On the Futaki invariant of Fano threefolds

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the K\"ahler 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 K\"ahler cone. In all cases, when the Picard rank is greater or equal to two, we exhibit explicit 2-dimensional differentiable families of K\"ahler classes containing the anti-canonical class and on which the Futaki invariant is identically zero. As a corollary, we deduce the existence of non K\"ahler-Einstein cscK metrics on all such Fano threefolds.


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).
Then, the Futaki invariant of X vanishes identically on its Kähler cone.
Note that when Aut(X) is finite or when the Picard rank ρ(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 ρ(X) ≥ 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.
Then, there is at least a 2-dimensional family of Kähler classes on X, containing c 1 (X), where the Futaki invariant vanishes.
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 Sk lodowska-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 Aut(X) (respectively Aut 0 (X)) its automorphism group (respectively the connected component of the identity of the reduced automorphism group of X), and by aut(X) the Lie algebra of Aut(X).If Z ⊂ X is a subvariety (not necessarily connected), Aut(X, Z) stands for elements in Aut(X) that leave Z globally invariant.We denote by K X the Kähler cone of X.We will identify a divisor D with O(D), and use the notation c 1 (D) for its first Chern class.

Preliminaries
Let X be a compact Kähler manifold, and Ω ∈ K X a Kähler class on X.We denote the Futaki invariant of (X, Ω) by where aut 0 (X) is the Lie algebra of the reduced automorphism group of X, g denotes a Kähler metric with Kähler form in Ω and volume form dµ g , f v,g is the normalised holomorphy potential of v with respect to g, and 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]).
By construction, Fut X vanishes on any class that admits a cscK metric, and it is then straightforward that Fut X ≡ 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 2 X ∈ {6, 7, 8, 9}.Moreover, it is K-polystable and of Picard rank ρ(X) ≥ 2 if and only if X = P 1 × P 1 or K 2 X = 6, i.e. when X is a blow-up of P 2 along three non-collinear points ( [TY87,Tia90]).In the first case, X admits a product cscK metric in each class, and Fut X ≡ 0, while in the latter case, the vanishing locus of Fut X is described in [WZ11, Section 5] (see Section 4.2 for the exact description).

Further properties of the Futaki invariant.
The key property that we will use is the invariance of Fut X under the Aut(X)-action.This was already used in [Fut88, Chapter 3] to show the vanishing of Fut X on specific examples.
We will use the following proposition repeatedly.
Proposition 2.1.Let (X, Ω) be a polarised Fano manifold.Assume that there is τ ∈ Aut(X) and v ∈ aut(X) such that Proof.This follows from the Ad-invariance of the Futaki invariant, which implies that Fut (X,(τ As an application, we have the following useful corollary : Corollary 2.3.Let π : X → Y be the blow-up of a smooth Fano manifold Y along smooth and disjoint subvarieties Z i ⊂ Y .Assume that there is a finite group G ⊂ Aut(Y ) such that : Proof.From hypothesis (i), the G-action on Y lifts to a G-action on X.The vector space H 1,1 (X, R) is spanned by the pullback of the classes in H 1,1 (Y, R) and the exceptional divisors of π.By hypothesis (i) and (ii), any class in H 1,1 (X, R) is then G-invariant.The Lie algebra aut(X) is spanned by lifts of elements in aut(Y ) that preserve the Z i 's.For any such element, the identity Ad τ (v) = c • v holds on X \ i π −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 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 and hence its class [ω F S ] ∈ H 1,1 (P n , R), is invariant under the S n+1 -action on P n by permutation of the homogeneous coordinates.4.1.Families 2.34 and 3.27.The unique members in these two families are P 1 × P 2 and P 1 × P 1 × 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 P 1 ×S 6 , where S 6 is the Del Pezzo surface with K 2 S6 = 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 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

2.3.
where a, b, c are positive constants satisfying a + b < 3 and c < 3 2 .As the Futaki invariant vanishes on P 1 , we easily deduce the vanishing locus of the Futaki invariant on X = P 1 × S 6 .In particular, as c 1 (X) = c 1 (P 1 ) + c 1 (S 6 ), and as )), we deduce the existence of differentiable families of Kähler classes on X containing c 1 (X) for which the Futaki invariant vanishes identically.

Blow-ups of projective space
In this section we address families N°N , with N ∈ {2.22, 3.12, 3.25}.
All the members of these families are obtained by blowing up certain curves in projective space P 3 .5.1.Family 2.22.Members of the family 2.22 of Fano threefolds are obtained as blowups of certain curves in P 3 .More precisely, let Φ : The image of Φ is the surface A Fano threefold X is in the family 2.22 if it is the blowup of the image via Φ of a curve Č with O( Č) = 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 Č to be Č0 = {ux 3 − vy 3 = 0}, so that preserves C 0 and so lifts to X 0 .This generates 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 Č0 ⊂ P 1 × P 1 of the projection to the second factor.An explicit parametrisation is given by . Note that the action of the involution τ given by on P 3 preserves C 0 and so lifts to X 0 .We can then apply Corollary 2.3 to the blow-up X 0 → P 3 with the group G = τ ≃ Z/2Z, 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 = Bl L∪C (P 3 ) the blow up of P 3 along the disjoint curves The reduced automorphism group of X is isomorphic to C * , and its action is given by the lift of the C * -action on P 3 described by Then, we can consider the Z/2Z-action given by τ ([x 0 : The group generated by τ in Aut(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 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} ⊂ P 3 and {x 3 = x 4 = 0} ⊂ P 3 .One has where the first (resp.second) GL 2 (C) factor acts linearly on the coordinates (x 1 , x 2 ) (resp. on (x 3 , x 4 )) while the C * -action corresponds to homotheties on C 4 (see [PCS19, Section 4]).The Lie algebra aut(X) of Aut 0 (X) fits in an exact sequence We also have the sequence induced by the trace map gl 2 (C) → C : from which we deduce the sequence of vector spaces From the discussion in Section 3, the Futaki invariant of X will vanish on the sl 2 (C)-factors that project to aut(X).Hence, it is enough to test the vanishing of the Futaki invariant on the generators of the remaining two C * -actions modulo homotheties, which are induced by: where (λ, µ) ∈ (C * ) 2 .We can consider the finite group G generated by the reflections This group preserves the two blown-up lines, while the adjoint action of τ (resp.σ) sends the generator of the λ-action (resp.the µ-action) to its inverse.Hence, from Corollary 2.3, we see that the Futaki invariant of X vanishes on its whole Kähler cone.

Blow-ups of products of projective spaces
In this section, we will consider families N°N , with N ∈ {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 = Bl C (P 1 ×P 2 ) the blow up of P 1 × P 2 along the curve C = ψ( Č) given by the image of Č = {ux 5 + vy 5 = 0} ⊂ P 1 × P 1 via the map ψ : Then, Aut 0 (X) ≃ C * , where the C * -action is generated by the lift to X of the action We also have a Z/2Z-action induced by Those actions come respectively from the actions on P 1 × P 1 , with respect to which ψ is equivariant.Then, we see that C is τinvariant, as well as the classes π * i [ω i F S ], where π 1 : P 1 × P 2 → P 1 and π 2 : P 1 × P 2 → P 2 denote the projections and ω i F S stands for the Fubini-Study metric on P i .Finally, identifying λ ∈ C * with its action, we have τ • λ • τ −1 = λ −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 P 1 × P 1 × P 1 along 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 Aut 0 (X) ≃ C * where the action is given by the lift of the C * -action on P 1 × P 1 × P 1 given by The involution preserves C and the (1, 1)-classes on C given by ι * j [ω F S ], for ι j the composition of the inclusion C ⊂ P 1 × P 1 × P 1 and the projection on the j-th factor.The adjoint action of τ maps the generator of the 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 = Bl C (P 1 × P 1 × P 1 ) the blow up of P 1 × P 1 × P 1 along the curve The reduced automorphism group of X is isomorphic to C * , and its action is given by the lift of the C * -action on P 1 × P 1 × P 1 described by Then, we can consider the Z/2Z-action given by Clearly, this action satisfies hypothesis (i) − (iii) from Corollary 2.3 (notice that τ • λ • τ −1 = λ −1 , identifying λ with the induced action), from which we deduce the vanishing of the Futaki invariant of X for any Kähler class.

Blow-ups of a smooth quadric
In this section, we consider families N°N , with N ∈ {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 C * and Z/2Z, one member which admits an effective PGL 2 -action and one member which has a reduced automorphism group 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 G m -action can be described as follows (see [ACC + 21, Section 5.9]).Let C be the quartic rational curve in P 4 given as the image of the map P 1 → P 4 given by [p : q] → [p 4 : p 3 q : p 2 q 2 : pq 3 : q 4 ].
For t / ∈ {0, ±1}, let Q t be the smooth hypersurface Then X t is one of the members that admit an effective C * -action (including the member with an effective PGL 2action, which corresponds to t = ± 1 2 ).Note that X t has Picard rank 2, generated by a hyperplane H and the exceptional divisor E of the blowup.
The C * -action given by preserves C and Q t , as does the involution The lifts of these generate the effective actions of C * ⋊ Z/2Z on X t .As τ preserves C, the class [ω F S ] |Xt , and sends a generator of the 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 = ± 1 2 , with aut(X t ) ≃ sl 2 (C), was dealt with in Section 3).

Family 2.29.
There is a unique smooth Fano threefold X in family 2.29.It is isomorphic to the blow-up of [Süß13,Süß14,IS17]) and the group Aut 0 (X) is isomorphic to From the discussion in Section 3, the Futaki invariant of (X, [ω]) vanishes on the sl 2 (C)-component of aut(X) for any Kähler class [ω].Thus, to check the vanishing of the Futaki invariant, it remains to check the vanishing on the C-component of aut(X).From [PCS19, Lemma 5.7], the C * -component of Aut 0 (X) can be identified with the pointwise stabiliser of C in Aut 0 (Q).This is then the C * -action induced by λ • ([x 0 : This automorphism of P 4 preserves Q and C and lifts to an automorphism of X.Its adjoint action maps a generator of the 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 Aut(X) is infinite.Then, from [ACC + 21, Section 5.17], up to isomorphism, we may assume that X = Bl C1∪C2 (Q a ) is the blow-up of the quadric along the two disjoint smooth irreducible conics C 1 ⊂ Q a and C 2 ⊂ Q a given by where [x, y, z, t, w] stand for the homogeneous coordinates on P 4 and where a ∈ C \ {−1, +1} is a complex parameter.Moreover, for a = 0, Aut 0 (X) ≃ (C * ) 2 and for a = 0, Aut 0 (X) ≃ C * .7.3.1.Case a = 0.In this situation, the (C * ) 2 -action on X is the lift of the action on Q 0 induced by the following formula, for (α, 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 The involution τ ([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 ι * [ω F S ] invariant, where ι : Q → P 4 is the inclusion.Then, Corollary 2.3 applies to X → Q and G = τ ≃ Z/2Z, so that the Futaki invariant of X identically vanishes on K X .7.5.Families 3.19 and 3.20.Consider the smooth quadric Fano threefold Q = {x 2 0 + x 1 x 2 + x 3 x 4 = 0} ⊂ 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 In both cases, the Fano threefold X i is K-polystable (see [Süß13,Süß14,IS17]) and the group Aut 0 (X i ) is isomorphic to C * × PGL 2 (C) (see [PCS19, Section 5]).We then have aut(X i ) = C ⊕ sl 2 (C).From the discussion in Section 3, the Futaki invariant of (X i , [ω i ]) vanishes on the sl 2 (C)-component of aut(X i ) for any Kähler class [ω i ].Therefore, to check the vanishing of the Futaki invariant on (X i , [ω i ]), it remains to check the vanishing on the C-component of aut(X i ).
To this aim we introduce the involution This automorphism of P 4 preserves Q, and swaps the two connected components of the blown-up locus in both cases.Therefore, τ lifts to an automorphism of X i , still denoted τ , for i ∈ {1, 2}.Note that on X i , any Kähler class of the form and the E i j 's denote the exceptional divisors of the blow-up X i → Q. Next, we investigate how this action interacts with the generator of the C *component in 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 P 4 that leaves Q and {P 1 } ∪ {P 2 } invariant.By linearity, such an automorphism preserves the line spanned by the two points, and thus its orthogonal complement Π = {x 3 = x 4 = 0}.It then leaves the conic C = Q ∩ Π invariant.From [PCS19, Lemma 5.7], the C * -component of Aut 0 (X 1 ) can be identified with the pointwise stabiliser of C in Aut 0 (Q).This is then the C * -action given by λ • ([x 0 : x 1 : x 2 : x 3 : The adjoint action of τ 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 , [ω ε ]).7.5.2.Family 3.20.Following the discussion in [PCS19, Lemma 5.14], the C *component of Aut 0 (X 2 ) is obtained as follows.An element in Aut 0 (Q, L 1 ∪ L 2 ) must preserve the linear span of L 1 and L 2 , that is Q ∩ {x 0 = 0}.It then leaves invariant The C * -component of Aut 0 (X) then corresponds to the stabiliser of the lines L 1 = ℓ ∞ and L 2 = ℓ 0 under this action.In coordinates, the action is given by λ As with family 3.19, using the τ -action and the Ad-invariance of the Futaki invariant, we can conclude that the Futaki invariant of (X 2 , [ω ε ]) 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 Aut 0 (X) ≃ (C * ) 2 , and it is Kpolystable 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 ⊂ 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 ⊂ P 4 to be Q = {x 2 0 + x 1 x 2 + x 3 x 4 = 0} ⊂ 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 Again, the involution 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 for (ε, δ) ∈ R 2 small enough and where E is the exceptional divisor of X → X 1 , while E 1 and E 2 are the strict transforms of the exceptional divisors of X 1 → 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 where we denote by [x 1 : x 2 : x 3 : x 4 : x 5 ] the homogeneous coordinates on P 4 .We then fix a smooth conic C = Q ∩ {x 4 = x 5 = 0} ⊂ Q and points P 1 = [1 : 0 : 0 : 0 : 0], P 2 = [0 : 1 : 0 : 0 : 0] and P 3 = [0 : 0 : 1 : 0 Then, X is obtained as the blow-up of Y along Č.Its automorphism group satisfies Aut 0 (X) ≃ C * , where the C * -action is the lift of the action defined on Q by The manifold X also admits an involution which is the lift of the involution τ defined on Q by We observe that τ preserves the Kähler class associated to the hyperplane section H ∩ 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 τ maps the generator of the 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 P 2 × P 2 and their blow-ups In this section, we consider families N°N , with N ∈ {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 , where the action of (α, β) ∈ (C * ) 2 is given by The group G = Z/2Z×Z/2Z acts on P 2 ×P 2 , with the action of (σ, τ ) ∈ G generated by σ([x : where ω i F S 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 C * -action α • ([x : Using Ad-invariance of the Futaki invariant, as discussed in Proposition 2.1, we deduce that for any Kähler class Ω on X : hence 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 = Bl C (Y ) the blow up of Y along the curve C, where is a smooth divisor of degree (1, 2) and where C = π −1 1 ([1 : 0 : 0]), with π 1 the projection onto the first factor of P 2 × P 2 .The variety Y is the only element in Family 2.24 with infinite automorphism group, and Identifying λ with the corresponding element in Aut(Y ), we have τ • λ • τ −1 = λ −1 , so that item (iii) in Corollary 2.3 is satisfied.The inclusion ι : Y → P 2 × P 2 is τequivariant, and then the classes ι * [ω i F S ] are τ -invariant, for ω i F S the Fubini-Study metric on each factor of P 2 × P 2 .This shows that hypothesis (ii) from Corollary 2.3 holds as well.Finally, the curve C is τ -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 P 2 × 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 where [x, y, z] and [u, v, w] stand for homogeneous coordinates on the first and second factors respectively.We will denote by π i : W → P 2 the natural projection on the i-th factor.We then let C i = π −1 i ([0 : 0 : 1]) ⊂ W .Then, X = Bl C1∪C2 (W ) and from [PCS19, Lemmas 7.1 and 7.7], we have The isomorphism is defined as follows.First, automorphisms of X are induced by automorphisms of W that leave C 1 ∪ C 2 invariant.Arguing as in [PCS19, Lemma 7.7], they correspond to lift of isomorphisms of P 2 that leave the set invariant.Those elements are easily identified to elements in GL 2 (C).From Section 3, the Futaki invariant vanishes on the sl 2 (C)-component in aut(X).We can identify a supplementary subspace of sl 2 (C) in aut(X) by considering the lift to X of the generators of the C * -action on P 2 given by λ • ([x : y : z]) = ([λx : y : z]).
The lift of this action to W is given by This preserves W , and swaps the curves C 1 and C 2 .It also swaps the (1, 1)-classes . Finally, its adjoint actions maps a generator of the 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 , where π : X → W denotes the blow-down map, E 1 and E 2 the exceptional divisors, and (ε, η) ∈ R 2 are chosen so that the class is positive.

Remaining cases
We finish with families N°N , with N ∈ {2.20, 3.9, 3.13, 4.2}.9.1.Family 2.20.Consider the Plücker embedding of Gr(2, 5) in 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 = 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, Aut(X) is a semidirect product C * ⋊ Z/2Z.
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 P 6 cut out by the equations x 2 x 4 − x 3 x 5 − x 1 x 6 = 0. We will then identify V 5 with this variety.Then, we can chose C to be the twisted cubic parametrised by ([r : s]) → ([r 3 : r 2 s : rs 2 : s 3 : 0 : 0 : 0]) ∈ V 5 .
Let S be either P 2 or P 1 × P 1 and let C ⊂ S be a smooth irreducible curve given by a quartic if S = P 2 and a (2, 2)-curve in the other case.Denote by pr i the projection of P 1 × S onto the i-th factor.We then set The G-action lifts to P 1 × S, with the involution τ swapping E and E ′ .We then introduce η : W → P 1 × S a double cover branched over E + E ′ + B, and E, E ′ and B the preimages on W of the surfaces E, E ′ and B respectively.Then, set X → W the blow-up of W along the curves E ∩ B and E ′ ∩ B with exceptional surfaces Ŝ and Ŝ′ .We denote the proper transforms of E, E ′ and B by Ê, Ê′ , B respectively.Finally, X is obtained as the image of a contraction X → X of B to a curve isomorphic to C. We set E, E ′ , S and S ′ the proper transforms on X of Ê, Ê′ , Ŝ and Ŝ′ respectively.
One can check that all the birational maps involved in producing X are Gequivariant, and we obtain Aut 0 (X) ≃ C * .Moreover, the involution on X induced by τ (that we will still denote τ ) swaps E and E ′ , and also swaps S and S ′ .Hence, the Kähler classes c 1 (E) + c 1 (E ′ ) and c 1 (S) + c 1 (S ′ ) are both τ -invariant.Clearly, on P 1 × S, the adjoint action of the involution τ maps a generator of the 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 where (ε, δ) ∈ 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 S = P 2 .X can then also be obtained as the blow-up φ : X → V of V along a curve C ⊂ V where π : V = P(O ⊕ O(2)) → P 2 = S is a P 1 -bundle, and C = π * C ∩ E V , where E V is the zero section of π.We also have that the strict transform of E V (resp. of the infinity section E ′ V , and of π * C) on X is E (resp.E ′ and S ′ ), while the exceptional divisor of φ is S. Hence we get the relation c 1 (E) + c 1 (E ′ ) = 0 in this case (but c 1 (S) + c 1 (S ′ ) = 0), and we obtain a 2-dimensional family of classes that admit cscK metrics given by (δ, r) → r(c 1 (X) + δ(c 1 (S) + c 1 (S ′ ))).9.2.2.Family 4.2.This is the case when S = P 1 × P 1 .Again, we can recover X from the maps π : X → V and φ : V → S, with π the contraction of S to a curve isomorphic to C and φ a P 1 -bundle over P 1 × P 1 .According to [Fuj16, Section 10], we have where H i = (π • φ) * (ℓ i ) and ℓ 1 , ℓ 2 denote two different rulings of P 1 × P 1 .There are relations −K X ∼ 2(H 1 + H 2 ) + E + E ′ , S ∼ H 1 + H 2 − E + E ′ and S ′ ∼ H 1 + H 2 + E − E ′ , so that the Kähler classes described above can be written 2(1 + δ)(c 1 (H 1 ) + c 1 (H 2 )) + (1 + ε)(c 1 (E) + c 1 (E ′ )).
Remark 9.1.We have used two different involutions τ x,z and τ y,z in the above to the deduce the vanishing of the Futaki invariant in the classes c 1 (X) + εα y and c 1 (X) + εα 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.