# On the Futaki invariant of Fano threefolds

Lars Martin Sektnan and Carl Tipler  Lars Martin Sektnan, Department of
Mathematical Sciences, University of Gothenburg, 412 96 Gothenburg, Sweden
<EMAIL_ADDRESS>Carl Tipler, Univ Brest, UMR CNRS 6205, Laboratoire de
Mathématiques de Bretagne Atlantique, France<EMAIL_ADDRESS>
###### Abstract.

We study the zero locus of the Futaki invariant on $K$-polystable Fano
threefolds, seen as a map from the Kähler cone to the dual of the Lie algebra
of the reduced automorphism group. We show that, apart from families
$3.9,3.13,3.19,3.20,4.2,4.4,4.7$ and $5.3$ of the Iskovskikh–Mori–Mukai
classification of Fano threefolds, the Futaki invariant of such manifolds
vanishes identically on their Kähler cone. In all cases, when the Picard rank
is greater or equal to two, we exhibit explicit $2$-dimensional differentiable
families of Kähler classes containing the anti-canonical class and on which
the Futaki invariant is identically zero. As a corollary, we deduce the
existence of non Kähler–Einstein cscK metrics on all such Fano threefolds.

##  1\. Introduction

The Futaki invariant was introduced by Akito Futaki ([Fut83, Fut88]) as an
obstruction to the existence of Kähler–Einstein metrics on Fano manifolds. Its
definition extends to any compact polarised Kähler manifold, and its vanishing
is a necessary condition for the existence of a constant scalar curvature
Kähler metric (cscK for short) in a given Kähler class.

In this note, we study the zero locus of the Futaki invariant, seen as a map
from the Kähler cone to the dual of the Lie algebra of the reduced
automorphism group (see Section 2 for the definitions). This locus is fully
understood for Fano surfaces from the works [TY87, Tia90, WZ11], which we
recall in Section 2.1. Here we will focus on $K$-polystable Fano threefolds.
The description of this class of manifolds has seen recently great progress,
in particular with [ACC+21] (see also references therein).

Relying on a case by case analysis, our little contribution to Fanography is
the following :

######  Theorem 1.1.

Let $(X,-K_{X})$ be a $K$-polystable Fano threefold that belongs to family
N°$\mathcal{N}$, with

$\mathcal{N}\notin\\{3.9,3.13,3.19,3.20,4.2,4.4,4.7,5.3\\}.$

Then, the Futaki invariant of $X$ vanishes identically on its Kähler cone.

Note that when $\mathrm{Aut}(X)$ is finite or when the Picard rank
$\rho(X)=1$, the Futaki invariant vanishes identically on the Kähler cone, as
soon as $X$ is $K$-polystable in the second case. From the classification in
[ACC+21], there exists $33$ families of Fano threefolds with $\rho(X)\geq 2$
that admit members which are $K$-polystable with respect to the anti-canonical
polarisation and which have infinite automorphism group. We verify that of
these, only $8$ families might have members with classes on which the Futaki
invariant does not vanish. Further, for these $8$ families, we provide
explicit $2$-dimensional families of Kähler classes that contain $c_{1}(X)$
and on which the Futaki invariant vanishes.

######  Theorem 1.2.

Let $(X,-K_{X})$ be a $K$-polystable Fano threefold that belongs to family
N°$\mathcal{N}$, with

$\mathcal{N}\in\\{3.9,3.13,3.19,3.20,4.2,4.4,4.7,5.3\\}.$

Then, there is at least a 2-dimensional family of Kähler classes on $X$,
containing $c_{1}(X)$, where the Futaki invariant vanishes.

From the LeBrun–Simanca openness theorem ([LS93]), we deduce the following
corollary.

######  Corollary 1.3.

Let $X$ be a $K$-polystable Fano threefold with Picard rank $\rho(X)\geq 2$.
Then $X$ admits a $2$-dimensional family of cscK metrics parametrised by a
$2$-dimensional family of Kähler classes containing $c_{1}(X)$.

######  Remark 1.4.

For $K$-polystable members of the families $\mathcal{N}\in\\{4.2,4.4,4.7\\}$
with infinite automorphism group, we actually show that there is a
$3$-dimensional family of Kähler classes near $-K_{X}$ with vanishing Futaki
invariant.

######  Remark 1.5.

Our results should be compared with the recent [OSY23], where the Futaki
invariant of Bott manifolds is studied. In contrast to our results, which
guarantee the vanishing of the Futaki invariant in many cases, it is shown in
[OSY23] that the only Bott manifolds for which the Futaki invariant vanishes
on the whole Kähler cone are isomorphic to products of projective lines. The
key observation to prove our results is the existence of enough discrete
symmetries that preserve every Kähler class on Fano threefolds, which in the
majority of the cases considered here will be responsible for the vanishing of
the Futaki invariant.

### Acknowledgments

The authors would like to thank Hendrik Süß for kindly answering our questions
on complexity one Fano threefolds. CT is partially supported by the grants
MARGE ANR-21-CE40-0011 and BRIDGES ANR–FAPESP ANR-21-CE40-0017. LMS is funded
by a Marie Skłodowska-Curie Individual Fellowship, funded from the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 101028041.

### Notations and conventions

Throughout the paper, for a compact Kähler manifold $X$, we will denote by
$\mathrm{Aut}(X)$ (respectively $\mathrm{Aut}_{0}(X)$) its automorphism group
(respectively the connected component of the identity of the reduced
automorphism group of $X$), and by $\mathfrak{aut}(X)$ the Lie algebra of
$\mathrm{Aut}(X)$. If $Z\subset X$ is a subvariety (not necessarily
connected), $\mathrm{Aut}(X,Z)$ stands for elements in $\mathrm{Aut}(X)$ that
leave $Z$ globally invariant. We denote by $\mathcal{K}_{X}$ the Kähler cone
of $X$. We will identify a divisor $D$ with $\mathcal{O}(D)$, and use the
notation $c_{1}(D)$ for its first Chern class.

##  2\. Preliminaries

Let $X$ be a compact Kähler manifold, and $\Omega\in\mathcal{K}_{X}$ a Kähler
class on $X$. We denote the Futaki invariant of $(X,\Omega)$ by

$\begin{array}[]{cccc}\mathrm{Fut}_{(X,\Omega)}:&\mathfrak{aut}_{0}(X)&\to&\mathbb{C}\\\
&v&\mapsto&-\int_{X}\,f_{v,g}\,\mathrm{scal}_{g}\,d\mu_{g},\end{array}$

where $\mathfrak{aut}_{0}(X)$ is the Lie algebra of the reduced automorphism
group of $X$, $g$ denotes a Kähler metric with Kähler form in $\Omega$ and
volume form $d\mu_{g}$, $f_{v,g}$ is the normalised holomorphy potential of
$v$ with respect to $g$, and $\mathrm{scal}_{g}$ denotes the scalar curvature
of $g$ (see e.g. [Cal85], [LS94, Section 3.1] or [Gau, Chapter 4] for this
formulation of the Futaki invariant, initially introduced in [Fut83]).

We will be interested in $K$-polystable Fano manifolds, or equivalently Fano
manifolds admitting a Kähler–Einstein metric of positive curvature by the
resolution of the Yau–Tian–Donaldson conjecture ([Yau93, Tia97, Don02,
CDS15]). For such manifolds, by Matsushima’s result ([Mat57]), and from
Bochner’s formula (see [Gau, Section 3.6]), we have
$\mathfrak{aut}_{0}(X)=\mathfrak{aut}(X)$. We will therefore consider the
Futaki invariant as a map

$\mathrm{Fut}_{X}:\mathcal{K}_{X}\to\mathfrak{aut}(X)^{*}.$

By construction, $\mathrm{Fut}_{X}$ vanishes on any class that admits a cscK
metric, and it is then straightforward that $\mathrm{Fut}_{X}\equiv 0$
whenever $X$ is a $K$-polystable Fano manifold with Picard rank $1$, or when
the automorphism group of $X$ is finite.

###  2.1. The case of smooth Del Pezzo surfaces

We refer here the reader to [PCS19, Section 2] and [ACC+21, Section 2]. If $X$
is a smooth Del Pezzo surface with infinite automorphism group, then
$K_{X}^{2}\in\\{6,7,8,9\\}$. Moreover, it is $K$-polystable and of Picard rank
$\rho(X)\geq 2$ if and only if $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$ or
$K_{X}^{2}=6$, i.e. when $X$ is a blow-up of $\mathbb{P}^{2}$ along three non-
collinear points ([TY87, Tia90]). In the first case, $X$ admits a product cscK
metric in each class, and $\mathrm{Fut}_{X}\equiv 0$, while in the latter
case, the vanishing locus of $\mathrm{Fut}_{X}$ is described in [WZ11, Section
5] (see Section 4.2 for the exact description).

###  2.2. Further properties of the Futaki invariant

The key property that we will use is the invariance of $\mathrm{Fut}_{X}$
under the $\mathrm{Aut}(X)$-action. This was already used in [Fut88, Chapter
3] to show the vanishing of $\mathrm{Fut}_{X}$ on specific examples.

We will use the following proposition repeatedly.

######  Proposition 2.1.

Let $(X,\Omega)$ be a polarised Fano manifold. Assume that there is
$\tau\in\mathrm{Aut}(X)$ and $v\in\mathfrak{aut}(X)$ such that

  * (i)

$\tau^{*}\Omega=\Omega$,

  * (ii)

there is $c\in\mathbb{C}^{*}\setminus\\{1\\}$ with
$\mathrm{Ad}_{\tau}(v)=c\cdot v$.

Then $\mathrm{Fut}_{(X,\Omega)}(v)=0$.

###### Proof.

This follows from the $\mathrm{Ad}$-invariance of the Futaki invariant, which
implies that

$\mathrm{Fut}_{(X,(\tau^{-1})^{*}\Omega)}(\mathrm{Ad}_{\tau}(v))=\mathrm{Fut}_{(X,\Omega)}(v),$

see [Fut88, Chapter 3] or [LS94, Section 3.1]. ∎

######  Remark 2.2.

The anti-canonical class $c_{1}(X)$ is always $\mathrm{Aut}(X)$-invariant.

As an application, we have the following useful corollary :

######  Corollary 2.3.

Let $\pi:X\to Y$ be the blow-up of a smooth Fano manifold $Y$ along smooth and
disjoint subvarieties $Z_{i}\subset Y$. Assume that there is a finite group
$G\subset\mathrm{Aut}(Y)$ such that :

  * (i)

Each $Z_{i}$ is $G$-invariant;

  * (ii)

Each class $\Omega\in H^{1,1}(Y,\mathbb{R})$ is $G$-invariant;

  * (iii)

For any $v\in\mathfrak{aut}(Y)$ that lifts to $X$, there is $\tau\in G$ and
$c\in\mathbb{C}^{*}\setminus\\{1\\}$ such that $\mathrm{Ad}_{\tau}(v)=c\cdot
v$.

Then $\mathrm{Fut}_{X}\equiv 0$.

###### Proof.

From hypothesis (i), the $G$-action on $Y$ lifts to a $G$-action on $X$. The
vector space $H^{1,1}(X,\mathbb{R})$ is spanned by the pullback of the classes
in $H^{1,1}(Y,\mathbb{R})$ and the exceptional divisors of $\pi$. By
hypothesis (i) and (ii), any class in $H^{1,1}(X,\mathbb{R})$ is then
$G$-invariant. The Lie algebra $\mathfrak{aut}(X)$ is spanned by lifts of
elements in $\mathfrak{aut}(Y)$ that preserve the $Z_{i}$’s. For any such
element, the identity $\mathrm{Ad}_{\tau}(v)=c\cdot v$ holds on
$X\setminus\bigcup_{i}\pi^{-1}(Z_{i})$, hence on $X$, by continuity. The
result follows from Proposition 2.1. ∎

######  Remark 2.4.

In practice, we will mainly use Corollary 2.3 with

$G\simeq\mathbb{Z}/2\mathbb{Z},\>\mathfrak{aut}(X)\simeq\mathbb{C},\>c=-1.$

To prove item $(i)$ of Proposition 2.1 or item $(ii)$ of Corollary 2.3, we
will use the fact that in homogeneous coordinates, the Fubini–Study metric

$\omega_{FS}=\frac{i}{2}\partial\overline{\partial}\log(|z|^{2}),$

and hence its class $[\omega_{FS}]\in H^{1,1}(\mathbb{P}^{n},\mathbb{R})$, is
invariant under the $\mathfrak{S}_{n+1}$-action on $\mathbb{P}^{n}$ by
permutation of the homogeneous coordinates.

###  2.3. The list to check

From the discussion in the beginning of this section, to prove Theorem 1.1, it
is enough to consider $K$-polystable Fano threefolds with infinite
automorphism group and Picard rank $\rho(X)\geq 2$. From [ACC+21, Section 6],
this reduces to Fano threefolds in family N°$\mathcal{N}$, for

$\mathcal{N}\in\left\\{\begin{array}[]{lll}2.20,2.21,2.22,2.24,2.27,2.29,2.32,2.34,3.5,3.8,3.9,\\\
3.10,3.12,3.13,3.15,3.17,3.19,3.20,3.25,3.27,4.2,4.3,\\\
4.4,4.6,4.7,4.13,5.1,5.3,6.1,7.1,8.1,9.1,10.1\end{array}\right\\}.$

The strategy of the proof is then direct – we will use the invariance of
$\mathrm{Fut}_{X}$ to show its vanishing on $\mathcal{K}_{X}$ using a case by
case study. For $X$ belonging to family N°$\mathcal{N}$ with

$\mathcal{N}\in\left\\{\begin{array}[]{lll}2.20,2.21,2.22,2.24,2.27,2.29,2.32,2.34,\\\
3.5,3.8,3.10,3.12,3.15,3.17,3.25,3.27,\\\
4.3,4.6,4.13,5.1,6.1,7.1,8.1,9.1,10.1\end{array}\right\\},$

we will see that $\mathrm{Fut}_{X}\equiv 0$.

For the remaining $8$ families, we will obtain explicit Kähler classes of the
form

$c_{1}(X)+\varepsilon c_{1}(D)\in\mathcal{K}_{X}$

with $D\subset X$ a divisor and $\varepsilon\in\mathbb{R}$ a parameter such
that $\mathrm{Fut}_{(X,c_{1}(X)+\varepsilon c_{1}(D))}=0$. As the vanishing of
the Futaki invariant is preserved under scaling of the Kähler metric, this
provides the $2$-dimensional families of Kähler classes with vanishing Futaki
invariant alluded to in the introduction. Corollary 1.3 then follows from
LeBrun–Simanca’s openness theorem [LS93], which asserts that the locus in the
Kähler cone of Kähler classes that admit an extremal metric in the sense of
Calabi ([Cal82]) is open, together with the characterisation of cscK metrics
amongst extremal metrics as the ones with zero Futaki invariant ([Cal85]).

##  3\. Families with $\mathfrak{aut}(X)\simeq\mathfrak{sl}_{n}(\mathbb{C})$

Here we will consider families N°$\mathcal{N}$, with

$\mathcal{N}\in\\{2.27,2.32,3.17,4.6,6.1,7.1,8.1,9.1,10.1\\}.$

The Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$ is simple, hence equal to its
derived ideal $[\mathfrak{sl}_{n}(\mathbb{C}),\mathfrak{sl}_{n}(\mathbb{C})]$.
As the Futaki invariant is a character from $\mathfrak{aut}(X)$ to
$\mathbb{C}$ (see e.g. [Fut88, Chapter 3] or [LS94, Section 3.1]), it vanishes
on the derived ideal $[\mathfrak{aut}(X),\mathfrak{aut}(X)]$. Hence, if
$\mathfrak{aut}(X)\simeq\mathfrak{sl}_{n}(\mathbb{C})$,
$[\mathfrak{aut}(X),\mathfrak{aut}(X)]=\mathfrak{aut}(X)$ and the Futaki
invariant vanishes identically on the whole Kähler cone of $X$. From

$\mathrm{PGL}_{n}(\mathbb{C})\simeq\mathrm{SL}_{n}(\mathbb{C})/\mu_{n},$

the Lie algebra of $\mathrm{PGL}_{n}(\mathbb{C})$ is
$\mathfrak{sl}_{n}(\mathbb{C})$. From [ACC+21, Section 6, Big Table], this
settles the case of all the $K$-polystable Fano threefolds in families
N°$\mathcal{N}$, with

$\mathcal{N}\in\\{2.27,2.32,3.17,4.6,6.1,7.1,8.1,9.1,10.1\\},$

and also some cases in families $\\{2.21,3.13\\}$.

##  4\. Products

Next, we consider families N°2.34, N°3.27 and N°5.3, which are products of
lower dimensional Fano manifolds.

###  4.1. Families 2.34 and 3.27

The unique members in these two families are
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ and
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$, which both carry a
product of cscK metrics in any class, and thus has vanishing Futaki character
for any Kähler class.

###  4.2. Family 5.3

The unique Fano threefold in family 5.3 is $\mathbb{P}^{1}\times S_{6}$, where
$S_{6}$ is the Del Pezzo surface with $K_{S_{6}}^{2}=6$. It is $K$-polystable
as a product of Kähler–Einstein manifolds from [TY87, Tia90]. The surface
$S_{6}$ is the unique (up to isomorphism) toric surface obtained by blowing-up
$\mathbb{P}^{2}$ in the three fixed points under the torus action. We denote
by $H$ (the strict transform of) a generic hyperplane and $D_{1},D_{2}$ and
$D_{3}$ the three exceptional divisors in $S_{6}$. From [WZ11, Section 5,
Proposition 5.2 and Remark 5.1.(iii)], the Futaki invariant of $S_{6}$
vanishes exactly in the following families of Kähler classes

$3c_{1}(H)-ac_{1}(D_{1})-bc_{1}(D_{2})-(3-a-b)c_{1}(D_{3})$

and

$3c_{1}(H)-c(c_{1}(D_{1})+c_{1}(D_{2})+c_{1}(D_{3})),$

where $a,b,c$ are positive constants satisfying $a+b<3$ and $c<\frac{3}{2}$.
As the Futaki invariant vanishes on $\mathbb{P}^{1}$, we easily deduce the
vanishing locus of the Futaki invariant on $X=\mathbb{P}^{1}\times S_{6}$. In
particular, as $c_{1}(X)=c_{1}(\mathbb{P}^{1})+c_{1}(S_{6})$, and as
$c_{1}(S_{6})=3c_{1}(H)-(c_{1}(D_{1})+c_{1}(D_{2})+c_{1}(D_{3}))$, we deduce
the existence of differentiable families of Kähler classes on $X$ containing
$c_{1}(X)$ for which the Futaki invariant vanishes identically.

##  5\. Blow-ups of projective space

In this section we address families N°$\mathcal{N}$, with

$\mathcal{N}\in\\{2.22,3.12,3.25\\}.$

All the members of these families are obtained by blowing up certain curves in
projective space $\mathbb{P}^{3}$.

###  5.1. Family 2.22

Members of the family 2.22 of Fano threefolds are obtained as blowups of
certain curves in $\mathbb{P}^{3}$. More precisely, let
$\Phi:\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{3}$ be the Segre
embedding

$([x:y],[u:v])\mapsto[xu:xv:yu:yv].$

The image of $\Phi$ is the surface

$S=\\{z_{0}z_{3}-z_{1}z_{2}=0\\}.$

A Fano threefold $X$ is in the family 2.22 if it is the blowup of the image
via $\Phi$ of a curve $\check{C}$ with
$\mathcal{O}(\check{C})=\mathcal{O}(3,1).$ Such $X$ have Picard rank $2$,
generated by the line bundle associated to the proper transform of a
hyperplane and of that generated by the exceptional divisor $E$ of the blowup.
The $K$-polystability of members of this family (with respect to the
anticanonical polarisation) is discussed in detail in [CP22].

Up to biholomorphism, there is a unique member $X_{0}$ of this family with
infinite automorphism group. It is $K$-polystable, and can be obtained by
picking the curve $\check{C}$ to be $\check{C}_{0}=\\{ux^{3}-vy^{3}=0\\}$, so
that

$X_{0}=\operatorname{Bl}_{C_{0}}\mathbb{P}^{3},$

where $C_{0}=\Phi(\check{C}_{0}).$ The $\mathbb{C}^{*}$-action

$\lambda\cdot([z_{0}:z_{1}:z_{2}:z_{3}])=[\lambda
z_{0}:\lambda^{4}z_{1}:z_{2}:\lambda^{3}z_{3}]$

preserves $C_{0}$ and so lifts to $X_{0}$. This generates
$\mathrm{Aut}_{0}(X_{0})$ (see [PCS19, Lemma 6.13]).

The curve $C_{0}$ is a rational curve, which can e.g. be seen by applying the
Riemann–Hurwitz formula to the restriction to
$\check{C}_{0}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ of the projection to
the second factor. An explicit parametrisation is given by

$[\tau_{0}:\tau_{1}]\mapsto[\tau_{0}\tau_{1}^{3}:\tau_{0}^{4}:\tau_{1}^{4}:\tau_{1}\tau_{0}^{3}].$

Note that the action of the involution $\tau$ given by

$\tau\cdot([z_{0}:z_{1}:z_{2}:z_{3}])=[z_{3}:z_{2}:z_{1}:z_{0}]$

on $\mathbb{P}^{3}$ preserves $C_{0}$ and so lifts to $X_{0}$. We can then
apply Corollary 2.3 to the blow-up $X_{0}\to\mathbb{P}^{3}$ with the group
$G=\langle\tau\rangle\simeq\mathbb{Z}/2\mathbb{Z}$, which implies the
vanishing of the Futaki invariant on the Kähler cone of $X_{0}$.

###  5.2. Family 3.12

From [ACC+21, Section 5.18], the only element in Family 3.12 with infinite
automorphism group is given, up to isomorphism, by $X=\mathrm{Bl}_{L\cup
C}(\mathbb{P}^{3})$ the blow up of $\mathbb{P}^{3}$ along the disjoint curves

$L=\\{x_{0}=x_{3}=0\\}\subset\mathbb{P}^{3}$

and

$C=\\{[s^{3}:s^{2}t:st^{2}:t^{3}],[s:t]\in\mathbb{P}^{1}\\}\subset\mathbb{P}^{3}.$

The reduced automorphism group of $X$ is isomorphic to $\mathbb{C}^{*}$, and
its action is given by the lift of the $\mathbb{C}^{*}$-action on
$\mathbb{P}^{3}$ described by

$\lambda\cdot([x_{0}:x_{1}:x_{2}:x_{3}])=[x_{0}:\lambda
x_{1}:\lambda^{2}x_{2}:\lambda^{3}x_{3}].$

Then, we can consider the $\mathbb{Z}/2\mathbb{Z}$-action given by

$\tau([x_{0}:x_{1}:x_{2}:x_{3}])=[x_{3}:x_{2}:x_{1}:x_{0}].$

The group generated by $\tau$ in $\mathrm{Aut}(\mathbb{P}^{3})$ satisfies
hypothesis $(i)-(iii)$ from Corollary 2.3, and we deduce that the Futaki
invariant of $X$ vanishes on the whole Kähler cone.

###  5.3. Family 3.25

The Fano threefold $X$ in family 3.25 is the blow-up of $\mathbb{P}^{3}$ in
two disjoint lines. It is $K$-polystable from [BS99, WZ04]. We can assume the
two blown-up lines are $\\{x_{1}=x_{2}=0\\}\subset\mathbb{P}^{3}$ and
$\\{x_{3}=x_{4}=0\\}\subset\mathbb{P}^{3}$. One has

$\mathrm{Aut}_{0}(X)\simeq\mathrm{PGL}_{(2,2)}(\mathbb{C})\simeq\operatorname{GL}_{2}(\mathbb{C})\times\operatorname{GL}_{2}(\mathbb{C})/\mathbb{C}^{*},$

where the first (resp. second) $\operatorname{GL}_{2}(\mathbb{C})$ factor acts
linearly on the coordinates $(x_{1},x_{2})$ (resp. on $(x_{3},x_{4})$) while
the $\mathbb{C}^{*}$-action corresponds to homotheties on $\mathbb{C}^{4}$
(see [PCS19, Section 4]). The Lie algebra $\mathfrak{aut}(X)$ of
$\mathrm{Aut}_{0}(X)$ fits in an exact sequence

$0\to\mathbb{C}\to\mathfrak{gl}_{2}(\mathbb{C})\oplus\mathfrak{gl}_{2}(\mathbb{C})\to\mathfrak{aut}(X)\to
0.$

We also have the sequence induced by the trace map
$\mathfrak{gl}_{2}(\mathbb{C})\to\mathbb{C}$ :

$0\to\mathfrak{sl}_{2}(\mathbb{C})\to\mathfrak{gl}_{2}(\mathbb{C})\to\mathbb{C}\to
0,$

from which we deduce the sequence of vector spaces

$0\to\mathbb{C}\to(\mathfrak{sl}_{2}(\mathbb{C})\oplus\mathbb{C})\oplus(\mathfrak{sl}_{2}(\mathbb{C})\oplus\mathbb{C})\to\mathfrak{aut}(X)\to
0.$

From the discussion in Section 3, the Futaki invariant of $X$ will vanish on
the $\mathfrak{sl}_{2}(\mathbb{C})$-factors that project to
$\mathfrak{aut}(X)$. Hence, it is enough to test the vanishing of the Futaki
invariant on the generators of the remaining two $\mathbb{C}^{*}$-actions
modulo homotheties, which are induced by:

$(\lambda,\mu)\cdot([x_{0}:x_{1}:x_{2}:x_{3}])=[\lambda x_{0}:x_{1}:\mu
x_{2}:x_{3}],$

where $(\lambda,\mu)\in(\mathbb{C}^{*})^{2}$. We can consider the finite group
$G$ generated by the reflections

$\tau([x_{0}:x_{1}:x_{2}:x_{3}])=[x_{1}:x_{0}:x_{2}:x_{3}]$

and

$\sigma([x_{0}:x_{1}:x_{2}:x_{3}])=[x_{0}:x_{1}:x_{3}:x_{2}].$

This group preserves the two blown-up lines, while the adjoint action of
$\tau$ (resp. $\sigma$) sends the generator of the $\lambda$-action (resp. the
$\mu$-action) to its inverse. Hence, from Corollary 2.3, we see that the
Futaki invariant of $X$ vanishes on its whole Kähler cone.

##  6\. Blow-ups of products of projective spaces

In this section, we will consider families N°$\mathcal{N}$, with

$\mathcal{N}\in\\{3.5,4.3,4.13\\}.$

These are obtained as blowups of products of projective spaces.

###  6.1. Family 3.5

From [ACC+21, Section 5.14], the only element in Family 3.5 with infinite
automorphism group is given, up to isomorphism, by
$X=\mathrm{Bl}_{C}(\mathbb{P}^{1}\times\mathbb{P}^{2})$ the blow up of
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ along the curve $C=\psi(\check{C})$ given
by the image of

$\check{C}=\\{ux^{5}+vy^{5}=0\\}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$

via the map

$\begin{array}[]{cccc}\psi:&\mathbb{P}^{1}\times\mathbb{P}^{1}&\to&\mathbb{P}^{1}\times\mathbb{P}^{2}\\\
&([u:v],[x:y])&\mapsto&([u:v],[x^{2}:xy:y^{2}]).\end{array}$

Then, $\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$, where the
$\mathbb{C}^{*}$-action is generated by the lift to $X$ of the action

$\lambda\cdot([u:v],[x_{0}:x_{1}:x_{2}])=([\lambda^{5}u:v],[x_{0}:\lambda
x_{1}:\lambda^{2}x_{2}]).$

We also have a $\mathbb{Z}/2\mathbb{Z}$-action induced by

$\tau([u:v],[x_{0}:x_{1}:x_{2}])=([v:u],[x_{2}:x_{1}:x_{0}]).$

Those actions come respectively from the actions

$\lambda\cdot([u:v],[x:y])=([\lambda^{5}u:v],[x:\lambda y])$

and

$\tau([u:v],[x:y])=([v:u],[y:x])$

on $\mathbb{P}^{1}\times\mathbb{P}^{1}$, with respect to which $\psi$ is
equivariant. Then, we see that $C$ is $\tau$-invariant, as well as the classes
$\pi_{i}^{*}[\omega_{FS}^{i}]$, where
$\pi_{1}:\mathbb{P}^{1}\times\mathbb{P}^{2}\to\mathbb{P}^{1}$ and
$\pi_{2}:\mathbb{P}^{1}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$ denote the
projections and $\omega_{FS}^{i}$ stands for the Fubini–Study metric on
$\mathbb{P}^{i}$. Finally, identifying $\lambda\in\mathbb{C}^{*}$ with its
action, we have $\tau\circ\lambda\circ\tau^{-1}=\lambda^{-1}$. Hence,
hypothesis $(i)-(iii)$ from Corollary 2.3 are satisfied, and the Futaki
invariant of $X$ vanishes for any Kähler class.

###  6.2. Family 4.3

Following [ACC+21, Section 5.21], up to isomorphism, the unique Fano threefold
in Family 4.3 is the blow-up of
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ along

$C=\\{x_{0}y_{1}-x_{1}y_{0}=x_{0}z_{1}^{2}+x_{1}z_{0}^{2}=0\\}$

where $[x_{0}:x_{1}]$, $[y_{0}:y_{1}]$ and $[z_{0}:z_{1}]$ denote the
homogeneous coordinates on the first, second and last factor respectively. We
have $\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$ where the action is given by
the lift of the $\mathbb{C}^{*}$-action on
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ given by

$\lambda\cdot([x_{0}:x_{1}],[y_{0}:y_{1}],[z_{0}:z_{1}])=([x_{0}:\lambda^{2}x_{1}],[y_{0}:\lambda^{2}y_{1}],[z_{0}:\lambda
z_{1}]).$

The involution

$\tau([x_{0}:x_{1}],[y_{0}:y_{1}],[z_{0}:z_{1}])=([x_{1}:x_{0}],[y_{1}:y_{0}],[z_{1}:z_{0}])$

preserves $C$ and the $(1,1)$-classes on $C$ given by
$\iota^{*}_{j}[\omega_{FS}]$, for $\iota_{j}$ the composition of the inclusion
$C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ and the
projection on the $j$-th factor. The adjoint action of $\tau$ maps the
generator of the $\mathbb{C}^{*}$-action to its inverse, so Proposition 2.1
applies and the Futaki invariant of $X$ vanishes identically.

###  6.3. Family 4.13

From [ACC+21, Section 5.22], the only element in Family 4.13 with infinite
automorphism group is given, up to isomorphism, by
$X=\mathrm{Bl}_{C}(\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1})$
the blow up of $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ along
the curve

$C=\\{x_{0}y_{1}-x_{1}y_{0}=x_{0}^{3}z_{0}+x_{1}^{3}z_{1}=0\\}.$

The reduced automorphism group of $X$ is isomorphic to $\mathbb{C}^{*}$, and
its action is given by the lift of the $\mathbb{C}^{*}$-action on
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ described by

$\lambda\cdot([x_{0}:x_{1}],[y_{0}:y_{1}],[z_{0}:z_{1}])=([\lambda
x_{0}:x_{1}],[\lambda y_{0}:y_{1}],[\lambda^{-3}z_{0}:z_{1}]).$

Then, we can consider the $\mathbb{Z}/2\mathbb{Z}$-action given by

$\tau([x_{0}:x_{1}],[y_{0}:y_{1}],[z_{0}:z_{1}])=([x_{1}:x_{0}],[y_{1}:y_{0}],[z_{1}:z_{0}]).$

Clearly, this action satisfies hypothesis $(i)-(iii)$ from Corollary 2.3
(notice that $\tau\circ\lambda\circ\tau^{-1}=\lambda^{-1}$, identifying
$\lambda$ with the induced action), from which we deduce the vanishing of the
Futaki invariant of $X$ for any Kähler class.

##  7\. Blow-ups of a smooth quadric

In this section, we consider families N°$\mathcal{N}$, with

$\mathcal{N}\in\\{2.21,2.29,3.10,3.15,3.19,3.20,4.4,5.1\\}.$

###  7.1. Family 2.21

This family is somewhat similar to the Mukai–Umemura family 1.10. In addition
to members of the family with discrete automorphism group, there is a one-
dimensional family with automorphism group containing a semi-direct product of
$\mathbb{C}^{*}$ and $\mathbb{Z}/2\mathbb{Z}$, one member which admits an
effective $\textnormal{PGL}_{2}$-action and one member which has a reduced
automorphism group $\mathbb{G}_{a}$. The first two of these are $K$-polystable
for the anti-canonical polarisation, whereas the last does not have a
reductive automorphism group and is therefore not $K$-polystable.

The members that admit an effective $\mathbb{G}_{m}$-action can be described
as follows (see [ACC+21, Section 5.9]). Let $C$ be the quartic rational curve
in $\mathbb{P}^{4}$ given as the image of the map
$\mathbb{P}^{1}\to\mathbb{P}^{4}$ given by

$[p:q]\mapsto[p^{4}:p^{3}q:p^{2}q^{2}:pq^{3}:q^{4}].$

For $t\notin\\{0,\pm 1\\}$, let $Q_{t}$ be the smooth hypersurface

$Q_{t}=V(z_{1}z_{3}-t^{2}z_{0}z_{4}+(t^{2}-1)z_{2}^{2}).$

Note that $C\subset Q_{t}$ for any $t$. Let
$X_{t}=\operatorname{Bl}_{C}(Q_{t})$. Then $X_{t}$ is one of the members that
admit an effective $\mathbb{C}^{*}$-action (including the member with an
effective $\textnormal{PGL}_{2}$-action, which corresponds to
$t=\pm\frac{1}{2}$). Note that $X_{t}$ has Picard rank $2$, generated by a
hyperplane $H$ and the exceptional divisor $E$ of the blowup.

The $\mathbb{C}^{*}$-action given by

$\lambda\cdot([z_{0}:z_{1}:z_{2}:z_{3}:z_{4}])=[z_{0}:\lambda
z_{1}:\lambda^{2}z_{2}:\lambda^{3}z_{3}:\lambda^{4}z_{4}]$

preserves $C$ and $Q_{t}$, as does the involution

$\tau([z_{0}:z_{1}:z_{2}:z_{3}:z_{4}])=[z_{4}:z_{3}:z_{2}:z_{1}:z_{0}].$

The lifts of these generate the effective actions of
$\mathbb{C}^{*}\rtimes\mathbb{Z}/2\mathbb{Z}$ on $X_{t}$. As $\tau$ preserves
$C$, the class $[\omega_{FS}]_{|X_{t}}$, and sends a generator of the
$\mathbb{C}^{*}$-action to its inverse by conjugation, Proposition 2.1 shows
that the Futaki invariant of $X_{t}$ vanishes on its whole Kähler cone (note
that the case $t=\pm\frac{1}{2}$, with
$\mathfrak{aut}(X_{t})\simeq\mathfrak{sl}_{2}(\mathbb{C})$, was dealt with in
Section 3).

###  7.2. Family 2.29

There is a unique smooth Fano threefold $X$ in family 2.29. It is isomorphic
to the blow-up of

$Q=\\{x_{0}^{2}+x_{1}x_{2}+x_{3}x_{4}=0\\}\subset\mathbb{P}^{4}.$

along the smooth conic

$C=\\{x_{0}^{2}+x_{1}x_{2}=x_{3}=x_{4}=0\\}\subset Q.$

It is $K$-polystable (see [Süß13, Süß14, IS17]) and the group
$\mathrm{Aut}_{0}(X)$ is isomorphic to
$\mathbb{C}^{*}\times\mathrm{PGL}_{2}(\mathbb{C})$ (see [PCS19, Lemma 5.8]).
We then have
$\mathfrak{aut}(X)\simeq\mathbb{C}\oplus\mathfrak{sl}_{2}(\mathbb{C})$. From
the discussion in Section 3, the Futaki invariant of $(X,[\omega])$ vanishes
on the $\mathfrak{sl}_{2}(\mathbb{C})$-component of $\mathfrak{aut}(X)$ for
any Kähler class $[\omega]$. Thus, to check the vanishing of the Futaki
invariant, it remains to check the vanishing on the $\mathbb{C}$-component of
$\mathfrak{aut}(X)$. From [PCS19, Lemma 5.7], the $\mathbb{C}^{*}$-component
of $\mathrm{Aut}_{0}(X)$ can be identified with the pointwise stabiliser of
$C$ in $\mathrm{Aut}_{0}(Q)$. This is then the $\mathbb{C}^{*}$-action induced
by

$\lambda\cdot([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[x_{0}:x_{1}:x_{2}:\lambda
x_{3}:\lambda^{-1}x_{4}].$

We then introduce the involution

$\tau([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[x_{0}:x_{2}:x_{1}:x_{4}:x_{3}].$

This automorphism of $\mathbb{P}^{4}$ preserves $Q$ and $C$ and lifts to an
automorphism of $X$. Its adjoint action maps a generator of the
$\mathbb{C}^{*}$-action of interest to its inverse, and by Corollary 2.3, we
deduce the vanishing of the Futaki invariant of $X$ for any Kähler class.

###  7.3. Family 3.10

Let $X$ be a $K$-polystable element in the family 3.10 such that
$\mathrm{Aut}(X)$ is infinite. Then, from [ACC+21, Section 5.17], up to
isomorphism, we may assume that $X=\operatorname{Bl}_{C_{1}\cup C_{2}}(Q_{a})$
is the blow-up of the quadric

$Q_{a}=\\{w^{2}+xy+zt+a(xt+yz)=0\\}\subset\mathbb{P}^{4}$

along the two disjoint smooth irreducible conics $C_{1}\subset Q_{a}$ and
$C_{2}\subset Q_{a}$ given by

$C_{1}=\\{w^{2}+zt=x=y=0\\}$

and

$C_{2}=\\{w^{2}+xy=z=t=0\\}$

where $[x,y,z,t,w]$ stand for the homogeneous coordinates on $\mathbb{P}^{4}$
and where $a\in\mathbb{C}\setminus\\{-1,+1\\}$ is a complex parameter.
Moreover, for $a=0$, $\mathrm{Aut}_{0}(X)\simeq(\mathbb{C}^{*})^{2}$ and for
$a\neq 0$, $\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$.

####  7.3.1. Case $a=0$

In this situation, the $(\mathbb{C}^{*})^{2}$-action on $X$ is the lift of the
action on $Q_{0}$ induced by the following formula, for
$(\alpha,\beta)\in(\mathbb{C}^{*})^{2}$ :

$(\alpha,\beta)\cdot([x:y:z:t:w])=[\alpha x:\alpha^{-1}y:\beta
z:\beta^{-1}t:w].$

Consider the group $G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$
generated by $(\sigma,\tau)$ defined by

$\sigma([x:y:z:t:w])=[y:x:z:t:w]$

and

$\tau([x:y:z:t:w])=[x:y:t:z:w].$

Then $G\subset\mathrm{Aut}(Q_{0})$, and $G$ preserves $C_{1}$ and $C_{2}$. It
also leaves invariant the class $\iota^{*}[\omega_{FS}]$ on $Q_{0}$, where
$\iota:Q_{0}\to\mathbb{P}^{4}$ denotes the inclusion and $\omega_{FS}$ the
Fubini–Study metric. Hence, hypothesis $(i)$ and $(ii)$ of Corollary 2.3 are
satisfied. Finally, $\mathrm{Ad}_{\sigma}(v_{1})=-v_{1}$ and
$\mathrm{Ad}_{\tau}(v_{2})=-v_{2}$, where $v_{1}$ generates the
$\mathbb{C}^{*}$-action $\alpha\mapsto[\alpha x:\alpha^{-1}y:z:t:w]$ while
$v_{2}$ generates the $\mathbb{C}^{*}$-action $\beta\mapsto[x:y:\beta
z:\beta^{-1}t:w]$. Then, Corollary 2.3 implies the vanishing of the Futaki
invariant on $X$ for any class.

####  7.3.2. Case $a\neq 0$

The same argument as in the previous case applies, where this time the
$\mathbb{C}^{*}$-action of $\mathrm{Aut}_{0}(X)$ is induced by the diagonal of
the above, given by

$\alpha\cdot([x:y:z:t:w])=([\alpha x:\alpha^{-1}y:\alpha z:\alpha^{-1}t:w]).$

and the group $G\simeq\mathbb{Z}/2\mathbb{Z}$ is generated by

$\varsigma([x:y:z:t:w])=([y:x:t:z:w]).$

###  7.4. Family 3.15

From [ACC+21, Section 5.20], the only smooth $K$-polystable Fano threefold in
family 3.15 is given by the blow-up $X=\operatorname{Bl}_{L\cup C}(Q)\to Q$ of
the quadric

$Q=\\{x_{0}^{2}+2x_{1}x_{2}+2x_{1}x_{4}+2x_{2}x_{3}=0\\}\subset\mathbb{P}^{4}$

along the line

$L=\\{x_{0}=x_{1}=x_{2}=0\\}$

and the smooth conic (disjoint from $L$)

$C=\\{x_{0}^{2}+2x_{1}x_{2}=x_{3}=x_{4}=0\\}.$

The automorphism group of $X$ satisfies
$\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$ with $\mathbb{C}^{*}$-action given,
for $\lambda\in\mathbb{C}^{*}$, by (the lift of)

$\lambda\cdot([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[\lambda
x_{0}:\lambda^{2}x_{1}:x_{2}:\lambda^{2}x_{3}:x_{4}].$

The involution

$\tau([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[x_{0}:x_{2}:x_{1}:x_{4}:x_{3}]$

preserves $Q$, $L$ and $C$. It also leaves the class $\iota^{*}[\omega_{FS}]$
invariant, where $\iota:Q\to\mathbb{P}^{4}$ is the inclusion. Then, Corollary
2.3 applies to $X\to Q$ and
$G=\langle\tau\rangle\simeq\mathbb{Z}/2\mathbb{Z}$, so that the Futaki
invariant of $X$ identically vanishes on $\mathcal{K}_{X}$.

###  7.5. Families 3.19 and 3.20

Consider the smooth quadric Fano threefold

$Q=\\{x_{0}^{2}+x_{1}x_{2}+x_{3}x_{4}=0\\}\subset\mathbb{P}^{4}.$

The family 3.19 (resp. 3.20) is obtained by blowing-up $Q$ in two points
(respectively two disjoint lines). More precisely, we can obtain the unique
Fano threefold in family 3.19 by considering $X_{1}$ to be the blow-up of $Q$
along the points

$P_{1}=[0:0:0:1:0]$

and

$P_{2}=[0:0:0:0:1].$

The unique Fano threefold $X_{2}$ in family 3.20 is the blow-up of $Q$ along
the two disjoint lines

$L_{1}=\\{x_{0}=x_{1}=x_{3}=0\\}$

and

$L_{2}=\\{x_{0}=x_{2}=x_{4}=0\\}.$

In both cases, the Fano threefold $X_{i}$ is $K$-polystable (see [Süß13,
Süß14, IS17]) and the group $\mathrm{Aut}_{0}(X_{i})$ is isomorphic to
$\mathbb{C}^{*}\times\mathrm{PGL}_{2}(\mathbb{C})$ (see [PCS19, Section 5]).
We then have
$\mathfrak{aut}(X_{i})=\mathbb{C}\oplus\mathfrak{sl}_{2}(\mathbb{C})$. From
the discussion in Section 3, the Futaki invariant of $(X_{i},[\omega_{i}])$
vanishes on the $\mathfrak{sl}_{2}(\mathbb{C})$-component of
$\mathfrak{aut}(X_{i})$ for any Kähler class $[\omega_{i}]$. Therefore, to
check the vanishing of the Futaki invariant on $(X_{i},[\omega_{i}])$, it
remains to check the vanishing on the $\mathbb{C}$-component of
$\mathfrak{aut}(X_{i})$.

To this aim we introduce the involution

$\tau([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[x_{0}:x_{2}:x_{1}:x_{4}:x_{3}].$

This automorphism of $\mathbb{P}^{4}$ preserves $Q$, and swaps the two
connected components of the blown-up locus in both cases. Therefore, $\tau$
lifts to an automorphism of $X_{i}$, still denoted $\tau$, for
$i\in\\{1,2\\}$. Note that on $X_{i}$, any Kähler class of the form

$[\omega_{\varepsilon}]:=c_{1}(X_{i})+\varepsilon(c_{1}(\mathcal{O}(E_{1}^{i}))+c_{1}(\mathcal{O}(E_{2}^{i})))$

is $\tau$-invariant, where $\varepsilon\in\mathbb{R}$ is chosen so that

$c_{1}(X_{i})+\varepsilon(c_{1}(\mathcal{O}(E_{1}^{i}))+c_{1}(\mathcal{O}(E_{2}^{i})))>0$

and the $E_{j}^{i}$’s denote the exceptional divisors of the blow-up $X_{i}\to
Q$.

Next, we investigate how this action interacts with the generator of the
$\mathbb{C}^{*}$-component in $\mathrm{Aut}_{0}(X_{i})$, to verify that we can
apply Proposition 2.1 to deduce the vanishing of the Futaki invariant. We do
this for the two families separately.

####  7.5.1. Family 3.19

We follow the discussion in [PCS19, Lemma 5.13]. An automorphism of $X_{1}$
comes from an automorphism of $\mathbb{P}^{4}$ that leaves $Q$ and
$\\{P_{1}\\}\cup\\{P_{2}\\}$ invariant. By linearity, such an automorphism
preserves the line spanned by the two points, and thus its orthogonal
complement $\Pi=\\{x_{3}=x_{4}=0\\}$. It then leaves the conic $C=Q\cap\Pi$
invariant. From [PCS19, Lemma 5.7], the $\mathbb{C}^{*}$-component of
$\mathrm{Aut}_{0}(X_{1})$ can be identified with the pointwise stabiliser of
$C$ in $\mathrm{Aut}_{0}(Q)$. This is then the $\mathbb{C}^{*}$-action given
by

$\lambda\cdot([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[x_{0}:x_{1}:x_{2}:\lambda
x_{3}:\lambda^{-1}x_{4}].$

The adjoint action of $\tau$ maps a generator of this action to its inverse,
and by Proposition 2.1, we deduce the vanishing of the Futaki invariant of
$(X_{1},[\omega_{\varepsilon}])$.

####  7.5.2. Family 3.20

Following the discussion in [PCS19, Lemma 5.14], the
$\mathbb{C}^{*}$-component of $\mathrm{Aut}_{0}(X_{2})$ is obtained as
follows. An element in $\mathrm{Aut}_{0}(Q,L_{1}\cup L_{2})$ must preserve the
linear span of $L_{1}$ and $L_{2}$, that is $Q\cap\\{x_{0}=0\\}$. It then
leaves invariant

$Q^{\prime}=\\{x_{0}=x_{1}x_{2}+x_{3}x_{4}=0\\}.$

The group $\mathrm{Aut}_{0}(Q,L_{1}\cup L_{2})$ acts on the family of lines
$(\ell_{t})_{t\in\mathbb{P}^{1}}$ in $Q^{\prime}$ given by

$[x_{1}:x_{3}]\mapsto[0:x_{1}:tx_{3}:x_{3}:-tx_{1}]\subset Q^{\prime}.$

The $\mathbb{C}^{*}$-component of $\mathrm{Aut}_{0}(X)$ then corresponds to
the stabiliser of the lines $L_{1}=\ell_{\infty}$ and $L_{2}=\ell_{0}$ under
this action. In coordinates, the action is given by

$\lambda\cdot([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[\lambda
x_{0}:x_{1}:\lambda^{2}x_{2}:x_{3}:\lambda^{2}x_{4}].$

As with family 3.19, using the $\tau$-action and the $\mathrm{Ad}$-invariance
of the Futaki invariant, we can conclude that the Futaki invariant of
$(X_{2},[\omega_{\varepsilon}])$ vanishes.

###  7.6. Family 4.4

Up to isomorphism, there is a unique smooth Fano threefold $X$ in family 4.4.
Its automorphism group satisfies
$\mathrm{Aut}_{0}(X)\simeq(\mathbb{C}^{*})^{2}$, and it is $K$-polystable from
[Süß13, Süß14, IS17]. Recall that the smooth Fano threefold $X_{1}$ in family
3.19 can be obtained as a blow-up along two points of a smooth quadric
$Q\subset\mathbb{P}^{4}$. We can then realise the manifold $X$ as the blow-up
of $X_{1}$ along the proper transform of the conic that passes through the
blown-up points in $Q$. Coming back to our parametrisation in Section 7.5.1,
we can take $Q\subset\mathbb{P}^{4}$ to be

$Q=\\{x_{0}^{2}+x_{1}x_{2}+x_{3}x_{4}=0\\}\subset\mathbb{P}^{4}$

and the blown-up points to be

$P_{1}=[0:0:0:1:0]$

and

$P_{2}=[0:0:0:0:1].$

Then, the conic in $Q$ joining $P_{1}$ and $P_{2}$ is

$\mathcal{C}_{1}=\\{x_{1}=x_{2}=x_{0}^{2}+x_{3}x_{4}=0\\}\subset Q.$

The $(\mathbb{C}^{*})^{2}$-action on $Q$ that lifts to $X$ through the two
blow-up maps $X\to X_{1}\to Q$ is given in coordinates by

$(\lambda,\mu)\cdot([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[x_{0}:\lambda
x_{1}:\lambda^{-1}x_{2}:\mu x_{3}:\mu^{-1}x_{4}].$

Again, the involution

$\tau([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}])=[x_{0}:x_{2}:x_{1}:x_{4}:x_{3}]$

preserves $C_{1}$ and swaps the blown-up points. Arguing as before, we see
that the Futaki invariant of $X$ will vanish in classes of the form

$c_{1}(X)+\varepsilon\,c_{1}(\mathcal{E})+\delta\,(c_{1}(E_{1})+c_{1}(E_{2}))$

for $(\varepsilon,\delta)\in\mathbb{R}^{2}$ small enough and where
$\mathcal{E}$ is the exceptional divisor of $X\to X_{1}$, while $E_{1}$ and
$E_{2}$ are the strict transforms of the exceptional divisors of $X_{1}\to Q$.
Note that after scaling, this gives a $3$-dimensional family in the Kähler
cone of $X$.

###  7.7. Family 5.1

From [ACC+21, Section 5.23], the unique smooth Fano threefold $X$ in family
5.1 is $K$-polystable. It can be described as follows. Consider first the
smooth quadric in $\mathbb{P}^{4}$

$Q=\\{x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1}+x_{4}x_{5}=0\\}\subset\mathbb{P}^{4}$

where we denote by $[x_{1}:x_{2}:x_{3}:x_{4}:x_{5}]$ the homogeneous
coordinates on $\mathbb{P}^{4}$. We then fix a smooth conic
$C=Q\cap\\{x_{4}=x_{5}=0\\}\subset Q$ and points $P_{1}=[1:0:0:0:0]$,
$P_{2}=[0:1:0:0:0]$ and $P_{3}=[0:0:1:0:0]$ in $Q$. Let $Y\to Q$ the blow-up
of $Q$ in the three points $(P_{i})_{1\leq i\leq 3}$ and $\check{C}$ the
strict transform of $C$ in $Y$. Then, $X$ is obtained as the blow-up of $Y$
along $\check{C}$. Its automorphism group satisfies
$\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$, where the $\mathbb{C}^{*}$-action
is the lift of the action defined on $Q$ by

$\lambda\cdot([x_{1}:x_{2}:x_{3}:x_{4}:x_{5}])=[\lambda x_{1}:\lambda
x_{2}:\lambda x_{3}:\lambda^{2}x_{4}:x_{5}].$

The manifold $X$ also admits an involution which is the lift of the involution
$\tau$ defined on $Q$ by

$\tau([x_{1}:x_{2}:x_{3}:x_{4}:x_{5}])=[x_{1}:x_{2}:x_{3}:x_{5}:x_{4}].$

We observe that $\tau$ preserves the Kähler class associated to the hyperplane
section $H\cap Q$ and fixes $C$, as well as the points $P_{1}$, $P_{2}$ and
$P_{3}$. Hence, all the $(1,1)$-classes on $X$ are invariant under the
(lifted) involution. As the adjoint action of $\tau$ maps the generator of the
$\mathbb{C}^{*}$-action to its inverse, we conclude as in 2.3 the vanishing of
the Futaki invariant of $X$ for all its Kähler classes.

##  8\. Hypersurfaces in $\mathbb{P}^{2}\times\mathbb{P}^{2}$ and their blow-
ups

In this section, we consider families N°$\mathcal{N}$, with

$\mathcal{N}\in\\{2.24,3.8,4.7\\}.$

###  8.1. Family 2.24

From [Süß13, Süß14, IS17] (see also [ACC+21, Section 4.7]), the only
$K$-polystable element in Family 2.24 with infinite automorphism group is
given, up to isomorphism, by

$X=\\{xu^{2}+yv^{2}+zw^{2}\\}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}.$

It has $\mathrm{Aut}_{0}(X)\simeq(\mathbb{C}^{*})^{2}$, where the action of
$(\alpha,\beta)\in(\mathbb{C}^{*})^{2}$ is given by

$(\alpha,\beta)\cdot([x:y:z],[u:v:w])=([\alpha^{2}x:\beta^{2}y:z],[\alpha^{-1}u:\beta^{-1}v:w]).$

The group $G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ acts on
$\mathbb{P}^{2}\times\mathbb{P}^{2}$, with the action of $(\sigma,\tau)\in G$
generated by

$\sigma([x:y:z],[u:v:w])=([z:y:x],[w:v:u])$

and

$\tau([x:y:z],[u:v:w])=([x:z:y],[u:w:v]).$

Note that $G\subset\mathrm{Aut}(X)$ and that the inclusion
$\iota:X\to\mathbb{P}^{2}\times\mathbb{P}^{2}$ is $G$-equivariant. Hence we
deduce that $\iota^{*}[\omega_{FS}^{i}]$ is $G$-invariant, where
$\omega_{FS}^{i}$ denote the Fubini–Study metric on the $i$-th factor. Then,
any Kähler class on $X$ is $G$-invariant. Denote by $v_{1}$ (resp. $v_{2}$)
the generator of the $\mathbb{C}^{*}$-action
$\alpha\cdot([x:y:z],[u:v:w])=([\alpha^{2}x:y:z],[\alpha^{-1}u:v:w])$ (resp.
$\beta\cdot([x:y:z],[u:v:w])=([x:\beta^{2}y:z],[u:\beta^{-1}v:w])$) on $X$. A
direct computation shows

$\left\\{\begin{array}[]{ccc}\mathrm{Ad}_{\sigma}(v_{1})&=&-(v_{1}+v_{2})\\\
\mathrm{Ad}_{\tau}(v_{2})&=&-(v_{1}+v_{2}).\end{array}\right.$

Using $\mathrm{Ad}$-invariance of the Futaki invariant, as discussed in
Proposition 2.1, we deduce that for any Kähler class $\Omega$ on $X$ :

$\left\\{\begin{array}[]{ccc}\mathrm{Fut}_{(X,\Omega)}(v_{1})&=&-\mathrm{Fut}_{(X,\Omega)}(v_{1})-\mathrm{Fut}_{(X,\Omega)}(v_{2})\\\
\mathrm{Fut}_{(X,\Omega)}(v_{2})&=&-\mathrm{Fut}_{(X,\Omega)}(v_{1})-\mathrm{Fut}_{(X,\Omega)}(v_{2}),\end{array}\right.$

hence $\mathrm{Fut}_{X}$ is identically zero.

###  8.2. Family 3.8

From [ACC+21, Section 5.16], the only element in Family 3.8 with infinite
automorphism group is given, up to isomorphism, by $X=\mathrm{Bl}_{C}(Y)$ the
blow up of $Y$ along the curve $C$, where

$Y=\\{(vw+u^{2})x+v^{2}y+w^{2}z=0\\}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$

is a smooth divisor of degree $(1,2)$ and where $C=\pi_{1}^{-1}([1:0:0])$,
with $\pi_{1}$ the projection onto the first factor of
$\mathbb{P}^{2}\times\mathbb{P}^{2}$. The variety $Y$ is the only element in
Family 2.24 with infinite automorphism group, and

$\mathrm{Aut}(X)\simeq\mathrm{Aut}(Y)\simeq\mathbb{C}^{*}\rtimes\mathbb{Z}/2\mathbb{Z}.$

More explicitly, the $\mathbb{C}^{*}$-action is for $\lambda\in\mathbb{C}^{*}$
given by

$\lambda\cdot([x:y:z],[u:v:w])=([x:\lambda^{-2}y:\lambda^{2}z],[\lambda
u:\lambda^{2}v:w]),$

while the $\mathbb{Z}/2\mathbb{Z}$-action is generated by $\tau$:

$\tau([x:y:z],[u:v:w])=([x:z:y],[u:w:v]).$

Identifying $\lambda$ with the corresponding element in $\mathrm{Aut}(Y)$, we
have $\tau\circ\lambda\circ\tau^{-1}=\lambda^{-1}$, so that item $(iii)$ in
Corollary 2.3 is satisfied. The inclusion
$\iota:Y\to\mathbb{P}^{2}\times\mathbb{P}^{2}$ is $\tau$-equivariant, and then
the classes $\iota^{*}[\omega_{FS}^{i}]$ are $\tau$-invariant, for
$\omega_{FS}^{i}$ the Fubini–Study metric on each factor of
$\mathbb{P}^{2}\times\mathbb{P}^{2}$. This shows that hypothesis $(ii)$ from
Corollary 2.3 holds as well. Finally, the curve $C$ is $\tau$-invariant, and
by Corollary 2.3, the Futaki character of $X$ is identically zero on its
Kähler cone.

###  8.3. Family 4.7

Let $X$ be a smooth Fano threefold in family 4.7. Then it is a blow-up of a
smooth divisor $W$ of bidegree $(1,1)$ on $\mathbb{P}^{2}\times\mathbb{P}^{2}$
along two disjoints curves of bidegrees $(1,0)$ and $(0,1)$, and it is
$K$-polystable [Süß13, Süß14, IS17]. To perform computations, we will assume
that

$W=\\{xu+yv+zw=0\\}\subset\mathbb{P}^{2}\times\mathbb{P}^{2},$

where $[x,y,z]$ and $[u,v,w]$ stand for homogeneous coordinates on the first
and second factors respectively. We will denote by
$\pi_{i}:W\to\mathbb{P}^{2}$ the natural projection on the $i$-th factor. We
then let $C_{i}=\pi_{i}^{-1}([0:0:1])\subset W$. Then,
$X=\operatorname{Bl}_{C_{1}\cup C_{2}}(W)$ and from [PCS19, Lemmas 7.1 and
7.7], we have

$\mathrm{Aut}_{0}(X)\simeq\operatorname{GL}_{2}(\mathbb{C}).$

The isomorphism is defined as follows. First, automorphisms of $X$ are induced
by automorphisms of $W$ that leave $C_{1}\cup C_{2}$ invariant. Arguing as in
[PCS19, Lemma 7.7], they correspond to lift of isomorphisms of
$\mathbb{P}^{2}$ that leave the set

$\pi_{1}(C_{1}\cup
C_{2})=\\{[0:0:1]\\}\cup\\{[x:y:0],(x,y)\in\mathbb{C}^{2}\setminus\\{0\\}\\}$

invariant. Those elements are easily identified to elements in
$\operatorname{GL}_{2}(\mathbb{C})$. From Section 3, the Futaki invariant
vanishes on the $\mathfrak{sl}_{2}(\mathbb{C})$-component in
$\mathfrak{aut}(X)$. We can identify a supplementary subspace of
$\mathfrak{sl}_{2}(\mathbb{C})$ in $\mathfrak{aut}(X)$ by considering the lift
to $X$ of the generators of the $\mathbb{C}^{*}$-action on $\mathbb{P}^{2}$
given by

$\lambda\cdot([x:y:z])=([\lambda x:y:z]).$

The lift of this action to $W$ is given by

(8.1) $\lambda\cdot([x:y:z],[u:v:w])=([\lambda x:y:z],[\lambda^{-1}u:v:w]).$

We introduce the involution

$\tau([x:y:z],[u:v:w])=([u:v:w],[x:y:z]).$

This preserves $W$, and swaps the curves $C_{1}$ and $C_{2}$. It also swaps
the $(1,1)$-classes $\pi_{1}^{*}[\omega_{FS}]$ and $\pi_{2}^{*}[\omega_{FS}]$.
Finally, its adjoint actions maps a generator of the $\mathbb{C}^{*}$-action
(8.1) to its inverse. Then, following Section 2, we deduce the vanishing of
the Futaki invariant on $X$ for any Kähler class of the form

$c_{1}(X)+\varepsilon\pi^{*}(\pi_{1}^{*}[\omega_{FS}]+\pi_{2}^{*}[\omega_{FS}])+\eta(c_{1}(\mathcal{O}(E_{1}))+c_{1}(\mathcal{O}(E_{2}))),$

where $\pi:X\to W$ denotes the blow-down map, $E_{1}$ and $E_{2}$ the
exceptional divisors, and $(\varepsilon,\eta)\in\mathbb{R}^{2}$ are chosen so
that the class is positive.

##  9\. Remaining cases

We finish with families N°$\mathcal{N}$, with

$\mathcal{N}\in\\{2.20,3.9,3.13,4.2\\}.$

###  9.1. Family 2.20

Consider the Plücker embedding of $\mathrm{Gr}(2,5)$ in $\mathbb{P}^{9}$. Any
smooth intersection of this embedded sixfold with a linear subspace of
codimension 3 is a Fano manifold. We call this Fano threefold $V_{5}$ and it
is the unique member of family 1.15 of Fano threefolds.

Now, let $C$ be a twisted cubic in $V_{5}$ and let
$X=\operatorname{Bl}_{C}(V_{5})$. Then $X$ is a member of the family 2.20 of
Fano threefolds. Up to isomorphism, there is a unique choice of curve such
that $X$ has infinite automorphism group [PCS19, Lemma 6.10]. In this case,
$\mathrm{Aut}(X)$ is a semidirect product
$\mathbb{C}^{*}\rtimes\mathbb{Z}/2\mathbb{Z}$.

In [ACC+21, Section 5.8], it is shown that the unique element in family 2.20
with infinite automorphism group is $K$-polystable. Moreover, the following
explicit description of $X$ is given. First, $V_{5}$ can be realised as the
subvariety of $\mathbb{P}^{6}$ cut out by the equations

$\left\\{\begin{array}[]{lllll}x_{4}x_{5}-x_{0}x_{2}+x_{1}^{2}=0\\\
x_{4}x_{6}-x_{1}x_{3}+x_{2}^{2}=0\\\ x_{4}^{2}-x_{0}x_{3}+x_{1}x_{2}=0\\\
x_{1}x_{4}-x_{0}x_{6}-x_{2}x_{5}=0\\\
x_{2}x_{4}-x_{3}x_{5}-x_{1}x_{6}=0.\end{array}\right.$

We will then identify $V_{5}$ with this variety. Then, we can chose $C$ to be
the twisted cubic parametrised by

$([r:s])\mapsto([r^{3}:r^{2}s:rs^{2}:s^{3}:0:0:0])\in V_{5}.$

We consider $X=\operatorname{Bl}_{C}(V_{5})$ with this parametrisation. The
$\mathbb{C}^{*}\rtimes\mathbb{Z}/2\mathbb{Z}$-action on $\mathbb{P}^{6}$
generated by

$\lambda\cdot([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}])=[\lambda^{3}x_{0}:\lambda^{5}x_{1}:\lambda^{7}x_{2}:\lambda^{9}x_{3}:\lambda^{6}x_{4}:\lambda^{4}x_{5}:\lambda^{8}x_{6}]$

for $\lambda\in\mathbb{C}^{*}$ and the involution

$\tau([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}:x_{5}:x_{6}])=[x_{3}:x_{2}:x_{1}:x_{0}:x_{4}:x_{6}:x_{5}]$

preserves $V_{5}$ and $C$, hence lifts to $X$. This provides the isomorphism

$\mathrm{Aut}(X)\simeq\mathbb{C}^{*}\rtimes\mathbb{Z}/2\mathbb{Z}.$

Note that conjugation by $\tau$ sends a generator of the
$\mathbb{C}^{*}$-action to its inverse. As $H^{1,1}(V_{5},\mathbb{R})$ is
generated by the class of a hyperplane section in $\mathbb{P}^{6}$, and as the
class of the Fubini–Study metric on $\mathbb{P}^{6}$ is $\tau$-invariant, we
can apply Corollary 2.3 to $X$, and we deduce the vanishing of the Futaki
invariant on the Kähler cone of $X$.

###  9.2. Families 3.9 and 4.2

We now consider families 3.9 and 4.2. Any member of one of these families have
$\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$ and is $K$-polystable by [ACC+21,
Section 4.6], which we will follow closely.

Let $\mathcal{S}$ be either $\mathbb{P}^{2}$ or
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and let $\mathcal{C}\subset\mathcal{S}$
be a smooth irreducible curve given by a quartic if
$\mathcal{S}=\mathbb{P}^{2}$ and a $(2,2)$-curve in the other case. Denote by
$\mathrm{pr}_{i}$ the projection of $\mathbb{P}^{1}\times\mathcal{S}$ onto the
$i$-th factor. We then set
$\mathcal{B}=\mathrm{pr}_{2}^{*}(\mathcal{C})\simeq\mathbb{P}^{1}\times\mathcal{C}$,
$\mathcal{E}=\mathrm{pr}_{1}^{*}([1:0])$ and
$\mathcal{E}^{\prime}=\mathrm{pr}_{1}^{*}([0:1])$. We consider

$G=\mathbb{C}^{*}\rtimes\mathbb{Z}/2\mathbb{Z}$

acting on $\mathbb{P}^{1}$ by

$\lambda\cdot[u:v]=[u:\lambda v]$

and

$\tau([u:v])=[v:u].$

The $G$-action lifts to $\mathbb{P}^{1}\times\mathcal{S}$, with the involution
$\tau$ swapping $\mathcal{E}$ and $\mathcal{E}^{\prime}$. We then introduce
$\eta:W\to\mathbb{P}^{1}\times\mathcal{S}$ a double cover branched over
$\mathcal{E}+\mathcal{E}^{\prime}+\mathcal{B}$, and $\overline{E}$,
$\overline{E}^{\prime}$ and $\overline{B}$ the preimages on $W$ of the
surfaces $\mathcal{E}$, $\mathcal{E}^{\prime}$ and $\mathcal{B}$ respectively.
Then, set $\hat{X}\to W$ the blow-up of $W$ along the curves
$\overline{E}\cap\overline{B}$ and $\overline{E}^{\prime}\cap\overline{B}$
with exceptional surfaces $\hat{S}$ and $\hat{S}^{\prime}$. We denote the
proper transforms of $\overline{E}$, $\overline{E}^{\prime}$ and
$\overline{B}$ by $\hat{E}$, $\hat{E}^{\prime}$, $\hat{B}$ respectively.
Finally, $X$ is obtained as the image of a contraction $\hat{X}\to X$ of
$\hat{B}$ to a curve isomorphic to $\mathcal{C}$. We set $E$, $E^{\prime}$,
$S$ and $S^{\prime}$ the proper transforms on $X$ of $\hat{E}$,
$\hat{E}^{\prime}$, $\hat{S}$ and $\hat{S}^{\prime}$ respectively.

One can check that all the birational maps involved in producing $X$ are
$G$-equivariant, and we obtain $\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$.
Moreover, the involution on $X$ induced by $\tau$ (that we will still denote
$\tau$) swaps $E$ and $E^{\prime}$, and also swaps $S$ and $S^{\prime}$.
Hence, the Kähler classes $c_{1}(E)+c_{1}(E^{\prime})$ and
$c_{1}(S)+c_{1}(S^{\prime})$ are both $\tau$-invariant. Clearly, on
$\mathbb{P}^{1}\times\mathcal{S}$, the adjoint action of the involution $\tau$
maps a generator of the $\mathbb{C}^{*}$-action to its inverse. This remains
true on $W$ by equivariance, and thus on $X$ that is birationally equivalent
to $W$ by continuity of holomorphic vector fields away from the exceptional
loci. Then, Proposition 2.1 applies to show that the Futaki invariant of $X$
vanishes in any Kähler class of the form

$c_{1}(X)+\varepsilon(c_{1}(E)+c_{1}(E^{\prime}))+\delta(c_{1}(S)+c_{1}(S^{\prime})),$

where $(\varepsilon,\delta)\in\mathbb{R}^{2}$ is chosen so that the class is
positive.

To understand the subset of the Kähler cone these classes generate, we use the
following alternative description of $X$, still following [ACC+21, Section
4.6].

####  9.2.1. Family 3.9

This is the case when $\mathcal{S}=\mathbb{P}^{2}$. $X$ can then also be
obtained as the blow-up $\phi:X\to V$ of $V$ along a curve $C\subset V$ where

$\pi:V=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(2))\to\mathbb{P}^{2}=\mathcal{S}$

is a $\mathbb{P}^{1}$-bundle, and $C=\pi^{*}\mathcal{C}\cap E_{V}$, where
$E_{V}$ is the zero section of $\pi$. We also have that the strict transform
of $E_{V}$ (resp. of the infinity section $E^{\prime}_{V}$, and of
$\pi^{*}\mathcal{C}$) on $X$ is $E$ (resp. $E^{\prime}$ and $S^{\prime}$),
while the exceptional divisor of $\phi$ is $S$. Hence we get the relation
$c_{1}(E)+c_{1}(E^{\prime})=0$ in this case (but
$c_{1}(S)+c_{1}(S^{\prime})\neq 0$), and we obtain a $2$-dimensional family of
classes that admit cscK metrics given by

$(\delta,r)\to r(c_{1}(X)+\delta(c_{1}(S)+c_{1}(S^{\prime}))).$

####  9.2.2. Family 4.2

This is the case when $\mathcal{S}=\mathbb{P}^{1}\times\mathbb{P}^{1}$. Again,
we can recover $X$ from the maps

$\pi:X\to V$

and

$\phi:V\to\mathcal{S},$

with $\pi$ the contraction of $S$ to a curve isomorphic to $\mathcal{C}$ and
$\phi$ a $\mathbb{P}^{1}$-bundle over $\mathbb{P}^{1}\times\mathbb{P}^{1}$.
According to [Fuj16, Section 10], we have

$\mathrm{Pic}(X)=\mathbb{Z}[H_{1}]\oplus\mathbb{Z}[H_{2}]\oplus\mathbb{Z}[E]\oplus\mathbb{Z}[E^{\prime}]$

where $H_{i}=(\pi\circ\phi)^{*}(\ell_{i})$ and $\ell_{1},\ell_{2}$ denote two
different rulings of $\mathbb{P}^{1}\times\mathbb{P}^{1}$. There are relations
$-K_{X}\sim 2(H_{1}+H_{2})+E+E^{\prime}$, $S\sim H_{1}+H_{2}-E+E^{\prime}$ and
$S^{\prime}\sim H_{1}+H_{2}+E-E^{\prime}$, so that the Kähler classes
described above can be written

$2(1+\delta)(c_{1}(H_{1})+c_{1}(H_{2}))+(1+\varepsilon)(c_{1}(E)+c_{1}(E^{\prime})).$

Together with scaling we therefore obtain a $3$-dimensional family of classes
with vanishing Futaki invariant.

###  9.3. Family 3.13

Let $X$ be a smooth $K$-polystable Fano threefold in family 3.13. From
[ACC+21, Section 5.19], either
$\mathrm{Aut}_{0}(X)\simeq\mathrm{PGL}_{2}(\mathbb{C})$, and so from Section 3
the Futaki invariant vanishes identically, or
$\mathrm{Aut}_{0}(X)\simeq\mathbb{C}^{*}$. In the latter case, denoting
$[x_{0}:x_{1}:x_{2}]$, $[y_{0}:y_{1}:y_{2}]$ and $[z_{0}:z_{1}:z_{3}]$ the
homogeneous coordinates on the first, second and third factors of
$\mathbb{P}^{2}\times\mathbb{P}^{2}\times\mathbb{P}^{2}$, $X$ is given by the
equations

$\left\\{\begin{array}[]{ccc}x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}&=&0\\\
y_{0}z_{0}+y_{1}z_{1}+y_{2}z_{2}&=&0\\\
(1+s)x_{0}z_{1}+(1-s)x_{1}z_{0}-2x_{2}z_{2}&=&0\end{array}\right.$

in $\mathbb{P}^{2}\times\mathbb{P}^{2}\times\mathbb{P}^{2}$, for
$s\notin\\{-1,0,1\\}$, and

$\mathrm{Aut}(X)\simeq\mathbb{C}^{*}\rtimes\mathfrak{S}_{3}.$

The $\mathbb{C}^{*}$-action for $\lambda\in\mathbb{C}^{*}$ is given on a point
$P$ with homogeneous coordinates
$([x_{0}:x_{1}:x_{2}],[y_{0}:y_{1}:y_{2}],[z_{0}:z_{1}:z_{3}])$ by

$\lambda\cdot(P)=([\lambda
x_{0}:\lambda^{-1}x_{1}:x_{2}],[\lambda^{-1}y_{0}:\lambda
y_{1}:y_{2}],[\lambda z_{0}:\lambda^{-1}z_{1}:z_{3}]).$

Further, there are two involutions $\tau_{x,z}$ and $\tau_{y,z}$ in
$\mathrm{Aut}(X)$, whose actions are given by

$\tau_{x,z}(P)=([z_{1}:z_{0}:z_{2}],[y_{1}:y_{0}:y_{2}],[x_{1}:x_{0}:x_{2}])$

and

$\tau_{y,z}(P)=([x_{1}:x_{0}:-x_{2}],[(1-s)z_{0}:(1+s)z_{1}:2z_{2}],[\frac{y_{0}}{1-s}:\frac{y_{1}}{1+s}:\frac{y_{2}}{2}]).$

Note that $\tau_{x,z}\circ\lambda\circ\tau_{x,z}^{-1}=\lambda^{-1}$ and
$\tau_{y,z}\circ\lambda\circ\tau_{y,z}^{-1}=\lambda^{-1}$ (where we identified
$\lambda$ with the corresponding element in $\mathrm{Aut}(X)$). From [ACC+21,
Diagram 5.19.1], the projection maps
$\eta_{x},\eta_{y},\eta_{z}:\mathbb{P}^{2}\times\mathbb{P}^{2}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$
induce holomorphic maps, still denoted $\eta_{x},\eta_{y}$ and $\eta_{z}$,
from $X$ to $\mathbb{P}^{2}$. If we denote
$\alpha_{i}:=\eta_{i}^{*}[\omega_{FS}]\in H^{1,1}(X,\mathbb{R})$ the pullback
of the class of the Fubini–Study form, for $i\in\\{x,y,z\\}$, by equivariance
of the projections, we see that $\alpha_{y}$ is $\tau_{x,z}$-invariant while
$\alpha_{x}$ is $\tau_{y,z}$-invariant. Hence, for any $\varepsilon>0$ small
enough, the class $c_{1}(X)+\varepsilon\alpha_{x}$ is $\tau_{y,z}$-invariant
and the class $c_{1}(X)+\varepsilon\alpha_{y}$ is $\tau_{x,z}$-invariant. From
Proposition 2.1, the Futaki invariants of $(X,c_{1}(X)+\varepsilon\alpha_{x})$
and $(X,c_{1}(X)+\varepsilon\alpha_{y})$ vanish. Hence, $X$ will carry cscK
deformations of its Kähler–Einstein metrics in the classes
$c_{1}(X)+\varepsilon\alpha_{y}$ and $c_{1}(X)+\varepsilon\alpha_{x}$ for
$\varepsilon$ small enough by LeBrun–Simanca’s openness theorem.

######  Remark 9.1.

We have used two different involutions $\tau_{x,z}$ and $\tau_{y,z}$ in the
above to the deduce the vanishing of the Futaki invariant in the classes
$c_{1}(X)+\varepsilon\alpha_{y}$ and $c_{1}(X)+\varepsilon\alpha_{x}$. We are
therefore not able from these arguments to deduce that the Futaki invariant
vanishes on the sums of these classes. Hence we still only get a
$2$-dimensional family of classes with vanishing Futaki invariant in this
case.

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