$p_{3}$ |  $3\alpha-\beta$, |  $-\alpha$, |  $\alpha$, | $-\beta$ | $3\alpha-2\beta$
$p_{4}$ |  $\alpha-\beta$, |  $-2\alpha$, |  $2\alpha-\beta$, | $-\alpha$ | $-2\beta$
Table 5.4. Weights at the fixed points of the effective $\mathbb{T}$ action on
$W$ and the corresponding value of $c_{1}^{\mathbb{T}}$.

In what follows we prove the existence of canonical classes with respect to a
certain choice of generic $\xi\in\mathfrak{t}$ (see Section 5).

Choose a generic $\xi$ in the cone determined by the inequalities
$\langle\beta,\xi\rangle>0$ and $\langle\alpha,\xi\rangle>0$, and so that the
$\xi$-component of the moment map $\varphi:=\psi^{\xi}$ satisfies $p_{2}\prec
p^{\prime}_{2}$: It can be easily checked, that this means choosing a $\xi$ in
the above cone and on the right of the line through $p_{0}$ with direction
$x+4y$. These choices, as well as the corresponding orientation on the
multigraph describing $W$, is illustrated in Figure 5.7.

$\xi$$(0,0)$$y$$x$$p_{0}$$p_{1}$$p_{2}$$p_{2}^{\prime}$$p_{3}$$p_{4}$$-3\alpha+\beta$$\alpha$$-2\alpha+\beta$$-\alpha+\beta$$\beta$
Figure 5.7. Oriented multigraph describing the action on $W$.

####  5.2.1. The canonical class $\tau_{1}\in
H^{2}_{\mathbb{T}}(W;\mathbb{Z})$:

Using Lemma 5.1 and the values of $c_{1}^{\mathbb{T}}$ in Table 5.4, we obtain
that the canonical class $\tau_{1}$ of $W$ at $p_{1}$ (with respect to the
chosen $\xi$) is given by

(5.9) $\tau_{1}=\frac{-6\alpha+4\beta-c_{1}^{\mathbb{T}}}{3}\,.$

Its values at all fixed points can therefore be computed from Table 5.4 and
can be found in Table 5.5.

| $\tau_{1}$ | $\tau_{2}$ | $\tau^{\prime}_{2}$ | $\tau_{3}$
---|---|---|---|---
$p_{0}$ | 0 | 0 | 0 | 0
$p_{1}$ | $-3\alpha+\beta$ | 0 | 0 | 0
$p_{2}$ | $\beta$ | $\alpha\beta$ | 0 | 0
$p^{\prime}_{2}$ | $-4\alpha+2\beta$ | $0$ | $(-2\alpha+\beta)(-\alpha+\beta)$ | 0
$p_{3}$ | $-3\alpha+2\beta$ | $\alpha\beta$ | $(-3\alpha+\beta)\beta$ | $\alpha\beta(-3\alpha+\beta)$
$p_{4}$ | $-2\alpha+2\beta$ | $2\alpha(-\alpha+\beta)$ | $(-2\alpha+\beta)(-\alpha+\beta)$ | $2\alpha(-2\alpha+\beta)(-\alpha+\beta)$
Table 5.5.  Restrictions to the fixed points of the “non-trivial” canonical
classes on $W$.

####  5.2.2. The canonical classes $\tau_{2}$ and $\tau^{\prime}_{2}$ in
$H^{4}_{\mathbb{T}}(W;\mathbb{Z})$:

Consider Kirwan classes $\widetilde{\tau}_{2}$ at $p_{2}$ and
$\widetilde{\tau}^{\prime}_{2}$ at $p^{\prime}_{2}$, which are elements of
$H_{\mathbb{T}}^{4}(W;\mathbb{Z})$. Since $\xi$ is chosen so that $p_{2}\prec
p^{\prime}_{2}$, by the properties of the Kirwan classes we have:

(5.10)
$\begin{cases}\widetilde{\tau}_{2}(p_{2})=\displaystyle\Lambda_{p_{2}}^{-}\\\
\widetilde{\tau}_{2}(p^{\prime}_{2})\text{ is not necessarily
zero}\end{cases}\quad\begin{cases}\widetilde{\tau}^{\prime}_{2}(p^{\prime}_{2})=\displaystyle\Lambda_{p^{\prime}_{2}}^{-}\\\
\widetilde{\tau}^{\prime}_{2}(p_{2})=0\,.\end{cases}$

Since the only points which, in the ordering induced by $\varphi$, lie above
$p^{\prime}_{2}$ are $p_{3}$ and $p_{4}$, we conclude that
$\widetilde{\tau}^{\prime}_{2}=:\tau^{\prime}_{2}$ is the canonical class at
$p^{\prime}_{2}$. What we need to prove is that, given a Kirwan class
$\widetilde{\tau}_{2}$, we can modify it so that it is zero at
$p^{\prime}_{2}$, obtaining the canonical class $\tau_{2}$.

As $\widetilde{\tau}_{2}$ is a Kirwan class, $p_{0}\prec p_{2}$ and
$p_{1}\prec p_{2}$, we have that
$\widetilde{\tau}_{2}(p_{0})=\widetilde{\tau}_{2}(p_{1})=0$. At
$p^{\prime}_{2}$, there is only one weight of the $\mathbb{T}$ action multiple
of $2\alpha-\beta$ (namely $2\alpha-\beta$), and only one weight multiple of
$\alpha-\beta$ (namely $\alpha-\beta$, see Figure 5.7). This implies that the
connected components of the isotropy submanifolds $M^{2\alpha-\beta}$ and
$M^{\alpha-\beta}$ containing $p^{\prime}_{2}$ are spheres, which we call
respectively $S_{0}^{2}$ and $S_{1}^{2}$. The induced action of $\mathbb{T}$
on $S^{2}_{0}$ is Hamiltonian and has exactly two fixed points,
$p^{\prime}_{2}$ and $q$. Since the image of the moment map for the
$\mathbb{T}$ action on $S^{2}_{0}$ must be on the line $p^{\prime}_{2}+\langle
2\alpha-\beta\rangle$, it is easy to see from Figure 5.7 that $q=p_{0}$. For
degree reasons, the integral $\int_{S^{2}_{0}}\widetilde{\tau}_{2}$ must be a
polynomial of degree one in $\mathbb{Z}[\alpha,\beta]$. Therefore the ABBV
formula applied to the submanifold $S^{2}_{0}$, together with the fact that
$\widetilde{\tau}_{2}(p_{0})=0$, gives

$\widetilde{\tau}_{2}(p^{\prime}_{2})=a(-2\alpha+\beta)\,\quad\text{for some
}a\in\mathbb{Z}[\alpha,\beta]\,.$

By repeating the same argument for the invariant sphere $S^{2}_{1}$, we see
that the $\mathbb{T}$ fixed points on $S^{2}_{1}$ are $p^{\prime}_{2}$ and
$p_{1}$, and that there exists an element $b$ of degree one
$\mathbb{Z}[\alpha,\beta]$ such that

$\widetilde{\tau}_{2}(p^{\prime}_{2})=b(-\alpha+\beta)\,.$

Since $-2\alpha+\beta$ and $-\alpha+\beta$ are coprime, if regarded as
polynomials in $\mathbb{Z}[\alpha,\beta]$, from the above displayed equations
we deduce that there exists $k\in\mathbb{Z}$ such that

$\widetilde{\tau}_{2}(p^{\prime}_{2})=k(-2\alpha+\beta)(-\alpha+\beta)=k\Lambda_{p^{\prime}_{2}}^{-}\,.$

It is now easy to see that

$\tau_{2}:=\widetilde{\tau}_{2}-k\widetilde{\tau}^{\prime}_{2}$

is the canonical class at $p_{2}$. Our aim is to compute the restriction of
$\tau_{2}$ at all the fixed points of $W$. Since $\tau_{2}$ is a canonical
class, the values that we already know are:

(5.11)
$\tau_{2}(p_{0})=\tau_{2}(p_{1})=\tau_{2}(p^{\prime}_{2})=0\quad\text{and}\quad\tau_{2}(p_{2})=\Lambda_{p_{2}}^{-}=\alpha\beta\,.$

What is left to compute is $\tau_{2}(p_{3})$ and $\tau_{2}(p_{4})$ . Consider
the connected component of $M^{\beta}$ containing $p_{3}$. This is an
invariant sphere in $M$ and therefore, for the same argument above, since
$\tau_{2}(p_{1})=0$, we must have that $\tau_{2}(p_{3})$ is a multiple of
$\beta$. Now consider the connected component of the isotropy submanifold
$M^{\alpha}$ containing $p^{\prime}_{2}$. For arguments similar to those
above, one can see that the only $\mathbb{T}$ fixed points on this component
are $p^{\prime}_{2},p_{3}$ and $p_{4}$. Since the $\mathbb{T}$ action on
$M^{\alpha}$ has three fixed points, it must be equivariantly symplectomorphic
to $\mathbb{C}P^{2}$ with suitably rescaled Fubini-Study form and standard
$S^{1}$-action (see [30]). (Here the equivariance is with respect to the
residual $\mathbb{T}/H_{\alpha}$ action). In this $\mathbb{C}P^{2}$, consider
the two invariant spheres containing $p^{\prime}_{2}$ and $p_{3}$, and
$p^{\prime}_{2}$ and $p_{4}$. Applying arguments similar to those above, as
well as the fact that $\tau_{2}(p^{\prime}_{2})=0$, we see that
$\tau_{2}(p_{3})$ and $\tau_{2}(p_{4})$ must be multiples of $\alpha$. Since
$\alpha$ and $\beta$ are coprime when regarded as polynomials in
$\mathbb{Z}[\alpha,\beta]$, we obtain that there exists $m_{1}\in\mathbb{Z}$
such that

$\tau_{2}(p_{3})=m_{1}\,\alpha\,\beta\,.$

Arguing as above for the connected component of the isotropy submanifold
$M^{-\alpha+\beta}$ containing $p_{0}$ and $p_{4}$, we conclude that there
exists $m_{2}\in\mathbb{Z}$ such that

$\tau_{2}(p_{4})=m_{2}\,\alpha\,(-\alpha+\beta)\,.$

What is left to do is to determine the integers $m_{1}$ and $m_{2}$. In order
to do so, we use two conditions, namely: Consider the 4-dimensional isotropy
submanifold $M^{\alpha}$ containing $p^{\prime}_{2},p_{3}$ and $p_{4}$, and
the whole manifold $W$. Then, by degree reasons, we must have

(5.12)
$\int_{M^{\alpha}}\tau_{2}\in\mathbb{Z}\,\quad\text{and}\quad\int_{W}\tau_{2}=0\,.$

For both equations we use the ABBV formula, which for the first integral gives

$\int_{M^{\alpha}}\tau_{2}=\frac{m_{1}\alpha\beta}{\alpha(-\alpha)}+\frac{m_{2}\alpha(-\alpha+\beta)}{2\alpha^{2}}\in\mathbb{Z}\,.$

This is satisfied if and only if $m_{2}=2m_{1}$ . The second condition in
(5.12), together with the ABBV formula, gives that

$\int_{W}\tau_{2}=\frac{1}{(-2\alpha+\beta)(-3\alpha+\beta)}+\frac{m_{1}\alpha\beta}{\alpha(-\alpha)(-3\alpha+\beta)\beta}+\frac{2m_{1}\alpha(-\alpha+\beta)}{2\alpha^{2}(-2\alpha+\beta)(-\alpha+\beta)}=0$

which is satisfied if and only if $m_{1}=1$. This completes the computation of
the values of $\tau_{2}$ at all the fixed points, which are summarized in
Table 5.5.

What is left to compute are the values of the canonical class
$\tau^{\prime}_{2}$ at the fixed points. We notice that the $\mathbb{T}$
action on the connected component $N$ of $M^{\alpha}$ containing
$p^{\prime}_{2}$ has fixed points $p^{\prime}_{2},p_{3}$ and $p_{4}$, which
are exactly the fixed points at which the canonical class $\tau^{\prime}_{2}$
can be nonzero. Therefore the equivariant Poincaré dual to $N$ in $W$ is the
canonical class at $p^{\prime}_{2}$, and its value at a fixed point $q\in N$
is exactly the product of the weights in the normal bundle of $N$ at $q$ in
$W$, giving exactly the values described in Table 5.5.

####  5.2.3. The canonical class $\tau_{3}\in
H^{6}_{\mathbb{T}}(W;\mathbb{Z})$:

The only fixed points at which $\tau_{3}$ is not zero are $p_{3}$ and $p_{4}$.
By definition of canonical class we have
$\tau_{3}(p_{3})=\alpha\,\beta(-3\alpha+\beta)$. To find the value of
$\tau_{3}$ at $p_{4}$, we first observe that, applying the similar arguments
as above, one obtains easily that there exists a polynomial
$a\in\mathbb{Z}[\alpha,\beta]$ such that
$\tau_{3}(p_{4})=a(-2\alpha+\beta)(-\alpha+\beta).$ Using that, for degree
reasons, $\int_{W}\tau_{3}=0$ and the ABBV formula to compute the last
integral, leads to $a=2\alpha$.

We end this section with the following

######  Theorem 5.8.

. Let $W$ be the quintic del Pezzo fourfold with the Hamiltonian
$\mathbb{T}$-action described in Theorem 5.6 and page 5.2. Let $\alpha,\beta$
be a $\mathbb{Z}$-basis of the dual lattice $\widetilde{\ell}^{*}$. Then there
exists a basis of the equivariant cohomology ring
$H_{\mathbb{T}}^{*}(W;\mathbb{Z})$ given by canonical classes
$\\{\tau_{i}\\}_{i=0}^{4}\cup\\{\tau^{\prime}_{2}\\}$, satisfying the
following properties and relations:
For each $i=0,\ldots,4$, $\tau_{i}\in H_{\mathbb{T}}^{2i}(W;\mathbb{Z})$,
$\tau^{\prime}_{2}\in H_{\mathbb{T}}^{4}(W;\mathbb{Z})$ and

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

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

(5.14) $\displaystyle\begin{aligned}
&c_{1}^{\mathbb{T}}=-3\tau_{1}+(-6\alpha+4\beta).\\\
&c_{2}^{\mathbb{T}}=13\tau_{2}+9\tau_{2}^{\prime}+(4a-7b)\tau_{1}+(11\alpha^{2}-18\alpha\beta+6\beta^{2}).\\\
&c_{3}^{\mathbb{T}}=-16\tau_{3}+9(2\beta-3\alpha)\tau_{2}+(7\alpha+6\beta)\tau_{2}^{\prime}-(\alpha-5\beta)(\alpha-\beta)\tau_{1}+(-6\alpha^{2}+22\alpha^{2}\beta\\\
&-18\alpha\beta^{2}+4\beta^{3}).\\\
&c_{4}^{\mathbb{T}}=6\tau_{4}+(-4b)\tau_{3}+(12\alpha^{2}-15\alpha\beta+5\beta^{2})\tau_{2}+(2\alpha^{2}+\beta\alpha+\beta^{2})\tau_{2}^{\prime}+\beta(\beta-\alpha)(\alpha-\beta)\tau_{1}\\\
&+(-3\alpha+\beta)(-2\alpha+\beta)(-\alpha+\beta)\beta.\\\
&\;\;\;\;\;\;\;\;\\\ \end{aligned}$

###### Proof.

The proof of this theorem follows, mutatis mutandis, that of Theorem 5.5. ∎

Corollary 1.2 is a consequence of Theorem 5.8 and the procedure explained in
Section 2.2.1.

###  5.3. Index 4: The Grassmannian $Q=\text{Gr}(2,4)$ and its equivariant
cohomology ring

Let $Q$ be the Grassmannian of complex 2-planes in $\mathbb{C}^{4}$. It is
well-known that $Q$ can be realized as a degenerate coadjoint orbit of
$SU(4)$, endowed with the action of a three-dimensional torus $T$, with 6
fixed points. For details on how this construction works, as well as for the
computation of the weights at the fixed points, we refer the reader to [16,
Section 5.3.1], in particular to Example 5.25 (2), and the references therein.
The Grassmannian $Q$ inherits a symplectic form preserved by the torus action,
with moment map $\psi\colon Q\to\mathfrak{t}^{*}$, the dual of the Lie algebra
of the torus $T$. The symplectic form can be chosen to be monotone and to
satisfy $c_{1}=[\omega]$ (see Lemma 5.2 as well as [16, Proposition 5.24] for
a combinatorial proof). Using the notation in [16] we have the following: Let
$\alpha=x_{1}-x_{2},\beta=x_{2}-x_{3}$ and $\gamma=x_{3}-x_{4}$ be a choice of
simple roots, which are a basis of $\mathfrak{t}^{*}$, and choose
$\xi\in\mathfrak{t}$ so that $(x_{i}-x_{j})(\xi)>0$ for all $1\leq i<j\leq 4$,
for instance $\xi=\alpha^{*}+\beta^{*}+\gamma^{*}$, where
$\alpha^{*},\beta^{*},\gamma^{*}$ is a basis of $\mathfrak{t}$ dual to
$\alpha,\beta,\gamma$. The $T$ action on $Q$ is a GKM action with GKM graph
given in Figure 5.8. Moreover, the weights at each fixed points and the values
of the first Chern class are listed in Table 5.6.

|  |  |  |  | $c_{1}^{T}(q)=-\psi(q)$
---|---|---|---|---|---
$q_{0}$ |  $\beta$, |  $\beta+\gamma$, |  $\alpha+\beta+\gamma$, | $\alpha+\beta$ | $2(\alpha+2\beta+\gamma)$
$q_{1}$ |  $-\beta$, |  $\alpha$, |  $\gamma$, | $\alpha+\beta+\gamma$ | $2(\alpha+\gamma)$
$q_{2}$ |  $-\gamma$, |  $-\beta-\gamma$, |  $\alpha$, | $\alpha+\beta$ | $2(\alpha-\gamma)$
$q^{\prime}_{2}$ |  $-\alpha$, |  $-\alpha-\beta$, |  $\beta+\gamma$, | $\gamma$ | $-2(\alpha-\gamma)$
$q_{3}$ |  $-\alpha-\beta-\gamma$, |  $-\alpha$, |  $-\gamma$, | $\beta$ | $-2(\alpha+\gamma)$
$q_{4}$ |  $-\alpha-\beta-\gamma$, |  $-\beta-\gamma$, |  $-\alpha-\beta$, | $-\beta$ | $-2(\alpha+2\beta+\gamma)$
Table 5.6.  Weights at the fixed points of $T$ action on $Q$ and the
corresponding value of $c_{1}^{T}$.

Even if the equivariant cohomology ring of $Q$ is well-known, for completeness
we carry on computations similar to those in the previous sections. The first
observation is that canonical classes exist, as the GKM graph of $Q$ is index
increasing (see [18, Theorem 1.6]). Therefore we compute their restrictions to
the fixed point set which, in turns, determine the (equivariant) structure
constants.

Let $\tau_{i}\in H^{2i}(Q;\mathbb{Z})$ be the canonical class at $q_{i}$, for
all $i=0,\ldots,4$ and $\tau^{\prime}_{2}\in H^{4}(Q;\mathbb{Z})$ the
canonical class at $q^{\prime}_{2}$.

By Lemma 5.1, $\tau_{1}$ is given by

$\tau_{1}=\frac{2\alpha+4\beta+2\gamma-c_{1}^{T}}{4}\,,$

and its values are as in Table 5.7

| $\tau_{1}$ | $\tau_{2}$ | $\tau^{\prime}_{2}$ | $\tau_{3}$
---|---|---|---|---
$q_{0}$ | 0 | 0 | 0 | 0
$q_{1}$ | $\beta$ | 0 | 0 | 0
$q_{2}$ | $\beta+\gamma$ | $\gamma(\beta+\gamma)$ | 0 | 0
$q^{\prime}_{2}$ | $\alpha+\beta$ | $0$ | $\alpha(\alpha+\beta)$ | 0
$q_{3}$ | $\alpha+\beta+\gamma$ | $\gamma(\alpha+\beta+\gamma)$ | $\alpha(\alpha+\beta+\gamma)$ | $\alpha\gamma(\alpha+\beta+\gamma)$
$q_{4}$ | $\alpha+2\beta+\gamma$ | $(\alpha+\beta+\gamma)(\beta+\gamma)$ | $(\alpha+\beta)(\alpha+\beta+\gamma)$ | $(\alpha+\beta)(\beta+\gamma)(\alpha+\beta+\gamma)$
Table 5.7.  Restrictions to the fixed points of the “non-trivial” canonical
classes on $Q$.

Now we compute the restrictions to the fixed points of the canonical classes
$\tau_{2}$ and $\tau^{\prime}_{2}$. In order to do this, we follow the same
procedure illustrated in Subsection 5.1 for computing canonical classes on
$V$. As for $\tau_{2}$, its value at $q_{2}$ is given –by definition– by
$\gamma(\beta+\gamma)$. In order to compute $\tau_{2}(q_{3})$ it is sufficient
to compute $\Theta(q_{2},q_{3})$ which, with the choice of $\xi$ given at the
beginning of this section, is easily seen to be 1. Therefore [18, (4.1)] gives
$\tau_{2}(q_{3})=\gamma(\alpha+\beta+\gamma)$. In order to compute
$\tau_{2}(q_{4})$ we see that the only path in the path-formula of [18,
Theorem 1.6] from $q_{2}$ to $q_{4}$ is the one containing the edges
$(q_{2},q_{3})$ and $(q_{3},q_{4})$. Moreover, $\Theta(q_{3},q_{4})=1$;
therefore the formula [18, (1.3)] gives that
$\tau_{2}(q_{4})=(\alpha+\beta+\gamma)(\beta+\gamma)$. Notice that there is no
path in the path formula from $q_{2}$ to $q^{\prime}_{2}$, so by [18, Theorem
1.6] it follows that $\tau_{2}(q^{\prime}_{2})=0$.

Similar computations can be carried out for $\tau^{\prime}_{2}$, and its
values at the fixed points are written in Table 5.7.

As for $\tau_{3}$, it is sufficient to observe that the equivariant Poincarè
dual to the $T$-invariant sphere connecting $q_{3}$ to $q_{4}$ satisfies the
defining properties of the canonical class at $q_{3}$. Therefore its values
can be immediately computed, and are written in Table 5.7.

Arguing as in the previous examples, we use Table 5.7 and the procedure
described in Section 2.2.1 to compute the equivariant structure constants,
obtaining the following:

######  Theorem 5.9.

. Let $Q$ be the Grassmannian with the Hamiltonian $T$-action described in
Section 5.3. Let $\alpha,\beta,\gamma$ be a $\mathbb{Z}$-basis of the dual
lattice $\widetilde{\ell}^{*}$. Then there exists a basis of the equivariant
cohomology ring $H_{T}^{*}(Q;\mathbb{Z})$ given by canonical classes
$\\{\tau_{i}\\}_{i=0}^{4}\cup\\{\tau^{\prime}_{2}\\}$, satisfying the
following properties and relations:
For each $i=0,\ldots,4$, $\tau_{i}\in H_{T}^{2i}(Q;\mathbb{Z})$,
$\tau^{\prime}_{2}\in H_{T}^{4}(Q;\mathbb{Z})$ and

(5.15) $\displaystyle\begin{aligned}
&\tau_{1}^{2}=\beta\tau_{1}+\tau_{2}+\tau^{\prime}_{2},&\tau_{1}\tau_{2}=(\beta+\gamma)\tau_{2}+\tau_{3}\\\
&\tau_{1}\tau^{\prime}_{2}=(\alpha+\beta)\tau^{\prime}_{2}+\tau_{3},&\tau_{1}\tau_{3}=(\alpha+\beta+\gamma)\tau_{3}+\tau_{4}\\\
&\tau_{2}^{2}=\gamma(\beta+\gamma)\tau_{2}+\gamma\,\tau_{3}+\tau_{4},&(\tau^{\prime}_{2})^{2}=\alpha(\alpha+\beta)\tau^{\prime}_{2}+\alpha\tau_{3}+\tau_{4}\\\
&\tau_{2}\tau^{\prime}_{2}=(\alpha+\beta+\gamma)\tau_{3}\,.\\\ \end{aligned}$

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

(5.16) $\displaystyle\begin{aligned}
&c_{1}^{T}=-4\tau_{1}+2(\alpha+2\beta+\gamma).\\\
&c_{2}^{T}=7\tau_{2}^{\prime}+7\tau_{2}-(7\alpha+7\beta+7\gamma)\tau_{1}+(\alpha^{2}+6\alpha\beta+3\alpha\gamma+6\beta^{2}+6\beta\gamma+\gamma^{2}).\\\
&c_{3}^{T}=-12\tau_{3}+(9\alpha+6\beta+3\gamma)\tau_{2}+(3\alpha+6\beta+9\gamma)\tau_{2}^{\prime}+(-3\alpha^{2}-7\alpha\beta-8\alpha\gamma-4\beta^{2}-7\beta\gamma-3\gamma^{2})\tau_{1}\\\
&+(\alpha+2\beta+\gamma)(2\alpha\beta+\alpha\gamma+2\beta^{2}+2\beta\gamma).\\\
&c_{4}^{T}=6\tau_{4}-3(\alpha+2\beta+\gamma)\tau_{3}+(3\alpha^{2}+4\alpha\beta+2\alpha\gamma+\beta^{2}+\beta\gamma)\tau_{2}+(\alpha\beta+2\alpha\gamma+\beta^{2}+4\beta\gamma+3\gamma^{2})\tau_{2}^{\prime}\\\
&-(\alpha+\beta+\gamma)(\alpha\beta+2\alpha\gamma+\beta^{2}+\beta\gamma)\tau_{1}+\beta(\beta+\gamma)(\alpha+\beta+\gamma)(\alpha+\beta).\\\
&\;\;\;\;\;\;\;\;\\\ \end{aligned}$

###### Proof.

The proof of this theorem follows, mutatis mutandis, that of Theorem 5.5. ∎

The following corollary is a consequence of Theorem 5.9 and the procedure
explained in Section 2.2.1.

We conclude this section with the following corollary, which is a consequence
of Theorem 5.9 and gives the well-known structure constants of the cohomology
ring of $Q$ with respect to the basis given by the restriction to ordinary
cohomology of the canonical classes, which indeed (modulo sign) correspond to
the ordinary Schubert classes (see [42, Proposition 6.5]):

######  Corollary 5.10.

Let $Q$ be the Grassmannian of complex planes in $\mathbb{C}^{4}$. Then there
exists a basis $\\{\zeta_{i}\\}_{i=0,\ldots,4}\cup\\{\zeta^{\prime}_{2}\\}$ of
the cohomology ring $H^{*}(Q;\mathbb{Z})$ satisfying the following properties
and relations:
For each $i=0,\ldots,4$, $\zeta_{i}\in H^{2i}(Q;\mathbb{Z})$,
$\zeta^{\prime}_{2}\in H^{4}(Q;\mathbb{Z})$ and

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

Moreover the Chern classes are given by

$c_{1}=-4\,\zeta_{1},\quad c_{2}=7\,\zeta_{2}+7\,\zeta_{2}^{\prime},\quad
c_{3}=-12\,\zeta_{3}\quad\text{and}\quad c_{4}=6\,\zeta_{4}.$

$q_{0}$$q_{4}$$q_{3}$$q_{1}$$q_{2}$$q_{2}^{\prime}$$\beta+\gamma$$\alpha+\beta+\gamma$$\beta$$\alpha+\beta$$\alpha$$\gamma$$\gamma$$\alpha$$\beta$$\beta+\gamma$$\alpha+\beta+\gamma$$\alpha+\beta$
Figure 5.8. The oriented GKM graph of $Q$.

###  5.4. Torus actions on $\mathbb{CP}^{2}\times\mathbb{CP}^{2}$

We conclude this section with the following well-known result, which will be
useful in Section 7.2.2

######  Theorem 5.11.

Consider the Fano $4$-fold
$\left(X=\mathbb{CP}^{2}\times\mathbb{CP}^{2},\omega=3\,\omega_{0}\oplus
3\,\omega_{0}\right)$, where $\omega_{0}$ is the Fubini-study form on
$\mathbb{CP}^{2}$. Then $c_{1}(X)=[\omega]$ and the weights for the standard
(toric) $T^{4}$-action on $X$ at the $9$ fixed points are

$\displaystyle\\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\\},\\{(1,0,0,0),(0,1,0,0),(0,0,-1,-1),(0,0,0,1)\\},$
$\displaystyle\\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,-1,-1)\\},\\{(-1,-1,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\\}$
$\displaystyle\\{(-1,-1,0,0),(0,1,0,0),(0,0,-1,-1),(0,0,0,1)\\},\\{(-1,-1,0,0),(0,1,0,0),(0,0,1,0),(0,0,-1,-1)\\}$
$\displaystyle\\{(1,0,0,0),(-1,-1,0,0),(0,0,1,0),(0,0,0,1)\\},\\{(1,0,0,0),(-1,-1,0,0),(0,0,-1,-1),(0,0,0,1)\\},$
$\displaystyle\\{(1,0,0,0),(-1,-1,0,0),(0,0,1,0),(0,0,-1,-1)\\}.$

The moment polytope is $\Delta^{2}\subset\mathbb{R}^{4}$ where
$\Delta\subset\mathbb{R}^{2}$ is the triangle with vertices
$(0,0),(3,0),(0,3)$.

##  6\. The algorithm

Let $M$ be a closed symplectic manifold of dimension $2n$ equipped with a
circle action with isolated fixed points. In this case the weights of the
action at a fixed point $p$ are nonzero integers $w_{1},\ldots,w_{n}$. Let
$W^{+}$ and $W^{-}$ be the multisets of positive and negative weights at all
fixed points of $M$

$W^{+}=\displaystyle{\bigsqcup_{p_{i}\in M^{S^{1}}}}\\{w_{ik}\mid
w_{ik}>0\\}\quad\text{and}\quad W^{-}=\displaystyle{\bigsqcup_{p_{i}\in
M^{S^{1}}}}\\{w_{ik}\mid w_{ik}<0\\},$

where $\bigsqcup$ denotes the disjoint union of multisets (cf. Remark 4.20).

We have $W_{+}=-W_{-}$ and so we can always choose a bijection $g:W^{+}\to
W^{-}$ taking each $w_{ik}$ to some $w_{jl}$ with $w_{ik}=-w_{jl}$, leading to
a multigraph $(V,E)$ as in Definition 4.3 (see also Remark 4.20).

We can choose a direction on the multigraph as in Remark 4.20. Moreover, we
will label the (directed) edges with a map $\mathsf{w}:E\to\mathbb{Z}_{>0}$
given by $\mathsf{w}(e_{ik})=\lvert w_{ik}\rvert$ and work with labeled
multigraphs $(V,E,\mathsf{w})$.

###### Remark 6.1

Since we are working with a Hamiltonian $S^{1}$-action on a closed symplectic
manifold $M$, we can take an $S^{1}$-invariant metric and consider the
gradient of the Hamiltonian function with respect to this metric. Then the
$\mathbb{R}$-action associated to the flow of this gradient commutes with the
$S^{1}$-action, yielding a $\mathbb{C}^{*}=(S^{1}\times\mathbb{R}^{+})$-action
on $M$. The closure of the $\mathbb{C}^{*}$-orbit through a point $p\in
M\setminus M^{S^{1}}$ is an embedded, symplectic ($S^{1}$-invariant)
$2$-sphere, not necessarily smooth at the poles. In some cases, when two fixed
points $p$ and $q$ are poles of one of these gradient spheres, we can pick one
sphere that is smoothly embedded and contains the two fixed points. These
special spheres, which are symplectic, $S^{1}$-invariant spheres fixed by some
subgroup $\mathbb{Z}_{k}$, with $k\in\mathbb{Z}_{>0}$, (where $S^{1}$ acts
with fixed points $p$ and $q$), are represented by edges in one of the
possible multigraphs describing $M$ and the $S^{1}$-action. These edges are
labeled by the integers $k$. Some of the multigraphs describing $M$ correspond
to the possible arrangements of gradient spheres for a circle action on $M$
with the same multisets of weights $W^{+}$ and $W^{-}$ (see [1, Section 3] for
more details on gradient spheres).

Jang and Tolman [28, Lemma 5] showed that, given a circle action on a closed
almost complex manifold $M$ with isolated fixed points, there always exists a
multigraph describing $M$ obtained from $W^{+}$ and $W^{-}$ with no self-
loops. Therefore we will consider the family $\mathcal{W}$ of all directed
multigraphs determined by $W^{+}$ and $W^{-}$ that have no self-loops.

Given a multigraph $\Gamma=(V,E)$ describing $M$, we define its magnitude
function $\mathsf{m}:E\to\mathbb{Z}$ as the evaluation of $c_{1}^{S^{1}}$ on
the edges of $\Gamma$,

$\mathsf{m}(e)=c_{1}^{S^{1}}[e]$

(cf. Definition 4.10), where $c_{1}^{S^{1}}$ is the first equivariant Chern
class.

###### Remark 6.2

Note that, for each fixed point $p\in M^{S^{1}}$, we have

$c_{1}^{S^{1}}(p)=\Big{(}\sum_{j=1}^{n}w_{j}\Big{)}x=\Sigma(p)\,x,$

where $\\{w_{1},\ldots,w_{n}\\}$ is the multiset of weights at $p$ and $x$ is
a degree-$2$ generator of $H_{S^{1}}({pt},\mathbb{Z})\simeq\mathbb{Z}[x]$.

The sum of the values of the magnitude function over all the edges of a
multigraph $\Gamma=(V,E)$ in $\mathcal{W}$ does not depend on $\Gamma$ (i.e.
it is independent of the choice of bijection between $W^{-}$ and $W^{+}$).

######  Proposition 6.3.

[15, Proposition 4.6] Let $(M,\omega)$ be a closed symplectic manifold
equipped with a Hamiltonian $S^{1}$ action with isolated fixed points and let
$\mathcal{W}$ be the family of directed multigraphs describing $M$. Then the
magnitude $\mathsf{m}\colon E\to\mathbb{Z}$ associated to a multigraph
$\Gamma=(V,E)\in\mathcal{W}$ satisfies

(6.1) $\sum_{e\in
E}\mathsf{m}(e)=c_{1}c_{n-1}[M]=\sum_{i=0}^{n}b_{2i}(M)\left(6\,i(i-1)+\frac{5n-3n^{2}}{2}\right),$

where $b_{2i}(M)$ is the $2i$-th Betti number of $M$.

Given a labeled directed multigraph $\Gamma=(V,E,\mathsf{w})$ we define the
map $\delta\colon V\times E\to\\{-1,0,1\\}$ given by

$\delta(p,e)=\left\\{\begin{array}[]{rl}1,&\text{if $i(e)=p$}\\\ -1,&\text{if
$t(e)=p$}\\\ 0,&\text{otherwise,}\end{array}\right.$

where $i(e)$ and $t(e)$ are the initial and terminal points of $e$. It is easy
to check that, fixing an edge $\tilde{e}\in E$, we always have

(6.2) $\sum_{e\in
E}\big{(}\delta\left(\mathsf{i}(\tilde{e}),e\right)-\delta\left(\mathsf{t}(\tilde{e}),e\right)\big{)}\mathsf{w}(e)-\mathsf{w}(\tilde{e})\mathsf{m}(\tilde{e})=0.$

Hence, considering an ordering $E=\\{e_{1},\ldots,e_{\lvert E\rvert}\\}$ on
the edge set and the $(\lvert E\rvert\times\lvert E\rvert)$-matrices
$A(\Gamma)=(a_{ij})$ and $\operatorname{diag}(\mathsf{m}(E))$ defined by

$a_{ij}:=\delta\left(\mathsf{i}(e_{i}),e_{j}\right)-\delta\left(\mathsf{t}(e_{i}),e_{j}\right)$

and

$\operatorname{diag}(\mathsf{m}(E)):=\operatorname{diag}\left(\mathsf{m}(e_{1}),\ldots,\mathsf{m}(e_{\lvert
E\rvert})\right),$

we have that the weight vector
$\mathsf{w}(E):=(\mathsf{w}(e_{1}),\ldots,\mathsf{w}(e_{\lvert E\rvert})$ must
be in the kernel of

(6.3) $A(\Gamma)-\operatorname{diag}(\mathsf{m}(E)).$

Therefore, given positive integers $n,N$ and integers $N_{i}\geq 0$ for
$i=0,\ldots,n$ such that $\sum_{i=0}^{n}N_{i}=N$ the possible isotropy weights
for a Hamiltonian $S^{1}$ action on a compact symplectic manifold $M$ of
dimension $n$ with $N$ isolated fixed points such that the number of fixed
points with $i$ negative weights is $N_{i}$ (i.e. such that $b_{2i}(M)=N_{i}$)
must satisfy several linear relations determined by the following algorithm
(see [15] for details):

  1. (1)

Consider all possible multigraphs $\Gamma=(V,E)$ with vertex set
$V=\\{p_{1},\ldots,p_{N}\\}$ and such that the number of vertices which are
the endpoints of $i$ edges is $N_{i}$.

  2. (2)

Determine the matrices $A(\Gamma)$ associated to the multigraphs $\Gamma$.

  3. (3)

Look for possible functions $\mathsf{m}:E\to\mathbb{Z}$ satisfying (6.1) and
such that

(6.4)
$\Big{(}\ker\left(A(\Gamma)-\operatorname{diag}\left(\mathsf{m}(E)\right)\right)\Big{)}\cap\mathbb{Z}_{>0}^{\lvert
E\rvert}\neq\emptyset,$

(these functions will be the possible magnitudes of the multigraphs).

  4. (4)

When they exist, the weight vectors $\mathsf{w}(E)$ associated to the edges of
the multigraphs $\Gamma$ must be in the kernel of the matrices

$A(\Gamma)-\operatorname{diag}(\mathsf{m}(E)).$

If we assume the functions $\mathsf{m}:E\to\mathbb{Z}$ to be positive, they
correspond to partitions of

(6.5)
$\sum_{i=0}^{n}N_{i}\left(6i(i-1)+\frac{5n-3n^{2}}{2}\right)\in\mathbb{Z}$

into $\lvert E\rvert$ positive integers $\mathsf{m}(e)$ and so there is only a
finite number of possibilities for $\mathsf{m}$ and the algorithm ends.

###### Remark 6.4

One cannot expect that, in general, the magnitudes associated to multigraphs
describing manifolds with circle actions are positive. For example, if the
number of fixed points is large and $b_{2i}=N_{i}>1$ for all $i$, then
$c_{1}c_{n-1}[M]$ can be negative (cf. Proposition 6.3). However, if for
example $M$ is monotone, as is the case of $\chi(M)=6$ where
$c_{1}(M)=[\omega]$ (cf. Lemma 4.25), positive magnitudes do exist.

##  7\. Classification Results

###  7.1. Classification results for $b_{1}(M)=1$ and $b_{4}(M)=2$:

We first run Part $1$ of the file
CircleActions$\\_$dim$8\\_$nfpt$6\\_$General.nb to generate the list of all
possible multigraphs $\Gamma=(E,V)$ describing a circle action on a
$8$-dimensional manifold with $6$ fixed points (these multigraphs can be seen
in the file Multigraphs.pdf).

Then we run Part $2$ of this file to obtain the complete list of determinants
of the matrices $A(\Gamma)-\operatorname{diag}(\mathsf{m}(E))$ in (6.3)
(listed in the file Determinants.pdf).

After this, we run the c++ files Partitions-$\\#$.cpp, where $\\#$ denotes the
order number of the multigraph in the list above, to obtain partitions of
$c_{1}c_{3}[M]$ into $12$ positive integers $\mathsf{m}(e)$ (according to the
algorithm described in Section 6), for which there exists a determinant of a
matrix $A(\Gamma_{i})-\operatorname{diag}(\mathsf{m}(E_{i}))$ that is zero.

####  7.1.1. Index 1

If we assume that the index of the manifold $M$ is $k_{0}=1$, the greatest
common divisor of all the $12$ elements of the partitions considered must be
$1$ and the first and last elements of the partitions are always equal to $1$
(cf. Lemma 5.1). The possible partitions can be found in the file
PartitionsIndex1.zip.

We run Part $3$ of the file Index1.nb to sort these partitions according to
the rank of the matrices (6.3), dividing them into two sets: those that
originate singular matrices with maximal rank, and those that originate
matrices of lower rank.

In the first case, we also select the partitions that originate matrices with
null spaces containing vectors with positive components, according to (6.4).
We store the first type of partitions in the files Part$\\#$toprk.txt and the
others in the files Part$\\#$smallrk.txt.

Then, Part $4$ of the file Index1.nb considers the first set of partitions,
producing a list of the corresponding isotropy weights, checking if they
satisfy the polynomial equations in [15, Equation (1.1)] obtained from the
Atiyah-Bott-Berline-Vergne Localization Theorem.

Finally, part $5$ of Index1.nb considers the second set of partitions and

  * •

selects those that originate matrices with null spaces containing vectors with
positive entries;

  * •

produces the list of the corresponding isotropy weights;

  * •

selects those isotropy weights that satisfy several necessary properties such
as

    1. (i)

at each fixed point, the isotropy weights must be coprime integers;

    2. (ii)

at a fixed point $P$ of index $2i$ the first $i$ weights are negative and the
others are positive;

    3. (iii)

if there are three multiple edges between two vertices then the weights
corresponding to these edges must be coprime [15, Lemma 8.2(1) ].

At each step, the resulting lists of isotropy weights are saved in different
files so that it is easy to verify which isotropy weights are discarded with
each test performed.

In the end, the program eliminates all possibilities. Indeed, no potential
multiset of isotropy weights satisfies all the necessary conditions and so we
conclude that there are no possible circle actions in the index-$1$ case.

####  7.1.2. Index 2

When the index is $2$, the greatest common divisor of the $12$ elements of the
partitions considered must be $2$ and the first and last elements of the
partitions are always $2$ (cf. Lemma 5.1). The possible partitions can be
found in the file PartitionsIndex2.zip. Running Part 3 of the file Index2.nb
we sort these partitions according to the rank of the matrices (6.3), dividing
them again in two sets: those that originate singular matrices with maximal
rank, and those that do not. In the first case, we also select the partitions
that originate matrices with null spaces which contain vectors with positive
components. We store the first type of partitions in the files
Part$\\#$toprk.txt and the others in the files Part$\\#$smallrk.txt.

Then, Parts $4$ and $5$ of Index2.nb consider these sets of partitions,
producing a list of the corresponding isotropy weights, checking if they
satisfy the polynomial equations in [15, Equation (1.1)] and the necessary
properties $(i)$ to $(iii)$ listed in Section 7.1.1. The isotropy weights and
the corresponding multigraphs that satisfy all the necessary conditions can be
found in the file ResultsIndex2.pdf and are shown in Figures 7.1 and 7.2.

Figure 7.1. Multigraphs for Index $2$ Figure 7.2. Isotropy weights for Index
$2$

In all cases the Chern numbers obtained from the Isotropy weights in Figure
7.2 are

$\displaystyle c_{1}^{4}[M]=288,\quad
c_{1}^{2}c_{2}[M]=168\quad\text{and}\quad c_{2}^{2}[M]=98.$

The first two multisets of weights give us a two-dimensional family of circle
actions that generate the $T$-action on the Fano-Mukai $4$-fold $V$ described
in Section 5.1.

$\bullet$ The first $2$-dimensional family of weights listed in Figure 7.2
generates the effective $T$-action on the Fano-Mukai $4$-fold $V$. Indeed, if
we consider the basis ${\emph{b}}_{1}=2x+y=(2,1)$ and
${\emph{b}}_{2}=x+y=(1,1)$ of the dual lattice $\ell^{*}$ of $T$, then the
weights for the action of $T$ listed in Table 5.1 become those in Table 7.1.

|  |  |  |
---|---|---|---|---
$q_{0}$ |  $-2\emph{b}_{1}+3\emph{b}_{2}$, |  $-\emph{b}_{1}+2\emph{b}_{2}$, |  $\emph{b}_{2}$, | $\emph{b}_{1}$
$q_{1}$ |  $2\emph{b}_{1}-3\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, |  $\emph{b}_{2}$, | $-\emph{b}_{1}+3\emph{b}_{2}$
$q_{2}$ |  $-\emph{b}_{1}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, |  $-\emph{b}_{1}+2\emph{b}_{2}$, | $-\emph{b}_{1}+3\emph{b}_{2}$
$q^{\prime}_{2}$ |  $\emph{b}_{1}-3\emph{b}_{2}$, |  $\emph{b}_{1}-2\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, | $\emph{b}_{1}$
$q_{3}$ |  $\emph{b}_{1}-3\emph{b}_{2}$, |  $-\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, | $-2\emph{b}_{1}+3\emph{b}_{2}$
$q_{4}$ |  $-\emph{b}_{1}$, |  $-\emph{b}_{2}$, |  $\emph{b}_{1}-2\emph{b}_{2}$, | $2\emph{b}_{1}-3\emph{b}_{2}$
Table 7.1.  Weights for the effective $T$ action on $V$ for the new basis of
$\ell^{*}$.

The multigraphs obtained for these circle actions from the projection of the
$T$-action to a generic subcircle are always as the one on the left of Figure
7.1. Moreover, considering the corresponding moment maps, we obtain the
following ordering of the fixed point set as defined in (2.2)

$q_{0}\prec q_{2}\prec q_{3}\prec q_{1}\prec q_{2}^{\prime}\prec q_{4}.$

Note that this ordering is different from the one chosen in Section 5.1 to
compute the equivariant cohomology for the $T$-action on $V$.

$\bullet$ The second $2$-dimensional family of weights listed in Figure 7.2
with $\text{b[1]},\text{b[2]}\in\mathbb{Z}^{+}$ satisfying
$\text{b[2]}>\text{b[1]}$, $2\text{b[1]}+\text{b[2]}=0\pmod{3}$ and
$\text{b[1]}+2\text{b[2]}=0\pmod{3}$ with

$\gcd\left(\text{b[1]},\text{b[2]},\frac{2\text{b[1]}+\text{b[2]}}{3},\frac{\text{b[1]}+2\text{b[2]}}{3}\right)=1,$

also generates the effective $T$-action on the Fano-Mukai $4$-fold $V$ since,
writing $u=b_{1}+b_{2}\in\mathbb{Z}$ and
$v=\frac{b_{1}+2b_{2}}{3}\in\mathbb{Z}$, we recover the first family of
weights. The multigraphs obtained from the corresponding projections of the
$T$-action are always as the one on the left of Figure 7.1 and, considering
the ordering on the fixed point set given by the associated moment maps, we
obtain that

$q_{1}\prec q_{0}\prec q_{2}^{\prime}\prec q_{2}\prec q_{4}\prec q_{3}.$

###### Remark 7.1

The third multiset of weights is a particular case of the first with $b[1]=1$
and $b[2]=2$. The multigraph for this action (on the right of Figure 7.1) is
different from the one of the previous action only for the edges that are
labeled $1$. These edges correspond to free gradient spheres which we know
depend on the metric considered. Therefore we can still obtain this multigraph
and this muitiset of weights from a subcircle of $T$.

####  7.1.3. Index 3

When the index is $3$, the greatest common divisor of the $12$ elements of the
partitions considered must be $3$ and the first and last elements of the
partitions are always $3$ (cf. Lemma 5.1). The possible partitions can be
found in the file PartitionsIndex3.zip. We run Part 3 of the file Index3.nb to
sort these partitions according to the rank of the matrices (6.3), dividing
them in two sets, stored in Part$\\#$toprk.txt and Part$\\#$smallrk.txt.

Part $4$ considers the first type of partitions producing a list of the
corresponding isotropy weights, checking if they satisfy the polynomial
equations in [15, Equation (1.1)]. Then Part $5$ considers the second type of
partitions, selecting those that originate matrices whose null spaces contain
vectors with positive entries and producing the list of the corresponding
isotropy weights. Just like in the previous cases, it then selects those that
satisfy the necessary conditions $(i)$ to $(iii)$ listed in Section 7.1.1. In
the index-3 case, it further selects the isotropy weights that originate a
possible set of Chern numbers according to Table 3.2.

The isotropy weights that satisfy all the necessary conditions can be found in
the file ResultsIndex3.pdf and are presented in Figure 7.4. The corresponding
multigraphs are shown in Figure 7.3, where the first two multisets of weights
correspond to the multigraph on the left of Figure 7.3 while the other four
possibilities correspond to the multigraph on the right. In all possible cases
the Chern numbers obtained from the weights are

$\displaystyle c_{1}^{4}[M]=405,\quad
c_{1}^{2}c_{2}[M]=198\quad\text{and}\quad c_{2}^{2}[M]=97.$

Figure 7.3. Multigraphs for Index $3$ Figure 7.4. Isotropy weights for Index
$3$

All the multisets of weights obtained give us a two-dimensional family of
circle actions that generate the effective $\mathbb{T}$-action on the quintic
del Pezzo fourfold $W$ described in Section 5.2.

$\bullet$ The first $2$-dimensional family of weights listed in Figure 7.4
with $\text{b[1]},\text{b[2]}\in\mathbb{Z}^{+}$ satisfying
$\text{b[2]}>\text{b[1]}$, $\text{b[1]}=\text{b[2]}\mod 2$ and

$\gcd\left(\text{b[1]},\text{b[2]},\frac{\text{b[1]}+\text{b[2]}}{2},\frac{\text{b[1]}+3\text{b[2]}}{2}\right)=1,$

generates the effective $\mathbb{T}$-action on the quintic del Pezzo fourfold
$W$ since, writing $\hat{\alpha}=\frac{b_{1}+b_{2}}{2}\in\mathbb{Z}$ and
$\hat{\beta}=\frac{b_{1}+3b_{2}}{2}\in\mathbb{Z}$, the $2$-dimensional family
of weights becomes

$\displaystyle\left\\{\\{3\hat{\alpha}-\hat{\beta},\hat{\alpha},-\hat{\alpha}+\hat{\beta},\hat{\beta}\\},\\{-3\hat{\alpha}+\hat{\beta},-2\hat{\alpha}+\hat{\beta},-\hat{\alpha}+\hat{\beta},\hat{\beta}\\},\\{2\hat{\alpha}-\hat{\beta},\hat{\alpha},\hat{\alpha}-\hat{\beta},2\hat{\alpha}\\},\right.$
$\displaystyle\left.\\{3\hat{\alpha}-\hat{\beta},-\hat{\beta},-\hat{\alpha},\hat{\alpha}\\},\\{-3\hat{\alpha}+\hat{\beta},-\hat{\beta},-\hat{\alpha},-2\hat{\alpha}+\hat{\beta}\\},\\{-2\hat{\alpha},-\hat{\alpha},2\hat{\alpha}-\hat{\beta},\hat{\alpha}-\hat{\beta}\\}\right\\}.$

The multigraphs obtained from the projections of the $\mathbb{T}$-action to a
generic subcircle defined by
$\hat{\alpha}\,v_{1}+\hat{\beta}\,v_{2}\in\widetilde{\ell}$ are always as the
one on the left of Figure 7.3 and, considering the ordering on the fixed point
set given by the associated moment maps, we obtain that

$p_{1}\prec p_{0}\prec p_{2}^{\prime}\prec p_{3}\prec p_{2}\prec p_{4}.$

$\bullet$ The second $2$-dimensional family of weights listed in Figure 7.4
also generates the effective $\mathbb{T}$-action on $W$ since, if we consider
the basis ${\emph{b}}_{1}=\alpha-\beta=(-1,-2)$ and
${\emph{b}}_{2}=\alpha=(-1,1)$ of the dual lattice $\tilde{\ell}^{*}$ of
$\mathbb{T}$, the $\mathbb{T}$-weights listed in Table 5.4 become those in
Table 7.2.

$p_{0}$ |  $-\emph{b}_{1}-2\emph{b}_{2}$, |  $-\emph{b}_{1}-\emph{b}_{2}$, |  $-\emph{b}_{1}$, | $-\emph{b}_{1}+\emph{b}_{2}$
---|---|---|---|---
$p_{1}$ |  $\emph{b}_{1}+2\emph{b}_{2}$, |  $\emph{b}_{2}$, |  $-\emph{b}_{1}$, | $-\emph{b}_{1}+\emph{b}_{2}$
$p_{2}$ |  $-\emph{b}_{1}-2\emph{b}_{2}$, |  -$\emph{b}_{2}$, |  $-\emph{b}_{1}-\emph{b}_{2}$, | $\emph{b}_{1}-\emph{b}_{2}$
$p^{\prime}_{2}$ |  $\emph{b}_{1}$, |  $\emph{b}_{1}+\emph{b}_{2}$, |  $\emph{b}_{2}$, | $2\emph{b}_{2}$
$p_{3}$ |  $\emph{b}_{1}+2\emph{b}_{2}$, |  $-\emph{b}_{2}$, |  $\emph{b}_{2}$, | $\emph{b}_{1}-\emph{b}_{2}$
$p_{4}$ |  $\emph{b}_{1}$, |  $-2\emph{b}_{2}$, |  $\emph{b}_{1}+\emph{b}_{2}$, | $-\emph{b}_{2}$
Table 7.2.  Weights for the effective $\mathbb{T}$-action on $W$ for the new
basis of $\tilde{\ell}^{*}$.

The multigraphs obtained for these circle actions from the projection of the
$\mathbb{T}$-action to a generic subcircle are always as the one on the left
of Figure 7.1. Moreover, considering the corresponding moment maps, we obtain
the following ordering of the fixed point set as defined in (2.2)

$p_{2}^{\prime}\prec p_{1}\prec p_{3}\prec p_{4}\prec p_{0}\prec p_{2}.$

$\bullet$ Similarly, the third $2$-dimensional family of weights listed in
Figure 7.4 generates the $\mathbb{T}$-action on $W$ since, if we consider the
basis ${\emph{b}}_{1}=\beta$ and ${\emph{b}}_{2}=-\alpha+\beta$ of the dual
lattice $\tilde{\ell}^{*}$ of $\mathbb{T}$, the weights for the action of
$\mathbb{T}$ listed in Table 5.4 become those in Table 7.3.

$p_{0}$ |  $-2\emph{b}_{1}+3\emph{b}_{2}$, |  $-\emph{b}_{1}+2\emph{b}_{2}$, |  $\emph{b}_{2}$, | $\emph{b}_{1}$
---|---|---|---|---
$p_{1}$ |  $2\emph{b}_{1}-3\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, |  $\emph{b}_{2}$, | $\emph{b}_{1}$
$p_{2}$ |  $-2\emph{b}_{1}+3\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, |  $-\emph{b}_{1}+2\emph{b}_{2}$, | $-\emph{b}_{1}$
$p^{\prime}_{2}$ |  $-\emph{b}_{2}$, |  $\emph{b}_{1}-2\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, | $2\emph{b}_{1}-2\emph{b}_{2}$
$p_{3}$ |  $2\emph{b}_{1}-3\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, | $-\emph{b}_{1}$
$p_{4}$ |  $-\emph{b}_{2}$, |  $-2\emph{b}_{1}+2\emph{b}_{2}$, |  $\emph{b}_{1}-2\emph{b}_{2}$, | $-\emph{b}_{1}+\emph{b}_{2}$
Table 7.3.  Weights for the effective $\mathbb{T}$-action on $W$ for the new
basis of $\tilde{\ell}^{*}$.

The multigraphs obtained for these circle actions from the projection of the
$\mathbb{T}$-action to a generic subcircle are always as the one on the right
of Figure 7.3 and the ordering on the fixed point set obtained from the
corresponding moment maps is

$p_{0}\prec p_{2}\prec p_{4}\prec p_{1}\prec p_{3}\prec p_{2}^{\prime}.$

$\bullet$ The forth $2$-dimensional family of weights in Figure 7.4 with
$\text{b[1]},\text{b[2]}\in\mathbb{Z}^{+}$ such that
$\text{b[2]}>\text{b[1]}$, $\text{b[1]}=\text{b[2]}\mod 2$ and

$\gcd\left(\text{b[1]},\text{b[2]},\frac{\text{b[1]}+\text{b[2]}}{2},\frac{3\text{b[1]}+\text{b[2]}}{2}\right)=1,$

also generates the $\mathbb{T}$-action on $W$. In particular, writing
$u=\frac{2b_{1}+b_{2}}{3}\in\mathbb{Z}$ and
$v=\frac{b_{1}+2b_{2}}{3}\in\mathbb{Z}$ and considering $\hat{\alpha}=u-v$ and
$\hat{\beta}=2u-v$, the $2$-dimensional family of weights becomes

$\displaystyle\left\\{\\{\hat{\beta},-\hat{\alpha}+\hat{\beta},-2\hat{\alpha}+\hat{\beta},-3\hat{\alpha}+\hat{\beta},\\},\\{-\hat{\beta},-\hat{\alpha},-2\hat{\alpha}+\hat{\beta},-3\hat{\alpha}+\hat{\beta}\\},\\{\hat{\beta},\hat{\alpha},-\hat{\alpha}+\hat{\beta},3\hat{\alpha}-\hat{\beta}\\},\right.$
$\displaystyle\left.\\{2\hat{\alpha}-\hat{\beta},\hat{\alpha}-\hat{\beta},-\hat{\alpha},-2\hat{\alpha},\\},\\{-\hat{\beta},\hat{\alpha},-\hat{\alpha},3\hat{\alpha}-\hat{\beta}\\},\\{2\hat{\alpha}-\hat{\beta},2\hat{\alpha},\hat{\alpha},\hat{\alpha}-\hat{\beta}\\}\right\\}.$

The multigraphs obtained from the projection of the $\mathbb{T}$-action to the
corresponding generic subcircles are always as the one on the right of Figure
7.3 and the ordering of the fixed point set obtained is

$p_{0}\prec p_{2}\prec p_{1}\prec p_{4}\prec p_{3}\prec p_{2}^{\prime}.$

$\bullet$ The fifth $2$-dimensional family of weights in Figure 7.4 with
$\text{b[2]}>\text{b[1]}>0$ and

$\text{b[1]}=\text{b[2]}\pmod{2}$

also generates the $\mathbb{T}$-action on $W$. In particular, writing
$u=\frac{-b_{1}+b_{2}}{2}\in\mathbb{Z}$ and
$v=\frac{b_{1}+b_{2}}{2}\in\mathbb{Z}$ and considering $\hat{\alpha}=v$ and
$\hat{\beta}=-u+2v$, the $2$-dimensional family of weights becomes

$\displaystyle\left\\{\\{-\hat{\alpha}+\hat{\beta},\hat{\alpha},\hat{\beta},3\hat{\alpha}-\hat{\beta}\\},\\{\hat{\alpha}-\hat{\beta},2\hat{\alpha}-\hat{\beta},\hat{\alpha},2\hat{\alpha}\\},\\{-\hat{\alpha}+\hat{\beta},-2\hat{\alpha}+\hat{\beta},\hat{\beta},-3\hat{\alpha}+\hat{\beta}\\},\right.$
$\displaystyle\left.\\{-\hat{\beta},-\hat{\alpha},\hat{\alpha},3\hat{\alpha}-\hat{\beta}\\},\\{\hat{\alpha}-\hat{\beta},-2\hat{\alpha},-\hat{\alpha},2\hat{\alpha}-\hat{\beta}\\},\\{-\hat{\beta},-\hat{\alpha},-2\hat{\alpha}+\hat{\beta},-3\hat{\alpha}+\hat{\beta}\\}\right\\}.$

The multigraphs obtained from the projection of the $\mathbb{T}$-action to the
generic subcircles are always as the one on the right of Figure 7.3 and the
ordering of the fixed point set obtained is

$p_{1}\prec p_{2}^{\prime}\prec p_{0}\prec p_{3}\prec p_{4}\prec p_{2}.$

$\bullet$ The sixth $2$-dimensional family of weights listed in Figure 7.4
with $\text{b[2]}>2\text{b[1]}>0$ and $\gcd(\text{b[1]},\text{b[2]})=1$
generates the $\mathbb{T}$-action on $W$ since, if we consider the basis
${\emph{b}}_{1}=-2\alpha+\beta$ and ${\emph{b}}_{2}=-3\alpha+\beta$ of the
dual lattice $\tilde{\ell}^{*}$ of $\mathbb{T}$, the weights for the action of
$\mathbb{T}$ listed in Table 5.4 become those in Table 7.4.

$p_{0}$ |  $\emph{b}_{2}$, |  $\emph{b}_{1}$, |  $2\emph{b}_{1}-\emph{b}_{2}$, | $3\emph{b}_{1}-2\emph{b}_{2}$
---|---|---|---|---
$p_{1}$ |  $-\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, |  $2\emph{b}_{1}-\emph{b}_{2}$, | $3\emph{b}_{1}-2\emph{b}_{2}$
$p_{2}$ |  $\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, |  $\emph{b}_{1}$, | $-3\emph{b}_{1}+2\emph{b}_{2}$
$p^{\prime}_{2}$ |  $-2\emph{b}_{1}+\emph{b}_{2}$, |  $-\emph{b}_{1}$, |  $\emph{b}_{1}-\emph{b}_{2}$, | $2\emph{b}_{1}-2\emph{b}_{2}$
$p_{3}$ |  $-\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, | $-3\emph{b}_{1}+2\emph{b}_{2}$
$p_{4}$ |  $-\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, |  $\emph{b}_{1}-\emph{b}_{2}$, | $-3\emph{b}_{1}+2\emph{b}_{2}$
Table 7.4.  Weights for the effective $\mathbb{T}$-action on $W$ for the new
basis of $\tilde{\ell}^{*}$.

The multigraphs obtained for these circle actions from the projection of the
$\mathbb{T}$-action to a generic subcircle are always as the one on the right
of Figure 7.3 and the ordering on the fixed point set obtained from the
corresponding moment maps is

$p_{2}\prec p_{4}\prec p_{3}\prec p_{0}\prec p_{2}^{\prime}\prec p_{1}.$

$\bullet$ The seventh $2$-dimensional family of weights listed in Figure 7.4
with $\text{b[2]}>\text{b[1]}>0$ and $\gcd(\text{b[1]},\text{b[2]})=1$
generates the $\mathbb{T}$-action on $W$ as we can easily see if we consider
the basis ${\emph{b}}_{1}=-\alpha$ and ${\emph{b}}_{2}=2\alpha-\beta$ of the
dual lattice $\tilde{\ell}^{*}$ of $\mathbb{T}$. The multigraphs obtained from
the projection of the $\mathbb{T}$-moment polytope are always as the one on
the right of Figure 7.3 and the ordering of the fixed point set obtained from
the corresponding moment maps is

$p_{4}\prec p_{3}\prec p_{2}\prec p_{2}^{\prime}\prec p_{1}\prec p_{0}.$

$\bullet$ The eighth $2$-dimensional family of weights listed in Figure 7.4
with $\text{b[2]}>\text{b[1]}>0$ and $\gcd(\text{b[1]},\text{b[2]})=1$ also
generates the $\mathbb{T}$-action on $W$. We can easily see this again by
considering the basis ${\emph{b}}_{1}=-\alpha$ and
${\emph{b}}_{2}=2\alpha-\beta$ of the dual lattice $\tilde{\ell}^{*}$ of
$\mathbb{T}$ as in the previous example.The multigraphs obtained from the
projection of the $\mathbb{T}$-moment polytope are again as the one on the
right of Figure 7.3 and the only thing different from the previous family is
the ordering of the fixed point set obtained from the corresponding moment
maps, which now interchanges $p_{2}$ and $p_{2}^{\prime}$:

$p_{4}\prec p_{3}\prec p_{2}^{\prime}\prec p_{2}\prec p_{1}\prec p_{0}.$

####  7.1.4. Index 4

For index $4$, since the greatest common divisor of the elements of the
partitions considered is $4$ there is only one possible partition of $48$: the
one with all elements equal to $4$. We run Part 3 of the file Index4.nb to
determine which matrices (6.3) are singular for this partition, dividing them
in two sets according to their ranks. Then Parts $4$ and $5$ of the file
Index4.nb produces a list of the corresponding isotropy weights, checking if
they satisfy the polynomial equations in [15, Equation (1.1)] and the
necessary properties $(i)$ to $(iii)$ listed in Section 7.1.1. In the end, the
only multiset of isotropy weights that satisfies all the necessary conditions
is listed in Figure 7.6. The corresponding multigraph is the one in Figure
7.5. The Chern numbers obtained from these isotropy weights are

$\displaystyle c_{1}^{4}[M]=512,\quad
c_{1}^{2}c_{2}[M]=224\quad\text{and}\quad c_{2}^{2}[M]=98.$

Figure 7.5. Multigraph for Index $4$ Figure 7.6. Isotropy weights for Index
$4$

The $3$-dimensional family of weights in Figure 7.6 with
$\text{b[1]},\text{b[2]},\text{b[3]}\in\mathbb{Z}^{+}$ satisfying
$\text{b[2]}>\text{b[1]}$, $\text{b[3]}>\text{b[1]}$ and

$\gcd\left(\text{b[1]},\text{b[2]},\text{b[3]}\right)=1,$

generates the action of the three-dimensional torus $\mathbb{T}$ on the
Grassmannian $Q=Gr(2,4)$ described in Section 5.3. Indeed, if we consider the
basis ${\emph{b}}_{1}=\beta$, ${\emph{b}}_{2}=\beta+\gamma$ and
${\emph{b}}_{3}=\alpha+\beta$ of $\mathfrak{t}^{*}$, the weights for the
action of $T$ listed in Table 5.6 become those in Table 7.5.

$q_{0}$ |  $\emph{b}_{1}$, |  $\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{2}+\emph{b}_{3}$, | $\emph{b}_{3}$
---|---|---|---|---
$q_{1}$ |  $-\emph{b}_{1}$, |  $-\emph{b}_{1}+\emph{b}_{3}$, |  $-\emph{b}_{1}+\emph{b}_{2}$, | $-\emph{b}_{1}+\emph{b}_{2}+\emph{b}_{3}$
$q_{2}$ |  $\emph{b}_{1}-\emph{b}_{2}$, |  $-\emph{b}_{2}$, |  $-\emph{b}_{1}+\emph{b}_{3}$, | $\emph{b}_{1}+\emph{b}_{2}$
$q^{\prime}_{2}$ |  $\emph{b}_{1}-\emph{b}_{3}$, |  $-\emph{b}_{3}$, |  $\emph{b}_{2}$, | $-\emph{b}_{1}+\emph{b}_{2}$
$q_{3}$ |  $\emph{b}_{1}-\emph{b}_{2}-\emph{b}_{3}$, |  $\emph{b}_{1}-\emph{b}_{3}$, |  $\emph{b}_{1}-\emph{b}_{2}$, | $\emph{b}_{1}$
$q_{4}$ |  $\emph{b}_{1}-\emph{b}_{2}-\emph{b}_{3}$, |  $-\emph{b}_{2}$, |  $-\emph{b}_{3}$, | $-\emph{b}_{1}$
Table 7.5.  Weights for the effective $T$-action on $Q$ for the new basis of
$\mathfrak{t}^{*}$.

The multigraphs obtained for these circle actions from the projection of the
$T$-action to a generic subcircle are always as the one in Figure 7.5 and the
ordering on the fixed point set obtained from the corresponding moment maps is

$q_{0}\prec q_{1}\prec q_{2}\prec q_{2}^{\prime}\prec q_{3}\prec q_{4}.$

###  7.2. Additional Results

Let us now consider additional cases that satisfy the conditions of
Proposition 4.38. For that let $(M,\omega)$ be a symplectic manifold admitting
a Hamiltonian $T^{2}$-action with isolated fixed points.

####  7.2.1. Index 2, $b_{2}(M)=1$ and $3\leq b_{4}(M)\leq 6$

If the index of $(M,\omega)$ is $2$ and we assume $b_{2}(M)=1$, we run Part 1
of the file index2b2_1_b4_i.nb, with $i=2,4,5,6$ to generate the list of
multigraphs $\Gamma=(E,V)$ describing a circle action when $3\leq b_{4}(M)\leq
6$. Note that by Theorem 3.4 the number of fixed points is, in these cases,
equal to $4+b_{4}(M)$ and that $c_{1}c_{3}[M]=52-2\,b_{4}(M)$. We obtain the
multigraphs in the files Multigraphsb2_1_b4_i.txt, with $i=2,4,5,6$.

Then, for each value of $b_{4}(M)$, we run Part 2 of the file
index2b2_1_b4_i.nb, considering partitions $m(e)$ of $52-2b_{4}(M)$ into
$8+2b_{4}(M)$ positive even integers, such that the first and last elements of
the partitions are always equal to $2$ (cf. Lemma 5.1), checking which of
these annihilate a determinant of a matrix
$A(\Gamma_{i})-\text{diag}(m(E_{i}))$, (according to the algorithm described
in Section 6). These partitions can be found in the files graph#.txt stored in
the folder b4_i for $i=3,4,5,6$.

We run Part 3 of the file index2b2_1_b4_i.nb to sort these partitions
according to the rank of the matrices, dividing them into two sets: those that
originate singular matrices with maximal rank, and those that originate
matrices of lower rank. In the first case, we also select the partitions that
originate matrices with null space containing vectors with positive entries.
We store the first type of partitions in the files Part$\\#$toprk.txt and the
others in the files Part$\\#$smallrk.txt.

Then, Part 4 of the file index2b2_1_b4_i.nb considers the first set of
partitions, producing a list of the corresponding isotropy weights, checking
if they satisfy the polynomial equations in [15, Equation (1.1)] (which are
obtained from the ABBV localization formula).

Finally, Part $5$ of index2b2_1_b4_i.nb considers the second set of
partitions, selects those that originate matrices with null spaces containing
vectors with positive entries, produces the list of the corresponding isotropy
weights and then selects those that satisfy [15, Equation (1.1)] as well as
the necessary conditions $(i)$ to $(iii)$ listed in Section 7.1.1. At each
step, the resulting lists of isotropy weights are saved in different files so
that it is easy to verify which isotropy weights are discarded with each test
performed. All these files can be found in the folders b4_i for $i=3,4,5,6$.

In the end, the program eliminates all possibilities. Indeed, no potential
multiset of isotropy weights satisfies all the necessary conditions and so, in
particular, we conclude that there are no possible $T^{2}$-actions with
isolated fixed points in the index-$2$ case when $b_{2}(M)=1$ and $3\leq
b_{4}(M)\leq 6$.

###### Remark 7.2

Note that this implies that $b_{2}=2$ since we know from the classification
results in [15] that, in this situation, if $b_{2}=b_{4}=1$ then the index
must be $k_{0}=5$.

####  7.2.2. Index $3$

If the index of $(M,\omega)$ is $3$ then we know from Proposition 4.38 that
$(b_{2}(M),b_{4}(M))$ is either $(1,2)$ or $(2,3)$ and the first case was
already studied in Section 7.1.3. Let us know describe the results obtained in
the second case. For this we run Part 1 of the file Index3b2_2_b4_3.nb to
generate the list of multigraphs $\Gamma=(E,V)$ describing a circle action
when $b_{2}(M)=2$ and $b_{4}(M)=3$. Note that the number of fixed points is,
in this case, equal to $9$ and that $c_{1}c_{3}[M]=54$. We obtain the
multigraphs listed in the file Multigraphsb2_2_b4_3.zip.

Since there is only one partition $m(e)$ of $54$ into $18$ positive integers
that are multiples of $3$, namely the partition $L$ with all $18$ elements
equal to $3$ (cf. Lemma 5.1), we run Part 2 of Index3b2_2_b4_3.nb to determine
which matrces $A(\Gamma_{i})-\text{diag}(L)$ are singular, (according to the
algorithm described in Section 6). The indices of the multigraphs that
originate thes matrices are stored in the file numberofgraphs.txt.

We run Part 3 of the file Index3b2_2_b4_3.nb to sort the singular matrices
according to their, dividing them into two sets. The indices of the matrices
with maximal rank are stored in the file Matricestoprk.txt and the remaining
ones in the file Matricessmallrk.txt.

Then, Part 4 of the file Index3b2_2_b4_3.nb considers the first set of
matrices, producing a list of the corresponding isotropy weights, checking if
they satisfy the polynomial equations in [15, Equation (1.1)].

Finally, Part $5$ of Index3b2_2_b4_3.nb considers the second set of matrices
and runs the necessary tests as in the previous sections. At each step, the
resulting lists of isotropy weights are saved in different files so that it is
easy to verify which isotropy weights are discarded with each test performed.
All these files can be found in the folder IsotropyWeights.

Up to a permutation of the vertices corresponding to fixed points of index
$4$, the $2$-dimensional and $4$-dimensional families of isotropy weights that
satisfy all the necessary conditions as in previous sections can be found in
the file Conlusion.pdf and are presented in Figure 7.7.

Figure 7.7. Weights

The corresponding multigraphs are shown in Figure 7.8. In all cases the Chern
numbers obtained are

$\displaystyle c_{1}^{4}[M]=486,\quad
c_{1}^{2}c_{2}[M]=216\quad\text{and}\quad c_{2}^{2}[M]=99.$

It is easy to check that the $4$-dimensional family of isotropy weights
corresponding to the seventh multigraph in Figure 7.8 generates an action of a
$4$-torus whose weights are the same as those of the toric action on
$\mathbb{C}P^{2}\times\mathbb{C}P^{2}$ described in Theorem 5.11. Moreover,
the remaining $2$-dimensional families of weights listed in Figure 7.7
generate actions of $2$-subtori of this $4$-dimensional torus. For example,
the first $2$-dimensional family of weights of Figure 7.8 can be obtained from
the seventh by taking ${\texttt{b}[2]=b[1]}$ and ${\texttt{b}[4]=2b[1]}$.

Figure 7.8. Multigraphs

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