# Compact monotone tall complexity one $T$-spaces

Isabelle Charton  Department of Mathematics, University of Haifa, 199 Abba
Khoushy Avenue, Mount Carmel, Haifa, 3498838, Israel.
<EMAIL_ADDRESS>,  Silvia Sabatini  Department
Mathematik/Informatik, Universität zu Köln, Weyertal 86-90, 50931 Köln,
Germany<EMAIL_ADDRESS>and  Daniele Sepe  Escuela de
Matemáticas, Universidad Nacional de Colombia sede Medellín, Calle 59A Norte
#63-20, Edifício 43, Medellín, Colombia.  Instituto de Matemática e
Estatística, Departamento de Matemática Aplicada (GMA), Universidade Federal
Fluminense, Campus Gragoatá, Rua Prof. Marcos Waldemar de Freitas Reis, s/n,
São Domingos, Niterói, RJ, 24210–201, Brazil<EMAIL_ADDRESS>
###### Abstract.

In this paper we study compact monotone tall complexity one $T$-spaces. We use
the classification of Karshon and Tolman, and the monotone condition, to prove
that any two such spaces are isomorphic if and only if they have equal
Duistermaat-Heckman measures. Moreover, we show that the moment polytope is
Delzant and reflexive, and provide a complete description of the possible
Duistermaat-Heckman measures. Whence we obtain a finiteness result that is
analogous to that for compact monotone symplectic toric manifolds.
Furthermore, we show that any such $T$-action can be extended to a toric
$(T\times S^{1})$-action. Motivated by a conjecture of Fine and Panov, we
prove that any compact monotone tall complexity one $T$-space is equivariantly
symplectomorphic to a Fano manifold endowed with a suitable symplectic form
and a complexity one $T$-action.

###### Contents

  1. 1 Introduction
    1. 1.1 The results
    2. 1.2 Structure of the paper
    3. 1.3 Conventions
  2. 2 (Compact) Hamiltonian $T$-spaces and their invariants
    1. 2.1 Definition and first properties
    2. 2.2 Invariants of compact Hamiltonian $T$-spaces
    3. 2.3 Compact complexity preserving Hamiltonian $T$-spaces
    4. 2.4 Compact tall complexity one $T$-spaces
  3. 3 Compact monotone Hamiltonian $T$-spaces
    1. 3.1 The weight sum formula
    2. 3.2 Moment polytopes of monotone complexity preserving Hamiltonian $T$-spaces
  4. 4 Complete invariants of compact monotone tall complexity one $T$-spaces
    1. 4.1 The genus and a minimal facet
    2. 4.2 A characterization of isolated fixed points
    3. 4.3 The space of exceptional orbits and the painting
    4. 4.4 The Duistermaat-Heckman function
    5. 4.5 The Duistermaat-Heckman function is a complete invariant
  5. 5 Realizability, extension to toric and finiteness results
    1. 5.1 Necessary conditions for the realization and a finiteness result
    2. 5.2 Sufficient conditions for the realization and extension to a toric action
    3. 5.3 Compact monotone tall complexity one spaces are equivariantly Fano

##  1\. Introduction

Fano manifolds play an important role in complex algebraic geometry and
beyond. A compact complex manifold is Fano if its anticanonical bundle is
ample. Any such manifold is simply connected (see [25, Corollary 6.2.18] and
[37, Remark 3.12]), and its Todd genus equals one (see [24, Section 1.8] for a
definition). Moreover, in any complex dimension there are finitely many
topological types of Fano manifolds (see [32]). The Fano condition can be
reformulated in Kähler terms: A compact complex manifold $(Y,J)$ is Fano if
and only if there exists a Kähler form $\omega\in\Omega^{1,1}(Y)$ such that
$c_{1}=[\omega]$, where $c_{1}$ is the first Chern class of $(Y,J)$ – see, for
instance, [2, Sections 5.4 and 7]. This motivates the following
definition111In the literature there are slight variations on this definition
and sometimes these manifolds are also called symplectic Fano (see [43, Remark
11.1.1]).: A symplectic manifold $(M,\omega)$ is (positive) monotone if there
exists (a positive) $\lambda\in\mathbb{R}$ such that

$c_{1}=\lambda[\omega],$

where $c_{1}$ is the first Chern class of $(M,J)$ and $J$ is any almost
complex structure that is compatible with $\omega$. If $(M,\omega)$ is
positive monotone, then $\omega$ can be rescaled so as to be equal to $c_{1}$
in cohomology.

A driving question in symplectic topology is to determine whether every
compact positive monotone symplectic manifold is diffeomorphic to a Fano
manifold. The answer is affirmative in (real) dimension up to four by work of
McDuff (see [41]), and is negative starting from dimension twelve by work of
Fine and Panov (see [14, 48]). To the best of our knowledge, this is an open
problem in the remaining dimensions.

Motivated by a conjecture due to Fine and Panov (see [15, Conjecture 1.4]),
which is supported by recent results (see [6, 7, 8, 36]), we study the above
question in the presence of a Hamiltonian torus action: An action of a compact
torus $T$ on a symplectic manifold $(M,\omega)$ by symplectomorphisms that is
codified by a smooth $T$-invariant map $\Phi:M\to\mathfrak{t}^{*}$, called
moment map (see Section 2.1 for details). If the action is effective and $M$
is connected, the triple $(M,\omega,\Phi)$ is called a Hamiltonian $T$-space.
We remark that a monotone Hamiltonian $T$-space is necessarily positive
monotone (see Proposition 3.3). The following is the long-term question behind
this paper.

######  Problem 1.1.

Find necessary and sufficient conditions for a compact monotone Hamiltonian
$T$-space to be diffeomorphic to a Fano variety.

A starting point to attack Problem 1.1 is to consider ‘large’ torus
symmetries. This is codified precisely by the complexity of a Hamiltonian
$T$-space $(M,\omega,\Phi)$, which is the non-negative integer

$k:=\frac{1}{2}\dim M-\dim T.$

Intuitively, the lower the complexity, the larger the symmetry. A Hamiltonian
$T$-space of complexity $k$ is called a complexity $k$ $T$-space. Problem 1.1
has been already solved in complexity zero, i.e., for compact monotone
symplectic toric manifolds. To recall the solution, we say that two
Hamiltonian $T$-spaces are isomorphic if they are symplectomorphic so that the
moment maps are intertwined (see Definition 2.1 for a precise statement). Let
$(M,\omega,\Phi)$ be a compact monotone symplectic toric manifold. By
Delzant’s classification (see [9]), the isomorphism class of $(M,\omega,\Phi)$
is determined by the moment polytope $\Phi(M)\subset\mathfrak{t}^{*}$, which
is Delzant (see Section 2.3.1). Moreover, if without loss of generality we
assume that $c_{1}=[\omega]$, then, up to translation, $\Phi(M)$ is also
reflexive (see Definition 3.12). Reflexive polytopes were introduced by
Batyrev in [3] in the study of toric Fano varieties and, like Fano manifolds,
enjoy special properties. For instance, if $\ell\subset\mathfrak{t}$ is the
standard lattice, then there are finitely many reflexive polytopes of full
dimension in $\mathfrak{t}^{*}$ up to the standard action of
$\mathrm{GL}(\ell^{*})$ – see Corollary 3.14.

Figure 1.1. The reflexive Delzant polytope in $\mathbb{R}^{2}$ that is the
moment map image of the standard $T^{2}$-action on
$\mathbb{C}P^{2}\times\mathbb{C}P^{2}$ endowed with the product of the Fubini-
Study form on each factor.

The combination of Delzant’s classification and this result above yields
finiteness for compact monotone symplectic toric manifolds up to the following
notion of equivalence: Two Hamiltonian $T$-spaces
$(M_{1},\omega_{1},\Phi_{1})$ and $(M_{2},\omega_{2},\Phi_{2})$ are equivalent
if there exists a symplectomorphism
$\Psi:(M_{1},\omega_{1})\to(M_{2},\omega_{2})$ and an affine transformation
$a\in\mathrm{GL}(\ell^{*})\ltimes\mathfrak{t}^{*}$ of $\mathfrak{t}^{*}$ such
that $\Phi_{2}\circ\Psi=a\circ\Phi_{1}$. More precisely, the following holds.

######  Theorem 1.2.

For each $n\in\mathbb{Z}_{\geq 0}$, there are finitely many equivalence
classes of compact symplectic toric manifolds of dimension $2n$ with first
Chern class equal to the class of the symplectic form.

Moreover, by Delzant’s classification and the Kähler description of the Fano
condition, the following result solves Problem 1.1 in complexity zero.

######  Theorem 1.3.

If $\left(M,\omega,\Phi\right)$ is a compact monotone symplectic toric
manifold, then there exists an integrable complex structure $J$ on $M$ that is
compatible with $\omega$ and invariant under the torus action such that the
Kähler manifold $(M,J)$ is Fano.

In fact, Theorem 1.3 proves the stronger result that any compact monotone
symplectic toric manifold is symplectomorphic to a Fano manifold (endowed with
a suitable symplectic form).

###  1.1. The results

In this paper we solve Problem 1.1 for tall complexity one $T$-spaces, i.e.,
those for which no reduced space is a point (see Definition 2.74). Such spaces
have been classified by Karshon and Tolman in a series of papers (see [28, 29,
30]). This classification is more involved than that of compact symplectic
toric manifolds: For instance, there are several invariants, namely the moment
polytope, the genus, the painting and the Duistermaat-Heckman measure (see
Section 2.4 for more details), and these invariants satisfy some compatibility
conditions (see [30]).

In order to attack Problem 1.1 in the above setting, first we study the
isomorphism classes of compact monotone tall complexity one $T$-spaces. Our
first main result states that, for these spaces, the Duistermaat-Heckman
measure determines all other invariants. For our purposes, we codify this
measure by the unique continuous function that represents its Radon-Nikodym
derivative with respect to the Lebesgue measure on $\mathfrak{t}^{*}$, which
we call the Duistermaat-Heckman function (see Theorem 2.49 and Definition
2.51).

######  Theorem 1.4.

Two compact monotone tall complexity one $T$-spaces are isomorphic if and only
if their Duistermaat-Heckman functions are equal.

Our second main result is finiteness of compact monotone tall complexity one
$T$-spaces up to equivalence, which is the analog of Theorem 1.2 . To this
end, we observe that the moment polytope of a space
$\left(M,\omega,\Phi\right)$ with $c_{1}=\lambda[\omega]$ is a reflexive
Delzant polytope if and only if $c_{1}=[\omega]$ and the moment map $\Phi$
satisfies the so-called weight sum formula (see Proposition 3.18 and Lemma
3.20). If $\left(M,\omega,\Phi\right)$ is a compact monotone tall complexity
one $T$-space, then we may rescale $\omega$ and translate $\Phi$ so as to
satisfy the above conditions (see Corollary 3.4 and Proposition 3.5). The next
result is a crucial step towards establishing finiteness.

######  Theorem 1.5.

Given a reflexive Delzant polytope $\Delta$, there exist finitely many
isomorphism classes of compact monotone tall complexity one $T$-spaces with
$\Phi(M)=\Delta$.

The following result is a simple consequence of Theorem 1.5 and answers a
question originally posed to us by Yael Karshon.

######  Corollary 1.6.

For each $n\in\mathbb{Z}_{\geq 0}$, there are finitely many equivalence
classes of compact tall complexity one $T$-spaces of dimension $2n$ with first
Chern class equal to the class of the symplectic form.

Our third main result concerns the extendability of a tall complexity one
$T$-action on a compact monotone symplectic manifold $(M,\omega)$ to a toric
$(T\times S^{1})$-action. To the best of our knowledge, there is no criterion
to ensure such extendability for compact tall complexity one $T$-spaces of
dimension at least six.

######  Theorem 1.7.

If $\left(M,\omega,\Phi\right)$ is a compact monotone tall complexity one
$T$-space $\left(M,\omega,\Phi\right)$, then the Hamiltonian $T$-action
extends to a symplectic toric $(T\times S^{1})$-action.

Figure 1.2. The reflexive Delzant polytopes that are the moment map images of
the toric extensions of a compact tall monotone complexity one $T$-space
$\left(M,\omega,\Phi\right)$ of dimension six whose moment map image is as in
Figure 1.1. We remark that the set consisting of the ‘height function’ of each
such polytope, i.e., the difference between the $z$-coordinate of the ‘top’
and ‘bottom’ boundary, is precisely the set of the possible for the
Duistermaat-Heckman functions of $\left(M,\omega,\Phi\right)$.

Finally, our last main result is a solution to Problem 1.1 for tall complexity
one $T$-spaces. We recall that, given a compact torus $T$, there exists a
unique complex Lie group $T_{\mathbb{C}}$ such that the Lie algebra of
$T_{\mathbb{C}}$ is the complexification of $\mathfrak{t}$ and $T$ is a
maximal compact subgroup of $T_{\mathbb{C}}$ (see [19, Proposition 4.1]). For
instance, if $T=(S^{1})^{d}$, then $T_{\mathbb{C}}=(\mathbb{C}^{*})^{d}$. The
following result is concerned with the existence of an integrable complex
structure.

######  Theorem 1.8.

If $\left(M,\omega,\Phi\right)$ is a compact monotone tall complexity one
$T$-space, then there exists a $T$-invariant integrable complex structure $J$
on $M$ that is compatible with $\omega$ such that the complex manifold $(M,J)$
is Fano and the $T$-action extends to an effective holomorphic
$T_{\mathbb{C}}$-action.

As an immediate consequence of Theorem 1.8, the following result solves a
stronger version of Problem 1.1 for compact tall complexity one $T$-spaces.

######  Corollary 1.9.

Any compact monotone tall complexity one $T$-space is equivariantly
symplectomorphic to a Fano manifold endowed with a suitable symplectic form
and a complexity one $T$-action.

###  1.2. Structure of the paper

In Section 2 we recall fundamental properties of (compact) Hamiltonian
$T$-spaces. While many of the notions and results presented therein are well-
known to experts, we also set the terminology and notation used throughout.
Some basic properties of Hamiltonian $T$-spaces are considered in Section 2.1,
which describes in detail the local models near any orbit, and introduces the
notion of exceptional and regular points and orbits. These are a
generalization of the corresponding concepts introduced by Karshon and Tolman
in complexity one, which play a key role in their classification. Moreover, we
also discuss the notion of exceptional and regular sheets, which are closely
related to the notion of x-ray (see [51, Definition 2.1]). In Section 2.2 we
take a closer look at the invariants of compact Hamiltonian $T$-spaces,
starting with the so-called Convexity Package and its consequences. A large
part of Section 2 is taken up by the existence of the Duistermaat-Heckman
function of a compact Hamiltonian $T$-space (see Theorem 2.49 and Definition
2.51). We could not find an appropriate reference for this result and, hence,
included the material for completeness. We focus on the complexity one case,
showing that in this case there is a polytope in
$\mathfrak{t}^{*}\times\mathbb{R}$ that encodes the Duistermaat-Heckman
function (see Corollary 2.64). In Section 2.3 we introduce compact complexity
preserving Hamiltonian $T$-spaces, a class of spaces that generalizes
simultaneously compact symplectic toric manifolds and compact tall complexity
one $T$-spaces. For instance, we prove that their moment polytopes are Delzant
polytopes (see Proposition 2.73), and their Duistermaat-Heckman functions
enjoy natural properties (see Corollary 2.71). These spaces may be of
independent interest. Finally, we review the classification of compact tall
complexity one $T$-spaces due to Karshon and Tolman in Section 2.4.

In Section 3 we prove some properties of Hamiltonian $T$-actions on compact
monotone symplectic manifolds. We show that the presence of a Hamiltonian
$T$-action forces monotonicity to be positive (see Proposition 3.3). Hence,
the symplectic form of any compact monotone Hamiltonian $T$-space
$\left(M,\omega,\Phi\right)$ can be rescaled so that it is equal to $c_{1}$ in
cohomology (see Corollary 3.4). Moreover, we show that the moment map of any
compact Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ with
$c_{1}=[\omega]$ can be translated to satisfy the so-called weight sum formula
(see Proposition 3.5). Hence, in order to study compact monotone Hamiltonian
$T$-spaces, it suffices to consider those that satisfy both aforementioned
conditions, which we call normalized monotone. That is the content of Section
3.1. In Section 3.2, we characterize normalized monotone complexity preserving
Hamiltonian $T$-spaces as being precisely those that are compact monotone and
have reflexive Delzant polytopes as moment map images (see Proposition 3.18
and Lemma 3.20).

Section 4 is the technical heart of the paper and is where we prove our first
main result, Theorem 1.4. In Section 4.1, we recall that the genus of a
compact monotone tall complexity one $T$-space is zero, a result proved in
[49]. Moreover, we show that there is a facet of the moment polytope along
which the Duistermaat-Heckman function is constant and equal to the minimal
value (see Proposition 4.2). Such a facet, which we call minimal, plays an
important role in the arguments of Sections 4 and 5. In Section 4.2, we
characterize isolated fixed points in normalized monotone tall complexity one
$T$-spaces (see Proposition 4.6). This is another fundamental result that we
use extensively in Sections 4 and 5. The space of exceptional orbits and the
painting of a normalized monotone tall complexity one $T$-space are studied in
Section 4.3. We show that the isotropy data associated to the former can be
reconstructed by ‘looking at the moment polytope’ (see Remark 4.12 for a
precise statement). Moreover, we prove that the painting of a normalized
monotone tall complexity one $T$-space is trivial (see Definition 2.79 and
Theorem 4.18). In Section 4.4, we provide an explicit formula for the
Duistermaat-Heckman function of a normalized monotone tall complexity one
$T$-space (see Theorem 4.20). It is given in terms of the number of connected
components of the space of exceptional orbits and an integer that can be
associated to the preimage of any vertex that lies on a given minimal facet
(see Lemma 4.5). Finally, in Section 4.5 we prove Theorem 1.4 by bringing the
above results together.

In Section 5 we prove all our remaining main results. Section 5.1 addresses
the question of finding necessary conditions for a function to be the
Duistermaat-Heckman function of a normalized monotone tall complexity one
$T$-space (see Proposition 5.1). This allows us to prove the finiteness
result, namely Theorem 1.5 and Corollary 1.6. In Section 5.2, we prove that
the aforementioned necessary conditions are, in fact, sufficient (see Theorem
5.12 and Corollary 5.15). Our method to prove this result leads us naturally
to obtain the extendability result, namely Theorem 1.7. Finally, in Section
5.3, we prove Theorem 1.8.

### Acknowledgments

We would like to thank Yael Karshon for posing the question of finiteness. The
authors were partially supported by SFB-TRR 191 grant Symplectic Structures in
Geometry, Algebra and Dynamics funded by the Deutsche Forschungsgemeinschaft.
D.S. 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. D. S. would like to thank Universität zu Köln for the
kind hospitality during a long stay. This study was financed in part by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brazil (CAPES) –
Finance code 001. I.C. would like to thank Instituto de Matemática Pura e
Aplicada (IMPA), Rio de Janeiro, for the support of a stay in Brazil.

###  1.3. Conventions

#### Tori

Throughout the paper, we identify the integral lattice of
$S^{1}\subset\mathbb{C}$ with $\mathbb{Z}$. This means that the exponential
map $\exp:\mathrm{Lie}(S^{1})=i\mathbb{R}\to S^{1}$ is given by
$\exp(ix)=e^{2\pi ix}$, where $z\mapsto e^{z}$ is the standard complex
exponential function. Moreover, we often identify $\mathrm{Lie}(S^{1})$ and
its dual with $\mathbb{R}$ tacitly; we trust that this does not cause
confusion.

In this paper, $T$ is a compact torus of dimension $d$ with Lie algebra
$\mathfrak{t}$. We denote its integral lattice by $\ell$, namely
$\ell=\ker(\exp\colon\mathfrak{t}\rightarrow T)$, and the dual of $\ell$ by
$\ell^{*}$. Moreover, we denote by $\langle\cdot,\cdot\rangle$ the standard
pairing between $\mathfrak{t}^{*}$ and $\mathfrak{t}$. Finally, we fix an
inner product on $\mathfrak{t}$ once and for all.

#### Convex polytopes

Let $\mathfrak{t}$ be a real vector space of dimension $d$ with full-rank
lattice $\ell\subset\mathfrak{t}$. A (convex) polytope $\Delta$ in
$\mathfrak{t}^{*}$ is a subset that satisfies either of the following two
equivalent conditions:

  * •

$\Delta$ is the convex hull of a finite set of points, or

  * •

$\Delta$ is the bounded intersection of a finite set of (closed) half-spaces
of $\mathfrak{t}^{*}$,

(see [52, Theorem 2.15]). Throughout the paper, we assume that a polytope
$\Delta$ has dimension equal to that of $\mathfrak{t}^{*}$. We often write
$\Delta$ in its minimal representation, i.e.,

(1.1) $\Delta=\bigcap_{i=1}^{l}\,\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{i}\rangle\geq c_{i}\\}\,$

where $\nu_{i}\in\mathfrak{t}$ is the inward normal, $c_{i}\in\mathbb{R}$, and
the affine hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{i}\rangle=c_{i}\\}$ supports a facet of $\Delta$ for $i=1,\ldots,l$. A
finite non-empty intersection of facets of $\Delta$ is a face of $\Delta$. For
convenience, we think of $\Delta$ also as a face. The dimension of a face
$\mathcal{F}$ of $\Delta$ is the dimension of the affine span of $\mathcal{F}$
in $\mathfrak{t}^{*}$. Faces that are 1-dimensional are called edges, while
0-dimensional faces are vertices.

##  2\. (Compact) Hamiltonian $T$-spaces and their invariants

###  2.1. Definition and first properties

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$. A smooth
$T$-action $\psi\colon T\times M\to M$ is Hamiltonian if it admits a moment
map, i.e., a smooth $T$-invariant map $\Phi\colon M\to\mathfrak{t}^{*}$ that
satisfies

(2.1) $d\langle\Phi,\xi\rangle=-\iota_{\xi^{\\#}}\omega\,\text{ for all
}\xi\in\mathfrak{t},$

where $\xi^{\\#}\in\mathfrak{X}(M)$ denotes the vector field associated to
$\xi$. In this case, the diffeomorphism $\psi(t,\cdot)\colon M\to M$ is a
symplectomorphism for each $t\in T$, i.e., it preserves $\omega$. For brevity
we denote $\psi(t,p)$ by $t\cdot p$.

######  Definition 2.1.

  * •

A (compact) Hamiltonian $T$-space is a (compact) connected symplectic manifold
$(M,\omega)$ endowed with an effective Hamiltonian $T$-action and a moment map
$\Phi\colon M\to\mathfrak{t}^{*}$. We denote such a space by
$(M,\omega,\Phi)$.

  * •

Two Hamiltonian $T$-spaces $(M_{1},\omega_{1},\Phi_{1})$ and
$(M_{2},\omega_{2},\Phi_{2})$ are isomorphic if there exists a
symplectomorphism $\Psi:(M_{1},\omega_{1})\to(M_{2},\omega_{2})$ such that
$\Phi_{2}\circ\Psi=\Phi_{1}$.

######  Remark 2.2.

Since $T$ is connected, an isomorphism of Hamiltonian $T$-spaces is
necessarily a $T$-equivariant diffeomorphism.

Definition 2.1 includes $\dim T=0$ (this is used, for instance, in Theorem
2.49). In this case, a Hamiltonian $T$-space is simply a symplectic manifold
and an isomorphism is simply a symplectomorphism.

####  2.1.1. Orbital moment map and reduced spaces

We endow the quotient space $M/T$ with the quotient topology. Since $\Phi$ is
$T$-invariant, it descends to a continuous map
$\bar{\Phi}:M/T\to\mathfrak{t}^{*}$ that is called the orbital moment map.

######  Remark 2.3.

If $\Psi$ is an isomorphism between $(M_{1},\omega_{1},\Phi_{1})$ and
$(M_{2},\omega_{2},\Phi_{2})$ , then there is a homeomorphism
$\bar{\Psi}:M_{1}/T\to M_{2}/T$ such that
$\bar{\Phi}_{1}=\bar{\Phi}_{2}\circ\bar{\Psi}$.

The fibers of $\bar{\Phi}$ can be canonically identified with the quotient of
the fibers of $\Phi$ by the $T$-action. These are known as reduced spaces. If
$\alpha\in\Phi(M)$ is a regular value of $\Phi$, then the reduced space at
$\alpha$, $\Phi^{-1}(\alpha)/T$, is an orbifold that inherits a symplectic
form $\omega_{\mathrm{red}}$ (see [39] and [34, Lemma 3.9]).

####  2.1.2. Complexity of a Hamiltonian $T$-space

Since the $T$-action on $M$ is effective and since orbits are isotropic
submanifolds of $(M,\omega)$, we have that $d\leq n$. The difference $n-d$ is
a simple, but important invariant of $\left(M,\omega,\Phi\right)$.

######  Definition 2.4.

The complexity of a Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ is

$k:=\frac{1}{2}\dim M-\dim T.$

######  Example 2.5.

Complexity zero Hamiltonian $T$-spaces are symplectic toric manifolds.
Throughout the paper, we refer to torus actions of complexity zero as toric.

Intuitively, the complexity of a Hamiltonian $T$-space is half of the
dimension of a reduced space at a regular value.

####  2.1.3. Local model and local normal form

Given $p\in M$, its stabilizer is the closed subgroup $H:=\\{t\in T\mid t\cdot
p=p\\}$. We set $h:=\dim H$. Since $T$ is abelian, any two points on the same
orbit have equal stabilizers. Hence, the stabilizer of an orbit is well-
defined. If $\mathcal{O}$ denotes the $T$-orbit containing $p$, the
infinitesimal symplectic linear action of $H$ on $(T_{p}M,\omega_{p})$ fixes
$T_{p}\mathcal{O}$. Thus there is a symplectic linear action of $H$ on the
quotient vector space $(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ endowed
with the quotient linear symplectic structure. We call the underlying Lie
group homomorphism the symplectic slice representation of $\boldsymbol{p}$.

######  Remark 2.6.

The symplectic slice representations of two points lying on the same orbit are
naturally isomorphic. Hence, the symplectic slice representation of an orbit
is well-defined. This allows us to ‘decorate’ the quotient space $M/T$ by
attaching the symplectic slice representation to every orbit (this data
includes the stabilizer of the orbit).

Let $\Psi$ be an isomorphism between $(M_{1},\omega_{1},\Phi_{1})$ and
$(M_{2},\omega_{2},\Phi_{2})$ and let $\bar{\Psi}:M_{1}/T\to M_{2}/T$ be the
homeomorphism given by Remark 2.3. For any $p\in M_{1}$, $\bar{\Psi}([p])$ and
$[p]$ have equal stabilizers and symplectic slice representations.

Fix a $T$-invariant almost complex structure on $(M,\omega)$; the existence of
such a structure is proved in [42, Lemma 5.52]. We observe that
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ has real dimension
$2(h+n-d)=2(h+k)$, where $k$ is the complexity of
$\left(M,\omega,\Phi\right)$. Hence, we use the above almost complex structure
to identify $(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ with
$\mathbb{C}^{h+k}$ endowed with the standard symplectic form

(2.2) $\frac{i}{2}\,\sum_{j=1}^{h+k}dz_{j}\wedge d\overline{z}_{j}.$

Moreover, under this identification, the linear $H$-action is by unitary
transformations.

Let $\rho:H\to U(\mathbb{C}^{h+k})$ be the associated homomorphism of Lie
groups. Since $H$ is abelian, $\rho(H)$ is contained in a maximal torus of
$U(\mathbb{C}^{h+k})$. We denote the maximal torus of $U(\mathbb{C}^{h+k})$
consisting of diagonal transformations by $(S^{1})^{h+k}$. Hence, we may
assume that $\rho$ factors through a Lie group homomorphism
$H\to(S^{1})^{h+k}$ that we also denote by $\rho$ by a slight abuse of
notation. We write $\rho_{j}$ for the $j$th component of $\rho$, where
$j=1,\ldots,h+k$. Let $d_{e}\rho_{j}$ denote the derivative at the identity of
$\rho_{j}$ and set $d_{e}\rho_{1}:=2\pi
i\alpha_{1},\ldots,d_{e}\rho_{h+k}:=2\pi i\alpha_{h+k}$. Hence,
$\alpha_{j}\in\ell^{*}_{\mathfrak{h}}$ for all $j=1,\ldots,h+k$, where
$\mathfrak{h}$ is the Lie algebra of $H$ and $\ell_{\mathfrak{h}}$ denotes the
integral lattice in $\mathfrak{h}$. We call $\alpha_{1},\ldots,\alpha_{h+k}$
the isotropy weights of $\boldsymbol{p}$ (for the $\boldsymbol{H}$-action).
The multiset of isotropy weights of $p$ does not depend on the choice of
$T$-invariant almost complex structure on $(M,\omega)$. Moreover, this
multiset encodes the action of the identity component of $H$ on
$\mathbb{C}^{h+k}$. Explicitly,

(2.3) $\exp(\xi)\cdot(z_{1},\ldots,z_{h+k})=(e^{2\pi
i\langle\alpha_{1},\xi\rangle}z_{1},\ldots,e^{2\pi
i\langle\alpha_{h+k},\xi\rangle}z_{h+k})\quad\text{for
every}\;\;\xi\in\mathfrak{h}\,.$

In particular, if $H$ is connected, then the multiset of isotropy weights of
$p$ determine the symplectic slice representation up to unitary isomorphisms.
Finally, by Remark 2.6, the multisets of isotropy weights of two points lying
on the same orbit are equal.

From the stabilizer $H\leq T$ of $p$ and the Lie group homomorphism
$\rho:H\to(S^{1})^{h+k}$, we construct a symplectic manifold together with a
Hamiltonian $T$-action and a moment map. This is the local model for a
$T$-invariant neighborhood of $\mathcal{O}$ in $\left(M,\omega,\Phi\right)$.
We do this in two equivalent ways, seeing as one is more convenient for proofs
and the other is more convenient for calculations.

#### The abstract construction

Let $\Omega$ denote the symplectic form on $T^{*}T\times\mathbb{C}^{h+k}$
given by taking the sum of the pullbacks of the canonical symplectic form on
$T^{*}T$ and the standard symplectic form on $\mathbb{C}^{h+k}$ (see equation
(2.2)). Let $H$ act (on the right) on $T^{*}T\times\mathbb{C}^{h+k}$ as
follows: On $T^{*}T$ it acts by the cotangent lift of (right) multiplication,
while on $\mathbb{C}^{h+k}$ it acts by $z\cdot h:=\rho(h^{-1})(z)$, for $h\in
H$ and $z\in\mathbb{C}^{h+k}$. By construction, the $H$-action on
$(T^{*}T\times\mathbb{C}^{h+k},\Omega)$ is Hamiltonian. Let
$\hat{\Phi}:T^{*}T\times\mathbb{C}^{h+k}\to\mathfrak{h}^{*}$ be the moment map
that sends $(0,0)\in T^{*}T\times\mathbb{C}^{h+k}$ to the origin in
$\mathfrak{h}^{*}$. Since this $H$-action is free and proper, the quotient

$(T^{*}T\times\mathbb{C}^{h+k})/\\!/H:=\hat{\Phi}^{-1}(0)/H$

is a smooth manifold that inherits a symplectic form $\omega_{\mathrm{red}}$
(see [42, Proposition 5.40]).

Let $T$ act (on the left) on $T^{*}T\times\mathbb{C}^{h+k}$ as follows: On
$T^{*}T$ it acts by the cotangent lift of (left) multiplication, while on
$\mathbb{C}^{h+k}$ it acts trivially. This $T$-action is Hamiltonian and
commutes with the above $H$-action. Hence, it induces a Hamiltonian $T$-action
on $((T^{*}T\times\mathbb{C}^{h+k})/\\!/H,\omega_{\mathrm{red}})$. As a moment
map for this $T$-action we take the one that sends
$[0,0]\in(T^{*}T\times\mathbb{C}^{h+k})/\\!/H$ to the origin in
$\mathfrak{t}^{*}$ and we denote it by $\Phi_{\mathrm{red}}$. The desired
local model is the triple
$((T^{*}T\times\mathbb{C}^{h+k})/\\!/H,\omega_{\mathrm{red}},\Phi_{\mathrm{red}})$.

#### The explicit construction

The choice of inner product on $\mathfrak{t}$ induces an inner product on
$\mathfrak{t}^{*}$ which, in turn, determines an isomorphism
$\mathfrak{t}^{*}\simeq\mathrm{Ann}(\mathfrak{h})\oplus\mathfrak{h}^{*}$.
Moreover, we choose a trivialization $T^{*}T\cong T\times\mathfrak{t}^{*}$.
With these choices, we fix an identification $T^{*}T\times\mathbb{C}^{h+k}$
with
$T\times\mathrm{Ann}(\mathfrak{h})\times\mathfrak{h}^{*}\times\mathbb{C}^{h+k}$.
Under this identification, the above (right) $H$-action is given by

$(t,\alpha,\beta,z)\cdot h=(th,\alpha,\beta,\rho(h^{-1})z),$

while the above moment map $\hat{\Phi}$ is given by

$(t,\alpha,\beta,z)\mapsto\beta-\Phi_{H}(z),$

where $\Phi_{H}:\mathbb{C}^{h+k}\to\mathfrak{h}^{*}$ is the homogeneous moment
map for the (left) $H$-action on $\mathbb{C}^{h+k}$ given by $h\cdot
z=\rho(h)z$. The map

$\begin{split}T\times\mathrm{Ann}(\mathfrak{h})\times\mathbb{C}^{h+k}&\to\hat{\Phi}^{-1}(0)\\\
(t,\alpha,z)&\mapsto(t,\alpha,\Phi_{H}(z),z)\end{split}$

is an $H$-equivariant diffeomorphism, where the left hand side is equipped
with the (right) $H$-action on
$T\times\mathrm{Ann}(\mathfrak{h})\times\mathbb{C}^{h+k}$ given by

(2.4) $(t,\alpha,z)\cdot h=(th,\alpha,\rho(h^{-1})z).$

Hence the quotient

(2.5) $Y:=T\times_{H}\mathrm{Ann}(\mathfrak{h})\times\mathbb{C}^{h+k}$

is diffeomorphic to $(T^{*}T\times\mathbb{C}^{h+k})/\\!/H$. We denote by
$\omega_{Y}$ (respectively $\Phi_{Y}$) the pullback of $\omega_{\mathrm{red}}$
(respectively $\Phi_{\mathrm{red}}$) under the above diffeomorphism. The
(left) $T$-action on $Y$ is given by

(2.6) $s\cdot[t,\alpha,z]=[st,\alpha,z],$

while the moment map $\Phi_{Y}$ takes the form

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

The stabilizer of $[1,0,0]\in Y$ is $H$ and the symplectic slice
representation of $[1,0,0]$ is $\rho:H\to(S^{1})^{h+k}$. Moreover, if the
$T$-action is effective, the complexity of $(Y,\omega_{Y},\Phi_{Y})$ is equal
to $k$. We refer to $(Y,\omega_{Y},\Phi_{Y})$ as the local model of
$\boldsymbol{p}$. By Remark 2.6, the local models at two points lying on the
same orbit are equal.

######  Remark 2.7.

By a slight abuse of terminology, we also refer to
$((T^{*}T\times\mathbb{C}^{h+k})/\\!/H,\omega_{\mathrm{red}},\Phi_{\mathrm{red}})$
as the local model of $p$. Moreover, throughout the paper we sometimes state
results in terms of $Y$ but use $(T^{*}T\times\mathbb{C}^{h+k})/\\!/H$ in the
proof. We trust that this does not cause confusion.

As a consequence of the local normal form theorem for Hamiltonian actions of
compact Lie groups due to Guillemin-Sternberg [20] and Marle [38], any
Hamiltonian $T$-space is isomorphic to a local model near an orbit. More
precisely, the following holds.

######  Theorem 2.8.

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space. Given $p\in M$,
let $(Y,\omega_{Y},\Phi_{Y})$ be the local model of $p$. There exist
$T$-invariant open neighborhoods $U\subset M$ of $p$ and $V\subset Y$ of
$[1,0,0]$ and an isomorphism between $(U,\omega,\Phi)$ and
$(V,\omega_{Y},\Phi_{Y}+\Phi(p))$ that maps $p$ to $[1,0,0]$.

######  Corollary 2.9.

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space. For any $p\in M$
with stabilizer $H$, the homomorphism $\rho:H\to(S^{1})^{h+k}$ is injective.

###### Proof.

Fix $p\in M$ and notation as above. Let $(Y,\omega_{Y},\Phi_{Y})$ be the local
model of $p$. Since the $T$-action on $M$ is assumed to be effective, so is
the $T$-action on $Y$ by Theorem 2.8 (see Remark 2.2). By definition of $Y$
and of the $T$-action on $Y$, the $T$-action on
$Y=T\times_{H}\mathrm{Ann}(\mathfrak{h})\times\mathbb{C}^{h+k}$ is effective
if and only if the Lie group homomorphism $\rho:H\to(S^{1})^{h+k}$ is
injective, as desired. ∎

######  Remark 2.10.

Given $p\in M$, let $H$ be its stabilizer and let $\\{\alpha_{j}\\}$ be the
multiset of isotropy weights of $p$. By Corollary 2.9, the $H$-action on
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}$ is effective. In particular, so
is the action by its identity component. Using equation (2.3), it follows that
the $\mathbb{Z}$-span of $\\{\alpha_{j}\\}$ equals $\ell^{*}_{\mathfrak{h}}$.

######  Remark 2.11.

The above discussion simplifies significantly if $p$ is a fixed point, i.e.,
if $H=T$. In this case, $h=d$ so that $h+k=n$, and the isotropy weights
$\alpha_{1},\ldots,\alpha_{n}$ lie in $\ell^{*}$. Moreover,
$Y=\mathbb{C}^{n}$, the symplectic form $\omega_{Y}$ is the standard
symplectic form on $\mathbb{C}^{n}$, the $T$-action on $Y$ is given by

(2.8) $\exp(\xi)\cdot(z_{1},\ldots,z_{n})=(e^{2\pi
i\langle\alpha_{1},\xi\rangle}z_{1},\ldots,e^{2\pi
i\langle\alpha_{n},\xi\rangle}z_{n})\quad\text{for
every}\;\;\xi\in\mathfrak{t}\,,$

and the moment map $\Phi_{Y}:Y\to\mathfrak{t}^{*}$ is given by

(2.9) $\Phi_{Y}(z)=\pi\sum_{j=1}^{n}\alpha_{j}|z_{j}|^{2},$

where $z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$.

####  2.1.4. Regular and exceptional local models

Theorem 2.8 allows us to understand a Hamiltonian $T$-space in a $T$-invariant
open neighborhood of a point $p$ by studying the local model of $p$. For this
reason, in this subsection we take a closer look at local models.

In what follows, we fix a non-negative integer $k$ and a closed subgroup
$H\leq T$. As above, we set $d:=\dim T$ and $h:=\dim H$. We also fix an
injective Lie group homomorphism $\rho:H\hookrightarrow(S^{1})^{h+k}$. We
denote the subspace of fixed points of the $H$-action induced by $\rho$ by
$(\mathbb{C}^{h+k})^{H}$. As in Section 2.1.3, we use $k$, $H$ and $\rho$ to
construct Hamiltonian $T$-spaces

$((T^{*}T\times\mathbb{C}^{h+k})/\\!/H,\omega_{\mathrm{red}},\Phi_{\mathrm{red}})\cong(Y,\omega_{Y},\Phi_{Y})$

that we refer to as the local model determined by $\boldsymbol{k}$,
$\boldsymbol{H}$ and $\boldsymbol{\rho}$ (see equations (2.5), (2.6) and (2.7)
for the definition of $Y$, the $T$-action and $\Phi_{Y}$ respectively). We
remark that $k$ is the complexity of $(Y,\omega_{Y},\Phi_{Y})$, that $H$ is
the stabilizer of $p:=[1,0,0]$ and that $\rho$ is the symplectic slice
representation of $p$.

######  Lemma 2.12.

Let $k$ be a non-negative integer, let $H\leq T$ be a closed subgroup and let
$\rho:H\hookrightarrow(S^{1})^{h+k}$ be an injective Lie group homomorphism,
where $h=\dim H$. Set $s:=\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}$. There
exists an isomorphism of Hermitian vector spaces
$\mathbb{C}^{h+k}\simeq\mathbb{C}^{h+k-s}\oplus\mathbb{C}^{s}$ such that

(2.10) $\rho=(\rho^{\prime},1):H\hookrightarrow(S^{1})^{h+k-s}\times
1\hookrightarrow(S^{1})^{h+k-s}\times(S^{1})^{s}\simeq(S^{1})^{h+k}$

and $(\mathbb{C}^{h+k-s})^{H}=\\{0\\}$. Moreover, $s\leq k$ and, if $s=k$,
then $\rho$ is an isomorphism between $H$ and $(S^{1})^{h}$ and the $H$-action
on $\mathbb{C}^{h}$ is toric.

###### Proof.

To simplify notation, set $V:=(\mathbb{C}^{h+k})^{H}$. The standard Hermitian
product on $\mathbb{C}^{h+k}$ induces an isomorphism of Hermitian vector
spaces $\mathbb{C}^{h+k}\simeq V^{\perp}\oplus V$, and both $V$ and
$V^{\perp}$ are endowed with the restriction of the standard Hermitian
product. By definition $\rho(H)$ fixes $V$ pointwise and is a subgroup of
$U(h+k)$. Hence, $\rho$ splits as the direct sum of two $H$-representations
that we denote by $\rho_{V}:H\to U(V)$ and $\rho^{\prime}:H\to U(V^{\perp})$.
By construction, $\rho_{V}$ is the trivial representation. Moreover,
$\rho^{\prime}(H)$ is contained in a maximal torus of $U(V^{\perp})$ (see
Section 2.1.3). Since the Hermitian vector spaces $V$ and $V^{\perp}$ are
isomorphic to $\mathbb{C}^{s}$ and to $\mathbb{C}^{h+k-s}$ respectively and
since $\\{0\\}=(\mathbb{C}^{h+k-s})^{H}$, this proves the first statement. To
prove the second, we observe that $\rho^{\prime}:H\to U(V^{\perp})$ is
injective, since $\rho$ is injective. Since maximal tori in $U(V^{\perp})$
have dimension equal to $h+k-s$ and since $h=\dim H$, it follows at once that
$s\leq k$. Finally, if $s=k$, then, since $\rho^{\prime}$ is injective, it
follows that the map $H\hookrightarrow(S^{1})^{h}$ is an isomorphism of Lie
groups. ∎

Lemma 2.12 allows us to ‘decompose’ local models as follows (see [28, Lemma
4.4] for a proof in the case $k=s=1$).

######  Proposition 2.13.

Let $k$ be a non-negative integer, let $H\leq T$ be a closed subgroup and let
$\rho:H\hookrightarrow(S^{1})^{h+k}$ be an injective Lie group homomorphism,
where $h=\dim H$. Let $s=\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}$ and
$\rho^{\prime}:H\hookrightarrow(S^{1})^{h+k-s}$ be as in the statement of
Lemma 2.12, so that $(\mathbb{C}^{h+k-s})^{H}=\\{0\\}$. The local model
determined by $k$, $H$ and $\rho$ is isomorphic to

$(Y^{\prime}\times\mathbb{C}^{s},\mathrm{pr}^{*}_{1}\omega_{Y^{\prime}}+\mathrm{pr}^{*}_{2}\omega_{0},\mathrm{pr}_{1}^{*}\Phi_{Y^{\prime}}),$

where $(Y^{\prime},\omega_{Y^{\prime}},\Phi_{Y^{\prime}})$ is the local model
determined by $k-s$, $H$ and $\rho^{\prime}$, $\omega_{0}$ is the standard
symplectic form $\mathbb{C}^{s}$, and $\mathrm{pr}_{j}$ is the projection from
$Y^{\prime}\times\mathbb{C}^{s}$ to its $j$th component, for $j=1,2$.

###### Proof.

In this proof we use the abstract construction of local models (see Section
2.1.3 and Remark 2.7). Fix an isomorphism
$\mathbb{C}^{h+k}\simeq\mathbb{C}^{s}\oplus\mathbb{C}^{h+k-s}$ as in the
statement of Lemma 2.12. This induces a symplectomorphism between
$(T^{*}T\times\mathbb{C}^{h+k},\Omega)$ and
$((T^{*}T\times\mathbb{C}^{h+k-s})\times\mathbb{C}^{s},\mathrm{pr}^{*}_{1}\Omega^{\prime}+\mathrm{pr}^{*}\omega_{0})$,
where $\Omega^{\prime}$ denotes the symplectic form on
$T^{*}T\times\mathbb{C}^{h+k-s}$ and, as above, $\mathrm{pr}_{j}$ is the
projection from $(T^{*}T\times\mathbb{C}^{h+k-s})\times\mathbb{C}^{s}$ to its
$j$th component, for $j=1,2$. We endow
$(T^{*}T\times\mathbb{C}^{h+k-s})\times\mathbb{C}^{s}$ with the following
(right) $H$-action and (left) $T$-action:

  * •

($H$-action): On $T^{*}T\times\mathbb{C}^{h+k-s}$ we consider the (right)
$H$-action used to construct the local model determined by $k-s$, $H$ and
$\rho^{\prime}$, while $H$ acts trivially on $\mathbb{C}^{s}$, and

  * •

($T$-action): On $T^{*}T\times\mathbb{C}^{h+k-s}$ we consider the (left)
$T$-action used to construct the local model determined by $k-s$, $H$ and
$\rho^{\prime}$, while $T$ acts trivially on $\mathbb{C}^{s}$.

The above symplectomorphism is both $H$\- and $T$-equivariant. Hence, there is
a $T$-equivariant symplectomorphism between the symplectic quotients with
respect to the $H$-actions, i.e., an isomorphism between the resulting
Hamiltonian $T$-spaces. Once the abstract constructions are identified with
the explicit ones as in Section 2.1.3, this yields the desired isomorphism of
Hamiltonian $T$-spaces. ∎

######  Remark 2.14.

Proposition 2.13 is particularly simple if $s=k$, i.e., if
$\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}$ is maximal. In this case,
$(Y^{\prime},\omega^{\prime},\Phi_{Y^{\prime}})$ is toric.

Motivated by Theorem 2.8 and by Remark 2.14, we introduce the following
dichotomy.

######  Definition 2.15.

Let $\left(M,\omega,\Phi\right)$ be a complexity $k$ $T$-space and let $p\in
M$ be a point with stabilizer $H$. Let $\mathcal{O}$ be the orbit containing
$p$. We say that $p$ is regular if $\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}=k$
and exceptional otherwise, where $H$ acts on
$(T_{p}\mathcal{O})^{\omega}/T_{p}\mathcal{O}\simeq\mathbb{C}^{h+k}$ via the
symplectic slice representation at $p$.

######  Remark 2.16.

We observe that Definition 2.15 extends also to orbits (see Remark 2.6);
hence, throughout the paper, we also refer to regular and exceptional orbits.

######  Definition 2.17.

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space. We define the set
of exceptional orbits $M_{\mathrm{exc}}$ as the subset of $M/T$ consisting of
exceptional orbits.

######  Example 2.18.

A fixed point $p$ in a complexity $k$ $T$-space $\left(M,\omega,\Phi\right)$
is regular if and only if it lies on a fixed submanifold of dimension $k$. We
observe that the latter condition is equivalent to the $T$-action on the
normal bundle to the fixed submanifold that contains $p$ being toric.

The techniques of Example 2.18 and Lemma 2.12, together with Theorem 2.8, can
be used to prove the following result.

######  Lemma 2.19.

  * •

Any point with trivial stabilizer, as well as any point in a complexity zero
$T$-space, is regular.

  * •

If a Hamiltonian $T$-space has positive complexity, then any isolated fixed
point is exceptional.

  * •

The subset of regular points in a Hamiltonian $T$-space is open.

  * •

If a point $p$ is regular, then the stabilizer of $p$ is connected.

We conclude this section with two results in complexity one.

######  Lemma 2.20.

Let $\left(M,\omega,\Phi\right)$ be a complexity one $T$-space. A point $p\in
M^{T}$ is isolated if and only if it is exceptional.

###### Proof.

By Lemma 2.19, if $p\in M^{T}$ is isolated, then it is exceptional.
Conversely, if $p\in M^{T}$ is regular, then, by Theorem 2.8 and Example 2.18,
$p$ is not isolated. ∎

Finally, as an immediate consequence of Theorem 2.8 and of Proposition 2.13,
the following characterization of exceptional points holds (cf. [28, Lemma
4.4]).

######  Corollary 2.21.

Let $\left(M,\omega,\Phi\right)$ be a complexity one $T$-space. A point $p\in
M$ is exceptional if and only if every nearby orbit in the same moment fiber
has a strictly smaller stabilizer.

In particular, Definition 2.15 agrees with the definition of exceptional
orbits of [28] in complexity one, and is the appropriate notion for our
purposes.

####  2.1.5. Regular and exceptional sheets

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space. For any closed
subgroup $H\leq T$, the action of $H$ on $M$ is also Hamiltonian: Let
$\mathfrak{h}$ be the Lie algebra of $H$ and let
$i:\mathfrak{h}\hookrightarrow\mathfrak{t}$ denote the inclusion. Then a
moment map for the $H$-action is given by the composition

(2.11)
$M\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}\mathfrak{t}^{*}\stackrel{{\scriptstyle
i^{*}}}{{\longrightarrow}}\mathfrak{h}^{*},$

where $i^{*}\colon\mathfrak{t}^{*}\to\mathfrak{h}^{*}$ is the dual of $i$. We
denote the set of fixed points of the $H$-action by

$M^{H}=\\{p\in M\mid h\cdot p=p\,\text{ for all }h\in H\\}.$

The following result is used throughout and a proof can be found in [42, Lemma
5.53].

######  Lemma 2.22.

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space and let $H\leq T$
be closed. Then any connected component of $M^{H}$ is an embedded symplectic
submanifold of $(M,\omega)$.

We refer to the connected components of $M^{T}$ as the fixed submanifolds of
$\left(M,\omega,\Phi\right)$.

###### Remark 2.23

If $N\subset M^{T}$ is a fixed submanifold, then for any $p,p^{\prime}\in N$,
the isotropy weights of $p$ are equal to those at $p^{\prime}$. Hence, the
isotropy weights of the fixed submanifold $\boldsymbol{N}$ are well-defined.
Moreover, they can be used to determine $\dim N$: By Theorem 2.8, $\dim N$
equals twice the number of isotropy weights of $N$ that are equal to 0.

Let $H\leq T$ be a closed subgroup that is the stabilizer of a point $p\in M$
and let $N\subset M^{H}$ denote the connected component that contains $p$. We
set $\omega_{N}:=\omega|_{N}$. By Lemma 2.22, $(N,\omega_{N})$ is an embedded
symplectic submanifold of $(M,\omega)$. Moreover, since $T$ is abelian and
connected, and since $N$ is connected, $N$ is $T$-invariant. The $T$-action on
$N$ induces a $T^{\prime}:=T/H$-action on $N$ that is effective because the
stabilizer of $p$ for the $T$-action equals $H$. This $T^{\prime}$-action on
$N$ is Hamiltonian. To see this, we construct a moment map starting from
$\Phi$ and the point $p$. To this end, let $\mathrm{pr}:T\to T^{\prime}$ be
the quotient map. We denote its derivative at the identity by
$\mathrm{pr}_{*}:\mathfrak{t}\to\mathfrak{t}^{\prime}$ and the dual of
$\mathrm{pr}_{*}$ by
$\mathrm{pr}^{*}:(\mathfrak{t}^{\prime})^{*}\to\mathfrak{t}^{*}$. By
construction, $\mathrm{pr}^{*}$ is an isomorphism between
$(\mathfrak{t}^{\prime})^{*}$ and $\mathrm{Ann}(\mathfrak{h})$. Since
$N\subset M^{H}$ is connected and since $\Phi$ is a moment map,
$\langle\Phi|_{N},\xi\rangle=\langle\Phi(p),\xi\rangle$ for all
$\xi\in\mathfrak{h}$. Consequently,
$\Phi(N)\subset\Phi(p)+\mathrm{Ann}(\mathfrak{h})$. Hence, there exists a
unique smooth map $\Phi_{N}:N\to(\mathfrak{t}^{\prime})^{*}$ such that

(2.12) $\mathrm{pr}^{*}\circ\Phi_{N}=\Phi|_{N}-\Phi(p).$

By (2.12), since $\Phi:M\to\mathfrak{t}^{*}$ is a moment map for the
$T$-action on $M$ and since $(N,\omega_{N})$ is a symplectic submanifold of
$(M,\omega)$, it follows shows that $\Phi_{N}$ is a moment map for the
$T^{\prime}$-action on $N$. Since the $T^{\prime}$-action on $N$ is effective,
this shows that $(N,\omega_{N},\Phi_{N})$ is a Hamiltonian $T^{\prime}$-space.

######  Definition 2.24.

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space and let $H\leq T$
be a subgroup that occurs as a stabilizer of some point $p\in M$. The
Hamiltonian $T^{\prime}$-space $(N,\omega_{N},\Phi_{N})$ constructed above is
called a sheet stabilized by $\boldsymbol{H}$. Whenever we wish to emphasize
the role of $p$, we say that $(N,\omega_{N},\Phi_{N})$ is the sheet through
$\boldsymbol{p}$.

Any fixed submanifold $N\subset M^{T}$ gives rise to a sheet that we denote
simply by $(N,\omega_{N})$.

######  Remark 2.25.

For our purposes, it is useful to allow some freedom in the choice of moment
map associated to a sheet. Modulo $\mathrm{pr}^{*}$, $\Phi_{N}$ and
$\Phi|_{N}$ only differ by a constant. Depending on the context, we use either
moment map. We trust that this does not cause confusion.

Given a sheet $(N,\omega_{N},\Phi_{N})$ of $\left(M,\omega,\Phi\right)$, we
are interested in understanding the complexity of $(N,\omega_{N},\Phi_{N})$ in
relation to that of $\left(M,\omega,\Phi\right)$. By Definition 2.4, the
complexity of $(N,\omega_{N},\Phi_{N})$ is $\frac{1}{2}\dim N-(d-h)$.

######  Proposition 2.26.

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space and let
$(N,\omega_{N},\Phi_{N})$ be a sheet stabilized by $H\leq T$. The complexity
of $(N,\omega_{N},\Phi_{N})$ is at most that of $\left(M,\omega,\Phi\right)$.
Moreover, the following are equivalent:

  1. (1)

The complexity of $(N,\omega_{N},\Phi_{N})$ is less than that of
$\left(M,\omega,\Phi\right)$.

  2. (2)

Any $p\in N$ with stabilizer $H$ is exceptional.

  3. (3)

The fiberwise $H$-action on the symplectic normal bundle to $N$ has positive
complexity.

###### Proof.

Let $p\in N$ be a point that is stabilized by $H$ and let
$(Y,\omega_{Y},\Phi_{Y})$ be the local model of $p$. Since the above
conditions can be checked locally, by Theorem 2.8, it suffices to prove the
result in the case $\left(M,\omega,\Phi\right)=(Y,\omega_{Y},\Phi_{Y})$ and
$p=[1,0,0]$. Recall that
$Y=T\times_{H}\mathrm{Ann}(\mathfrak{h})\times\mathbb{C}^{h+k}$, where $k$ is
the complexity of $(Y,\omega_{Y},\Phi_{Y})$, and that the $T$-action on $Y$ is
given by equation (2.6). The submanifold $Y^{H}$ is given by

(2.13)
$T\times_{H}\mathrm{Ann}(\mathfrak{h})\times(\mathbb{C}^{h+k})^{H}\simeq
T/H\times\mathrm{Ann}(\mathfrak{h})\times(\mathbb{C}^{h+k})^{H},$

where $(\mathbb{C}^{h+k})^{H}$ is the fixed point set for the $H$-action on
$\mathbb{C}^{h+k}$. Since $(\mathbb{C}^{h+k})^{H}$ is a subspace of
$\mathbb{C}^{h+k}$, it follows that $Y^{H}$ is connected. Since $N\subseteq
Y^{H}$ is a connected component of $Y^{H}$, $N=Y^{H}$. Moreover, by (2.13),
the dimension of $N$ equals $2(d-h)+2\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}$.
By Lemma 2.12, $\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}\leq k$. Hence, the
complexity of $(N,\omega_{N},\Phi_{N})$ is at most $k$. Moreover, since
$(\mathbb{C}^{h+k})^{H}$ is a complex subspace of $\mathbb{C}^{h+k}$, the
$H$-action on the symplectic normal vector space to $N$ at $p$ can be
identified with the linear $H$-action on
$\mathbb{C}^{h+k}/(\mathbb{C}^{h+k})^{H}$.

Suppose that the complexity of $(N,\omega_{N},\Phi_{N})$ is less than that of
$(Y,\omega_{Y},\Phi_{Y})$. Since $\dim
N=2(d-h)+2\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}$, it follows that
$\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}<k$, so that $p$ is exceptional.
Therefore, (1) implies (2). If $p$ is exceptional, so that
$\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}<k$, then
$\dim_{\mathbb{C}}\mathbb{C}^{h+k}/(\mathbb{C}^{h+k})^{H}>h$. Hence, the
linear $H$-action on $\mathbb{C}^{h+k}/(\mathbb{C}^{h+k})^{H}$ has positive
complexity. Therefore, (2) implies (3). Finally, if the $H$-action on the
symplectic normal vector space to $N$ at $p$ has positive complexity, then
$\dim_{\mathbb{C}}\mathbb{C}^{h+k}/(\mathbb{C}^{h+k})^{H}>h$. The latter is
equivalent to $\dim_{\mathbb{C}}(\mathbb{C}^{h+k})^{H}<k$, so that $\dim
N<2(d-h)+2k$. Hence, the complexity of $(N,\omega_{N},\Phi_{N})$ is less than
that of $(Y,\omega_{Y},\Phi_{Y})$. Therefore, (3) implies (1). ∎

######  Remark 2.27.

The proof of Proposition 2.26 can be adapted to prove equivalence of the
following conditions:

  1. (1’)

The complexity of $(N,\omega_{N},\Phi_{N})$ is equal to that of
$\left(M,\omega,\Phi\right)$.

  2. (2’)

Any $p\in N$ with stabilizer $H$ is regular.

  3. (3’)

The fiberwise $H$-action on the symplectic normal bundle to $N$ has complexity
zero.

We observe that, by Proposition 2.26, either (1) or (1’) needs to hold for any
sheet.

######  Remark 2.28.

If $(N,\omega_{N})$ is a fixed submanifold in $\left(M,\omega,\Phi\right)$,
properties (1) and (2) of Proposition 2.26 (respectively properties (1’) and
(2’) of Remark 2.27) simplify as follows: The dimension of $N$ is less than
(respectively equal to) twice the complexity of $\left(M,\omega,\Phi\right)$,
and all points in $N$ are exceptional (respectively regular). Moreover, $\dim
N\leq 2k$, where $k$ is the complexity of $\left(M,\omega,\Phi\right)$.

Motivated by Proposition 2.26 and Remark 2.27, we introduce the following
terminology.

######  Definition 2.29.

A sheet $(N,\omega_{N},\Phi_{N})$ in $\left(M,\omega,\Phi\right)$ is
exceptional if it satisfies any of the conditions of Proposition 2.26, and
regular otherwise.

Exceptional sheets enjoy the following stronger characterization.

######  Lemma 2.30.

A sheet $(N,\omega_{N},\Phi_{N})$ in $\left(M,\omega,\Phi\right)$ is
exceptional if and only if every point in $N$ is exceptional.

###### Proof.

If every point in $N$ is exceptional, then $(N,\omega_{N},\Phi_{N})$ is
exceptional by Proposition 2.26. Conversely, suppose that
$(N,\omega_{N},\Phi_{N})$ is exceptional. By contradiction, suppose that $p\in
N$ is regular. By Lemma 2.19, every point in $N$ that is sufficiently close to
$p$ is also regular. By the principal orbit theorem (see [12, Theorem 2.8.5]),
the set of points in $N$ that is stabilized by $H$ is dense. Hence, there
exist points in $N$ that are stabilized by $H$ that are arbitrarily close to
$p$. However, by Proposition 2.26, any such point is exceptional, a
contradiction. ∎

######  Remark 2.31.

It is not necessarily true that, if $(N,\omega_{N},\Phi_{N})$ is a regular
sheet, then all points in $N$ are regular. A counterexample is as follows: Let
$\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space of positive complexity
that contains an isolated fixed point $p$. Since $T$ is compact and abelian,
and since $M$ is connected, the principal orbit theorem (see [12, Theorem
2.8.5]) implies that there exists an open and dense subset of $M$ whose points
have trivial stabilizer. Hence, taking $H=\\{e\\}$, it follows that
$\left(M,\omega,\Phi\right)$ is a regular sheet. However, by Lemma 2.19, $p$
is exceptional.

###  2.2. Invariants of compact Hamiltonian $T$-spaces

In this section we recall some fundamental results about compact222In fact,
many of the theorems presented in this section hold under the weaker
assumption that the moment map be proper as a map to a convex open subset of
$\mathfrak{t}^{*}$. However, this degree of generality goes beyond the scope
of this paper. Hamiltonian $T$-spaces.

####  2.2.1. Convexity package and its consequences

We start with the following foundational result (see [1, 18, 50]).

######  Theorem 2.32 (Convexity Package).

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space.

  1. (i)

(Connectedness) The fibers of the moment map are connected.

  2. (ii)

(Stability) The moment map is open as a map onto its image.

  3. (iii)

(Convexity) The moment map image is the convex hull of the images of the fixed
submanifolds.

We remark that, since the action is effective, the moment map image of a
compact Hamiltonian $T$-space is a polytope that has dimension equal to
$\mathfrak{t}^{*}$.

######  Definition 2.33.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space. The image
$\Phi(M)$ is called the moment polytope.

By Theorem 2.32, the moment polytope of a compact Hamiltonian $T$-space is
convex. In fact, more is true and, in order to prove this, we need to recall a
few notions. We say that a polytope $\Delta\subset\mathfrak{t}^{*}$ is
rational if any edge $e\subset\Delta$ is of the form

(2.14) $e=\\{v+t\alpha\mid t\in[0,l]\\}\quad\text{for some
}v\in\mathfrak{t}^{*},\,\alpha\in\ell^{*}\text{ and }l\in\mathbb{R}_{>0}.$

Moreover, a subset $C\subseteq\mathfrak{t}^{*}$ is a cone if, for all $v\in C$
and all $\lambda\in\mathbb{R}_{\geq 0}$, $\lambda v\in C$. A cone in
$\mathfrak{t}^{*}$ is proper if it does not contain any subspace of
$\mathfrak{t}^{*}$ of positive dimension. The following result provides a
local description of the moment polytope of a compact Hamiltonian $T$-space
near the image of a fixed submanifold.

######  Corollary 2.34.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space of
dimension $2n$. Let $N$ be a fixed submanifold and let
$\alpha_{1},\dots\alpha_{n}$ be the isotropy weights of $N$ (see Remark 2.23).
Consider the cone

(2.15) $\mathcal{C}_{N}=\mathbb{R}_{\geq
0}\text{-span}\left\\{\alpha_{1},\dots\alpha_{n}\right\\}\subseteq\mathfrak{t}^{*},$

and let $\mathcal{H}_{N}\subseteq\mathfrak{t}^{*}$ be the maximal subspace
that is contained in $\mathcal{C}_{N}$.

  1. (i)

There exist an open neighborhood $V$ of $\Phi(N)$ in $\Phi(M)$ and an open
neighborhood $W$ of $0$ in $\mathcal{C}_{N}$ such that $V=W+\Phi(N)$. In
particular, $\Phi(M)$ is rational.

  2. (ii)

The intersection

$\displaystyle\left(\mathcal{H}_{N}+\Phi(N)\right)\cap\Phi(M)$

is a face of $\Phi(M)$ and the dimension of this face equals the dimension of
$\mathcal{H}_{N}$.

In particular, $\Phi(N)$ is a vertex of $\Phi(M)$ if and only if the cone
$\mathcal{C}_{N}$ is proper.

Figure 2.1. The arrows are the non-zero isotropy weights of the fixed
submanifold $N$, the shaded area equals $\mathcal{C}_{N}$ and the straight
line spanned by the collinear isotropy weights is $\mathcal{H}_{N}$.

###### Proof.

For any $p\in N$, by Theorem 2.8, there exist a $T$-invariant open
neighborhood $U_{p}$ of $p$ and an open neighborhood $W_{p}$ of $0$ in
$\mathcal{C}_{N}$ such that $\Phi(U_{p})=W_{p}+\Phi(N)$. Moreover, by Theorem
2.32, $\Phi(U_{p})$ is an open neighborhood of $\Phi(p)$ in $\Phi(M)$. Since
$N$ is compact, there exist finitely many $p_{1},\ldots,p_{r}\in N$ such that
$N$ is contained in $\bigcup_{j=1}^{r}U_{p_{j}}$. Set
$W:=\bigcap_{j=1}^{r}W_{p_{j}}$. By construction,
$W+\Phi(N)=\bigcap_{j=1}^{r}\Phi(U_{p_{j}})$ is an open neighborhood of
$\Phi(N)$ in $\Phi(M)$. Since the cone $\mathcal{C}_{N}$ is convex and
rational, and since vertices of the moment polytope are the image of fixed
submanifolds by Theorem 2.32, the moment polytope is rational. This proves
part (i).

To prove part (ii), observe that any cone is the product of the maximal
subspace that it contains with a proper cone. Thus we can write
$\mathcal{C}_{N}=\mathcal{H}_{N}\times\mathcal{C}^{\prime}_{N}$ for some
proper cone $\mathcal{C}^{\prime}_{N}$. Hence, without loss of generality, we
may take an open neighborhood $W$ in $\mathcal{C}_{N}$ as in the statement of
part (i) to be of the form $W_{\mathcal{H}}\times W^{\prime}$, where
$W_{\mathcal{H}}$ (respectively $W^{\prime}$) is an open neighborhood of $0$
in $\mathcal{H}_{N}$ (respectively $\mathcal{C}^{\prime}_{N}$). Therefore, the
desired result holds ‘locally’ by part (i); convexity of $\Phi(M)$ (see
Theorem 2.32) implies that it is true ‘globally’. ∎

Until the end of the section, we deduce some consequences of Theorem 2.32 that
we use throughout the paper. We start with the following sufficient condition
for a sheet to be exceptional (see Definition 2.29).

######  Lemma 2.35.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space and let
$(N,\omega_{N},\Phi_{N})$ be a sheet stabilized by a non-trivial subgroup $H$.
If $\Phi(N)$ is not contained in the boundary of $\Phi(M)$, then
$(N,\omega_{N},\Phi_{N})$ is exceptional.

###### Proof.

We prove the contrapositive. Let $(N,\omega_{N},\Phi_{N})$ be a regular sheet.
It suffices to show that the set of regular values of $\Phi|_{N}$ that are
contained in $\Phi(N)$ is contained in the boundary of $\Phi(M)$. Let
$x\in\Phi(N)$ be a regular value for $\Phi|_{N}$. By Theorem 2.8, there exists
$p\in\Phi|_{N}^{-1}(x)$ that has trivial stabilizer for the $T/H$-action,
i.e., the stabilizer of $p$ for the $T$-action is $H$. By Remark 2.27, $p$ is
regular and, hence, the $T/H$-action on the symplectic normal bundle to $N$ is
toric. By the local normal form (Theorem 2.8), and by openness of the moment
map (Theorem 2.32), it follows that $\Phi(x)$ lies in the boundary of
$\Phi(M)$. ∎

For compact Hamiltonian $T$-spaces, the existence of exceptional sheets is
intimately connected to the existence of exceptional fixed points. More
precisely, the following holds.

######  Lemma 2.36.

A compact Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ contains an
exceptional sheet if and only if it contains an exceptional fixed point.

###### Proof.

Suppose first that $(N,\omega_{N},\Phi_{N})$ is an exceptional sheet of
$\left(M,\omega,\Phi\right)$. Since $N$ is compact, it contains a fixed point
$p\in M^{T}$. By Lemma 2.30, $p$ is exceptional. Conversely, suppose that
$p\in M^{T}$ is exceptional. By definition, the sheet
$(N,\omega_{N},\Phi_{N})$ through $p$ is exceptional. ∎

Next we deduce some general results about isotropy weights of isolated fixed
points that are used in one of the key results of our paper, Proposition 4.6.

######  Lemma 2.37.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity $k$ $T$-space and let
$p\in M^{T}$ be isolated. For any isotropy weight $\alpha$ of $p$, there
exists a sheet $(N_{\alpha},\omega_{\alpha},\Phi_{\alpha})$ with the following
properties:

  * •

the point $p$ lies in $N_{\alpha}$,

  * •

the sheet $(N_{\alpha},\omega_{\alpha},\Phi_{\alpha})$ is stabilized by the
codimension 1 subgroup

(2.16)
$H_{\alpha}:=\exp(\\{\xi\in\mathfrak{t}\mid\langle\alpha,\xi\rangle\in\mathbb{Z}\\}),$

  * •

the dimension of $N_{\alpha}$ is at most $2(k+1)$,

  * •

the moment map image $\Phi_{\alpha}(N_{\alpha})$ is contained in the affine
line $\Phi(p)+\mathbb{R}\langle\alpha\rangle$ and intersects the open half-ray
$\Phi(p)+\mathbb{R}_{>0}\langle\alpha\rangle$, and

  * •

there exists $q_{\alpha}\in M^{T}\cap N_{\alpha}$ such that
$\Phi(q_{\alpha})=\Phi_{\alpha}(q_{\alpha})$ is a global extremum of
$\Phi_{\alpha}(N_{\alpha})$ with
$\Phi(q_{\alpha})\in\Phi(p)+\mathbb{R}_{>0}\langle\alpha\rangle$, and
$-\alpha$ is an isotropy weight of $q_{\alpha}$.

###### Proof.

Let $\alpha_{1},\ldots,\alpha_{n}\in\ell^{*}$ be the isotropy weights of $p$.
Without loss of generality, we may assume that $\alpha=\alpha_{n}$. By the
local normal form of Theorem 2.8, we may identify a $T$-invariant open
neighborhood of $p$ in $M$ with a $T$-invariant open neighborhood of
$0\in\mathbb{C}^{n}$ so that the action becomes that of (2.8) and the moment
map is given by (2.9). Since $p\in M^{T}$ is isolated, $\alpha_{j}\neq 0$ for
all $j$. Hence, since
$\mathbb{Z}\langle\alpha_{1},\ldots,\alpha_{n}\rangle=\ell^{*}$, there can be
at most $k$ isotropy weights that are multiples of $\alpha_{n}$. Therefore the
subgroup $H_{\alpha}$ of (2.16) stabilizes a subspace of $\mathbb{C}^{n}$ that
is of real dimension at most $2(k+1)$. Moreover, by (2.8), the subgroup
$H_{\alpha}$ is the stabilizer of some point $p^{\prime}\in M$. The sheet
$(N,\omega_{N},\Phi_{N})$ through $p^{\prime}$ in the sense of Definition 2.24
satisfies the desired conditions. ∎

For our purposes, it is useful to introduce the following terminology.

######  Definition 2.38.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space and let
$p\in M^{T}$ be isolated. Given an isotropy weight $\alpha$ of $p$, we say
that the sheet $(N_{\alpha},\omega_{\alpha},\Phi_{\alpha})$ of Lemma 2.37 is
along $\boldsymbol{\alpha}$.

######  Lemma 2.39.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space. Let
$\mathcal{F}$ be the facet of $\Phi(M)$ supported on
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=c\\}$. If $p\in M^{T}$
satisfies $\langle\Phi(p),\nu\rangle>c$, then there exists an isotropy weight
$\alpha$ of $p$ with $\langle\alpha,\nu\rangle<0$.

###### Proof.

Let $\alpha_{1},\ldots,\alpha_{n}$ be the isotropy weights of $p$ and suppose
that $\langle\alpha_{j},\nu\rangle\geq 0$ for all $j=1,\ldots,n$. By part (i)
of Corollary 2.34, an open neighborhood of $\Phi(p)$ in $\Phi(M)$ equals an
open neighborhood of $\Phi(p)$ in $\Phi(p)+\mathbb{R}_{\geq
0}\langle\alpha_{1},\ldots,\alpha_{n}\rangle$. Hence, since
$\langle\alpha_{j},\nu\rangle\geq 0$ for all $j$, an open neighborhood $V$ of
$\Phi(p)$ in $\Phi(M)$ is contained in the half-space
$\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu\rangle\geq\langle\Phi(p),\nu\rangle\\}$. Since
$\langle\Phi(p),\nu\rangle>c$ and since $\Phi(M)$ has a facet supported on
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=c\\}$, this is a
contradiction. ∎

Motivated by Lemma 2.39, we introduce the following terminology.

######  Definition 2.40.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space and let
$\mathcal{F}$ be the facet of $\Phi(M)$ supported on
$\\{x\in\mathfrak{t}^{*}\mid\langle x,\nu\rangle=c\\}$. For any $p\in M^{T}$
with $\langle\Phi(p),\nu\rangle>c$, we say that the isotropy weight $\alpha$
of $p$ of Lemma 2.39 is ($\boldsymbol{\mathcal{F}}$-)downward pointing.

Combining Lemmas 2.37 and 2.39, we obtain the following result.

######  Corollary 2.41.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space and let
$\mathcal{F}$ be the facet of $\Phi(M)$ supported on
$\\{x\in\mathfrak{t}^{*}\mid\langle x,\nu\rangle=c\\}$. Let $p\in M^{T}$ be
isolated with $\langle\Phi(p),\nu\rangle>c$ and let $\alpha$ be an isotropy
weight of $p$ that is $\mathcal{F}$-downward pointing. Let
$(N_{\alpha},\omega_{\alpha},\Phi_{\alpha})$ be the sheet along $\alpha$.
There exists $q_{\alpha}\in M^{T}\cap N_{\alpha}$ such that

  * •

$\Phi(q)=\Phi_{\alpha}(q_{\alpha})$ is a global extremum of $\Phi_{\alpha}$,

  * •

$-\alpha$ is an isotropy weight of $q_{\alpha}$, and

  * •

$\langle\Phi(q_{\alpha}),\nu\rangle<\langle\Phi(p),\nu\rangle$.

###### Proof.

Taking $q_{\alpha}$ as in Lemma 2.37, we need to prove only the last property.
This follows immediately by observing that
$\Phi(q)\in\Phi(p)+\mathbb{R}_{>0}\langle\alpha\rangle$ and that $\alpha$ is
$\mathcal{F}$-downward pointing. ∎

To conclude this section, we look at the preimage of faces of the moment
polytope. Given a face $\mathcal{F}$ of $\Phi(M)$, we set

(2.17) $\mathfrak{h}_{\mathcal{F}}:=\left\\{\xi\in\mathfrak{t}\mid\left\langle
x-y,\xi\right\rangle=0\text{ for all }x,y\in\mathcal{F}\right\\}.$

By part (i) of Corollary 2.34, $\mathfrak{h}_{\mathcal{F}}$ is a rational
subspace of $\mathfrak{t}$ of dimension equal to the codimension of
$\mathcal{F}$ in $\Phi(M)$. Hence $\exp(\mathfrak{h}_{\mathcal{F}})$ is
subtorus of $T$. The subset $M_{\mathcal{F}}:=\Phi^{-1}(\mathcal{F})\subset M$
is $T$-invariant; we set

(2.18) $H_{\mathcal{F}}:=\\{t\in T\mid t\cdot p=p\text{ for all }p\in
M_{\mathcal{F}}\\}.$

######  Lemma 2.42 (Theorem 3.5.12 in [45]).

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space and let
$\mathcal{F}$ be a face of $\Phi(M)$. Then
$M_{\mathcal{F}}=\Phi^{-1}(\mathcal{F})$ is a connected component of
$M^{H_{\mathcal{F}}}$ and the Lie algebra of $H_{\mathcal{F}}$ equals
$\mathfrak{h}_{\mathcal{F}}$.

######  Remark 2.43.

Connectedness of $M_{\mathcal{F}}$ can also be proved using the fact that the
Convexity Package implies that the preimage of any convex set is connected
(see [30, Section 2]).

######  Corollary 2.44.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space and let
$\mathcal{F}$ be a face of $\Phi(M)$. There exists a connected, open and dense
subset of $M_{\mathcal{F}}$ whose points have stabilizer equal to
$H_{\mathcal{F}}$.

###### Proof.

By the principal orbit theorem (see [12, Theorem 2.8.5]) and since $T$ is
abelian, there exists a subgroup $H$ of $T$ and a connected, open and dense
subset of $M_{\mathcal{F}}$ such that $H$ is the stabilizer of any point in
this subset. Hence, $H_{\mathcal{F}}\subseteq H$. However, since $H$ is the
stabilizer of orbits of principal type and since $T$ is abelian, $H$ is
contained in the stabilizer of any point in $M_{\mathcal{F}}$. Therefore,
$H\subseteq H_{\mathcal{F}}$. ∎

By Corollary 2.44, the preimage of a face $\mathcal{F}$ of $\Phi(M)$ gives
rise to a sheet in the sense of Definition 2.24 that is stabilized by
$H_{\mathcal{F}}$. We denote it by

(2.19) $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}}).$

In particular, since the complexity of
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is at most that of
$\left(M,\omega,\Phi\right)$ by Proposition 2.26, if the codimension of
$\mathcal{F}$ is $r$, then $\dim M_{\mathcal{F}}\leq 2n-2r$.

####  2.2.2. The Duistermaat-Heckman measure and its density function

In this section we take a close look at an invariant of compact Hamiltonian
$T$-spaces that is central to this paper. We start by recalling the following
notion.

######  Definition 2.45.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space of
dimension $2n$. The Duistermaat-Heckman measure of
$\boldsymbol{\left(M,\omega,\Phi\right)}$ is the pushforward of the
(normalized) Liouville measure, i.e., for any Borel set
$U\subset\mathfrak{t}^{*}$,

(2.20)
$m_{DH}(U)\colon=\frac{1}{(2\pi)^{n}}\int_{\Phi^{-1}(U)}\frac{\omega^{n}}{n!}.$

The Duistermaat-Heckman measure is absolutely continuous with respect to the
Lebesgue measure on $\mathfrak{t}^{*}$ (see [11, equation (3.4)]). Therefore
its Radon-Nikodym derivative with respect to the Lebesgue measure is a
Lebesgue integrable function
$f_{\text{DH}}\colon\mathfrak{t}^{*}\rightarrow\mathbb{R}$ that is uniquely
defined up to a set of measure zero. Without loss of generality, we henceforth
assume that $f_{\text{DH}}$ vanishes identically on
$\mathfrak{t}^{*}\smallsetminus\Phi(M)$.

In [11], Duistermaat and Heckman give an explicit representative of the
restriction of $f_{\text{DH}}$ to the intersection of the moment polytope with
the set of regular values of $\Phi$. In order to state this result, we denote
the set of regular values of $\Phi$ contained in $\Phi(M)$ by
$\Phi(M)_{\mathrm{reg}}$. Moreover, we recall that for any
$x\in\Phi(M)_{\mathrm{reg}}$, the reduced space $M_{x}$ is an orbifold that
inherits a symplectic form that we denote by $\omega_{x}$ (see Section 2.1.1).

######  Theorem 2.46 (Proposition 3.2 in [11]).

Let $\left(M,\omega,\Phi\right)$ be a compact complexity $k$ $T$-space. The
restriction of the Radon-Nikodym derivative of the Duistermaat-Heckman measure
of $\left(M,\omega,\Phi\right)$ to $\Phi(M)_{\mathrm{reg}}$ can be chosen to
be equal to the function $\Phi(M)_{\mathrm{reg}}\to\mathbb{R}$

(2.21)
$x\mapsto\frac{1}{(2\pi)^{k}}\int_{M_{x}}\frac{\omega_{x}^{k}}{k!}=:\frac{1}{(2\pi)^{k}}\mathrm{Vol}(M_{x}),\quad
x\in\Phi(M)_{\mathrm{reg}}.$

######  Remark 2.47.

By [11, Corollary 3.3], the restriction of the function (2.21) to each
connected component of $\Phi(M)_{\mathrm{reg}}$ is a polynomial of degree at
most $k$. Moreover, if this polynomial has positive degree, the coefficients
of the monomials of top degree are integral. This is because the cohomology
classes $[\omega_{x}]$ vary linearly with $x$ on such a connected component
and the variation is controlled by a cohomology class with integral
coefficients (see [11, Theorem 1.1]).

###### Remark 2.48

Theorem 2.46 is sufficient to calculate the Duistermaat-Heckman measure, as
the set of singular values has measure zero by Sard’s theorem.

By Remark 2.47, the function of (2.21) is continuous. The aim of this section
is to prove Theorem 2.49 below, which is probably well-known to experts (cf.
[17, Theorem 3.2.11] for linear symplectic actions on vector spaces and [46,
Theorem A and Remark 1.2.10]). However, since we use it extensively throughout
the paper, we include a proof for completeness.

######  Theorem 2.49.

Given a compact Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$, there
exists a unique continuous function
$DH\left(M,\omega,\Phi\right):\Phi(M)\to\mathbb{R}$ that extends the function
of (2.21).

######  Remark 2.50.

By Remark 2.47, if the interior of the moment polytope consists solely of
regular values, then Theorem 2.49 is trivial. In this case,
$DH\left(M,\omega,\Phi\right)$ is the restriction of a polynomial of degree at
most the complexity of $\left(M,\omega,\Phi\right)$. For instance, the desired
function in the case of compact symplectic toric manifolds is the indicator
function of the moment polytope, since reduced spaces are connected by Theorem
2.32, and since $k=\dim M_{x}=0$ for all $x\in\Phi(M)_{\mathrm{reg}}$.

Theorem 2.49 allows us to introduce the following notion.

######  Definition 2.51.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space. We call
the continuous map $DH\left(M,\omega,\Phi\right):\Phi(M)\to\mathbb{R}$ given
by Theorem 2.49 the Duistermaat-Heckman function of
$\left(M,\omega,\Phi\right)$.

######  Example 2.52.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space and let
$H\subset T$ be a subtorus. Choose a complementary subtorus $K$ of $T$ so that
$T=H\times K$. This induces an identification
$\mathfrak{t}^{*}\simeq\mathfrak{h}^{*}\oplus\mathfrak{k}^{*}$. We write the
Lebesgue measure on $\mathfrak{t}^{*}$ as $dxdy$, where $dx$ (respectively
$dy$) is the Lebesgue measure on $\mathfrak{h}^{*}$ (respectively
$\mathfrak{k}^{*}$). Let $\pi:\mathfrak{t}^{*}\to\mathfrak{h}^{*}$ be the
projection induced by the inclusion $H\subset T$. The $H$-action is
Hamiltonian with moment map $\Phi^{\prime}:=\pi\circ\Phi$. Since
$DH\left(M,\omega,\Phi\right)$ is continuous, by Fubini’s theorem, the
Duistermaat-Heckman function of $(M,\omega,\Phi^{\prime})$ is given by

(2.22)
$DH(M,\omega,\Phi^{\prime})(x)=\int\limits_{\Delta_{x}}DH\left(M,\omega,\Phi\right)(x,y)dy\quad\text{for
all }x\in\Phi^{\prime}(M),$

where $\Delta_{x}=\pi^{-1}(x)\cap\Phi(M)$.

Suppose further that $\left(M,\omega,\Phi\right)$ is a compact symplectic
toric manifold and that $H$ has codimension 1. For any $x\in\Phi^{\prime}(M)$,
$\Delta_{x}$ is an interval in
$\\{x\\}\times\mathfrak{k}^{*}\simeq\\{x\\}\times\mathbb{R}$ that we write as
$\\{(x,y)\in\mathfrak{h}^{*}\oplus\mathbb{R}\mid
y\in[p_{min}(x),p_{max}(x)]\\}$. By Remark 2.50,

(2.23) $DH(M,\omega,\Phi^{\prime})(x)=p_{\max}(x)-p_{\min}(x)\quad\text{for
all }x\in\Phi^{\prime}(M).$

We observe that, since $\Phi(M)$ is a convex polytope, the difference
$p_{max}-p_{min}$ is concave (cf. Proposition 2.59 below).

Our proof of Theorem 2.49 uses extensively ideas from [17, Section 3.5].
Before proceeding with the proof, we need to recall some facts about singular
values of the moment map.

#### Intermezzo 1: chambers of the moment map

We begin by relating singular points of the moment map of a Hamiltonian
$T$-space (that is not necessarily compact), to sheets arising from one
dimensional stabilizers (see Definition 2.24). If $K\leq T$ is a closed one-
dimensional subgroup that occurs as the stabilizer of some point in $M$, then
every $p\in M^{K}$ is a singular point of $\Phi$. In fact, the converse also
holds.

######  Lemma 2.53.

Let $\left(M,\omega,\Phi\right)$ be a Hamiltonian $T$-space. If $K\leq T$ is a
one-dimensional closed subgroup that occurs as the stabilizer of some point in
$M$ then $\Phi(M^{K})$ is contained in the set of singular values of $\Phi$.
Conversely, if $x\in\mathfrak{t}^{*}$ is a singular value of $\Phi$, then
there exists a one-dimensional closed subgroup $K\leq T$ that occurs as the
stabilizer of some point in $M$ such that $x\in\Phi(M^{K})$.

###### Proof.

Since the action is Hamiltonian, the first statement holds. Conversely, let
$p\in M$ be a singular point with $\Phi(p)=x$. Let $H$ be the stabilizer of
$p$, let $h=\dim H$ and let $k$ be the complexity of
$\left(M,\omega,\Phi\right)$. Since the $T$-action is Hamiltonian and since
$p$ is a singular point of $\Phi$, $\dim H\geq 1$. If $\dim H=1$, there is
nothing to prove. Hence, suppose that $\dim H\geq 2$. By Theorem 2.8, it
suffices to prove the result for the Hamiltonian $T$-action on the local model
of $p$. Hence, we may assume that $\left(M,\omega,\Phi\right)$ is the local
model $(Y,\omega_{Y},\Phi_{Y})$ at $p$ and that $p=[1,0,0]$. Moreover, by
Corollary 2.9, the symplectic slice representation $\rho:H\to(S^{1})^{h+k}$ of
$p$ is injective. The stabilizer of any point in
$Y=T\times_{H}\mathrm{Ann}(\mathfrak{h})\times\mathbb{C}^{h+k}$ is a subgroup
of $H$. In fact, if $\tilde{H}\leq H$ is the stabilizer of some point in $Y$,
we have that

$Y^{\tilde{H}}=T\times_{H}\mathrm{Ann}(\mathfrak{h})\times(\mathbb{C}^{h+k})^{\tilde{H}},$

where $H$ acts on $\mathbb{C}^{h+k}$ via the symplectic slice representation
of $p$. Since the $H$-action on $\mathbb{C}^{h+k}$ is linear,
$(\mathbb{C}^{r})^{\tilde{H}}$ is a subspace for any subgroup $\tilde{H}\leq
H$. Hence, to prove the result, it suffices to show that there exists a one-
dimensional closed subgroup $K\leq H$ that occurs as a stabilizer of a point
in $\mathbb{C}^{h+k}$.

Let $\eta_{1},\ldots,\eta_{h+k}\in\mathfrak{h}^{*}$ be the isotropy weights of
$p$. Since the symplectic slice representation of $p$ is injective, the
$H$-action on $\mathbb{C}^{h+k}$ is effective. This implies that
$\eta_{1},\ldots,\eta_{h+k}$ span $\mathfrak{h}^{*}$. Hence we may assume that
there exists $s\geq 1$ such that the span of $\eta_{s+1},\ldots,\eta_{h+k}$
has codimension 1\. Since the $H$-action on $\mathbb{C}^{h+k}$ is Hamiltonian
with moment map
$\Phi_{H}(z)=\frac{1}{2}\sum\limits_{j=1}^{h+k}|z_{j}|^{2}\eta_{j}$, it can be
checked directly that all points in

$\\{z=(z_{1},\ldots,z_{r})\in\mathbb{C}^{h+k}\mid z_{j}=0\text{ if
}j=1,\ldots,s,\,z_{j}\neq 0\text{ if }j\geq s+1\\},$

have stabilizers of dimension one, as desired. ∎

In other words, the set of singular values of the moment map equals the union
of the moment map image of all sheets that are stabilized by some one
dimensional closed subgroup of $T$ (see Definition 2.24). Each such image is
contained in some affine hyperplane (see Remark 2.25 and the discussion
preceding it). If, in addition, $M$ is compact there are two important
consequences. First, there are only finitely many subgroups of $T$ that occur
as the stabilizer of some point in $M$ (see [22, Theorem 10.9.1]). Second,
since the action is Hamiltonian, $\Phi$ has some singular point; hence, by
Lemma 2.53, there exists a one dimensional closed subgroup of $T$ that that
occurs as the stabilizer of some point in $M$. Let $K_{1},\ldots,K_{r}\leq T$
be the collection of such one dimensional closed subgroups. Since $M$ is
compact, by Lemma 2.22, $M^{K_{i}}$ is a compact submanifold of $M$ for each
$i=1,\ldots,r$ and, therefore, has finitely many connected components
$N_{i1},\ldots,N_{is_{i}}$. For each $i,j$, we denote the corresponding sheet
by $(N_{ij},\omega_{ij},\Phi_{ij})$ and we set
$\Delta_{ij}:=\Phi_{ij}(N_{ij})$. We observe that the union of the
$\Delta_{ij}$’s includes the union of all facets. More precisely, the
following holds.

######  Lemma 2.54.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space. For any
facet $\mathcal{F}$ of $\Phi(M)$, there exist indices $i,j$ as above so that
$\mathcal{F}=\Delta_{ij}$.

###### Proof.

Let $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ be the sheet
corresponding to $\mathcal{F}$ (see Corollary 2.44 and (2.19)), and let
$H_{\mathcal{F}}$ its stabilizer. By Lemma 2.42, since the codimension of
$\mathcal{F}$ in $\mathfrak{t}^{*}$ is one, the dimension of $H_{\mathcal{F}}$
is also one. By Corollary 2.44, it follows that
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is one of the
sheets constructed above. ∎

The complement of the union of the $\Delta_{ij}$’s in the moment polytope is
precisely $\Phi(M)_{\mathrm{reg}}$. We call the closure of a connected
component of this complement a chamber of $\boldsymbol{\Phi(M)}$. These
chambers partition the moment polytope into subpolytopes, i.e., the following
properties hold (see Figure 2.2):

  1. (1)

Each chamber is a polytope in $\mathfrak{t}^{*}$ of full dimension and any two
chambers intersect in a common face.

  2. (2)

Let $F$ be a facet of a chamber. The set of points $x\in F$ that are regular
values for all the moment maps $\Phi_{ij}$ of the sheets
$(N_{ij},\omega_{ij},\Phi_{ij})$ constructed above is dense in $F$. Moreover
this set is contained in the interior of $F$. Conversely, if $x$ is a singular
value that is a regular value for all the moment maps $\Phi_{ij}$ of the
sheets $(N_{ij},\omega_{ij},\Phi_{ij})$ constructed above, then $x$ lies in
the interior of a facet of a chamber.

  3. (3)

If two chambers $\mathfrak{c}$ and $\mathfrak{c}^{\prime}$ intersect in a face
$F$, then there exists a sequence of chambers
$\mathfrak{c}_{0}=\mathfrak{c},\mathfrak{c}_{1},\ldots,\mathfrak{c}_{s}=\mathfrak{c}^{\prime}$
with the property that the intersection of $\mathfrak{c}_{l}$ and
$\mathfrak{c}_{l+1}$ is a facet that contains $F$ for all $l=0,\ldots,s-1$.

Figure 2.2. The six triangles above are the chambers for the moment polytope
for the standard $(S^{1})^{2}$-action on the complete flag variety in
$\mathbb{C}^{3}$ with first Chern class equal to the class of the symplectic
form.

Properties (1) – (3) follow from the above definition of the $N_{ij}$’s and
from Theorems 2.8 and 2.32 (see also [21, Section 14]).

By Remark 2.50, Theorem 2.49 follows at once if the interior of $\Phi(M)$
consists entirely of regular values. The following result describes precisely
when this happens in terms of exceptional sheets.

######  Lemma 2.55.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space. There is
precisely one chamber of $\Phi(M)$ if and only if there are no exceptional
sheets.

###### Proof.

If $\Phi(M)$ has no exceptional sheets then all sheets are contained in the
boundary of $\Phi(M)$ by Lemma 2.35. This implies that the union of the
$\Delta_{ij}$’s constructed above is contained in the boundary of $\Phi(M)$,
which equals the union of the facets of $\Phi(M)$ since $\Phi(M)$ is a convex
polytope. By Lemma 2.54, the union of all the facets of $\Phi(M)$ is contained
in the union of the $\Delta_{ij}$’s. Hence, the union of the $\Delta_{ij}$’s
equals the boundary of $\Phi(M)$. Since $\Phi(M)$ is a convex polytope of full
dimension, the complement of the boundary of $\Phi(M)$ in $\Phi(M)$ equals the
interior of $\Phi(M)$, which is connected. Hence, there is precisely one
chamber of $\Phi(M)$.

Conversely, if $\Phi(M)$ has at least two chambers, then at least one of the
$\Delta_{ij}$’s constructed above cannot be contained in the boundary of
$\Phi(M)$. By Lemma 2.35, the corresponding sheet is exceptional. ∎

Seeing as the case of only one chamber has already been proved, in what
follows (namely, in Intermezzo 2 and in the proof of Theorem 2.49 below), we
assume that there are at least two chambers.

#### Intermezzo 2: the wall-crossing formula

The main tool that we use in the proof of Theorem 2.49 is the so-called wall-
crossing formula for compact Hamiltonian $T$-spaces (see [4], [17, Section
3.5] and [46]). We recall it here for completeness and we draw on the above
Intermezzo for notation. Moreover, in this subsection we consider the closure
of the complement of the moment map as a chamber.

Let $\mathfrak{c}_{\pm}$ be two chambers in $\Phi(M)$ that intersect in a
facet $F$. Let $\xi\in\ell$ be the primitive element that is normal to the
hyperplane supporting $F$ and that points out of $\mathfrak{c}_{-}$ (see
Figure 2.3). We fix a point $x\in F$ that has the property that is a regular
value for all for all the moment maps $\Phi_{ij}$ of the sheets
$(N_{ij},\omega_{ij},\Phi_{ij})$ constructed in the above Intermezzo; such a
point exists by property (2).

Figure 2.3. Two chambers, $\mathfrak{c}_{-}$ and $\mathfrak{c}_{+}$, separated
by a hyperplane with normal $\xi$.

We use the fixed inner product on $\mathfrak{t}$ to choose a complementary
subspace $\mathfrak{k}$ to the span of $\xi$, i.e.,
$\mathfrak{t}=\langle\xi\rangle\oplus\mathfrak{k}$. Hence, viewing
$\mathfrak{t}$ as the space of homogeneous polynomials of degree one on
$\mathfrak{t}^{*}$, we can view polynomials on $\mathfrak{t}^{*}$ as being
generated by $\xi$ and polynomials on $\mathfrak{k}^{*}$. Moreover, since
$\xi\in\ell$, we have that $\exp(\langle\xi\rangle)$ is a circle in $T$ that
we denote by $S^{1}$. The subspace $\mathfrak{k}$ is isomorphic to the Lie
algebra of the quotient $T/S^{1}$. In what follows, we use this identification
tacitly since it is compatible with the identification of Remark 2.25.

Let $f_{\pm}:\mathfrak{t}^{*}\to\mathbb{R}$ be the polynomial that, when
restricted to the interior of $\mathfrak{c}_{\pm}$, equals (2.21). Since $x$
is a regular value of $\Phi_{ij}$, it lies in the interior of a chamber
$\mathfrak{c}_{ij}$ of $\Phi_{ij}(N_{ij})$ for all $i,j$. Hence, there is a
corresponding polynomial $f_{ij}:\mathfrak{k}^{*}\to\mathbb{R}$ that, when
restricted to the interior of the $\mathfrak{c}_{ij}$, equals (2.21). (If
$x\notin\Phi_{ij}(N_{ij})$, this polynomial is identically zero.)

Finally, for each $i,j$, we let $\kappa_{ij}$ be half the codimension of
$N_{ij}$ in $M$ and, if $i,j$ are such that $x\in\Phi_{ij}(N_{ij})$, we let
$\alpha_{ij1},\ldots,\alpha_{ij\kappa_{ij}}\in\mathbb{Z}$ be the isotropy
weights for the $S^{1}$-action on the normal bundle to $N_{ij}$.

######  Theorem 2.56 (Wall-crossing formula).

Set the notation as above. For all $y\in\mathfrak{t}^{*}$ we have that

(2.24) $f_{+}(y)-f_{-}(y)=\sum\limits_{\\{i,j\mid
x\in\Phi_{ij}(N_{ij})\\}}\xi^{\kappa_{ij}-1}(y-x)\left(\prod\limits_{s=1}^{\kappa_{ij}}\alpha_{ijs}\right)^{-1}\left[\frac{f_{ij}(y-x)}{(\kappa_{ij}-1)!}+P_{ij}(y-x)\right],$

where $P_{ij}$ is a polynomial depending on $i,j$ that is divisible by $\xi$.

######  Remark 2.57.

  * •

We stress that Theorem 2.56 also holds if, say, $\mathfrak{c}_{-}$ is the
closure of the complement of the moment map image.

  * •

The polynomials $P_{ij}$ have been computed explicitly (see [46, Theorem
1.2.9]). They depend on the symplectic reduction of $N_{ij}$ at $x$ by the
$T/S^{1}$-action.

#### Back to the Duistermaat-Heckman function of $\left(M,\omega,\Phi\right)$

With the above Intermezzos, we have all the ingredients to proceed with the
proof of Theorem 2.49.

###### Proof of Theorem 2.49.

First, we prove existence of a continuous extension. Let
$\mathfrak{c}_{1},\ldots,\mathfrak{c}_{l}$ be the chambers of $\Phi(M)$ and,
for each $i=1,\ldots,l$, let $f_{i}:\mathfrak{t}^{*}\to\mathbb{R}$ be the
polynomial that equals (2.21) when restricted to the interior of
$\mathfrak{c}_{i}$. The result is proved if we show that the map given by

(2.25) $x\mapsto f_{i}(x)\qquad\text{if }x\in\mathfrak{c}_{i}$

is well-defined, i.e., given two chambers $\mathfrak{c}_{i}$ and
$\mathfrak{c}_{j}$ such that
$\mathfrak{c}_{i}\cap\mathfrak{c}_{j}\neq\emptyset$, the following holds:

(2.26) $f_{i}(x)=f_{j}(x)\text{ for all
}x\in\mathfrak{c}_{i}\cap\mathfrak{c}_{j}.$

Clearly equation (2.26) holds if $i=j$, so suppose that $i\neq j$. Set
$F:=\mathfrak{c}_{i}\cap\mathfrak{c}_{j}$; this is a face of both
$\mathfrak{c}_{i}$ and $\mathfrak{c}_{j}$ (see property (1)).

We consider first the special case that $F$ is a facet. Since the set of
points in $F$ that are regular values for all the sheets
$(N,\omega_{N},\Phi_{N})$ constructed in Intermezzo 1 is dense in $F$ (see
property (2)), and since both $f_{i}$ and $f_{j}$ are continuous, it suffices
to check that equation (2.26) holds for those points. Let $x\in F$ be such a
point. If we show that the codimension of each $N$ such that $x\in\Phi(N)$ is
at least four, then we are done by Theorem 2.56 (cf. [46, Remark 1.2.10]).
Since $F$ is a facet of two distinct chambers, it is not contained in the
boundary of $\Phi(M)$. In particular, since $x$ lies in the interior of $F$,
it does not lie in the boundary of $\Phi(M)$. Therefore, if $N$ is such that
$x\in\Phi(N)$, then $\Phi(N)$ is not contained in the boundary of $\Phi(M)$.
By Lemma 2.35, $(N,\omega,\Phi)$ is an exceptional sheet. Thus the complexity
of $(N,\omega,\Phi)$ is strictly less than that of
$\left(M,\omega,\Phi\right)$ by Proposition 2.26. Since the dimension of the
torus that acts effectively on $N$ is one less than that of $T$, it follows
that the codimension of $N$ in $M$ is at least four, as desired.

If $F$ is not a facet, then by property (3) above we can find chambers
$\mathfrak{c}_{0}=\mathfrak{c}_{i},\mathfrak{c}_{1},\ldots,\mathfrak{c}_{s}=\mathfrak{c}_{j}$
such that $\mathfrak{c}_{l}$ and $\mathfrak{c}_{l+1}$ intersect in a facet
that contains $F$ for all $l=0,\ldots,s-1$ (see property (3)). The result then
follows immediately by the above special case.

Uniqueness of the continuous extension follows immediately by observing that
$\Phi(M)_{\mathrm{reg}}$ is a dense subset of $\Phi(M)$ and that the function
of (2.21) is continuous. ∎

The Duistermaat-Heckman function is an invariant of the isomorphism class of a
compact Hamiltonian $T$-space that plays an important role in this paper. As
an illustration, the following result describes the restriction of the
Duistermaat-Heckman function of $\left(M,\omega,\Phi\right)$ to any facet of
the moment polytope.

######  Proposition 2.58.

Let $\left(M,\omega,\Phi\right)$ be a compact Hamiltonian $T$-space. If
$\mathcal{F}$ is a facet of $\Phi(M)$, then

(2.27)
$DH\left(M,\omega,\Phi\right)|_{\mathcal{F}}=\begin{cases}DH(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})&\text{if
}\dim M_{\mathcal{F}}=\dim M-2,\\\ 0&\text{otherwise,}\end{cases}$

where $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is as in
(2.19).

###### Proof.

In this proof we denote the closure of the complement of the moment map image
by $\mathfrak{c}_{-}$ (cf. the paragraph preceding Theorem 2.56 and Remark
2.57). By Lemma 2.54,
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is one of the
sheets constructed in Intermezzo 1; in particular, the quotient
$T/H_{\mathcal{F}}$ is isomorphic to $S^{1}$, where $H_{\mathcal{F}}$ is the
stabilizer of $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$. Let
$x\in\mathcal{F}$ be a regular value of $\Phi_{\mathcal{F}}$. Since
$M_{\mathcal{F}}=\Phi^{-1}(\mathcal{F})$ (see Lemma 2.42), it follows that
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is the only sheet
constructed in Intermezzo 1 that contains $x$ in its moment map image. In
particular, $x$ is a regular value for all the moment maps of all the sheets
constructed in Intermezzo 1. Hence, there exists a chamber $\mathfrak{c}_{+}$
and a facet $F$ of $\mathfrak{c}_{+}$ such that $x$ lies in the interior of
$F$.

Let $\xi\in\ell$ be the primitive normal to the hyperplane supporting $F$ that
points out of $\mathfrak{c}_{-}$. Let $f_{+}:\mathfrak{t}^{*}\to\mathbb{R}$
denotes the polynomial that, when restricted to $\mathfrak{c}_{+}$, equals
(2.21). Since $f_{-}\equiv 0$, by Theorem 2.56,

(2.28)
$f_{+}(y)=\xi^{\kappa_{\mathcal{F}}-1}(y-x)\left(\prod\limits_{s=1}^{\kappa_{\mathcal{F}}}\alpha_{s}\right)^{-1}\left[\frac{f_{\mathcal{F}}(y-x)}{(\kappa_{\mathcal{F}}-1)!}+P_{\mathcal{F}}(y-x)\right]\quad\text{for
all }y\in\mathfrak{t}^{*},$

where $\kappa_{\mathcal{F}}$ is half of the codimension of $M_{\mathcal{F}}$
in $M$, $\alpha_{1},\ldots,\alpha_{\kappa_{\mathcal{F}}}\in\mathbb{Z}$ are the
isotropy weights of the $S^{1}$-action on the normal bundle to
$M_{\mathcal{F}}$, $f_{\mathcal{F}}$ is the polynomial that equals (2.21) when
restricted to the chamber of $\Phi_{\mathcal{F}}$ containing $x$, and
$P_{\mathcal{F}}$ is the polynomial associated to
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ in (2.24). By
(2.28), if $\kappa_{\mathcal{F}}\geq 2$, then $f_{+}(x)=0$. On the other hand,
if $\kappa_{\mathcal{F}}=1$, since $P_{\mathcal{F}}$ is divisible by $\xi$
(see Theorem 2.56, then the right hand side of (2.28) evaluated at $x$ equals
$f_{\mathcal{F}}(0)/\alpha_{1}$. Since the $S^{1}$-action is effective and
$\xi$ is chosen to point out of $\mathfrak{c}_{-}$, it follows that
$\alpha_{1}=1$ and, hence, $f_{+}(x)=f_{\mathcal{F}}(0)$. Since $f_{+}$ and
$f_{\mathcal{F}}$ are restrictions of $DH\left(M,\omega,\Phi\right)$ and
$DH(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ respectively, we
have shown that, if $x$ is a regular value of $\Phi_{\mathcal{F}}$, then

(2.29)
$DH\left(M,\omega,\Phi\right)(x)=\begin{cases}DH(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})(x)&\text{if
}\dim M_{\mathcal{F}}=\dim M-2\\\ 0&\text{otherwise.}\end{cases}$

Since $\Phi_{\mathcal{F}}(M_{\mathcal{F}})_{\mathrm{reg}}$ is dense in
$\Phi_{\mathcal{F}}(M_{\mathcal{F}})$, and since
$DH\left(M,\omega,\Phi\right)$ and
$DH(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ are continuous
and defined on $\mathcal{F}$, equation (2.29) implies the desired result. ∎

#### The Duistermaat-Heckman function of a complexity one $T$-space

In this section we prove some properties of the Duistermaat Heckman function
of a compact complexity one $T$-space $\left(M,\omega,\Phi\right)$. We start
with the following special property that fails to hold in higher complexity
(see [26]).

######  Proposition 2.59.

The Duistermaat-Heckman function
$DH\left(M,\omega,\Phi\right):\Phi(M)\to\mathbb{R}$ of a compact complexity
one $T$-space $\left(M,\omega,\Phi\right)$ is concave.

###### Proof.

By Theorem 2.49, $DH\left(M,\omega,\Phi\right)$ is continuous. Hence, it
suffices to check that the restriction of $DH\left(M,\omega,\Phi\right)$ to
the interior of $\Phi(M)$ is concave. Since the complexity of
$\left(M,\omega,\Phi\right)$ is one, by Remark 2.47,
$DH\left(M,\omega,\Phi\right)$ is piecewise linear. Thus the restriction of
$DH\left(M,\omega,\Phi\right)$ to the interior of $\Phi(M)$ is concave if and
only if it is log-concave. The result then follows immediately from [5,
Theorem 1.1]. ∎

As observed in [49, Corollary 3.8], continuity and concavity of
$DH\left(M,\omega,\Phi\right)$ (see Theorem 2.49 and Proposition 2.59),
together with convexity of $\Phi(M)$ (see Theorem 2.32), immediately imply the
following result.

######  Corollary 2.60.

The minimum of the Duistermaat-Heckman function of a compact complexity one
$T$-space is attained at a vertex.

Next we prove two results relating $DH\left(M,\omega,\Phi\right)$ with
exceptional orbits and singular values of $\Phi$.

######  Lemma 2.61.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity one $T$-space. If
there are no isolated fixed points, then $M_{\mathrm{exc}}=\emptyset$ and the
Duistermaat-Heckman function
$DH\left(M,\omega,\Phi\right):\Phi(M)\to\mathbb{R}$ is the restriction of an
affine function.

###### Proof.

Since there are no isolated fixed points, Lemma 2.20 implies that there are no
exceptional fixed points. Hence, by Lemma 2.36, there are no exceptional
sheets. By Lemma 2.30, $M_{\mathrm{exc}}=\emptyset$. Moreover, by Lemma 2.55,
there is only one chamber of $\Phi(M)$. The result then follows by Remark
2.50. ∎

######  Lemma 2.62.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity one $T$-space. If the
Duistermaat-Heckman function $DH\left(M,\omega,\Phi\right)$ is constant, then
there are no singular values in the interior of $\Phi(M)$.

###### Proof.

We prove the contrapositive. Suppose that there is a singular value in the
interior of $\Phi(M)$. Hence there exist two chambers $\mathfrak{c}_{-}$ and
$\mathfrak{c}_{+}$ of $\Phi(M)$ that intersect in a facet $F$ that is not
contained in the boundary of $\Phi(M)$. As in Intermezzo 2, we let
$\xi\in\ell$ be a primitive element that is normal to the hyperplane
supporting $F$ and points out of $\mathfrak{c}_{-}$. Moreover, we fix $x\in F$
that is a regular value for all the sheets $(N_{ij},\omega_{ij},\Phi_{ij})$
constructed in Intermezzo 1; we observe that $x$ lies in the interior of $F$
(see property (2)).

Since $F$ is not contained in the boundary of $\Phi(M)$, neither is $x$. In
particular, if $x\in\Phi_{ij}(N_{ij})$, then $\Phi_{ij}(N_{ij})$ is not
contained in the boundary of $\Phi(M)$. Hence,
$(N_{ij},\omega_{ij},\Phi_{ij})$ is exceptional by Lemma 2.35. By Proposition
2.26, the complexity of $(N_{ij},\omega_{ij},\Phi_{ij})$ is strictly less than
that of $\left(M,\omega,\Phi\right)$. Since the complexity of
$\left(M,\omega,\Phi\right)$ is one, the complexity of
$(N_{ij},\omega_{ij},\Phi_{ij})$ is zero. Hence, the codimension of $N_{ij}$
in $M$ equals 4. Therefore, the lowest order term in $\xi$ in the right hand
side of equation (2.24) is

(2.30) $\xi\left(\sum\limits_{\\{i,j\mid
x\in\Phi_{ij}(N_{ij})\\}}(\alpha_{ij1}\alpha_{ij2})^{-1}f_{ij}\right),$

where, for each $i,j$, the polynomial$f_{ij}$ and the integers
$\alpha_{ij1},\alpha_{ij2}$ are as in the discussion leading up to Theorem
2.56. Since $x$ lies in the interior of $\Phi(M)$, it follows that
$\alpha_{ij1}\alpha_{ij2}<0$ and $f_{ij}(x)>0$ for each $i,j$. In particular,
the polynomial in equation (2.30) is not identically zero and, therefore,
neither is the right hand side of equation (2.24).

Let $f_{\pm}:\mathfrak{t}^{*}\to\mathbb{R}$ be the polynomial that, when
restricted to $\mathfrak{c}_{\pm}$, equals (2.21). By (2.30) and Theorem 2.56,
$f_{+}$ and $f_{-}$ are not equal. Hence, since $f_{\pm}$ is the restriction
of the Duistermaat-Heckman function $DH\left(M,\omega,\Phi\right)$ to
$\mathfrak{c}_{\pm}$, $DH\left(M,\omega,\Phi\right)$ is not constant, as
desired. ∎

The next result describes $DH\left(M,\omega,\Phi\right)$ near a vertex of
$\Phi(M)$ that corresponds to a fixed surface. To this end, let $v\in\Phi(M)$
be a vertex and let $\Sigma=\Phi^{-1}(v)$ be a fixed surface. Let
$\alpha_{1},\ldots,\alpha_{n}$ be the isotropy weights of $\Sigma$ (see Remark
2.23), labeled so that $\alpha_{n}=0$. Since $v$ is a vertex, by part (i) of
Corollary 2.34, a sufficiently small neighborhood of $v$ in $\Phi(M)$ is of
the form

$\Big{\\{}v+\sum\limits_{i=1}^{n-1}t_{i}\alpha_{i}\mid t_{i}\geq 0\text{
sufficiently small}\Big{\\}}.$

Moreover, by part (ii) of Corollary 2.34, for each $i=1,\ldots,n-1$, the edge
$e_{i}$ of $\Phi(M)$ that is incident to $v$ is contained in the half-line
$\\{v+t_{i}\alpha_{i}\mid t_{i}\geq 0\\}$. Let $(M_{i},\omega_{i},\Phi_{i})$
be the sheet corresponding to the edge $e_{i}$ as in (2.19) with stabilizer
$H_{i}$. By Proposition 2.67, $\dim M_{i}=4$ and $(M_{i},\omega_{i},\Phi_{i})$
is a compact complexity one Hamiltonian $T/H_{i}$-space. In what follows, we
identify $T/H_{i}\simeq S^{1}$. Let $N$ be the normal bundle of $\Sigma$ in
$M$. Then $N$ splits $T$-equivariantly as a direct sum

(2.31) $N=L_{1}\oplus\ldots\oplus L_{n-1},$

where $L_{i}$ denotes the normal bundle of $\Sigma$ in $M_{i}$.

######  Lemma 2.63.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity one $T$-space of
dimension $2n$ and let $v\in\Phi(M)$ be a vertex such that
$\Sigma=\Phi^{-1}(v)$ is a fixed surface. Let $\alpha_{1},\ldots,\alpha_{n-1}$
be the non-zero isotropy weights of $\Sigma$. For all
$t_{1},\ldots,t_{n-1}\geq 0$ sufficiently small,

(2.32)
$DH\left(M,\omega,\Phi\right)\left(v+\sum\limits_{i=1}^{n-1}t_{i}\alpha_{i}\right)=\int\limits_{\Sigma}\omega-\sum\limits_{i=1}^{n-1}t_{i}c_{1}(L_{i})\left[\Sigma\right],$

where $L_{1},\ldots,L_{n-1}$ are as in (2.31) and $c_{1}(L_{i})$ is the first
Chern class of $L_{i}$ for $i=1,\ldots,n-1$. In particular, if
$DH\left(M,\omega,\Phi\right)$ attains its minimum at $v$, then

(2.33) $c_{1}(L_{i})[\Sigma]\leq 0\qquad\text{for all }i=1,\ldots,n-1.$

###### Proof.

Since $v$ is a vertex and the complexity of $\left(M,\omega,\Phi\right)$ is
one, the restriction of $DH\left(M,\omega,\Phi\right)$ to a sufficiently small
neighborhood of $v$ is an affine function. Thus there exist real numbers
$\beta_{0},\beta_{1},\ldots,\beta_{n-1}$ such that, for all
$t_{1},\ldots,t_{n-1}\geq 0$ sufficiently small,

(2.34)
$DH\left(M,\omega,\Phi\right)\left(v+\sum\limits_{i=1}^{n-1}t_{i}\alpha_{i}\right)=\beta_{0}+\sum\limits_{i=1}^{n-1}t_{i}\beta_{i}.$

In order to determine the constants $\beta_{0},\beta_{1},\ldots,\beta_{n-1}$,
it suffices to understand the restriction of $DH\left(M,\omega,\Phi\right)$ to
elements of the form $v+t_{j}\alpha_{j}$ for $j=1,\ldots,n-1$. Fix
$i=1,\ldots,n-1$. By Corollary 2.71,

(2.35)
$DH\left(M,\omega,\Phi\right)(v+t_{i}\alpha_{i})=DH(M_{i},\omega_{i},\Phi_{i})(v+t_{i}\alpha_{i}).$

By [27, Lemma 2.19], for all $t_{i}\geq 0$ sufficiently small,

(2.36)
$DH(M_{i},\omega_{i},\Phi_{i})(v+t_{i}\alpha_{i})=\int\limits_{\Sigma}\omega_{i}-t_{i}c_{1}(L_{i})[\Sigma].$

The result follows immediately by comparing equations (2.34), (2.35) and
(2.36). ∎

The following result is an immediate consequence of piecewise linearity of the
Duistermaat-Heckman function of a compact complexity one $T$\- space and of
Proposition 2.59.

######  Corollary 2.64.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity one $T$-space. The
subset

(2.37) $\\{(x,t)\in\mathfrak{t}^{*}\times\mathbb{R}\mid
x\in\Phi(M)\,,\,t\in[0,DH\left(M,\omega,\Phi\right)(x)]\\}$

is a convex polytope in $\mathfrak{t}^{*}\times\mathbb{R}$.

The convex polytope of Corollary 2.64 and the Duistermaat-Heckman function of
a compact complexity one $T$-space are equivalent in the sense that knowing
one allows to reconstruct the other. To emphasize the combinatorial nature of
the problem we study, we introduce the following notion.

######  Definition 2.65.

The convex polytope (2.37) of a compact complexity one $T$-space
$\left(M,\omega,\Phi\right)$ is called the Duistermaat-Heckman polytope of
$\left(M,\omega,\Phi\right)$.

######  Remark 2.66.

Suppose that $\left(M,\omega,\Phi\right)$ is a compact complexity one
$T$-space obtained by restricting a complexity zero $T\times S^{1}$-action on
$(M,\omega)$ to the subtorus $T\times\\{1\\}$. The moment polytope of the
complexity zero action need not agree with the Duistermaat-Heckman polytope of
$\left(M,\omega,\Phi\right)$. However, the two are related as follows. If
$\Delta$ denotes the moment polytope of the complexity zero $T\times
S^{1}$-action, we have that

$\Delta=\\{(x,y)\in\Phi^{\prime}(M)\times\mathbb{R}\mid
y\in[p_{min}(x),p_{max}(x)]\\},$

whereas, combining Example 2.52 and (2.37), the Duistermaat-Heckman polytope
of $\left(M,\omega,\Phi\right)$ equals

$\\{(x,t)\in\mathfrak{t}^{*}\times\mathbb{R}\mid
x\in\Phi(M)\,,\,t\in[0,p_{max}(x)-p_{min}(x)]\\}.$

Finally, we observe that the latter can be obtained from the former by
applying a piecewise integral affine transformation of
$\mathfrak{t}^{*}\oplus\mathbb{R}$, where the lattice is
$\ell^{*}\oplus\mathbb{Z}$ (see Figure 2.4).

Figure 2.4. An example in which the moment polytope of the compact symplectic
toric manifold does not agree with the Duistermaat-Heckman polytope of the
complexity one space constructed as in Remark 2.66.

###  2.3. Compact complexity preserving Hamiltonian $T$-spaces

In this section, we introduce a class of Hamiltonian $T$-spaces that enjoy
special properties, which are also enjoyed by compact symplectic toric
manifolds (see Corollary 2.68, Proposition 2.73 and Corollary 2.71). We begin
with the following result, which extends [49, Corollary 3.1, Lemma 3.3 and
Remark 3.4].

######  Proposition 2.67.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity $k$ $T$-space. If
$N\subset M^{T}$ is a fixed submanifold with $\dim N=2k$, then $\Phi(N)$ is a
vertex of $\Phi(M)$. Moreover, for every face $\mathcal{F}$ of $\Phi(M)$ that
contains $\Phi(N)$, the sheet
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is stabilized by a
connected subgroup and has complexity equal to $k$.

###### Proof.

Since $\dim N=2k$, by Remark 2.28, $(N,\omega_{N})$ is regular and every point
in $N$ is regular. Hence, by Remark 2.27, the $T$-action on the normal bundle
to $N$ is toric. By Corollary 2.34, $\Phi(N)$ is a vertex of $\Phi(M)$.

Let $\mathcal{F}$ be a face of $\Phi(M)$ that contains $\Phi(N)$ and let $p\in
N\cap M_{\mathcal{F}}$. By Corollary 2.44, there exist points arbitrarily
close to $p$ that have stabilizer equal to $H_{\mathcal{F}}$, the stabilizer
of $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$. Since $p$ is
regular and all stabilizers in a regular local model are connected,
$H_{\mathcal{F}}$ is connected by Theorem 2.8. Finally, we observe that, since
$(N,\omega_{N})$ is a fixed submanifold and $N\subseteq M_{\mathcal{F}}$,
$(N,\omega_{N})$ is a sheet in
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$. Since $\dim N=2k$
and $N\subset M^{T/H_{\mathcal{F}}}_{\mathcal{F}}$, by Proposition 2.26, the
complexity of $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is
at least $k$. On the other hand,
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is a sheet in
$\left(M,\omega,\Phi\right)$ and the complexity of the latter is $k$. Hence,
by Proposition 2.26, the complexity of
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is at most $k$. ∎

######  Corollary 2.68.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity $k$ $T$-space. The
following are equivalent:

  1. (i)

for each face $\mathcal{F}$ of $\Phi(M)$, the complexity of the sheet
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ equals $k$;

  2. (ii)

for each face $\mathcal{F}$ of $\Phi(M)$ of codimension $r$, $M_{\mathcal{F}}$
has maximal dimension, i.e., $\dim M_{\mathcal{F}}=2n-2r$;

  3. (iii)

for each vertex $v$ of $\Phi(M)$, $\Phi^{-1}(v)$ has maximal dimension, i.e.,
$\dim\Phi^{-1}(v)=2k$.

###### Proof.

For any face $\mathcal{F}$ of $\Phi(M)$ of codimension $r$, the complexity of
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ equals that of
$\left(M,\omega,\Phi\right)$ if and only if $\dim M_{\mathcal{F}}=\dim M-2r$.
This shows that (i) and (ii) are equivalent. Since vertices are faces of
maximal codimension, (ii) implies (iii). The converse follows from Proposition
2.67. ∎

Motivated by Corollary 2.68, we introduce the following terminology.

######  Definition 2.69.

A compact Hamiltonian $T$-space is said to be complexity preserving if it
satisfies any (and hence all) of the conditions (i) – (iii) in Corollary 2.68.

compact complexity preserving Hamiltonian $T$-spaces generalize compact
symplectic toric manifolds.

######  Remark 2.70.

By Corollary 2.68, if $\left(M,\omega,\Phi\right)$ is a compact complexity
preserving Hamiltonian $T$-space, then, for every face $\mathcal{F}$ of
$\Phi(M)$, so is $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$.

The following result describes a property of the Duistermaat-Heckman function
of compact complexity preserving Hamiltonian $T$-spaces and is an immediate
consequence of Propositions 2.58, 2.67 and Remark 2.70.

######  Corollary 2.71.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity $k$ $T$-space and
suppose that $N\subset M^{T}$ is a fixed submanifold with $\dim N=2k$. For any
face $\mathcal{F}$ of $\Phi(M)$ containing $\Phi(N)$,

(2.38)
$DH\left(M,\omega,\Phi\right)|_{\mathcal{F}}=DH(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}}),$

where $(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is the sheet
given in (2.19). In particular, if $\left(M,\omega,\Phi\right)$ is complexity
preserving, then (2.38) holds for all faces of $\Phi(M)$.

To conclude this section, we prove the following result, which we need in
Section 4.1.

######  Lemma 2.72.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity preserving
Hamiltonian $T$-space of positive complexity. If there are no singular values
of $\Phi$ contained in the interior of $\Phi(M)$, then the action has no
isolated fixed points.

###### Proof.

Suppose that $p\in M^{T}$ is isolated. By condition (iii) in Corollary 2.68,
$\Phi(p)$ is not a vertex. Hence, if $\mathcal{F}$ is the face of smallest
dimension in which $\Phi(p)$ lies, then $\dim\mathcal{F}\geq 1$. By part (i)
of Corollary 2.34, an open neighborhood of $\Phi(p)$ in $\Phi(M)$ can be
identified with an open neighborhood of
$(0,0)\in\mathbb{R}^{\dim\mathcal{F}}\times\mathbb{R}^{d-\dim\mathcal{F}}$ in
$\mathbb{R}^{\dim\mathcal{F}}\times\mathcal{C}^{\prime}_{p}$, where
$\mathcal{C}^{\prime}_{p}$ is the proper cone in the proof of part (ii) of
Corollary 2.34. We observe that, since $\dim\mathcal{F}\geq 1$, the subset
$\\{0\\}\times\mathcal{C}^{\prime}_{p}$ intersects the interior of
$\mathbb{R}^{\dim\mathcal{F}}\times\mathcal{C}^{\prime}_{p}$. Choose
$d-\dim\mathcal{F}$ linearly independent isotropy weights of $p$ that span
$\mathcal{C}^{\prime}_{p}$ and, if needed, complete this set with
$\dim\mathcal{F}-1$ linearly independent isotropy weights of $p$ whose span is
contained in $\mathbb{R}^{\dim\mathcal{F}}$. The span of these isotropy
weights $\alpha_{1},\ldots,\alpha_{d-1}$ satisfies

(2.39) $\left(\Phi(p)+\mathbb{R}_{\geq
0}\langle\alpha_{1},\ldots,\alpha_{d-1}\rangle\right)\cap\mathrm{Int}(\Phi(M))\neq\emptyset,$

where $\mathrm{Int}(\Phi(M))$ denotes the interior of $\Phi(M)$.

By Theorem 2.8, we may identify a $T$-invariant neighborhood of $p$ with a
$T$-invariant neighborhood of $0\in\mathbb{C}^{n}$ so that $\Phi$ becomes the
map

(2.40)
$(z_{1},\ldots,z_{n})\mapsto\pi\sum\limits_{i=1}^{n}\alpha_{i}|z_{i}|^{2}+\Phi(p).$

Moreover, by part (i) of Corollary 2.34, an open neighborhood of $\Phi(p)$ in
$\Phi(M)$ can be identified with an open neighborhood of $\Phi(p)$ in the
image of the map of equation (2.40).

By (2.39), the affine hyperplane
$\Phi(p)+\mathbb{R}\langle\alpha_{1},\ldots,\alpha_{d-1}\rangle$ intersects
$\mathrm{Int}(\Phi(M))$. All values in this intersection have a one-
dimensional stabilizer: this is because

$\dim\left(\mathrm{Ann}\left(\mathbb{R}\langle\alpha_{1}\rangle\right)\cap\ldots\cap\mathrm{Ann}\left(\mathbb{R}\langle\alpha_{d-1}\rangle\right)\right)=1,$

since $\alpha_{1},\ldots,\alpha_{d-1}$ are linearly independent. Hence, there
is a singular value of $\Phi$ in $\mathrm{Int}(\Phi(M))$, a contradiction. ∎

####  2.3.1. Moment polytopes for compact complexity preserving Hamiltonian
$T$-spaces

In this subsection, we characterize the moment map image of complexity
preserving compact Hamiltonian $T$-spaces (see Proposition 2.73 below). To
this end, given a polytope $\Delta\subset\mathfrak{t}^{*}$ and a vertex
$v\in\Delta$, we say that $\Delta$ is smooth at $v$ if

  * •

there are exactly $d$ edges $e_{1},\ldots,e_{d}$ that are incident to $v$, and

  * •

there exists a basis $\alpha_{1},\ldots,\alpha_{d}$ of $\ell^{*}$ such that
$\alpha_{i}$ is a tangent vector to the edge $e_{i}$ for all $i=1,\ldots,d$.

A polytope $\Delta\subset\mathfrak{t}^{*}$ is smooth at $v$ if and only if the
collection of inward (or outward) normals to the facets of $\Delta$ that
contain $v$ can be chosen to be a basis of $\ell$. We say that a polytope
$\Delta$ is Delzant if it is smooth at every vertex.

The moment map image of a compact symplectic toric manifold is a Delzant
polytope and, conversely, every Delzant polytope arises as such an image (see
[9]). In general, this fails to be true in higher complexity. However, under
the additional hypothesis of complexity preserving, the following result
holds.

######  Proposition 2.73.

The moment map image of a compact complexity preserving Hamiltonian $T$-space
is a Delzant polytope in $\mathfrak{t}^{*}$.

Conversely, for every Delzant polytope $\Delta$ in $\mathfrak{t}^{*}$ and for
every integer $k\geq 0$, there exists a compact complexity preserving
Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ of complexity $k$ such that
$\Phi(M)=\Delta$.

###### Proof.

Let $\left(M,\omega,\Phi\right)$ be a compact complexity preserving
Hamiltonian $T$-space of complexity $k$. Let $v\in\Phi(M)$ be a vertex. By
Corollary 2.68, $\dim\Phi^{-1}(v)=2k$. Let $\alpha_{1},\ldots,\alpha_{n}$ be
the isotropy weights of $\Phi^{-1}(v)$ (see Remark 2.23). Since
$\dim\Phi^{-1}(v)=2k$, precisely $k$ weights are zero. Without loss of
generality, we may assume that $\alpha_{n-k+1},\ldots,\alpha_{n}=0$. By part
(i) of Corollary 2.34 an open neighborhood of $v$ in $\Phi(M)$ looks like an
open neighborhood of $0$ in

(2.41) $\mathbb{R}_{\geq
0}\text{-span}\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}=\mathbb{R}_{\geq
0}\text{-span}\left\\{\alpha_{1},\dots\alpha_{n-k}\right\\}.$

Since the complexity of $\left(M,\omega,\Phi\right)$ is $k$, $d=\dim T=n-k$.
Hence, by equation (2.41), there are exactly $d$ edges that are incident to
$v$. Moreover, by Remark 2.10, the $\mathbb{Z}$-span of
$\alpha_{1},\ldots,\alpha_{d}$ equals $\ell^{*}$. Hence, $\Phi(M)$ is smooth
at $v$ and the first statement follows.

Conversely, fix an integer $k\geq 0$ and suppose that $\Delta$ is a Delzant
polytope in $\mathfrak{t}^{*}$. By the classification of compact symplectic
toric manifolds (see [9]), there exists a compact complexity zero $T$-space
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ such that
$\Phi^{\prime}(M^{\prime})=\Delta$. Let
$(M^{\prime\prime},\omega^{\prime\prime})$ be a closed symplectic manifold of
dimension $2k$. Consider the $T$-action on $M:=M^{\prime}\times
M^{\prime\prime}$ given by taking the product of the above $T$-action on
$M^{\prime}$ with the trivial $T$-action on $M^{\prime\prime}$. This action is
Hamiltonian for the symplectic form $\omega$ obtained by summing the pullbacks
to $M$ of $\omega^{\prime}$ and $\omega^{\prime\prime}$ along the projections.
A moment map for this $T$-action is given by the pullback to $M$ of
$\Phi^{\prime}$ along the projection $M\to M^{\prime}$; we denote this moment
map by $\Phi$. Then $(M,\omega,\Phi)$ is a compact complexity preserving
complexity $k$ $T$-space with moment map image given by $\Delta$, as desired.
∎

###  2.4. Compact tall complexity one $T$-spaces

In this section we introduce an important class of compact complexity one
$T$-spaces.

######  Definition 2.74.

A compact complexity one $T$-space $\left(M,\omega,\Phi\right)$ is called tall
if no reduced space is a point.

To shed light on Definition 2.74 we observe that, if
$\left(M,\omega,\Phi\right)$ is a compact complexity one $T$-space, then the
reduced space $M_{x}$ is homeomorphic to a closed, connected orientable
surface for any $x\in\Phi(M)_{\mathrm{reg}}$ (see Section 2.1.1). If
$\left(M,\omega,\Phi\right)$ is tall, then this holds for all $x\in\Phi(M)$
(see [28, Proposition 6.1]). Moreover, the following result holds.

######  Proposition 2.75.

A compact complexity one $T$-space is tall if and only if it is complexity
preserving.

###### Proof.

Let $\left(M,\omega,\Phi\right)$ be a compact tall complexity one $T$-space
$\left(M,\omega,\Phi\right)$ and let $v\in\Phi(M)$ be a vertex. By Remark
2.28, $\Phi^{-1}(v)$ is either a fixed point or a fixed surface. Since the
reduced space at $v$ can be identified with $\Phi^{-1}(v)$ and since
$\left(M,\omega,\Phi\right)$ is tall, $\Phi^{-1}(v)$ has dimension two. Hence,
since $\left(M,\omega,\Phi\right)$ has complexity one, it satisfies property
(iii) in Corollary 2.68; therefore, it is complexity preserving. Conversely,
if $\left(M,\omega,\Phi\right)$ is complexity preserving, then it satisfies
property (i) in Corollary 2.68. Hence, by [30, Corollary 2.6], no reduced
space is a point and $\left(M,\omega,\Phi\right)$ is tall. ∎

In [28, 29, 30] the authors classify tall complexity one $T$-spaces333In loc.
cit. the authors consider a more general class of tall complexity one spaces,
namely those for which $M$ is connected but not necessarily compact and such
that there exists an open convex set $\mathcal{T}\subseteq\mathfrak{t}^{*}$
containing the image of the moment map with the property that $\Phi\colon
M\to\mathcal{T}$ is proper. However, we state all results in loc. cit. only in
the compact case.. Below we recall this classification. Henceforth, we fix a
compact complexity one $T$-space $\left(M,\omega,\Phi\right)$. As a
consequence of [28, Corollary 9.7] or [35], any two reduced spaces of
$\left(M,\omega,\Phi\right)$ are homeomorphic. This motivates introducing the
following notion.

######  Definition 2.76.

The genus of a compact tall complexity one $T$-space
$\left(M,\omega,\Phi\right)$ is the genus of the reduced space $M_{x}$ for any
$x\in\Phi(M)$.

The following result is a stepping stone for the classification of compact
tall complexity one $T$-spaces (see Proposition 2.2 in [29], Proposition 1.2
and Remark 1.9 in [30])

######  Proposition 2.77.

If $\left(M,\omega,\Phi\right)$ is a compact tall complexity one $T$-space,
then there exist a closed oriented surface $\Sigma$ and a map $f\colon
M/T\to\Sigma$ such that

(2.42) $(\overline{\Phi},f)\colon M/T\longrightarrow\Phi(M)\times\Sigma$

is a homeomorphism and the restriction $f:\Phi^{-1}(x)/T\to\Sigma$ is
orientation-preserving for any $x\in\Phi(M)$. Given two such maps $f$ and
$f^{\prime}$, there exists an orientation-preserving homeomorphism
$\xi\colon\Sigma^{\prime}\to\Sigma$ such that $f$ is homotopic to $\xi\circ
f^{\prime}$ through maps that induce homeomorphisms
$M/T\to\Phi(M)\times\Sigma$.

By Proposition 2.77, the genus of $\left(M,\omega,\Phi\right)$ is the genus of
$\Sigma$.

The next invariant of tall complexity one $T$-spaces is related to the
exceptional orbits (see Remark 2.16), and is introduced below. To this end, we
observe that given a closed surface $\Sigma$ and a map $f:M/T\to\Sigma$ as in
Proposition 2.77, its restriction to $M_{\mathrm{exc}}$ makes
$(\overline{\Phi},f):M_{\mathrm{exc}}\to\Phi(M)\times\Sigma$ injective.

######  Definition 2.78.

Let $(M,\omega,\Phi),(M^{\prime},\omega^{\prime},\Phi^{\prime})$ be compact
tall complexity one $T$-spaces and let $\Sigma,\Sigma^{\prime}$ be closed
oriented surfaces.

  * •

A painting of $\boldsymbol{(M,\omega,\Phi)}$ is a map
$f:M_{\mathrm{exc}}\to\Sigma$ such that
$(\overline{\Phi},f):M_{\mathrm{exc}}\to\Phi(M)\times\Sigma$ is injective.

  * •

An isomorphism of exceptional orbits is a homeomorphism $i:M_{\mathrm{exc}}\to
M^{\prime}_{\mathrm{exc}}$ satisfying
$\overline{\Phi}=\overline{\Phi}^{\prime}\circ i$ that sends each orbit to an
orbit with the same symplectic slice representation.

  * •

A painting $f:M_{\mathrm{exc}}\to\Sigma$ of $(M,\omega,\Phi)$ is equivalent to
a painting $f^{\prime}:M^{\prime}_{\mathrm{exc}}\to\Sigma^{\prime}$ of
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ if there exists an isomorphism of
exceptional orbits $i:M_{\mathrm{exc}}\to M_{\mathrm{exc}^{\prime}}$ and an
orientation-preserving homeomorphism $\xi:\Sigma\to\Sigma^{\prime}$ such that
$f^{\prime}\circ i$ and $\xi\circ f$ are homotopic through paintings.

By Proposition 2.77, we can associate an equivalence class of paintings to a
compact tall complexity one $T$-space (see [29, Section 2]). For our purposes,
it is useful to introduce the following terminology.

######  Definition 2.79.

Let $\left(M,\omega,\Phi\right)$ be a tall, compact complexity one $T$-space.
The equivalence class of paintings $[f]$ associated to
$\left(M,\omega,\Phi\right)$ is trivial if there exists a painting
$f:M_{\mathrm{exc}}\to\Sigma$ representing $[f]$ that is constant on each
connected component of $M_{\mathrm{exc}}$.

The classification of compact tall complexity one $T$-spaces is as follows.

######  Theorem 2.80.

(Karshon–Tolman, Theorem 1 in [29], and Theorem 1.8 and Remark 1.9 [30]) Two
compact tall complexity one $T$-spaces are isomorphic if and only if they have
equal genera, equal Duistermaat-Heckman measures, and equivalent paintings.

The invariants of a compact tall complexity one $T$-space determine the moment
map image, as it is the support of the Duistermaat-Heckman measure (cf. [30,
Theorem 1.8]).

##  3\. Compact monotone Hamiltonian $T$-spaces

In this section we use ideas and techniques from equivariant cohomology,
referring the reader to [22] for details and background.

###  3.1. The weight sum formula

In this paper we are mostly concerned with compact Hamiltonian $T$-spaces
satisfying the following condition.

######  Definition 3.1.

A symplectic manifold $(M,\omega)$ is monotone if there exists
$\lambda\in\mathbb{R}$ such that $c_{1}=\lambda[\omega]$, where $c_{1}$ is the
first Chern class of $(M,\omega)$. It is positive monotone if $\lambda>0$.

######  Remark 3.2.

  1. (1)

If $(M,\omega)$ is compact and monotone, since $[\omega]\neq 0$, then
$\lambda$ in Definition 3.1 is unique.

  2. (2)

Let $(M,\omega)$ be a monotone symplectic manifold and let
$\Psi:(M^{\prime},\omega^{\prime})\to(M,\omega)$ be a symplectomorphism. Since
$\Psi$ pulls back almost complex structures that are compatible with $\omega$
to almost complex structures that are compatible with $\omega^{\prime}$,
$(M^{\prime},\omega^{\prime})$ is monotone. Moreover, if $(M,\omega)$ is
compact and if $\lambda,\lambda^{\prime}\in\mathbb{R}$ are such that
$c_{1}=\lambda[\omega]$ and
$c_{1}^{\prime}=\lambda^{\prime}[\omega^{\prime}]$, then
$\lambda=\lambda^{\prime}$.

If $(M,\omega)$ is such that $H^{2}(M;\mathbb{R})=\mathbb{R}$, then it is
monotone (e.g. $\mathbb{C}P^{n}$). In general, (positive) monotonicity is very
restrictive. In the presence of a Hamiltonian torus action, the following
result holds.

######  Proposition 3.3.

If $(M,\omega)$ is compact and monotone, and admits an effective Hamiltonian
$T$-action, then $(M,\omega)$ is positive monotone.

###### Proof.

The proof follows mutatis mutandis that of [16, Lemma 5.2], in which it is
assumed that $M^{S^{1}}$ is discrete (see [16, Definition 3.1]). Let $H\leq T$
be a one dimensional subtorus and let $\phi:M\to\mathfrak{h}^{*}$ be the
induced moment map. We identify $H\simeq S^{1}$ and consider $(M,\omega,\phi)$
as a Hamiltonian $S^{1}$-space. We observe that $\phi\colon
M\to(\text{Lie}(S^{1}))^{*}$ is a Morse-Bott function; moreover, by (2.1), the
isotropy weights in the positive normal bundle to a fixed point are positive
(cf. [16, Proof of Lemma 5.2]). Therefore, if $F_{\min}$ (respectively
$F_{\max}$) denotes a fixed component on which $\phi$ attains its minimum
(respectively maximum), all the isotropy weights in the normal bundle to
$F_{\min}$ (respectively $F_{\max}$) are positive (respectively negative).
Moreover, even if some of the isotropy weights of $p_{\min}\in F_{\min}$
(respectively at $p_{\max}\in F_{\max}$) are zero, by the effectiveness of the
action some of them must be different from zero. Hence, the sum of the
isotropy weights of $p_{\min}$ (resp. $p_{\max}$) is strictly positive
(respectively strictly negative). To complete the proof, it is enough to
consider the equivariant extensions of $[\omega]$ and $c_{1}$ in the
equivariant cohomology ring of $M$, which are respectively $[\omega-\phi]$ and
$c_{1}^{S^{1}}$, and to compare them at $p_{\min}$ and $p_{\max}$ to deduce
that $\lambda$ must be positive (see equation (5.1) in [16, Proof of Lemma
5.2]). ∎

Throughout this paper, a Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ is
monotone if $(M,\omega)$ is. The following result is an immediate consequence
of Remark 3.2 and Proposition 3.3.

######  Corollary 3.4.

If $(M,\omega,\Phi)$ is a compact monotone Hamiltonian $T$-space, then there
exists a unique $\lambda>0$ such that $c_{1}=[\lambda\omega]$.

The next proposition extends [16, Remark 3.11].

######  Proposition 3.5.

If $\left(M,\omega,\Phi\right)$ is a compact Hamiltonian $T$-space with
$c_{1}=[\omega]$, then there exists a unique $w\in\mathfrak{t}^{*}$ such that
the moment map $\widetilde{\Phi}:=\Phi+w$ satisfies the weight sum formula,
i.e.,

(3.1) $\widetilde{\Phi}(p)=-\sum_{j=1}^{n}\alpha_{j},\quad\text{for all }p\in
M^{T}\,,$

where $\alpha_{1},\ldots,\alpha_{n}\in\ell^{*}$ are the isotropy weights of
$p$.

###### Proof.

Since $c_{1}=[\omega]$ and since the action is Hamiltonian, there exists a
unique $w\in\mathfrak{t}^{*}$ such that

$c_{1}^{T}+w=[\omega-\Phi].$

Thus the moment map $\widetilde{\Phi}=\Phi+w$ satisfies
$c_{1}^{T}=[\omega-\widetilde{\Phi}]$. Since $M$ is compact and the action is
Hamiltonian, there exists $p\in M^{T}$. The equality in (3.1) is obtained by
comparing these two equivariant cohomology classes at $p\in M^{T}$ and
observing that $c_{1}^{T}(p)=\sum_{j=1}^{n}\alpha_{j}$. ∎

######  Remark 3.6.

Let $(M,\omega,\Phi)$ be a compact Hamiltonian $T$-space with
$c_{1}=[\omega]$. If $(M^{\prime},\omega^{\prime},\Phi^{\prime})$ is
isomorphic to $(M,\omega,\Phi)$, then $c_{1}^{\prime}=[\omega^{\prime}]$ by
Remark 3.2. Let
$\Psi:(M,\omega,\Phi)\to(M^{\prime},\omega^{\prime},\Phi^{\prime})$ be an
isomorphism. By Remark 2.2, $\Psi$ is equivariant. Hence, by (3.1), if
$w,w^{\prime}\in\mathfrak{t}^{*}$ are as in Proposition 3.5 for $\Phi$ and
$\Phi^{\prime}$ respectively, then $w=w^{\prime}$.

######  Definition 3.7.

A compact monotone Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ is
normalized if

  1. (i)

$c_{1}=[\omega]$, and

  2. (ii)

the moment map $\Phi$ satisfies the weight sum formula (3.1).

In this case we call $\left(M,\omega,\Phi\right)$ a normalized monotone
Hamiltonian $T$-space.

Since the isotropy weights of a fixed point lie in $\ell^{*}$, Proposition 3.5
has the following immediate consequence.

######  Corollary 3.8.

If $\left(M,\omega,\Phi\right)$ is a normalized monotone Hamiltonian
$T$-space, then $[\omega]\in H^{2}(M;\mathbb{Z})$ and, for any $p\in M^{T}$,
$\Phi(p)\in\ell^{*}$.

The following result is an immediate consequence of Corollary 3.4 and
Proposition 3.5.

######  Corollary 3.9.

If $\left(M,\omega,\Phi\right)$ is a compact monotone Hamiltonian $T$-space,
then there exist unique $\lambda>0$ and $w\in\mathfrak{t}^{*}$ such that
$(M,\lambda\omega,\lambda\Phi+w)$ is normalized monotone.

Classifying compact monotone Hamiltonian $T$-space is almost equivalent to
classifying normalized monotone Hamiltonian $T$-spaces. More precisely, the
following holds.

######  Lemma 3.10.

Let $(M,\omega,\Phi)$, $(M^{\prime},\omega^{\prime},\Phi^{\prime})$ be compact
monotone Hamiltonian $T$-spaces. Let $\lambda,\lambda^{\prime}>0$ and
$v,v^{\prime}\in\mathfrak{t}^{*}$ be as in Corollary 3.9. Then
$(M,\omega,\Phi)$ and $(M^{\prime},\omega^{\prime},\Phi^{\prime})$ are
isomorphic if and only if $\lambda=\lambda^{\prime}$, $v=v^{\prime}$ and
$(M,\lambda\omega,\lambda\Phi+v)$ is isomorphic to
$(M^{\prime},\lambda^{\prime}\omega^{\prime},\lambda^{\prime}\Phi^{\prime}+v^{\prime})$.

###### Proof.

Suppose that $(M,\omega,\Phi)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ are isomorphic. Let
$\Psi:(M,\omega)\to(M^{\prime},\omega^{\prime})$ be a symplectomorphism such
that $\Phi^{\prime}\circ\Psi=\Phi$. By part (2) of Remark 3.2 and by Remark
3.6, $\lambda=\lambda^{\prime}$ and $v=v^{\prime}$. Hence,
$\Psi:(M,\lambda\omega)\to(M^{\prime},\lambda^{\prime}\omega^{\prime})$ is a
symplectomorphism and
$(\lambda^{\prime}\Phi^{\prime}+v^{\prime})\circ\Psi=\lambda\Phi+v$, i.e.,
$\Psi$ is an isomorphism between $(M,\lambda\omega,\lambda\Phi+v)$ and
$(M^{\prime},\lambda^{\prime}\omega^{\prime},\lambda^{\prime}\Phi^{\prime}+v^{\prime})$.
The converse is entirely analogous and is left to the reader. ∎

###  3.2. Moment polytopes of monotone complexity preserving Hamiltonian
$T$-spaces

We recall that a polytope $\Delta$ in $\mathfrak{t}^{*}$ can be described by
its minimal representation (see Section 1.3):

$\Delta=\bigcap_{i=1}^{l}\,\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{i}\rangle\geq c_{i}\\}$

for some inward normals $\nu_{1},\ldots,\nu_{l}\in\mathfrak{t}$ and
$c_{1},\ldots,c_{l}\in\mathbb{R}$. Such a polytope $\Delta$ is integral if its
vertices belong to $\ell^{*}$.

######  Remark 3.11.

If $\Delta$ is integral, then it is possible to choose the inward normal
$\nu_{i}$ so that it is a primitive element of $\ell$, for every
$i=1,\ldots,l$. The corresponding constants $c_{i}$’s are therefore uniquely
determined by this choice of $\nu_{i}$’s.

######  Definition 3.12.

A polytope $\Delta\subset\mathfrak{t}^{*}$ is reflexive if it is integral and
$\nu_{i}\in\ell$ in its minimal representation is primitive with corresponding
$c_{i}=-1$, for all $i=1,\ldots,l$.

The following result is an immediate consequence of Definition 3.12 and is
stated below without proof (see [23, Proposition 7.3]).

######  Lemma 3.13.

For any reflexive polytope the origin is the only interior lattice point.