# Monotone Symplectic Six-Manifolds that admit a Hamiltonian GKM Action are
diffeomorphic to Smooth Fano Threefolds

Isabelle Charton1 and Liat Kessler2

(1University of Haifa, Department of Mathematics
Campus at Mount Carmel, Mount Carmel 3498838, Haifa, Israel
<EMAIL_ADDRESS>
2University of Haifa, Department of Mathematics, Physics and Computer Science
Campus at Oranim, Tivon 36006, Haifa, Israel
<EMAIL_ADDRESS>
###### Abstract

Let $(M,\omega)$ be a compact symplectic manifold with a Hamiltonian GKM
action of a compact torus. We formulate a positive condition on the space;
this condition is satisfied if the underlying symplectic manifold is monotone.
The main result of this article is that the underlying manifold of a positive
Hamiltonian GKM space of dimension six is diffeomorphic to a smooth Fano
threefold. We prove the main result in two steps.

In the first step, we analyze the equivariant cohomology of (maybe non-
positive) Hamiltonian GKM spaces in any dimension with tools provided by
Kirwan: the so-called Kirwan classes. We show that if the complexity of the
action is zero or one, or if the dimension of the manifold is six, then the
equivariant and the ordinary cohomology with integer coefficients are
determined by the GKM graph. This analysis, in combination with a
classification result by Jupp, Wall and Z̆ubr for certain six-manifolds, shows
that the diffeomorphism type of a symplectic six-manifold with a Hamiltonian
GKM action is determined by the associated GKM graph.

In the second step, based on results by Godinho and Sabatini, we give a
complete list of the GKM graphs of positive Hamiltonian GKM spaces of
dimension six. We deduce that any such GKM graph is isomorphic to a GKM graph
of a smooth Fano threefold.

###### Contents

  1. 1 Introduction
    1. 1.1 Results
    2. 1.2 Structure of the Article
  2. 2 Preliminaries on Equivariant Cohomology and Hamiltonian GKM Spaces
    1. 2.1 Equivariant Cohomology
      1. 2.1.1 Equivariant Chern Classes
      2. 2.1.2 The ABBV Localization Formula
    2. 2.2 Hamiltonian GKM Spaces
      1. 2.2.1 GKM Graphs of Symplectic Toric Manifolds
      2. 2.2.2 Projections of GKM Graphs
      3. 2.2.3 Generic Vectors and Morse Theory
  3. 3 Relations between the GKM Graph and the Equivariant Cohomology
    1. 3.0.1 Intermezzo: Kirwan Classes
    2. 3.1 Proof of Theorem 1.6
    3. 3.2 Proof of Theorem 1.7
  4. 4 Positive Hamiltonian GKM Spaces
    1. 4.1 An Upper Bound for the Number of Fixed Points in Dimension Six
    2. 4.2 Special Kirwan Classes
  5. 5 Constructing (Abstract) GKM Graphs
    1. 5.1 Abstract GKM Graphs
      1. 5.1.1 Relations between Hamiltonian and Abstract GKM Graphs
    2. 5.2 GKM Skeletons
      1. 5.2.1 Isomorphic GKM Skeletons
      2. 5.2.2 The Kernel Condition (K1)
      3. 5.2.3 The Kernel Condition (K2)
      4. 5.2.4 If the Kernel Condition (K2) is not satisfied
    3. 5.3 Positive Hamiltonian Abstract $(3,d)$-GKM Graphs
  6. 6 About the Computer Programs
    1. 6.1 Proof of Theorem 1.8
  7. A Positive Hamiltonian GKM Graphs in Dimension Six
  8. B Remarks about the Technical Condition (1.2)

##  1 Introduction

A symplectic manifold $(M,\omega)$ admits an almost complex structure $J\colon
TM\rightarrow TM$, $J^{2}=-\operatorname{Id}$, that is compatible with the
symplectic form $\omega$, i.e., $\omega(\cdot,J\cdot)$ is a Riemannian metric.
By picking a compatible almost complex structure, we can consider the tangent
bundle $TM$ as a complex vector bundle over $M$. We denote by $c_{i}(M)$ the
$i$-th Chern class of $(M,\omega)$; the Chern classes are well defined since
the space of compatible almost complex structures is contractible. A compact
symplectic manifold $(M,\omega)$ is called monotone if
$c_{1}(M)=r\cdot[\omega]$ for some $r\in\mathbb{R}$; $(M,\omega)$ is called
positive monotone if $r>0$.

The algebraic counterparts of positive monotone symplectic manifolds are
smooth Fano varieties. A smooth Fano variety $X$ is a compact complex manifold
whose anticanonical line bundle $K_{X}^{-1}$ is ample. The ampleness of
$K_{X}^{-1}$ implies that there exists a holomorphic embedding
$i:X\hookrightarrow\mathbb{C}P^{N}$ such that
$(K_{X}^{-1})^{k}=i^{*}\mathcal{O}(1)$ for some $N>0$ and $k>0$, known as
polarisation of $X$ by $K_{X}^{-1}$. The form $i^{*}(\omega_{FS})$ is a
symplectic form on $X$, where $\omega_{FS}$ is the Fubini-Study form on
$\mathbb{C}P^{N}$. Moreover, the almost complex structure $J\colon TM\to TM$
induced from the complex structure on $X$ is compatible with
$i^{*}(\omega_{FS})$. Since $c_{1}(X)=c_{1}(K_{X}^{-1})$ and
$c_{1}(i^{*}\mathcal{O}(1))=[i^{*}(\omega_{FS})]$, the symplectic manifold
$(X,i^{*}(\omega_{FS}))$ is positive monotone.

Since smooth Fano varieties carry a lot of geometric and algebraic structures
it is important to understand in which content positive monotone symplectic
manifolds are similar to smooth Fano varieties. In real dimensions two and
four, it is known that any positive monotone symplectic manifold is
diffeomorphic to a smooth Fano variety. In dimension two, this fact follows
from the work of Morse [35]. In dimension four, this fact is a result of Ohta
and Ono [36] based on works of Gromov [20], McDuff [31] and Taubes [39]. In
dimension greater or equal to twelve, Fine and Panov [13] provide examples of
positive monotone symplectic manifolds that are not diffeomorphic to a smooth
Fano variety. In dimensions six, eight, and ten, the question of whether any
positive monotone symplectic manifold is diffeomorphic to a Fano variety is
open.

If a positive monotone symplectic manifold $(M,\omega)$ admits an integrable
almost complex structure $J\colon TM\rightarrow TM$ that is compatible with
$\omega$, then $M$ endowed with the complex atlas indicated by $J$ is a smooth
Fano variety. This is a consequence of the Kodaira Embedding Theorem, see [32,
Sect. 14.4]. In particular, if a compact symplectic manifold $(M,\omega)$ of
dimension $2n$ is endowed with an effective and Hamiltonian action of a
compact torus $T={(S^{1})}^{n}$ of half the dimension, then $(M,\omega)$
admits a $T$-invariant and integrable almost complex structure $J$ that is
compatible with $\omega$ [9]; if $(M,\omega)$ is positive monotone, then $M$
with the induced complex atlas is a smooth Fano variety. Moreover, it is
enough then that $(M,\omega)$ is monotone; a monotone symplectic manifold that
admits an effective and Hamiltonian action of a compact torus is positive
monotone, see [5, Proposition 3.3] and [17, Lemma 5.2]. Recall that an action
of a compact torus $T={(S^{1})}^{d}$ on a symplectic manifold is Hamiltonian
if there exists a moment map $\phi:M\rightarrow\mathfrak{t}^{*}$, i.e., a
smooth and $T$-invariant map whose codomain is the dual of the Lie algebra
$\mathfrak{t}$ of $T$ such that

$\displaystyle\iota_{X^{\xi}}\omega=-\operatorname{d}\left\langle\phi,\xi\right\rangle\quad\text{for
all }\,\xi\in\mathfrak{t},$

where $X^{\xi}$ is the vector field on $M$ generated by $\xi$ and
$\left\langle,\right\rangle$ is the natural pairing between $\mathfrak{t}^{*}$
and $\mathfrak{t}$. We note that on a simply connected manifold, (as is a
smooth Fano variety), a smooth $T$-action is symplectic iff it is Hamiltonian:
the action is symplectic iff $\iota_{X^{\xi}}\omega$ is closed for all
$\xi\in\mathfrak{t}$ (by Cartan’s formula) and Hamiltonian iff
$\iota_{X^{\xi}}\omega$ is exact for all $\xi\in\mathfrak{t}$. Given a
connected symplectic manifold $(M,\omega)$ endowed with an effective
Hamiltonian $T$-action with moment map $\phi$, we call the quadruple
$(M,\omega,T,\phi)$ a Hamiltonian $\mathbf{T}$-space. Since the orbits of a
Hamiltonian $T$-action on $(M,\omega)$ are isotropic,
$\operatorname{dim}(T)\leq\frac{1}{2}\operatorname{dim}(M)$; we call
$k:=\frac{1}{2}\operatorname{dim}(M)-\operatorname{dim}(T)$ the complexity of
$(M,\omega,T,\phi)$. We also call $(M,\omega,T,\phi)$ a complexity
$\mathbf{k}$ space.

The algebraic counterparts of Hamiltonian $T$-actions on symplectic manifolds
are holomorphic actions of algebraic tori on compact complex manifolds. Recall
that an algebraic torus $T_{\mathbb{C}}=(\mathbb{C}^{*})^{d}$ is the
complexification of a compact torus $T=(S^{1})^{d}$; in particular, $T$ is
contained in $T_{\mathbb{C}}$. If $X$ is a smooth Fano variety endowed with a
holomorphic $T_{\mathbb{C}}$-action, then its anticanonical line bundle is
$T_{\mathbb{C}}$-invariant. Hence, the polarisation of $X$ by $K_{X}^{-1}$ is
$T_{\mathbb{C}}$-equivariant with respect to a $T_{\mathbb{C}}$-representation
on $\mathbb{C}P^{N}$. The induced $T$-action on $\mathbb{C}P^{N}$ is
Hamiltonian with respect to the Fubini-Study symplectic form $\omega_{FS}$.
Hence, also the induced $T$-action on $X$ is Hamiltonian with respect to
$i^{*}(\omega_{FS})$.

Since the underlying manifold of a compact monotone Hamiltonian $T$-space of
complexity zero is diffeomorphic to a smooth Fano variety, it is natural to
investigate the following question.

######  Question 1.1.

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space of positive complexity
whose underlying symplectic manifold is monotone.

Is $M$ diffeomorphic to a smooth Fano variety?

In dimension six, i.e., the lowest dimension in which it is not known if any
positive monotone symplectic manifold is diffeomorphic to a smooth Fano
variety, Fine and Panov stated in [14] the following conjecture.

######  Conjecture 1.2.

Let $(M,\omega)$ be a monotone symplectic manifold of dimension six that
admits an effective and Hamiltonian $S^{1}$-action. Then $M$ is diffeomorphic
to a smooth Fano threefold.

A recent work by Lindsay and Panov [30] provides results that support this
conjecture. In a series [6, 7, 8] of papers, Cho classifies monotone
symplectic manifolds of dimension six admitting an effective and Hamiltonian
$S^{1}$-action that is semifree. He shows that such a space admits an
$S^{1}$-invariant integrable almost complex structure that is compatible with
the symplectic form. Hence, Conjecture 1.2 is true if the action is in
addition semifree. But it is still open if Conjecture 1.2 is true in general.

In this paper, we look at Question 1.1 in case the complexity is one.
Karshon [25] classifies compact complexity one spaces of dimension four, and
shows that such a space admits an invariant compatible integrable almost
complex structure. In dimension six and above, there are complexity one spaces
that do not admit an invariant compatible integrable almost complex structure
[40]. We note that these spaces might admit a non-invariant compatible
integrable almost complex structure [19]. In any dimension, a special class of
complexity one spaces, namely the tall ones, are classified by Karshon and
Tolman [26, 27, 28]. Tall means that for any $x$ in the dual Lie algebra
$\mathfrak{t}^{*}$ the reduced space $\phi^{-1}(x)/T$ is not a point. In [5],
Sabatini, Sepe and the first author study compact monotone tall complexity one
spaces, and show that in such a space the Hamiltonian action extends to an
effective Hamiltonian action of complexity zero, and hence the space admits an
invariant compatible integrable almost complex structure. This result is based
on the classification of Karshon and Tolman. However, so far there is no
classification result for non-tall Hamiltonian $T$-spaces of complexity one in
dimension greater than four. Sabatini and Sepe [37] prove that a compact
complexity one monotone space shares topological properties with smooth Fano
varieties; namely, it is simply connected and its Todd genus is one.

The following is a special case of Conjecture 1.2.

######  Goal 1.3.

Let $(M,\omega,T,\phi)$ be a complexity one space where $(M,\omega)$ is a
monotone symplectic manifold of dimension six. Then $M$ is diffeomorphic to a
smooth Fano threefold.

By [5], this is true if the space is tall. In this article, we prove Goal 1.3
under the assumption that the $T$-action is GKM. A Hamiltonian $T$-space is
GKM if the set $M^{T}$ of fixed points is finite, for each codimensional one
subtorus $H\subset T$ any connected component of $M^{H}$ has at most dimension
two, and the underlying manifold is compact. Note that a Hamiltonian GKM space
is not tall.

In order to achieve Goal 1.3 in the GKM case, we first analyze the equivariant
and the ordinary cohomology rings with integer coefficients of Hamiltonian GKM
spaces in any dimension. In this article we apply the results of this analysis
to Hamiltonian GKM spaces of dimension six to prove that the diffeomorphism
type of the underlying manifold is determined by a combinatorial object
associated to the space, namely the GKM graph. Beyond this application in
dimension six, the results of this analysis might be used in other contexts
for studying Hamiltonian GKM spaces.

We present our results in the following subsection.

###  1.1 Results

Given a Hamiltonian GKM space, one can associate to it canonically a graph,
the so-called GKM graph; see Subsection 2.2. Moreover, any edge $e$ of the GKM
graph is associated in a canonical way to a $2$-dimensional symplectic two-
sphere $S_{e}$ in the underlying symplectic manifold $(M,\omega)$. We call a
Hamiltonian GKM space positive if for each edge $e$ the evaluation of the
first Chern class of $(M,\omega)$ on $S_{e}$ is positive. We introduce the
notion of a positive Hamiltonian GKM space in a careful way at the beginning
of Section 4. In particular, if the underlying symplectic manifold is monotone
then the Hamiltonian GKM space is positive (see Lemma 4.6). The main result of
this article is stated in the following theorem.

######  Theorem 1.4.

Let $(M,\omega,T,\phi)$ be a positive and six-dimensional Hamiltonian GKM
space of complexity one. Then $M$ is diffeomorphic to a smooth Fano threefold.

This theorem supports Goal 1.3. Indeed a direct consequence of Theorem 1.4 and
Lemma 4.6 is the following corollary.

######  Corollary 1.5.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space of complexity one where
$(M,\omega)$ is a monotone symplectic manifold of dimension six. Then $M$ is
diffeomorphic to a smooth Fano threefold.

The proof of Theorem 1.4 relies on two steps, as follows.

In the first step, we relate the equivariant cohomology of such a space to its
GKM graph. For this step, the positive condition is not needed. Due to the
work of Goresky, Kottwitz and MacPherson [21], it is known that the
equivariant cohomology with rational coefficients of a GKM space can be
recovered, in a simple way, from its GKM graph. But in general the analogue
for integer coefficients is not true. We give a sufficient condition for a
Hamiltonian GKM space that ensures that the isomorphism class of its GKM graph
determines its equivariant cohomology with integer coefficients. This
sufficient condition is a technical condition. Let $(M,\omega,T,\phi)$ be a
Hamiltonian GKM space of dimension $2n$. The manifold $M$ admits an almost
complex structure $J:TM\rightarrow TM$ that is $T$-invariant and compatible
with the symplectic form (but not necessarily integrable). Let $p\in M^{T}$ be
a fixed point. The $T$-action induces a $\mathbb{C}$-linear $T$-representation
on the complex vector space $(T_{p}M,J)\cong\mathbb{C}^{n}$. Let
$\alpha_{p,1},\dots,\alpha_{p,n}\in\ell_{T}^{*}\cong\mathbb{Z}^{d}$ be the
weights of this $T$-representation, where $\ell_{T}^{*}$ is the dual lattice
of $T$ and $d$ is the dimension of $T$. These weights are pairwise coprime if
for any two different indices $i,j\in\\{1,\dots,n\\}$ there exists no integer
$k\in\mathbb{Z}\setminus\\{0,\pm 1\\}$ that divides $\alpha_{i}$ and
$\alpha_{j}$ in $\ell_{T}^{*}$. The technical condition which we are assuming
is that

$\displaystyle\text{for each}\,p\in M^{T},$ $\displaystyle\text{the weights of
the $T$-representation on}\,\,T_{p}M$ (1.1) are pairwise coprime.

######  Theorem 1.6.

Let $(M_{1},\omega_{1},T,\phi_{1})$ and $(M_{2},\omega_{2},T,\phi_{2})$ be two
Hamiltonian GKM spaces that satisfy the technical condition (1.1). If there
exists an isomorphism from the GKM graph of $(M_{1},\omega_{1},T,\phi_{1})$ to
the one of $(M_{2},\omega_{2},T,\phi_{2})$, then this isomorphism induces

  * (a)

a ring isomorphism in equivariant cohomology

$\displaystyle H_{T}^{*}(M_{1};\mathbb{Z})\rightarrow
H_{T}^{*}(M_{2};\mathbb{Z})$

that maps the equivariant Chern classes of $(M_{1},\omega_{1},T,\phi_{1})$ to
the ones of $(M_{2},\omega_{2},T,\phi_{2})$ and

  * (b)

a ring isomorphism in cohomology

$\displaystyle H^{*}(M_{1};\mathbb{Z})\rightarrow H^{*}(M_{2};\mathbb{Z})$

that maps the Chern classes of $(M_{1},\omega_{1})$ to the ones of
$(M_{2},\omega_{2})$.

The technical condition (1.1) is always satisfied if the complexity of the
action is zero or one. We deduce that condition (1.1) is satisfied if the
dimension of $M$ is equal to six (see Lemma 3.4). Hence, by combining Theorem
1.6 with a classification result for certain six-dimensional manifolds by Jupp
[24], Wall [41] and Z̆ubr [42] we obtain the following theorem.

######  Theorem 1.7.

Let $(M_{1},\omega_{1},T,\phi_{1})$ and $(M_{2},\omega_{2},T,\phi_{2})$ be two
Hamiltonian GKM spaces of dimension six. If there exists an isomorphism from
the GKM graph of $(M_{1},\omega_{1},T,\phi_{1})$ to the one of
$(M_{2},\omega_{2},T,\phi_{2})$, then this isomorphism induces a (non-
equivariant) diffeomorphism $M_{2}\rightarrow M_{1}$.

Theorems 1.6 and 1.7 are related to recent results of Goertsches, Konstantis,
and Zoller [18]. In [18], GKM spaces, which are not necessarily Hamiltonian,
are considered. In particular, the analogue of Theorems 1.6 and 1.7 for (non
Hamiltonian) GKM spaces is true if the spaces are simply connected, the odd
integer cohomology vanishes, and the spaces satisfy the technical condition
(1.2), as described below ([18, Theorem 3.1]). The one skeleton of a $T$-space
is

$M_{(1)}:=\left\\{x\in M\,|\,\operatorname{dim}(T\cdot x)\leq 1\right\\}.$

The technical condition is that

$\displaystyle\text{for each}\,p\notin$ $\displaystyle M_{(1)},\,\text{the
isotropy group}\,T_{p}=\left\\{t\in T\,|\,t\cdot p=p\right\\}$ (1.2)
$\displaystyle\text{is contained in a proper subtorus of }T.$

Hamiltonian GKM spaces are more restrictive than non-Hamiltonian ones. But the
technical condition (1.2) is more restrictive than the technical condition
(1.1). In Appendix B we give an example of a six-dimensional Hamiltonian GKM
$T$-space of complexity one that satisfies the technical condition (1.1), but
not the technical condition (1.2). Hence, Theorem 1.6 resp. 1.7 can not be
deduced form [18, Theorem 3.1 (a)] resp. [18, Theorem 3.1 (c)] .

The ideas behind our proof of Theorem 1.6 and the proof of [18, Theorem 3.1
(a)] are different. The proof of [18, Theorem 3.1 (a)] relies on a result of
Franz and Puppe [15, Corollary 2.2] for (maybe non Hamiltonian) $T$-spaces. In
contrast, in the proof of Theorem 1.6 we rely on the existence of so-called
(integer) Kirwan classes that form a basis of the equivariant cohomology of a
compact Hamiltonian $T$-space with isolated fixed points as a module over
$H^{*}(BT;\mathbb{Z})$. The existence of Kirwan classes for compact
Hamiltonian $T$-spaces relies on (equivariant) Morse theoretical arguments.
Namely, since a compact Hamiltonian $T$-space with isolated fixed points is
endowed with a moment map, there exists a $T$-invariant Morse function whose
set of critical points coincides with the set of fixed points of the
$T$-action. Beyond these contrasts, our deduction of Theorem 1.7 and the
deduction of [18, Theorem 3.1 (c)] both rely on a classification result for
compact and simply connected six-manifolds by Jupp [24], Wall [41] and Z̆ubr
[42].

In the second step, we give a complete list of the GKM graphs that are coming
from positive six-dimensional Hamiltonian GKM spaces. This result is based on
a work by Godinho and Sabatini [16]. Let $(\Gamma_{GKM},\eta)$ be the GKM
graph of such a Hamiltonian GKM space. So $\Gamma_{GKM}=(V_{GKM},E_{GKM})$ is
a $3$-valent graph and the weight map $\eta:E_{GKM}\rightarrow\ell_{T}^{*}$ is
a map from the edge set $E_{GKM}$ to the dual lattice of the torus $T$. The
results of [16] imply that the cardinality of the vertex set $V_{GKM}$ is at
most $16$ and that the map $\eta$ satisfies certain linear relations. By an
analysis of these linear relations and a classification result for $3$-valent
graphs with a low number of vertices [4], we create a computer program. This
program gives us a finite list of (abstract) GKM graphs that must contain all
GKM graphs that are coming from positive six-dimensional Hamiltonian GKM
spaces. It turns out that each (abstract) GKM graph in this list is a GKM
graph that is coming from a holomorphic GKM $T$-action on a Fano threefold
$X$, i.e., it is the GKM graph of a GKM Hamiltonian $T$-action on
$(X,i^{*}\omega_{FS})$ that is induced from a holomorphic
$T_{\mathbb{C}}$-action on $X$. (We show this directly; see also [38]).

We will prove the following theorem.

######  Theorem 1.8.

Let $(M,\omega,T,\phi)$ be a positive six-dimensional Hamiltonian GKM space of
complexity one. Then its GKM graph is isomorphic to a GKM graph that is coming
from a holomorphic GKM $T$-action on a Fano threefold.

Combining Theorem 1.7 and Theorem 1.8 gives Theorem 1.4.

###  1.2 Structure of the Article

  * Section 2

We introduce notations and recall preliminary results about equivariant
cohomology and Hamiltonian GKM spaces that are needed in this paper.

  * Section 3

We relate the (equivariant) cohomology ring of a Hamiltonian GKM space with
integer coefficients to its GKM graph and prove Theorems 1.6 and 1.7.

  * Section 4

We introduce positive Hamiltonian GKM spaces and prove basic properties of six
dimensional ones. These results are needed to compute the list of graphs
associated to such spaces.

  * Section 5

We analyze the linear relations between the first Chern class maps and the
weight maps of (abstract) GKM graphs.

  * Section 6

We sum up the results of Section 5 and explain how we use computer programs to
classify the GKM graphs of positive six-dimensional Hamiltonian GKM spaces. We
conclude by a proof of Theorem 1.8.

  * Appendix A

We give the complete list of GKM graphs of positive six-dimensional
Hamiltonian GKM spaces that are not projections of GKM graphs coming from
smooth reflexive polytopes.

  * Appendix B

We give an example of a six-dimensional Hamiltonian GKM space that does not
satisfy Condition (1.2).

Acknowledgments. The first author would like to thank Silvia Sabatini for
drawing her attention to the question of in which content smooth Fano
threefolds and monotone symplectic six-manifolds with a Hamiltonian GKM action
are different. The second author is supported by the Israel Science
Foundation, grant no. 570/20.

##  2 Preliminaries on Equivariant Cohomology and Hamiltonian GKM Spaces

###  2.1 Equivariant Cohomology

Let $M$ be a topological space endowed with a continuous action of a torus
$T=(S^{1})^{d}$. In the Borel model, the $T$-equivariant cohomology of $M$ is
defined as follows. Let $ET$ be a contractible space on which $T$ acts freely
and let $BT=ET/T$ be the classifying space of $T$. The diagonal action of $T$
on $M\times ET$ is free. $M\times_{T}ET$ denotes the orbit space. The
$T$-equivariant cohomology ring of $M$ is

$\displaystyle H_{T}^{*}(M;R)\colon=H^{*}(M\times_{T}ET;R),$

where $R$ is the coefficients ring. In particular, if the $T$-action on $M$ is
trivial, then

$\displaystyle H_{T}^{*}(M;R)=H^{*}(M;R)\otimes H^{*}(BT;R).$

If $T=S^{1}$, then $ES^{1}$ is the unit sphere $S^{\infty}$ in
$\mathbb{C}^{\infty}$ and $BS^{1}$ is $\mathbb{C}P^{\infty}$. Hence

$\displaystyle
H_{S^{1}}^{*}(\left\\{\operatorname{point}\right\\};R)=H^{*}(\mathbb{C}P^{\infty};R)=R[x],$

where $x$ has degree $2$. Moreover, if $T$ is a $d$-dimensional torus, then
$BT$ is the $d$-times product of $\mathbb{C}P^{\infty}$, and so

$\displaystyle
H_{T}^{*}(\left\\{\operatorname{point}\right\\};R)=H^{*}(BT;R)=R[x_{1},\dots,x_{d}],$

where $\left\\{x_{1},\dots,x_{d}\right\\}$ is a basis of the dual lattice
$\ell_{T}^{*}$ of $T$ and $\operatorname{deg}(x_{i})=2$ for all $i=1,...,d$.
If the coefficient ring is $\mathbb{Z}$ then $H^{2}(BT;\mathbb{Z})$ is equal
to $\ell_{T}^{*}$ and if the coefficient ring is $\mathbb{R}$ then
$H^{2}(BT;\mathbb{R})$ is equal to $\mathfrak{t}^{*}$. The projection map
$M\times ET\rightarrow ET$ is $T$-equivariant, so we obtain a map

$\displaystyle\pi:\,M\times_{T}ET\rightarrow BT.$

This makes $M\times_{T}ET$ an $M$-bundle over $BT$,

$\displaystyle
M\overset{r}{\longrightarrow}M\times_{T}ET\overset{\pi}{\longrightarrow}BT.$

So it induces a sequence of ring homomorphisms

$\displaystyle
H^{*}(BT;R)\overset{\pi^{*}}{\longrightarrow}H_{T}^{*}(M;R)\overset{r^{*}}{\longrightarrow}H^{*}(M;R).$

The map $\pi^{*}$ gives $H_{T}^{*}(M;R)$ an $H^{*}(BT;R)$-module structure by

$\displaystyle\alpha\cdot\beta=\pi^{*}(\alpha)\cup\beta$

for $\alpha\in H^{*}(BT;R)$ and $\beta\in H_{T}^{*}(M;R)$.

####  2.1.1 Equivariant Chern Classes

Let $V\rightarrow M$ be a $T$-invariant vector bundle. Its equivariant Euler
class $e^{T}(V)$ is the Euler class of the vector bundle

$\displaystyle V\times_{T}ET\rightarrow M\times_{T}ET.$ (2.1)

If $V\rightarrow M$ is a complex $T$-invariant vector bundle, then (2.1) is a
complex vector bundle, and its $i$-th equivariant Chern class $c_{i}^{T}(V)$
is defined in the same way. These equivariant classes are mapped to the
ordinary Euler class resp. ordinary Chern classes of $V\rightarrow M$ under
the map

$\displaystyle r^{*}:H_{T}^{*}(M;\mathbb{Z})\rightarrow H^{*}(M;\mathbb{Z})$

induced from $r:M\rightarrow M\times_{T}ET$. In case $M$ is a point, a complex
$T$-invariant vector bundle $V\rightarrow M=\\{p\\}$ of rank $k$ is just a
complex vector space of complex dimension $k$ with a $T$-representation. Let
$\alpha_{1},...,\alpha_{k}\in\ell_{T}^{*}$ be the weights of this
representation, then the total equivariant Chern class of the complex vector
bundle $V\rightarrow\\{p\\}$

$\displaystyle c^{T}(V)=\prod_{i=1}^{k}\left(1+\alpha_{i}\right)\in
H_{T}^{*}(\\{p\\};\mathbb{Z})\cong H^{*}(BT;\mathbb{Z}).$

Hence, the $i$-th equivariant Chern class is

$\displaystyle c_{i}^{T}(V)=\sigma_{k,i}(\alpha_{1},\dots\alpha_{k})\in
H^{2i}(BT;\mathbb{Z}),$

where $\sigma_{k,i}$ is the elementary symmetric polynomial in $k$ variables
of degree $i$. 111This means $\sigma_{k,0}=1$, $\sigma_{k,i}=0$ for $i>k$ and
for $i=1,...,k$ $\sigma_{k,i}(X_{1},...,X_{k})=\sum_{1\leq j_{1}<...<j_{i}\leq
k}X_{j_{1}}\cdot...\cdot X_{j_{k}}.$

In particular, the first equivariant Chern class is
$c_{1}^{T}(V)=\sum_{i=1}^{k}\alpha_{i}$ and the equivariant Euler class is
$e^{T}(V)=c_{k}^{T}(V)=\prod_{i=1}^{k}\alpha_{i}$.

####  2.1.2 The ABBV Localization Formula

Let $M^{T}$ be the set of fixed points and assume that it is not empty. Let
$F$ be one of its connected components. The inclusion map $i_{F}\colon
F\rightarrow M$ is a $T$-equivariant map, so it induces a map

$\displaystyle i_{F}^{*}\colon H_{T}^{*}(M;R)\rightarrow H_{T}^{*}(F;R).$

Moreover, the map $\pi\colon M\times_{T}ET\rightarrow BT$ induces a push-
forward map in equivariant cohomology

$\displaystyle H_{T}^{*}(M;R)\rightarrow H^{*-\operatorname{dim}(M)}(BT;R),$

which can be seen as integration along the fibers. So we denote it by
$\int_{M}$. The following theorem, due to Atiyah-Bott [2] and Berline-Vergne
[3], gives a formula for the map $\int_{M}$ in terms of the fixed point set
data.

######  Theorem 2.1.

(ABBV Localization formula) Let $M$ be a compact oriented manifold endowed
with a smooth $T$-action. For $\mu^{T}\in H_{T}^{*}(M;\mathbb{Q})$

$\displaystyle\int_{M}\mu^{T}\,=\,\sum_{F\subset
M^{T}}\int_{F}\dfrac{i^{*}_{F}\mu^{T}}{\text{e}^{T}(N_{F})},$

where the sum is over all the connected components $F$ of $M^{T}$ and
$\text{e}^{T}(N_{F})$ is the equivariant Euler class of the normal bundle
$N_{F}\to F$.

###  2.2 Hamiltonian GKM Spaces

Given a $T$-action on a compact connected symplectic manifold of dimension
$2n$, there exists a $T$-invariant almost complex structure $J:TM\rightarrow
TM$ that is compatible with the symplectic form $\omega$; moreover, the space
of $T$-invariant almost complex structures on $(M,\omega)$ is contractible
(see e.g., [32, Proposition 4.1.1]). For each fixed point $p\in M^{T}$, the
$T$-action induces a $T$-representation on
$(T_{p}M,J_{p})\cong\mathbb{C}^{n}$. This $T$-representation splits into a
direct sum of one-dimensional $T$-representations, i.e.,
$T_{p}M=\oplus_{i=1}^{n}L_{i}$, where $L_{i}$ is a one-dimensional
$T$-representation with weight $\alpha_{p,i}\in\ell_{T}^{*}$ for $i=1,...,n$.
The elements $\alpha_{p,1},...,\alpha_{p,n}$ are called the weights of the
$T$-representation on $T_{p}M$. These weights do not depend on the choice of
such an almost complex structure since the space of $T$-invariant almost
complex structures on $(M,\omega)$ is contractible. Hence, the weights are
well-defined. Since the $T$-action on $M$ is effective, the following lemma
holds.

######  Lemma 2.2.

Given an effective $T$-action on a connected symplectic $(M,\omega)$, let
$p\in M^{T}$ be a fixed point and let
$\alpha_{p,1},...,\alpha_{p,n}\in\ell_{T}^{*}$ be the weights of the
$T$-representation on $T_{p}M$. Then the $\mathbb{Z}$-span

$\displaystyle\left\\{\kappa_{1}\cdot\alpha_{p,1}+\dots+\kappa_{n}\cdot\alpha_{p,n}\,|\,\kappa_{1},...,\kappa_{n}\in\mathbb{Z}\right\\}$

of the weights $\alpha_{p,1},...,\alpha_{p,n}$ is equal to $\ell_{T}^{*}$.

###### Proof.

Since $T$ is compact there exist $T$-invariant neighborhoods $U$ resp. $V$ of
$\mathbf{0}$ in $T_{p}M$ resp. of $p$ in $M$ that are $T$-equivariantly
diffeomorphic. Since the $T$-action on $M$ is effective and $M$ is connected,
by the Principal Orbit Theorem [11, Theorem 2.8.5], the $T$-action on $V$ is
effective. Hence, the $T$-action on $U$ is also effective. The $T$-action on
$U$ is determined by the $T$-representation on $T_{p}M$ and is effective if
and only if the $\mathbb{Z}$-span of the weights
$\alpha_{p,1},...,\alpha_{p,n}$ is equal to $\ell_{T}^{*}$. ∎

######  Definition 2.3.

A Hamiltonian GKM space is a compact Hamiltonian $T$-space $(M,\omega,T,\phi)$
such that the following hold.

  * (i)

The set $M^{T}$ of fixed points of the $T$-action on $M$ is finite.

  * (ii)

For each $p\in M^{T}$, the weights
$\alpha_{p,1},...,\alpha_{p,n}\in\ell_{T}^{*}$ of the $T$-representation on
$T_{p}M$ are pairwise linearly independent.

We say that the Hamiltonian $T$-action is GKM.

In the following lemma, we point out an obstruction for the dimension of the
torus of a Hamiltonian GKM space.

######  Lemma 2.4.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space of dimension $2n\geq 4$.
Then the dimension of the torus $T$ is greater or equal to $2$.

###### Proof.

Since $M$ is compact, the set of fixed points $M^{T}$ is not empty. Let $p\in
M^{T}$ be a fixed point and let $\alpha_{p,1},...,\alpha_{p,n}\in\ell_{T}^{*}$
be the weights of the $T$-representation on $T_{p}M$. Since, by (ii) of
Definition 2.3, these weights are pairwise linearly independent, the dimension
of the torus must be greater or equal to two. ∎

Given a Hamiltonian GKM space $(M,\omega,T,\phi)$, the second condition of
Definition 2.3 implies that for each codimensional one subtorus $H$ of $T$,
any connected component of

$\displaystyle M^{H}:=\left\\{p\in M\,\mid\,t\cdot p=p\,\text{ for all }t\in
H\right\\}$

has at most dimension two. Let $S$ be a two-dimensional component of $M^{H}$.
Then $S$ is a symplectic embedded two-sphere and there exist exactly two
points $p,q\in M^{T}$ such that $S\cap M^{T}=\\{p,q\\}$. Moreover, the
inclusion $H\hookrightarrow T$ induces an $H$-representation on $T_{p}M$ and
the tangent $T_{p}S$ is equal to the subspace of $T_{p}M$ that is fixed by the
$H$-representation. Hence, there exists an element $\alpha\in\ell_{T}^{*}$
that is a weight of the $T$-representation on $T_{p}M$ such that

$\displaystyle H=\exp(\operatorname{ker}\,\alpha),$ (2.2)

where $\exp:\mathfrak{t}\rightarrow T$ is the exponential map and

$\displaystyle\operatorname{ker}\,\alpha=\left\\{\xi\in\mathfrak{t}\,|\,\left\langle\alpha,\xi\right\rangle=0\right\\},$

where $\left\langle,\right\rangle$ is the natural pairing between
$\mathfrak{t}^{*}$ and $\mathfrak{t}$. Vice versa, $-\alpha$ is a weight of
the $T$-representation on $T_{q}M$. Given $p,q\in M^{T}$ there exists at most
one two-sphere $S$ that is fixed by a codimensional one subtorus such that
$S\cap M^{T}=\\{p,q\\}$. This is since a point fixed by two different
codimensional one subtori is in $M^{T}$. On the other hand, for $p\in M^{T}$
and a weight $\alpha$ of the $T$-representation on $T_{p}M$, there is a
codimensional one subtorus $H$ such that the connected component of $M^{H}$
that contains $p$ is a symplectic embedded two-sphere $S$ and the weight of
the $T$-representation on $T_{p}S$ is $\alpha$; moreover, the subtorus $H$
must be as in (2.2), so this sphere is unique.
The information about the $T$-representations on the tangent spaces of the
fixed points and about the two-dimensional components fixed by a codimensional
one subtorus can be stored in a graph, the so-called GKM graph, which is
defined as follows.

######  Definition 2.5.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space. Its GKM graph
$(\Gamma_{GKM},\eta)$ is the graph $\Gamma_{GKM}=(V_{GKM},E_{GKM})$ with
directed edges together with the map $\eta:E_{GKM}\rightarrow\ell_{T}^{*}$
defined as follows.

  * (i)

The set of vertices $V_{GKM}$ is equal to the set of fixed points $M^{T}$.

  * (ii)

The set of edges is a subset of $V_{GKM}\times V_{GKM}$. There exists a
directed edge

$e=(p,q)\in V_{GKM}\times V_{GKM}$

if and only if there exists a two-sphere $S$ that is fixed by a codimensional
one subtorus $H$ of $T$ such that $S\cap M^{T}=\\{p,q\\}$.

  * (iii)

For each edge $e=(p,q)\in E_{GKM}$, we denote by $S_{(p,q)}$ the unique two-
sphere as described in $(ii)$. Then $\eta(p,q)$ is the weight of the
$T$-representation on $T_{p}S_{(p,q)}$.

In the following remark, we sum up some properties of GKM graphs.

######  Remark 2.6.

Let $(\Gamma_{GKM},\eta)$ be the GKM graph of a Hamiltonian GKM space
$(M,\omega,T,\phi)$, then the following properties hold.

  * (i)

Let $2n$ be the dimension of $M$. Then the graph
$\Gamma_{GKM}=(M^{T},E_{GKM})$ is $n$-valent, i.e., for each $p\in M^{T}$
there exist exactly $n$ points $p_{1},\dots,p_{n}\in M^{T}\setminus\\{p\\}$
such that $(p,p_{i})\in E_{GKM}$ for $i=1,...,n$. Moreover,
$\eta(p,p_{1}),...,\eta(p,p_{n})$ are the weights of the $T$-representation on
$T_{p}M$.

  * (ii)

Given $p,q\in M^{T}$, then $(p,q)$ belongs to $E_{GKM}$ if and only if $(q,p)$
belongs to $E_{GKM}$. If $(p,q)\in E_{GKM}$, then $\eta(q,p)=-\eta(p,q)$.

######  Definition 2.7.

Given two Hamiltonian GKM spaces $(M_{1},\omega_{1},T,\phi_{1})$ and
$(M_{2},\omega_{2},T,\phi_{2})$ of the same dimension, let
$(\Gamma_{1,GKM},\eta_{1})$ and $(\Gamma_{2,GKM},\eta_{2})$ be their GKM
graphs. An isomorphism between these GKM graphs is a pair $(F,\theta)$ such
that

  * •

$F$ is an isomorphism between the underlying graphs

$\Gamma_{1,GKM}=(M_{1}^{T},E_{1,GKM})\quad\text{and}\quad\Gamma_{2,GKM}=(M_{2}^{T},E_{2,GKM}),$

i.e., $F:M_{1}^{T}\rightarrow M_{2}^{T}$ is a bijection such that for any two
points $p,q\in M_{1}^{T}$

$(p,q)\in E_{1,GMK}\quad\text{if and only if}\quad(F(p),F(q))\in E_{2,GKM},$

  * •

$\theta:\ell_{T}^{*}\rightarrow\ell_{T}^{*}$ is a linear isomorphism such that

$\displaystyle\theta\left(\eta_{1}(p,q)\right)=\eta_{2}(F(p),F(q)),$

for all $(p,q)\in E_{GKM}$.

Given a Hamiltonian GKM space $(M,\omega,T,\phi)$, let $(\Gamma_{GKM},\eta)$
be its GKM graph. The initial map $i:E_{GKM}\rightarrow V_{GKM}$ and the
terminal map $t:E_{GKM}\rightarrow V_{GKM}$ are given by

$\displaystyle i(p,q)=p\quad\text{and}\quad t(p,q)=q\quad\text{for every edge
}e=(p,q)\in E_{GKM}.$

We associate to each point $p\in M^{T}=V_{GKM}$ the following two subsets of
$E_{GKM}$,

$\displaystyle E_{GKM}^{p,i}:=\\{e\in E_{GKM}\mid
i(e)=p\\}\quad\quad\text{and}\quad\quad E_{GKM}^{p,t}:=\\{e\in E_{GKM}\mid
t(e)=p\\}.$

A connection along an edge $e=(p,q)\in E_{GKM}$ is a bijection

$\displaystyle\nabla_{e}:E_{GKM}^{p,i}\longrightarrow E_{GKM}^{q,i}$

such that $\nabla_{e}(p,q)=(q,p)$. The connection is compatible if for all
$e^{\prime}\in E_{GKM}^{p,i}$ there exists an integer
$a_{e,e^{\prime}}\in\mathbb{Z}$, such that

$\displaystyle\eta(e^{\prime})-\eta(\nabla_{e}(e^{\prime}))=a_{e,e^{\prime}}\cdot\eta(e).$

Due to the next lemma, there always exists a compatible connection.

######  Lemma 2.8.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let
$(\Gamma_{GKM},\eta)$ be its GKM graph. For each edge $e\in E_{GKM}$ of
$\Gamma_{GKM}$, there exists a connection along $e$ that is compatible.

This lemma is by [23, Theorem 1.1.2].

######  Remark 2.9.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space. Its GKM graph does not
depend on the moment map $\phi$. Moreover, if $\lambda\in\mathbb{R}$ is non
zero, then the $T$-action on $(M,\lambda\cdot\omega)$ is also Hamiltonian and
GKM and $\phi^{\prime}=\lambda\cdot\phi$ is a moment map. If $\lambda>0$, then
the GKM graphs of $(M,\omega,T,\phi)$ and
$(M,\lambda\cdot\omega,T,\lambda\cdot\phi)$ are the same.

######  Remark 2.10.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let
$(\Gamma_{GKM},\eta)$ be its GKM graph. Let $e=(p,q)\in E_{GKM}$ be an edge.
The moment map images $\phi(p)$ and $\phi(q)$ determine $\eta(p,q)$ up to
positive integer multiple. Namely, let $S$ be the unique $2$-sphere in $M$
that is fixed by a codimensional one subtorus of $T$ such that $S\cap
M^{T}=\\{p,q\\}$. The moment map image $\phi(S)$ is given by the compact line
segment through $\phi(p)$ and $\phi(q)$ in the vector space
$\mathfrak{t}^{*}$. This line segment is rational, i.e., there exists a unique
primitive vector
$\alpha\in\ell_{T}^{*}\setminus\\{0\\}\subset\mathfrak{t}^{*}\setminus\\{0\\}$
such that $\phi(q)-\phi(p)=r\cdot\alpha$ for some real number $r>0$. In
particular,

$\eta(p,q)=m\cdot\alpha\quad\text{for some positive integer }m$

and

$\eta(p,q)=r\cdot m\cdot\left(\phi(q)-\phi(p)\right).$

####  2.2.1 GKM Graphs of Symplectic Toric Manifolds

By the famous Convexity Theorem of Atiyah [1] and Guillemin-Sternberg [22],
the moment map image of a compact Hamiltonian $T$-space is a convex polytope.
So the moment map image is called the moment map polytope. Moreover, compact
symplectic toric manifolds, i.e., compact complexity zero spaces, are
determined by their moment map polytopes (up to isomorphism) and the moment
map polytope of such a space is smooth [9]. We recall this and the definition
of smooth polytopes in the following.

######  Definition 2.11.

Let $\Delta$ be a $d$-dimensional polytope in $\mathbb{R}^{d}$. A vertex $v$
of $\Delta$ is called smooth if there exist exactly $d$ edges of $\Delta$ that
contain $v$ and a $\mathbb{Z}$-basis $\alpha_{v,1},...,\alpha_{v,d}$ of
$\mathbb{Z}^{d}$ such that the edges that contain $v$ are given by

$\left(v+\mathbb{R}_{\geq 0}\cdot\alpha_{v,j}\right)\cap\Delta$

for $j=1,...,d$. The polytope $\Delta$ is called smooth if all of its vertices
are smooth.

######  Remark 2.12.

Let $T$ be a $d$-dimensional torus. By considering its dual Lie algebra
$\mathfrak{t}^{*}\cong\mathbb{R}^{d}$ together with the dual lattice
$\ell^{*}_{T}\cong\mathbb{Z}^{d}$, the definition of smooth polytopes in
$\mathbb{R}^{d}$ naturally extends to polytopes in $\mathfrak{t}^{*}$.

######  Theorem 2.13.

[9, Delzant] Let $(M,\omega,T,\phi)$ be a compact symplectic toric manifold.
The space admits an integrable almost complex structure that is T-invariant
and compatible with $\omega$ and the moment map polytope is a smooth polytope.
Moreover, if $\Delta$ is a smooth polytope in $\mathfrak{t}^{*}$, then there
exists a unique (up to isomorphism) compact symplectic toric manifold whose
moment map image is $\Delta$.

Let $(M,\omega,T,\phi)$ be a compact symplectic toric manifold. Then the space
is also GKM and its GKM graph $(\Gamma_{GKM},\eta)$ is determined by the
moment map polytope $\Delta_{M}\colon=\phi(M)$, as follows. The moment map
induces a bijection from the set of fixed points to the set of vertices of
$\Delta_{M}$. Hence, the set $V_{GKM}$ of vertices of the graph $\Gamma_{GKM}$
can be identified with the set of vertices of the polytope $\Delta_{M}$. There
exists an edge $(p,q)\in E_{GKM}\subset M^{T}\times M^{T}$ if and only if the
corresponding vertices $\phi(p)$ and $\phi(q)$ of $\Delta_{M}$ are connected
by an edge of the polytope $\Delta_{M}$. If $(p,q)\in E_{GKM}$, then
$\eta(p,q)$ is the unique primitive element in $\ell_{T}^{*}$ such that
$\phi(q)-\phi(p)=r\cdot\eta(p,q)$ for some $r>0$. In the following, we say
that the GKM graph of a compact symplectic toric manifold $(M,\omega,T,\phi)$
is coming from the smooth polytope $\Delta_{M}\colon=\phi(M)$. We note that
the GKM graph does not determine the moment map polytope. In particular, the
GKM graph does not contain information about the symplectic form.

######  Example 2.14.

In Figure 1, the polytopes (A), (B) and (C) are smooth.

  * •

The polytope (A) is the moment map polytope of the standard
$(S^{1})^{2}$-action on $(\mathbb{C}P^{2},\omega_{FS})$.

  * •

The polytope (B) is the moment map polytope of the standard
$(S^{1})^{2}$-action on $(S^{2}\times S^{2},\omega\oplus\omega)$, where
$\omega$ is volume form on $S^{2}$ with $\int_{S^{2}}\omega=2$.

  * •

The polytope (C) is the moment map polytope of the standard
$(S^{1})^{2}$-action on $(S^{2}\times S^{2},\omega\oplus\frac{5}{4}\omega)$.

The polytopes (A) and (B) induce the same GKM graph.

######  Definition 2.15.

Let $\Delta$ be a $d$-dimensional polytope in $\mathbb{R}^{d}$ that is
integral, i.e., all of its vertices lie in $\mathbb{Z}^{d}$. For each facet
$F$ of $\Delta$, let $l_{F}\in\mathbb{Z}^{d}$ be the primitive outward normal
vector to the hyperplane defining $F$, i.e.,

$F=\left\\{x\in\mathbb{R}^{d}\,|\,\left\langle
l_{F},x\right\rangle_{\mathbb{R}^{d}}=c_{F}\right\\}\cap\Delta\quad\text{and}\quad\Delta\subseteq\left\\{x\in\mathbb{R}^{d}\,|\,\left\langle
l_{F},x\right\rangle_{\mathbb{R}^{d}}\leq c_{F}\right\\}$

for some $c_{F}\in\mathbb{R}$, where
$\left\langle,\right\rangle_{\mathbb{R}^{d}}$ is the standard scalar product
on $\mathbb{R}^{d}$. The polytope $\Delta$ is reflexive if $c_{F}=1$ for any
facet $F$ of $\Delta$.

Figure 1: The polytopes (A) and (B) are smooth and reflexive. The polytope (C)
is smooth, but not reflexive. The polytope (D) is reflexive, but not smooth.

Monotone symplectic toric manifolds and reflexive polytopes are related as
follows.

######  Proposition 2.16.

Let $\Delta$ be a smooth polytope in $\mathfrak{t}^{*}$ and let
$(M,\omega,T,\phi)$ the compact symplectic toric manifold whose moment map
image is $\Delta$. Then the following two conditions are equivalent.

  * •

There exists $\nu\in\mathfrak{t}^{*}$ such that $\Delta+\nu$ is reflexive.

  * •

$(M,\omega)$ is monotone with $c_{1}(M)=[\omega]$.

A proof of this proposition can be found in [12, Proposition 1.8] and [33,
Sect.3]

####  2.2.2 Projections of GKM Graphs

Given a Hamiltonian GKM space $(M,\omega,T,\phi)$, let $T^{\prime}$ be a
subtorus of $T$. The restriction of the $T$-action to $T^{\prime}$ is again a
Hamiltonian action on $(M,\omega)$ with moment map

$\displaystyle i^{*}\circ\phi:M\rightarrow(\mathfrak{t^{\prime}})^{*},$

where $i^{*}:\mathfrak{t}^{*}\rightarrow(\mathfrak{t}^{\prime})^{*}$ is the
dual of the inclusion $i:\mathfrak{t}^{\prime}\hookrightarrow\mathfrak{t}$
from the Lie algebra of $T^{\prime}$ to the one of $T$. So the quadruple

$\displaystyle(M,\omega,T^{\prime},i^{*}\circ\phi)$

is a compact Hamiltonian $T$-space. The map $i^{*}$ maps the dual lattice
$\ell_{T}^{*}$ of $T$ to the dual lattice $\ell_{T^{\prime}}^{*}$ of
$T^{\prime}$. Note that the quadruple is a Hamiltonian GKM space if and only
if for each fixed point $p\in M^{T}$, the weights of the $T$-representation on
$T_{p}M$ are mapped to pairwise linearly independent weights. In this case the
GKM graph of $(M,\omega,T^{\prime},i^{*}\circ\phi)$ is
$(\Gamma_{GKM},i^{*}\circ\eta)$, where $(\Gamma_{GKM},\eta)$ is the GKM graph
of $(M,\omega,T,\phi)$. We say that the GKM graph
$(\Gamma_{GKM},i^{*}\circ\eta)$ is a projection of $(\Gamma_{GKM},\eta)$.

####  2.2.3 Generic Vectors and Morse Theory

Let $(M,\omega,T,\phi)$ be a compact Hamiltonian $T$-space with only isolated
fixed points. A vector $\xi\in\mathfrak{t}$ is called generic if
$\left\langle\alpha,\xi\right\rangle\neq 0$ for each weight
$\alpha\in\ell_{T}^{*}$ of the $T$-representation of $T_{p}M$ for every $p\in
M^{T}$. If $\xi\in\mathfrak{t}$ is generic, then the $\xi$-component of the
moment map

$\displaystyle\phi^{\xi}:M\rightarrow\mathbb{R}\,,\quad
p\mapsto\left\langle\phi(p),\xi\right\rangle$

is a Morse function and its set of critical points coincides with the set of
fixed points $M^{T}$.

Given $p\in M^{T}$, the Morse index of $p$ with respect to $\phi^{\xi}$ is
equal to twice the number of weights of the $T$-representation on $T_{p}M$
that satisfy $\left\langle\alpha,\xi\right\rangle<0$.

For any fixed point $p\in M^{T}$ we define the index $\lambda(p)$ of $p$ as
follows.

$\displaystyle\lambda(p)\,\text{is the half of the Morse index of
}p\,\text{with respect to }\phi^{\xi},$

or equivalently,

$\lambda(p)$ is the number of weights of the $T$-representation on $T_{p}M$
that satisfy $\left\langle\alpha,\xi\right\rangle<0.$

Since for each $p\in M^{T}$ the Morse index is even and $M$ is connected, $M$
is homotopy equivalent to a CW complex with only even cells. This implies that
$M$ is simply connected and $H^{\operatorname{odd}}(M;\mathbb{Z})=0$. The
$2i$-th Betti number $b_{2i}(M)$ of $M$ is equal to the number of fixed points
with $\lambda(p)=i$ and $H^{2i}(M;\mathbb{Z})\cong\mathbb{Z}^{b_{2i}(M)}$. Let
$2n$ be the dimension of $M$. Since $M$ is compact and connected and
$\omega^{n}$ is a volume form, we have that $b_{0}(M)$ and $b_{2n}(M)$ are
equal to one. Hence, there exists a unique fixed point $p_{min}$ resp.
$p_{max}$ with $\lambda(p_{min})=0$ resp. $\lambda(p_{max})=n$. Moreover,
$p_{min}$ resp. $p_{max}$ is the unique point in $M$ on which $\phi^{\xi}$
attains its minimum resp. maximum.

We close this chapter with two remarks about Hamiltonian GKM spaces.

######  Remark 2.17.

Assume that $(M,\omega,T,\phi)$ is a Hamiltonian GKM space and let
$(\Gamma_{GKM},\eta)$ be its GKM graph. Given a fixed point $p\in M^{T}$, let
$p_{1},\dots,p_{n}\in M^{T}\setminus\\{p\\}$ be the unique fixed points such
that $(p,p_{i})\in E_{GKM}$. Since $\eta(p,p_{1}),...,\eta(p,p_{n})$ are the
weights of the $T$-representation on $T_{p}M$, we have that $\lambda(p)$ is
equal to the number of edges of $(p,p_{1}),...,(p,p_{n})$, which satisfy

$\displaystyle\phi^{\xi}(p_{i})<\phi^{\xi}(p),\,\text{or equivalently,
}\,\left\langle\eta(p,p_{i}),\xi\right\rangle<0.$

######  Remark 2.18.

Note that since the underlying manifold of a Hamiltonian GKM space is
connected, its GKM graph $(\Gamma_{GKM},\eta)$ is also connected. The latter
means that for two different points $p,q\in M^{T}$, there exists a sequence
$p_{0},...,p_{k}\in M^{T}$ such that $p_{0}=p$, $p_{k}=q$ and
$(p_{i},p_{i+1})\in E_{GKM}$ for $i=0,\dots,k-1$. We say that $p$ and $q$ are
_connected by a path_. To see this fix a generic vector $\xi\in\mathfrak{t}$
and let $p_{min}$ be the unique fixed point on which $\phi^{\xi}$ attains its
minimum. Let $p\in M^{T}\setminus\\{p_{min}\\}$ be another fixed point. We set
$p_{0}=p$. By Remark 2.17 there exists a fixed point $p_{1}\in
M^{T}\setminus\\{p_{0}\\}$ with $\phi^{\xi}(p_{1})<\phi^{\xi}(p_{0})$. It
might be that $p_{1}=p_{min}$; otherwise, we repeat this step. Note that since
$M^{T}$ is finite, we conclude that there exist $p_{0},...,p_{k}$ such that
$p_{0}=p$, $p_{k}=p_{min}$ and $(p_{i},p_{i+1})\in E_{GKM}$ for
$i=0,\dots,k-1$. Hence, each $p\in M^{T}\setminus\\{p_{min}\\}$ is connected
with $p_{min}$. Moreover, since $(p,q)\in E_{GKM}$ implies $(q,p)\in E_{GKM}$,
it follows that any two fixed points are connected by a path.

##  3 Relations between the GKM Graph and the Equivariant Cohomology

Due to the work of Goresky, Kottwitz and MacPherson [21], it is known that the
equivariant cohomology ring with rational coefficients of a Hamiltonian GKM
space can be easily recovered from its GKM graph. In this section, we prove
that the analogue of this holds for integer coefficients if for each fixed
point the weights of the $T$-representation on $T_{p}M$ are pairwise coprime.
Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let $R$ be $\mathbb{Z}$
or $\mathbb{Q}$. By the Kirwan Injectivity Theorem [29], the inclusion
$i:M^{T}\hookrightarrow M$ induces an injective map
$i^{*}:H_{T}^{*}(M;R)\rightarrow H_{T}^{*}(M^{T};R)$. Therefore, we consider
$H_{T}^{*}(M;R)$ as a subring of $H_{T}^{*}(M^{T};R)$. Since
$H^{*}_{T}(\\{p\\};R)\cong H^{*}(BT;R)$, we consider the ring
$H_{T}^{*}(M^{T};R)$ as the ring of maps from $M^{T}$ to $H^{*}(BT;R)$,
denoted by

$\operatorname{Maps}\left(M^{T},H^{*}(BT;R)\right).$

So any class $\alpha\in H_{T}^{*}(M;R)$ is completely determined by the
restrictions

$\displaystyle i^{*}_{\\{p\\}}(\alpha)\in H^{*}_{T}(\\{p\\};R)\cong
H^{*}(BT;R)$

for all $p\in M^{T}$, where $i^{*}_{\\{p\\}}$ is the map induced by the
inclusion $p\hookrightarrow M$. For simplicity, in the following we write
$\alpha(p)$ instead of $i^{*}_{\\{p\\}}(\alpha)$. Since we can consider
$H^{*}_{T}(M;R)$ as a subring of
$\operatorname{Maps}\left(M^{T},H^{*}(BT;R)\right)$, we have that

$\displaystyle
H^{2i}_{T}(M;R)=H_{T}^{*}(M;R)\cap\operatorname{Maps}\left(M^{T},H^{2i}(BT;R)\right).$

The following lemma is known and gives a necessary condition for a map
$M^{T}\rightarrow H^{*}(BT;R)$ to be an element of $H_{T}^{*}(M;R)$.

######  Lemma 3.1.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let
$(\Gamma_{GKM},\eta)$ be its GKM graph. Let $\alpha$ be a class in
$H^{*}_{T}(M;R)$. Then for each edge $e=(p,q)\in E_{GKM}$ the difference
$\alpha(p)-\alpha(q)$ is divisible by $\eta(p,q)$ in $H^{*}(BT;R)$.

###### Proof.

Let $e=(p,q)\in E_{GKM}$ be an edge and let $S_{e}$ be the unique two-sphere
with $S_{e}\cap M^{T}=\\{p,q\\}$ that is fixed by a codimensional one subtorus
of $T$. The sphere $S_{e}$ is $T$-invariant and the weight of the
$T$-representation of $T_{p}S_{e}$ is $\eta(p,q)$ and the weight of the
$T$-representation of $T_{q}S_{e}$ is $\eta(q,p)=-\eta(p,q)$. The equivariant
Euler class of the normal bundle of $p$ resp. $q$ in $S_{e}$ is $\eta(p,q)$
resp. $-\eta(p,q)$. Let $\beta$ be a class in $H^{*}_{T}(S_{e};R)$. Due to the
ABBV localization formula (Theorem 2.1), we have

$\displaystyle\int_{S_{e}}\beta\,=\,\dfrac{\beta(p)-\beta(q)}{\eta(p,q)}.$

The left-hand side of this equation is an element in $H^{*}(BT;R)$. Therefore,
the difference $\beta(p)-\beta(q)$ is divisible by $\eta(p,q)$ in
$H^{*}(BT;R)$.
The commutative diagram of inclusion maps

$\\{p,q\\}$$S_{e}$$M$

induces a commutative diagram of ring homomorphisms

$H^{*}_{T}(M;R)$$H^{*}(S_{e};R)$$H^{*}_{T}(\\{p,q\\};R)$

.

Let $\alpha\in H_{T}^{*}(M;R)$ and let $\beta\in H^{*}_{T}(S_{e};R)$ be its
image under the map $H^{*}_{T}(M;R)\rightarrow H^{*}_{T}(S_{e};R)$. Since the
images of $\alpha$ and $\beta$ under $H^{*}_{T}(M;R)\rightarrow
H^{*}_{T}(\\{p,q\\};R)$ resp. $H^{*}_{T}(S_{e};R)\rightarrow
H^{*}_{T}(\\{p,q\\};R)$ are equal, we have $\alpha(p)=\beta(p)$ and
$\alpha(q)=\beta(q)$. Hence,

$\alpha(p)-\alpha(q)=\beta(p)-\beta(q)$

is divisible by $\eta(p,q)$ in $H^{*}(BT;R)$. ∎

The converse of this lemma holds for rational coefficients.

######  Theorem 3.2.

(Goresky, Kottwitz and MacPherson [21].) Let $(M,\omega,T,\phi)$ be a
Hamiltonian GKM space and let $(\Gamma_{GKM},\eta)$ be its GKM graph. Let
$\alpha:M^{T}\rightarrow H^{*}(BT;\mathbb{Q})$ be a map such that for each
edge $(p,q)\in E_{GKM}$ the difference $\alpha(p)-\alpha(q)$ is divisible by
$\eta(p,q)$ in $H^{*}(BT;\mathbb{Q})$. Then $\alpha$ belongs to
$H^{*}_{T}(M;\mathbb{Q})$.

In the following, we prove an integer version of this theorem under the
assumption that for each fixed point $p$ the weights of the $T$-representation
on $T_{p}M$ are pairwise coprime.

######  Proposition 3.3.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space such that for each fixed
point $p\in M^{T}$ the weights of the $T$-representation on $T_{p}M$ are
pairwise coprime. Let $(\Gamma_{GKM},\eta)$ be its GKM graph. Given a map
$\alpha\colon M^{T}\rightarrow H^{*}(BT;\mathbb{Z})$, the following conditions
are equivalent.

  * (i)

The map $\alpha$ belongs to $H_{T}^{*}(M;\mathbb{Z})$.

  * (ii)

For each edge $(p,q)\in E_{GKM}$, the difference $\alpha(p)-\alpha(q)$ is
divisible by $\eta(p,q)$ in $H^{*}(BT;\mathbb{Z})$.

Before we prove Proposition 3.3, we show that the technical condition that the
weights of the $T$-representation on the tangent space are pairwise coprime
for each fixed point is satisfied whenever the complexity or the dimension of
the space is low.

######  Lemma 3.4.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space. Assume that one of the
following conditions is true.

  * •

The complexity of the $T$-action is equal to zero or one.

  * •

The dimension of the manifold $M$ is equal to four or six.

Then for each fixed point $p\in M^{T}$ the weights of the $T$-representation
on $T_{p}M$ are pairwise coprime in $H^{*}(BT;\mathbb{Z})$.

###### Proof.

Let $2n$ be the dimension of the manifold $M$ and let $d$ be the dimension of
the torus $T$. So the complexity of the $T$-action is $k=n-d$. Let $p\in
M^{T}$ be a fixed point and let $\alpha_{p,1},\dots,\alpha_{p,n}$ be the
weights of the $T$-representation on $T_{p}M$. By Lemma 2.2, the
$\mathbb{Z}$-span of the weights $\alpha_{p,1},\dots,\alpha_{p,n}$ is equal to
$\ell_{T}^{*}\cong\mathbb{Z}^{d}$. If the complexity is equal to zero, i.e.,
$d=n$, then the $n$ weights form a $\mathbb{Z}$-basis of $\ell_{T}^{*}$ and
they are pairwise coprime. Now assume that the complexity is equal to one,
i.e., $d=n-1$. Let $\beta_{1},...,\beta_{n-1}$ be a $\mathbb{Z}$-basis of
$\ell_{T}^{*}\cong\mathbb{Z}^{n-1}$ and let

$\displaystyle\operatorname{det}\colon\left(\ell_{T}^{*}\right)^{n-1}\rightarrow\mathbb{Z}$

be the determinant map such that

$\displaystyle\operatorname{det}\left(\beta_{1},...,\beta_{n-1}\right)=1.$

For each $i=1,...,n-1$ there exist integers $A_{i,1},...,A_{i,n}$ such that
$\beta_{i}=\sum_{j=1}^{n}A_{i,j}\cdot\alpha_{p,j}$. Therefore, we have

$\displaystyle\operatorname{det}\left(\beta_{1},...,\beta_{n-1}\right)\,=\,\sum_{j=1}^{n}C_{j}\cdot\operatorname{det}\left(\alpha_{p,1},...,\widehat{\alpha_{p,j}},...,\alpha_{n}\right),$
(3.1)

where $C_{1},...,C_{n}$ are integers. Now let $m\in\mathbb{Z}\setminus\\{0\\}$
be an integer that divides at least two of the weights
$\alpha_{p,1},\dots,\alpha_{p,n}$ in $H^{2}(BT;\mathbb{Z})$. Then each of the
$n$ terms in the right-hand side of (3.1) is an integer multiple of $m$. Since
the left-hand side of (3.1) is equal to $1$, we have $m=\pm 1$. Hence, the
weights are indeed pairwise coprime.
Now assume that the dimension of $M$ is four or six. By Lemma 2.4 the
dimension $d$ of the torus is greater or equal to $2$. Therefore, the
complexity of the space is zero or one, and so again the weights
$\alpha_{p,1},\dots,\alpha_{p,n}$ are pairwise coprime. ∎

####  3.0.1 Intermezzo: Kirwan Classes

To prove Proposition 3.3, we use results by Kirwan [29] on the existence of
so-called Kirwan classes. We repeat the definition of Kirwan classes. Before
we concentrate on the GKM case, we consider a more general setting, the case
that the fixed points are isolated. Let $(M,\omega,T,\phi)$ be a compact
Hamiltonian $T$-space with only isolated fixed points and let
$\xi\in\mathfrak{t}$ be a generic vector. For each $p\in M^{T}$ the
equivariant Euler class of negative tangent bundle at $p$ with respect to
$\phi^{\xi}$ is

$\displaystyle\Lambda_{p}^{-}=\prod_{\begin{subarray}{c}i=1,...,n\\\
\left\langle a_{p_{i}},\xi\right\rangle<0\end{subarray}}\alpha_{p,i},$

where $\alpha_{p,1},...,\alpha_{p,n}$ are the weights of the
$T$-representation on $T_{p}M$. Here, the empty product is equal to the
multiplicative identity of $H^{*}(BT;\mathbb{Z})$.

######  Lemma 3.5.

(Kirwan [29]) Let $(M,\omega,T,\phi)$ be a compact Hamiltonian $T$-space with
only isolated fixed points and let $\xi\in\mathfrak{t}$ be a generic vector.
For every fixed point $p\in M^{T}$, there exists an equivariant class
$\gamma_{p}\in H_{T}^{2\lambda(p)}(M;\mathbb{Z})$ such that

  * (i)

$\gamma_{p}(p)=\Lambda_{p}^{-}$ and

  * (ii)

$\gamma_{p}(q)=0$ for every fixed point $q\in M^{T}\setminus\\{p\\}$ with
$\phi^{\xi}(q)\leq\phi^{\xi}(p)$.

Moreover, for any choice of such classes, the set $\\{\gamma_{p}\\}_{p\in
M^{T}}$ is a basis for $H_{T}^{*}(M;\mathbb{Z})$ as a module over
$H^{*}(BT;\mathbb{Z})$.

A class that satisfies properties (i) and (ii) of Lemma 3.5 is called a Kirwan
class at $p$.

In general, Kirwan classes are not unique. Let $p$ and $q$ be fixed points and
let $\gamma_{p}$ and $\gamma_{q}$ be Kirwan classes at $p$ and $q$. If
$\lambda(p)\leq\lambda(q)$ and $\phi^{\xi}(p)\leq\phi^{\xi}(q)$, then
$\gamma_{p}+f\cdot\gamma_{q}$ is also a Kirwan class at $p$ for any $f\in
H^{2(\lambda(p)-\lambda(q))}(BT;\mathbb{Z})$. In order to recover the
multiplicative structure of $H_{T}^{*}(M;\mathbb{Z})$ it is enough to compute
the restrictions $\gamma_{p}(q)$ for all $p,q\in M^{T}$ for some set of Kirwan
classes $\\{\gamma_{p}\\}_{p\in M^{T}}$. In this case one can easily compute
the equivariant structure constants with respect to this basis; these are, for
all $q,p,r\in M^{T}$, the unique elements $c^{r}_{p,q}\in
H^{*}(BT;\mathbb{Z})$ such that $\gamma_{p}\cdot\gamma_{q}=\sum_{r\in
M^{T}}c^{r}_{p,q}\gamma_{r}$. Furthermore, if one knows these equivariant
structure constants, then the ordinary cohomology ring is also known. Indeed,
by the Kirwan Surjectivity Theorem [29], the restriction map

$\displaystyle r^{*}:H_{T}^{*}(M;\mathbb{Z})\rightarrow H^{*}(M;\mathbb{Z})$

is surjective and its kernel is the ideal generated by $H^{2}(BT;\mathbb{Z})$.
If we denote by $\tau_{p}$ the image of $\gamma_{p}$ for $p\in M^{T}$, then
$H^{2i}(M;\mathbb{Z})\cong\mathbb{Z}^{b_{2i}(M)}$ is the free abelian group
generated by the elements $\tau_{p}$ with $\lambda(p)=i$ and
$H^{odd}(M;\mathbb{Z})=0$. The multiplicative structure of
$H^{*}(M;\mathbb{Z})$ is given by

$\displaystyle\tau_{p}\cdot\tau_{q}=\sum_{\begin{subarray}{c}r\in M^{T}\\\
\lambda(r)=\lambda(p)+\lambda(q)\end{subarray}}c_{p,q}^{r}\tau_{r}\,,$

where the empty sum is equal to zero.

######  Remark 3.6.

Let $\alpha\in H_{T}^{*}(M;\mathbb{Z})$ be a class and let
$\\{\gamma_{p}\\}_{p\in M^{T}}$ be a set of Kirwan classes. For each $p\in
M^{T}$, there exists a unique element $c_{p}\in H^{*}(BT;\mathbb{Z})$ such
that $\alpha=\sum_{p\in M^{T}}c_{p}\cdot\gamma_{p}$. These coefficients can be
computed as follows. Given $p\in M^{T}$ and suppose that we already know
$c_{q}$ for all $q\in M^{T}$ with $\phi^{\xi}(q)<\phi^{\xi}(p)$. We have

$\displaystyle\alpha(p)=c_{p}\cdot\Lambda_{p}^{-}+\sum_{\begin{subarray}{c}q\in
M^{T}\\\ \phi^{\xi}(q)<\phi^{\xi}(p)\end{subarray}}c_{q}\cdot\gamma_{q}(p).$
(3.2)

Since $\Lambda_{p}^{-}\neq 0$, $c_{p}$ can be simply recovered from equation
(3.2).

A simple consequence of (3.2) is the following.

######  Lemma 3.7.

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space with only isolated fixed
points, let $\xi\in\mathfrak{t}$ be a generic vector and let $\alpha\in
H_{T}^{*}(M;\mathbb{Z})$.

  * (i)

If $p$ is a fixed point such that $\alpha(q)=0$ for all $q\in M^{T}$ with
$\phi^{\xi}(q)<\phi^{\xi}(p)$, then $\alpha(p)=f\cdot\Lambda_{p}^{-}$ for some
$f\in H^{*}(BT;\mathbb{Z})$.

  * (ii)

If $\alpha\in H^{2i}_{T}(M;\mathbb{Z})$ and $\alpha(q)=0$ for all $q\in M^{T}$
with $\lambda(q)\leq i$, then $\alpha=0$.

We now return to the GKM case.

######  Remark 3.8.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let
$\xi\in\mathfrak{t}$ be a generic vector. For any $p\in M^{T}$ the equivariant
Euler class of the negative tangent bundle at $p$ with respect to $\phi^{\xi}$
can be simply recovered from the GKM graph $(\Gamma_{GKM},\eta)$, as follows.
Note that if $p$ is the unique fixed point on which $\phi^{\xi}$ attains its
minimum (,i.e., $\lambda(p)=0$), then

$\displaystyle\Lambda_{p}^{-}\in
H_{T}^{2\lambda(p)}(p;\mathbb{Z})=H^{2\lambda(p)}(BT;\mathbb{Z})$

is the multiplicative identity of $H^{*}(BT;\mathbb{Z})$. If $p$ is not the
unique fixed point on which $\phi^{\xi}$ attains its minimum, then
$\lambda(p)\geq 1$ holds. There exist exactly $\lambda(p)$ fixed points
$q_{1},...,q_{\lambda(p)}$ such that $(p,q_{i})\in E_{GKM}$ and
$\phi^{\xi}(q_{i})<\phi^{\xi}(p)$ for $i=1,...,\lambda(p)$. We have

$\displaystyle\Lambda_{p}^{-}=\prod_{i=1}^{\lambda(p)}\eta(p,q_{i}).$

Now we are able to give the proof Proposition 3.3.

###### Proof of Proposition 3.3 .

That condition $(i)$ implies condition $(ii)$ follows directly from Lemma 3.1.
In order to prove the converse, let $\alpha:M^{T}\rightarrow
H^{*}(BT;\mathbb{Z})$ be a map that satisfies condition $(ii)$. Let
$\xi\in\mathfrak{t}$ be a generic vector and let
$p_{1},\dots,p_{\left|M^{T}\right|}\in M^{T}$ be the fixed points, ordered
such that

$\displaystyle\phi^{\xi}(p_{1})<\phi^{\xi}(p_{2})\leq\dots\leq\phi^{\xi}(p_{\left|M^{T}\right|-1})<\phi^{\xi}(p_{\left|M^{T}\right|}).$
(3.3)

We show by induction that for all $i=1,\dots\left|M^{T}\right|$, there exists
a class $\beta_{i}\in H_{T}^{*}(M;\mathbb{Z})$ such that

$\displaystyle\beta_{i}(p_{j})=\alpha(p_{j})\text{ for all }j=1,\dots,i.$
(3.4)

Note that $p_{1}$ is the unique fixed point on which $\phi^{\xi}$ attains its
minimum. The unique Kirwan class $\gamma_{p_{1}}\in H^{0}_{T}(M;\mathbb{Z})$
that belongs to $p_{1}$ is the constant map $M^{T}\rightarrow
H^{*}(BT;\mathbb{Z})$ that maps each fixed point to the multiplicative
identity of $H^{*}(BT;\mathbb{Z})$. Hence,
$\beta_{1}:=\alpha(p_{1})\cdot\gamma_{p_{1}}$ is a class in
$H_{T}^{*}(M;\mathbb{Z})$ that satisfies (3.4). For the induction step,
suppose that for a fixed $i\in\left\\{1,\dots,\left|M^{T}\right|-1\right\\}$,
there exists a class $\beta_{i}$ in $H_{T}^{*}(M;\mathbb{Z})$ that satisfies
(3.4). Consider the fixed point $p_{i+1}$. Since $\phi^{\xi}$ does not attain
its minimum at $p_{i+1}$, we have $\lambda(p_{i+1})\geq 1$. Let
$r_{1},\dots,r_{\lambda(p_{i+1})}$ be the unique fixed points such that for
all $k=1,\dots,\lambda(p_{i+1})$

$\displaystyle(p_{i+1},r_{k})\in E_{GKM}\,\text{ and
}\phi^{\xi}(r_{k})<\phi^{\xi}(p_{i+1}).$

Sine the fixed points are ordered such that (3.3) holds, we have that
$r_{1},\dots,r_{\lambda(p_{i+1})}\in\\{p_{1},\dots,p_{i}\\}$. Consider the map

$\displaystyle\tau:=\alpha-\beta_{i}:M^{T}\rightarrow H^{*}(BT;\mathbb{Z}).$

Note that $\tau(r_{k})=0$ for all $k=1,\dots,\lambda(p_{i+1})$. Since
$\beta_{i}\in H_{T}^{*}(M;\mathbb{Z})$, by Lemma 3.1

$\displaystyle\beta_{i}(p_{i+1})-\beta_{i}(r_{k})$

is divisible by $\eta(p_{i+1},r_{k})$ for all $k=1,\dots,\lambda(p_{i+1})$.
Hence, since $\alpha$ satisfies condition $(ii)$, we have

$\displaystyle\tau(p_{i+1})-\tau(r_{k})=\tau(p_{i+1})$

is divisible by $\eta(p_{i+1},r_{k})$ for all $k=1,\dots,\lambda(p_{i+1})$.
Since the weights $\eta(p_{i+1},r_{k})$ are pairwise coprime in
$H^{*}(BT;\mathbb{Z})$, there exists an element $f\in H^{*}(BT;\mathbb{Z})$
such that

$\displaystyle\tau(p_{i+1})=f\cdot\prod_{k=1}^{\lambda(p_{i+1})}\eta(p_{i+1},r_{k})=f\cdot\Lambda_{p_{i+1}}^{-}.$

Now let $\gamma_{p_{i+1}}$ be a Kirwan class at $p_{i+1}$, then

$\displaystyle\beta_{i+1}:=\beta_{i}+f\cdot\gamma_{p_{i+1}}$

is a class in $H_{T}^{*}(M;\mathbb{Z})$ that satisfies (3.4). ∎

###  3.1 Proof of Theorem 1.6

In this subsection, we prove Theorem 1.6. We first introduce a notation.

######  Definition 3.9.

Given two Hamiltonian GKM spaces $(M_{1},\omega_{1},T,\phi_{1})$ and
$(M_{2},\omega_{2},T,\phi_{2})$, let $(F,\theta)$ be an isomorphism between
the GKM graphs $(\Gamma_{1,GKM},\eta_{1})$ and $(\Gamma_{2,GKM},\eta_{2})$ of
these spaces. Then the isomorphism induces a ring isomorphism

$\displaystyle\Psi_{(F,\theta)}\,:\,\operatorname{Maps}(M_{1}^{T}\rightarrow
H^{*}(BT;\mathbb{Z}))\longrightarrow\operatorname{Maps}(M_{2}^{T}\rightarrow
H^{*}(BT;\mathbb{Z})),$

as follows. Since $\ell_{T}^{*}=H^{2}(BT;\mathbb{Z})$, the linear isomorphism
$\theta:\ell_{T}^{*}\rightarrow\ell_{T}^{*}$ extends in a canonical way to a
ring isomorphism $\widetilde{\theta}:H^{*}(BT;\mathbb{Z})\longrightarrow
H^{*}(BT;\mathbb{Z})$. The image of a map $\alpha:M_{1}^{T}\rightarrow
H^{*}(BT;\mathbb{Z})$ under $\Psi_{(F,\theta)}$ is the map
$\Psi_{(F,\theta)}(\alpha):M_{2}^{T}\rightarrow H^{*}(BT;\mathbb{Z})$ given by

$\displaystyle\Psi_{(F,\theta)}(\alpha)(q)=\widetilde{\theta}\left(\alpha(F^{-1}(q))\right)\quad\text{for
all }q\in M_{2}^{T}.$

######  Remark 3.10.

It is clear that the map $\Psi_{(F,\theta)}$ is a ring homomorphism. Moreover,
if $(F^{-1},\theta^{-1})$ is the inverse isomorphism to $(F,\theta)$, then

$\displaystyle\Psi_{(F^{-1},\theta^{-1})}\,:\,\operatorname{Maps}(M_{2}^{T}\rightarrow
H^{*}(BT;\mathbb{Z}))\longrightarrow\operatorname{Maps}(M_{1}^{T}\rightarrow
H^{*}(BT;\mathbb{Z})),$

is the inverse of $\Psi_{(F,\theta)}$. Hence, $\Psi_{(F,\theta)}$ is indeed a
ring isomorphism.

######  Lemma 3.11.

Given two Hamiltonian GKM spaces $(M_{1},\omega_{1},T,\phi_{1})$ and
$(M_{2},\omega_{2},T,\phi_{2})$, let $(F,\theta)$ be an isomorphism between
the GKM graphs $(\Gamma_{1,GKM},\eta_{1})$ and $(\Gamma_{2,GKM},\eta_{2})$ of
these spaces. The following holds.

  * (i)

For a map $\alpha:M_{1}^{T}\rightarrow H^{*}(BT;\mathbb{Z})$, the following
two conditions are equivalent.

    * (a)

For all $(p,q)\in E_{1,GKM}$, the difference $\alpha(p)-\alpha(q)$ is
divisible by $\eta_{1}(p,q)$ in $H^{*}(BT;\mathbb{Z})$.

    * (b)

For all $(p,q)\in E_{2,GKM}$, the difference
$\Psi_{(F,\theta)}\left(\alpha\right)(p)-\Psi_{(F,\theta)}\left(\alpha\right)(q)$
is divisible by $\eta_{2}(p,q)$ in $H^{*}(BT;\mathbb{Z})$.

  * (ii)

$\Psi_{(F,\theta)}$ maps the equivariant Chern classes of
$(M_{1},\omega_{1},T,\phi_{1})$ to the ones of
$(M_{2},\omega_{2},T,\phi_{2})$.

###### Proof.

By definition, $F:M_{1}^{T}\rightarrow M_{2}^{T}$ induces a bijection

$\displaystyle E_{1,GKM}\ni(p,q)\longmapsto(F(p),F(q))\in E_{2,GKM}$ (3.5)

such that

$\displaystyle\eta_{2}(F(p),F(q))=\theta(\eta_{1}(p,q)).$

  * (i)

Let $\alpha:M_{1}^{T}\rightarrow H^{*}(BT;\mathbb{Z})$ be a map and $(p,q)\in
E_{1,GKM}$ be an edge such that

$\displaystyle\alpha(p)-\alpha(q)=\eta_{1}(p,q)\cdot f$

for some $f\in H^{*}(BT;\mathbb{Z})$. We have

$\displaystyle\Psi_{(F,\theta)}\left(\alpha\right)(F(p))-\Psi_{(F,\theta)}\left(\alpha\right)(F(q))$
$\displaystyle=\widetilde{\theta}\left(\alpha(p)\right)-\widetilde{\theta}\left(\alpha(q)\right)$
$\displaystyle=\widetilde{\theta}\left(\alpha(p)-\alpha(q)\right)$
$\displaystyle=\widetilde{\theta}\left(\eta_{1}(p,q)\cdot f\right)$
$\displaystyle=\theta\left(\eta_{1}(p,q)\right)\cdot\widetilde{\theta}\left(f\right)$
$\displaystyle=\eta_{2}(F(p),F(q))\cdot\widetilde{\theta}\left(f\right),$

where $\widetilde{\theta}:H^{*}(BT;\mathbb{Z})\rightarrow
H^{*}(BT;\mathbb{Z})$ is the ring isomorphism induced by
$\theta:\ell_{T}^{*}\rightarrow\ell_{T}^{*}$. Since (3.5) is a bijection, we
conclude that condition (a) implies (b). By applying the same argument to
$\Psi_{(F^{-1},\theta^{-1})}$, we have that condition (b) implies (a).

  * (ii)

Fix a non-negative integer $k$ and let $c_{k}^{T}(M_{1})$ resp.
$c_{k}^{T}(M_{2})$ be the $k$-th equivariant Chern class of
$(M_{1},\omega_{1},T,\phi_{1})$ resp. $(M_{2},\omega_{2},T,\phi_{2})$.
Considered as a map $M_{1}^{T}\rightarrow H^{2k}(BT;\mathbb{Z})$, the class
$c_{k}^{T}(M_{1})$ is given by

$\displaystyle M_{1}^{T}\ni
p\longmapsto\sigma_{n,k}(\eta_{1}(p,p_{1}),\dots,\eta_{1}(p,p_{n})),$

where $\sigma_{n,k}$ is the elementary symmetric polynomial in $n$ variables
of degree $k$ and $p_{1},...,p_{n}$ are the fixed points with $(p,p_{i})\in
E_{1,GKM}$. The same holds for $c_{k}^{T}(M_{2})$. Let $q\in M_{2}^{T}$ be the
fixed point with $F(p)=q$. Then $q_{1}=F(p_{1}),...,q_{n}=F(p_{n})$ are the
fixed points with $(q,q_{i})\in E_{2,GKM}$. So we have

$\displaystyle\Psi_{(F,\theta)}\left(c_{k}^{T}(M_{1})\right)(q)$
$\displaystyle=\widetilde{\theta}\left(\sigma_{n,k}(\eta_{1}(p,p_{1}),\dots,\eta_{1}(p,p_{n}))\right)$
$\displaystyle=\sigma_{n,k}\left(\widetilde{\theta}(\eta_{1}(p,p_{1})),\dots,\widetilde{\theta}(\eta_{2}(p,p_{n}))\right)$
$\displaystyle=\sigma_{n,k}(\eta_{2}(F(p),F(p_{1})),\dots,\eta_{2}(F(p),F(p_{n})))$
$\displaystyle=\sigma_{n,k}(\eta_{2}(q,q_{1}),\dots,\eta_{2}(q,q_{n}))$
$\displaystyle=c_{k}^{T}(M_{2})(q).$

Hence, $\Psi_{(F,\theta)}\left(c_{k}^{T}(M_{1})\right)=c_{k}^{T}(M_{2})$
holds.

∎

We can now prove Theorem 1.6.

###### Proof of Theorem 1.6.

Let $(M_{1},\omega_{1},T,\phi_{1})$ and $(M_{2},\omega_{2},T,\phi_{2})$ be two
Hamiltonian GKM spaces such that for each fixed point $p\in M_{1}^{T}$ resp.
$q\in M_{2}^{T}$ the weights of the $T$-representation on $T_{p}M_{1}$ resp.
on $T_{q}M_{2}$ are pairwise coprime. Let $(F,\theta)$ be an isomorphism
between the GKM graphs $(\Gamma_{1,GKM},\eta_{1})$ and
$(\Gamma_{2,GKM},\eta_{1})$ of these spaces.
(a): By Proposition 3.3, we have that a map $\alpha:M_{1}^{T}\rightarrow
H^{*}(BT;\mathbb{Z})$ belongs to $H_{T}^{*}(M_{1};\mathbb{Z})$ if and only if
for each edge $(p,q)\in E_{1,GKM}$ the difference $\alpha(p)-\alpha(q)$ is
divisible by $\eta_{1}(p,q)$ in $H^{*}(BT;\mathbb{Z})$. The same holds for
$H_{T}^{*}(M_{2};\mathbb{Z})$. Let

$\displaystyle\Psi_{(F,\theta)}\,:\,\operatorname{Maps}(M_{1}^{T}\rightarrow
H^{*}(BT;\mathbb{Z}))\longrightarrow\operatorname{Maps}(M_{2}^{T}\rightarrow
H^{*}(BT;\mathbb{Z}))$

be the ring isomorphism that is induced by $(F,\theta)$. By Lemma 3.11, the
restriction of $\Psi_{(F,\theta)}$ to $H_{T}^{*}(M_{1};\mathbb{Z})$ gives a
ring isomorphism from $H_{T}^{*}(M_{1};\mathbb{Z})$ to
$H_{T}^{*}(M_{2};\mathbb{Z})$ that maps the equivariant Chern classes of
$(M_{1},\omega_{1},T,\phi_{1})$ to the ones of
$(M_{2},\omega_{2},T,\phi_{2})$. We denote this ring isomorphism by

$\mathbf{\varphi}:H_{T}^{*}(M_{1};\mathbb{Z})\rightarrow
H_{T}^{*}(M_{2};\mathbb{Z}).$

(b): Let $\alpha\in H^{*}_{T}(M_{1};\mathbb{Z})$ and let $\beta\in
H^{*}_{T}(M_{2};\mathbb{Z})$ be the image of $\alpha$ under $\varphi$. Let
$\widetilde{\theta}:H^{*}(BT;\mathbb{Z})\rightarrow H^{*}(BT;\mathbb{Z})$ be
the ring isomorphism that is induced by
$\theta:\ell_{T}^{*}\rightarrow\ell_{T}^{*}$. Since $\varphi$ is the
restriction of $\Psi_{(F,\theta)}$ to $H_{T}^{*}(M_{1};\mathbb{Z})$, for any
$f\in H^{*}(BT;\mathbb{Z})$ the following conditions are equivalent.

  * (1)

For each fixed $p\in M_{1}^{T}$, $\alpha(p)$ is divisible by $f$.

  * (2)

For each fixed $p\in M_{2}^{T}$, $\beta(p)$ is divisible by
$\widetilde{\theta}(f)$.

Let

$r_{1}^{*}:H_{T}^{*}(M_{1};\mathbb{Z})\rightarrow
H^{*}(M_{1};\mathbb{Z})\quad\text{and}\quad
r_{2}^{*}:H_{T}^{*}(M_{2};\mathbb{Z})\rightarrow H^{*}(M_{2};\mathbb{Z})$

be the restriction maps. Due to the Kirwan Surjectivity Theorem [29], the
kernel of $r_{1}^{*}$ resp. $r_{2}^{*}$ is the ideal in
$H_{T}^{*}(M_{1};\mathbb{Z})$ resp. $H_{T}^{*}(M_{2};\mathbb{Z})$ generated by
$H^{2}(BT;\mathbb{Z})$. This means that a class $\alpha\in
H_{T}^{*}(M_{1};\mathbb{Z})$ belongs to the kernel of $r_{1}^{*}$ if and only
if there exists an element $f\in H^{2k}(BT;\mathbb{Z})$ for some $k>0$ such
that for any $p\in M_{1}^{T}$, $\alpha(p)$ is divisible by $f$. The same holds
for the kernel of $r_{2}^{*}$. Since the conditions (1) and (2) are
equivalent, we conclude that $\varphi$ gives a bijection from the kernel of
$r_{1}^{*}$ to the one of $r_{2}^{*}$.

So there exists a ring isomorphism
$\widetilde{\varphi}:H^{*}(M_{1};\mathbb{Z})\rightarrow
H^{*}(M_{2};\mathbb{Z})$ such that the following diagram commutes.

$H^{*}_{T}(M_{1};\mathbb{Z})$$H^{*}(M_{1};\mathbb{Z})$$H^{*}_{T}(M_{2};\mathbb{Z})$$H^{*}(M_{2};\mathbb{Z})$$r_{1}^{*}$$\widetilde{\varphi}$$\varphi$$r_{2}^{*}$

Note that the restriction maps $r_{1}^{*}$ and $r_{2}^{*}$ map the equivariant
Chern classes to the ordinary Chern classes. Hence, since $\varphi$ maps
equivariant Chern classes to equivariant Chern classes, we have that
$\widetilde{\varphi}$ maps the Chern classes of $(M_{1},\omega_{1})$ to the
ones of $(M_{2},\omega_{2})$. ∎

###  3.2 Proof of Theorem 1.7

In this subsection we prove Theorem 1.7. We use a classification result for
certain six-manifolds by Jupp [24], Wall [41] and Z̆ubr [42]. First we need to
introduce several definitions.

######  Definition 3.12.

Let $M$ be a compact oriented and simply connected manifold of dimension six
with $H^{\operatorname{odd}}(M;\mathbb{Z})=0$. The system of invariants of $M$
is the quadruple $(H,\mu,\operatorname{w}_{2}(M),p_{1}(M))$ defined as
follows.

  * •

$H=H^{2}(M;\mathbb{Z})$ is the second cohomology group of $M$.

  * •

$\mu:H\otimes H\otimes H\rightarrow\mathbb{Z}$ is the map given by

$\mu(x,y,z)=\left\langle x\cup y\cup z,[M]\right\rangle_{M},$

where $[M]\in H_{6}(M;\mathbb{Z})$ is the fundamental class of $M$ and
$\left\langle,\right\rangle_{M}$ is the Kronecker pairing

$H^{6}(M;\mathbb{Z})\times H_{6}(M;\mathbb{Z})\rightarrow\mathbb{Z}.$

  * •

$\operatorname{w}_{2}(M)\in H^{2}(M;\mathbb{Z}_{2})\cong
H\times\mathbb{Z}_{2}$ is the second Stiefel-Whitney class, where
$H^{2}(M;\mathbb{Z}_{2})$ is identified with $H\times\mathbb{Z}_{2}$ via the
isomorphism induced by the homomorphism of coefficients
$\mathbb{Z}\rightarrow\mathbb{Z}_{2}$.

  * •

$p_{1}(M)\in
H^{4}(M;\mathbb{Z})\cong\operatorname{Hom}_{\mathbb{Z}}(H,\mathbb{Z})$ is the
first Pontrjagin class.

Let $M_{1}$ and $M_{2}$ be compact oriented and simply connected manifolds of
dimension six with
$H^{\operatorname{odd}}(M_{1};\mathbb{Z})=H^{\operatorname{odd}}(M_{2};\mathbb{Z})=0$.
Let $(H_{1},\mu_{1},\operatorname{w}_{2}(M_{1}),p_{1}(M_{1}))$ and
$(H_{2},\mu_{2},\operatorname{w}_{2}(M_{2}),p_{1}(M_{2}))$ be the systems of
invariants of $M_{1}$ and $M_{2}$. An isomorphism between these two systems of
invariants is a group isomorphism $\Phi:H_{1}\rightarrow H_{2}$ such that

$\Phi(\operatorname{w}_{2}(M_{1}))=\operatorname{w}_{2}(M_{2}),\quad\Phi^{*}(\mu_{2})=\mu_{1},\quad\text{and}\quad\Phi^{*}(p_{1}(M_{2}))=p_{1}(M_{1}).$

######  Theorem 3.13.

(Jupp [24], Wall [41] and Z̆ubr [42].) Let $M_{1}$ and $M_{2}$ be compact
oriented and simply connected manifolds of dimension six with
$H^{\operatorname{odd}}(M_{1};\mathbb{Z})=H^{\operatorname{odd}}(M_{2};\mathbb{Z})=0$.
An isomorphism from the system of invariants of $M_{1}$ to the one of $M_{2}$
induces an orientation-preserving diffeomorphism $M_{2}\rightarrow M_{1}$.

We can now prove Theorem 1.7.

###### Proof of Theorem 1.7.

Let $(M_{1},\omega_{1},T,\phi_{1})$ and $(M_{2},\omega_{2},T,\phi_{2})$ be
Hamiltonian GKM spaces of dimension six and assume that there exists an
isomorphism $(F,\theta)$ from the GKM graph of $(M_{1},\omega_{1},T,\phi_{1})$
to the one of $(M_{2},\omega_{2},T,\phi_{2})$. We need to show that this
isomorphism induces a (non-equivariant) diffeomorphism $M_{2}\rightarrow
M_{1}$.

As explained in subsection 2.2.3, since the set $M_{1}^{T}$ contains only
isolated points, $M_{1}$ is simply connected and
$H^{odd}(M_{1};\mathbb{Z})=0$. Moreover, the Euler characteristic of $M_{1}$
is not zero. So the Euler class of $M_{1}$, which is equal to the third Chern
class $c_{3}(M_{1})$ of $(M_{1},\omega_{1})$, is a non-zero element in
$H^{6}(M_{1};\mathbb{Z})\cong\mathbb{Z}$. We consider $M_{1}$ together with
the orientation such that $c_{3}(M_{1})$ is a positive multiple of the
fundamental class $[M_{1}]$. The same hold for $M_{2}$; we consider $M_{2}$
together with the orientation such that $c_{3}(M_{2})$ is a positive multiple
of the fundamental class $[M_{2}]$.

By Lemma 3.4 and Theorem 1.6 (b), the isomorphism $(F,\theta)$ between the GKM
graphs induces an isomorphism in cohomology

$\displaystyle\widetilde{\varphi}:H^{*}(M_{1};\mathbb{Z})\rightarrow
H^{*}(M_{2};\mathbb{Z}),$ (3.6)

that maps the Chern classes of $(M_{1},\omega_{1})$ to the one of
$(M_{2},\omega_{2})$.

Let $(H_{1},\mu_{1},\operatorname{w}_{2}(M_{1}),p_{1}(M_{1}))$ resp.
$(H_{2},\mu_{2},\ w_{2}(M_{2}),p_{1}(M_{2}))$ be the system of invariants of
$M_{1}$ resp. $M_{2}$. The restriction of $\widetilde{\varphi}$ to $H_{1}$
gives us a linear isomorphism

$\Phi\,:H_{1}\longrightarrow H_{2}.$

By Theorem 3.13 it enough to prove that $\Phi$ is an isomorphism between the
systems of invariants.

  * •

For $i=1,2$, the first Chern class of $(M_{i},\omega_{i})$ is mapped under the
homomorphism $H^{2}(M_{i};\mathbb{Z})\rightarrow H^{2}(M_{i};\mathbb{Z}_{2})$
to the second Stiefel-Whitney class of $M_{i}$. Since $\Phi$ maps
$c_{1}(M_{1})$ to $c_{1}(M_{2})$ we have
$\Phi(\operatorname{w}_{2}(M_{1}))=\operatorname{w}_{2}(M_{2})$.

  * •

Since (3.6) is an isomorphism that maps $c_{3}(M_{1})$ to $c_{3}(M_{2})$, we
have that $\Phi^{*}(\mu_{2})=\mu_{1}$.

  * •

For $i=1,2$, the first Pontrjagin class $p_{1}(M_{i})$ of $M_{i}$ is given by
$c_{1}^{2}(M_{i})-2c_{2}(M_{i})$, where $c_{1}(M_{i})$ and $c_{2}(M_{i})$ are
the first and second Chern class of $(M_{i},\omega_{i})$. Since $\Phi$ maps
the Chern classes $(M_{1},\omega_{1})$ to the ones of $(M_{2},\omega_{2})$, we
have $\Phi^{*}(p_{1}(M_{2}))=p_{1}(M_{1}))$.

∎

##  4 Positive Hamiltonian GKM Spaces

In this section, we introduce the notion of being positive for Hamiltonian GKM
spaces. In particular, we prove some properties of six-dimensional positive
Hamiltonian GKM spaces. First, we define the first Chern Class map of a
Hamiltonian GKM space.

######  Definition 4.1.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space, $(\Gamma_{GKM},\eta)$ its
GKM graph, and $c_{1}(M)$ the first Chern class of $(M,\omega)$. The first
Chern class map of $(M,\omega,T,\phi)$ is the map
$\mathcal{C}_{1}:E_{GKM}\rightarrow\mathbb{Z}$ given by

$\displaystyle\mathcal{C}_{1}(e)=\int_{S_{(p,q)}}i^{*}_{S_{(p,q)}}\left(c_{1}(M)\right)$

for each edge $e=(p,q)\in E_{GKM}$, where $S_{(p,q)}$ is the unique
$T$-invariant two-sphere fixed by a codimensional one subtorus of $T$ that
contains $p$ and $q$, and $i^{*}_{S_{(p,q)}}$ is the map induced on
$H^{2}(\,\cdot\,;\mathbb{Z})$ by the inclusion $i:S_{(p,q)}\hookrightarrow M$.

######  Definition 4.2.

A Hamiltonian GKM space is called positive if its first Chern class map
$\mathcal{C}_{1}:E_{GKM}\rightarrow\mathbb{Z}$ is positive, i.e.,
$\mathcal{C}_{1}(e)>0$ for all $e\in E_{GKM}$.

Due to the ABBV localization formula, the first Chern Class map can be
computed from the GKM graph. This is the content of the following lemma. In
particular, the GKM graph contains the information of whenever the Hamiltonian
GKM space is positive.

######  Lemma 4.3.

Consider a Hamiltonian GKM space $(M,\omega,T,\phi)$ with GKM graph
$(\Gamma_{GKM},\eta)$. For each $e=(p,q)\in E_{GKM}$ the following holds,

$\displaystyle\mathcal{C}_{1}(e)=\frac{\sum_{e^{\prime}\in
E_{GKM}^{p,i}}\eta(e^{\prime})-\sum_{e^{\prime}\in
E_{GKM}^{q,i}}\eta(e^{\prime})}{\eta(e)}.$ (4.1)

###### Proof.

Let $c_{1}^{T}(M)\in H^{2}_{T}(M,\mathbb{Z})$ be the equivariant extension of
the first Chern class $c_{1}(M)$ of $(M,\omega)$. Since $S_{(p,q)}$ is a
$T$-invariant two-sphere, we have

$\displaystyle\mathcal{C}_{1}(e)=\int_{S_{(p,q)}}i^{*}_{S_{(p,q)}}\left(c_{1}(M)\right)=\int_{S_{(p,q)}}i^{*}_{S_{(p,q)}}\left(c_{1}^{T}(M)\right).$
(4.2)

The torus $T$ acts on $S_{(p,q)}$ with isolated fixed points $p$ and $q$. The
weight of the $T$-representation on $T_{p}S_{(p,q)}$ resp. $T_{q}S_{(p,q)}$ is
$\eta(e)$ resp. $-\eta(e)$. Hence, the equivariant Euler class of the normal
bundle of $p$ resp. $q$ is $\eta(e)$ resp. $-\eta(e)$. Moreover, the
restriction $i_{\\{p\\}}^{*}(c_{1}^{T}(M))$ is equal to the sum of the weights
of the $T$-representation on $T_{p}M$. Due to Remark 2.6 $(i)$, the latter is
equal to $\sum_{e^{\prime}\in E_{GKM}^{p,i}}\eta(e^{\prime})$. For the same
reason $i_{\\{q\\}}^{*}(c_{1}^{T}(M))=\sum_{e^{\prime}\in
E_{GKM}^{q,i}}\eta(e^{\prime})$. Therefore, the ABBV localization formula
(Theorem 2.1) implies that the right-hand sides of equations (4.1) and (4.2)
coincide. ∎

######  Remark 4.4.

Since $\mathcal{C}_{1}$ is completely determined by the GKM graph
$(\Gamma_{GKM},\eta)$, we call $\mathcal{C}_{1}$ also the first Chern class
map of $(\Gamma_{GKM},\eta)$. Moreover, we say that $(\Gamma_{GKM},\eta)$ is
positive if $\mathcal{C}_{1}(e)>0$ for all $e\in E_{GKM}$.

######  Remark 4.5.

Let $e=(p,q)\in E_{GKM}$ be an edge. By Lemma 2.8 there exists a compatible
connection

$\nabla_{e}:E_{GKM}^{p,i}\longrightarrow E_{GKM}^{q,i}$

along $e$. Let $2n$ be the dimension of $M$ and let $p_{1},\dots,p_{n-1}$
resp. $q_{1},\dots,q_{n-1}$ be the unique $n-1$ fixed points in
$M^{T}\setminus\\{p,q\\}$ such that

$(p,p_{i})\in E_{GKM}^{p,i}\quad\text{and}\quad(q,q_{i})\in E_{GKM}^{q,i},$

ordered so that

$\nabla_{e}(p,p_{i})=(q,q_{i})$

for all $i=1,...,n-1.$ So there exist integers $a_{1},...,a_{n-1}$ such that

$\displaystyle\eta(p,p_{i})-\eta(q,q_{i})=a_{i}\cdot\eta(p,q)\quad\text{for
all }i=1,...,n-1.$

By the formula in Lemma 4.3 we have

$\displaystyle\mathcal{C}_{1}(e)$ $\displaystyle=\frac{\sum_{e^{\prime}\in
E_{GKM}^{p,i}}\eta(e^{\prime})-\sum_{e^{\prime}\in
E_{GKM}^{q,i}}\eta(e^{\prime})}{\eta(e)}$
$\displaystyle=\frac{\eta(p,q)-\eta(q,p)}{\eta(p,q)}+\sum_{i=1}^{n-1}\frac{\eta(p,p_{i})-\eta(q,q_{i})}{\eta(p,q)}$
$\displaystyle=2+a_{1}+\dots a_{n-1}.$

######  Lemma 4.6.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space, where $(M,\omega)$ is a
monotone symplectic manifold. Then this space is positive.

###### Proof.

Since $(M,\omega,T,\phi)$ is GKM the fixed point set $M^{T}$ is finite. Since
$(M,\omega)$ is monotone and the torus action is effective and Hamiltonian, by
[17, Proposition 5.2], $(M,\omega)$ is positive monotone, i.e., there exists
an $r\in\mathbb{R}_{>0}$ such that $c_{1}(M)=r\cdot[\omega]$. Let $e=(p,q)\in
E_{GKM}$ be an edge. We have that

$\displaystyle\mathcal{C}_{1}(e)=\int_{S_{(p,q)}}i^{*}_{S_{(p,q)}}\left(c_{1}(M)\right)=r\cdot\int_{S_{(p,q)}}i^{*}_{S_{(p,q)}}\omega.$
(4.3)

Since $S_{(p,q)}$ is an embedded symplectic submanifold of $(M,\omega)$, the
right hand side of Equation (4.3) is positive. ∎

###  4.1 An Upper Bound for the Number of Fixed Points in Dimension Six

In this subsection we prove that the number of fixed points of a positive six-
dimensional Hamiltonian GKM space is at most $16$ and we give an obstruction
for the first Chern class map. This is the content of Corollary 4.12. This
corollary is a direct consequence of results by Godinho-Sabatini [16] and
Godinho-von Heymann-Sabatini [17], which we recall here.

######  Lemma 4.7.

[16, Corollary 3.1] Let $(M,\omega,T,\phi)$ be a compact Hamiltonian $T$-space
of dimension $2n$ with only isolated fixed points. For $p=0,...,n$ let
$b_{2p}$ be the $2p$-th Betti number of $M$. Then

$\displaystyle\int_{M}c_{1}(M)c_{n-1}(M)=\sum_{p=0}^{n}b_{2p}\left[6p(p-1)+\frac{5n-3n^{2}}{2}\right],$

where $c_{1}(M)$ resp. $c_{n-1}(M)$ is the first resp. $(n-1)$-th Chern class
of $(M,\omega)$.

######  Remark 4.8.

That Lemma 4.7 holds in case the torus $T$ has dimension one, i.e., $T\cong
S^{1}$, is the content of [16, Corollary 3.1]. Since each Hamiltonian
$T$-space admits a subcircle $S^{1}\subset T$ such that $M^{S^{1}}=M^{T}$
holds, it follows that Lemma 4.7 is also true whenever the dimension of the
torus is greater than one.

Before we state the next lemma, we need to introduce a notation.

######  Definition 4.9.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let
$(\Gamma_{GKM},\eta)$ be its GKM graph. An orientation $\sigma$ for the edge
set $E_{GKM}$ is a subset $E^{\sigma}_{GKM}$ of $E_{GKM}$ such that for each
$(p,q)\in E_{GKM}$ exactly one of the following two conditions is true.

  * •

$(p,q)\in E^{\sigma}_{GKM}$ and $(q,p)\notin E^{\sigma}_{GKM}$

  * •

$(q,p)\in E^{\sigma}_{GKM}$ and $(p,q)\notin E^{\sigma}_{GKM}$

######  Remark 4.10.

Let $2n$ be the dimension of $M$. The graph $\Gamma_{GKM}=(M^{T},E_{GKM})$ is
$n$-valent, i.e., for each fixed point $p$ there exist exactly $n$ edges whose
initial point is $p$. Therefore, the cardinalities of $M^{T}$ and $E_{GKM}$
are related by

$\displaystyle\left|E_{GKM}\right|=n\cdot\left|M^{T}\right|.$

Whenever $\sigma$ is an orientation of the edge set, then the cardinality of
$E_{GKM}^{\sigma}$ is equal to half of the one of $E_{GKM}$. Hence,

$\displaystyle\left|E_{GKM}^{\sigma}\right|=\frac{n}{2}\cdot\left|M^{T}\right|.$

######  Lemma 4.11.

(c.f. [17, Lemma 4.13]) Choose an orientation $\sigma$ of the edge set
$E_{GKM}$. Then

$\displaystyle\sum_{e\in
E^{\sigma}_{GKM}}\mathcal{C}_{1}(e)=\int_{M}c_{1}(M)c_{n-1}(M).$

Lemma 4.11 follows directly from [17, Lemma 4.13]. Since our setting is
slightly different from the one in [17], we give the proof of Lemma 4.11.

###### Proof of Lemma 4.11.

The proof is a simple application of the ABBV Localization Formula (Theorem
2.1). For each fixed point $p\in M^{T}$, let $\alpha_{p,1},\dots,\alpha_{p,n}$
be the weights of the $T$-representation on $T_{p}M$. Note that the set of
these weights is equal to the set $\\{\eta(e^{\prime})\\}_{e^{\prime}\in
E_{GKM}^{p,i}}$. Let $e=(p,q)\in E_{GKM}$ be an edge. By Lemma 4.3 we have

$\displaystyle\mathcal{C}_{1}(e)=\dfrac{\alpha_{p,1}+\dots+\alpha_{p,n}}{\eta(e)}-\dfrac{\alpha_{q,n}+\dots+\alpha_{q,n}}{\eta(e)}.$

For each fixed point $p$ and each weight $\alpha_{p,j}$ there exists exactly
one edge $e\in E_{GKM}$ such that $i(e)=p$ and $\eta(e)=\alpha_{p,j}$. Note
that $\eta(\bar{e})=-\alpha_{p,j}$ and either $e\in E^{\sigma}_{GKM}$ or
$\bar{e}\in E^{\sigma}_{GKM}$, where $\bar{e}$ is the unique edge with
$(i(\bar{e}),t(\bar{e}))=(t(e),i(e))$. We conclude that

$\displaystyle\sum_{e\in E^{\sigma}_{GKM}}\mathcal{C}_{1}(e)=\sum_{p\in
M^{T}}\left[\sum_{j=1}^{n}\dfrac{\alpha_{p,1}+...+\alpha_{p,n}}{\alpha_{p,j}}\right].$

For the equivariant extensions $c_{1}^{T}(M)$ and $c_{n-1}^{T}(M)$ of
$c_{1}(M)$ and $c_{n-1}(M)$ we have

$\displaystyle\int_{M}c_{1}(M)c_{n-1}(M)=\int_{M}c_{1}^{T}(M)c_{n-1}^{T}(M).$

For each fixed point $p\in M^{T}$ we have

$\displaystyle
i^{*}_{p}\left(c_{1}^{T}(M)c_{n-1}^{T}(M)\right)=\left(\sum_{j=1}^{n}\alpha_{p,j}\right)\cdot\left(\sum_{k=1}^{n}\prod_{l=1,l\neq
k}^{n}\alpha_{p,l}\right)$

and the equivariant Euler class of $T_{p}M$ is $\prod_{j=1}^{n}\alpha_{p,j}$.
Therefore, the ABBV Localization Formula gives

$\displaystyle\int_{M}c_{1}^{T}(M)c_{n-1}^{T}(M)=\sum_{p\in
M^{T}}\left[\frac{i^{*}_{p}\left(c_{1}^{T}(M)c_{n-1}^{T}(M)\right)}{\prod_{j=1}^{n}\alpha_{p,j}}\right]=\sum_{p\in
M^{T}}\left[\sum_{j=1}^{n}\dfrac{\alpha_{p,1}+...+\alpha_{p,n}}{\alpha_{p,j}}\right].$

The lemma follows. ∎

The following corollary is a direct consequence of Lemma 4.7 and Lemma 4.11.

######  Corollary 4.12.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space of dimension $2n$ and let
$(\Gamma_{GKM},\eta)$ be its GKM graph. Choose an orientation $\sigma$ of the
edge set $E_{GKM}$. Then the following hold.

  * (i)

$\displaystyle\sum_{e\in
E^{\sigma}_{GKM}}\mathcal{C}_{1}(e)=\sum_{p=0}^{n}b_{2p}\left[6p(p-1)+\frac{5n-3n^{2}}{2}\right],$

where $b_{0},\dots,b_{2n}$ are the even Betti numbers of $M$.

  * (ii)

If the dimension of $M$ is equal to six, then

$\displaystyle\sum_{e\in E^{\sigma}_{GKM}}\mathcal{C}_{1}(e)=24.$

  * (iii)

If $(M,\omega,T,\phi)$ is positive and the dimension of $M$ is equal to six,
then the number of fixed points is at most $16$.

###### Proof.

  * (i)

This statement follows directly from Lemma 4.7 and Lemma 4.11.

  * (ii)

Assume that the dimension of $M$ is equal to six, i.e., $n$ is equal to $3$.
Since $(M,\omega,T,\phi)$ is a Hamiltonian GKM space, the underlying
symplectic manifold is compact and connected. Therefore, $b_{0}=b_{6}=1$
holds. Moreover, by the Poincaré Duality Theorem we have that $b_{2}=b_{4}$
holds. Hence, the statement follows from $(i)$.

  * (iii)

Assume that the dimension of $M$ is equal to six and that the space is
positive, i.e., $\mathcal{C}_{1}(e)$ is a positive integer for all $e\in
E_{GKM}$. Then it follows from $(ii)$ that

$\displaystyle\left|E_{GKM}^{\sigma}\right|\leq 24.$

By Remark 4.10 we have
$\left|E^{\sigma}_{GKM}\right|=\frac{3}{2}\left|M^{T}\right|$. Hence, the
number of fixed points is at most $16$.

∎

###  4.2 Special Kirwan Classes

In this subsection we consider Hamiltonian GKM spaces that are weak index
increasing with respect to a generic vector $\xi$ and we prove the existence
of special Kirwan classes for such a space. We also show that six-dimensional
positive Hamiltonian GKM spaces are weak index increasing with respect to any
generic vector.

######  Definition 4.13.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let
$\xi\in\mathfrak{t}$ be a generic vector. Its GKM graph $(\Gamma_{GKM},\eta)$
is called index increasing resp. weak index increasing with respect to $\xi$
if

$\displaystyle\lambda(p)<\lambda(q)\quad\text{resp.}\quad\lambda(p)\leq\lambda(q)$

holds for any edge $(p,q)\in E_{GKM}$ with $\phi^{\xi}(p)<\phi(q)^{\xi}$.

######  Lemma 4.14.

Let $(M,\omega,T,\phi)$ be a six-dimensional positive Hamiltonian GKM space.
Then its GKM graph $(\Gamma_{GKM},\eta)$ is weak index increasing with respect
to each generic vector $\xi\in\mathfrak{t}$.

###### Proof.

Let $\xi\in\mathfrak{t}$ be a generic vector and assume that
$(\Gamma_{GKM},\eta)$ is not weak index increasing with respect to $\xi$. We
show that $(M,\omega,T,\phi)$ is not positive. Indeed by this assumption there
exists an edge $e=(p,q)\in E_{GKM}$ such that $\lambda(p)>\lambda(q)$ and
$\phi^{\xi}(p)<\phi^{\xi}(q)$. By Remark 2.10 there exists an $\tilde{r}>0$
such that $\eta(p,q)=\tilde{r}\cdot\left(\phi(q)-\phi(p)\right)$. So we have

$\displaystyle\left\langle\eta(p,q)\,,\,\xi\right\rangle=\tilde{r}\cdot\left(\left\langle\phi(q),\xi\right\rangle-\left\langle\phi(p),\xi\right\rangle\right)=\tilde{r}\cdot\left(\phi^{\xi}(q)-\phi^{\xi}(p)\right)>0.$
(4.4)

Since $M$ is six-dimensional, we have that

$\lambda(p),\lambda(q)\in\\{0,1,2,3\\}.$

Note that $\lambda(p)=3$ can not happen, because in this case $\phi^{\xi}$
attains its maximum at $p$, which contradicts $\phi^{\xi}(p)<\phi^{\xi}(q)$.
For the same reason $\lambda(q)=0$ can not happen, because in this case
$\phi^{\xi}$ attains its minimum at $q$. Hence, we have $\lambda(p)=2$ and
$\lambda(q)=1$. Let

$\nabla_{e}:E_{GKM}^{p,i}\rightarrow E_{GKM}^{q,i}$

be a compatible connection along the edge $e=(p,q)$ and let $p_{1},p_{2}$ and
$q_{1},q_{2}$ be the fixed points in $M^{T}\setminus\\{p,q\\}$ with
$(p,p_{i}),(q,q_{i})\in E_{GKM}$ for $i=1,2$, ordered so that

$\displaystyle\nabla_{e}(p,p_{i})=(q,q_{i})$

for $i=1,2$. So there exist integers $a_{1}$ and $a_{2}$ such that

$\displaystyle\eta(p,p_{i})-\eta(q,q_{i})=a_{i}\cdot\eta(p,q)$ (4.5)

for $i=1,2$. Recall that $\lambda(p)$ resp. $\lambda(q)$ is the number of
weights of the $T$-representation on $T_{p}M$ resp. $T_{q}M$ such that
$\left\langle\cdot\,,\xi\right\rangle$ is negative. Since $\lambda(p)=2$,
$\lambda(q)=1$ and $\left\langle\eta(p,q),\xi\right\rangle>0$, we have

$\displaystyle\left\langle\eta(p,p_{i})\,,\,\xi\right\rangle<0\quad\text{and
}\quad\left\langle\eta(q,q_{i})\,,\,\xi\right\rangle>0$ (4.6)

for $i=1,2$. By combining (4.4), (4.5) and (4.6), we conclude that $a_{1}$ and
$a_{2}$ are negative integers. So by Lemma 4.3 and its Remark 4.5 we have that

$\displaystyle\mathcal{C}_{1}(e)=2+a_{1}+a_{2}\leq 0.$

Hence, the space is indeed not positive. ∎

######  Definition 4.15.

Given a Hamiltonian GKM space $(M,\omega,T,\phi)$, let $(\Gamma_{GKM},\eta)$
be its GKM graph and let $\xi\in\mathfrak{t}$ be a generic vector. An
ascending path from a fixed point $p$ to another fixed point $q$ is a
$(k+1)$-tuple $\nu=(p_{0},...,p_{k})$ of points in $M^{T}$ such that
$p_{0}=p$, $p_{k}=q$ and

$(p_{i-1},p_{i})\in
E_{GKM}\,\,\text{and}\,\,\phi^{\xi}(p_{i-1})<\phi^{\xi}(p_{i})$

for $i=1,\dots,k$. Moreover, for each fixed point $p\in M^{T}$, the stable set
of $p$, denoted by $\Xi_{p}$ is the set of points $q\in M^{T}$ such that there
exists an ascending path from $p$ to $q$, including $p$ itself.

In the following lemma we sum up some properties about ascending paths and the
stable sets. These properties follow directly from the definitions.

######  Lemma 4.16.

Let $p,q\in M^{T}$ be two different fixed points. Then the following hold.

  * (i)

If $q\in\Xi_{p}$ then $\phi^{\xi}(p)<\phi^{\xi}(q)$.

  * (ii)

If $\lambda(q)=0$, then $q\notin\Xi_{p}$. If $\lambda(q)>0$, then
$q\notin\Xi_{p}$ if and only if for all $r\in M^{T}$ with $(r,q)\in E_{GKM}$
and $\phi^{\xi}(r)<\phi^{\xi}(q)$ we have $r\notin\Xi_{p}$.

  * (iii)

If the GKM graph of $(M,\omega,T,\phi)$ is index increasing resp. weak index
increasing with respect to $\xi$ and $q\in\Xi_{p}$ , then
$\lambda(p)<\lambda(q)$ resp. $\lambda(p)\leq\lambda(q)$.

######  Proposition 4.17.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space and let
$\xi\in\mathfrak{t}$ be a generic vector such that the GKM graph
$(\Gamma_{GKM},\eta)$ is weak index increasing with respect to $\xi$. Let
$p\in M^{T}$ with $\lambda(p)=1$. Then there exists a unique Kirwan class
$\gamma_{p}\in H_{T}^{2}(M;\mathbb{Z})$ at $p$ such that for $q\in M^{T}$ the
following hold.

  * (i)

$\gamma_{p}(q)=\Lambda_{p}^{-}$ if $\lambda(q)=1$ and $q\in\Xi_{p}$.

  * (ii)

$\gamma_{p}(q)=0$ if $q\notin\Xi_{p}$.

###### Proof.

Note that $p\in\Xi_{p}$. By Lemma 4.16 $(i)$ for all $q\in
M^{T}\setminus\\{p\\}$ with $\phi^{\xi}(q)\leq\phi^{\xi}(p)$ we have
$q\notin\Xi_{p}$. Hence, a class $\gamma_{p}$ with properties $(i)$ and $(ii)$
is indeed a Kirwan class at $p$. Moreover, that such a class is unique is easy
to see. Namely, let $\gamma_{p}$ and $\widetilde{\gamma}_{p}$ be two such
Kirwan classes at $p$. For all $q\in M^{T}$ with $\lambda(q)\leq 1$,
$\gamma_{p}(q)=\widetilde{\gamma}_{p}(q)$ holds and the degree of each of
these classes is two. Hence, by Lemma 3.7 (ii), we have
$\gamma_{p}=\widetilde{\gamma}_{p}$.
Now we prove the existence of such a class. Consider the set

$\displaystyle\mathcal{L}$ $\displaystyle:=\left\\{q\in
M^{T}\,\,|\,\,\phi^{\xi}(p)\leq\phi^{\xi}(q)\right\\}.$

Note that by Lemma 4.16 $(i)$ we have $\Xi_{p}\subset\mathcal{L}$. Let

$\displaystyle
p=q_{1},q_{2},...,q_{\left|\mathcal{L}\right|-1},q_{\left|\mathcal{L}\right|}$

be the points of $\mathcal{L}$ ordered so that

$\displaystyle\phi^{\xi}(q_{1})\leq\phi^{\xi}(q_{2})\leq...\leq\phi^{\xi}(q_{\left|\mathcal{L}\right|-1})\leq\phi^{\xi}(q_{\left|\mathcal{L}\right|}).$

By induction over $i=1,...,\left|\mathcal{L}\right|$ we show that for all such
$i$ there exists a class $\beta_{i}\in H_{T}^{2}(M;\mathbb{Z})$ that satisfies
the following properties (a.$i$) and (b).

  * (a.$i$)

For all $j=1,...,i$

$\displaystyle\beta_{i}(q_{j})=\Lambda_{q}^{-}\quad\text{if}\,\,\lambda(q_{j})=1\,\,\text{and}\,\,q_{j}\in\Xi_{p}$
$\displaystyle\beta_{i}(q_{j})=0\quad\,\,\,\,\,\text{if}\,\,q_{j}\notin\Xi_{p}.$

  * (b)

$\beta_{i}(r)=0$ if $r\in M^{T}$ and $\phi^{\xi}(r)<\phi^{\xi}(p)$.

The induction base is true. Any Kirwan class at $p$ satisfies the properties
(a.$1$) and (b). Now assume that for a fixed
$i=1,...,\left|\mathcal{L}\right|-1$ there exists a class $\beta_{i}\in
H_{T}^{2}(M;\mathbb{Z})$ that satisfies (a.$i$) and (b). Consider the fixed
point $q_{i+1}$. We have four cases.

  * 1.case : $\lambda(q_{i+1})\geq 2$ and $q_{i+1}\in\Xi_{p}$

  * 2.case : $\lambda(q_{i+1})\geq 2$ and $q_{i+1}\notin\Xi_{p}$

  * 3.case : $\lambda(q_{i+1})=1$ and $q_{i+1}\in\Xi_{p}$

  * 4.case : $\lambda(q_{i+1})=1$ and $q_{i+1}\notin\Xi_{p}$

1.case : Since (a.${i+1}$) and (b) do not force any obstruction for
$\beta_{i+1}(q_{i+1})$, we can choose $\beta_{i+1}=\beta_{i}$.
2.case : We show that $\beta_{i}(q_{i+1})=0$. Hence, we can choose
$\beta_{i+1}=\beta_{i}$. Since $\lambda(q_{i+1})\geq 2$ there exist two
different fixed points $r_{1},r_{2}\in M^{T}\setminus\\{q_{i+1}\\}$ such that

$(r_{1},q_{i+1}),(r_{2},q_{i+1})\in
E_{GKM}\quad\text{and}\quad\phi^{\xi}(r_{1}),\phi^{\xi}(r_{2})<\phi^{\xi}(q_{i+1}).$

If $\phi^{\xi}(r_{1})<\phi^{\xi}(p)$, then (b) implies $\beta_{i}(r_{1})=0$.
If $\phi^{\xi}(p)\leq\phi^{\xi}(r_{1})$, then $r_{1}\in\mathcal{L}$ and
$\phi^{\xi}(r_{1})<\phi^{\xi}(q_{i+1})$. This implies that $r_{1}=q_{j}$ for
some $j=1,...,i$. Moreover, since $q_{i+1}\notin\Xi_{p}$, Lemma 4.16 $(ii)$
implies that $r_{1}=q_{j}\notin\Xi_{p}$. Therefore, (a.$i$) implies
$\beta_{i}(r_{i})=0$. Hence, in both cases $\phi^{\xi}(r_{1})<\phi^{\xi}(p)$
or $\phi^{\xi}(p)\leq\phi^{\xi}(r_{1})$, we have $\beta_{i}(r_{1})=0$. For the
same reason we have also $\beta_{i}(r_{2})=0$. Since
$(r_{1},q_{i+1}),(r_{2},q_{i+1})\in E_{GKM}$, by Lemma 3.1 there exist
integers $A_{1}$ and $A_{2}$ such that

$\displaystyle\beta_{i}(q_{i+1})=\beta_{i}(q_{i+1})-\beta_{i}(r_{1})=A_{1}\cdot\eta(q_{i+1},r_{1})\quad\text{and}$
$\displaystyle\beta_{i}(q_{i+1})=\beta_{i}(q_{i+1})-\beta_{i}(r_{2})=A_{2}\cdot\eta(q_{i+1},r_{2}).$

Since $\eta(q_{i+1},r_{1})$ and $\eta(q_{i+1},r_{2})$ are linearly
independent, we conclude that $A_{1}=A_{2}=0$ and $\beta_{i}(q_{i+1})=0$.
3.case : Since $\lambda(q_{i+1})=1$ there exists a unique fixed point $r\in
M^{T}$ such that

$(r,q_{i+1})\in E_{GKM}\quad\text{and}\quad\phi^{\xi}(r)<\phi^{\xi}(q_{i+1}).$

Since $q_{i+1}\in\Xi_{p}$, by Lemma 4.16 $(ii)$ we have $r\in\Xi_{p}$.
Moreover, since $\Xi_{p}\subset\mathcal{L}$ we have $r=q_{j}$ for some
$j=1,...,i$. Note that also $q_{i+1}\in\Xi_{r}$. Since the GKM graph of
$(M,\omega,T,\phi)$ is weak index increasing with respect to $\xi$, by Lemma
4.16 $(iii)$ we have

$1=\lambda(p)\leq\lambda(r)\leq\lambda(q_{i+1})=1.$

So we have that $\lambda(r)=1$. Since (a.$i$) holds for the class $\beta_{i}$,
we conclude that $\beta_{i}(r)=\Lambda_{p}^{-}$. Since $(r,q_{i+1})\in
E_{GKM}$, by Lemma 3.1 there exists an integer $A$ such that

$\beta_{i}(q_{i+1})-\Lambda_{p}^{-}=\beta_{i}(q_{i+1})-\beta_{i}(r)=A\cdot\eta(q_{i+1},r).$

Note that since $\lambda(q_{i+1})=1$ holds, we have
$\eta(q_{i+1},r)=\Lambda_{q_{i+1}}^{-}$. So we have

$\beta_{i}(q_{i+1})=\Lambda_{p}^{-}+A\cdot\Lambda_{q_{i+1}}^{-}.$

Let $\alpha_{q_{i+1}}\in H_{T}^{2}(M;\mathbb{Z})$ be a Kirwan class at
$q_{i+1}$. So the class

$\beta_{i+1}=\beta_{i}-A\cdot\alpha_{q_{i+1}}$

satisfies the properties (a.$i+1$) and (b).
4.case : Since $\lambda(q_{i+1})=1$, there exists a unique fixed point $r\in
M^{T}$ such that

$(r,q_{i+1})\in E_{GKM}\quad\text{and}\quad\phi^{\xi}(r)<\phi^{\xi}(q_{i+1}).$

Since $q_{i+1}\notin\Xi_{p}$, by Lemma 4.16 $(ii)$ we have $r\notin\Xi_{p}$.
So if $r\in\mathcal{L}$ then we have $r=q_{j}$ for some $j=1,...,i$. So
$(a.$i$)$ and $r\notin\Xi_{p}$ implies that $\beta_{i}(r)=0$. If
$r\notin\mathcal{L}$ then (b) also implies $\beta_{i}(r)=0$. Since
$(r,q_{i+1})\in E_{GKM}$, there exists an integer $A$ such that

$\beta_{i}(q_{i+1})=\beta_{i}(q_{i+1})-\beta_{i}(r)=A\cdot\eta(q_{i+1,r}).$

Note that since $\lambda(q_{i+1})=1$ holds, we have
$\eta(q_{i+1},r)=\Lambda_{q_{i+1}}^{-}$. So we have

$\beta_{i}(q_{i+1})=A\cdot\Lambda_{q_{i+1}}^{-}.$

Let $\alpha_{q_{i+1}}\in H_{T}^{2}(M;\mathbb{Z})$ be a Kirwan class at
$q_{i+1}$. So the class

$\beta_{i+1}=\beta_{i}-A\cdot\alpha_{q_{i+1}}$

satisfies the properties (a.$i+1$) and (b).
We conclude that there exists a class $\beta_{\left|\mathcal{L}\right|}\in
H^{2}_{T}(M;\mathbb{Z})$ that satisfies the properties
(a.$\left|\mathcal{L}\right|$) and (b). In fact, such a class
$\beta_{\left|\mathcal{L}\right|}$ satisfies the desired properties $(i)$ and
$(ii)$. ∎

##  5 Constructing (Abstract) GKM Graphs

In this section, we prove further statements that are needed for the
classification of GKM graphs of positive Hamiltonian GKM spaces of dimension
six. Let $(\Gamma_{GKM},\eta)$ be the GKM graph of such a space. Then the
graph $\Gamma_{GKM}$ is a simple, connected and $3$-valent (see Definition
5.2). By Corollary 4.12, the graph has at most $16$ vertices and we have only
finitely many possibilities for the first Chern class map
$\mathcal{C}_{1}:E_{GKM}\rightarrow\mathbb{Z}$. Since simple and connected
$3$-valent graphs with at most $16$ vertices are classified [4], the
classification problem is strongly related to the following question.

######  Question 5.1.

Let $T$ be a torus with dual lattice $\ell_{T}^{*}$. Let $\Gamma=(V,E)$ be a
simple, connected and $n$-valent graph and let $\mathcal{D}\colon
E\rightarrow\mathbb{Z}$ be a map.

  * •

Does the pair $(\Gamma,\mathcal{D})$ support a Hamiltonian GKM graph, i.e.,
does there exist a map $\eta:E\rightarrow\ell_{T}^{*}$ such that the pair
$(\Gamma,\eta)$ is the GKM graph of a Hamiltonian GKM space
$(M,\omega,T,\phi)$ of dimension $2n$ and for all edges $e\in E$

$\displaystyle\mathcal{C}_{1}(e)=\mathcal{D}(e),$

where $\mathcal{C}_{1}\colon E\rightarrow\mathbb{Z}$ is the first Chern class
map?

  * •

And if so, how many such maps $\eta:E\rightarrow\ell_{T}^{*}$ exist (up to
isomorphisms and projections of GKM graphs), and can we compute such maps?

In the first part of this section, we introduce abstract GKM graphs and
formulate Question 5.1 for abstract GKM graphs. In the second part of this
section, we show that Question 5.1 for abstract GKM graphs can be solved by
methods of linear algebra. In the last part of this section, we consider
positive (abstract) GKM graphs that are coming from six-dimensional
Hamiltonian GKM spaces.

###  5.1 Abstract GKM Graphs

Let $\Gamma=(V,E)$ be a graph with directed edges. This means that there exist
an initial map $i:E\rightarrow V$ and a terminal map $t:E\rightarrow V$. We
associate to each vertex $v\in V$ the following two sets

$\displaystyle E_{v}^{i}:=\\{e\in E\mid
i(e)=v\\}\quad\quad\text{and}\quad\quad E_{v}^{t}:=\\{e\in E\mid t(e)=v\\}.$

######  Definition 5.2.

A graph $\Gamma=(V,E)$ with directed edges is called simple if the following
three conditions are true.

  * •

The graph has no loops, i.e., $i(e)\neq t(e)$ for all $e\in E$.

  * •

The graph has no double edges, i.e., if $i(e)=i(e^{\prime})$ and
$t(e)=t(e^{\prime})$ for $e,e^{\prime}\in E$ then $e=e^{\prime}$.

  * •

For each edge $e\in E$ there exists a unique edge $\bar{e}$ such that
$i(\bar{e})=t(e)$ and $t(\bar{e})=i(e)$.

Moreover, such a graph $\Gamma=(V,E)$ is called $n$-valent if for each vertex
$v\in V$ the cardinality of $E_{v}^{i}$ (or equivalently, the cardinality of
$E_{v}^{t}$) is equal to $n$.

Note that by the second item in Definition 5.2 we can consider the edge set of
a simple graph $\Gamma=(V,E)$ as a subset of $V\times V$. Hence, we write an
edge $e\in E$ also as $(v,w)$, where $i(e)=v$ and $t(e)=w$. Such a graph is
called connected if for any two different vertices $v,w\in V$, there exists a
sequence $v_{0},...,v_{k}$ in $V$ such that $v_{0}=v$, $v_{k}=w$ and
$(v_{i},v_{i+1})\in E$ for $i=0,\dots,k-1$. A connection along an edge $e\in
E$ is a bijection

$\displaystyle\nabla_{e}:E_{i(e)}^{i}\longrightarrow E_{t(e)}^{i}$

such that $\nabla_{e}(e)=\bar{e}$, where $\bar{e}$ is the edge with
$(i(\bar{e}),t(\bar{e}))=(t(e),i(e))$.

######  Definition 5.3.

Let $n$ and $d$ be positive integers. An abstract $(n,d)$-GKM graph
$(\Gamma,\operatorname{w})$ is a connected simple and $n$-valent graph
$\Gamma=(V,E)$ together with a weight map
$\operatorname{w}:E\rightarrow\mathbb{Z}^{d}\setminus\\{0\\}$ that is
antisymmetric, i.e., for all $e\in E$ we have
$\operatorname{w}(\bar{e})=-\operatorname{w}(e)$, such that the following
hold.

  * (i)

For each vertex $v\in V$, the $\mathbb{Z}$-span of vectors
$\operatorname{w}(e)$ for $e\in E^{i}_{v}$ is equal to $\mathbb{Z}^{d}$.

  * (ii)

For each vertex $v\in V$, the vectors $\operatorname{w}(e)$ for $e\in
E^{i}_{v}$ are pairwise linearly independent in $\mathbb{Z}^{d}$ over
$\mathbb{Z}$.

  * (iii)

For each $e\in E$, there exists a connection

$\displaystyle\nabla_{e}:E_{i(e)}^{i}\longrightarrow E_{t(e)}^{i}$

that is compatible with the weight map $\operatorname{w}$. The latter means
that for each $e^{\prime}\in E_{i(e)}^{i}$ there exists an integer
$a_{e,e^{\prime}}$ such that

$\displaystyle\operatorname{w}(e^{\prime})-\operatorname{w}(\nabla_{e}(e^{\prime}))=a_{e,e^{\prime}}\cdot\operatorname{w}(e);$

the former means that $\nabla_{e}$ is a bijection and $\nabla_{e}(e)=\bar{e}$.

######  Remark 5.4.

If $n\geq 2$, then for an abstract $(n,d)$-GKM graph we have $2\leq d\leq n$,
as follows from items $(i)$ and $(ii)$ of Definition 5.3. If $n=1$ then $d=1$
as well.

######  Definition 5.5.

Let $\Gamma_{1}=(E_{1},V_{1})$ and $\Gamma_{2}=(E_{2},V_{2})$ be two simple
graphs. An isomorphism between $\Gamma_{1}$ and $\Gamma_{2}$ is a bijection
$F:V_{1}\rightarrow V_{2}$ such that for each two vertices $v,w\in V_{1}$, we
have that $(v,w)\in E_{1}$ if and only if $(F(v),F(w))\in E_{2}$.

######  Remark 5.6.

An isomorphism $F:V_{1}\rightarrow V_{2}$ between simple graphs
$\Gamma_{1}=(E_{1},V_{1})$ and $\Gamma_{2}=(E_{2},V_{2})$ induces a bijection
$E_{1}\rightarrow E_{2}$, $(v,w)\mapsto(F(v),F(w)).$

######  Definition 5.7.

Let $(\Gamma_{1}=(V_{1},E_{1}),\operatorname{w_{1}})$ and
$(\Gamma_{2}=(V_{2},E_{2}),\operatorname{w_{2}})$ be two abstract $(n,d)$\-
GKM graphs. An isomorphism $(F,\theta)$ between
$(\Gamma_{1},\operatorname{w_{1}})$ and $(\Gamma_{2},\operatorname{w_{2}})$ is
an isomorphism $F$ between the simple graphs $\Gamma_{1}$ and $\Gamma_{2}$
together with a linear isomorphism
$\theta:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d}$ such that for all $(v,w)\in
E_{1}$

$\displaystyle\theta(\operatorname{w_{1}(v,w)})=\operatorname{w_{2}}(F(v),F(w)).$

####  5.1.1 Relations between Hamiltonian and Abstract GKM Graphs

The GKM graphs of Hamiltonian GKM spaces give in a natural way examples of
abstract GKM graphs. Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space of
dimension $2n$ and let $(\Gamma_{GKM},\eta)$ be its GKM graph. The graph
$\Gamma_{GKM}$ is a simple and connected $n$-valent graph. Let
$\chi:\ell_{T}^{*}\rightarrow\mathbb{Z}^{d}$ be a linear isomorphism, where
$d$ is the dimension of the torus $T$. Then the graph $\Gamma_{GKM}$ together
with the composition $\chi\circ\eta:E_{GKM}\rightarrow\mathbb{Z}^{d}$ is an
abstract $(n,d)$-GKM graph. In particular, $\chi$ induces a map

$\displaystyle\mathcal{L}_{\chi}\colon\left\\{\begin{array}[]{r}\text{GKM
graphs of Hamiltonian }\\\ \text{ GKM $T$-spaces of dimension $2n$}\\\
\end{array}\right\\}\longrightarrow\left\\{\begin{array}[]{r}\text{abstract
$(n,d)$ }\\\ \text{ -GKM graphs}\\\ \end{array}\right\\}.$

Note that two GKM graphs of Hamiltonian GKM $T$-spaces are isomorphic (in the
sense of Definition 2.7) if and only if their images under
$\mathcal{L}_{\chi}$ are isomorphic abstract GKM graphs (in the sense of
Definition 5.7). Hence, $\mathcal{L}_{\chi}$ induces an injective map

$\displaystyle\mathcal{L}_{n,d}\colon\left\\{\begin{array}[]{r}\text{isomorphism
classes of}\\\ \text{GKM graphs of Hamiltonian }\\\ \text{ GKM $T$-spaces of
dimension $2n$}\\\
\end{array}\right\\}\longrightarrow\left\\{\begin{array}[]{r}\text{isomorphism
classes of}\\\ \text{abstract $(n,d)$ }\\\ \text{ -GKM graphs}\\\
\end{array}\right\\}.$ (5.7)

The map $\mathcal{L}_{n,d}$ is canonical. Namely, it does not depend on the
choice of the linear isomorphism $\chi:\ell_{T}^{*}\rightarrow\mathbb{Z}^{d}$.
Since $\mathcal{L}_{n,d}$ is injective, we can consider the set of isomorphism
classes of the GKM graphs of Hamiltonian GKM $T$-space of dimension $2n$ as a
subset of the isomorphism classes of abstract $(n,d)$-GKM graphs. An abstract
$(n,d)$-GKM graph resp. its isomorphism class is called Hamiltonian if its
isomorphism class lies in the image of $\mathcal{L}_{n,d}$.

Moreover, the concepts of projections and of the first Chern class maps
defined for GKM graphs of Hamiltonian GKM spaces generalize to abstract GKM
graphs.

######  Definition 5.8.

Let $n$, $d$ and $d^{\prime}$ be positive integers with $d^{\prime}<d\leq n$.
Let $\Gamma=(V,E)$ be a simple and connected $n$-valent graph and let
$\operatorname{w}:E\rightarrow\mathbb{Z}^{d}\setminus\\{0\\}$ and
$\operatorname{w}^{\prime}:E\rightarrow\mathbb{Z}^{d^{\prime}}\setminus\\{0\\}$
maps such that $(\Gamma,\operatorname{w})$ and
$(\Gamma,\operatorname{w}^{\prime})$ are abstract GKM graphs. Then
$(\Gamma,\operatorname{w}^{\prime})$ is a projection of
$(\Gamma,\operatorname{w})$ if there exists a linear surjection
$\theta:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d^{\prime}}$ such that

$\displaystyle\operatorname{w}^{\prime}(e)=\theta\left(\operatorname{w}(e)\right)$

for all $e\in E$.

The concept of projection of an abstract GKM graph can be considered as a
generalization of projection of a Hamiltonian GKM graph. Let
$(M,\omega,T,\phi)$ be a Hamiltonian GKM space with GKM graph
$(\Gamma_{GKM},\eta)$ and let $T^{\prime}$ be a subtorus of $T$ such that the
$T^{\prime}$-action on $(M,\omega)$ is also GKM. Note that
$(M,\omega,T^{\prime},i^{*}\circ\phi)$ is a Hamiltonian GKM space, where
$i^{*}:\mathfrak{t}^{*}\rightarrow(\mathfrak{t}^{\prime})^{*}$ is the dual map
of the inclusion $i:\mathfrak{t}^{\prime}\rightarrow\mathfrak{t}$ from the Lie
algebra of $T^{\prime}$ to the one of $T$. The GKM graph of this Hamiltonian
GKM space is $(\Gamma_{GKM},\eta^{\prime})$, where
$\eta^{\prime}=i^{*}\circ\eta$. So the Hamiltonian GKM graph
$(\Gamma_{GKM},\eta^{\prime})$ is a projection of $(\Gamma_{GKM},\eta)$. Let

$\displaystyle\chi:\ell_{T}^{*}\rightarrow\mathbb{Z}^{d}\quad\text{and}\quad\chi^{\prime}:\ell_{T^{\prime}}^{*}\rightarrow\mathbb{Z}^{d^{\prime}}$

be linear isomorphisms, where $d$ resp. $d^{\prime}$ is the dimension of $T$
resp. $T^{\prime}$.
Then the abstract GKM graph $(\Gamma_{GKM},\chi^{\prime}\circ\eta^{\prime})$
is a projection of $(\Gamma_{GKM},\chi\circ\eta)$. Indeed,

$\displaystyle\theta=\chi^{\prime}\circ
i^{*}\circ\chi^{-1}\colon\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d^{\prime}}$

is a linear surjection such that

$\displaystyle\theta\left(\chi\circ\eta(e)\right)=\chi^{\prime}\circ\eta^{\prime}(e)$

for all $e\in E_{GKM}$. That the converse is true is the statement of the
following lemma.

######  Lemma 5.9.

Let $(\Gamma,\operatorname{w})$ be an abstract $(n,d)$-GKM graph that is
Hamiltonian and let $(\Gamma,\operatorname{w}^{\prime})$ be an abstract
$(n,d^{\prime})$-GKM graph that is a projection of
$(\Gamma,\operatorname{w})$. Then $(\Gamma,\operatorname{w}^{\prime})$ is
Hamiltonian. Explicitly, if $(M,\omega,T,\phi)$ is a Hamiltonian GKM space
with GKM graph $(\Gamma_{GKM},\eta)$ such that

  * •

$\Gamma_{GKM}=\Gamma$, and

  * •

$\operatorname{w}=\chi\circ\eta$, where
$\chi:\ell_{T}^{*}\rightarrow\mathbb{Z}^{d}$ is a linear isomorphism,

then there exists a subtorus $T^{\prime}$ of $T$ such that

  * •

$(M,\omega,T^{\prime},i^{*}\circ\phi)$ is a Hamiltonian GKM space, and

  * •

$\operatorname{w}^{\prime}=\chi^{\prime}\circ\eta^{\prime}$, where
$(\Gamma,\eta^{\prime})$ is the GKM graph of
$(M,\omega,T^{\prime},i^{*}\circ\phi)$ and
$\chi^{\prime}:\ell_{T^{\prime}}^{*}\rightarrow\mathbb{Z}^{d^{\prime}}$ is a
linear isomorphism.

###### Proof.

Since $(\Gamma,\operatorname{w}^{\prime})$ is a projection of
$(\Gamma,\operatorname{w})$, there exists a linear surjection
$\theta:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d^{\prime}}$ such that
$\operatorname{w}^{\prime}=\theta\circ\operatorname{w}$. The kernel of
$\theta\circ\chi$ is a subgroup of $\ell_{T}^{*}$ of rank $d-d^{\prime}$. Let
$T^{\prime}$ be the subtorus of $T$ whose Lie algebra is

$\displaystyle\left\\{\xi\in\mathfrak{t}\,|\,\left\langle
x,\xi\right\rangle=0\text{ for all }x\in\ker(\theta\circ\chi)\right\\}.$

The dimension of $T^{\prime}$ is $d^{\prime}$. Let
$i:\mathfrak{t}^{\prime}\rightarrow\mathfrak{t}$ be the inclusion from the Lie
algebra of $T^{\prime}$ to the one of $T$ and let
$i^{*}:\mathfrak{t}^{*}\rightarrow(\mathfrak{t}^{\prime})^{*}$ be its dual
map. We have that

$\displaystyle\ker(i^{*})\cap\ell_{T}^{*}=\ker(\theta\circ\chi).$

In particular, there exists a unique linear map
$\chi^{\prime}:\ell_{T^{\prime}}^{*}\rightarrow\mathbb{Z}^{d^{\prime}}$ such
that the following diagram commutes.

$\ell_{T}^{*}$$\ell_{T^{\prime}}^{*}$$\,\,\mathbb{Z}^{d}$$\mathbb{Z}^{d^{\prime}}$$i^{*}$$\chi^{\prime}$$\chi$$\theta$

Since $\theta\circ\chi$ is surjective, we have that $\chi^{\prime}$ is a
linear isomorphism. Moreover, $i^{*}\circ\phi$ is a moment map for the
$T^{\prime}$-action on $(M,\omega)$. So $(M,\omega,T^{\prime},i^{*}\circ\phi)$
is a Hamiltonian $T^{\prime}$-space. We show that this space is GKM. Let $p\in
M^{T}$ be a fixed point of the $T$-action and let
$\alpha_{p,1},...,\alpha_{p,n}\in\ell_{T}^{*}$ be the weights of the
$T$-representation on $T_{p}M$. We need to show that
$i^{*}\left(\alpha_{p,1}\right),\dots,i^{*}\left(\alpha_{p,n}\right)\in\ell_{T^{\prime}}^{*}$
are pairwise linearly independent. By Remark 2.6 $(i)$ the weights
$\alpha_{p,1},...,\alpha_{p,n}$ are equal to
$\eta(p,p_{1}),...,\eta(p,p_{n})$, where $p_{1},...,p_{n}\in
M^{T}\setminus\\{p\\}$ are the $n$ fixed points such that $(p,p_{i})\in E$ for
$i=1,...,n$. Since $\operatorname{w}^{\prime}=\theta\circ\operatorname{w}$,
$\operatorname{w}=\chi\circ\eta$ and $\theta\circ\chi=\chi^{\prime}\circ
i^{*}$, we have that

$\operatorname{w}^{\prime}(p,p_{i})=\theta\left(\operatorname{w}(p,p_{i})\right)=\theta\circ\chi\left(\eta(p,p_{i})\right)=\chi^{\prime}\circ
i^{*}\left(\eta(p,p_{i})\right)$

for all $i$. Since $(\Gamma,\operatorname{w}^{\prime})$ is an abstract GKM
graph, we have that $\operatorname{w}^{\prime}(p,p_{i})$ and
$\operatorname{w}^{\prime}(p,p_{j})$ are linearly independent for $i\neq j$.
So $i^{*}\left(\eta(p,p_{i})\right)$ and $i^{*}\left(\eta(p,p_{j})\right)$ are
linearly independent for $i\neq j$. We conclude that
$(M,\omega,T^{\prime},i^{*}\circ\phi)$ is GKM. Its GKM graph is
$(\Gamma,\eta^{\prime})$, where $\eta^{\prime}=i^{*}\circ\eta$ and
$\operatorname{w}^{\prime}=\chi^{\prime}\circ\eta^{\prime}$. Hence,
$(\Gamma,\operatorname{w}^{\prime})$ is Hamiltonian. ∎

The first Chern class map of a Hamiltonian GKM space is the map
$\mathcal{C}_{1}:E_{GKM}\rightarrow\mathbb{Z}$ that maps an edge $e\in
E_{GKM}$ of the GKM graph to the evaluation of the first Chern class on the
symplectic two-sphere that belongs to the edge $e$. By Lemma 4.3 this map can
be computed from the GKM graph. Therefore, we can extend the definition of the
first Chern class map to abstract GKM graphs.

######  Definition 5.10.

Let $(\Gamma=(V,E),\operatorname{w})$ be an abstract $(n,d)$-GKM graph. Its
first Chern class map $\mathcal{C}_{1}:E\rightarrow\mathbb{Z}$ is the map such
that

$\displaystyle\sum_{e^{\prime}\in
E_{i(e)}^{i}}\operatorname{w}(e^{\prime})\,-\,\sum_{e^{\prime}\in
E_{t(e)}^{i}}\operatorname{w}(e^{\prime})=\mathcal{C}_{1}(e)\cdot\operatorname{w}(e)$
(5.8)

holds for all $e\in E$. The first Chern class map is well defined, because
$\operatorname{w}(e)\neq 0$ for all $e\in E$ and by Definition 5.3 $(iii)$ the
left hand side of Equation (5.8) is an integer multiple of
$\operatorname{w}(e)$.

######  Remark 5.11.

Let $(M,\omega,T,\phi)$ be a Hamiltonian GKM space with GKM graph
$(\Gamma_{GKM},\eta)$ and let $\chi:\ell_{T}^{*}\rightarrow\mathbb{Z}^{d}$ be
a linear isomorphism. Then by Lemma 4.3 the first Chern class map
$\mathcal{C}_{1}^{Ham}:E_{GKM}\rightarrow\mathbb{Z}$ of $(\Gamma_{GKM},\eta)$
and the first Chern class map
$\mathcal{C}_{1}^{Abs}:E_{GKM}\rightarrow\mathbb{Z}$ of the abstract GKM graph
$(\Gamma_{GKM},\chi\circ\eta)$ coincide. Hence, Definition 5.10 extends the
definition of the first Chern class (Definition 4.1) for abstract GKM graphs.

We close this subsection with two lemmas about properties of the first Chern
class map.

######  Lemma 5.12.

Let $(\Gamma_{1},\operatorname{w}_{1})$ and
$(\Gamma_{2},\operatorname{w}_{2})$ be two abstract GKM graphs and let
$\mathcal{C}_{1}^{1}:E_{1}\rightarrow\mathbb{Z}$ and
$\mathcal{C}_{1}^{2}:E_{2}\rightarrow\mathbb{Z}$ be their first Chern class
maps. Then the following hold.

  * •

The first Chern class map is invariant under isomorphism, i.e., if there
exists an isomorphism $(F,\theta)$ between $(\Gamma_{1},\operatorname{w}_{1})$
and $(\Gamma_{2},\operatorname{w}_{2})$, then

$\mathcal{C}_{1}^{1}(v,w)=\mathcal{C}_{1}^{2}((F(v),F(w)))$

for all $(v,w)\in E_{1}$.

  * •

The first Chern class map is invariant under projections, i.e., if
$(\Gamma_{2},\operatorname{w}_{2})$ is a projection of
$(\Gamma_{1},\operatorname{w}_{1})$, so $\Gamma_{1}=\Gamma_{2}$ and
$E_{1}=E_{2}$, then

$\mathcal{C}_{1}^{1}(e)=\mathcal{C}_{1}^{2}(e)$

for all $e\in E_{1}=E_{2}$.

###### Proof.

The proof of this lemma follows directly from the definitions. ∎

######  Lemma 5.13.

Let $(\Gamma,\operatorname{w})$ be an abstract GKM graph. Then its first Chern
class map $\mathcal{C}_{1}:E\rightarrow\mathbb{Z}$ is symmetric, i.e.,
$\mathcal{C}_{1}(\bar{e})=\mathcal{C}_{1}(e)$ for all $e\in E$.

###### Proof.

By the definition of the first Chern class map we have

$\displaystyle\sum_{e^{\prime}\in
E_{i(\bar{e})}^{i}}\operatorname{w}(e^{\prime})\,-\,\sum_{e^{\prime}\in
E_{t(\bar{e})}^{i}}\operatorname{w}(e^{\prime})=\mathcal{C}_{1}(\bar{e})\cdot\operatorname{w}(\bar{e}).$

The left hand side of this equation is equal to

$\displaystyle-\left(\sum_{e^{\prime}\in
E_{i({e})}^{i}}\operatorname{w}(e^{\prime})\,-\,\sum_{e^{\prime}\in
E_{t({e})}^{i}}\operatorname{w}(e^{\prime})\right),$

because $i(\bar{e})=t(e)$ and $t(\bar{e})=i(e)$. Again by the definition of
first the Chern class map this latter term is equal to
$-\mathcal{C}_{1}(e)\cdot\operatorname{w}(e)$. We obtain
$\mathcal{C}_{1}(\bar{e})\cdot\operatorname{w}(\bar{e})=-\mathcal{C}_{1}(e)\cdot\operatorname{w}(e)$.
Since $\operatorname{w}$ is antisymmetric, i.e.,
$\operatorname{w}(\bar{e})=-\operatorname{w}(e)$ and $\operatorname{w}(e)\neq
0$, the claim of the lemma follows. ∎

###  5.2 GKM Skeletons

We return to Question 5.1 for abstract GKM graphs. First we generalize the
notation of orientation as in Definition 4.9 to simple graphs.

######  Definition 5.14.

Let $\Gamma=(V,E)$ be a simple $n$-valent graph. An orientation $\sigma$ of
the edge set $E$ is a subset $E^{\sigma}$ of $E$ such that for each $e\in E$
exactly one of the following two conditions is true.

  * •

$e\in E^{\sigma}$ and $\bar{e}\notin E^{\sigma}$

  * •

$\bar{e}\in E^{\sigma}$ and $e\notin E^{\sigma}$

Here, $\bar{e}$ is the edge such that $i(\bar{e})=t(e)$ and $t(\bar{e})=i(e)$.

######  Remark 5.15.

Let $\Gamma=(V,E)$ be a simple $n$-valent graph and let $\sigma$ be an
orientation for the edge set $E$. The cardinalities of the vertex set $V$, the
edge set $E$ and the set $E^{\sigma}$ are related by the equations

$\displaystyle\left|E\right|=n\cdot\left|V\right|\quad\text{and}\quad\left|E^{\sigma}\right|=\frac{1}{2}\left|E\right|.$

Let $\Gamma=(V,E)$ be a connected, simple and $n$-valent graph and let
$\mathcal{D}:E\rightarrow\mathbb{Z}$ be a map. The analogue of Question 5.1
for abstract GKM graphs is to ask whenever there exists a map
$\operatorname{w}:E\rightarrow\mathbb{Z}^{d}\setminus\\{0\\}$ such that
$(\Gamma,\operatorname{w})$ is an abstract $(n,d)$-GKM graph and
$\mathcal{C}_{1}(e)=\mathcal{D}(e)$ for all $e\in E$. Note that, by definition
of the first Chern class map, such a map $\operatorname{w}$ must satisfy the
linear equation

$\displaystyle\sum_{e^{\prime}\in
E_{i(e)}^{i}}\operatorname{w}(e^{\prime})\,-\,\sum_{e^{\prime}\in
E_{t(e)}^{i}}\operatorname{w}(e^{\prime})=\mathcal{D}(e)\cdot\operatorname{w}(e)$
(5.9)

for all $e\in E$. So this question can be considered as a task of linear
algebra. Note that if such a map $\operatorname{w}$ exists, then it is
antisymmetric, i.e., $\operatorname{w}(\bar{e})=-\operatorname{w}(e)$ for all
$e\in E$. Moreover, since the first Chern class map is symmetric (Lemma 5.13),
we have that $\mathcal{D}$ is symmetric. In particular, (5.9) holds for $e\in
E$ if and only if it holds for $\bar{e}$. Let us fix an orientation $\sigma$
of the edge set $E$. So we have that (5.9) holds for all $e\in E$ if and only
if it holds for all $e\in E^{\sigma}$. In order to analyze equations (5.9) for
all $e\in E^{\sigma}$, it makes sense to fix an order for the elements of
$E^{\sigma}$. For this reason we introduce GKM skeletons.

######  Definition 5.16.

An $n$-GKM skeleton
$(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ is a connected
simple $n$-valent graph $\Gamma=(V,E)$ together with an orientation $\sigma$
of the edge set, an ordering222By Remark 5.15 the cardinality of $E^{\sigma}$
is equal to half of the one of $E$ $\operatorname{Ord}(E^{\sigma})$

$\displaystyle e_{1},\dots,e_{\frac{1}{2}\left|E\right|}$

of $E^{\sigma}$ and a vector
$\mathbf{d}=(d_{1},...,d_{\frac{1}{2}\left|E\right|})$ with integers entries.
We say that $(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$
supports an abstract $(n,d)$-GKM graph if there exists a map
$\operatorname{w}:E\rightarrow\mathbb{Z}^{d}\setminus\\{0\\}$ such that
$(\Gamma,\operatorname{w})$ is an abstract $(n,d)$-GKM graph and such that
$\mathcal{C}_{1}(e_{i})=d_{i}$ for all $i=1,...,\frac{1}{2}\left|E\right|$,
where $\mathcal{C}_{1}:E\rightarrow\mathbb{Z}$ is the first Chern class map of
$(\Gamma,\operatorname{w})$.

By the discussion above it is clear that the analog of Question 5.1 for
abstract GKM graphs is equivalent to the question of whenever a given GKM
skeleton supports an abstract GKM graph.

Next we rewrite equations (5.9) so that the terms
$\operatorname{w}(e^{\prime})$ for $e^{\prime}\in E_{i(e)}^{i},E_{t(e)}^{i}$
are replaced by the terms $\operatorname{w}(e_{i})$ for
$i=1,\dots.\frac{1}{2}\left|E\right|$. In order to do so we introduce the
structure matrix.

######  Definition 5.17.

The structure matrix of an $n$-GKM skeleton
$(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ is the
$(\frac{1}{2}\left|E\right|\times\frac{1}{2}\left|E\right|)$-matrix
$\mathbf{A}=(a_{j,k})_{j,k}$ given by

$\displaystyle a_{j,k}=\begin{cases}2\quad\quad\,\,\,\,\text{if}\,j=k,\\\
1\quad\quad\,\,\,\,\text{if}\,j\neq
k\,\text{and}\,i(e_{j})=i(e_{k})\,\text{or}\,\,t(e_{j})=t(e_{k}),\\\
-1\quad\quad\text{if}\,j\neq k\,\text{and}\
i(e_{j})=t(e_{k})\,\text{or}\,\,t(e_{j})=i(e_{k}),\\\
0\quad\quad\,\,\,\,\text{otherwise}.\end{cases}$

######  Remark 5.18.

The structure matrix $\mathbf{A}$ of an $n$-GKM skeleton
$(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ does not depend on
the vector $\mathbf{d}$. Since the graph $\Gamma$ is simple, for two indices
with $j\neq k$ only the condition
$i(e_{j})=i(e_{k})\,\text{or}\,t(e_{j})=t(e_{k})$ or only the condition
$i(e_{j})=t(e_{k})\,\text{or}\,t(e_{j})=i(e_{k})$ can be satisfied. Hence, the
structure matrix $\mathbf{A}$ is well defined. It is clear that $\mathbf{A}$
is symmetric. Moreover, since the graph $\Gamma$ is $n$-valent, for each $j$
there exists exactly $n-1$ indices $k\neq j$ such that
$i(e_{j})=i(e_{k})\,\text{or}\,i(e_{j})=t(e_{k})$ and there exists exactly
$n-1$ indices $k\neq j$ such that
$t(e_{j})=i(e_{k})\,\text{or}\,t(e_{j})=t(e_{k})$. Hence, each row vector
resp. column vector of $\mathbf{A}$ has one entry that is equal to $2$,
$2(n-1)$ entries that are equal to $1$ or $-1$ and the remaining other
$\frac{1}{2}\left|E\right|-2(n-1)-1$ entries are equal to zero.

######  Lemma 5.19.

Let $(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ be a GKM
skeleton and let $\operatorname{w}:E\rightarrow\mathbb{Z}^{d}$ be an
antisymmetric map, i.e., $\operatorname{w}(\bar{e})=-\operatorname{w}(e)$.
Then for each $j=1,\dots,\frac{1}{2}\left|E\right|$

$\displaystyle\sum_{e\in E_{i(e_{j})}^{i}}\operatorname{w}(e)\,-\,\sum_{e\in
E_{t(e_{j})}^{i}}\operatorname{w}(e)=\sum_{k=1}^{\frac{1}{2}\left|E\right|}a_{j,k}\operatorname{w}(e_{k}),$

where $\mathbf{A}=(a_{j,k})_{j,k=1,...,\frac{1}{2}\left|E\right|}$ is the
structure matrix of the GKM skeleton.

###### Proof.

Note that for each $e\in E$ there exists a unique index
$k\in\\{1,\dots,\frac{1}{2}\left|E\right|\\}$ with either $e=e_{k}$ or
$\bar{e}=e_{k}$. Since $\Gamma$ is a simple graph we have

$\displaystyle e\in E_{i(e_{j})}^{i}\,\,\text{and}\,\,\bar{e}\in
E_{t(e_{j})}^{t}\quad\implies\quad e=e_{j}.$

Moreover, since the map $\operatorname{w}$ is antisymmetric, by the definition
of the structure matrix it is clear that the lemma is true. ∎

######  Corollary 5.20.

Let $(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ be a GKM
skeleton and let $\operatorname{w}:E\rightarrow\mathbb{Z}^{d}\setminus\\{0\\}$
be a map such that $(\Gamma,\operatorname{w})$ is an abstract GKM graph. Then
the following conditions are equivalent.

  * •

The abstract GKM graph $(\Gamma,\operatorname{w})$ is supported by the GKM
skeleton $(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$.

  * •

For each $j=1,\dots,\frac{1}{2}\left|E\right|$

$\displaystyle\sum_{k=1}^{\frac{1}{2}\left|E\right|}a_{j,k}\operatorname{w}(e_{k})\,=\,d_{j}\cdot\operatorname{w}(e_{j}),$
(5.10)

where $\mathbf{A}=(a_{j,k})_{j,k=1,...,\frac{1}{2}\left|E\right|}$ is the
structure matrix of the GKM skeleton.

###### Proof.

If $(\Gamma,\operatorname{w})$ is supported by
$(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$, then
$d_{j}=\mathcal{C}_{1}(e_{j})$ for all $j=1,...,\frac{1}{2}\left|E\right|$. By
the definition of the first Chern class map, we have

$\displaystyle\sum_{e\in E_{i(e_{j})}^{i}}\operatorname{w}(e)\,-\,\sum_{e\in
E_{t(e_{j})}^{i}}\operatorname{w}(e)=d_{j}\cdot\operatorname{w}(e_{j}),$
(5.11)

for all $j=1,...,\frac{1}{2}\left|E\right|$. Since the weight map
$\operatorname{w}$ is antisymmetric, by Lemma 5.19 the left-hand sides of
(5.10) and (5.11) coincide for all $j=1,...,\frac{1}{2}\left|E\right|$. Hence,
(5.10) holds for all $j=1,...,\frac{1}{2}\left|E\right|$.
On the other hand, if (5.10) holds for all
$j=1,...,\frac{1}{2}\left|E\right|$, then by Lemma 5.19 also (5.11) holds for
all $j=1,...,\frac{1}{2}\left|E\right|$. So by the definition of the first
Chern class map $d_{j}=\mathcal{C}_{1}(e_{j})$ for all
$j=1,...,\frac{1}{2}\left|E\right|$, i.e., $(\Gamma,\operatorname{w})$ is
supported by $(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$. ∎

######  Definition 5.21.

Let $(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ be an $n$-GKM
skeleton. The defect $\delta$ of the $n$-GKM skeleton is the dimension of the
kernel of $\mathbf{A-D}$, where $\mathbf{A}$ is the structure matrix and
$\mathbf{D}$ is the diagonal matrix given by

$\displaystyle\mathbf{D}=\operatorname{Diag}(d_{1},\dots,d_{\frac{1}{2}\left|E\right|}).$

If the defect $\delta$ is positive, a fundamental system of
$(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ is a collection of
$\frac{1}{2}\left|E\right|$ vectors
$f_{1},\dots,f_{\frac{1}{2}\left|E\right|}$ in $\mathbb{Q}^{\delta}$ such that
the transposes of the row vectors of the
$\left(\delta\times\frac{1}{2}\left|E\right|\right)$-matrix

$\displaystyle\mathbf{F}=\left(\begin{matrix}|&&|\\\
f_{1}&\dots&f_{\frac{1}{2}\left|E\right|}\\\ |&&|\end{matrix}\right)$

form a basis of the kernel of $\mathbf{A-D}$.

######  Remark 5.22.

Let $(\Gamma,\sigma,\operatorname{Ord}(E^{\sigma}),\mathbf{d})$ be an $n$-GKM
skeleton with a positive defect $\delta$. Since the matrices $\mathbf{A}$ and
$\mathbf{D}$ have only integer entries, there exist vectors with integer
entries that form a basis of the kernel of $\mathbf{A-D}$. Hence, an $n$-GKM