbb{C}}^{h+1})/\\!/H$ discussed in
Remark 4.2.

By a straightforward computation, the map
$T\times{\mathfrak{t}}^{*}\times{\mathbb{C}}^{h+1}\to
T\times{\mathfrak{t}}^{*}\times{\mathbb{C}}^{h+1}$ given by
$\textstyle(t,\beta,z)\mapsto\left(t,\beta-\Lambda^{*}_{2}\left(\frac{1}{2}|z_{0}|^{2},\ldots,\frac{1}{2}|z_{h}|^{2}\right),\Lambda_{2}(t)z\right)$
is a symplectomorphism. It induces a symplectomorphism between the reduced
space $(T\times{\mathfrak{t}}^{*}\times{\mathbb{C}}^{h+1})/\\!/H$ and
$T/H\times{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ that is given by

$\textstyle[t,\beta,z]\mapsto(tH,\beta-\Lambda^{*}_{2}\left(\frac{1}{2}|z_{0}|^{2},\ldots,\frac{1}{2}|z_{h}|^{2}\right),\Lambda_{2}(t)z).$

Here the symplectic form on
$T/H\times{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ is the sum of the
pullbacks of the canonical symplectic form on $T^{*}(T/H)\cong
T/H\times{\mathfrak{h}}^{\circ}$ and of the standard symplectic form on
${\mathbb{C}}^{h+1}$.

By a dimension count, $\Lambda_{1}$ induces an isomorphism between $T/H$ and
$(S^{1})^{n-h-1}$. Hence it induces a symplectomorphism between
$T/H\times{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ and
$(S^{1})^{n-h-1}\times\left({\mathbb{R}}^{n-h-1}\right)^{*}\times{\mathbb{C}}^{h+1}$
that is given by

$(tH,\alpha,w)\mapsto(\Lambda_{1}(t),(\Lambda^{*}_{1})^{-1}(\alpha),w).$

Composing these maps, we obtain a symplectomorphism $\Psi\colon
Y\to(S^{1})^{n-h-1}\times\left({\mathbb{R}}^{n-h-1}\right)^{*}\times{\mathbb{C}}^{h+1}$.
Since $\Lambda_{2}|_{H}=\rho$, it is given by

$\textstyle\Psi([t,\alpha,z])=\left(\Lambda_{1}(t),(\Lambda_{1}^{*})^{-1}\left(\alpha-\mathrm{pr}_{{\mathfrak{h}}^{\circ}}\left(\Lambda^{*}_{2}\left(\frac{1}{2}|z_{0}|^{2},\ldots,\frac{1}{2}|z_{h}|^{2}\right)\right)\right),\Lambda_{2}(t)z\right).$

Here
$\mathrm{pr}_{{\mathfrak{h}}^{\circ}}\colon{\mathfrak{t}}^{*}\to{\mathfrak{h}}^{\circ}$
denotes the projection induced by the fixed choice of metric on
${\mathfrak{t}}$. Clearly this map is $T$-equivariant and so $\Phi\circ\Psi$
is a moment map for the $T$-action on $Y$. Since $\Phi\circ\Psi(1,0,0)=0$, it
follows that $\Phi\circ\Psi$ is the homogeneous moment map $\Phi_{Y}$.
Moreover, it is easy check that $g\circ\Psi=g_{b}$. By construction, the
integrable system
$(Y,\omega_{Y},{\mathfrak{t}}\times{\mathbb{R}},f\circ\Psi=(\Phi_{Y},g_{b}))$
is isomorphic to the original one, as desired. ∎

We are finally ready to prove the main result of this section.

######  Proposition 4.5.

Let $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=(\Phi,g))$ be an integrable
system so that $(M,\omega,\Phi)$ is a complexity one $T$-space. Let
$p\in\Phi^{-1}(0)\cap g^{-1}(0)$ be a tall critical point of $g$ modulo $\Phi$
that has purely elliptic type. After possibly replacing $g$ by $-g$, there
exist an open neighborhood $U$ of $[p]$ in ${M/\\!/T}$ and a map $\Psi\colon
U\to{\mathbb{R}}^{2}$ taking $[p]$ to $(0,0)$ satisfying:

  1. (1)

$\Psi\colon U\to\Psi(U)$ is a homeomorphism onto an open set;

  2. (2)

$\Psi$ restricts to a diffeomorphism on $U\smallsetminus\\{[p]\\}$; and

  3. (3)

$(\overline{g}\circ\Psi^{-1})(x,y)=x^{2}+y^{2}$ for all $(x,y)\in\Psi(U)$.

###### Proof.

Since $p$ has tall local model
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, Remark 2.3
implies that $(1,\ldots,1)\notin\rho_{*}({\mathfrak{h}})$, where $\rho\colon
H\to(S^{1})^{h+1}$ is the symplectic slice representation at $p$. Hence, by
Lemma 4.4 and the Morse Lemma, after possibly replacing $g$ by $-g$, we may
assume that $M$ is a local model
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, that
$\Phi=\Phi_{Y}$, that $g([t,\alpha,z])=|z|^{2}$, and that $p=[1,0,0]$. Let $N$
be the degree of the defining polynomial
$P\colon{\mathbb{C}}^{h+1}\to{\mathbb{C}}$. By Lemma 2.7, the map
$\overline{P}\colon{Y/\\!/T}\to{\mathbb{C}}$ is a homeomorphism; moreover, it
restricts to a diffeomorphism from $({Y/\\!/T})\smallsetminus\\{[t,0,0]\\}$ to
${\mathbb{C}}\smallsetminus\\{0\\}$. By Lemma 2.9, there exists a positive
real number $C$ such that $g([t,\alpha,z])=C|P(z)|^{\frac{2}{N}}$ for all
$[t,\alpha,z]\in\Phi_{Y}^{-1}(0)$. Let $S\colon{\mathbb{C}}\to{\mathbb{C}}$ be
the homeomorphism that, in polar coordinates, takes $(r,\theta)$ to
$(\sqrt{C}\,r^{\frac{1}{N}},\theta)$; it restricts to a diffeomorphism
${\mathbb{C}}\smallsetminus\\{0\\}\to{\mathbb{C}}\smallsetminus\\{0\\}$.
Define $\Psi\colon{Y/\\!/T}\to{\mathbb{C}}\simeq{\mathbb{R}}^{2}$ by
$\Psi([t,\alpha,z])=S(P(z))$. Since

$|\Psi([t,\alpha,z])|^{2}=C|P(z)|^{\frac{2}{N}}=\overline{g}([t,\alpha,z]),$

for all $[t,\alpha,z]\in{Y/\\!/T}$, the map $\Psi$ satisfies properties (1) –
(3). ∎

######  Remark 4.6.

We observe that, by Lemma 2.8, if $N=1$ the map $\Psi$ constructed in the
above proof is a diffeomorphism.

##  5\. The semitoric case

The goal of this section is to prove that focus-focus points satisfy the
hypotheses of Proposition 3.4 (see Lemma 5.4 below). This allows us to provide
a short and direct new proof of Vu Ngoc’s theorem that the level sets of
semitoric systems are connected [VN] – see Theorem 5.7. In both proofs, the
key arguments follow from Morse theory; the most significant difference is
that we construct a new smooth structure on the reduced spaces, while he uses
the natural smooth structure on $M$.

######  Definition 5.1.

An integrable system
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f=\left(\Phi,g\right)\right)$
is semitoric if

  * •

$\left(M,\omega,\Phi\right)$ is a 4-dimensional complexity one $T$-space, and

  * •

every singular point of
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f\right)$ is non-degenerate
with no hyperbolic blocks.

Note that, by definition, every singular point of a semitoric system either
has purely elliptic type or is focus-focus, i.e., it has one focus-focus
block. We begin with an important example of a semitoric system containing one
focus-focus point.

######  Example 5.2.

Let $Y={\mathbb{C}}^{2}$, let $\omega_{Y}$ be the standard symplectic form,
and set
$\Phi_{Y}(z_{1},z_{2}):=\frac{1}{2}\left(|z_{1}|^{2}-|z_{2}|^{2}\right)$ and
$g_{Y}(z_{1},z_{2}):={\mathfrak{I}}(z_{1}z_{2})={\mathfrak{I}}(P(z_{1},z_{2}))$,
where $P\colon Y\to{\mathbb{C}}$ is the defining polynomial for the
$S^{1}$-action induced by $\Phi_{Y}$ (cf. Example 2.6).

Locally, near a focus-focus point, every semitoric system is modeled on
Example 5.2.

######  Proposition 5.3.

Let $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f=(\Phi,g))$ be an integrable
system such that $(M,\omega,\Phi)$ is a 4-dimensional complexity one
$T$-space. Assume that $p\in M$ is focus-focus. After identifying $T$ with
$S^{1}$, a neighborhood of $p$ is isomorphic to a neighborhood of the origin
in the semitoric system $(Y,\omega_{Y},{\mathbb{R}}\times{\mathbb{R}},f_{Y})$
described above.

###### Proof.

Identify $T$ with $S^{1}$. By the local normal form for focus-focus points, we
may assume that the integrable system is
$(Y,\omega_{Y},{\mathbb{R}}\times{\mathbb{R}},\varphi\circ f_{Y})$ and $p$ is
the origin, where
$(Y,\omega_{Y},{\mathbb{R}}\times{\mathbb{R}},f_{Y}=(\Phi_{Y},g_{Y}))$ is as
in Example 5.2 and
$\varphi\colon{\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}\times{\mathbb{R}}$
is a diffeomorphism [C, VW]. Since by assumption the first component of
$\varphi\circ f_{Y}$ induces an effective $S^{1}$-action on $Y$, and since the
level sets of $f_{Y}$ are not compact, up to a translation $\Phi_{Y}$ and the
first component of $\varphi\circ f_{Y}$ are equal up to sign (cf. [HSS,
Proposition 3.9]). Since the integrable systems
$(Y,\omega_{Y},{\mathbb{R}}\times{\mathbb{R}},f_{Y}=(\Phi_{Y},g_{Y}))$ and
$(Y,\omega_{Y},{\mathbb{R}}\times{\mathbb{R}},(-\Phi_{Y},g_{Y}))$ are
isomorphic, the claim follows. ∎

Hence, focus-focus points satisfy the hypotheses of Proposition 3.4.

######  Lemma 5.4.

Let $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f=(\Phi,g))$ be an integrable
system so that $(M,\omega,\Phi)$ is a 4-dimensional complexity one $T$-space.
Assume that $p\in\Phi^{-1}(0)\cap g^{-1}(0)$ is focus-focus. Then there exist
an open neighborhood $U$ of $[p]\in{M/\\!/T}$ and a map $\Psi\colon
U\to{\mathbb{R}}^{2}$ taking $[p]$ to $(0,0)$ satisfying:

  1. (1)

$\Psi\colon U\to\Psi(U)$ is a homeomorphism onto an open set;

  2. (2)

$\Psi$ restricts to a diffeomorphism on $U\smallsetminus\\{[p]\\}$; and

  3. (3)

$(\overline{g}\circ\Psi^{-1})(x,y)=y$ for all $(x,y)\in\Psi(U)$.

###### Proof.

By Proposition 5.3 we may assume that the integrable system is the system
$(Y,\omega_{Y},{\mathbb{R}}\times{\mathbb{R}},f_{Y}:=(\Phi_{Y},g_{Y}))$
described in Example 5.2, and that $p$ is the origin. Set
$\Psi[z_{1},z_{2}]:=z_{1}z_{2}=P(z_{1},z_{2})$. Then the claim follows
immediately from Lemma 2.7. ∎

We are now almost ready to give our proof of Vu Ngoc’s theorem on the
connectedness of the level sets of semitoric systems; however, we also need
the following important result, which is proved in [At, GS2, LMTW].

######  Theorem 5.5.

Let $T$ act on a connected symplectic manifold $(M,\omega)$ with moment map
$\Phi\colon M\to{\mathfrak{t}}^{*}$. If $\Phi$ is proper then the level set
$\Phi^{-1}(\alpha)$ is connected for all $\alpha\in{\mathfrak{t}}^{*}.$

We also need the following proposition. Atiyah proved the compact case (see
[At]), and used it to prove the theorem above; it was later extended to the
proper case; see [PRV, Proposition 4.5]. (For completeness, we provide a
variation on the arguments in [PRV] in Section 7.)

######  Proposition 5.6.

Let $N$ be a connected manifold and let $\overline{g}\colon N\to{\mathbb{R}}$
be a proper Morse-Bott function. If no critical point of $\overline{g}$ has
index or coindex one, then the level set $\overline{g}^{-1}(c)$ is connected
for all $c\in{\mathbb{R}}$.

We observe that the result below is a special case of Theorem 1.6, and the
proof below illustrates our general strategy.

######  Theorem 5.7 (Vũ Ngoc).

Let $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f=(\Phi,g))$ be a semitoric
system so that $\Phi$ is proper. Then the level set
$f^{-1}(\alpha,c)=\Phi^{-1}(\alpha)\cap g^{-1}(c)$ is connected for all
$(\alpha,c)\in{\mathfrak{t}}^{*}\times{\mathbb{R}}$.

###### Proof.

Fix $\alpha\in{\mathfrak{t}}^{*}$. Assume without loss of generality that
$\alpha=0$. Since $\Phi$ is proper, the reduced space ${M/\\!/T}$ is compact
and connected by Theorem 5.5.

If the reduced space ${M/\\!/T}$ is a single point then obviously
$\Phi^{-1}(0)$ consists of a single $T$-orbit. Hence, the level set
$f^{-1}(0,c)$ is connected for all $c\in{\mathbb{R}}$.

Otherwise, since the orbit of a short point in $\Phi^{-1}(0)$ is an isolated
point in $\Phi^{-1}(0)/T$, every point in $\Phi^{-1}(0)$ is tall. If
$p\in\Phi^{-1}(0)$ is a critical point of $g$ modulo $\Phi$, then $p$ is also
a singular point of
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f\right)$. So, by assumption,
$p$ either has purely elliptic type or is focus-focus. By Propositions 3.4 and
4.5, and Lemma 5.4, ${M/\\!/T}$ can be given the structure of a smooth closed,
connected, oriented surface such that $\overline{g}:{M/\\!/T}\to{\mathbb{R}}$
is a Morse function with no critical point of index 1. By Proposition 5.6, the
level set $\overline{g}^{-1}(c)$ is connected for all $c\in{\mathbb{R}}$.
Since $T$ is connected, it follows that $f^{-1}(0,c)$ is also connected. ∎

##  6\. Non-degenerate singular points

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. Let $p\in\Phi^{-1}(0)$ be a tall non-degenerate singular point.
By Lemma 3.2 and Proposition 4.5, if $p$ has purely elliptic type, then we can
put a smooth structure on ${M/\\!/T}$ near $[p]$ so that $\overline{g}$ is
locally Morse, and hence $p$ satisfies the hypotheses of Proposition 3.4. The
aim of this section is to extend this result to the remaining cases (see
Proposition 6.2). In particular, if $p$ either has a focus-focus block, or has
a hyperbolic block and disconnected $T$-stabilizer, then $[p]$ is a regular
point of $\overline{g}$, while if $p$ has a hyperbolic block and connected
$T$-stabilizer, then $[p]$ is a singular point with index one.

We start by recalling some facts about non-degenerate singular points.

######  Remark 6.1.

Let $p$ be a point in an integrable system $(M,\omega,V,f)$ and let
$W_{0}\subseteq V$ be the Lie algebra of the stabilizer of $p$. Let
$\mathcal{O}$ be the $V$-orbit through $p$. The linearized action at $p$
induces a Lie algebra homomorphism
$W_{0}\to\mathfrak{sp}((T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O})$. Given
$\xi\in W_{0}$, we can calculate the associated symplectic endomorphism as
follows: Let $f_{\mathrm{lin}}$ be the homogeneous moment map for the
linearized action. We view $\langle f_{\mathrm{lin}},\xi\rangle$ as a
homogeneous quadratic polynomial on
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$. We recall that there is a
standard isomorphism between homogeneous quadratic polynomials and
$\mathfrak{sp}((T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O})$ that, upon
choosing a standard basis for the symplectic vector space
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$, sends a symmetric matrix $A$
to $JA$, where $J$ is the standard complex structure on
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$. The above Lie algebra
homomorphism sends $\xi$ to $J\langle f_{\mathrm{lin}},\xi\rangle$.

If, in addition, $p$ is non-degenerate, then for a generic element $\xi$ in
the Lie algebra of $W$, the eigenvalues of the above endomorphism of
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ can be characterized as
follows: If $f_{\mathrm{lin}}$ can be written as the product of $k_{e}$
elliptic blocks, $k_{h}$ hyperbolic blocks and $k_{f}$ focus-focus blocks,
then the above endomorphism has exactly $2k_{e}$ distinct non-zero imaginary
eigenvalues, $2k_{h}$ distinct non-zero real eigenvalues, and $4k_{f}$
distinct non-zero complex eigenvalues that are neither real nor imaginary. In
particular, the numbers $k_{e},k_{h}$ and $k_{f}$ are a well-defined invariant
of $p$.

We can now state our main proposition.

######  Proposition 6.2.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. If $p\in\Phi^{-1}(0)\cap g^{-1}(0)$ is non-degenerate it has at
most one non-elliptic block; moreover,

  * •

If $p$ has a hyperbolic block and connected $T$-stabilizer, then there exist
an open neighborhood $U$ of $[p]\in{M/\\!/T}$ and a map $\Psi\colon
U\to{\mathbb{R}}^{2}$ taking $[p]$ to $(0,0)$ satisfying:

    1. (a)

$\Psi\colon U\to\Psi(U)$ is a diffeomorphism onto an open set; and

    2. (b)

$\overline{g}\circ\Psi^{-1}(x,y)=x^{2}-y^{2}$ for all $(x,y)\in\Psi(U)$.

  * •

If $p$ either has a focus-focus block, or has a hyperbolic block and
disconnected $T$-stabilizer, then there exist an open neighborhood $U$ of
$[p]\in{M/\\!/T}$ and a map $\Psi\colon U\to{\mathbb{R}}^{2}$ taking $[p]$ to
$(0,0)$ satisfying:

    1. (1)

$\Psi\colon U\to\Psi(U)$ is a homeomorphism onto an open set;

    2. (2)

$\Psi$ restricts to a diffeomorphism on $U\smallsetminus\\{[p]\\}$; and

    3. (3)

$(\overline{g}\circ\Psi^{-1})(x,y)=y$ for all $(x,y)\in\Psi(U)$.

Let $Y$ be the local model for $p$. We split the proof of Proposition 6.2 in
four cases, depending on whether $p$ is short (see Lemma 6.3), or is tall and
the degree $N$ of the defining polynomial of $Y$ is one (see Proposition 6.4),
two (see Proposition 6.6), or greater than two (see Proposition 6.9).

######  Lemma 6.3.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. If $p\in M$ is short and non-degenerate, then $p$ has purely
elliptic type.

###### Proof.

For simplicity, assume that $g(p)=0$. By the Marle-Guillemin-Sternberg local
normal form theorem, we may assume that $M$ is a local model
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, that
$\Phi=\Phi_{Y}$, and that $p=[1,0,0]$. Let $j\colon{\mathbb{C}}^{h+1}\to
T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$ denote the
inclusion $z\mapsto[1,0,z]$. By Lemma 3.2, $p$ is a critical point of $g$
modulo $\Phi$. Hence, the $(T\times{\mathbb{R}})$-orbit $\mathcal{O}$ through
$p$ coincides with the $T$-orbit $\mathfrak{O}$ through $p$ and the Taylor
polynomial $T^{1}_{0}(j^{*}g)$ of degree one at $0\in{\mathbb{C}}^{h+1}$
vanishes. Thus we may identify
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}=(T_{p}\mathfrak{O})^{\omega}/T_{p}\mathfrak{O}$
with ${\mathbb{C}}^{h+1}$ so that the homogeneous moment map for the
linearized action at $p$ is the Taylor polynomial of degree two
$T^{2}_{0}(j^{*}f)=(T^{2}_{0}(j^{*}\Phi),T^{2}_{0}(j^{*}g))=(\Phi_{H},T^{2}_{0}(j^{*}g))$,
where $\Phi_{H}\colon{\mathbb{C}}^{h+1}\to{\mathfrak{h}}^{*}$ is the
homogeneous moment map for the $H$-action. .

By [KT1, Lemma 5.4], $\Phi$ is proper; hence, so is
$(\Phi_{H},T_{0}^{2}(j^{*}g))$. If $p$ has a hyperbolic or a focus-focus
block, then the homogeneous moment map for the linearized
$(H\times{\mathbb{R}})$-action on ${\mathbb{C}}^{h+1}$ is not proper. Hence,
$p$ has purely elliptic type. ∎

######  Proposition 6.4.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. Assume that $p\in\Phi^{-1}(0)\cap g^{-1}(0)$ is tall and non-
degenerate but does not have purely elliptic type. If the defining polynomial
of the local model for $p$ has degree 1, then there exist an open neighborhood
$U$ of $[p]$ in the reduced space ${M/\\!/T}$ and a map $\Psi\colon
U\to{\mathbb{R}}^{2}$ taking $[p]$ to $(0,0)$ satisfying:

  1. (a)

$\Psi\colon U\to\Psi(U)$ is a diffeomorphism onto an open set; and

  2. (b)

$\overline{g}\circ\Psi^{-1}(x,y)=x^{2}-y^{2}$ for all $(x,y)\in\Psi(U)$.

Moreover, $p$ has one hyperbolic block and connected $T$-stabilizer; every
other block is elliptic.

###### Proof.

By the Marle-Guillemin-Sternberg local normal form theorem, we may assume that
$M$ is a tall local model
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, that
$\Phi=\Phi_{Y}$, and that $p=[1,0,0]$. As in the proof of Lemma 6.3, by Lemma
3.2 we may identify $(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ with
${\mathbb{C}}^{h+1}$ so that the homogeneous moment map for the linearized
action at $p$ is the Taylor polynomial of degree two
$(\Phi_{H},T^{2}_{0}(j^{*}g))$.

Since the defining polynomial $P\colon{\mathbb{C}}^{h+1}\to{\mathbb{C}}$ has
degree 1, we may assume that $P(z)=z_{0}$. Hence,
${\mathbb{C}}^{h+1}\simeq{\mathbb{C}}\times{\mathbb{C}}^{h}$ and
$H\simeq(S^{1})^{h}$, where $(S^{1})^{h}$ acts on ${\mathbb{C}}^{h}$ in the
standard fashion and trivially on ${\mathbb{C}}$. Thus the span of
$\\{\Phi_{H}^{\xi}\mid\xi\in{\mathfrak{h}}\\}$ equals the span of
$\\{|z_{k}|^{2}\mid k\geq 1\\}$. Moreover, since the Taylor polynomial
$T^{2}_{0}(j^{*}g)$ is $H$-invariant, by Remark 2.12, it lies in the span of
$\\{{\mathfrak{R}}(z_{0}^{2}),{\mathfrak{I}}(z_{0}^{2}),|z_{i}|^{2}\colon
i=0,\ldots,h\\}$. Therefore there exist constants
$a_{x},a_{y},a_{\rho}\in{\mathbb{R}}$ and $\xi\in{\mathfrak{h}}$ such that
$T^{2}_{0}(j^{*}g)$ is given by

(6.5)
$a_{x}{\mathfrak{R}}(z_{0}^{2})+a_{y}{\mathfrak{I}}(z_{0}^{2})+a_{\rho}|z_{0}|^{2}+\Phi_{H}^{\xi}.$

Moreover, a generic element in the span of
$\\{T^{2}_{0}(j^{*}g),\Phi_{H}^{\xi}\mid\xi\in{\mathfrak{h}}\\}$ is a multiple
of

$a_{x}{\mathfrak{R}}(z^{2}_{0})+a_{y}{\mathfrak{I}}(z^{2}_{0})+a_{\rho}|z_{0}|^{2}+\sum\limits_{k=1}^{h}\lambda_{k}|z_{k}|^{2},$

where $\lambda_{k}\in{\mathbb{R}}\smallsetminus\\{0\\}$ for $k=1,\ldots,h$.
The eigenvalues of the corresponding symplectic endomorphism are given by

$\pm 2\sqrt{a_{x}^{2}+a_{y}^{2}-a_{\rho}^{2}},\,\,\pm
2\lambda_{1}\sqrt{-1},\,\,\ldots,\,\,\pm 2\lambda_{h}\sqrt{-1}.$

Hence, since $p$ is non-degenerate and does not have purely elliptic type,
Remark 6.1 implies that $a_{x}^{2}+a_{y}^{2}-a_{\rho}^{2}>0$, that $p$ has one
hyperbolic block, and that the remaining blocks are all elliptic.

By Lemma 2.8, the inclusion $i\colon{\mathbb{C}}\to Y$ sending $z_{0}$ to
$[1,0,(z_{0},0,\ldots,0)]$ induces a diffeomorphism
$\overline{i}\colon{\mathbb{C}}\to{Y/\\!/T}$. By definition of the inclusion
$j$ and by (6.5),

$T^{2}_{0}(\overline{i}^{*}\overline{g})(z_{0})=a_{x}{\mathfrak{R}}(z_{0}^{2})+a_{y}{\mathfrak{I}}(z_{0}^{2})+a_{\rho}|z_{0}|^{2},$

and so $0$ is a non-degenerate critical point of
$\overline{i}^{*}\overline{g}$ of index 1. Hence, by the Morse lemma, there
exists an open neighborhood $V$ of $(0,0)\in{\mathbb{R}}^{2}$ and a map
$\psi\colon V\to{\mathbb{C}}$ taking $(0,0)$ to $0$ that is a diffeomorphism
onto an open set such that
$(\overline{i}\circ\psi)^{*}\overline{g}(x,y)=x^{2}-y^{2}$ for all $(x,y)\in
V$. Setting $U:=(\overline{i}\circ\psi)(V)$ and $\Psi\colon
U\to{\mathbb{R}}^{2}$ to be $(\overline{i}\circ\psi)^{-1},$ the result
follows. ∎

######  Proposition 6.6.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. Assume that $p\in\Phi^{-1}(0)\cap g^{-1}(0)$ is tall and non-
degenerate but does not have purely elliptic type. If the defining polynomial
of the local model for $p$ has degree 2, then there exist an open neighborhood
$U$ of $[p]$ in the reduced space ${M/\\!/T}$ and a map $\Psi\colon
U\to{\mathbb{R}}^{2}$ taking $[p]$ to $(0,0)$ satisfying:

  1. (1)

$\Psi\colon U\to\Psi(U)$ is a homeomorphism onto an open set;

  2. (2)

$\Psi$ restricts to a diffeomorphism on $U\smallsetminus\\{[p]\\}$; and

  3. (3)

$(\overline{g}\circ\Psi^{-1})(x,y)=y$ for all $(x,y)\in\Psi(U)$.

Moreover, either $p$ has one focus-focus block, or it has one hyperbolic block
and disconnected $T$-stabilizer; in either case, every other block is
elliptic.

###### Proof.

By the Marle-Guillemin-Sternberg local normal form theorem, we may assume that
$M$ is a tall local model
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, that
$\Phi=\Phi_{Y}$, and that $p=[1,0,0]$. As in the proof of Lemma 6.3, by Lemma
3.2 we may identify $(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ with
${\mathbb{C}}^{h+1}$ so that the homogeneous moment map for the linearized
action at $p$ is the Taylor polynomial of degree two
$(\Phi_{H},T^{2}_{0}(j^{*}g))$. Since the defining polynomial
$P\colon{\mathbb{C}}^{h+1}\to{\mathbb{C}}$ has degree 2, we may assume that we
are in one of the following two cases (cf. Examples 2.5 and 2.6).

Case 1: $\mathbf{P(z)=z_{0}^{2}}$. In this case,
${\mathbb{C}}^{h+1}\simeq{\mathbb{C}}\times{\mathbb{C}}^{h}$ and
$H\simeq\mathbb{Z}_{2}\times(S^{1})^{h}$, where $\mathbb{Z}_{2}$ acts on
${\mathbb{C}}$ by multiplication with $-1$ and trivially on
${\mathbb{C}}^{h}$, and $(S^{1})^{h}$ acts on ${\mathbb{C}}^{h}$ in the
standard fashion and trivially on ${\mathbb{C}}$. Thus the span of
$\\{\Phi_{H}^{\xi}\mid\xi\in{\mathfrak{h}}\\}$ equals the span of
$\\{|z_{k}|^{2}\mid k\geq 1\\}$. Moreover, since the Taylor polynomial
$T^{2}_{0}(j^{*}g)$ is $H$-invariant, by Remark 2.12, it lies in the span of
$\\{{\mathfrak{R}}(z_{0}^{2}),{\mathfrak{I}}(z_{0}^{2}),|z_{i}|^{2}\colon
i=0,\ldots,h\\}$. As in the proof of Proposition 6.4, this implies that there
exist constants $a_{x},a_{y},a_{\rho}\in{\mathbb{R}}$ satisfying
$a_{x}^{2}+a_{y}^{2}-a_{\rho}^{2}>0$ and $\xi\in{\mathfrak{h}}$ such that
$T^{2}_{0}(j^{*}g)$ is given by

(6.7)
$a_{x}{\mathfrak{R}}(z_{0}^{2})+a_{y}{\mathfrak{I}}(z_{0}^{2})+a_{\rho}|z_{0}|^{2}+\Phi^{\xi}.$

Moreover, $p$ has one hyperbolic block and the remaining blocks are all
elliptic.

By Lemma 2.8, the inclusion $i\colon{\mathbb{C}}\hookrightarrow Y$ sending
$z_{0}$ to $[1,0,(z_{0},0,\ldots,0)]$ induces a diffeomorphism $\overline{i}$
between ${\mathbb{C}}/\\!/{\mathbb{Z}}_{2}={\mathbb{C}}/{\mathbb{Z}}_{2}$ and
${Y/\\!/T}$. By definition of the inclusion $j$ and by (6.7),

$T^{2}_{0}(i^{*}g)(z_{0})=a_{x}{\mathfrak{R}}(z_{0}^{2})+a_{y}{\mathfrak{I}}(z_{0}^{2})+a_{\rho}|z_{0}|^{2},$

and so $0$ is a non-degenerate critical point of $i^{*}g$ of index 1. Hence,
by the equivariant Morse lemma, there exists an open neighborhood $V$ of
$(0,0)\in{\mathbb{R}}^{2}$ and a map $\psi\colon V\to{\mathbb{C}}$ taking
$(0,0)$ to $0$ that is an equivariant diffeomorphism onto an open set such
that $(i\circ\psi)^{*}g(x,y)=xy$ for all $(x,y)\in V$. Let
$\pi\colon{\mathbb{R}}^{2}/{\mathbb{Z}}_{2}\to{\mathbb{R}}^{2}$ be the map
that sends $[x,y]$ to $(xy,x^{2}-y^{2})$; it is a homeomorphism onto
${\mathbb{R}}^{2}$ and a diffeomorphism on the complement on $(0,0)$. Setting
$U:=(\overline{i}\circ\overline{\psi})(V/{\mathbb{Z}}_{2})$ and $\Psi\colon
U\to{\mathbb{R}}^{2}$ to be $\pi\circ(\overline{i}\circ\overline{\psi})^{-1},$
the result follows.

Case 2: $\mathbf{P(z)=z_{0}z_{1}}$. In this case,
${\mathbb{C}}^{h+1}\simeq{\mathbb{C}}^{2}\times{\mathbb{C}}^{h-1}$ and
$H\simeq S^{1}\times(S^{1})^{h-1}$, where $S^{1}$ acts on ${\mathbb{C}}^{2}$
with weights $\pm 1$ and trivially on ${\mathbb{C}}^{h-1}$, and
$(S^{1})^{h-1}$ acts on ${\mathbb{C}}^{h-1}$ in the standard fashion and
trivially on ${\mathbb{C}}^{2}$. Hence, the span of
$\\{\Phi_{H}^{\xi}\mid\xi\in{\mathfrak{h}}\\}$ equals the span of
$\\{|z_{k}|^{2}\mid k\geq 2\\}$ and
$\Phi_{1}(z):=\frac{1}{2}(|z_{0}|^{2}-|z_{1}|^{2})$. Moreover, since the
Taylor polynomial $T^{2}_{0}(j^{*}g)$ is $H$-invariant, by Remark 2.12, it
lies in the span of
$\\{{\mathfrak{R}}(z_{0}z_{1}),{\mathfrak{I}}(z_{0}z_{1}),|z_{i}|^{2}\colon
i=0,\ldots,h\\}$. Therefore, there exist constants
$a_{x},a_{y},a_{\rho}\in{\mathbb{R}}$ and $\xi\in{\mathfrak{h}}$ such that
$T^{2}_{0}(j^{*}g)$ is given by

(6.8)
$a_{x}{\mathfrak{R}}(z_{0}z_{1})+a_{y}{\mathfrak{I}}(z_{0}z_{1})+a_{\rho}(|z_{0}|^{2}+|z_{1}|^{2})+\Phi^{\xi}.$

Moreover, a generic element in the span of
$\\{T^{2}_{0}(j^{*}g),\Phi_{H}^{\xi}\mid\xi\in{\mathfrak{h}}\\}$ is a multiple
of

$a_{x}{\mathfrak{R}}(z_{0}z_{1})+a_{y}{\mathfrak{I}}(z_{0}z_{1})+a_{\rho}(|z_{0}|^{2}+|z_{1}|^{2})+\lambda_{1}\Phi_{1}+\sum\limits_{k=2}^{h}\lambda_{k}|z_{k}|^{2},$

where $\lambda_{j}\in{\mathbb{R}}\smallsetminus\\{0\\}$ for $j=1,\ldots,h$.
The eigenvalues of the corresponding symplectic endomorphism are given by

$\pm 2\lambda_{1}\sqrt{-1}\pm\sqrt{a_{x}^{2}+a_{y}^{2}-4a_{\rho}^{2}}\,,\,\pm
2\lambda_{2}\sqrt{-1},\,\ldots,\,\pm 2\lambda_{h}\sqrt{-1},$

where the signs in the first term are independent. Hence, since $p$ is non-
degenerate and does not have purely elliptic type, Remark 6.1 implies that
$a_{x}^{2}+a_{y}^{2}-4a_{\rho}^{2}>0$, that $p$ has one focus-focus block, and
that the remaining blocks are elliptic.

Let $i\colon{\mathbb{C}}^{2}\hookrightarrow Y$ be the inclusion sending
$(z_{0},z_{1})$ to $[1,0,(z_{0},z_{1},0,\ldots,0)]$. We observe that
$i^{*}\omega_{Y}=\omega_{0}$, the standard symplectic form on
${\mathbb{C}}^{2}$; see Remark 4.2. Moreover, since $i({\mathbb{C}}^{2})$ is
invariant under the $S^{1}$-action generated by $\Phi_{1}$, the functions
$i^{*}\Phi_{1}$ and $i^{*}g$ commute with respect to the induced Poisson
bracket. Moreover, by definition of the inclusion $j$ and by (6.8),

$T^{2}_{0}(i^{*}g)(z_{0},z_{1})=a_{x}{\mathfrak{R}}(z_{0}z_{1})+a_{y}{\mathfrak{I}}(z_{0}z_{1})+a_{\rho}(|z_{0}|^{2}+|z_{1}|^{2})+\lambda
i^{*}\Phi_{1}(z_{0},z_{1})$

for some $\lambda\in{\mathbb{R}}$. Since $a_{x},a_{y},$ and $a_{\rho}$ are not
all zero, the quadratic forms $i^{*}\Phi_{1}$ and $T^{2}_{0}(i^{*}g)$ are
linearly independent. Hence the map
$(i^{*}\Phi_{1},T^{2}_{0}(i^{*}g))\colon{\mathbb{C}}^{2}\to{\mathbb{R}}^{2}$
is regular on a dense subset. Since the Taylor polynomial of
$i^{*}g-T^{2}_{0}(i^{*}g)$ of degree two at zero vanishes, the 2-form
$d(i^{*}\Phi_{1})\wedge d(i^{*}g-T^{2}_{0}(i^{*}g))$ goes to 0 more quickly
than $d(i^{*}\Phi_{1})\wedge d(T^{2}_{0}(i^{*}g))$ as $z=(z_{0},z_{1})$
approaches $0$. Hence, the map $(i^{*}\Phi_{1},i^{*}g)$ is regular on a dense
subset of an open neighborhood $W$ of $0$ and so
$(W,\omega_{0},{\mathbb{R}}\times{\mathbb{R}},(i^{*}\Phi_{1},i^{*}g))$ is an
integrable system. Moreover, since $a_{x}^{2}+a_{y}^{2}-4a_{\rho}^{2}>0$, the
point $0\in{\mathbb{C}}^{2}$ is a non-degenerate singular point with one
focus-focus block. So by Lemma 5.4, there exist an open neighborhood $V$ of
$[0]\in{\mathbb{C}}^{2}/\\!/S^{1}$ and a map $\psi\colon V\to{\mathbb{R}}^{2}$
taking $[0]$ to $(0,0)$ satisfying (1) – (3). Moreover, by Lemma 2.8, the
inclusion $i$ induces a diffeomorphism
$\overline{i}\colon{\mathbb{C}}^{2}/\\!/S^{1}\to{M/\\!/T}$. Setting
$U:=\overline{i}(V)$ and $\Psi\colon U\to{\mathbb{R}}^{2}$ to be
$\psi\circ\overline{i}^{-1}$, the result follows. ∎

######  Proposition 6.9.

Let
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f:=\left(\Phi,g\right)\right)$
be an integrable system such that $\left(M,\omega,\Phi\right)$ is a complexity
one $T$-space. Assume that $p\in M$ is tall and non-degenerate. If the
defining polynomial of the local model for $p$ has degree $N>2$, then $p$ has
purely elliptic type.

###### Proof.

For simplicity, assume that $g(p)=0$. By the Marle-Guillemin-Sternberg local
normal form theorem, we may assume that $M$ is a tall local model
$Y=T\times_{H}{\mathfrak{h}}^{\circ}\times{\mathbb{C}}^{h+1}$, that
$\Phi=\Phi_{Y}$, and that $p=[1,0,0]$. As in the proof of Lemma 6.3, by Lemma
3.2 we may identify $(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ with
${\mathbb{C}}^{h+1}$ so that the homogeneous moment map for the linearized
action at $p$ is the restriction of $(\Phi,T^{2}_{p}g)$ to
${\mathbb{C}}^{h+1}$.

Since $N>2$ and since the Taylor polynomial $T^{2}_{0}(j^{*}g)$ is
$H$-invariant, by Remark 2.12, it lies in the span of $\\{|z_{i}|^{2}\colon
i=0,\dots,h\\}$, as does $\Phi_{H}^{\xi}$ for any $\xi\in{\mathfrak{h}}$.
Hence, a generic element in the span of
$\\{T^{2}_{0}(j^{*}g),\Phi_{H}^{\xi}\mid\xi\in{\mathfrak{h}}\\}$ can be
written as

$\sum\limits_{k=0}^{h}\lambda_{k}|z_{k}|^{2},$

where $\lambda_{k}\in{\mathbb{R}}\smallsetminus\\{0\\}$ for $k=0,\ldots,h$.
The eigenvalues of the corresponding symplectic endomorphism are given by

$\pm 2\lambda_{0}\sqrt{-1},\,\ldots,\,\pm 2\lambda_{h}\sqrt{-1}.$

Hence, since $p$ is non-degenerate, Remark 6.1 implies that $p$ has purely
elliptic type. ∎

Finally, we prove the main result of this section.

###### Proof of Proposition 6.2.

By Lemma 6.3, every short non-degenerate point has purely elliptic type, we
may assume that $p$ is tall. Hence, the result is an immediate consequence of
Propositions 6.4, 6.6 and 6.9 below. ∎

##  7\. Proof of the main result

In this section, we prove of the main results, Theorems 1.6 and 1.10. To
begin, we explain how certain questions about the topology of the integrable
systems we consider can be reduced to understanding Morse theory for oriented
surfaces.

######  Proposition 7.1.

Let $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f=(\Phi,g))$ be an integrable
system such that $(M,\omega,\Phi)$ is a complexity one $T$-space. Assume that
each tall singular point in $(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f)$ is
non-degenerate. Fix $\alpha\in{\mathfrak{t}}^{*}$. Each component of the
reduced space $\Phi^{-1}(\alpha)/T$ containing more than one point can be
given the structure of a smooth oriented 2-dimensional manifold such that
$\overline{g}\colon\Phi^{-1}(\alpha)/T\to{\mathbb{R}}$ is a Morse function.
Moreover, given $p\in\Phi^{-1}(\alpha)$:

  1. (1)

$[p]$ is a critical point of $\overline{g}$ of index $1$ exactly if $p$ has a
hyperbolic block and connected $T$-stabilizer, and

  2. (2)

$[p]$ is a critical points of index $0$ or $2$ exactly if $p$ has purely
elliptic type and is a critical point of $g$ modulo $\Phi$.

In particular, if $p$ has a focus-focus block, has a hyperbolic block with
disconnected $T$-stabilizer, or is a regular point of $g$ modulo $\Phi$, then
$[p]$ is a regular point of $\overline{g}.$

###### Proof.

Assume without loss of generality that $\alpha=0$. For simplicity, assume that
$\Phi^{-1}(0)/T$ is connected and contains more than one point. Since the
orbit of a short point in $\Phi^{-1}(0)$ is an isolated point in
$\Phi^{-1}(0)/T$, every point in $\Phi^{-1}(0)$ is tall. If $p\in\Phi^{-1}(0)$
is a critical point of $g$ modulo $\Phi$, then $p$ is also a singular point of
$\left(M,\omega,{\mathfrak{t}}\times{\mathbb{R}},f\right)$. So, by assumption,
$p$ is non-degenerate. Therefore, the claim follows immediately from
Propositions 3.4, 4.5 and 6.2. ∎

Therefore, we review a few lemmas from Morse theory.

######  Lemma 7.2.

Let $N$ be a connected manifold and let $\overline{g}\colon N\to{\mathbb{R}}$
be a proper Morse-Bott function. If the level set $\overline{g}^{-1}(c)$ is
connected for each $c\in{\mathbb{R}}$, then any local extremum is a global
extremum.

###### Proof.

Let $C$ be a critical manifold that is a local extremum. We may assume that it
has index $0$. By the Morse lemma, there exists a neighborhood $U$ of $p$ such
that $\overline{g}^{-1}(0)\cap U=\\{p\\}$ and
$\overline{g}(U)\subseteq[0,\infty)$. Since the level sets are connected, the
former implies that $\overline{g}^{-1}(0)=p$. Hence, the latter implies that
$\overline{g}^{-1}[0,\infty)$ is a nonempty subset of $N$ that is both open
and closed. Since $N$ is connected, this implies that $\overline{g}$ achieves
its minimum at $p$. The argument for the case of coindex zero is entirely
analogous. ∎

######  Lemma 7.3.

Let $N$ be a closed, connected 2-dimensional manifold and let
$\overline{g}\colon N\to{\mathbb{R}}$ be a Morse function. The following are
equivalent:

  1. (1)

no critical point of $\overline{g}$ has index one; and

  2. (2)

the level set $\overline{g}^{-1}(c)$ is connected for all $c\in{\mathbb{R}}$
and $N$ is simply connected.

###### Proof.

Assume that no critical point of $\overline{g}$ has index one. Since $N$ is
2-dimensional, by Proposition 5.6 the level set $\overline{g}^{-1}(c)$ is
connected for all $c\in{\mathbb{R}}$. Hence, since $N$ is closed, by Lemma 7.2
$\overline{g}$ has exactly two critical points. By [Mil, Theorem 4.1], $N$ is
homeomorphic to a sphere and, therefore, simply connected.

Conversely, assume that the level set $\overline{g}^{-1}(c)$ is connected for
all $c\in{\mathbb{R}}$ and that $N$ is simply connected. Since $N$ is closed,
by Lemma 7.2 $\overline{g}$ has exactly one point of index zero and one point
of index two. Hence, by [Mil, Theorem 5.2], the Euler characteristic of $N$ is
equal to $2-C_{1}$, where $C_{1}$ is the number of critical points of
$\overline{g}$ of index one. Since $N$ is closed and simply connected, it
follows that $C_{1}=0$. ∎

We can now prove our main result.

###### Proof of Theorem 1.6.

Fix $\alpha\in{\mathfrak{t}}^{*}$. Assume without loss of generality that
$\alpha=0$. Since $\Phi$ is proper, the reduced space ${M/\\!/T}$ is compact
and connected by Theorem 5.5.

If the reduced space ${M/\\!/T}$ is a single point then obviously it is simply
connected and the level set $f^{-1}(0,c)$ is connected for all
$c\in{\mathbb{R}}$. Otherwise, by Proposition 7.1, ${M/\\!/T}$ is a smooth
closed, connected oriented surface so that
$\overline{g}\colon{M/\\!/T}\to{\mathbb{R}}$ is a Morse function; moreover,
given $p\in\Phi^{-1}(0)$, $[p]$ is a critical point of $\overline{g}$ of index
$1$ exactly if $p$ has a hyperbolic block and connected $T$-stabilizer. Since
$T$ is connected, the level set $f^{-1}(0,c)$ is connected if and only if
$\overline{g}^{-1}(c)$ is connected. Hence, the result follows from Lemma 7.3.
∎

To prove Theorem 1.10, we need the following result.

######  Lemma 7.4.

Let $N$ be a closed, connected oriented $2$-dimensional manifold and let
$\overline{g}\colon N\to{\mathbb{R}}$ be a Morse function. If the level set
$\overline{g}^{-1}(c)$ is connected for each $c\in{\mathbb{R}}$, then each
level set contains an even number of critical points of index $1$.

###### Proof.

By Lemma 7.2, $N$ has a unique critical point of index $0$ (at the minimum),
and a unique critical point of index $2$ (at the maximum). Assume that the
level set $\overline{g}^{-1}(c)$ contains $k\geq 1$ critical points of index
$1$. By the Morse Lemma $c$ is in the interior of $\overline{g}(N)$, and so
there are no critical points of index $0$ or index $2$ in
$\overline{g}^{-1}(c)$. Fix $\epsilon>0$ so that $[c-\epsilon,c+\epsilon]$
contains no other critical values. Since there is only one point of index $0$
we know that the preimage $\overline{g}^{-1}((-\infty,c-\epsilon])$ is
connected. Hence, the first Betti number of
$\overline{g}^{-1}((-\infty,c+\epsilon])$ is the first Betti number of
$\overline{g}^{-1}((-\infty,c-\epsilon])$ plus $k$. Since every compact
oriented $2$-dimensional manifold with boundary with an odd first Betti number
has at least two boundary components, both of these Betti number must be even.
Hence $k$ is even. ∎

###### Proof of Theorem 1.10.

If there are no non-degenerate singular points with a hyperbolic block and
connected $T$-stabilizer, then by Theorem 1.6 $f^{-1}(\alpha,c)$ is connected
for all $(\alpha,c)\in{\mathfrak{t}}^{*}\times{\mathbb{R}}$. So assume instead
that $f^{-1}(\alpha,c)$ is connected for all
$(\alpha,c)\in{\mathfrak{t}}^{*}\times{\mathbb{R}}$. Fix
$\alpha\in{\mathfrak{t}}^{*}$. Assume without loss of generality that
$\alpha=0$. Since $\Phi$ is proper, the reduced space ${M/\\!/T}$ is compact
and connected by Theorem 5.5.

If the reduced space ${M/\\!/T}$ is a single point then every point in the
orbit $\Phi^{-1}(0)$ is short, and so by Lemma 6.3, every non-degenerate
singular point in $\Phi^{-1}(0)$ has purely elliptic type; in particular, it
has no hyperbolic block.

Otherwise, by Proposition 7.1, ${M/\\!/T}$ is a smooth closed, connected
oriented surface so that $\overline{g}\colon{M/\\!/T}\to{\mathbb{R}}$ is a
Morse function; moreover, given $p\in\Phi^{-1}(0)$, $[p]$ is a critical point
of $\overline{g}$ of index $1$ exactly if $p$ has a hyperbolic block and
connected $T$-stabilizer. If $\Phi^{-1}(\beta)/T$ is simply connected for some
$\beta\in\Phi(M)$, then $\Phi^{-1}(\alpha)/T$ is simply connected by [KT1,
Lemma 5.7 and Corollary 9.7]. Therefore, there are no non-degenerate singular
points with a hyperbolic block and connected $T$-stabilizer by Lemma 7.3.
Similarly, if $f^{-1}(\alpha,c)$ contains at most one orbit of non-degenerate
singular points with a hyperbolic block for all $c\in{\mathbb{R}}$, then it
contains no non-degenerate singular points with hyperbolic block and connected
$T$-stabilizer by Lemma 7.4.

∎

Since Proposition 5.6 is of central importance, we conclude this section with
a slight variation of the proof given in [PRV]. We begin with a preliminary
lemma (cf. [PRV, Lemma 4.3]).

######  Lemma 7.5.

Let $N$ be a connected manifold and let $h\colon N\to{\mathbb{R}}$ be a proper
function such that $h(N)\subseteq[0,\infty)$. Assume that the restriction of
$h$ to $h^{-1}(0,\infty)$ is a Morse-Bott function with no points of index
$1$, and that $0$ is an isolated critical value. Then $h^{-1}(0)$ is
connected.

###### Proof.

Consider points $x$ and $y$ in $h^{-1}(0)$. Since $N$ is path-connected, there
is a path $\gamma$ from $x$ to $y$. Hence, $x$ and $y$ are in the same
component of $h^{-1}([0,b))$ for some $b\in{\mathbb{R}}$. If $C\subseteq
h^{-1}((0,\infty))$ is a critical submanifold, then since the index of $C$ is
not $1$, the restriction $H^{0}(h^{-1}([0,f(C)+\epsilon)))\to
H^{0}(h^{-1}([0,f(C)-\epsilon)))$ is a surjection for sufficiently small
$\epsilon>0$. In particular, if $x$ and $y$ are in the same component of
$h^{-1}([0,h(C)+\epsilon))$, then they are also in the same component of
$h^{-1}([0,h(C)-\epsilon))$. Moreover, since $h$ is proper and $0$ is an
isolated critical value, there are a finite number of such critical
submanifolds in $h^{-1}([0,b))$. Hence, by induction, $x$ and $y$ are in the
same component of $h^{-1}(0).$ ∎

###### Proof of Proposition 5.6.

Fix $c\in{\mathbb{R}}$ and define $h\colon N\to{\mathbb{R}}$ by
$h=(\overline{g}-c)^{2}$. On the set $\overline{g}^{-1}(c,\infty)$ the
function $h$ is Morse-Bott; the critical components of $h$ and $\overline{g}$
coincide and have the same index. On the set $\overline{g}^{-1}(-\infty,c)$
the function $h$ is Morse-Bott; the critical components of $h$ and
$\overline{g}$ coincide but the index and coindex of each critical manifold
are exchanged. Finally, since $h$ is a proper Morse-Bott function there exists
a neighborhood of $c$ that contains no other critical values, so $0$ is an
isolated critical value. Therefore, $h^{-1}(0)=\overline{g}^{-1}(c)$ is
connected by Lemma 7.5. ∎

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