# On a symplectic generalization of a Hirzebruch problem

Leonor Godinho, Nicholas Lindsay and Silvia Sabatini

###### Abstract.

Motivated by a problem of Hirzebruch, we study closed, symplectic,
$8$-dimensional manifolds having a Hamiltonian torus action with isolated
fixed points and second Betti number equal to $1$. Such manifolds are
automatically positive monotone. Our main result concerns those endowed with a
Hamiltonian $T^{2}$-action and fourth Betti number equal to $2$. We classify
their isotropy data, (equivariant) cohomology rings and (equivariant) Chern
classes, and prove that they agree with those of certain explicit Fano
$4$-folds with torus actions.

Moreover, under more general assumptions, we prove several finiteness results
concerning Betti and Chern numbers of positive monotone, $8$-dimensional
symplectic manifolds with a Hamiltonian torus action.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
  2. 2 Preliminaries
    1. 2.1 Equivariant cohomology: generalities
    2. 2.2 Hamiltonian actions and their properties
      1. 2.2.1 Restrictions to the fixed points and equivariant structure constants
    3. 2.3 Hamiltonian GKM spaces
    4. 2.4 Monotone Hamiltonian $T$ spaces
  3. 3 The Chern Numbers
    1. 3.1 The case $\chi(M)=6$
  4. 4 Multigraphs describing the action and their positivity properties
    1. 4.1 Existence of positive multigraphs
    2. 4.2 Positive multigraphs and Betti numbers
  5. 5 Examples
    1. 5.1 Index 2: The manifold Fano-Mukai fourfold $V$ and its equivariant cohomology ring
      1. 5.1.1 Torus actions on the generalized flag variety $\Omega=G_{2}/P$
      2. 5.1.2 The Fano-Mukai $4$-fold $V$ and its torus action.
    2. 5.2 Index 3: The quintic del Pezzo fourfold $W$ and its equivariant cohomology ring
      1. 5.2.1 The canonical class $\tau_{1}\in H^{2}_{\mathbb{T}}(W;\mathbb{Z})$:
      2. 5.2.2 The canonical classes $\tau_{2}$ and $\tau^{\prime}_{2}$ in $H^{4}_{\mathbb{T}}(W;\mathbb{Z})$:
      3. 5.2.3 The canonical class $\tau_{3}\in H^{6}_{\mathbb{T}}(W;\mathbb{Z})$:
    3. 5.3 Index 4: The Grassmannian $Q=\text{Gr}(2,4)$ and its equivariant cohomology ring
    4. 5.4 Torus actions on $\mathbb{CP}^{2}\times\mathbb{CP}^{2}$
  6. 6 The algorithm
  7. 7 Classification Results
    1. 7.1 Classification results for $b_{1}(M)=1$ and $b_{4}(M)=2$:
      1. 7.1.1 Index 1
      2. 7.1.2 Index 2
      3. 7.1.3 Index 3
      4. 7.1.4 Index 4
    2. 7.2 Additional Results
      1. 7.2.1 Index 2, $b_{2}(M)=1$ and $3\leq b_{4}(M)\leq 6$
      2. 7.2.2 Index $3$

##  1\. Introduction

A complex manifold $X$ of complex dimension $n$ is called a compactification
of $\mathbb{C}^{n}$, if there exists a complex analytic subset $V\subset X$
such that $X\setminus V$ is biholomorphically equivalent to $\mathbb{C}^{n}$.
A celebrated problem of Hirzebruch aims at classifying compactifications of
$\mathbb{C}^{n}$ with second Betti number $b_{2}$ equal to one [26, Problem
27].

A projective compactification $X$ with $b_{2}(X)=1$ is automatically a Fano
variety (see [38, page 2]). When the compactification is projective, there has
been steady progress on Hirzebruch’s problem since its formulation. In
particular, combined work of several authors gives a complete solution when
$n\leq 3$ [36, 13, 39, 14]. Moreover, when $n=4$, and the Fano index is equal
to $3$ the problem was solved by Prokhorov [40]. In this case $X$ is unique up
to isomorphism, and there are four possibilities for $V\subset X$.

A rich subclass of projective compactifications of $\mathbb{C}^{n}$ is the
class of smooth projective varieties admitting a $\mathbb{C}^{*}$-action with
isolated fixed points. The fact that such varieties are compactifications of
$\mathbb{C}^{n}$ follows from the decomposition theorem of Bialynicki-Birula
[4, Theorem 4.4]. Note that all of the known projective compactifications of
$\mathbb{C}^{n}$ with second Betti number equal to one and $n\leq 4$ admit a
$\mathbb{C}^{*}$-action with isolated fixed points. Moreover, an algebraic
torus action on a smooth Fano variety is Hamiltonian with respect to a certain
invariant Kähler form (see Lemma 5.2 for a proof).

Motivated by these facts, we study Hamiltonian torus actions on compact,
positive monotone symplectic manifolds $(M,\omega)$. We recall that a
symplectic manifold $(M,\omega)$ is called positive monotone if there exists a
positive constant $\lambda>0$ such that $c_{1}(M)=\lambda[\omega]$. Observe
that a smooth Fano variety is automatically a positive monotone symplectic
manifold. Therefore this condition is the symplectic analogue of the Fano
condition in algebraic geometry.

In the past few years there has been intensive work trying to determine
whether these two categories are indeed distinct. In [11], Fine and Panov gave
an example of a non simply connected 12-dimensional compact positive monotone
symplectic manifold $(\widetilde{M},\widetilde{\omega})$ which, therefore,
cannot be homotopy equivalent to a Fano variety, as the latter is always
simply connected. In a subsequent paper [12] they conjecture that the
existence of a Hamiltonian circle action on a 6-dimensional compact, positive
monotone symplectic manifold implies the existence of a diffeomorphism with a
Fano 3-fold. There has already been intense work towards proving that this
conjecture holds (see for instance [43, 34, 7, 8]). However in higher
dimensions the existence of a Hamiltonian circle action is not enough to
ensure the existence of a diffeomorphism with a Fano variety (e. g. consider
$\widetilde{M}\times S^{2}$ with a Hamiltonian circle action on the second
factor that fixes the first). Nevertheless the problem remains open when the
circle action has isolated fixed points. (For some special actions of higher
dimensional tori see also [9]).

In this paper, motivated by Hirzebruch’s problem, we consider $8$-dimensional
compact symplectic manifolds $(M,\omega)$ with second Betti number equal to
one, admitting a Hamiltonian torus action with isolated fixed points. Observe
that such a manifold is automatically positive monotone, as $c_{1}$ must be a
multiple of $[\omega]$, and by [16, Lemma 5.2], this multiple must be
positive.

If we further assume that $b_{4}(M)=1$, or equivalently that the action has 5
fixed points (which, in dimension 8, is the minimal number of fixed points),
then the fixed point representations have been completely classified [15, 28].
They are all isomorphic to the ones induced by a linear action on
$\mathbb{CP}^{4}$. Note that, in this case, the fixed point representations
also determine the equivariant cohomology ring and Chern classes of
$(M,\omega)$ [43].

If $b_{4}(M)=2$ there are three known $8$-dimensional examples satisfying
$b_{2}(M)=1$, which may be distinguished by their index $k_{0}$ (the largest
integer such that $c_{1}=k_{0}\eta$, for some nonzero $\eta\in
H^{2}(M;\mathbb{Z})$, modulo torsion elements). These are the Fano-Mukai
fourfold $V$ (with $k_{0}=2$), the quintic del Pezzo fourfold $W$ (with
$k_{0}=3$) and the Grassmannian $Q=Gr(2,4)$ (with $k_{0}=4$). These smooth
Fano varieties are described in detail in Section 5. Some of their invariants
are summarized in Table 1.1.

$M$ | $(b_{2}(M),b_{4}(M))$ | $c_{1}^{4}[M]$ | $c_{1}^{2}c_{2}[M]$ | $c_{2}^{2}[M]$ | $c_{1}c_{3}[M]$ | $k_{0}$
---|---|---|---|---|---|---
$V$ | $(1,2)$ | 288 | 168 | 98 | 48 | 2
$W$ | $(1,2)$ | 405 | 198 | 97 | 48 | 3
$Q$ | $(1,2)$ | 512 | 224 | 98 | 48 | 4
Table 1.1. Some of the invariants of $V,W$ and $Q$.

Each of these manifolds admits a Hamiltonian torus action which is carefully
described in Section 5. The corresponding moment map polytopes, enriched with
the moment map images of their isotropy submanifolds $M^{w}$, for every
isotropy weight $w$ (see Remark 4.2), are shown in Figure 1.1. These can be
thought of as the ‘moment map image’ of the associated multigraphs defined in
Section 4.

Figure 1.1. The ‘enriched’ moment map polytopes corresponding to $V,W$ and
$Q$.

Observe that the first and third examples correspond to Hamiltonian GKM spaces
(see Section 2.3), whereas the second example does not. Indeed, the thicker
edge is the moment map image of an isotropy submanifold symplectomorphic to
$\mathbb{C}P^{2}$. Hence the first and third graphs in the picture are indeed
the GKM graphs associated with the torus action on $V$ and $Q$.

In [18] Goldin and Tolman introduce a special basis for the equivariant
cohomology ring with $\mathbb{Z}$-coefficients of a Hamiltonian $T$-space and
the elements of this basis are called canonical classes. In particular, the
restriction of these classes to the fixed point set, as well as the
equivariant structure constants, can be computed explicitly for Hamiltonian
GKM spaces admitting an index increasing component of the moment map (see
Section 2.2). We use this method to compute the equivariant cohomology ring
and Chern classes of $V$ and $Q$111Note that the computations for $Q$ are
already known, as this is just the Grassmannian of complex planes in
$\mathbb{C}^{4}$ endowed with the standard $3$-dimensional torus action. (see
Theorems 5.5 and 5.9). Even though the torus action on $W$ is not GKM, we can
still prove the existence of canonical classes and compute its equivariant
cohomology ring and Chern classes (see Theorem 5.8). As a byproduct, in the
next two corollaries we obtain the ordinary cohomology rings and Chern classes
of $V$ and $W$ (those of $Q$ are well-known):

######  Corollary 1.1.

Let $V$ be the Fano-Mukai 4-fold of genus 10 and degree 18. Then there exists
a basis $\\{\zeta_{i}\\}_{i=0,\ldots,4}\cup\\{\zeta^{\prime}_{2}\\}$ of the
cohomology ring $H^{*}(V;\mathbb{Z})$ satisfying the following properties and
relations:
For each $i=0,\ldots,4$, $\zeta_{i}\in H^{2i}(V;\mathbb{Z})$,
$\zeta^{\prime}_{2}\in H^{4}(V;\mathbb{Z})$ and

(1.1) $\displaystyle\begin{aligned}
&\zeta_{1}^{2}=3\zeta_{2}+3\zeta^{\prime}_{2}&\quad\zeta_{1}\zeta_{3}=\zeta_{2}^{2}=(\zeta^{\prime}_{2})^{2}=\zeta_{4}\\\
&\zeta_{1}\zeta_{2}=\zeta_{1}\zeta^{\prime}_{2}=3\zeta_{3}&\quad\zeta_{2}\zeta^{\prime}_{2}=0\,.\\\
\end{aligned}$

Moreover the Chern classes of the tangent bundle of $V$ are given by

(1.2) $c_{1}=-2\zeta_{1},\quad c_{2}=7(\zeta_{2}+\zeta^{\prime}_{2}),\quad
c_{3}=-24\zeta_{3},\quad c_{4}=6\zeta_{4}\,.$

######  Corollary 1.2.

Let $W$ be the quintic del Pezzo fourfold defined in Subsection 5.2. Then
there exists a basis
$\\{\zeta_{i}\\}_{i=0,\ldots,4}\cup\\{\zeta^{\prime}_{2}\\}$ of the cohomology
ring $H^{*}(W;\mathbb{Z})$ satisfying the following properties and relations:
For each $i=0,\ldots,4$, $\zeta_{i}\in H^{2i}(W;\mathbb{Z})$,
$\zeta^{\prime}_{2}\in H^{4}(W;\mathbb{Z})$ and

(1.3) $\displaystyle\begin{aligned}
&\zeta_{1}^{2}=3\zeta_{2}+2\zeta^{\prime}_{2},&\quad\zeta_{1}\zeta_{3}=\zeta_{2}^{2}=\zeta_{4}\\\
&\zeta_{1}\zeta_{2}=\zeta_{1}\zeta^{\prime}_{2}=\zeta_{3},&\quad(\zeta^{\prime}_{2})^{2}=2\zeta_{4}\\\
&\zeta_{2}\zeta^{\prime}_{2}=-\zeta_{4}&\quad\\\ \end{aligned}$

Moreover, the Chern classes of the tangent bundle of $W$ are given by

(1.4) $c_{1}=-3\zeta_{1},\quad c_{2}=13\,\zeta_{1}+9\,\zeta_{2}^{\prime},\quad
c_{3}=-16\zeta_{3},\quad c_{4}=6\zeta_{4}.$

###  1.1. Main results

We begin by proving several ‘finiteness’ results. More precisely, in Theorem
3.4 we show that, if $(M,\omega)$ is an 8-dimensional closed symplectic
manifold with $b_{2}(M)=1$ and $k_{0}>1$, admitting Hamiltonian circle action
with isolated fixed points, then for each value of $b_{4}(M)$ there are
finitely many possibilities for the Chern numbers of $(M,\omega)$. As a
consequence, for each value of $b_{4}(M)$, there are finitely many complex
cobordism classes of such $(M,\omega)$. In Theorem 3.7 we assume that the
Euler characteristic is 6 (which is equivalent to saying that $b_{2}(M)=1$ and
$b_{4}(M)=2$), and use number theoretical arguments to refine the
possibilities for the Chern numbers.

The Fano manifolds $V$, $W$ and $Q$ are examples of compact positive monotone
symplectic manifolds of dimension 8, with a Hamiltonian torus action and 6
fixed points, thus implying that 6 is also their Euler characteristic. Their
isotropy data, namely the isotropy weights at each fixed point, is given
respectively in Tables 5.1, 5.4 and 5.6. Our main result says that, in a
certain sense, these are the only possible examples. More precisely:

######  Main Theorem.

Let $(M,\omega)$ be a closed symplectic $8$-dimensional manifold admitting a
Hamiltonian $T^{2}$-action with 6 isolated fixed points. Then its index
$k_{0}$ satisfies $2\leq k_{0}\leq 4$. Moreover the isotropy data,
(equivariant) cohomology ring and (equivariant) Chern classes agree with those
of

  * $\bullet$

$V$ with the action described in Section 5.1, if $k_{0}=2$;

  * $\bullet$

$W$ with the action described in Section 5.2, if $k_{0}=3$;

  * $\bullet$

$Q$ with a subtorus action of the 3-dimensional torus action described in
Section 5.3, if $k_{0}=4$.

Note that the Euler characteristic of such $(M,\omega)$ is 6, and that, by
Lemma 4.25, $(M,\omega)$ is positive monotone with symplectic form that can be
rescaled to satisfy $c_{1}=[\omega]$. Moreover the moment map can be
translated so that the weight sum formula (2.5) holds. Therefore our main
theorem can be regarded as a classification theorem for positive monotone
symplectic manifolds of dimension $8$ and Euler characteristic $6$. The weight
sum formula also implies that, modulo $GL(2;\mathbb{Z})$-transformations, the
only enriched moment map polytopes are those in Figure 1.1.

To prove this theorem we use an adaptation of the algorithm described in [15]
(see Section 6). This requires the existence of a combinatorial object (a
multigraph) that describes the action, satisfying a particular positivity
condition involving the first equivariant Chern class. For instance, this
holds for all positive monotone Hamiltonian GKM spaces and we prove, in
Theorem 4.32, that it also holds under the conditions of the main theorem.

Hence, to prove the main theorem, we run a program based on this algorithm
producing all the possible isotropy data. For $k_{0}=3$ we need to use the
restrictions on the Chern numbers given by Theorem 3.7. Then it is enough to
show that this data is the same as that of $V$, $W$ and $Q$. This is done in
Section 7.1.

The accompanying software based on the algorithms presented in this paper and
all the accompanying files, can be found at

$\text{http://www.math.ist.utl.pt/~{}lgodin/HirzebruchProblem.html}.$

Another finiteness result concerns the Betti numbers. Indeed, in Proposition
4.38, we provide a finite list of the possible Betti numbers of a closed,
symplectic 8-dimensional manifold with $k_{0}>1$, that admits a Hamiltonian
action of a circle with isolated fixed points, and a multigraph that is
positive w.r.t. the first Chern class. More precisely we obtain

  1. (1)

If $k_{0}=5$ then $b_{2}=b_{4}=1$;

  2. (2)

If $k_{0}=4$ then $b_{2}=1$ and $b_{4}=2$;

  3. (3)

If $k_{0}=3$ then $(b_{2},b_{4})$ can be either $(1,2)$ or $(2,3)$;

  4. (4)

If $k_{0}=2$ then $1\leq b_{2}\leq b_{4}\leq 6$.

Therefore these restrictions hold for any 8-dimensional positive monotone
Hamiltonian GKM space.

All of the above possibilities for the Betti numbers with $k_{0}\geq 3$ are
realized by explicit examples. For $k_{0}=5$ the only possible example is
$\mathbb{C}P^{4}$ (see [15]). The cases with $(b_{2},b_{4})=(1,2)$ are
discussed in Section 5. An example with $k_{0}=3$ and $(b_{2},b_{4})=(2,3)$ is
given by the toric Fano $4$-fold $\mathbb{CP}^{2}\times\mathbb{CP}^{2}$. We
also note that the bound on $b_{4}$ in the $k_{0}=2$ case is sharp, by
considering the toric Fano $4$-fold $(\mathbb{CP}^{1})^{4}$, having $k_{0}=2$,
and $(b_{2},b_{4})=(4,6)$.

Motivated by these results, we again use the algorithm from [15] to obtain the
following theorem.

######  Theorem 1.3.

Let $(M,\omega)$ be a closed, symplectic 8-dimensional manifold with
$k_{0}>1$, that admits a Hamiltonian action of a $2$-dimensional torus with
isolated fixed points with a multigraph that is positive w.r.t. the first
Chern class.

  * •

If $k_{0}=3$ and $(b_{2},b_{4})=(2,3)$ then the isotropy data agrees with that
of a subtorus of the standard toric action on
$\mathbb{C}P^{2}\times\mathbb{C}P^{2}$.

  * •

If $k_{0}=2$ and $b_{2}=1$ then $b_{4}=2$ and the isotropy data agrees with
that of $V$.

Motivated by these results, we finish this introduction, with some questions
that we believe are interesting to the community of equivariant symplectic
geometers.

######  Question 1.

Let $(M,\omega)$ be a closed, positive monotone symplectic manifold, admitting
a Hamiltonian action of a torus with isolated fixed points.

  1. i)

Are there finitely many (equivariant) complex cobordism classes in each
dimension?

  2. ii)

Are there finitely many
homotopy/homeomorphism/diffeomorphism/symplectomorphism types in each
dimension?

##  2\. Preliminaries

###  2.1. Equivariant cohomology: generalities

We recall that the equivariant cohomology of a topological space $X$ acted on
by a torus $T$ of dimension $d$ is given by

$H_{T}(X;R):=H(X\times_{T}ET;R)\,,$

where $R$ is a ring with unit and $ET$ is a contractible space on which $T$
acts freely. As in this case $T$ is a torus of dimension $d$, we have that
$ET$ can be taken to be a product of $d$ copies of
$S^{\infty}\subset\mathbb{C}^{\infty}$ with the diagonal action of $T\simeq
S^{1}\times\cdots\times S^{1}$, where each factor acts on $S^{\infty}$ freely,
and the quotient $BT:=ET/T$ is therefore a product of $d$ copies of
$\mathbb{C}P^{\infty}$. We remark that $H(BT;R)$ is isomorphic to the
polynomial ring $R[x_{1},\ldots,x_{d}]$, where each $x_{i}$ has degree 2, and
that the constant map $X\to\\{pt\\}$ gives $H_{T}(X;R)$ the structure of an
$R[x_{1},\ldots,x_{d}]$-module. Moreover, for any subgroup $K$ of $T$, the
natural map $M\times_{K}ET\to M\times_{T}ET$ sending a $K$-orbit to a
$T$-orbit induces a restriction map $r_{K}\colon H_{T}(M;R)\to H_{K}(M;R)$
(observe that $ET$ is a classifying space for $K$ too). In particular, when
$K$ is the trivial group, we obtain a restriction

$r\colon H_{T}(M;R)\to H(M;R)\,.$

Suppose that the set of $T$-fixed points is nonempty and denote it by $M^{T}$.
Let $p\in M^{T}$, consider the (equivariant) inclusion map
$\iota_{p}\colon\\{p\\}\to M$ and the corresponding restriction

$\iota_{p}^{*}\colon H_{T}^{*}(M;R)\to H_{T}^{*}(\\{p\\};R)\simeq
R[x_{1},\ldots,x_{d}]\,.$

For a class $\alpha\in H_{T}^{*}(M;R)$, we denote $\iota_{p}^{*}(\alpha)$
simply by $\alpha(p)$. When $T$ is a circle and $R$ is $\mathbb{Z}$ or
$\mathbb{R}$, then $H_{S^{1}}^{*}(\\{p\\};R)=R[x]$ and it makes sense to say
that a class $\alpha\in H_{S^{1}}^{2k}(M;R)$ is positive (resp. negative),
namely we say that

$\alpha(p)=m\,x^{k}>0\quad\text{(resp. }<0):\iff
m>0\quad\text{(resp.}\;\;m<0)\,.$

###  2.2. Hamiltonian actions and their properties

Let $(M,\omega)$ be a compact symplectic manifold of dimension $2n$ that is
acted on by a torus $T$ of dimension $d$, preserving the symplectic structure.
Such action is called Hamiltonian if it admits a moment map, namely a map
$\psi\colon M\to\mathfrak{t}^{*}$ satisfying

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

where $\langle\cdot,\cdot\rangle$ denotes the natural pairing between
$\mathfrak{t}^{*}$ and $\mathfrak{t}$, and $\xi^{\\#}$ is the vector field on
$M$ corresponding to $\xi$. Assume that the fixed point set $M^{T}$ is
discrete. Consider a $\widetilde{\xi}\in\mathfrak{t}$ such that
$\alpha(\widetilde{\xi})\neq 0$ for all the elements $\alpha\in\ell^{*}$ that
occur as weights of the $T$ action at a fixed point, and the corresponding
component of the moment map given by
$\varphi:=\langle\psi,\widetilde{\xi}\rangle$, which, for the above properties
of $\widetilde{\xi}$, is called generic. We say that a weight $\alpha$ is
positive if $\alpha(\widetilde{\xi})>0$. For every $p\in M^{T}$, let
$\lambda_{p}$ be the number of negative weights at $p$ and $N_{k}$ be the
number of fixed points with $k$ negative weights. Then it is well known that
$\varphi$ is a perfect Morse function with critical set given by $M^{T}$,
that, for all $j=0,\ldots,n$, $H^{2j}(M;\mathbb{Z})=\mathbb{Z}^{N_{j}}$ and
that $H^{j}(M;\mathbb{Z})=0$ for every odd $j$. Observe that, since $M$ is
compact, $[\omega^{j}]\in H^{2j}(M;\mathbb{R})=\mathbb{R}^{N_{j}}$ is a
nontrivial cohomology class for every $j=0,\ldots,n$, and so $N_{j}>0$ for all
$j=0,\ldots,n$.

Once the information on the fixed points is known, computing the cohomology
groups is therefore trivial. However, the computation of the ring structure of
$H^{*}(M;\mathbb{Z})$ can be more complicated. Thanks to the results recalled
below, the existence of a Hamiltonian action can help to understand the
(equivariant) cohomology ring structure (see in particular Section 2.2.1).

Let $\Lambda_{p}^{-}$ be the product of the negative weights at $p$. The
following result is well-known and due to Kirwan [31], who provides a basis of
$H_{T}(M;\mathbb{Z})$ related to a generic component of the moment map. This
basis depends on an ordering of the fixed points that is induced by $\varphi$,
namely we say that

(2.2) $p\prec q\quad\text{(resp. }p\preceq
q)\quad:\iff\varphi(p)<\varphi(q)\quad\text{(resp.
}\varphi(p)\leq\varphi(q))\,.$

######  Proposition 2.1 (Kirwan).

Let $(M,\omega)$ be a closed symplectic manifold equipped with a Hamiltonian
$T$-action with discrete fixed point set $M^{T}$ and moment map $\psi\colon
M\to\mathfrak{t}^{*}$. Let $\varphi\colon M\to\mathbb{R}$ be a generic
component of the moment map and $\preceq$ the corresponding ordering on the
fixed points.

Then for every fixed point $p$ there exists a class $\gamma_{p}\in
H_{T}^{2\lambda_{p}}(M;\mathbb{Z})$ such that

  1. (1)

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

  2. (2)

$\gamma_{p}(q)=0$ for every $q\in M^{T}\setminus\\{p\\}$ such that $q\preceq
p$ .

Moreover, for any such classes, the $\\{\gamma_{p}\\}_{p\in M^{T}}$ are a
basis for $H_{T}^{*}(M;\mathbb{Z})$ as a module over $H^{*}(BT;\mathbb{Z})$
and their restrictions to the ordinary cohomology ring,
$\\{r(\gamma_{p})\\}_{p\in M^{T}}$, are a basis for $H^{*}(M;\mathbb{Z})$ as a
module over $\mathbb{Z}$.

Note that Proposition 2.1 also says that $H_{T}(M;\mathbb{Z})$ is a free
module over $H(BT;\mathbb{Z})$.

A class $\gamma_{p}$ satisfying (1) and (2) above is called a Kirwan class at
$p$. Although Kirwan classes always exist, they are not unique. For instance,
if there exists a $q\in M^{T}$ such that $p\prec q$ and
$\lambda_{p}=\lambda_{q}$, then $\gamma_{p}+k\gamma_{q}$ is also a Kirwan
class at $p$, for every $k\in\mathbb{Z}$.

Following [18], we recall another set of classes, called canonical classes,
that do not always exist (see [18, Example 2.2]) but, if they do, they are
unique (see [18, Lemma 2.7]).

###### Definition 2.2

Let $(M,\omega)$ be a closed symplectic manifold equipped with a Hamiltonian
$T$-action with discrete fixed point set $M^{T}$ and moment map $\psi\colon
M\to\mathfrak{t}^{*}$. Let $\varphi\colon M\to\mathbb{R}$ be a generic
component of the moment map.

Let $p\in M^{T}$. Then a class $\tau_{p}\in
H_{T}^{2\lambda_{p}}(M;\mathbb{Z})$ is called a canonical class at $p$ if

  1. (1)

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

  2. (2)

$\tau_{p}(q)=0$ for every $q\in M^{T}\setminus\\{p\\}$ such that
$\lambda_{q}\leq\lambda_{p}$ .

In [18, Lemma 2.8] it is proved that if a canonical class $\tau_{p}$ exists,
then it is also a Kirwan class. Therefore, if a canonical class $\tau_{p}$
exists for every fixed point $p$, the set $\\{\tau_{p}\\}_{p\in M^{T}}$ is a
basis of $H_{T}(M;\mathbb{Z})$ as a module over $H^{*}(BT;\mathbb{Z})$ and
their restrictions to the ordinary cohomology ring, $\\{r(\tau_{p})\\}_{p\in
M^{T}}$, give a basis for $H^{*}(M;\mathbb{Z})$ as a module over $\mathbb{Z}$.

###### Remark 2.3

There are two canonical classes that always exist, namely those corresponding
to the minimum $p_{0}$ and maximum $p_{\text{max}}$ of $\varphi$. In this case
it follows from Definition 2.2 that $\tau_{p_{0}}(p)=1$ for every $p\in
M^{T}$, and $\tau_{p_{\text{max}}}(p)=0$ for all $p\in
M^{T}\setminus\\{p_{\text{max}}\\}$.

####  2.2.1. Restrictions to the fixed points and equivariant structure
constants

Let $(M,\omega)$ be a closed symplectic manifold equipped with a Hamiltonian
$T$-action with discrete fixed point set $M^{T}$ and moment map $\psi\colon
M\to\mathfrak{t}^{*}$. Consider a basis of $H_{T}(M;\mathbb{Z})$ as a module
over $H(BT;\mathbb{Z})$, for instance the basis $\\{\gamma_{p}\\}_{p\in
M^{T}}$ given by the Kirwan classes with respect to a generic component
$\varphi\colon M\to\mathbb{R}$ of the moment map or, when they exist, that of
canonical classes. The equivariant structure constants with respect to the
basis $\\{\gamma_{p}\\}_{p\in M^{T}}$ are defined to be the elements
$a^{s}_{p,q}\in H(BT;\mathbb{Z})$ such that

(2.3) $\gamma_{p}\gamma_{q}=\sum_{s\in M^{T}}a^{s}_{p,q}\gamma_{s}\,.$

If we identify the dual lattice of $\mathfrak{t}$ with $\mathbb{Z}\langle
x_{1},\ldots,x_{d}\rangle$, then
$H(BT;\mathbb{Z})=\mathbb{Z}[x_{1},\ldots,x_{d}]$ and therefore $a^{s}_{p,q}$
is a polynomial of degree $\lambda_{p}+\lambda_{q}-\lambda_{s}$. In order to
obtain the ordinary structure constants with respect to the basis
$\\{r(\gamma_{p})\\}_{p\in M^{T}}$, namely the constants
$b_{p,q}^{s}\in\mathbb{Z}$ such that

(2.4) $r(\gamma_{p})r(\gamma_{q})=\sum_{s\in
M^{T}}b^{s}_{p,q}r(\gamma_{s})\,,$

it is then sufficient to take $r(a^{s}_{p,q})$, namely $b^{s}_{p,q}$ is the
constant term of $a^{s}_{p,q}$.

Computing the equivariant structure constants with respect to a basis
$\\{\gamma_{p}\\}_{p\in M^{T}}$ is indeed equivalent to computing the
restriction of each $\gamma_{p}$ to the fixed point set, the latter problem
being long studied in the literature (see for instance [22, 18, 42]). For
every $p^{\prime}\in M^{T}$ consider the equivariant inclusion
$\iota_{p^{\prime}}:\\{p^{\prime}\\}\hookrightarrow M$ and the induced map
$\iota^{*}_{p^{\prime}}\colon
H_{T}(M;\mathbb{Z})\to\mathbb{Z}[x_{1},\ldots,x_{d}]$; we denote
$\iota^{*}_{p^{\prime}}(\gamma)$ simply by $\gamma(p^{\prime})$. In order to
show that the set of polynomials given by $\gamma_{p}(p^{\prime})$, for all
$p,p^{\prime}\in M^{T}$, determines the equivariant structure constants, it is
enough to prove that for every equivariant cohomology class $\alpha$, its
coefficients with respect to the basis $\\{\gamma_{p}\\}_{p\in M^{T}}$ are
determined by the restrictions $\gamma_{p}(p^{\prime})$ and
$\alpha(p^{\prime})$, for all $p,p^{\prime}\in M^{T}$. This is illustrated in
the following lemma, whose proof is left to the reader (see [22] and [15,
Lemma 2.4]).

######  Lemma 2.4.

Let $(M,\omega)$ be a closed symplectic manifold equipped with a Hamiltonian
$T$-action with discrete fixed point set $M^{T}$ and moment map $\psi\colon
M\to\mathfrak{t}^{*}$. Let $\varphi\colon M\to\mathbb{R}$ be a generic
component of the moment map and $p_{0},p_{1},\ldots,p_{m}$ be the fixed points
ordered so that

$p_{0}\prec p_{1}\preceq p_{2}\preceq\cdots\preceq p_{m-1}\prec p_{m}\,.$

Consider a basis of $H_{T}(M;\mathbb{Z})$ given by Kirwan classes
$\\{\gamma_{i}\\}_{i=0}^{m}$ and let
$\alpha=\sum_{i=0}^{m}\alpha^{i}\gamma_{i}$, where $\alpha^{i}\in
H(BT;\mathbb{Z})$. Then the coefficients $\alpha^{i}$ can be computed from the
restriction of $\alpha$ to the fixed points. More precisely, they can be
computed recursively as

$\alpha^{i}=\frac{\alpha(p_{i})-\sum_{h\colon\varphi(p_{h})<\varphi(p_{i})}\alpha^{h}\gamma_{h}(p_{i})}{\Lambda_{i}^{-}}\,.$

A set of equivariant cohomology classes, for which the restrictions to the
fixed points can be computed easily from the isotropy data, is that of
equivariant Chern classes. The next lemma is a well known fact.

######  Lemma 2.5.

Let $(M,J)$ be a compact almost complex manifold acted on by a torus $T$
preserving $J$ and with discrete fixed point set $M^{T}$. Let
$w_{1},\ldots,w_{n}\in\ell^{*}$ be the isotropy weights of the $T$ action at a
fixed point $p$, where $\ell^{*}$ denotes the dual lattice in
$\mathfrak{t}^{*}$. Consider the equivariant Chern classes $c_{j}^{T}\in
H_{T}^{2j}(M;\mathbb{Z})$ of the tangent bundle of $(M,J)$. Then for every
$p\in M^{T}$

$c_{j}^{T}(p)=\sigma_{j}(w_{1},\ldots,w_{n})\,,$

where $\sigma_{j}(x_{1},\ldots,x_{n})$ is the $j$-th elementary symmetric
polynomial.

###### Remark 2.6

From Lemmas 2.4 and 2.5 it follows that, given a basis of Kirwan classes and
given their restrictions to the fixed point set, the coefficients of the
equivariant Chern classes with respect to this basis can be computed easily
from the isotropy data at the fixed points.

###### Remark 2.7

(Convention on canonical classes) Let $(M,\omega)$ be a closed symplectic
manifold equipped with a Hamiltonian $T$-action with discrete fixed point set
$M^{T}$ and moment map $\psi\colon M\to\mathfrak{t}^{*}$. Assume that for
every $p\in M^{T}$ a canonical class $\tau_{p}$ exists. In this paper we
consider a slight modification of the canonical classes, namely those given by
$(-1)^{\lambda_{p}}\tau_{p}$. We still denote the “new canonical class” at $p$
by $\tau_{p}$. Note that, with this convention, $\tau_{p}(p)$ is a product of
positive weights.

###  2.3. Hamiltonian GKM spaces

One special subclass of compact symplectic manifolds acted on by a torus in a
Hamiltonian way is that of Hamiltonian GKM spaces. A Hamiltonian action of a
torus $T$ of dimension greater or equal to 2 is called GKM (after Goresky-
Kottwitz-MacPherson [19]) if the fixed points are isolated and for each fixed
point $p$, the weights of the isotropy action at $p$ are pairwise linearly
independent. This implies that for every isotropy weight $\alpha\in\ell^{*}$,
the manifold fixed by the subgroup $T_{\alpha}:=\exp\\{\text{Ann}(\alpha)\\}$
is a 2 dimensional sphere, called _isotropy sphere_. Each sphere is indeed
symplectic and acted on effectively by the quotient circle $T/T_{\alpha}$,
with two fixed points, $p$ and $q$. Observe that, if $\alpha$ is the weight of
the isotropy $T$ action on $S^{2}$ at $p$, then $-\alpha$ is the weight of the
isotropy $T$ action on $S^{2}$ at $q$.

One can assign a labeled graph $\Gamma=(V,E)$ to each (Hamiltonian) GKM space,
where the vertices are in bijection with the fixed points, and the edges with
the isotropy spheres. If one chooses a direction on each edge, for instance
given by a generic component of the moment map, then the label of the edge
$e=(p,q)$ (the edge connecting the fixed points $p$ and $q$ of the quotient
action on the corresponding $S^{2}$) is given by the weight of the $T$ action
at $p$, and we denote this weight by $\eta(p,q)$. Since we are assuming that
this action comes from a Hamiltonian action with moment map $\psi\colon
M\to\mathfrak{t}^{*}$, it is natural to see the GKM graph in
$\mathfrak{t}^{*}$, mapping each vertex $v$, corresponding to the fixed point
$p$, to $\psi(p)$, and each isotropy sphere to its image through the moment
map $\psi$. Examples of such graphs are given in Figures 5.1, 5.2 and 5.8 (see
the corresponding sections for a description of the manifold and the action).

Suppose that there exists a generic component $\varphi$ of the moment map
$\psi$ that is _index increasing_ , namely

$p\prec q\implies\lambda_{p}<\lambda_{q}\quad\text{for all the edges
}e=(p,q)\text{ of the GKM graph.}$

Then Goldin and Tolman [18] give a recipe to compute the equivariant
cohomology ring with $\mathbb{Z}$ coefficients. This method will be used later
in Sections 5.1 and 5.3 to compute the equivariant cohomology rings of the
corresponding manifolds.

###  2.4. Monotone Hamiltonian $T$ spaces

Let $(M,\omega)$ be a compact symplectic manifold of dimension $2n$ endowed
with a Hamiltonian action of a torus $T$. We say that $(M,\omega)$ is monotone
if there exists $\lambda\in\mathbb{R}$ such that $c_{1}=\lambda[\omega]$.
Since the action is Hamiltonian, it is known that $\lambda$ must be positive
(see for instance [16, Lemma 5.2] or [9, Proposition 3.3]). We can therefore
rescale the symplectic form so that $c_{1}=[\omega]$ holds.

At the level of equivariant cohomology classes, the above equation implies
that there exists $a\in\mathfrak{t}^{*}$ such that
$c_{1}^{T}=[\omega-\psi]+a$. Since the moment map is defined up to a constant,
by replacing the moment map $\psi$ with the moment map $\psi-a$ we obtain that

$c_{1}^{T}=[\omega-\psi]\,.$

Therefore, restricting the above equality to a fixed point $p$, we obtain the
so called weight sum formula

(2.5) $\sum_{i=1}^{n}w_{i}^{p}=-\psi(p)\quad\text{for all fixed points }p\in
M^{T},$

where $w_{1}^{p},\ldots,w_{n}^{p}$ denote the weights of the isotropy action
at $p\in M^{T}$. (See [9, Section 3.1].)

##  3\. The Chern Numbers

In this section we investigate the Chern numbers of an 8-dimensional closed
symplectic manifold that can be endowed with a Hamiltonian circle action and
isolated fixed points.

An important role in this investigation is played by the index of
$(M,\omega)$, which is defined as follows. Given an almost complex manifold
$(M,J)$, the index of $(M,J)$ is the largest integer such that, modulo
torsion, $c_{1}=k_{0}\,\eta$ for some nonzero $\eta\in H^{2}(M;\mathbb{Z})$.
If we consider a symplectic manifold $(M,\omega)$, we regard it as an almost
complex manifold $(M,J)$, where $J$ is an almost complex structure compatible
with $\omega$. Since the set of such structures is contractible, complex
invariants of the tangent bundle of $(M,\omega)$, such as Chern classes, Chern
numbers and the index, are well-defined.

If $(M,\omega)$ can be endowed with a Hamiltonian $S^{1}$-action with isolated
fixed points, then the index $k_{0}$ coincides with the minimal Chern number
and satisfies the following inequalities

(3.1) $1\leq k_{0}\leq n+1$

(see [41, Corollary 1.1]).

The problem of determining which numbers can arise as the Chern numbers of a
compact symplectic manifold acted on a by torus in a Hamiltonian way is wide
open. Some information about them can be recovered from the Todd genus and the
index of $(M,\omega)$ (for an extensive discussion see, for instance, [41]).
We recall that the Todd genus of $(M,\omega)$, simply denoted by
$\text{Td}(M)$, is the genus associated to the power series given by
$\displaystyle\frac{x}{1-e^{-x}}$ and it can be computed in terms of the Chern
numbers. For instance, in dimension 8 (which is the dimension we are mostly
interested in in this paper) it is given by

(3.2)
$\text{Td}(M)=\frac{-c_{1}^{4}+4c_{1}^{2}c_{2}+3c_{2}^{2}+c_{1}c_{3}-c_{4}}{720}[M]\,.$

If $(M,\omega)$ is acted on by a circle in a Hamiltonian way, then it is well-
known that $\text{Td}(M)=\text{Td}(M_{\text{min}})$, where $M_{\text{min}}$
denotes the submanifold on which the moment map attains its minimum (see for
instance [25, Section 5.7]). Therefore, if the fixed points are isolated, one
obtains immediately that $\text{Td}(M)=1$.

Now consider an 8-dimensional closed symplectic manifold that can be endowed
with a Hamiltonian circle action and isolated fixed points. First we recall
that, in this case, two of the Chern numbers, namely $c_{4}[M]$ and
$c_{1}c_{3}[M]$, can be expressed as linear combinations of the even Betti
numbers (see [15, Section 3] and in particular [15, Corollary 3.1]).

######  Lemma 3.1 (Godinho–Sabatini, [15]).

Let $(M,\omega)$ be a closed, $8$-dimensional symplectic manifold admitting a
Hamiltonian $S^{1}$-action with isolated fixed points. Let $b_{i}(M)$ be the
Betti numbers of $M$. Then

(3.3) $c_{4}[M]=2+2b_{2}(M)+b_{4}(M)\quad\text{and}\quad
c_{1}c_{3}[M]=44+8b_{2}(M)-2b_{4}(M)\,.$

Before stating our next results, we also recall the following equations for
the other Chern numbers, that are a direct consequence of [41, Propositions
6.3, 6.4] applied to closed, connected symplectic manifolds that can be
endowed with a Hamiltonian $S^{1}$-action and isolated fixed points, where
$N_{0}=1$ and $b_{2i}(M)=N_{i}$ for every $i$.

######  Proposition 3.2 (Sabatini, [41]).

Let $(M,\omega)$ be a closed symplectic $8$-dimensional manifold, admitting a
Hamiltonian $S^{1}$-action with isolated fixed points. Let $b_{i}(M)$ be the
Betti numbers of $M$ and $k_{0}$ the index of $(M,\omega)$. Then:

$\bullet$ If $k_{0}=5$ then

(3.4)
$c_{1}^{4}[M]=625,\;\;c_{1}^{2}c_{2}[M]=250,\;\;c_{2}^{2}[M]=101-2b_{2}(M)+b_{4}(M)\,.$

$\bullet$ If $k_{0}=4$ then

(3.5)
$c_{1}^{4}[M]=512,\;\;c_{1}^{2}c_{2}[M]=224,\;\;c_{2}^{2}[M]=98-2b_{2}(M)+b_{4}(M)\,.$

$\bullet$ If $k_{0}=3$ then

(3.6)
$c_{1}^{2}c_{2}[M]=108+\frac{2}{9}c_{1}^{4}[M],\;\;c_{2}^{2}[M]=82-2b_{2}(M)+b_{4}(M)+\frac{1}{27}c_{1}^{4}[M]\,.$

$\bullet$ If $k_{0}=2$ then

(3.7)
$c_{1}^{2}c_{2}[M]=96+\frac{1}{4}c_{1}^{4}[M],\;\;c_{2}^{2}[M]=98-2b_{2}(M)+b_{4}(M)\,.$

Let $M$ be a closed, orientable manifold of dimension $4k$, for some
$k\in\mathbb{Z}_{>0}$, with chosen orientation class $[M]\in
H_{4k}(M;\mathbb{Z})$. Consider the intersection form
$\langle\cdot,\cdot\rangle\colon H^{2k}(M;\mathbb{Z})\times
H^{2k}(M;\mathbb{Z})\to\mathbb{Z}$ given by

$\langle\alpha,\beta\rangle=(\alpha\cup\beta)[M].$

If we consider the intersection form with $\mathbb{R}$-coefficients, then it
is bilinear, symmetric form on $H^{2k}(M;\mathbb{R})$. Let $b_{2k}^{+}(M)$
(resp. $b_{2k}^{-}(M)$) be the dimension of the maximal subspace on which
$\langle\cdot,\cdot\rangle$ is positive (resp. negative) definite. The
signature $\sigma(M)$ of $M$ is the signature of $\langle\cdot,\cdot\rangle$,
namely $\sigma(M)=b_{2k}^{+}(M)-b_{2k}^{-}(M)$. If the manifold is symplectic
and there is a Hamiltonian circle action with isolated fixed points, the
signature of $M$ can be computed from its Betti numbers [29]. The next
proposition, originally proved in [33], uses this fact to characterize under
which conditions $\langle\cdot,\cdot\rangle$ is positive definite in dimension
8:

######  Proposition 3.3.

Let $(M,\omega)$ be a closed, $8$-dimensional symplectic manifold admitting a
Hamiltonian $S^{1}$-action with isolated fixed points. Then the intersection
form $\langle\cdot,\cdot\rangle$ of $M$ is positive definite $\iff$
$b_{2}(M)=1$.

###### Proof.

By [29, Theorem 1.1] and Poicaré duality, we have

(3.8) $\sigma(M)=2-2b_{2}(M)+b_{4}(M).$

The condition that the bilinear form is positive definite is equivalent to
$\sigma(M)=b_{4}(M)$. To see this, write $b_{4}(M)$ as
$b_{4}^{+}(M)+b_{4}^{-}(M)$ and $\sigma(M)$ as $b_{4}^{+}(M)-b_{4}^{-}(M)$,
which immediately implies that $b_{4}^{+}(M)=b_{4}(M)\iff\sigma(M)=b_{4}(M)$.
From Equation 3.8, $\sigma(M)=b_{4}(M)$ is equivalent to $b_{2}(M)=1$. ∎

Consider a closed symplectic $8$-dimensional manifold admitting a Hamiltonian
$S^{1}$-action with isolated fixed points. From Proposition 3.2 it is clear
that, if the index $k_{0}$ is either 4 or 5, then the Betti numbers $b_{2}(M)$
and $b_{4}(M)$ determine all the Chern numbers, as the latter depend linearly
on the Betti numbers. The cases in which $k_{0}$ is equal to 2 or 3 are more
complicated, as equations (3.6) and (3.7) only give some Chern numbers as
functions of others. In the following, we prove that, once we assume
$b_{2}(M)=1$, then there are only finitely many possible Chern numbers for
each value of $b_{4}(M)$ and give an explicit range in which they vary. The
key role is played by the fact that, for $b_{2}(M)=1$, the intersection
product is positive definite (see Proposition 3.3).

######  Theorem 3.4.

Let $(M,\omega)$ be a closed symplectic $8$-dimensional manifold admitting a
Hamiltonian $S^{1}$-action with isolated fixed points. Assume that the index
$k_{0}$ is not one and that $b_{2}(M)=1$. Then, for each value of
$b_{4}:=b_{4}(M)$, there are finitely many possibilities for the Chern numbers
of $(M,\omega)$.

More precisely we have

(3.9) $c_{4}[M]=4+b_{4}\,,\quad c_{1}c_{3}[M]=52-2b_{4}\,,$

and for the Chern numbers $c_{1}^{4}[M]$, $c_{1}^{2}c_{2}[M]$ and
$c_{2}^{2}[M]$ we have that:

  * •

If $k_{0}=4,5$ then they are determined by the Betti numbers and their values
are given by Proposition 3.2 (3.4) and (3.5).

  * •

If $k_{0}=3$, then

$c_{2}^{2}[M]=80+b_{4}+\frac{1}{27}c_{1}^{4}[M],\quad
c_{1}^{2}c_{2}[M]=108+\frac{2}{9}c_{1}^{4}[M]$

and

$\frac{c_{1}^{4}[M]}{81}\in\left[\frac{b_{4}+32-\sqrt{b_{4}^{2}+64\,b_{4}+448}}{2},\,\frac{b_{4}+32+\sqrt{b_{4}^{2}+64\,b_{4}+448}}{2}\right]\cap\mathbb{Z}.$

  * •

If $k_{0}=2$ then

$c_{2}^{2}[M]=96+b_{4},\quad c_{1}^{2}c_{2}[M]=96+\frac{1}{4}c_{1}^{4}[M]$

and

$\frac{c_{1}^{4}[M]}{16}\in\left[\frac{(\sqrt{96+b_{4}}-\sqrt{b_{4}})^{2}}{4},\,\frac{(\sqrt{96+b_{4}}+\sqrt{b_{4}})^{2}}{4}\right]\cap\mathbb{Z}.$

###### Proof.

Equation (3.9) is direct consequence of Lemma 3.1. Then we observe that
$k_{0}$ is bounded above by 5 ([41, Corollary 1.1]) and that for $k_{0}=4,5$
the claim follows as a special case of Proposition 3.2 (3.4) and (3.5). What
is left to prove are the claims for $k_{0}=2$ and $3$.

Let $b_{4}=N$ and fix an integral basis of
$H^{4}(X,\mathbb{Z})=\mathbb{Z}^{N}$. Let $X=(x_{1},\ldots,x_{N})$ and
$Y=(y_{1},\ldots,y_{N})\in\mathbb{Z}^{N}$ be the coordinates of $c_{1}^{2}$
and $c_{2}$ with respect to this basis. Since $b_{2}(M)=1$, we have from
Proposition 3.3 that the intersection product on $H^{4}(M,\mathbb{Z})$ is a
positive definite inner product. Let $\lVert\cdot\rVert$ be the associated
norm. Then we have

$\lVert X\rVert^{2}=c_{1}^{4},\quad\langle
X,Y\rangle=c_{1}^{2}c_{2}\quad\text{and}\quad\lVert Y\rVert^{2}=c_{2}^{2}.$

Using the formula for $c_{1}c_{3}$ in [15, Corollary 3.1] and the fact that
$c_{4}[M]$ is the Euler characteristic of $M$, the equation $\text{Td}(M)=1$
gives

(3.10) $720=-\lVert X\rVert^{2}+4\langle X,Y\rangle+3\lVert
Y\rVert^{2}+48-3N.$

$\bullet$ $k_{0}=3$: From [41, Proposition 6.4, equation (6.23)], we have

(3.11) $\lVert Y\rVert^{2}=80+N+\frac{\lVert X\rVert^{2}}{27}.$

Then, substituting (3.11) into (3.10) gives

(3.12) $9\langle X,Y\rangle=2\lVert X\rVert^{2}+972.$

Applying the Cauchy-Schwarz inequality we obtain

$2\lVert X\rVert^{2}+972\leq 9\lVert X\rVert\lVert Y\rVert.$

Squaring both sides of the inequality, we get

$4\lVert X\rVert^{4}+3888\lVert X\rVert^{2}+944784\,\leq 81\,\lVert
X\rVert^{2}\left(80+N+\frac{\lVert X\rVert^{2}}{27}\right)$

and so

$\lVert X\rVert^{4}-\left(81N+2592\right)\lVert X\rVert^{2}+944784\leq 0.$

So, for each $N$, there are finitely many possibilities for $\lVert
X\rVert^{2}$, and consequently, by (3.11) and (3.12), for $\langle X,Y\rangle$
and $\lVert Y\rVert^{2}$, thus proving that for each $b_{4}$ there are
finitely many possibilities for the Chern numbers $c_{1}^{4}[M]$,
$c_{1}^{2}c_{2}[M]$ and $c_{2}^{2}[M]$.

$\bullet$ $k_{0}=2$: From [41, Corollary 6.1], we have

(3.13) $\lVert Y\rVert^{2}=96+N.$

Substituting (3.13) into (3.10) and rearranging gives

(3.14) $4\langle X,Y\rangle=384+\lVert X\rVert^{2}$

and so, from the Cauchy-Schwarz inequality, we get

$\lVert X\rVert^{2}-4\lVert X\rVert\lVert Y\rVert+384\leq 0.$

Solving the quadratic inequality and substituting the expression for $\lVert
Y\rVert$ in (3.13), we obtain that

(3.15) $\lVert
X\rVert\in[2\,(\sqrt{96+N}-\sqrt{N}),\,2\,(\sqrt{96+N}+\sqrt{N})]$

So, for each $N$, there are finitely many possibilities for $\lVert
X\rVert^{2}$, and consequently, by (3.13) and (3.14), for $\langle X,Y\rangle$
and $\lVert Y\rVert^{2}$, thus proving that for each $b_{4}$ there are
finitely many possibilities for the Chern numbers $c_{1}^{4}[M]$,
$c_{1}^{2}c_{2}[M]$ and $c_{2}^{2}[M]$.

∎

As a direct consequence of Theorem 3.4 we have the following

######  Corollary 3.5.

For each $N\in\mathbb{Z}_{>0}$, there is a finite number of complex cobordism
classes of 8-dimensional closed symplectic manifolds that admit a Hamiltonian
$S^{1}$-action with isolated fixed points, with index $k_{0}$ not equal to
one, $b_{2}(M)=1$ and $b_{4}(M)=N$.

Observe that any such manifold is automatically positive monotone. Indeed,
$b_{2}(M)=1$ forces $c_{1}=\lambda[\omega]$ for some $\lambda\in\mathbb{R}$,
and the existence of a Hamiltonian action forces $\lambda$ to be positive (see
[16, Lemma 5.2]). Therefore this corollary goes in the direction of proving
that there are finitely many complex cobordism classes of a positive monotone
symplectic manifold admitting a Hamiltonian group action.

###  3.1. The case $\chi(M)=6$

In this subsection we analyze the special case in which the Euler
characteristic $\chi(M)$ is equal to $6$, which will be further investigated
in the following sections. We begin with the following

######  Proposition 3.6.

Let $(M,\omega)$ be a closed, $8$-dimensional symplectic manifold admitting a
Hamiltonian $S^{1}$-action with isolated fixed points. Then the following
statements hold:

  1. (1)

$\chi(M)=6$ if and only if $b_{2}(M)=1,b_{4}(M)=2$.

  2. (2)

If $\chi(M)=6$, then there exists a basis $v_{1},v_{2}$ of
$H^{4}(M;\mathbb{Z})$ in which the restriction of $\langle\cdot,\cdot\rangle$
to $H^{4}(M;\mathbb{Z})\simeq\mathbb{Z}^{2}$ is the standard Euclidean form,
namely:

(3.16) $\langle v_{i},v_{j}\rangle=\delta_{i}^{\,j},\quad\text{for all
}i,j=1,2\,,$

where $\delta_{i}^{\,j}$ denotes the Kronecker delta.

###### Proof.

By Lemma 3.1, the Euler characteristic satisfies
$\chi(M)=c_{4}[M]=2+2b_{2}(M)+b_{4}(M)$, which, together with the fact that
$b_{2i}(M)\geq 1$ for all $i=0,\ldots,4$, is equal to $6$ if and only if
$b_{2}(M)=1$ and $b_{4}(M)=2$. This proves (1).

By Proposition 3.3 and (1) we know that the intersection form is positive
definite. Since there is exactly one isomorphism class of symmetric,
unimodular, positive definite, integral bilinear forms of rank $2$, that is
the Euclidean inner product on $\mathbb{Z}^{2}$ [35, Theorem 2.2], the result
follows. ∎

The next theorem is a refinement of Theorem 3.4 to the case in which
$\chi(M)=6$. Using Proposition 3.6 (2) and number theoretical arguments, we
are able to restrict further the possible values of the Chern numbers of
$(M,\omega)$, when $k_{0}>1$.

######  Theorem 3.7.

Let $(M,\omega)$ be a closed $8$-dimensional symplectic manifold with
$\chi(M)=6$, admitting a Hamiltonian $S^{1}$-action with isolated fixed
points. Then the index $k_{0}$ satisfies $1\leq k_{0}\leq 4$. If the index
$k_{0}$ is not one, then the possible Chern numbers are listed in Tables 3.1
and 3.2. More precisely:

$\bullet$ If $k_{0}=2$ or 4, then the possible Chern numbers are listed in
Table 3.1; in particular, when $k_{0}=4$, the Chern numbers are those of Case
1.

| $c_{1}^{4}[M]$ | $c_{1}^{2}c_{2}[M]$ | $c_{2}^{2}[M]$ | $c_{1}c_{3}[M]$ | $c_{4}[M]$
---|---|---|---|---|---
Case 1 | 512 | 224 | 98 | 48 | 6
Case 2 | 288 | 168 | 98 | 48 | 6
Case 3 | 400 | 196 | 98 | 48 | 6
Table 3.1.  Possible Chern numbers of $(M,\omega)$ if $k_{0}=2,4$.

$\bullet$ If $k_{0}=3$, then the possible Chern numbers are listed in Table
3.2.

| $c_{1}^{4}[M]$ | $c_{1}^{2}c_{2}[M]$ | $c_{2}^{2}[M]$ | $c_{1}c_{3}[M]$ | $c_{4}[M]$
---|---|---|---|---|---
Case 1 | 405 | 198 | 97 | 48 | 6
Case 2 | 648 | 252 | 106 | 48 | 6
Case 3 | 729 | 270 | 109 | 48 | 6
Case 4 | 1296 | 396 | 130 | 48 | 6
Case 5 | 1458 | 432 | 136 | 48 | 6
Case 6 | 2025 | 558 | 157 | 48 | 6
Case 7 | 2106 | 576 | 160 | 48 | 6
Case 8 | 2349 | 630 | 169 | 48 | 6
Table 3.2.  Possible Chern numbers of $(M,\omega)$ if $k_{0}=3$.

###### Proof.

First of all we observe that, by Proposition 3.6 (1) and Lemma 3.1, the Chern
numbers $c_{1}c_{3}[M]$ and $c_{4}[M]$ are always respectively equal to $48$
and $6$. Since, by [41, Corollary 1.1], the index $k_{0}$ of $(M,\omega)$
satisfies $1\leq k_{0}\leq 5$, to prove the first statement we need to prove
that $k_{0}$ cannot be 5. But this follows immediately from the fact that
$k_{0}$ should always divide the Chern number $c_{1}c_{3}[M]$, which, in this
case, is 48. Therefore $1\leq k_{0}\leq 4$.

$\bullet$ _Suppose that $k_{0}=2$ or 4._ Then, by Proposition 3.2, we have
that $c_{2}^{2}[M]=98-2b_{2}(M)+b_{4}(M)$, which, together with Proposition
3.6 (1), implies that $c_{2}^{2}[M]=98$. Hence we just have to determine the
possible values of $c_{1}^{4}[M]$ and $c_{1}^{2}c_{2}[M]$.

If $k_{0}=4$, then Propositions 3.2 (3.5) and 3.6 (1) imply that the Chern
numbers are exactly those of Case 1 in Table 3.1.

If $k_{0}=2$, by Proposition 3.2 (3.7) we have

(3.17) $c_{1}^{4}[M]+384=4c_{1}^{2}c_{2}[M].$

Let $v_{1},v_{2}$ be the basis of $H^{4}(M;\mathbb{Z})$ given by Proposition
3.6 (2). Then, writing $c_{2}=y_{1}v_{1}+y_{2}v_{2}$ with
$y_{1},y_{2}\in\mathbb{Z}$, we have $y_{1}^{2}+y_{2}^{2}=98$. It is easy to
check that this implies $y_{1}=\pm 7$ and $y_{2}=\pm 7$.

Since the index is 2, we have that $c_{1}=2u$, for some (primitive element)
$u\in H^{2}(M,\mathbb{Z})$. Hence $c_{1}^{2}=4u^{2}$ and writing
$u^{2}=k_{1}v_{1}+k_{2}v_{2}$ with $k_{1},k_{2}\in\mathbb{Z}$, gives
$c_{1}^{2}=4k_{1}v_{1}+4k_{2}v_{2}$ and
$c_{1}^{4}[M]=16(k_{1}^{2}+k_{2}^{2})$. Using (3.17), we obtain

$16(k_{1}^{2}+k_{2}^{2})+384=16(y_{1}k_{1}+y_{2}k_{2})$

and so, by Lemma 3.8, we have $(|k_{1}|,|k_{2}|)=(3,3),(3,4)$ or $(4,4)$,
implying that $c_{1}^{4}[M]$ is $288$, $400$ or $512$. The corresponding
values of $c_{1}^{2}c_{2}[M]$ can be obtained using (3.17).

$\bullet$ _Suppose that $k_{0}=3$._ We will first prove that there exists
$(a,b,c,d)\in\mathbb{Z}^{4}$ such that:

(3.18) $c_{1}^{4}[M]=81(c^{2}+d^{2}),\quad
c_{1}^{2}c_{2}[M]=9(ac+bd)\quad\text{and}\quad c_{2}^{2}[M]=a^{2}+b^{2},$

where $(a,b,c,d)$ satisfy

  1. (i)

$a^{2}+b^{2}-(82+3c^{2}+3d^{2})=0$

  2. (ii)

$ac+bd=12+2(c^{2}+d^{2})$. We begin by expressing $c_{2}$ and $c_{1}^{2}/9$ in
terms of the integral basis given by Proposition 3.6 (2). Let
$a,b\in\mathbb{Z}$ be such that $c_{2}=av_{1}+bv_{2}$. Since $k_{0}=3$, we
have $c_{1}=3u$, for some $u\in H^{2}(M,\mathbb{Z})$ and $c_{1}^{2}=9u^{2}$.
Writing $u^{2}=cv_{1}+dv_{2}$ with $c,d\in\mathbb{Z}$, gives
$c_{1}^{2}=9cv_{1}+9dv_{2}$.

On the other hand, by Proposition 3.6 (1) and Proposition 3.2 (3.6), we have

$c_{2}^{2}[M]=82+\frac{c_{1}^{4}[M]}{27}\quad\text{and}\quad
c_{1}^{2}c_{2}[M]=108+\frac{2}{9}c_{1}^{4}[M].$

Substituting the above expressions in terms of $v_{1}$ and $v_{2}$, we obtain
(i) and (ii).

To finish the proof we just need to determine the possible integer solutions
$(a,b,c,d)$ of (i) and (ii). This is done in Lemma 3.9. The possible values of
the Chern numbers $c_{1}^{4}[M],c_{1}^{2}c_{2}[M]$ and $c_{2}^{2}[M]$ can then
be obtained using (3.18).

∎

######  Lemma 3.8.

Let $x,y\in\mathbb{Z}$ and $y_{1},y_{2}\in\\{-7,7\\}$ be such that

$x^{2}+y^{2}+24=y_{1}x+y_{2}y.$

Then, $(|x|,|y|)$ is one of $(3,3),(3,4)$ or $(4,4)$.

###### Proof.

First we note that we can reduce to solving

(3.19) $x^{2}+y^{2}+24=7x+7y.$

Indeed, if $(x,y)\in\mathbb{Z}^{2}$ is a solution of

$x^{2}+y^{2}+24=y_{1}x+y_{2}y$

for some $y_{1},y_{2}\in\\{\pm 7\\}$, then any pair of the form
$(x^{\prime},y^{\prime})=(\pm x,\pm y)$ is a solution of (3.19) for some other
values of $y_{1},y_{2}\in\\{\pm 7\\}$.

Since the left hand side of (3.19) is positive, we have

$(x^{2}+y^{2})+24=7\lvert x+y\rvert\leq 7\left(\lvert x\rvert+\lvert
y\rvert\right)$

and so

(3.20) $(x^{2}-7|x|)+(y^{2}-7|y|)\leq-24.$

Suppose, without loss of generality, that $(x^{2}-7|x|)\leq(y^{2}-7|y|)$. Then
by (3.20) we have $x^{2}-7|x|+12\leq 0$. It is easy to check that the only
integer values of $|x|$ satisfying this inequality are $3$ or $4$ and that for
both of these values we have $x^{2}-7|x|=-12$. Consequently, by (3.20), we get
that $(y^{2}-7|y|)\leq-12$. The same argument then gives $|y|=3$ or $4$.

∎

######  Lemma 3.9.

Let $u:=(a,b)\in\mathbb{Z}^{2}$ and $v:=(c,d)\in\mathbb{Z}^{2}$, where we
endow $\mathbb{Z}^{2}$ with standard scalar product. Then there are finitely
many integer vectors $u$ and $v$ satisfying

(3.21) $\left\\{\begin{array}[]{l}\lVert u\rVert^{2}=3\lVert v\rVert^{2}+82\\\
\\\ \langle u,v\rangle=2\lVert v\rVert^{2}+12.\end{array}\right.$

Moreover the possible values of $\|u\|^{2},\langle u,v\rangle$ and $\|v\|^{2}$
are given in the following table:

$\,\,\lVert u\rVert^{2}$ | 97 | 106 | 109 | 130 | 136 | 157 | 160 | 169
---|---|---|---|---|---|---|---|---
$\langle u,v\rangle$ | 22 | 28 | 30 | 44 | 48 | 62 | 64 | 70
$\,\,\lVert v\rVert^{2}$ | 5 | 8 | 9 | 16 | 18 | 25 | 26 | 29
Table 3.3. Possible values

###### Proof.

By the Cauchy-Schwarz inequality and the first equation of (3.21) we have

(3.22) $\langle u,v\rangle^{2}\leq\lVert u\rVert^{2}\lVert
v\rVert^{2}=\left(3\lVert v\rVert^{2}+82\right)\lVert v\rVert^{2},$

and so, by the second equation of (3.21), we obtain

$\langle u,v\rangle^{2}=(2\lVert v\rVert^{2}+12)^{2}\leq 3\lVert
v\rVert^{4}+82\lVert v\rVert^{2}.$

We conclude that

$\left(\lVert v\rVert^{2}-17\right)^{2}\leq 145,$

and then $\lVert v\rVert^{2}\in[5,29]\cap\mathbb{Z}$. Moreover, since $\lVert
v\rVert^{2}$ and $\lVert u\rVert^{2}$ are sums of two squares and, by (3.21),
we have $\lVert u\rVert^{2}=3\lVert v\rVert^{2}+82$, we conclude that

(3.23) $\lVert v\rVert^{2}\in\\{5,8,9,13,16,18,25,26,29\\}.$

Note that $\lVert v\rVert^{2}$ cannot admit the values $10,17$ and $20$
despite the fact that they are sums of two squares, since the corresponding
values of $\lVert u\rVert^{2}$ would be $112,133$ and $143$ and these are not
sums of two squares222The nonnegative integers that are sums of two squares
were characterized by Euler [32, p.230] and form the sequence A001481 in OEIS.
The first elements of this sequence are 0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17,
18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64,
65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109,
113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145, 146, 148, 149,
153, 157, 160,….

We further note that $\lVert v\rVert^{2}\neq 13$. Indeed, if $\lVert
v\rVert^{2}=13$, then, by (3.21), we have $\lVert u\rVert^{2}=121$ and
$\langle u,v\rangle=38$. However, since $13=9+4$ and $121=11^{2}+0$ are the
only possible partitions of $13$ and $121$ as a sum of two squares, we have
that $\langle u,v\rangle$ is either $\pm 33$ or $\pm 22$.

It is easy to check that all other values of $\lVert v\rVert^{2}$ in (3.23)
are possible and so we conclude that there are only eight possibilities for
$\lVert v\rVert^{2}$ and that these determine $\lVert u\rVert^{2}$ and
$\langle u,v\rangle$ (see Table 3.3).

∎

##  4\. Multigraphs describing the action and their positivity properties

Let $(M,J)$ be a closed almost complex manifold of dimension $2n$ acted on by
a torus $T$ of dimension $k$ that preserves $J$. Let $\mathfrak{t}$ (resp.
$\mathfrak{t}^{*}$) be the Lie algebra (resp. the dual of the Lie algebra) of
$T$ and $\ell\subset\mathfrak{t}$ the integral lattice (resp.
$\ell^{*}\subset\mathfrak{t}^{*}$ the dual integral lattice). Let $M^{T}$ be
the set of fixed points of the action. In this section we review the notion of
multigraph describing the $T$ action on $(M,J)$, assuming that there are
finitely many fixed points, and describe some of its properties. This concept
is not new and has been introduced and studied by several authors (see for
instance [43, 15, 28, 27]).

To begin with, it is well-known that around each fixed point $p$ there exist
complex coordinates $(z_{1},\ldots,z_{n})$ and elements
$w_{1},\ldots,w_{n}\in\ell^{*}$, called the (isotropy) weights of the action
at $p$, that model the action around $p=(0,\ldots,0)$. Namely

(4.1) $T\ni\lambda=\exp(\xi)*(z_{1},\ldots,z_{n})=(e^{2\pi
iw_{1}(\xi)}z_{1},\ldots,e^{2\pi iw_{n}(\xi)}z_{n}).$

If $p$ is an isolated fixed point, then none of the weights at $p$ can be
zero.

Let $W$ be the multiset of all the weights at the fixed points of a $T$ action
on $(M,J)$. Note that if a weight $w\in\ell^{*}$ appears $m$ times as an
isotropy weight at (possibly different) fixed points, it will appear $m$ times
in $W$. This integer $m$ is called the multiplicity of $w$ and will be denoted
here by $N(w)$. The next lemma is a straightforward generalization of [24,
Proposition 2.11] and, although it has been proved in [27, Proposition 2.2],
we present here a different proof.

######  Lemma 4.1.

Let $(M,J)$ be a closed almost complex manifold equipped with an effective
action of a torus $T$ of dimension $k$ with isolated fixed points. Let $W$ be
the multiset of weights and $N(w)$ the multiplicity of a weight $w$ in $W$.
Then

(4.2) $N(w)=N(-w)\quad\text{for every }w\in W\,.$

###### Proof.

In [24, Proposition 2.11] Hattori proves (4.2) for circle actions with
isolated fixed points. Hence, in order to prove it for torus actions, it is
sufficient to show that, if (4.2) does not hold for the action of a torus $T$
of dimension $k$ with isolated fixed points, then there exists a circle
subgroup of $T$ acting with isolated fixed points whose weights do not satisfy
(4.2).

We begin with the following observations:

  1. $(i)$

Let $G$ be a circle subgroup of $T$ and assume that $\xi$ is a primitive
element of the lattice $\ell_{G}$ of $\text{Lie}(G)$ that determines the
orientation of $G$, namely $G=\exp\langle\xi\rangle$. Then the set $M^{G}$ of
fixed points of $G$ contains $M^{T}$ and, for each $p\in M^{T}$ the weights of
the $G$-action at $p$ are given by the integers $w_{i}(\xi)$, for
$i=1,\ldots,n$, where $w_{1},\ldots,w_{n}$ are the weights of the $T$-action
at $p$. Moreover, if $N(w(\xi))$ denotes the multiplicity of the weight
$N(w(\xi))$ of the $G$-action at $p\in M^{T}$, then

$N(w(\xi))\geq N(w),$

and equality holds iff $w(\xi)\neq w^{\prime}(\xi)$ for all the weights
$w^{\prime}$ of the $T$-action different from $w$.

  2. $(ii)$

The subset $D\subset\mathfrak{t}$ of $\xi$’s for which $h\xi\in\ell$ for some
$h\in\mathbb{Z}$ and such that the corresponding circle subgroup
$G:=\\{\exp(t\xi),t\in\mathbb{R}\\}$ satisfies $M^{G}=M^{T}$ is dense in
$\mathfrak{t}$.

Suppose that there exists $\widetilde{w}\in W$ such that $N(\widetilde{w})\neq
N(-\widetilde{w})$. The set

$U:=\\{\xi\in\mathfrak{t}\mid\widetilde{w}(\xi)\neq\pm w(\xi)\text{ for all
}w\in W\setminus\\{\widetilde{w}\\}\\}$

is open in $\mathfrak{t}$ and is invariant under multiplication by
$h\in\mathbb{R}$. As $D$ is dense in $\mathfrak{t}$ and invariant under
multiplication by $h\in\mathbb{Z}$, we have that $D\cap
U\cap\ell\neq\varnothing$. Consider a primitive element $\xi$ in this set and
the action of the corresponding circle subgroup $G$. Then, by $(i)$ and the
definition of $U$, it is easy to check that
$N(\widetilde{w}(\xi))=N(\widetilde{w})$ and
$N(-\widetilde{w}(\xi))=N(-\widetilde{w})$. As we assumed that
$N(\widetilde{w})\neq N(-\widetilde{w})$, the existence of such
$\widetilde{w}$ would then contradict [24, Proposition 2.11]. ∎

###### Remark 4.2

For any fixed point $p\in M^{T}$ and every weight $w\in\ell^{*}$ of the
$T$-action at $p$, the set $M^{w}$ of fixed points of the subgroup

(4.3) $H_{w}:=\exp\\{\xi\in\mathfrak{t}\mid w(\xi)\in\mathbb{Z}\\}$

of $T$ is a closed almost complex manifold, invariant under the action of $T$.
The weights of the $T$-action on $M^{w}$ at the points in $M^{T}\cap M^{w}$
are all integral multiples of $w$. By Lemma 4.1, we have that for every $p\in
M^{T}$ and every weight $w$ at $p$, there exists another fixed point $q\in
M^{T}$ such that one of the weights at $q$ is $-w$, and such that $p$ and $q$
belong to $M^{w}$. Moreover, let $p\in M^{T}\cap M^{w}$ such that $k\,w$ is a
weight at $p$ for some $k\in\mathbb{Z}$. Since $M^{k\,w}\subseteq M^{w}$ for
every $k\in\mathbb{Z}$, the connected component $N$ of $M^{k\,w}$ containing
$p$ is contained in $M^{w}$. Therefore there exists another fixed point $q\in
N\subset M^{w}$ such that $-k\,w$ is a weight at $q$.

As a consequence of Lemma 4.1, the multiset $W$ can be written as a disjoint
union

(4.4) $W=W^{+}\sqcup W^{-}$

of two multisets $W^{+}$ and $W^{-}$ of the same cardinality, where $w\in
W^{+}$ if and only if $-w\in W^{-}$. In the following, the triple $(M,J,T)$
denotes a closed almost complex manifold equipped with an effective action of
a torus $T$ preserving $J$ and isolated fixed points.

###### Definition 4.3

We say that a multigraph $\Gamma=(V,E)$ describes $(M,J,T)$ if

  * •

The vertex set $V$ is in bijection with the fixed point set $M^{T}$;

  * •

There exists a bijection $g\colon W^{+}\to W^{-}$, such that $g(w)=-w$ for
every $w\in W^{+}$ so that for each edge $e\in E$ with endpoints $p,q$, there
exists a weight $w_{p,e}$ at $p$ and a weight $w_{q,e}$ at $q$, such that
$g(w_{p,e})=w_{q,e}$ or $g(w_{q,e})=w_{p,e}$. Moreover, $p$ and $q$ must
belong to the same connected component of $M^{w}$, where $w$ is either
$w_{p,e}$ or $w_{q,e}$ and $M^{w}$ is the fixed point set of the subgroup
$H_{w}$ as in (4.3).

Note that the edges of the multigraph are _undirected_ and that, in the
notation above, $M^{w_{p,e}}=M^{w_{q,e}}$.

###### Remark 4.4

When $T\simeq S^{1}$ is a circle the weights at fixed points are integers and
so we can actually take the multisets $W^{+}$ and $W^{-}$ to be the multisets
of positive and negative weights. Moreover, for any weight $w$ of $S^{1}$ we
have that $M^{\mathbb{Z}_{\lvert w\rvert}}:=M^{w}$ is the submanifold of $M$
formed by all of the points that are fixed by the subgroup $\mathbb{Z}_{\lvert
w\rvert}$ of $S^{1}$.

###### Remark 4.5

Let $(M,\omega)$ be a closed, symplectic manifold admitting a symplectic
action of a torus $T$ with isolated fixed points. In this case we choose $J$
to be a compatible, invariant almost complex structure and say that a
multigraph describing $(M,J,T)$ is a multigraph describing $(M,\omega,T)$. If
the action is Hamiltonian with moment map $\psi$, we say that a multigraph
describes $(M,\omega,\psi)$.

###### Remark 4.6

Observe that for Hamiltonian GKM spaces, there is exactly one multigraph
describing the action and is given exactly by the GKM graph (see Section 2.3).

Lemma 4.1 and Remark 4.2 assert that every action of a torus on a closed
almost complex manifold with isolated fixed points can be described by a
multigraph.

###### Remark 4.7

For every $w\in W$, consider the almost complex manifold $(M^{w},J)$ with the
restriction of the $T$ action. Then it is clear that any multigraph describing
$(M,J,T)$ restricts to a multigraph describing $(M^{w},J,T)$.

The next lemma is important for the definition of a subclass of multigraphs
that we are interested in.

######  Lemma 4.8.

Let $\Gamma=(V,E)$ be a multigraph describing $(M,J,T)$. Let $e\in E$ be en
edge with endpoints $p,q$ and let $w:=w_{p,e}=-w_{q,e}$. Then for every class
$\alpha\in H^{2}_{T}(M;\mathbb{Z})$ we have

(4.5) $\alpha(p)-\alpha(q)=k\,w,\quad\text{for some }k\in\mathbb{Z}.$

###### Proof.

Each class $\alpha\in H^{2}_{T}(M;\mathbb{Z})$ defines a (unique)
$T$-equivariant line bundle $V$ on $M$ such that $c_{1}^{T}(V)=\alpha$ (see
for instance [23, Theorem A.1]). Moreover, for each fixed point $p\in M^{T}$,
the complex line $V(p)$ inherits a representation of $T$ with weight given
exactly by $\alpha(p)\in\ell^{*}$. By definition of multigraph, the endpoints
$p,q$ of each edge are in the same connected component of $M^{w}$ and so the
$H_{w}$-representations at $V(p)$ and $V(q)$ are isomorphic (see Remark 4.2),
thus implying (4.5).

∎

Consider the field of fractions $Q(\mathfrak{t}^{*})$ of the symmetric algebra
$\mathbb{S}(\mathfrak{t}^{*})$ of $\mathfrak{t}^{*}$. Then a priori, using the
above notation,

$\frac{\alpha(p)}{w_{p,e}}+\frac{\alpha(q)}{w_{q,e}}\in Q(\mathfrak{t}^{*}).$

However, an immediate consequence of Lemma 4.8 is the following result.

######  Corollary 4.9.

Let $\Gamma=(V,E)$ be a multigraph describing $(M,J,T)$. Then, for each edge
$e\in E$ with endpoints $p,q$ and each cohomology class $\alpha\in
H^{2}_{T}(M;\mathbb{Z})$, we have

(4.6)
$\alpha[e]:=\frac{\alpha(p)}{w_{p,e}}+\frac{\alpha(q)}{w_{q,e}}\in\mathbb{Z}\,.$

###### Proof.

This is an immediate consequence of Lemma 4.8, observing that
$w_{p,e}=-w_{q,e}$. ∎

###### Definition 4.10

Let $\Gamma=(V,E)$ be a multigraph describing $(M,J,T)$ and let $\alpha\in
H^{2}_{T}(M;\mathbb{Z})$. For each edge $e\in E$, the integer defined in (4.6)
is called the evaluation of $\alpha$ on the edge $e$.

###### Remark 4.11

Results similar to those in Lemma 4.8 and Corollary 4.9 hold for equivariant
cohomology classes with real coefficients. In particular, given $\alpha\in
H^{2}_{T}(M;\mathbb{R})$, an edge $e$ with endpoints $p,q$, and
$w=w_{p,e}=-w_{q,e}$, one has that

(4.7) $\alpha(p)-\alpha(q)=k\,w,\quad\text{for some }k\in\mathbb{R},$

and so the evaluation $\alpha[e]$ of a 2-form $\alpha\in
H_{T}^{2}(M;\mathbb{R})$ on $e$ can be defined as

(4.8)
$\alpha[e]:=\frac{\alpha(p)}{w_{p,e}}+\frac{\alpha(q)}{w_{q,e}}\in\mathbb{R}\,.$

A simple proof of (4.7) can be obtained using the Cartan model for equivariant
cohomology, which is the idea behind the introduction of polynomial
assignments in [20, Section 3].

###### Remark 4.12

There is a geometric interpretation of the evaluation of a class $\alpha\in
H^{2}_{T}(M;\mathbb{R})$ on an edge. Indeed, if there exists a $J$-invariant
2-sphere $S^{2}$ in $(M,J)$, which is $T$-invariant and such that the
$T$-action on $S^{2}$ has exactly $p$ and $q$ as fixed points, respectively
with weights $w_{p,e}$ and $w_{q,e}$, then by the Atiyah-Bott-Berline-Vergne
localization theorem ([2, 3]), the evaluation of $\alpha$ on $e$ is exactly
the integral of $\alpha$ on $S^{2}$, namely

$\int_{S^{2}}\alpha=\frac{\alpha(p)}{w_{p,e}}+\frac{\alpha(q)}{w_{q,e}}=\alpha[e]\,.$

Remark 4.12 leads to a generalization of the concept of toric one-skeleton as
defined in [16, Definition 4.10] for Hamiltonian $S^{1}$-spaces.

###### Definition 4.13

Let $(M,J)$ be a closed almost complex manifold equipped with an effective
action of a torus $T$ with isolated fixed points and let $\Gamma=(V,E)$ be a
multigraph describing $(M,J,T)$. Assume that for each edge $e\in E$ with
endpoints $p,q$, there exists a smoothly embedded $J$-holomorphic sphere
$S^{2}_{e}$ which is $T$-invariant and such that the fixed points are exactly
$p,q$ with weights (respectively) $w_{p,e},\,w_{q,e}$. In this case we say
that $\Gamma$ admits a toric 1-skeleton, and the set
$\mathcal{S}=\\{S^{2}_{e}\\}_{e\in E}$ is the toric 1-skeleton associated to
$\Gamma$333See also [6] for a generalization of toric $1$-skeletons to
complexity-one spaces whose fixed point sets are not necessarily discrete. .

###### Example 4.14

There are several examples of spaces $(M,J,T)$ that admit a toric $1$-skeleton
(see [16, Lemma 4.11]). Suppose that $J$ is a compatible almost complex
structure of a symplectic manifold $(M,\omega)$ and that the $T$-action is
Hamiltonian with moment map $\psi\colon M\to\mathfrak{t}^{*}$, with isolated
fixed points. Then, $(M,\omega,\psi)$ can be described by a multigraph that
admits a toric one skeleton if the $T$-action is, or extends to,

  * (i)

a GKM action (and in particular to a symplectic toric action), see Section
2.3,

  * (ii)

an $S^{1}$-action with isolated fixed points such that at each fixed point the
weights are pairwise coprime and not equal to 1,

  * (iii)

an $S^{1}$-action with isolated fixed points and $M$ is of dimension at most
$4$.

An important geometric consequence of the existence of a toric $1$-skeleton is
the following result, which is a straightforward generalization of [16, Lemma
4.13].

######  Lemma 4.15.

Let $(M,J)$ be a closed almost complex manifold of dimension $2n$ equipped
with an effective action of a torus $T$ with isolated fixed points and let
$\Gamma=(V,E)$ be a multigraph describing $(M,J,T)$. Consider the natural
restriction map

$r\colon H^{2}_{T}(M;\mathbb{Z})\to H^{2}(M;\mathbb{Z}).$

Then, for any equivariant cohomology class $\alpha\in
H^{2}_{T}(M;\mathbb{Z})$, we have

(4.9) $\sum_{e\in E}\alpha[e]=\int_{M}r(\alpha)\cup c_{n-1}\,.$

Moreover, if $r$ is surjective and $\Gamma$ admits a toric 1-skeleton
$\mathcal{S}=\\{S^{2}_{e}\\}_{e\in E}$, then the Chern class $c_{n-1}\in
H^{2(n-1)}(M;\mathbb{Z})$ is the Poincaré dual to the class of $\mathcal{S}\in
H_{2}(M;\mathbb{Z})$.

###### Proof.

This is a generalization of [16, Lemma 4.13], where the only fact used about
the existence of a Hamiltonian $T$-action is the surjectivity of $r$. The
details are left to the reader. ∎

###### Remark 4.16

The condition that $r\colon H^{2}_{T}(M;\mathbb{Z})\to H^{2}(M;\mathbb{Z})$ is
surjective is equivalent to the equivariant formality of the $T$ action. For a
nice account of these facts see [17, Section 7].

###### Definition 4.17

Let $(M,J)$ be a closed almost complex manifold equipped with an effective
action of a torus $T$ with isolated fixed points and let $\Gamma=(V,E)$ be a
multigraph describing $(M,J,T)$. Let $\alpha\in H^{2}_{T}(M;R)$, where $R$ is
either $\mathbb{Z}$ or $\mathbb{R}$. We say that an edge $e\in E$ is positive
w.r.t. $\alpha$ if the evaluation of $\alpha$ on $e$ is positive, i.e.

(4.10) $\alpha[e]=\frac{\alpha(p)}{w_{p,e}}+\frac{\alpha(q)}{w_{q,e}}>0,$

where $p,q\in V$ are the endpoints of $e$. The multigraph $\Gamma=(V,E)$ is
said to be positive w.r.t. $\alpha$ if all the edges are positive w.r.t.
$\alpha$.

###### Remark 4.18

The condition for an edge to be positive w.r.t. an equivariant cohomology
class $\alpha$ can be interpreted as follows: An edge $e$ with endpoints $p,q$
and weights $w_{p,e},w_{q,e}$ is positive w.r.t. $\alpha$ if and only if

$\alpha(p)-\alpha(q)$, regarded as a vector in $\mathfrak{t}^{*}$, is a
positive multiple of $w_{p,e}$.

Note that this condition is equivalent to saying that $\alpha(q)-\alpha(p)$
has the same direction as $w_{q,e}$. Therefore the concept of “positive edge”
(w.r.t. $\alpha$) belongs to undirected edges.

###### Example 4.19

For Hamiltonian $T$-actions with isolated fixed points, the positivity of an
edge $e$ with endpoints $p,q$ w.r.t. the class of the equivariant symplectic
form $[\omega-\psi]$ says that the vector $\psi(q)-\psi(p)$ is a positive
multiple of $w_{p,e}$.

###### Remark 4.20

(Orientation on $\Gamma$) In the particular case of a circle action and taking
the sets $W^{+}$ and $W^{-}$ to be respectively those of positive and negative
weights (see Remark 4.4), we can direct the edges of any multigraph $\Gamma$
describing $(M,J,S^{1})$, constructed using a bijection $g:W^{+}\to W^{-}$ as
in Definition 4.3. Namely, we orient an edge $e$ with endpoints $p,q$ from $p$
to $q$ if $g(w_{p,e})=w_{q,e}$, therefore if $w_{p,e}>0$. In this case $i(e)$
is the initial point (also known as tail) of $e$ and $t(e)$ is the terminal
point (also known as head) of $e$.

Note that, according to Definition 4.17, an oriented multigraph $\Gamma=(V,E)$
describing $(M,J,S^{1})$ is said to be positive w.r.t. some $\alpha\in
H^{2}_{S^{1}}(M;R)$, where $R$ is either $\mathbb{Z}$ or $\mathbb{R}$, if and
only if for every oriented edge $e\in E$ we have

$\alpha(i(e))>\alpha(t(e)).$

When $T$ is a torus of higher dimension, we can orient the graph by
considering a generic $\xi\in\mathfrak{t}$ (namely $w(\xi)\neq 0$ for any
isotropy weight $w\in\ell^{*}$) such that $\xi$ generates a circle
$S^{1}=\exp\langle\xi\rangle$, and use this circle to orient the edges of
$\Gamma$, as described above.

There is another natural consequence of the existence of a toric one-skeleton
for symplectic manifolds with a Hamiltonian action.

######  Proposition 4.21.

Let $(M,\omega)$ be a Hamiltonian $T$-space with isolated fixed points and
moment map $\psi\colon M\to\mathfrak{t}^{*}$. Assume that there exists a
multigraph $\Gamma$ describing $(M,\omega,\psi)$ which admits a toric one
skeleton. Then $\Gamma$ is positive w.r.t. $[\omega-\psi]$.

###### Proof.

Since the spheres of the toric one skeleton are, by definition, $J$-invariant,
they are symplectic surfaces in $M$, and therefore $\int_{S^{2}}\omega>0$. The
result then follows from Remark 4.12 and from the fact that, for degree
reasons, we have $\int_{S^{2}}\omega=\int_{S^{2}}(\omega-\psi)$. ∎

###### Remark 4.22

Let $(M,\omega)$ be a Hamiltonian $T$-space with isolated fixed points and
moment map $\psi\colon M\to\mathfrak{t}^{*}$. To the best of our knowledge,
every known example of such $(M,\omega,\psi)$ admits a multigraph describing
it which is positive w.r.t. $[\omega-\psi]$. If a multigraph describing
$(M,\omega,\psi)$ admits a toric one skeleton (which is the case, for
instance, for Hamiltonian GKM spaces, and in particular for symplectic toric
manifold, see Example 4.14), then its positivity w.r.t. $[\omega-\psi]$ comes
from Proposition 4.21. However, for other examples of $(M,\omega,\psi)$ for
which the existence of a toric one skeleton is not known, the existence of a
positive multigraph w.r.t. $[\omega-\psi]$ can be checked directly.

Given the remark above, we believe that this question is of interest to the
community working in equivariant symplectic geometry:

######  Question 2.

Let $(M,\omega)$ be a Hamiltonian $T$-space with isolated fixed points and
moment map $\psi\colon M\to\mathfrak{t}^{*}$.

Does there always exist a multigraph describing $(M,\omega,\psi)$ that is
positive w.r.t. $[\omega-\psi]$?

Given the existence of such multigraph, is there a toric one-skeleton
associated to it?

Before stating the next lemma, we introduce the following notation. We say
that a subtorus $K$ of a torus $T$ is generic if, for every
$w\in\mathfrak{t}^{*}$ that is a weight of the isotropy $T$ action at some
fixed point, we have

(4.11) $w\neq 0\iff\pi_{K}(w)\neq 0,$

where $\pi_{K}\colon\mathfrak{t}^{*}\to\mathfrak{k}^{*}$ is the dual of the
inclusion $\iota_{K}\colon\mathfrak{k}\to\mathfrak{t}$, and $\mathfrak{k}$ is
the Lie algebra of $K$. Note that since $K$ is generic, we have $M^{K}=M^{T}$.
The projection $\pi_{K}$ can be extended to be a map from
$\mathbb{S}(\mathfrak{t}^{*})$ to $\mathbb{S}(\mathfrak{k}^{*})$ by imposing
that it is an algebra morphism.

######  Lemma 4.23.

Let $(M,J)$ be a closed almost complex manifold equipped with an effective
action of a torus $T$ with isolated fixed points and let $\alpha\in
H^{2}_{T}(M;R)$, where $R$ is either $\mathbb{Z}$ or $\mathbb{R}$. If $K$ is a
subtorus of $T$, let $r_{K}\colon H_{T}^{2}(M;R)\to H_{K}^{2}(M;R)$ be the
restriction map in equivariant cohomology. Then the following statements are
equivalent:

  * (i)

There exists a multigraph describing $(M,J,T)$ that is positive w.r.t.
$\alpha$;

  * (ii)

For all generic subtori $K\subset T$, there exists a multigraph describing
$(M,J,K)$ that is positive w.r.t. $r_{K}(\alpha)$;

  * (iii)

There exists a generic $K\subset T$ and a multigraph describing $(M,J,K)$ that
is positive w.r.t. $r_{K}(\alpha)$.

###### Proof.

The equivalences follow easily from the fact that
$(r_{K}(\alpha))(p)=\pi_{K}(\alpha(p))$, that if $w$ is a weight of the $T$
representation at $p\in M^{T}$, then $\pi_{K}(w)$ is the weight of the $K$
representation at $p\in M^{K}$ and that

$R\ni\frac{\alpha(p)}{w_{p,e}}+\frac{\alpha(q)}{w_{q,e}}=\pi_{k}\left(\frac{\alpha(p)}{w_{p,e}}+\frac{\alpha(q)}{w_{q,e}}\right)\,.$

∎

The following lemma will be useful in the next section.

######  Lemma 4.24.

Let $(M,J)$ be a closed almost complex manifold equipped with an action of a
torus $T$ with isolated fixed points and let $\alpha\in H^{2}_{T}(M;R)$, where
$R$ is either $\mathbb{Z}$ or $\mathbb{R}$. Let $W$ be the multiset of all
isotropy weights and, for any $w\in W$, let $M^{w}$ be the fixed point set of
the group $H_{w}$ defined in (4.3). Let $\Gamma=(V,E)$ be a multigraph
describing $(M,J,T)$ and, for every $w\in W$, let $\Gamma_{w}=(V_{w},E_{w})$
be the restriction of $\Gamma$ to $M^{w}$, which is a multigraph describing
$(M^{w},J,T)$ (see Remark 4.7).

Then $\Gamma$ is positive w.r.t. $\alpha$ if and only if, for every $w\in W$,
$\Gamma_{w}$ is positive w.r.t. $\alpha$.

###### Proof.

In order to prove the statement it is enough to observe that any edge $e\in E$
is an edge of $E_{w}$, for some $w\in W$. ∎

###  4.1. Existence of positive multigraphs

The main goal of this section is to prove Theorem 4.32. We begin with the
following observations.

######  Lemma 4.25.

Let $(M,\omega)$ be a closed, connected 8-dimensional symplectic manifold with
a Hamiltonian action of a torus $T$ with isolated fixed points. Let
$\psi\colon M\to\mathfrak{t}^{*}$ be the corresponding moment map.

If $\chi(M)=6$, then the symplectic form can be rescaled and the moment map
can be translated so that

(4.12) $c_{1}^{T}=[\omega-\psi].$

In particular, with the aforementioned rescaling, the weight sum formula (2.5)
and $c_{1}=[\omega]$ hold.

###### Proof.

By Proposition 3.6 (1), the second Betti number is one. Therefore
$c_{1}=\lambda[\omega]$ for some $\lambda\in\mathbb{R}$ and, by [16, Lemma
5.2] applied to a generic subcircle $S^{1}\subset T$, we have that
$\lambda>0$. Therefore the symplectic form can be rescaled so that
$c_{1}=[\omega]$. As a consequence, if we consider the equivariant extensions
of $c_{1}$ and $\omega$, namely $c_{1}^{T}$ and $[\omega-\psi]$, there must be
an element $a$ of $\mathfrak{t}^{*}$ such that $c_{1}^{T}=[\omega-\psi]+a$. If
we replace the moment map $\psi$ with $\psi^{\prime}:=\psi-a$, we have that
$c_{1}^{T}=[\omega-\psi^{\prime}]$. Restricting the previous equality to a
fixed point $p$ yields the weight sum formula (2.5). ∎

###### Remark 4.26

Henceforth, if the manifold we are working with satisfies the hypotheses of
Lemma 4.25, we will always assume that the symplectic form is rescaled and the
moment map translated so that (4.12) holds.

In order to prove the main result of this section we need the following
lemmas. For circle actions, we orient the multigraph as in Remark 4.20.

######  Lemma 4.27 (Jang, Tolman [28]).

Let $(M,J)$ be a closed almost complex manifold acted on by a circle $S^{1}$
with isolated fixed points. Then there exists an oriented multigraph
$\Gamma=(V,E)$ describing $(M,J,S^{1})$ such that, for each $e\in E$, the
index of $i(e)$ in the isotropy submanifold $M^{\mathbb{Z}_{w(e)}}$ is two
less than the index of $t(e)$ in $M^{\mathbb{Z}_{w(e)}}$.

The following is an immediate consequence of [43, Lemma 3.1].

######  Lemma 4.28.

Suppose that $(M,\omega)$ is a closed symplectic manifold equipped with a
Hamiltonian $S^{1}$-action with isolated fixed points and moment map
$\psi:M\rightarrow\mathbb{R}$. Let $p_{\min}$ be the fixed point where $\psi$
attains its minimum. If $q$ is a fixed point such that there is no fixed point
in $\psi^{-1}((\psi(p_{\min}),\psi(q)))$, then $\lambda(q)=1$. Furthermore,
every fixed point $q^{\prime}$ with $\psi(q^{\prime})=\psi(q)$ has
$\lambda(q^{\prime})=1$.

###### Remark 4.29

Suppose that the dimension of $M$ is equal to $2n$. Reversing the circle
action it is easy to check that if $p_{\max}$ is the fixed point where $\psi$
attains its maximum and $q$ is a fixed point of the original $S^{1}$-action
for which there is no fixed point in $\psi^{-1}((\psi(q),\psi(p_{\max})))$,
then $\lambda(q)=n-1$. Furthermore, every fixed point $q^{\prime}$ with
$\psi(q^{\prime})=\psi(q)$ must have $\lambda(q^{\prime})=n-1$.

When the manifold is $8$-dimensional and the number of fixed points is $6$,
then the Euler characteristic of $M$ is $6$ and so, by Proposition 3.6 (1) we
have exactly one fixed point $p$ with $\lambda(p)=1$ and two fixed points with
$\lambda(p)=2$. Then Lemma 4.28 and Remark 4.29 imply the following statement.

######  Lemma 4.30.

Let $(M,\omega)$ be a closed symplectic $8$-manifold equipped with a
Hamiltonian $S^{1}$-action with $6$ isolated fixed points and moment map
$\psi:M\rightarrow\mathbb{R}$. Name the fixed points
$p_{0},p_{1},p_{2},p_{2}^{\prime},p_{3},p_{4}$ so that $\lambda(p_{i})=i$ for
each $i$ and $\lambda(p_{2}^{\prime})=2$. Then we have

$\psi(p_{0})<\psi(p_{1})<\psi(p_{2})\leq\psi(p_{2}^{\prime})<\psi(p_{3})<\psi(p_{4}).$

Suppose that $(M,\omega,\psi)$ is a Hamiltonian $T$-space of dimension $2n$
with $b_{2}(M)=1$ and discrete fixed point set $M^{T}$. Consider a generic
element $\xi$ of $\mathfrak{t}$ generating a circle
$\exp\langle\xi\rangle=S^{1}$. Then, by the perfection of the associated
moment map $\varphi=\langle\psi,\xi\rangle$, there is exactly one fixed point
$p_{1}$ with $\lambda(p_{1})=1$ and, by Poincarè duality, there is exactly one
fixed point $p_{n-1}$ with $\lambda(p_{n-1})=n-1$. The next lemma illustrates
another easy feature of multigraphs associated to Hamiltonian actions of this
kind.

######  Lemma 4.31.

Let $(M,\omega,\psi)$ be a Hamiltonian $T$-space of dimension $2n$ with
$b_{2}(M)=1$ and discrete fixed point set $M^{T}$. Let $\xi$ be a generic
element of $\mathfrak{t}$ generating a circle $\exp\langle\xi\rangle=S^{1}$.
Let $p_{i}$ be the only fixed point with $\lambda(p_{i})=i$, for
$i\in\\{0,1,n-1,n\\}$.

Then there exists a multigraph describing $(M,\omega,\psi)$ that has one edge
between $p_{0}$ and $p_{1}$, and one between $p_{n-1}$ and $p_{n}$.

###### Proof.

Consider the moment map $\varphi=\langle\psi,\xi\rangle$ of the $S^{1}$ action
on $(M,\omega)$. As there is exactly one negative isotropy weight at $p_{1}$,
by the definition of multigraph, there is exactly one edge $e\in E$ with
$t(e)=p_{1}$. By Lemma 4.27, there exists a multigraph such that the index of
$i(e)$ is zero on the isotropy submanifold of $M^{\mathbb{Z}_{w(e)}}$,
therefore $i(e)$ is a minimum for the restriction of $\varphi$ to
$M^{\mathbb{Z}_{w(e)}}$. Since $p_{0}$ is the only fixed point of the $S^{1}$
action on $M$ with $\varphi(p_{0})<\varphi(p_{1})$, it follows that
$i(e)=p_{0}$.

A similar argument applies to the unique fixed point $p_{n-1}$ with
$\lambda(p_{n-1})=n-1$. ∎

######  Theorem 4.32.

Let $(M,\omega)$ be a closed, connected 8-dimensional symplectic manifold with
an effective Hamiltonian action of a torus $T$ with moment map
$\psi:M\to\mathfrak{t}^{*}$ and isolated fixed points. Assume that
$\chi(M)=6$. Then $(M,\omega,\psi)$ can be described by a multigraph
$\Gamma=(V,E)$ that is positive w.r.t. $[\omega-\psi]$ if

  * (a)

the dimension of $T$ is at least 2, or

  * (b)

the dimension of $T$ is one and there is no 6-dimensional isotropy submanifold
with $6$ fixed points.

Moreover the multigraph has one edge connecting the minimum fixed point
$p_{0}$ to the unique fixed point $p_{1}$ with $\lambda(p_{1})=1$, and one
edge connecting the unique fixed point $p_{3}$ with $\lambda(p_{3})=3$ to the
maximum fixed point $p_{4}$.

###### Proof.

In both cases (a) and (b), the strategy to prove the theorem is the same: For
every weight $w\in W$, consider an edge $e$ with endpoints $p,q$ such that
$w_{p,e}=w$ or $-w$. This edge is also an edge for the restriction
$\Gamma^{w}$ of the multigraph $\Gamma$ on the isotropy submanifold $M^{w}$
with the restricted action of $T$ (see Remark 4.7). Therefore it is enough to
prove that the edge $e$, regarded as an edge of $\Gamma^{w}$, is positive
w.r.t. the restriction of $[\omega-\psi]$ on $M^{w}$. (See Lemma 4.24.)

We first observe that $M^{w}=M$ only if $\dim(T)=1$ and $w=\pm 1$. This
follows from the effectiveness of the action of $T$, the fact that $M$ is
connected, and that $M^{w}$ is fixed by $H_{w}$, which is the trivial subgroup
only if $\dim(T)=1$ and $w=\pm 1$. So assume first that $w=\pm 1$ and consider
the corresponding edge $e\in\Gamma$. Then the fact that $e$ is positive w.r.t.
$[\omega-\psi]$ follows from Lemmas 4.27 and 4.30.

Now suppose that $H_{w}$ is not the trivial subgroup, and therefore $M^{w}$ is
a proper submanifold of $M$. Consider the effective action of
$T^{\prime}:=T/H_{w}$ on $M^{w}$. This action is Hamiltonian w.r.t. the
restriction of the symplectic form, which we still call $\omega$, and the
restriction of the moment map $\psi$ to $M^{w}$ (here we identify
$(\text{Lie}(T^{\prime}))^{*}$ with
$\text{Ann(Lie}(H_{w})\subset\mathfrak{t}^{*}$). Moreover, it has isolated
fixed points.

If $\dim(M^{w})\leq 4$, then $(M^{w},\omega,T^{\prime})$ admits a toric
$1$-skeleton (see Example 4.14 (iii)), and then, by Proposition 4.21, it can
be described by a positive multigraph w.r.t. $[\omega-\psi]$.

If $\dim(M^{w})=6$, then we claim that the number of fixed points of the
$T^{\prime}$-action is exactly $4$. Indeed, we first note that, since the
action of $T^{\prime}$ on $M^{w}$ is Hamiltonian, the number of fixed points
if at least $4$. Moreover, by Poincaré duality, the number of fixed points
must be even. Since we are assuming that $\chi(M)=6$, we just need to prove
that, in case (a), the action cannot have $6$ fixed points. Therefore assume
that $\dim(T)\geq 2$. In this case $H_{w}$ is not discrete, so
$\dim(T^{\prime})=1$ and the image of $M^{w}$ by the $T^{\prime}$-moment map
is on an affine line in $\mathfrak{t}^{*}$ (see identifications above). If the
number of fixed points of the $T^{\prime}$-action on $M^{w}$ were six, then
the images of all the $T$-fixed points by the $T$-moment map would be on the
same affine line in $\mathfrak{t}^{*}$. However, as $\psi(M)$ is the convex
hull of the image of these points, this would imply that the $T$-moment
polytope were $1$–dimensional, which is impossible since we are assuming
$\dim(T)\geq 2$ and the $T$-action is effective. It follows that the number of
fixed points of the $T^{\prime}$-action on the six dimensional isotropy
submanifold $M^{w}$ is 4. Now consider a generic subcircle $S^{1}$ of
$T^{\prime}$ acting on $M^{w}$. As this $S^{1}$-action has 4 fixed points and
$M^{w}$ is six dimensional, the possible weights were classified by Tolman in
[43]. In all of the possible cases the manifold $M^{w}$ and the $S^{1}$-action
can be described by a multigraph that is positive w.r.t. (the restriction of)
$[\omega-\psi]$. Therefore, by Lemma 4.23, the action of $T^{\prime}$ on
$M^{w}$ can be described by a multigraph that is positive w.r.t. (the
restriction of) $[\omega-\psi]$ and therefore so does the action of $T$ on
$M^{w}$.

In order to prove the last claim, it is enough to observe that, by Proposition
3.6 (1), $b_{2}(M)=1$; so the conclusion follows from Lemma 4.31. ∎

###### Remark 4.33

If $c_{1}=\lambda[\omega]$ for some $\lambda>0$, then the existence of a
multigraph which is positive w.r.t. $[\omega-\psi]$ is equivalent to that of a
multigraph which is positive w.r.t. $c_{1}^{T}$. Therefore, by Lemma 4.25,
this is the case when $\chi(M)=6$.

If $T$ is a circle $S^{1}$, then for every fixed point $p$,
$c_{1}^{S^{1}}(p)=\Sigma(p)\,x$ where $\Sigma(p)\in\mathbb{Z}$ denotes the sum
of the weights at $p$ and $H_{S^{1}}(\\{p\\};\mathbb{Z})=\mathbb{Z}[x]$. We
conclude that, if $\chi(M)=6$, a multigraph $\Gamma=(V,E)$ describing
$(M,\omega,\psi)$ is positive w.r.t. $[\omega-\psi]$ if and only if

(4.13) $\Sigma(i(e))<\Sigma(t(e))\,\quad\text{for every edge }e\in E\,.$

(see also Remark 4.20).

The next proposition shows that the only case in Theorem 4.32 in which the
existence of a positive multigraph cannot be ensured concerns actions in which
the isotropy submanifold of dimension 6 containing all the fixed points is
fixed by $\mathbb{Z}_{2^{j}}$, for some $j\in\mathbb{Z}_{\geq 1}$.

######  Proposition 4.34.

Let $(M,\omega)$ be a closed symplectic manifold of dimension $2n$ equipped
with an effective Hamiltonian $S^{1}$-action with discrete fixed point set
$M^{S^{1}}$. If there exists a connected isotropy submanifold
$M^{{\mathbb{Z}}_{m}}$ of dimension $2n-2$ containing $M^{S^{1}}$, then
$m=2^{j}$ for some $j\in\mathbb{Z}_{\geq 1}$.

###### Proof.

First of all we observe that, for every prime $m^{\prime}$ dividing $m$, the
isotropy submanifold $M^{\mathbb{Z}_{m^{\prime}}}$ contains
$M^{{\mathbb{Z}}_{m}}$, and therefore it contains $M^{S^{1}}$. In order to
prove the claim is then enough to prove that $m^{\prime}$ can only be 2.

Therefore suppose that $M^{S^{1}}$ is contained in some connected component of
$M^{\mathbb{Z}_{m}}$ for some prime integer $m\geq 2$. Since the
$S^{1}$-action is effective, at least one of the isotropy weights at the
minimum is not divisible by $m$. Let $w$ be one of these isotropy weights.
Then, by [24, Proposition 2.11], there exists another fixed point which has
$-w$ as one of its isotropy weights. Since by assumption, both fixed points
are in the same connected component of $M^{\mathbb{Z}_{m}}$, we have by [43,
Lemma 2.6] that

$w=-w\mod m.$

It follows that $2w=0\mod m$, and then, since $m$ is prime and $w$ is not
divisible by $m$, we have that $m=2$ as required.

∎

###### Remark 4.35

Here we give an example of a Hamiltonian $S^{1}$-action on a $6$-dimensional
symplectic manifold with isolated fixed points, and with a connected isotropy
submanifold of dimension $4$ containing all the fixed points. Consider the
$S^{1}$-action on $M=Gr^{+}_{\mathbb{R}}(2,5)$ induced by the linear circle
action on $\mathbb{R}^{5}=\mathbb{R}\oplus\mathbb{C}^{2}$ given by

$z.(x,w_{1},w_{2})=(x,z^{-1}w_{1},zw_{2}).$

It is easy to check that this action has $4$ isolated fixed points. Moreover,
the subgroup $\mathbb{Z}_{2}\subset S^{1}$ fixes all of the oriented planes in
the subspace $\\{0\\}\times\mathbb{C}^{2}\subset\mathbb{R}^{5}$, and so
$M^{\mathbb{Z}_{2}}=Gr^{+}_{\mathbb{R}}(2,4)\subset Gr^{+}_{\mathbb{R}}(2,5)$
is $4$-dimensional and contains the four fixed points.

###  4.2. Positive multigraphs and Betti numbers

In this section we investigate the consequences of the existence of a positive
multigraph on an 8-dimensional closed symplectic manifold admitting a
Hamiltonian $S^{1}$-action with finitely many fixed points, proving two
finiteness results, one for the Betti numbers (Proposition 4.38) and one for
the Chern numbers (Corollary 4.41).

We begin with the following.

######  Lemma 4.36.

Let $(M,\omega)$ be a closed symplectic manifold of dimension $2n$ with index
$k_{0}$, and let $\mathbf{b}=(b_{0},\ldots,b_{2n})$ be the vector of its even
Betti numbers. Consider the integer $C(k_{0},n,\mathbf{b})$ defined as

$C(k_{0},n,\mathbf{b}):=\begin{cases}\displaystyle\sum_{k=1}^{\frac{n}{2}}\Big{[}12k^{2}-n(k_{0}+1)\Big{]}b_{n-2k}-\frac{n}{2}(k_{0}+1)\,b_{n}\,,\quad\mbox{
for }n\mbox{ even}\\\ &\\\
\displaystyle\sum_{k=1}^{\frac{n-1}{2}}\Big{[}12k(k+1)+3-n(k_{0}+1)\Big{]}b_{n-1-2k}-\big{[}n(k_{0}+1)-3\big{]}b_{n-1},\mbox{
for }n\mbox{ odd.}\end{cases}$

If $(M,\omega,\psi)$ can be described by a multigraph $\Gamma=(V,E)$ that is
positive w.r.t. $c_{1}^{S^{1}}$, then $C(k_{0},n,\mathbf{b})$ is a _non-
negative_ multiple of $k_{0}$.

Moreover $C(k_{0},n,\mathbf{b})$ vanishes if and only if $c_{1}[e]=k_{0}$ for
all $e\in E$.

###### Proof.

This lemma is a straightforward generalization of [16, Corollary 5.5].

We first note that, under the assumption of this lemma, for every $e\in E$,
$c_{1}^{S^{1}}[e]$ is a positive multiple of the index
$k_{0}\in\mathbb{Z}_{>0}$. Indeed, by definition of index, there exists a
(primitive) class $\eta\in H^{2}(M;\mathbb{Z})$ such that $c_{1}=k_{0}\eta$.
Since the restriction map $r\colon H_{S^{1}}^{2}(M;\mathbb{Z})\to
H^{2}(M;\mathbb{Z})$ is surjective, consider an equivariant extension
$\eta^{S^{1}}$ of $\eta$. Then

(4.14) $c_{1}^{S^{1}}=k_{0}\eta^{S^{1}}+\beta,\text{ for some }\beta\in
H^{2}_{S^{1}}(\\{pt\\};\mathbb{Z})\,.$

Since by assumption $\Gamma$ is positive w.r.t. $c_{1}^{S^{1}}$, and since
$k_{0}>0$, it follows that $\Gamma$ is positive w.r.t. $\eta^{S^{1}}$.
Therefore $\eta^{S^{1}}[e]$ is a positive integer for every $e\in E$, and from
(4.14) it follows that $c_{1}^{S^{1}}[e]$ is a positive multiple of $k_{0}$,
for every $e\in E$. Therefore

(4.15) $\sum_{e\in E}c_{1}^{S^{1}}[e]-k_{0}|E|=\sum_{e\in
E}c_{1}^{S^{1}}[e]-k_{0}\frac{n}{2}\chi(M)$

is a non-negative multiple of $k_{0}$ and it is zero precisely if
$c_{1}[e]=k_{0}$ for all $e\in E$.

Then, by (4.9), $\sum_{e\in E}c_{1}^{S^{1}}[e]=\int_{M}c_{1}c_{n-1}$ which, by
[15, Corollary 3.1], can be expressed as a linear combination of even Betti
numbers. Combining this expression with the definition of $\chi(M)$, it is
easy to see that $C(k_{0},n,\mathbf{b})$ is exactly $\sum_{e\in
E}c_{1}^{S^{1}}[e]-k_{0}|E|$, and the conclusion follows. ∎

###### Remark 4.37

Here below we write the values of $C(k_{0},4,\mathbf{b})$, as in [16, Table
1]:

$C(k_{0},4,\mathbf{b})=\begin{cases}6(6-b_{4})&\text{if }k_{0}=2\\\
4(8-b_{2}-2b_{4})&\text{if }k_{0}=3\\\ 2(14-4b_{2}-5b_{4})&\text{if
}k_{0}=4\\\ 12(2-b_{2}-b_{4})&\text{if }k_{0}=5\\\ \end{cases}$

We are ready to prove the following

######  Proposition 4.38.

Let $(M,\omega)$ be a closed 8-dimensional symplectic manifold admitting a
Hamiltonian $S^{1}$-action with isolated fixed points and moment map
$\psi\colon M\to\mathbb{R}$. Suppose that $(M,\omega,\psi)$ can be described
by a multigraph that is positive with respect to $c_{1}^{S^{1}}$. If the index
$k_{0}$ of $(M,\omega)$ is not $1$, then there are finitely many possibilities
for the Betti numbers of $M$. More precisely,

  1. (1)

If $k_{0}=5$ then $b_{2}=b_{4}=1$;

  2. (2)

If $k_{0}=4$ then $b_{2}=1$ and $b_{4}=2$;

  3. (3)

If $k_{0}=3$ then $(b_{2},b_{4})$ can be either $(1,2)$ or $(2,3)$;

  4. (4)

If $k_{0}=2$ then $1\leq b_{2}\leq b_{4}\leq 6$.

###### Proof.

Firstly, since $(M,\omega)$ is a closed symplectic $8$-manifold admitting a
Hamiltonian $S^{1}$-action with isolated fixed points, its even Betti numbers
are unimodal [10, Theorem 1.3], that is $b_{2}(M)\leq b_{4}(M)$.

Then we use Lemma 4.36 and the expression for $C(k_{0},4,\mathbf{b})$ in
Remark 4.37 to deduce that the only possible Betti numbers are those listed
above.

∎

###### Remark 4.39

We remark that for the $k_{0}=1$ case, some non-trivial information can be
extracted from Lemma 4.36. Indeed $C(1,4,\mathbf{b})$ is given by
$4(10+b_{2}-b_{4})$, and if $(M,\omega,\psi)$ can be described by a multigraph
that is positive w.r.t. $c_{1}^{S^{1}}$, then Lemma 4.36 gives that

$b_{4}(M)-b_{2}(M)<10.$

Moreover, since the Betti numbers are unimodal [10, Theorem 1.3], the left
hand side is non-negative, hence bounded in the interval [0,9]. Therefore, the
(interesting) question of whether $b_{2}(M),b_{4}(M)$ or $\chi(M)$ is bounded
from above are equivalent.

Let $(M,\omega)$ be a closed symplectic manifold which can be endowed with a
Hamiltonian $T$ action with moment map $\psi$ and isolated fixed points. If
$(M,\omega,\psi)$ admits a toric one-skeleton w.r.t. a multigraph $\Gamma$
describing $(M,\omega,\psi)$, and if $(M,\omega)$ is monotone, namely
$c_{1}=\lambda[\omega]$ for some positive $\lambda$, then, by Proposition
4.21, $\Gamma$ is also positive w.r.t. $c_{1}^{T}$, as the equation
$c_{1}=\lambda[\omega]$ implies $c_{1}^{T}=\lambda[\omega-\psi]+\beta$, for
some $\beta\in H_{T}(\\{pt\\};\mathbb{R})$. Since (Hamiltonian) GKM spaces
always admit a toric one-skeleton, the above argument proves the following

######  Corollary 4.40.

Let $(M,\omega)$ be a closed 8-dimensional symplectic manifold which can be
endowed with a Hamiltonian GKM action with moment map $\psi$. Assume that
$(M,\omega)$ is monotone and that the index $k_{0}$ of $(M,\omega)$ is not
$1$. Then there are finitely many possibilities for the Betti numbers of $M$.
More precisely,

  1. (1)

If $k_{0}=5$ then $b_{2}=b_{4}=1$;

  2. (2)

If $k_{0}=4$ then $b_{2}=1$ and $b_{4}=2$;

  3. (3)

If $k_{0}=3$ then $(b_{2},b_{4})$ can be either $(1,2)$ or $(2,3)$.

  4. (4)

If $k_{0}=2$ then $1\leq b_{2}\leq b_{4}\leq 6$.

Indeed we will prove in the following that there is no Hamiltonian GKM space
of dimension $8$ with $k_{0}=3$ and $(b_{2},b_{4})=(1,2)$ (see Section 7.1.3).

We recall that in Theorem 3.4 we proved that, for a closed symplectic
$8$-dimensional manifold with $b_{2}(M)=1$ that admits a Hamiltonian
$S^{1}$-action with isolated fixed points, and such that the index $k_{0}$ is
not one, then, for each value of $b_{4}:=b_{4}(M)$, there are finitely many
possibilities for the Chern numbers. Combining this result with Proposition
4.38, we immediately get the following

######  Corollary 4.41.

Let $(M,\omega)$ be a compact, connected 8-dimensional symplectic manifold
admitting a Hamiltonian $S^{1}$-action with moment map $\psi$ and isolated
fixed points. Suppose that $(M,\omega,\psi)$ can be described by a multigraph
that is positive with respect to $c_{1}^{S^{1}}$. If $b_{2}(M)=1$ and the
index $k_{0}$ of $(M,\omega)$ is not $1$, then there are finitely many
possibilities for the Chern numbers of $(M,\omega)$.

Following the argument preceding Corollary 4.40, we immediately obtain the
following

######  Corollary 4.42.

Let $(M,\omega)$ be a closed 8-dimensional symplectic manifold which can be
endowed with a Hamiltonian GKM action with moment map $\psi$. Assume that
$(M,\omega)$ is monotone, that the index $k_{0}$ of $(M,\omega)$ is not $1$
and that $b_{2}=1$. Then there are finitely many possibilities for the Chern
numbers of $(M,\omega)$.

###### Remark 4.43

If the index of $(M,\omega)$ is equal to $2$ and $b_{2}=1$, then (3.13),
(3.14) and (3.15) in the proof of Theorem 3.7 imply that

$c_{2}^{2}=96+b_{4},\quad c_{1}^{4}[M]=16k\quad\text{and}\quad
c_{1}^{2}c_{2}[M]=96+4k\,.$

The following table describes the possible values of $k$ (depending on the
value of $b_{4}$).

$b_{4}=2$ | $k\in\\{18,25,32\\}$
---|---
$b_{4}=3$ | $k\in\\{17,\ldots,34\\}$
$b_{4}=4$ | $k\in\\{16,\ldots,36\\}$
$b_{4}=5$ | $k\in\\{16,\ldots,37\\}$
$b_{4}=6$ | $k\in\\{15,\ldots,39\\}$
Table 4.1. Index 2: Possible values of $k$ s.t. $c_{1}^{4}[M]=16k$ and
$c_{1}^{2}c_{2}[M]=96+4k$

##  5\. Examples

In this section our aim is to describe explicitly the torus action and to
compute the equivariant and ordinary cohomology ring and Chern classes of some
special 8-dimensional positive monotone symplectic manifolds, $V,W$ and $Q$.
These are exactly the examples appearing in our classification result Main
Theorem. They are all smooth algebraic varieties with holomorphic
$\mathbf{T}=(\mathbb{C}^{*})^{k}$ actions and, as we will see in Lemma 5.2,
they also have invariant Kähler structures for which the actions of
$(S^{1})^{k}<\mathbf{T}$ are Hamiltonian.

In order to describe their equivariant cohomology rings and Chern classes, we
will follow the strategy described in Section 2.2, prove the existence of
canonical classes and compute their equivariant structure constants. Similar
computations will be done for the equivariant Chern classes.

We begin with the following observation, which is useful when the manifold
satisfies $b_{2}(M)=1$. This condition holds for the aforementioned manifolds
$V,W$ and $Q$.

######  Lemma 5.1.

Let $(M,\omega,\psi)$ be a Hamiltonian $T$-space with $b_{2}(M)=1$ and
discrete fixed point set $M^{T}$. Let $k_{0}$ be the index of $(M,\omega)$.
Consider a generic element $\xi$ of $\mathfrak{t}$ and the corresponding
component of the moment map $\varphi:=\psi^{\xi}$. Let $p_{i}$ be the only
fixed point of index $i$ w.r.t. $\varphi$, for $i=0,1$. Let $\Gamma=(V,E)$ be
a multigraph describing $(M,\omega,\psi)$ and let $e$ (resp. $e^{\prime}$) be
the only edge between $p_{0}$ and $p_{1}$ (resp. the only edge between
$p_{n-1}$ and $p_{n}$, see Lemma 4.31). Then there exists a canonical class
$\tau_{1}\in H^{2}_{T}(M;\mathbb{Z})$ w.r.t. $\xi$ and it is given by

$\tau_{1}=\frac{c_{1}^{T}(p_{0})-c_{1}^{T}}{k_{0}}\,.$

Moreover the index satisfies $k_{0}=c_{1}^{T}[e]=c_{1}^{T}[e^{\prime}]\,.$

###### Proof.

First of all we observe that, by [43, Lemma 3.1], the fixed points $P$ of
index greater than 1 satisfy $\varphi(P)>\varphi(p_{1})>\varphi(p_{0})$. Since
there is no other fixed point of index $1$, it follows that the Kirwan class
$\tau_{1}$ at $p_{1}$ is also a canonical class. Since Kirwan classes are a
basis of $H^{*}_{T}(M;\mathbb{Z})$ as an $H_{T}^{*}(pt;\mathbb{Z})$-module,
for degree reasons we have that

(5.1) $c_{1}^{T}=a\tau_{1}+b\,,\quad\text{for some }a\in\mathbb{Z}\text{ and
}b\in H^{2}_{T}(\\{pt\\};\mathbb{Z})\,.$

Since $\tau_{1}(p_{0})=0$, it follows that $b=c_{1}^{T}(p_{0})$.

Let $-w_{1}$ be the weight in the negative normal bundle at $p_{1}$. Since
$\tau_{1}(p_{1})=w_{1}$ (see Remark 2.7), and since there is an edge between
$p_{0}$ and $p_{1}$, (5.1) gives that $a=c_{1}^{T}[e]\,.$

Since $r(\tau_{1})$ generates $H^{2}(M;\mathbb{Z})\simeq\mathbb{Z}$, it must
be a primitive element, and by definition of index, it follows that
$k_{0}=c_{1}^{T}[e]$. In order to prove that
$c_{1}^{T}[e]=c_{1}^{T}[e^{\prime}]$ it is sufficient to consider the
component of the moment map corresponding to $-\xi$; the details are left to
the reader. ∎

The next lemma explains the link between algebraic torus actions and
Hamiltonian $T$-actions on smooth Fano varieties.

######  Lemma 5.2.

Let $\mathbf{T}=(\mathbb{C}^{*})^{k}$ be an algebraic torus acting
algebraically on a smooth Fano variety $X$.

Then there exists a $\mathbf{T}$-invariant Kähler form $\omega$ on $X$
satisfying $c_{1}(X)=[\omega]$, for which the action of
$(S^{1})^{k}<\mathbf{T}$ is Hamiltonian.

###### Proof.

As the tangent bundle $TX$ of $X$ is $\mathbf{T}$-invariant, the anticanonical
line bundle

$L=-K_{X}=\det TX=\wedge^{n}(TX)$

is also $\mathbf{T}$-invariant. Then there exists $m>0$ such that $L^{\otimes
m}$ is $\mathbf{T}$-linearizable444A line bundle $L$ over a complex algebraic
manifold $X$ with an algebraic $G$-action is called $G$-linearizable if the
$G$-action on $X$ lifts to an action on $L$ which is linear on the fibers of
$L$ [5, Definition 3.2.3]. (see [5, Proposition 3.4.5]). Moreover, since $X$
is Fano, we have that $L$ is ample and so $L^{\otimes m}$ is also ample (as
$(c_{1}(L^{\otimes m}))^{k}=m^{k}c_{1}(L)^{k}$ for each $k>0$).

By [5, Proposition 3.2.6] we have that there exists a $\mathbf{T}$-equivariant
embedding $i:X\to\mathbb{CP}^{N}$ into some projective space
$\mathbb{CP}^{N}$, for some linear action on $\mathbb{CP}^{N}$ (Hamiltonian
with respect to the Fubini-Study Kähler form). Hence, $X$ has a
$\mathbf{T}$-invariant Kähler structure $\omega$ induced by the Fubini-Study
Kähler form in $\mathbb{CP}^{N}$ and the $(S^{1})^{k}<\mathbf{T}$ action on
$X$ is Hamiltonian.

The pull-back of the Fubini-Study metric satisfies $[\omega]=c_{1}(L^{k})$ for
some integer $k>0$, therefore $[\omega]=\lambda c_{1}(X)$ for some positive
constant $\lambda$. Since the form remains invariant and Kähler after
rescaling by a positive constant, the result follows.

∎

###  5.1. Index 2: The manifold Fano-Mukai fourfold $V$ and its equivariant
cohomology ring

In this section, we describe an example of closed, monotone symplectic
manifold of dimension $8$ with index $2$, which is acted on in a Hamiltonian
way by a 2-dimensional torus with 6 fixed points. In particular we give the
isotropy weights at each fixed point and compute its equivariant cohomology
ring and Chern classes.

####  5.1.1. Torus actions on the generalized flag variety $\Omega=G_{2}/P$

Let $G_{2}$ be the complexification of the exceptional simple Lie group (of
complex dimension $14$) and let $P$ be a parabolic subgroup (of complex
dimension $9$) associated to a long root. The generalized flag manifold

$\Omega:=G_{2}/P\subset\mathbb{P}(\mathfrak{g}_{2})\subset\mathbb{CP}^{13}$

is known to be GKM for a $T^{2}$-action with GKM graph as in Figure 5.1 (see
[21]). From this graph and from Lemma 5.2 we obtain the following result.

######  Theorem 5.3.

There exists a Kähler form $\omega$ on $\Omega$ for which
$c_{1}(\Omega)=[\omega]$ and a holomorphic $(\mathbb{C}^{*})^{2}$-action on
$\Omega$ such that the action of $T^{2}<(\mathbb{C}^{*})^{2}$ is Hamiltonian
with respect to $\omega$ and has $6$ isolated fixed points, with weights given
in the following table.

| Weights
---|---
$q_{0}$ | $\\{(2,1),(1,1),(0,1),(-1,1),(1,2)\\}$
$q_{1}$ | $\\{(1,-1),(1,0),(1,1),(1,2),(2,1)\\}$
$q_{2}$ | $\\{(1,2),(0,1),(-1,0),(-2,-1),(-1,1)\\}$
$q_{2}^{\prime}$ | $\\{(-1,-2),(0,-1),(1,0),(2,1),(1,-1)\\}$
$q_{3}$ | $\\{(-1,1),(-1,0),(-1,-1),(-1,-2),(-2,-1)\\}$
$q_{4}$ | $\\{(-2,-1),(-1,-1),(0,-1),(1,-1),(-1,-2)\\}$
$q_{1}$$q_{0}$$q_{2}$$q_{2}^{\prime}$$q_{3}$$q_{4}$ Figure 5.1. GKM graph of
$\Omega$.

(See also [16] Section 5.3.1, and in particular Proposition 5.24, for how to
construct the GKM graph corresponding to a coadjoint orbit with
$c_{1}=[\omega]$.)

####  5.1.2. The Fano-Mukai $4$-fold $V$ and its torus action.

Let us now consider smooth hyperplane sections

$V_{18}:=\Omega\cap\mathbb{CP}^{12}\subset\mathbb{CP}^{13}$

of $\Omega$, known as Fano-Mukai fourfolds of genus $10$ and degree $18$. As
shown by Prokhorov and Zaidenberg [38, Remark 13.4], the moduli space of these
manifolds is $1$-dimensional. It contains a unique element $V_{18}^{a}$ with
an action of $\mathbb{C}\times\mathbb{C}^{*}$ and a unique element
$V_{18}^{s}$ with an effective action of $GL(2,\mathbb{C})$. A general element
$V_{18}^{g}\notin\\{V_{18}^{a},V_{18}^{s}\\}$ admits an action of
$(\mathbb{C}^{*})^{2}$ [38, Theorem 1.3] which is the restriction to
$V_{18}^{g}$ of the $(\mathbb{C}^{*})^{2}$-action on $\Omega$ [38, Theorem
1.2]. Let $V$ be one of these general elements. Then we have the following
result.

######  Theorem 5.4.

There exists a Kähler form $\omega$ on $V$ for which $c_{1}(V)=[\omega]$ and a
holomorphic $(\mathbb{C}^{*})^{2}$-action, such that the action of
$T^{2}<(\mathbb{C}^{*})^{2}$ is Hamiltonian with respect to $\omega$ and has
$6$ isolated fixed points, with weights given in the following table.

| Weights
---|---
$q_{0}$ | $\\{(2,1),(1,1),(0,1),(-1,1)\\}$
$q_{1}$ | $\\{(1,-1),(1,0),(1,1),(1,2)\\}$
$q_{2}$ | $\\{(1,2),(0,1),(-1,0),(-2,-1)\\}$
$q_{2}^{\prime}$ | $\\{(-1,-2),(0,-1),(1,0),(2,1)\\}$
$q_{3}$ | $\\{(-1,1),(-1,0),(-1,-1),(-1,-2)\\}$
$q_{4}$ | $\\{(-2,-1),(-1,-1),(0,-1),(1,-1)\\}$

$q_{1}$$q_{0}$$q_{2}$$q_{2}^{\prime}$$q_{3}$$q_{4}$$(0,0)$ Figure 5.2.  The
multigraph describing the $T^{2}$ action on $V$.

###### Proof.

Let $\omega$ be the invariant Kähler form on $\Omega$ from Theorem 5.3. Since
$V\subset\Omega$ is a $T^{2}$-invariant symplectic submanifold of $\Omega$, it
follows that $V$ is GKM for the restricted $T^{2}$-action and that the
corresponding GKM graph of $V$ is a subgraph of the GKM graph of $\Omega$.
Consequently, to obtain this last GKM graph we just have to determine which
edges in the GKM graph of $\Omega$ (corresponding to $T^{2}$-invariant spheres
in $\Omega$) are not in the GKM graph of $V$.

Note that any $T^{2}$-invariant sphere $S\subset\Omega$ that is not in $V$
must intersect $V$ transversely in two fixed points. Moreover, since $\Omega$
is a Fano variety of index $3$, we have that $c_{1}(\Omega)=3\tau$, where
$\tau$ is the generator of $H^{2}(\Omega,\mathbb{Z})$. Because $V$ is a
hyperplane section of the embedding associated to line bundle with first Chern
class equal to $\tau$, $\tau$ is Poincaré dual to $[V]\in
H_{8}(\Omega,\mathbb{Z})$. From the fact that $c_{1}=[\omega]$, we obtain

$S\cdot[V]=\int_{S}i^{*}\tau=\frac{1}{3}\int_{S}i^{*}\omega\,.$

where $i:S\to\Omega$ is the inclusion map. Therefore
$S\cdot[V]=2\iff\int_{S}i^{*}\omega=6$.

Given a GKM graph of a Hamiltonian $T$-space, and considering its image in
$\text{Lie}(T)^{*}$ via the moment map, using the ABBV formula it is easy to
see that the symplectic area of an invariant sphere corresponding to an edge
$e$ is exactly the integral length of $e$ (namely, the length of $e$ measured
with respect to the dual lattice). From Figure 5.1 we can see that the only
$T^{2}$-invariant spheres with area equal to $6$ are exactly the ones
corresponding to the blue edges. We conclude that the GKM graph of $V$ is the
one obtianed by deleting the blue edges in the GKM graph of $\Omega$ and the
result follows. ∎

Our next aim is to compute the equivariant cohomology ring of $V$ and its
Chern classes. Let $x:=(1,0)$ and $y:=(0,1)$ be a basis of $\ell^{*}$, the
dual lattice of $T$. Then the weights of the action computed in Theorem 5.4,
as well as the values of the equivariant first Chern class at the fixed
points, are written in Table 5.1.

Observe that we choose a moment map $\psi\colon V\to\mathfrak{t}^{*}$ so that
$c_{1}^{T}=[\omega-\psi]$, thus implying the weight sum formula (2.5).

|  |  |  |  | $c_{1}^{T}(q)=-\psi(q)$
---|---|---|---|---|---
$q_{0}$ |  $-x+y$, |  $y$, |  $x+y$, | $2x+y$ | $2x+4y$
$q_{1}$ |  $x-y$, |  $x$, |  $x+y$, | $x+2y$ | $4x+2y$
$q_{2}$ |  $-2x-y$, |  $-x$, |  $y$, | $x+2y$ | $-2x+2y$
$q^{\prime}_{2}$ |  $-x-2y$, |  $-y$, |  $x$, | $2x+y$ | $2x-2y$
$q_{3}$ |  $-x-2y$, |  $-x-y$, |  $-x$, | $-x+y$ | $-4x-2y$
$q_{4}$ |  $-2x-y$, |  $-x-y$, |  $-y$, | $x-y$ | $-2x-4y$
Table 5.1.  Weights at the fixed points of the effective $T$ action on $V$ and
the corresponding value of $c_{1}^{T}$.
$(0,0)$$y$$x$$-x+y$$y$$x+y$$x+2y$$q_{1}$$q_{0}$$q_{2}$$q_{2}^{\prime}$$q_{3}$$q_{4}$
Figure 5.3. The oriented GKM graph describing the action on $V$ and its
weights.

Let $\xi:=(1,2)\in\mathfrak{t}$ be the generic vector that we use to orient
the GKM graph of the $T$ action on $V$, as in Figure 5.3. Then it is easy to
see that the GKM graph associated this action is index increasing, namely it
holds:

$p\prec q\implies\lambda_{p}<\lambda_{q}\quad\text{for all the edges
}e=(p,q)\text{ of the GKM graph.}$

Therefore, as an immediate consequence of [18, Theorem 1.6], we have that for
each fixed point $q$ in $V^{T}$, there exists a canonical class $\tau_{q}\in
H^{2\lambda(q)}_{T}(V;\mathbb{Z})$ (see Remark 2.7 for our conventions on
canonical classes). Moreover, its values at the fixed points can be computed
with the recipe given in the same theorem. (In our case we will only need to
use [18, Theorem 1.6] to compute the values of the restrictions of the
canonical classes at $q_{2}$ and $q^{\prime}_{2}$, for reasons that will be
clear from the following.)

Observe that we named the fixed points in such a way that $q_{i}$ has index
exactly $i$ and $q^{\prime}_{2}$ has index $2$. Let $\tau_{i}\in
H^{2i}_{T}(V;\mathbb{Z})$ be the canonical class at $q_{i}$ and
$\tau^{\prime}_{2}\in H_{T}^{4}(V;\mathbb{Z})$ the canonical class at
$q^{\prime}_{2}$.

Using Lemma 5.1 and the values of $c_{1}^{T}$ in Table 5.1, we obtain that the
canonical class $\tau_{1}$ of $V$ (with respect to the chosen $\xi$) is given
by

(5.2) $\tau_{1}=\frac{2x+4y-c_{1}^{T}}{2}\,.$

Its values at all fixed points can therefore be computed from Table 5.1 and
can be found in Table 5.2. Next we compute the restrictions to the fixed
points of the canonical classes of degree 4, namely $\tau_{2}$ and
$\tau^{\prime}_{2}$. For these computations we follow the recipe in [18,
Theorem 1.6]. In order to do so, for any $\eta\in\mathfrak{t}^{*}$, consider
the projection $\rho_{\eta}\colon\mathfrak{t}^{*}\to\xi^{\perp}$ given by

$\rho_{\eta}(\alpha):=\alpha-\displaystyle\frac{\langle\alpha,\xi\rangle}{\langle\eta,\xi\rangle}\eta$

and extend it to $\mathbb{S}(\mathfrak{t}^{*})$ by imposing that it is an
(algebra) endomorphism. For every edge $e=(p,q)$ of the GKM graph of $V$ such
that $\lambda(q)-\lambda(p)=1$ we need to compute the following

(5.3)
$\Theta(p,q)=\frac{\rho_{\eta(p,q)}(\Lambda_{p}^{-})}{\rho_{\eta(p,q)}\left(\frac{\Lambda_{q}^{-}}{\eta(p,q)}\right)}$

which, by [18, Theorem 1.6], is a non-zero integer. Here $\eta(p,q)$ denotes
the weight labeling the edge $(p,q)$ of the GKM graph. According to the same
theorem, in order to compute the restrictions to the fixed points of
$\tau_{2}$ and $\tau^{\prime}_{2}$, we need to compute
$\Theta(q^{\prime}_{2},q_{3})$, $\Theta(q_{3},q_{4})$ and
$\Theta(q_{2},q_{3})$. We begin with $\Theta(q^{\prime}_{2},q_{3})$. One can
check that

$\eta(q^{\prime}_{2},q_{3})=x,\;\;\Lambda_{q^{\prime}_{2}}^{-}=y(x+2y)\;\;\text{
and }\;\;\Lambda_{q_{3}}^{-}/x=(x+y)(x+2y).$

Therefore, since $\xi=(1,2)$, using (5.3) it is easy to check that
$\Theta(q^{\prime}_{2},q_{3})=1$. Analogously it can be checked that
$\Theta(q_{3},q_{4})=1$ and $\Theta(q_{2},q_{3})=3$. For computing
$\tau_{2}(q_{3})$ it is sufficient to use [18, (4.1)], which gives immediately
that $\tau_{2}(q_{3})=3x(x+y)$. In order to compute $\tau_{2}(q_{4})$ we
observe that, using the notation of [18], the only path in
$\Sigma_{q_{2}}^{q_{4}}$ is the one with edges $(q_{2},q_{3})$ and
$(q_{3},q_{4})$. Therefore we have

$\tau_{2}(q_{4})=\Lambda_{q_{4}}^{-}\cdot\frac{\psi(q_{3})-\psi(q_{2})}{\psi(q_{4})-\psi(q_{2})}\cdot\frac{\Theta(q_{2},q_{3})}{x+2y}\cdot\frac{\Theta(q_{3},q_{4})}{-x+y}=(2x+y)(x+y)\,.$

Analogous computations can be done for $\tau^{\prime}_{2}$, and its values at
the fixed points can be found in Table 5.2.

In order to compute the values of $\tau_{3}$, it is not necessary to use [18,
Theorem 1.6]. Indeed, the only fixed points where $\tau_{3}$ does not vanish
are $q_{3}$ and $q_{4}$; moreover there is a smooth invariant sphere in $V$
that has $q_{3}$ and $q_{4}$ as fixed points. Therefore the canonical class at
$q_{3}$ coincides with the equivariant Poincarè dual to the sphere, and its
values are given in Table 5.2. In this table we have not given the values of
$\tau_{0}$ and $\tau_{4}$, as computing their restrictions to the fixed points
is trivial (see Remark 2.3).

| $\tau_{1}$ | $\tau_{2}$ | $\tau^{\prime}_{2}$ | $\tau_{3}$
---|---|---|---|---
$q_{0}$ | 0 | 0 | 0 | 0
$q_{1}$ | $-x+y$ | 0 | 0 | 0
$q_{2}$ | $2x+y$ | $x(2x+y)$ | 0 | 0
$q^{\prime}_{2}$ | $3y$ | $0$ | $y(x+2y)$ | 0
$q_{3}$ | $3x+3y$ | $3x(x+y)$ | $(x+y)(x+2y)$ | $x(x+y)(x+2y)$
$q_{4}$ | $2x+4y$ | $(2x+y)(x+y)$ | $3y(x+y)$ | $y(2x+y)(x+y)$
Table 5.2.  Restrictions to the fixed points of the “non-trivial” canonical
classes on $V$.

######  Theorem 5.5.

Let $V$ be the Fano-Mukai 4-fold of genus 10 and degree 18 endowed with the
Hamiltonian $T$-action described in Theorem 5.4. Let $x,y$ be a
$\mathbb{Z}$-basis of the dual lattice $\ell^{*}$. Then there exists a basis
of the equivariant cohomology ring $H_{T}^{*}(V;\mathbb{Z})$ given by
canonical classes $\\{\tau_{i}\\}_{i=0}^{4}\cup\\{\tau^{\prime}_{2}\\}$,
satisfying the following properties and relations:
For each $i=0,\ldots,4$, $\tau_{i}\in H_{T}^{2i}(V;\mathbb{Z})$,
$\tau^{\prime}_{2}\in H_{T}^{4}(V;\mathbb{Z})$ and

(5.4) $\displaystyle\begin{aligned}
&\tau_{1}^{2}=(-x+y)\tau_{1}+3\tau_{2}+3\tau^{\prime}_{2}\,,&\tau_{1}\tau_{2}=(2x+y)\tau_{2}+3\tau_{3}\\\
&\tau_{1}\tau^{\prime}_{2}=3y\,\tau^{\prime}_{2}+3\tau_{3}\,,&\tau_{1}\tau_{3}=3(x+y)\tau_{3}+\tau_{4}\\\
&\tau_{2}^{2}=x(2x+y)\tau_{2}+3x\,\tau_{3}+\tau_{4}\,,&\tau_{2}\tau^{\prime}_{2}=3(x+y)\tau_{3}\\\
&(\tau^{\prime}_{2})^{2}=y(x+2y)\tau^{\prime}_{2}+(x+2y)\tau_{3}+\tau_{4}\,.\\\
\end{aligned}$

Moreover, the equivariant Chern classes $c_{j}^{T}\in
H_{T}^{2j}(V;\mathbb{Z})$ of the tangent bundle satisfy

(5.5) $\displaystyle\begin{aligned} &c_{1}^{T}=-2\tau_{1}+2x+4y\\\
&c_{2}^{T}=7(\tau_{2}+\tau^{\prime}_{2})-7(x+y)\tau_{1}-x^{2}+6xy+6y^{2}\\\
&c_{3}^{T}=-24\tau_{3}+8(x+2y)\tau_{2}+8(2x+y)\tau^{\prime}_{2}-2(3x^{2}+7xy+3y^{2})\tau_{1}-2(x^{3}+x^{2}y-3xy^{2}-2y^{3})\\\
&c_{4}^{T}=6\tau_{4}-6(x+2y)\tau_{3}+(x^{2}+7xy+7y^{2})\tau_{2}+(7x^{2}+7xy+y^{2})\tau^{\prime}_{2}-(x^{3}+5x^{2}y+5xy^{2}+y^{3})\tau_{1}+\\\
&\;\;\;\;\;\;\;\;(-x+y)y(x+y)(2x+y).\\\ \end{aligned}$

###### Proof.

We have already proved that canonical classes exist and their values at the
fixed points are given in Table 5.2. Therefore, with the procedure described
in Section 2.2.1, the equivariant structure constants with respect to the
basis can be computed from this table. In a similar fashion, the coefficients
of the equivariant Chern classes can be computed from their restrictions to
the fixed points, which just depend on the isotropy weights (see Remark 2.6);
the details are left to the reader. ∎

Corollary 1.1 in the introduction is a consequence of Theorem 5.5 and the
procedure explained in Section 2.2.1.

###  5.2. Index 3: The quintic del Pezzo fourfold $W$ and its equivariant
cohomology ring

Let us consider a del Pezzo $4$-fold of degree-$5$ obtained as the
intersection

$W=W_{5}=Gr(2,5)\cap L$

of the Grassmanian $Gr(2,5)$ (under the Plücker embedding in
$\mathbb{P}(\Lambda^{2}\mathbb{C}^{5})=\mathbb{C}\mathbb{P}^{9}$) and a
codimension-$2$ projective subspace $L$ of $\mathbb{C}\mathbb{P}^{9}$.

Identifying $Gr(2,5)$ with the set of lines in $\mathbb{CP}^{4}$, we see that
there are two types of projective planes in $Gr(2,5)$. Namely the vertex type
planes $\sigma_{3,1}$ consisting of lines in $\mathbb{CP}^{4}$ through a given
point and contained in a fixed projective hyperplane of $\mathbb{CP}^{4}$, and
the non-vertex type planes $\sigma_{2,2}$ consisting of lines contained in a
fixed projective plane of $\mathbb{CP}^{4}$.

The manifold $W$ contains exactly one $\sigma_{2,2}$ plane $\Xi$ and a
$1$-parameter family of $\sigma_{3,1}$ planes. Each of these planes intersects
$\Xi$ along a line tangent to a fixed conic $C\subset\Xi$. Let $R$ be the
union of all the $\sigma_{3,1}$ planes in $W$. It is shown in [37, Proposition
3.4] that $R$ is the intersection of $W$ with a projective hyperplane and that
$R$ is singular along $\Xi$. In particular, we have

$C\subset\Xi\subset R\subset W$

with $\Xi\simeq\mathbb{CP}^{2}$ and $\dim_{\mathbb{C}}R=3$.

Let $Bl_{\Sigma}(W)$ be the blow up of $W$ along $\Xi$. It is shown in [38,
Proposition 2.2] that $Bl_{\Sigma}(W)$ can be identified with the blow up
$Bl_{\Gamma}(\mathbb{CP}^{4})$ of $\mathbb{CP}^{4}$ along a twisted cubic
curve $\Gamma\subset\mathbb{CP}^{4}$ and that we have the following
commutative diagram,

(5.6) ${{\widetilde{H}\subset Bl_{\Xi}(W)\simeq
Bl_{\Gamma}(\mathbb{CP}^{4})\supset\widetilde{R}}}$${{\Xi\subset R\subset
W}}$${{\mathbb{CP}^{4}\supset
H=\langle\Gamma\rangle\supset\Gamma}}$$\scriptstyle{\rho}$$\scriptstyle{\varphi}$$\scriptstyle{\tau}$

where $\rho:Bl_{\Xi}(W)\to W$ and
$\varphi:Bl_{\Gamma}(\mathbb{CP}^{4})\to\mathbb{CP}^{4}$ are the blow up maps
and $\tau:W\subset L\simeq\mathbb{CP}^{7}\to\mathbb{CP}^{4}$ is the projection
with center $\Xi$ (a birational map). Let $\widetilde{H}:=\rho^{-1}(\Xi)$ and
$\widetilde{R}:=\varphi^{-1}(\Gamma)$ be the corresponding exceptional
divisors. Then $\widetilde{H}$ can be identified with the proper transform of
the projective hyperplane $H=\langle\Gamma\rangle\simeq\mathbb{CP}^{3}$
spanned by $\Gamma$ and $\widetilde{R}$ can be identified with the proper
transform of $R$ in $Bl_{\Xi}(W)$ (see [38, Proposition 2.2]).

It is shown in [38, Corollary 2.2.2] that $W$ is a compactification of
$\mathbb{C}^{4}$ and that

$W\setminus R\simeq
Bl_{\Xi}(W)\setminus\left(\widetilde{R}\cup\widetilde{H}\right)\simeq\mathbb{CP}^{4}\setminus
H\simeq\mathbb{CP}^{4}\setminus\mathbb{CP}^{3}\simeq\mathbb{C}^{4}.$

Moreover, by [38, Corollary 2.2.4], the automorphism group $Aut(W)$ of $W$
leaves $\Xi$ invariant and can be identified with the group of automorphisms
of $\mathbb{CP}^{4}$ that leaves $\Gamma$ invariant

$Aut(W)=Aut(W,\Xi)=Aut(\mathbb{CP}^{4},\Gamma).$

Note that $R$ is also $Aut(W)$-invariant and that the action of $Aut(W)$ on
$R$ is effective (cf [38, Proposition 5.6]). Let $T^{2}$ be a maximal torus in
$Aut(W)$. In the following result we show that the action of $T^{2}$ on $W$
has $6$ fixed points and we compute the corresponding weights.

######  Theorem 5.6.

Let $W$ be the quintic del Pezzo fourfold defined as above and let $T^{2}$ be
a maximal torus of $Aut(W)$. Then the action of $T^{2}$ on $W$ has $6$
isolated fixed points and the weights at these points are:

| Weights
---|---
$p_{0}$ | $\\{(3,0),(2,1),(1,2),(0,3)\\}$
$p_{1}$ | $\\{(-3,0),(-1,1),(1,2),(0,3)\\}$
$p_{2}$ | $\\{(3,0),(1,-1),(2,1),(0,-3)\\}$
$p_{2}^{\prime}$ | $\\{(-1,-2),(-2,-1),(-1,1),(-2,2)\\}$
$p_{3}$ | $\\{(-3,0),(1,-1),(-1,1),(0,-3)\\}$
$p_{4}$ | $\\{(-1,-2),(2,-2),(-2,-1),(1,-1)\\}$

Moreover, this action is Hamiltonian with respect to a Kähler form $\omega$ on
$W$ satisfying $c_{1}(W)=[\omega]$. The image of the moment map, including the
invariant spheres is given in Figure 5.4.

$p_{0}$$p_{1}$$p_{2}$$p_{2}^{\prime}$$p_{3}$$p_{4}$ Figure 5.4.  The moment
image of the invariant spheres in $W$.

###### Proof.

Up to conjugation we can assume $T^{2}$ to be the diagonal torus in
$GL(2,\mathbb{C})\subset Aut(W)$. It has a unique fixed point $P_{0}$ in

$W\setminus R\simeq\mathbb{C}^{4}\simeq\mathbb{CP}^{4}\setminus H$

(cf. [38, Lemma 5.7]). Up to a change in coordinates, the curve
$\Gamma\subset\mathbb{CP}^{4}$ can be parametrized as

(5.7) $\phi([X:Y])=[0:X^{3}:X^{2}Y:XY^{2}:Y^{3}],$

so that $H=\langle\Gamma\rangle$ is identified with the projective hyperplane
$\\{z_{0}=0\\}$ (cf. [38, Notation 5.1]). In these coordinates $P_{0}$ is
identified with $q_{0}=[1:0:0:0]\in\mathbb{CP}^{4}$ and $W\setminus
R\simeq\mathbb{C}^{4}$ is identified with the set of points of the form
$[1:z_{1}:z_{2}:z_{3}:z_{4}]$. As it is shown in [38, Notation 5.1 and Lemma
5.7], the $T^{2}$-action on $W\setminus R$ is written in these coordinates as

(5.8)
$(\lambda,\mu)\cdot(z_{1},z_{2},z_{3},z_{4})=(\lambda^{3}z_{1},\lambda_{2}\mu
z_{2},\lambda\mu^{2}z_{3},\mu^{3}z_{4}).$

The corresponding action on $\mathbb{CP}^{4}\setminus E$ extends naturally to
a $T^{2}$-action on $\mathbb{CP}^{4}$ with $5$ fixed points: $q_{0}$ and the
points

$q_{1}:=[0:1:0:0:0],\,q_{2}=[0:0:1:0:0],\,q_{3}=[0:0:0:1:0]\,\text{and}\,q_{4}=[0:0:0:0:1].$

Note that this action has exactly two fixed points in $\Gamma$: the points
$q_{1}$ and $q_{4}$. The weights of the $T^{2}$-action on $\mathbb{CP}^{4}$ at
the five fixed points are shown in Table 5.3.

| Weights
---|---
$q_{0}$ | $\\{(3,0),(2,1),(1,2),(0,3)\\}$
$q_{1}$ | $\\{(-3,0),(-1,1),(-2,2),(-3,3)\\}$
$q_{2}$ | $\\{(-2,-1),(1,-1),(-1,1),(-2,2)\\}$
$q_{3}$ | $\\{(-1,-2),(2,-2),(1,-1),(-1,1)\\}$
$q_{4}$ | $\\{(0,-3),(3,-3),(2,-2),(1,-1)\\}$
Table 5.3. Weights for the $T^{2}$-action on $\mathbb{CP}^{4}$

Since $\Xi$ is $T^{2}$-invariant, we can perform the blow up equivariantly and
obtain a natural $T^{2}$-action on the blow up $Bl_{\Xi}(W)$. Through the
identification in (5.6) we obtain a $T^{2}$-action on
$Bl_{\Gamma}(\mathbb{CP}^{4})$.

$q_{1}$$q_{2}$$q_{4}$$q_{0}$$q_{3}$$(-1,1)$$H$$(-3,3)$$(-3,0)$$(-2,2)$$q_{4}$$q_{3}$$q_{1}$$q_{0}$$q_{2}$$\Gamma$$(1,-1)$$(3,-3)$$(0,-3)$$(2,-2)$$H$$\Gamma$$q_{0}$$A$$B$$q_{3}$$q_{4}$$(-1,1)$$(-1,1)$$(-1,1)$$(-3,0)$$(-3,3)$$(-2,2)$$(-1,1)$$(-1,-2)$$(0,-3)$$(0,3)$$(1,2)$$(1,-1)$$(-1,-2)$$\widetilde{H}$$q_{0}$$C$$D$$q_{2}$$q_{1}$$(1,-1)$$(1,-1)$$(1,-1)$$(0,-3)$$(3,-3)$$\tilde{H}$$(2,-2)$$(1,-1)$$(-2,-1)$$(-3,0)$$(3,0)$$(2,1)$$(1,-1)$
Figure 5.5.  Determining the weights on the blow-up
$Bl_{\Gamma}(\mathbb{CP}^{4})$ along $\Gamma$: The two fixed points
$q_{1},q_{4}\in\mathbb{CP}^{4}$ that are in $\Gamma\subset\mathbb{CP}^{4}$ are
shown on top. The edges corresponding to $T^{2}$-invariant spheres in
$H\simeq\mathbb{CP}^{3}$ are colored blue, and those that are not in $H$ are
colored red. At the bottom we have the weights at the fixed points in
$Bl_{\Gamma}(\mathbb{CP}^{4})$. The edges corresponding to invariant spheres
in the proper transform $\widetilde{H}\simeq Bl_{\Gamma}(\mathbb{CP}^{3})$ are
colored blue, and those that are not in $\widetilde{H}$ are colored red.

Note that this action has nine fixed points and that six of these points are
on the exceptional divisor $\widetilde{R}=\varphi^{-1}(\Gamma)$. The weights
at these fixed points are presented in Figure 5.5. Of these six points only
the points represented by $A,B,C$ and $D$ in Figure 5.5 are in the proper
transform $\widetilde{H}\simeq\rho^{-1}(\Xi)$ of $H$ and so the points in
$Bl_{\Xi}(W)$ corresponding to the remaining two fixed points in
$\widetilde{R}$ are not in the exceptional divisor of $Bl_{\Xi}(W)$.
Consequently, they are also fixed points for the $T^{2}$-action on $W$ and
their weights (for the $T^{2}$-action on $W$) remain the same after the blow
down to $W$. Note that the same is true for $P_{0}$ (the fixed point in $W$
corresponding to $q_{0}$), since this point is not in the exceptional divisor
$\varphi^{-1}(\Gamma)$, as $q_{0}\notin\widetilde{H}$.

Hence, we already have the weights of the $T^{2}$-action on $W$ at three fixed
points. We will now determine the weights at the remaining fixed points (i.e.
those that are in $\Xi$). These have to be obtained from the weights at the
fixed points $A,B,C$ and $D$ by contracting the exceptional divisor. By Lemma
5.7 we just have to see which pairs of points in $\\{A,B,C,D,q_{2},q_{3}\\}$
have two weights in common and one weight that is symmetric. We can easily see
that the only possibility is to identify $A$ with $q_{2}$, $B$ with $D$ and
$C$ with $q_{3}$ in the blow down.

Again by Lemma 5.7 we see that the weights of the $T^{2}$-action at the
resulting fixed points in $W$ are

$\displaystyle\\{(-1,-2),(-2,-1),$
$\displaystyle(-1,1),(-2,2)\\},\quad\\{(-3,0),(1,-1),(-1,1),(0,-3)\\}\quad\text{and}$
$\displaystyle\\{(-1,-2),(2,-2),(-2,-1),(1,-1)\\}.$

We conclude that the weights at the fixed points for the $T^{2}$-action on $W$
are those in the statement of the Theorem.

Finally, since $W$ is a Fano variety, we have by Lemma 5.2 that there exists a
$T^{2}$-invariant Kähler form $\omega$ for which the $T^{2}$-action is
Hamiltonian and $c_{1}=[\omega]$. The corresponding moment map image is shown
in Figure 5.4. ∎

######  Lemma 5.7.

Let $M$ be an $8$-dimensional symplectic manifold equipped with a Hamiltonian
$T^{2}$-action, and let $\Sigma\subset M$ be a $T^{2}$-invariant
$4$-dimensional submanifold of $M$. Let $P\in\Sigma$ be a fixed point for the
$T^{2}$-action on $M$ and let $w_{1},w_{2},w_{3},w_{4}$ be the weights for
this action at $P$, where $w_{1},w_{2}$ are the weights on $T_{P}\Sigma$ and
$w_{3},w_{4}$ are the weights on the normal bundle of $\Sigma$.

If $Bl_{\Sigma}(M)$ is a $T^{2}$-equivariant blow up of $M$ along $\Sigma$
with blow up map $\pi:Bl_{\Sigma}(M)\to M$, then the weights of the induced
$T^{2}$-action at the two fixed points in the fiber $\pi^{-1}(P)$ are

$\\{w_{1},w_{2},w_{3},w_{4}-w_{3}\\}\quad\text{and}\quad\\{w_{1},w_{2},w_{4},w_{3}-w_{4}\\}.$

###### Proof.

Since the exceptional divisor $\widetilde{\Sigma}$ is the projectivization of
the normal bundle of $\Sigma$, the fiber $\pi^{-1}(P)$ is a $T^{2}$-invariant
$\mathbb{CP}^{1}$ where $T^{2}$ acts by

$\lambda\cdot[z_{3}:z_{4}]=[\lambda^{w_{3}}z_{3}:\lambda^{w_{4}}z_{4}]=[z_{3}:\lambda^{w_{4}-w_{3}}z_{4}],\quad\lambda\in
T^{2}.$

(Here, for $\lambda=(\lambda_{1},\lambda_{2})\in T^{2}$ and a weight
$w=(w^{1},w^{2})$, we write
$\lambda^{w}:=\lambda_{1}^{w^{1}}\lambda_{2}^{w^{2}}$). ∎

Figure 5.6. Weights and Blow up along $\Sigma$

As it is easily seen, the action of the torus $T$ on $W$ is not effective.
Indeed, the subgroup $\mathbb{Z}_{3}$ of the diagonal circle in $T$ acts
trivially on $W$, as it acts trivially on an open neighborhood of $p_{0}$. In
the following we compute the weights of the effective action of
$\mathbb{T}:=T/\mathbb{Z}_{3}$.

Let $\xi_{1}:=(1,0)$ and $\xi_{2}:=(0,1)$ be a basis of the lattice $\ell$ of
$T$ and let $x=(1,0)$, $y=(0,1)$ be the dual basis; in this notation, the
$\mathbb{Z}_{3}$ that acts trivially is exactly
$\exp\mathbb{Z}\Big{\langle}\displaystyle\frac{\xi_{1}+\xi_{2}}{3}\Big{\rangle}$.
The lattice of $\mathbb{T}$ is then given by
$\widetilde{\ell}:=\mathbb{Z}\langle\xi_{1},\frac{\xi_{1}+\xi_{2}}{3}\rangle$;
a basis of the dual lattice $\widetilde{\ell}^{*}$ is given for example by

$\alpha:=-x+y\quad\text{and}\quad\beta:=3y\,.$

Observe that, in this new notation, the weights of the $\mathbb{T}$ action at
$p_{0}$ are given by

$3x=-3\alpha+\beta,\;\;3y=\beta,\;\;2x+y=-2\alpha+\beta,\;\;x+2y=-\alpha+\beta\,.$

|  |  |  |  | $c_{1}(P)=-\psi(P)$
---|---|---|---|---|---
$p_{0}$ |  $-3\alpha+\beta$, |  $-2\alpha+\beta$, |  $-\alpha+\beta$, | $\beta$ | $-6\alpha+4\beta$
$p_{1}$ |  $3\alpha-\beta$, |  $\alpha$, |  $-\alpha+\beta$, | $\beta$ | $3\alpha+\beta$
$p_{2}$ |  $-3\alpha+\beta$, |  $-\alpha$, |  $-2\alpha+\beta$, | $-\beta$ | $-6\alpha+\beta$
$p^{\prime}_{2}$ |  $\alpha-\beta$, |  $2\alpha-\beta$, |  $\alpha$, | $2\alpha$ | $6\alpha-2\beta$