Lie Fu

CLASSIFICATION OF POLARIZED SYMPLECTIC AUTOMORPHISMS OF FANO VARIETIES OF CUBIC FOURFOLDS

We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.

Introduction

The purpose of this paper is to classify the polarized symplectic automorphisms of the irreducible holomorphic symplectic projective varieties constructed by Beauville and Donagi [START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF], namely, the Fano varieties of (smooth) cubic fourfolds.

Finite order symplectic automorphisms of K3 surfaces have been studied in detail by Nikulin in [START_REF] Nikulin | Finite groups of automorphisms of Kählerian K3 surfaces[END_REF]. A natural generalization of K3 surfaces to higher dimensions is the notion of irreducible holomorphic symplectic manifolds or hyper-Kähler manifolds (cf. [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]), which by definition is a simply connected compact Kähler manifold with H 2,0 generated by a symplectic form (i.e. nowhere degenerate holomorphic 2-form). Initiated by Beauville [START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0, Classification of algebraic and analytic manifolds[END_REF], some results have been obtained in the study of automorphisms of such manifolds. Let us mention [START_REF]Antisymplectic involutions of holomorphic symplectic manifolds[END_REF], [START_REF] Boissière | A note on automorphisms and birational transformations of holomorphic symplectic manifolds[END_REF], [START_REF] Boissière | Automorphismes naturels de l'espace de Douady de points sur une surface[END_REF], [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF].

In [START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF], Beauville and Donagi show that the Fano varieties of lines of smooth cubic fourfolds provide an example of a 20-dimensional family of irreducible holomorphic symplectic projective fourfolds. We propose to classify the polarized symplectic automorphisms of this family.

Our result of classification is shown in the table below 1 . We firstly make several remarks concerning this table :   • As is remarked in §1, such an automorphism comes from a (finite order) automorphism of the cubic fourfold itself. Hence we express the automorphism in the fourth column as an element f in PGL 6 . • In the third column, n is the order of f , which is primary (i.e. a power of a prime number). The reason why we only listed the automorphisms with primary order is that every finite order automorphism is a product of commuting automorphisms with primary orders, by the structure of cyclic groups. See Remark 3.3. • We give an explicit basis of the family in the fifth column. • In the last column, we work out the fixed loci for a generic member. For geometric descriptions of the fixed loci, see §4. • The Family I in our classification has been discovered in [START_REF] Mongardi | On symplectic automorphisms of Hyperkähler fourfolds[END_REF].

• The Family V-(1) in our classification has been studied in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], where the fixed locus and the number of moduli are calculated. • The classification of prime order automorphisms of cubic fourfolds has been done in [START_REF] González | Automorphisms of prime order of smooth cubic n-folds[END_REF]. I am also informed by G. Mongardi that he classifies the prime order symplectic automorphisms of hyper-Kähler varieties which are of K3 [n] -deformation type in his upcoming thesis.

Theorem 0.1 (Classification). Here is the list of all families of cubic fourfolds equipped with an automorphism of primary order whose general member is smooth, such that the induced actions on the Fano varieties of lines are symplectic. Family p n = p m automorphism basis for B fixed loci 0 1 1 f = id P 5 degree 3 monomials F(X)

I 11 11 f = diag(1, ζ, ζ -1 , ζ 3 , ζ -5 , ζ 4 ) x 2 0 x 1 5 points ζ = e r 11 •2π √ -1 , 1 ≤ r ≤ 10 x 2 1 x 2 x 2 2 x 3 x 2 3 x 4 x 2 4 x 0 x 3 5 II 7 7 f = diag(1, ζ, ζ -1 , ζ 3 , ζ 2 , ζ -3 ) x 2 0 x 1 9 points ζ = e r 7 •2π √ -1 , 1 ≤ r ≤ 6 x 2 1 x 2 x 2 2 x 3 x 2 3 x 4 x 2 4 x 5 x 2 5 x 0 x 0 x 2 x 4 x 1 x 3 x 5 III 5 5 f = diag(1, ζ, ζ -1 , ζ -2 , ζ 2 , ζ 2 ) x 2 0 x 1 14 points ζ = e r 5 •2π √ -1 , 1 ≤ r ≤ 4 x 2 1 x 2 x 2 2 x 3 x 2 3 x 0 x 2 4 x
ω = e 2π √ -1 3 degree 3 monomials on x 2 , x 3 degree 3 monomials on x 4 , x 5 x 0 x 2 x 4 x 0 x 2 x 5 x 0 x 3 x 4 x 0 x 3 x 5 x 1 x 2 x 4 x 1 x 2 x 5 x 1 x 3 x 4 x 1 x 3 x 5 IV-(4) 3 f = diag(1, ζ -3 , ζ 3 , ζ, ζ 4 , ζ -2 ) x 2 0 x 1 9 points ζ = e r 9 •2π √ -1 , r = 1, 2, 4, 5, 7, 8 x 2 1 x 2 x 2 2 x 0 x 2 3 x 4 x 2 4 x 5 x 2 5 x 3 IV-(5) 3 f = diag(1, ζ 3 , ζ -3 , ζ, ζ, ζ 4 ) x 2 0 x 1 9 points ζ = e r 9 •2π √ -1 , r = 1, 2, 4, 5, 7, 8 x 2 1 x 2 x 2 2 x 0 x 2 3 x 4 x 3 x 2 4 x 3 3 x 3 4 x 3 5 LIE FU V-(1) 2 2 f = diag(1, 1, 1, 1, -1, -1) degree 3 monomials on x 0 , • • • , x 3 28 points and x 0 x 2 5 , x 1 x 2 5 , x 2 x 2 5 , x 3 x 2 5 a K3 surface x 0 x 2 4 , x 1 x 2 4 , x 2 x 2 4 , x 3 x 2 4 x 0 x 4 x 5 , x 1 x 4 x 5 , x 2 x 4 x 5 , x 3 x 4 x 5 V-(2)(a) 2 4 f = diag(1, 1, -1, -1, √ -1, - √ -1) x 3 0 , x 2 0 x 1 , x 0 x 2 1 , x 3 1 15 points x 0 x 2 2 x 1 x 2 2 x 0 x 2 3 x 1 x 2 3 x 0 x 2 x 3 x 1 x 2 x 3 x 2 x 2 4 x 3 x 2 4 x 2 x 2 5 x 3 x 2 5 x 0 x 4 x 5 x 1 x 4 x 5 V-(2)(b) 2 4 f = diag(1, 1, -1, -1, √ -1, - √ -1) x 2 • degree 2 monomials on x 0 , x 1 15 points x 3 • degree 2 monomials on x 0 , x 1 x 3 2 x 3 3 x 2 2 x 3 x 2 x 2 3 x 0 x 2 4 x 1 x 2 4 x 0 x 2 5 x 1 x 2 5 x 2 x 4 x 5 x 3 x 4 x 5 V-(3) 2 8 f = diag(1, -1, ζ 2 , ζ -2 , ζ, ζ 3 ) x 3 0 6 points ζ = e r 8 •2π √ -1 , r = ±1 mod 8 x 0 x 2 1 x 1 x 2 2 x 1 x 2 3 x 0 x 2 x 3 x 3 x 2 4 x 2 x 2 5 x 1 x 4 x 5
The structure of this paper is as follows. In §1 we set up the basic notation, and show that any polarized automorphism of the Fano variety comes from a finite order automorphism of the cubic fourfold. Then in §2 we reinterpret the assumption of being symplectic into a numerical equation by using Griffiths' theory of residue. Finally we do the classification in §3. The basic observation is that the generic smoothness of the family of cubics imposes strong combinatoric constrains.

Throughout this paper, we work over the field of complex numbers with a fixed choice of √ -1.

Acknowledgements: I would like to express my gratitude to Olivier Benoist, Zhi Jiang and Junyi Xie for their careful reading of the preliminary version of the paper as well as many helpful suggestions. I also want to thank Chiara Camere for a nice conversation at Luminy and for pointing out to me the paper [START_REF] González | Automorphisms of prime order of smooth cubic n-folds[END_REF]. Finally I wish to thank Giovanni Mongardi for informing me about his related classification work in his upcoming thesis.

Fano varieties of lines of cubic fourfolds

First of all, let us fix the notation and make some basic constructions. Let V be a 6-dimensional C-vector space, and P 5 := P(V) be the corresponding projective space of 1-dimensional subspaces of V. Let X ⊂ P 5 be a smooth cubic fourfold. The following subvariety of the Grassmannian Gr(P 1 , P 5 ) ( 1)

F(X) := [L] ∈ Gr(P 1 , P 5 ) | L ⊂ X
is called the Fano variety of lines2 of X. It is well-known that F(X) is a 4-dimensional smooth projective variety. Throughout this paper, we always equip F(X) with the polarization L , which is by definition the restriction to it of the Plücker line bundle on the ambient Grassmannian Gr(P 1 , P 5 ).

Consider the incidence variety (i.e. the universal projective line over F(X)):

P(X) := {(x, [L]) ∈ X × F(X) | x ∈ L} ,
and then the following natural correspondence:

P(X) q / / p X F(X)
we have the following Theorem 1.1 ). Keeping the above notation, (i) F(X) is a 4-dimensional irreducible holomorphic symplectic projective variety, i.e. F(X) is simplyconnected and H 2,0 (F(X)) = C • ω with ω a no-where degenerate holomorphic 2-form.

(ii) The correspondence p * q * : H 4 (X, Z) → H 2 (F(X), Z)
is an isomorphism of Hodge structures.

By definition, an automorphism ψ of F(X) is called polarized, if it preserves the Plücker polarization: ψ * L ≃ L . Now we investigate the meaning for an automorphism of F(X) to be polarized.

Lemma 1.2. An automorphism ψ of F(X) is polarized if and only if it is induced from an automorphism of the cubic fourfold X itself.

Proof. See [START_REF] Franc | A remark on the Torelli theorem for cubic fourfolds[END_REF]Proposition 4].

Define Aut(X) to be the automorphism group of X, and Aut pol (F(X), L ) or simply Aut pol (F(X)) to be the group of polarized automorphisms of F(X). Then Lemma 1.2 says that the image of the natural homomorphism Aut(X) → Aut(F(X)) is exactly Aut pol (F(X)). This homomorphism of groups is clearly injective (since for each point of X there passes a 1-dimensional family of lines), hence we have

Corollary 1.3. The natural morphism Aut(X) ≃ -→ Aut pol (F(X))
which sends an automorphism f of X to the induced (polarized) automorphism f of F(X) is an isomorphism. Remark 1.4. This group is a finite group. Indeed, since Pic(X) = Z • O X (1), all its automorphisms come from linear automorphisms of P 5 , hence Aut(X) is a closed subgroup of PGL 6 thus of finite type. On the other hand, H 0 (F(X), T F(X) ) = H 1,0 (F(X)) = 0, which implies that the group considered is also discrete, therefore finite. By Corollary 1.3, the classification of polarized symplectic automorphisms of F(X) is equivalent to the classification of automorphism of cubic fourfolds such that the induced action satisfies the symplectic condition. The first thing to do is to find a reformulation of this symplectic condition purely in terms of the action on the cubic fourfold:

The symplectic condition

The content of this section has been done in my paper [10, Section 1]. For the sake of completeness, we briefly reproduce it here. Keeping the notation in the previous section. Suppose the cubic fourfold X ⊂ P 5 is defined by a polynomial T ∈ H 0 (P 5 , O(3)) = Sym 3 V ∨ . Let f be an automorphism of X. By Remark 1.4, f is the restriction of a finite order linear automorphism of P 5 preserving X, still denoted by f . Let n ∈ N + be its order. We can assume without loss of generality that f : P 5 → P 5 is given by:

(2)

f : [x 0 :

x 1 : • • • : x 5 ] → [ζ e 0 x 0 : ζ e 1 x 1 : • • • : ζ e 5 x 5 ],
where

ζ = e 2π √ -1 n is a primitive n-th root of unity and e i ∈ Z/nZ for i = 0, • • • , 5.
It is clear that X is preserved by f if and only if the defining equation T is contained in an eigenspace of Sym 3 V ∨ . More precisely: let the coordinates x 0 , x 1 , • • • , x 5 of P 5 be a basis of V ∨ , then x α α∈Λ is a basis of Sym 3 V ∨ = H 0 (P 5 , O(3)), where

x α denotes x α 0 0 x α 1 1 • • • x α 5 5 . Define (3) Λ := α = (α 0 , • • • , α 5 ) ∈ N 5 | α 0 + • • • + α 5 = 3 .
We write the eigenspace decomposition of Sym 3 V ∨ :

Sym 3 V ∨ = j∈Z/nZ          α∈Λ j C • x α          ,
where for each j ∈ Z/nZ, we define the subset of Λ (4)

Λ j := α = (α 0 , • • • , α 5 ) ∈ N 5 | α 0 +•••+α 5 =3
e 0 α 0 +•••+e 5 α 5 = j mod n . and the eigenvalue of α∈Λ j C • x α is thus ζ j . Therefore, explicitly speaking, we have: Lemma 2.1. A cubic fourfold X is preserved by the f in [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF] if and only if there exists a j ∈ Z/nZ such that its defining polynomial T ∈ α∈Λ j C • x α .

Then we deal with the symplectic condition. Note that Theorem 1.1 (ii) says in particular that

p * q * : H 3,1 (X) ≃ -→ H 2,0 (F(X))
is an isomorphism. If X is equipped with an action f as before, we denote by f the induced automorphism of F(X). Since the construction of F(X) as well as the correspondence p * q * are both functorial with respect to X, the condition that f is symplectic i.e. f * acts on H 2,0 (F(X)) as identity, is equivalent to the condition that f * acts as identity on H 3,1 (X). Work it out explicitly, we arrive at the congruence equation ( 5) in the following Lemma 2.2 (Symplectic condition). Let f be the linear automorphism in [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF], and X be a cubic fourfold defined by equation T . Then the followings are equivalent:

• f preserves X and the induced action f on F(X) is symplectic;

• There exists a j ∈ Z/nZ satisfying the equation

(5) e 0 + e 1 + • • • + e 5 = 2 j mod n,
such that the defining polynomial T ∈ α∈Λ j C • x α , where as in ( 4)

Λ j := α = (α 0 , • • • , α 5 ) ∈ N 5 | α 0 +•••+α 5 =3 e 0 α 0 +•••+e 5 α 5 = j mod n .
Proof. Firstly, the condition that f preserves X is given in Lemma 2.1. As is remarked before the lemma, f is symplectic if and only if f * acts as identity on H 3,1 (X). On the other hand, by Griffiths' theory of the Hodge structures of hypersurfaces (cf. [START_REF] Voisin | Hodge theory and complex algebraic geometry[END_REF]Chapter 18]), H 3,1 (X) is generated by the residue Res Ω T 2 , where

Ω := 5 i=0 (-1) i x i dx 0 ∧ • • • ∧ dx i ∧ • • • ∧ dx 5 is a generator of H 0 (P 5 , K P 5 (6)). f being defined in (2), we find f * Ω = ζ e 0 +•••+e 5 Ω and f * (T ) = ζ j T . Hence the action of f * on H 3,1 (X) is the multiplication by ζ e 0 +•••+e 5 -2 j
, from where we obtain the equation ( 5).

Classification

We now turn to the classification. Retaining the notation of §2: (2),( 3),( 4), [START_REF] Boissière | Automorphismes naturels de l'espace de Douady de points sur une surface[END_REF]. We define the parameter space ( 6)

B := P          α∈Λ j C • x α          .
Let B ⊂ B be the open subset parameterizing the smooth ones.

In this paper we are only interested in the smooth cubic fourfolds, that is the case when B ∅, or equivalently, when a general member of B is smooth. The easy observation below (see Lemma 3.1) which makes the classification feasible is that this non-emptiness condition imposes strong combinatoric constrains on the defining equations.

Lemma 3.1. If a general member in B is smooth then for

each i ∈ {0, 1, • • • , 5}, there exists i ′ ∈ {0, 1, • • • , 5}, such that x i 2 x i ′ ∈ B.
Proof. Suppose on the contrary that, without loss of generality, for i = 0, none of the monomials x 3 0 ,

x 2 0 x 1 , x 2 0 x 2 , x 2 0 x 3 , x 2 0 x 4 , x 2 0
x 5 are contained in B, then every equation in this family can be written in the following form:

x 0 Q(x 1 , • • • , x 5 ) + C(x 1 , • • • , x 5 )
, where Q (resp. C) is a homogeneous polynomial of degree 2 (resp. 3). It is clear that [1, 0, 0, 0, 0, 0] is always a singular point, which is a contradiction.

Since a finite-order automorphism amounts to the action of a finite cyclic group, which is the product of some finite cyclic groups with order equals to a power of a prime number, we only have to classify automorphisms of primary order, that is n = p m for p a prime number and m ∈ N + . To get general results for any order from the classification of primary order case, see Remark 3.3. We thus assume n = p m in the sequel.

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For the convenience of the reader, we summarize all the relevant equations:

(7)                    e 0 + e 1 + • • • + e 5 = 2 j mod p m ; α 0 + • • • + α 5 = 3; α i ∈ N; e 0 α 0 + • • • + e 5 α 5 = j mod p m ; ( * ) ∀i, ∃i ′ such that 2e i + e i ′ = j mod p m
where the last condition ( * ) comes from Lemma 3.1.

We associate to each solution of (7) a diagram, i.e. a finite oriented graph, as follows:

(i) The vertex set is the quotient set of {0, • • • , 5} with respect to the equivalence relation defined by: i 1 ∼ i 2 if and only if e i 1 = e i 2 mod p m . (ii) For each pair (i, i ′ ) satisfying 2e i + e i ′ = j mod p m , there is an arrow from i to i ′ .

Clearly the arrows in (ii) are well-defined cause we have taken into account of the equivalence relation in (i). It is also obvious that each vertex can have at most one arrow going out. Thanks to the condition ( * ) in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], we know that each vertex has exactly one arrow going out.

If p 2, it is easy to see that each vertex has at most one arrow coming in. Since the total going-out degree should coincide with the total coming-in degree, each vertex has exactly one arrow coming-in. As a result, the diagram is in fact a disjoint union of several cycles3 in this case.

Before the detailed case-by-case analysis, let us point out that a cycle in the diagram would have some congruence implications:

Lemma 3.2.

(i) There cannot be cycles of length 2. (ii) If p 3, there are at most one cycle of length 1. (iii) If there is a cycle of length l =3, 4, 5 or 6, then p divides (-2) l -1 3 .

Proof. (i) It is because 2e i + e i ′ = 2e i ′ + e i mod p m will imply e i = e ′ i mod p m , contradicts to the definition of a cycle. (ii) A cycle of length 1 means 3e i = j mod p m , and when p 3, e i is determined by j. (iii) Without loss of generality, we can assume that the cycle is given by:

2e 0 + e 1 = 2e 1 + e 2 = • • • = 2e l-2 + e l-1 = 2e l-1 + e 0 = j mod p m .
This system of congruence equations implies that As in the previous case, by Lemma 3.2, cycles of length 2,3,4 or 6 cannot occur. Thus the only possible lengths of cycles are 1 and 5. If there is no cycle of length 5, then as before, since p 3, all e i 's will be equal and f will be the identity. Let the 5-cycle be 2e 0 + e 1 = 2e 1 + e 2 = 2e 2 + e 3 = 2e 3 + e 4 = 2e 4 + e 0 = j mod 11 m .

(8) (-2) l -1 e 0 = (-2) l -1 3 • j mod p m . If p does not divide (-2) l -1
As in [START_REF] Franc | A remark on the Torelli theorem for cubic fourfolds[END_REF], 33e 0 = 11 j mod 11 m . There thus exists r ∈ Z/11Z, such that j = 3e 0 + r • 11 m-1 mod 11 m , and

                               e 0 = e 0 ; e 1 = e 0 + r • 11 m-1 ; e 2 = e 0 -r • 11 m-1 ; e 3 = e 0 + 3r • 11 m-1 ; e 4 = e 0 -5r • 11 m-1 ; e 5 = e 0 + 4r • 11 m-1 ; mod 11 m
where the last equality comes from the first equation in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF]. Clearly, r cannot be 0, otherwise f = id P 5 . One verifies easily that it is indeed a solution: 3e 5 = j. As a result, remembering that we are in the projective space P 5 , [START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF]. Going back to [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], we easily work out all solutions for α i 's:

f =                          1 ζ ζ -1 ζ 3 ζ -5 ζ 4                          where ζ = e r 11 •2π √ -1 and 1 ≤ r ≤ 10. That is, p = 11, m = 1, (e 0 , • • • , e 5 ) = (0, 1, -1, 3, -5,
(α 0 , • • • , α 5 ) = (2,
1, 0, 0, 0, 0), (0, 2, 1, 0, 0, 0), (0, 0, 2, 1, 0, 0), (0, 0, 0, 2, 1, 0), (1, 0, 0, 0, 2, 0), (0, 0, 0, 0, 0, 3).

Thus the corresponding family

B = P Span x 2 0 x 1 , x 2 1 x 2 , x 2 2 x 3 , x 2 3 x 4 , x 2 4 x 0 , x 3 5 .
In order to verify the smoothness of a general member, it suffices to give one smooth cubic fourfold in B.

For example,

x 2 0 x 1 + x 2 1 x 2 + x 2 2 x 3 + x 2 3 x 4 + x 2 4 x 0 + x 3 5
is smooth. This is Family I in Theorem 0.1. We would like to mention that this example has been discovered in [START_REF] Mongardi | On symplectic automorphisms of Hyperkähler fourfolds[END_REF].

Case II. When p = 7. As before, by Lemma 3.2, cycles of length 2,3,4 or 5 cannot occur. Thus the only possible lengths of cycles are 1 and 6; and except the trivial Family 0, there must be a 6-cycle: As in [START_REF] Franc | A remark on the Torelli theorem for cubic fourfolds[END_REF], 63e 0 = 21 j mod 7 m .

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There thus exists r ∈ Z/7Z, such that j = 3e 0 + r • 7 m-1 mod 7 m , and Clearly, r cannot be 0, otherwise f = id P 5 . One verifies easily that it is indeed a solution:

                               e 0 =
e 0 + • • • + e 5 = 2 j.
As a result,

f =                          1 ζ ζ -1 ζ 3 ζ 2 ζ -3                          where ζ = e r 7 •2π √ -1 and 1 ≤ r ≤ 6. That is, p = 7, m = 1, (e 0 , • • • , e 5 ) = (0, 1, -1, 3, 2, -3).
Going back to (7), we easily work out all solutions for α i 's:

(α 0 , • • • , α 5 ) = (2,
1, 0, 0, 0, 0), (0, 2, 1, 0, 0, 0), (0, 0, 2, 1, 0, 0), (0, 0, 0, 2, 1, 0), (0, 0, 0, 0, 2, 1), (1, 0, 0, 0, 0,

Thus the corresponding family

B = P Span x 2 0 x 1 , x 2 1 x 2 , x 2 2 x 3 , x 2 3 x 4 , x 2 4 x 5 , x 2 5 x 0 , x 0 x 2 x 4 , x 1 x 3 x 5 .
As before, to show that a general member of this family is smooth, we only need to remark that

x 2 0 x 1 + x 2 1 x 2 + x 2 2 x 3 + x 2 3 x 4 + x 2 4 x 5 + x 2 5
x 0 is smooth. This accomplishes Family II in Theorem 0.1.

Case III. When p = 5.

As before, by Lemma 3.2, cycles of length 2,3,5 or 6 cannot occur. Thus the only possible lengths of cycles are 1 and 4; and except the trivial Family 0, there must be a 4-cycle:

2e 0 + e 1 = 2e 1 + e 2 = 2e 2 + e 3 = 2e 3 + e 0 = j mod 5 m .
As before, by [START_REF] Franc | A remark on the Torelli theorem for cubic fourfolds[END_REF] we get 15e 0 = 5 j mod 5 m .

There thus exists r ∈ Z/5Z, such that j = 3e 0 + r • 5 m-1 mod 5 m , and As a result,

                   e 0 =
f =                          1 ζ ζ -1 ζ -2 ζ a ζ 4-a                          where ζ = e r 5 •2π √ -1 for 1 ≤ r ≤ 4 and a ∈ Z/5Z. That is, p = 5, m = 1, (e 0 , • • • , e 5 ) = (0, 1, -1, -2, a, 4 -a).
Going back to [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], we work out the solutions for α i 's depending on the value of a:

Subcase III (i). When a = 0. p = 5, m = 1, (e 0 , • • • , e 5 ) = (0, 1, -1, -2, 0, -1), and

f =                          1 ζ ζ -1 ζ -2 1 ζ -1                          where ζ = e r 5 •2π √ -1 for 1 ≤ r ≤ 4.
Solving α i 's from the equation ( 7):

(α 0 , • • • , α 5 ) = (2, 1, 0, 0, 0, 0), (0, 2, 1, 0, 0, 0), (0, 0, 2, 1, 0, 0), (1, 0, 0, 2, 0, 0), (0, 1, 0, 0, 2, 0), (0, 0, 0, 2, 1, 0), (0, 2, 0, 0, 0, 1), (0, 0, 0, 1, 0, 2), (1, 1, 0, 0, 1, 0), (0, 0, 1, 1, 0, 1).

Thus the corresponding family

B = P Span x 2 0 x 1 , x 2 1 x 2 , x 2 2 x 3 , x 2 3 x 0 , x 2 4 x 1 , x 2 3 x 4 , x 2 1 x 5 , x 2 5 x 3 , x 0 x 1 x 4 , x 2 x 3 x 5 .
However, there is no smooth cubic fourfolds in this family: in fact each member would have two singular points in the line (

x 0 = x 1 = x 3 = x 4 = 0).

Subcase III (ii). When

a = 1. p = 5, m = 1, (e 0 , • • • , e 5 ) = (0, 1, -1, -2, 1, -2), and ζ = e r 5 •2π √ -1 for 1 ≤ r ≤ 4 f =                          1 ζ ζ -1 ζ -2 ζ ζ -2                         
By the transformation ζ → ζ 3 (which amounts to let r → 3r), this f is exactly the one in Subcase III(i).

Subcase III (iii).

When a = 2. p = 5, m = 1, (e 0 , • • • , e 5 ) = (0, 1, -1, -2, 2, 2), and

f =                          1 ζ ζ -1 ζ -2 ζ 2 ζ 2                          where ζ = e r 5 •2π √ -1 for 1 ≤ r ≤ 4.
Solving α i 's from the equation ( 7):

(α 0 , • • • , α 5 ) = (2, 1, 0, 0, 0, 0), (0, 2, 1, 0, 0, 0), (0, 0, 2, 1, 0, 0), (1, 0, 0, 2, 0, 0), (0, 0, 0, 0, 2, 1), (0, 0, 0, 0, 1, 2), (0, 0, 0, 0, 0, 3), (0, 0, 0, 0, 3, 0), (1, 0, 1, 0, 1, 0), (1, 0, 1, 0, 0, 1), (0, 1, 0, 1, 1, 0), (0, 1, 0, 1, 0, 1).

Thus the corresponding family

B = P Span x 2 0 x 1 , x 2 1 x 2 , x 2 2 x 3 , x 2 3 x 0 , x 2 4 x 5 , x 2 5 x 4 , x 3 5 , x 3 4 , x 0 x 2 x 4 , x 0 x 2 x 5 , x 1 x 3 x 4 , x 1 x 3 x 5 .
Moreover, a general cubic fourfold in this family is smooth. Indeed, we give a particular smooth member:

x 2 0 x 1 + x 2 1 x 2 + x 2 2 x 3 + x 2 3 x 0 + x 3 4 + x 3 5
. This is Family III in Theorem 0.1. Subcase III (iv). When a = 3. By the symmetry between a and b, it is the same case as Subcase III(ii), hence as Subcase III(i).

Subcase III (v).

When a = 4. By the symmetry between a and b, it is the same case as Subcase III(i).

Case IV. When p = 3. Still by Lemma 3.2, we know that cycles of length 2,4 or 5 cannot occur. Thus the only possible lengths of cycles are 1, 3 and 6. We first claim that 6-cycle cannot exist. Suppose on the contrary that the diagram is a 6-cycle: 2e 0 + e 1 = 2e 1 + e 2 = 2e 2 + e 3 = 2e 3 + e 4 = 2e 4 + e 5 = 2e 5 + e 0 = j mod 3 m , then we have as in (8) that 63e 0 = 21 j mod 3 m . There thus exists r ∈ Z/3Z, such that j = 3e 0 + r • 3 m-1 mod 3 m , and

                               e 0 = e 0 ; e 1 = e 0 + r • 3 m-1 ; e 2 = e 0 -r • 3 m-1 ; e 3 = e 0 ; e 4 = e 0 + r • 3 m-1 ; e 5 = e 0 -r • 3 m-1 . mod 3 m
This contradicts to the assumption that e i 's are distinct. Therefore, there are only 1-cycles and 3-cycles. A 1-cycle means 3e i = j mod 3 m . On the other hand, a 3-cycle 2e 0 + e 1 = 2e 1 + e 2 = 2e 2 + e 0 = j mod 3 m would imply 9e 0 = 3 j. In particular, 9e 0 = 9e 1 = • • • = 9e 5 = 3 j mod 3 m . Without loss of generality, we can demand e 0 = 0. As a result, f has the form

f = diag(1, ζ a 1 , • • • , ζ a 5 ) where ζ = e 2π √ - 1 9
. In particular, f is of order 3 or 9.

Subcases IV (i). If

f is of order 3. Let ω := e 2π √ - 1 3
. Then up to isomorphism, f is one of the following automorphisms:

• diag(1, 1, 1, 1, 1, ω): this case does not satisfy condition ( * ).

• diag(1, 1, 1, 1, 1, ω 2 ): this case does not satisfy condition ( * ).

• diag(1, 1, 1, 1, ω, ω 2 ): we find Family IV-(1) in Theorem 0.1. We remark that its general member is indeed smooth because in particular the Fermat cubic fourfold (which is smooth) is contained in this family. • diag(1, 1, 1, 1, ω, ω): this case does not satisfy condition ( * ).

• diag(1, 1, 1, 1, ω 2 , ω 2 ): this case does not satisfy condition ( * ).

• diag(1, 1, 1, ω, ω, ω 2 ): Here we find B has a basis:

x 5 • degree 2 monomials on x 0 , x 1 and x 2 ;

x 4 x 2 5 , x 3 x 2 5 , x 0 x 3 x 4 , x 1 x 3 x 4 , x 2 x 3 x 4 , x 0 x 2 3 , x 1 x 2 3 , x 2 x 2 3 , x 0 x 2 4 , x 1 x 2 4 , x 2 x 2 4
, However, any cubic fourfold in this family is singular along a conic curve in the projective plane (x 3 = x 4 = x 5 = 0).

• diag(1, 1, 1, ω 2 , ω 2 , ω): this is as in the previous case, with ω be replaced by ω 2 .

• diag(1, 1, 1, ω, ω, ω): By solving [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], we find a the following basis for B: degree 3 monomials on x 0 , x 1 and x 2 ; degree 3 monomials on x 3 , x 4 and x 5 .

As the Fermat cubic fourfold is in this family, the general member is also smooth. This is Family IV-(2) in Theorem 0.1.

• diag(1, 1, ω, ω, ω 2 , ω 2 ): The basis of B is degree 3 monomials on x 0 and x 1 ; degree 3 monomials on x 2 and x 3 ; degree 3 monomials on x 4 and x 5 ;

x 0 x 2 x 4 , x 0 x 2 x 5 , x 0 x 3 x 4 , x 0 x 3 x 5 , x 1 x 2 x 4 , x 1 x 2 x 5 , x 1 x 3 x 4 , x 1 x 3 x 5 .
Because B contains the Fermat cubic fourfold, its general member is smooth. This is Family IV-(3) in Theorem 0.1. From which we have 3 j = 9e 0 = 9e 3 mod 3 m . Hence there exists t = ±1 such that j = 3e 0 + t • 3 m-1 , and

Subcase IV (ii). If the diagram consists of

                               e 0 = e 0 ; e 1 = e 0 + t • 3 m-1 ; e 2 = e 0 -t • 3 m-1 ; e 3 = e 0 + r • 3 m-2 ; e 4 = e 0 + t • 3 m-1 -2r • 3 m-2 ; e 5 = e 0 -t • 3 m-1 + 4r • 3 m-2 . mod 3 m
where r ∈ Z/9Z. Note that r 0, 3, 6 mod 9, since otherwise e i 's cannot be distinct. By the first equation in ( 7), t = -r mod 3. Putting this back into the previous system of equations, we obtain: p = 3, m = 2, n = 9, (e 0 , • • • , e 5 ) = (0, -3, 3, 1, 4, -2), j = -3 mod 9, and

f =                          1 ζ -3 ζ 3 ζ ζ 4 ζ -2                         
where ζ = e r 9 •2π √ -1 for r ∈ {1, 2, 4, 5, 7, 8}. Solving α i 's from the equation ( 7):

(α 0 , • • • , α 5 ) = (2, 1, 0, 0, 0, 0), (0, 2, 1, 0, 0, 0), (1, 0, 2, 0, 0, 0), (0, 0, 0, 2, 1, 0), (0, 0, 0, 0, 2, 1), (0, 0, 0, 1, 0, 2).

Thus the corresponding family

B = P Span x 2 0 x 1 , x 2 1 x 2 , x 2 2 x 0 , x 2 3 x 4 , x 2 4 x 5 , x 2 5 x 3 . Clearly, the cubic x 2 0 x 1 + x 2 1 x 2 + x 2 2 x 0 + x 2 3 x 4 + x 2 4 x 5 + x 2 5
x 3 is smooth, hence so is the general cubic fourfold in this family. This is Family IV-(4) in Theorem 0.1.

Subcase IV (iii). If the diagram contains only one 3-cycle:

2e 0 + e 1 = 2e 1 + e 2 = 2e 2 + e 0 = j mod 9.

As before, we can assume e 0 = 0, then e 1 = j, e 2 = -j and 3 j = 0 mod 9. In particular, 3| j. Since j 0 mod 9 (otherwise e 0 = e 1 = e 2 is a contradiction), j = ±3. For i = 3, 4, 5, e i either takes value in {e 0 , e 1 , e 2 }, or 3e i = j.

If j = 3, then f has the form f = diag(1, ζ 3 , ζ -3 , ζ a , ζ b , ζ c
), where a, b, c ∈ {0, 3, 6, 1, 4, 7}. By the first equation in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF],

a + b + c = 6 mod 9.
Thus either a, b, c ∈ {0, 3, 6}, or a, b, c ∈ {1, 4, 7}. While the former will make f of order 3, which has been treated in Subcases IV(i). Therefore a, b, c ∈ {1, 4, 7} and a + b + c = 6. There are only three possibilities (up to permutations of a, b, c): (a, b, c) = (1, 1, 4) or [START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF][START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF][START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF] or [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF][START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF][START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0, Classification of algebraic and analytic manifolds[END_REF]. However these three corresponds to the following same automorphism

f =                          1 ζ 3 ζ -3 ζ ζ ζ 4                         
Back to [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], we solve the corresponding α i 's to get the following basis for B:

B = P Span x 2 0 x 1 , x 2 1 x 2 , x 2 2 x 0 , x 2 3 x 4 , x 2 4 x 3 , x 3 3 , x 3 4 , x 3 5 .
As we have a smooth member

x 2 0 x 1 + x 2 1 x 2 + x 2 2 x 0 + x 3 3 + x 3 4 + x 3
5 is this family, the general one is also smooth. This is Family IV-( 5

Subcase IV (iv).

If the diagram has only 1-cycles, i.e. for any 0 ≤ i ≤ 5, 3e i = j mod 9.

In particular, 3| j. First of all, j 0, otherwise, f is of order 3, which is treated in Subcases IV(i). If j = 3. Then e i ∈ {1, 4, 7} for any i. Taking into account the first equation of [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], we find all the solutions for (e 0 , • • • , e 5 ), up to permutations: (e 0 , • • • , e 5 ) = (1, 1, 1, 4, 4, 4), (1, 1, 1, 7, 7, 7), [START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF][START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF][START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF][START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF][START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF][START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF] (

where the automorphisms in the first line is equal to diag(1, 1, 1, ω, ω, ω), which has been done in Family IV-(2); the automorphisms in the second line is equal to diag(1, 1, 1, 1, ω, ω 2 ), which has been done in Family IV-(1); the last automorphism is equal to diag(1, 1, ω, ω, ω 2 , ω 2 ), which has been done in Family IV-(3) in Theorem 0.1.

Case V. When p = 2. By Lemma 3.2, we find that the associated diagram has only 1-cycles. The new phenomenon is that the coming-in degree in this case is not necessarily 1. Firstly, we claim that the order of f divides 32. Indeed, for any 1-cycle, say, 3e 0 = j mod 2 m , then e 0 is in fact well-determined mod 2 m . Without loss of generality, we can assume that all the 1-cycles are 0, i.e. j = 0. As a result, a vertex pointing to a 1-cycle is divisible by 2 m-1 , and a vertex pointing to a vertex pointing to a 1-cycle is divisible by 2 m-2 , etc. . As there is no cycle of length ≥ 2, every vertex, after at most 5 steps, arrives at some 1-cycle vertex. Therefore, every vertex is divisible by 2 m-5 , hence we can reduce everything modulo 32: namely n = 32 and e i ∈ Z/32Z.

7 -9 3 -13 -5 11 - 15 1 -7 9 -3 13 5 -11 8 -8 -4 12 -12 2 -14 -6 10 14 6 -10 -1 15 0 16 -2 4
Let us put Z/32Z into the above complete binary tree, then clearly, our associated diagram is a subdiagram of this tree, satisfying two properties:

• If a vertex belongs to the diagram then so do its ancestors;

• The sum of vertices (multiplicities counted) is zero modulo 32.

It is immediate that the leaves (vertices on the bottom sixth level) cannot appear in the diagram: since by the parity of their sum, if there are leaves, there are at least two. But we already have five ancestors to include, while we have only six places in total. Next, we remark that the vertices in the fifth level cannot belong to the diagram neither: since the sum is divisible by 4, there are at least two vertices from the fifth level if there is any, and they should have the same father (otherwise we need to include at least five ancestors, and it will be out of place). Therefore we only have four possibilities, and it is straightforward to check that none of them has sum zero as demanded.

As a result, we have a further reduction: since only the first four levels can appear, the order of f always divides 8. We can assume now n = 8 and e i ∈ Z/8Z. Similarly, we put Z/8Z into the following complete binary tree:

Then our diagram is a sub-diagram of this tree which is as before 'ancestor-closed' and of multiplicities counted sum zero modulo 8. We easily work out all the possibilities as follows. It is worthy to point out that when p = 2, for a given automorphism f , there may be two possible values of j, which would correspond to two different families of cubic fourfolds.

• (e 0 , • • • , e 5 ) = (0, 0, 0, 0, 4, 4) or (0, 0, 4, 4, 4, 4). In this case, f is the involution diag(1, 1, 1, 1, -1, -1)

and we reduce to n = p = 2 with (e 0 , • • • , e 5 ) = (0, 0, 0, 0, 1, 1). It splits into two cases depending

• (e 0 , • • • , e 5 ) = (0, 0, 4, 4, 2, -2). It is the following automorphism of order n = 4:

f = diag(1, 1, -1, -1, √ -1, - √ -1).
We then reduce to n = 4 and (e 0 , • • • , e 5 ) = (0, 0, 2, 2, 1, -1). Therefore 2 j = 0 mod 4, hence we have two cases. If j = 0 mod 4, the equation for α i 's is 2α 2 + 2α 3 + α 4 -α 5 = 0 mod 4.It corresponds Family V-(2)(a) in Theorem 0.1. It is easy to find a smooth member, for example,

x 3 0 + x 3 1 + x 2 2 x 0 + x 2 3 x 1 + x 2 4
x 2 + x 2 5 x 3 . If j = 2 mod 4, the equation for α i 's is 2α 2 + 2α 3 + α 4 -α 5 = 2 mod 4, we have Family V-(2)(b) in Theorem 0.1. We give an example of smooth cubic fourfold in this family:

x 3 2 + x 3 3 + x 2 x 2 0 + x 3 x 2 1 + x 0 x 2 4 + x 1 x 2 5 .
• (e 0 , • • • , e 5 ) = (0, 4, 4, 2, -1, -1) or (0, 4, 4, 2, 3, 3) or (0, 4, 4, -2, 1, 1) or (0, 4, 4, -2, -3, -3). Although they are different automorphisms of order n = 8, they correspond to four possible choices of the primitive eighth root of unity ζ in the automorphism:

f = diag(1, -1, -1, ζ -2 , ζ, ζ),
where ζ = e r 8 •2π √ -1 for r = ±1, ±3. For each choice of r, one of the two possible values of j does not satisfies the last condition ( * ) in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF]. The remaining one gives the equation

4α 1 + 4α 2 -2α 3 + α 4 + α 5 = 0 mod 8.
The solutions form a basis for B:

B = P Span x 3 0 , x 0 x 2 1 , x 0 x 2 2 , x 0 x 1 x 2 , x 2 x 2 3 , x 1 x 2 3 , x 3 x 2 4 , x 3 x 2 5 , x 3 x 4 x 5 .
However any cubic fourfold in this family is singular on two points on the line defined by (x 0 = x 1 = x 2 = x 3 = 0).

• (e 0 , • • • , e 5 ) = (0, 0, 4, 2, -1, 3) or (0, 0, 4, -2, 1, -3). They are different automorphisms of order n = 8. In fact the four possible choices of the primitive eighth root of unity ζ collapse into two cases. The automorphism:

f = diag(1, 1, -1, ζ 2 , ζ -1 , ζ 3 ),
where ζ = e r 8 •2π √ -1 for r = ±1 (here r = ±3 will give the same two automorphisms). In this case, one of the two possible values of j does not satisfies the last condition ( * ) in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF]. The remaining one corresponds to the equation 4α 2 + 2α 3 -α 4 + 3α 5 = 0 mod 8.

We easily resolve it to obtain B = P Span degree 3 monomials on x 0 and x 1 ,

x 0 x 2 2 , x 1 x 2 2 , x 2 x 2 3 , x 3 x 2 4 , x 3 x 2 5
. But each member in this family is singular at least on two points of the line defined by (x 0 = x 1 = x 2 = x 3 = 0).

• (e 0 , • • • , e 5 ) = (0, 4, 2, -2, 1, 3) or (0, 4, 2, -2, -1, -3). As in the previous case, although they are different automorphisms of order n = 8, each corresponds to two possible choices of the primitive eighth root of unity ζ. The automorphism is

f = diag(1, -1, ζ 2 , ζ -2 , ζ, ζ 3 ),
where ζ = e r 8 •2π √ -1 for r = ±1 (here r = ±3 will give the same two automorphisms). In this case, one of the two possible values of j does not satisfies the last condition ( * ) in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF]. The remaining one is Family V-(3) in Theorem 0.1, where the smoothness of the general member is affirmed by the example:

x 3 0 + x 0 x 2 1 + x 1 x 2 2 + x 1 x 2 3 + x 0 x 2 x 3 + x 3 x 2 4 + x 2 x 2 5 .
Since the verification of the smoothness multiplications by third roots of unity on coordinates and permutations of coordinates. Using Griffiths' residue description of Hodge structure as in the proof of Lemma 2.2, we find that

Aut pol,symp (F(X)) = f ∈ Aut(X) | f * | H 3,1 (X) = id = (Z/3Z) 4 ⋊ A 6 ,
where each element has the form:

[x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] → [x σ(0) , ω i 1 x σ(1) , • • • , ω i 5 x σ(5) ],
where ω = e

• in V-(1), V-(2)(a) and V-(2)(b), because [x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] → [x 1 , x 2 , x 3 , x 0 , x 5 , x 4
] is an order 4 automorphism which induces a symplectic automorphism of its Fano variety. The eigenvalues are 1, 1, -1, -1, √ -1, -√ -1, therefore the automorphism is the one given in V-(2) (up to a linear automorphism of P 5 ).

Fixed loci

We calculate the fixed loci of a generic member for each example in the list of Theorem 0.1. Firstly, we make several general remarks concerning the fixed loci:

• For a smooth variety, the fixed locus of any automorphism of finite order is a (not necessarily connected) smooth subvariety. For a proof, cf. [9, Lemma 4.1]. • If furthermore the variety is symplectic and the finite-order automorphism preserves the symplectic form, then the components of the fixed locus are symplectic subvarieties. Indeed, for a given fixed point, the automorphism acts also on the tangent space at this fixed point, preserving the symplectic form, where the tangent space of the component of the fixed locus passing through this point is exactly the fixed subspace. However, since the fixed subspace is orthogonal to the other eigenspaces with respect to the symplectic form, it must be a symplectic subspace. In consequence, the fixed locus is a (smooth) symplectic subvariety.

Therefore, in the case of this paper, the fixed loci must be disjoint unions of (isolated) points, K3 surfaces and abelian surfaces, and we will see that all three types do occur in the list in Theorem 0.1.

We now turn to the calculation of the fixed loci of the examples in our classification. For a cubic fourfold X with an action f , we denote f the induced action on F(X). Then the fixed points of f in F(X) are the lines contained in X which are preserved by f . Since any automorphism of P 1 admits two (not necessarily distinct) fixed points, it suffices to check for each line joining two fixed points of f in X whether it is contained in X. In the following, we choose some typical examples in our list to give the argument in detail, while the complete result is presented in the last column of the table in Theorem 0.1. Denote P 0 := [1, 0, 0, 0, 0, 0], P 1 := [0, 1, 0, 0, 0, 0], • • • , P 5 := [0, 0, 0, 0, 0, 1]. We have explicit description of the fixed loci: LIE FU For Family I, a cubic fourfold in this family has equation of the following form

a 0 x 2 0 x 1 + a 1 x 2 1 x 2 + a 2 x 2 2 x 3 + a 3 x 2 3 x 4 + a 4 x 2 4
x 0 + a 5 x 3 5 . The fixed points of f in X are P 0 , P 1 , P 2 , P 3 , P 4 . We check the 10 possible lines joining two of them and find that only the following 5 are contained in X: P 0 P 2 , P 0 P 3 , P 1 P 3 , P 1 P 4 , P 2 P 4 , where PQ means the line joining two points P and Q. Similarly, the same argument applies in the following families and gives: Family II: the fixed points in F(X) are given by the following nine lines: P 0 P 2 , P 0 P 3 , P 0 P 4 , P 1 P 3 , P 1 P 4 , P 1 P 5 , P 2 P 4 , P 2 P 5 , P 3 P 5 . Family IV-(4): the fixed points of F(X) correspond to the following nine lines: P 0 P 3 , P 0 P 4 , P 0 P 5 , P 1 P 3 , P 1 P 4 , P 1 P 5 , P 2 P 3 , P 2 P 4 , P 2 P 5 . Family V-(3): the fixed points are given by the six lines: P 1 P 4 , P 1 P 5 , P 2 P 3 , P 2 P 4 , P 3 P 5 , P 4 P 5 .

For Family III, the equation has the following form

C(x 4 , x 5 ) + R(x 0 , • • • , x 5 ),
where C is a homogeneous polynomial of degree 3, and R is a polynomial with the degrees of x 4 and x 5 at most 1. The fixed points of f in X are P 0 , P 1 , P 2 , P 3 and the line P 4 P 5 . On one hand, among the six possible lines joining P 0 , P 1 , P 2 , P 3 , only P 0 P 2 and P 1 P 3 are contained in X; on the other hand, for 0 ≤ i ≤ 4, the line P i , [0, 0, 0, 0, λ, µ] is contained in X if and only if [λ, µ] satisfies the cubic equation C, and therefore we have three for each i. Altogether, the fixed locus in F(X) consists of 2 + 4 × 3 = 14 lines. Similar arguments gives the results of the following: Family IV-(3): the equation has the form C 1 (x 0 , x 1 ) + C 2 (x 2 , x 3 ) + C 3 (x 4 , x 5 ) + R, where C i are of degree 3 while each term of R is square-free. Then the fixed locus of in F(X) corresponds to the 27 lines Q ik Q jl for 0 ≤ i < j ≤ 3 and k, l = 1, 2, 3, where Q i1 , Q i2 , Q i3 are the three points satisfying the equation C i . Family IV-(5): the equation writes C(x 3 , x 4 ) + a 0 x 2 0 x 1 + a 1 x 2 1 x 2 + a 2 x 2 2 x 0 + a 5 x 3 5 , where C is of degree 3. Let Q 1 , Q 2 , Q 3 be the three points on the line P 3 P 4 satisfying C. Then the fixed locus in F(X) correspond to the 9 lines: P i Q j for i = 0, 1, 2 and j = 1, 2, 3.

For Family IV-(1), let defining equation of the cubic fourfold be

C(x 0 , • • • , x 3 ) + R
where C is of degree 3 and each term of R contains x 4 or x 5 . Clearly, the fixed locus of f in X is P 3 = (x 4 = x 5 = 0). A line in this P 3 is contained in X if and only if it satisfies C, namely, it is contained in the cubic surface defined by C. It is well-known that there are 27 such lines.

For Family IV-(2), let cubic fourfold be defined by C 1 (x 0 , x 1 , x 2 ) + C 2 (x 3 , x 4 , x 5 ), where C 1 , C 2 are of degree 3. The fixed locus in X is two disjoint planes: W 1 = (x 0 = x 1 = x 2 = 0) and W 2 = (x 3 = x 4 = x 5 = 0). On one hand, inside each plane it is impossible to have a line contained in X. On the other hand, a line joining a point Q 1 ∈ W 1 and a point Q 2 ∈ W 2 is contained is X if and only if Q i satisfies the equation C i for i = 1, 2, i.e. Q i is in the elliptic curve E i defined by C i . Thus such lines are parameterized by E 1 × E 2 , which is an abelian surface.

Family V-(1) is done also in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF], but we reproduce the argument for the sake of completeness. The equation has the following form:

C(x 0 , • • • , x 3 ) + x 2 4 L 1 + x 2 5 L 2 + x 4 x 5 L 3
, where C is of degree 3, and L 1 , L 2 , L 3 are linear forms in x 0 , • • • , x 3 . The fixed points of f in X is clearly the disjoint union of P 3 = (x 4 = x 5 = 0) and the line P 4 P 5 . First of all, the line P 4 P 5 is contained in X, giving a isolated fixed point in F(X); secondly a line of this P 3 is contained in X if and only if it satisfies C, i.e. it is contained in the cubic surface defined by C, and we thus obtain another 27 isolated fixed points in F(X); finally, the condition that a line joining a point Q 1 ∈ P 3 and another point Q 2 ∈ P 4 P 5 is contained in X is given by a double cover of the cubic surface (C = 0) ramified along the degree 6 curve (C = L 2 3 -L 1 L 2 = 0), which is a K3 surface. Altogether, the fixed locus of f in F(X) is 28 points with a K3 surface.

For Family V-(2)(a), the fixed point set of f consists of the disjoint union of P 0 P 1 , P 2 P 3 and P 4 , P 5 . The line P 4 P 5 is contained in X; there are three points Q 1 , Q 2 , Q 3 ∈ P 0 P 1 such that Q i P j is contained in X, for i = 1, 2, 3 and j = 4, 5; there are two points Q 4 , Q 5 ∈ P 2 P 3 such that Q 4 P 4 and Q 5 P 5 are contained in X; finally for each Q i ∈ P 0 P 1 , 1 ≤ i ≤ 3, there exists two points on P 2 P 3 such that the joining line is contained in X. Thus f has altogether 1 + 3 × 2 + 2 + 3 × 2 = 15 isolated fixed points. Similarly, for Family V-(2)(b), the fixed locus in F(X) also consists of 15 isolated points.

4 x 0 x 2 x 4 x 0 x 2 x 5 x 1 x 3 x 4 x 1 x 3 x 5 IV-( 1 ) 3 fx 0 x 4 x 5 x 1 x 4 x 5 x 2 x 4 x 5 x 3 x 4 x 5

 45135 = diag(1, 1, 1, 1, ω, ω 2 ) degree 3 monomials on x 0 , • • • , IV-(2) 3 f = diag(1, 1, 1, ω, ω, ω) degree 3 monomials on x 0 , x 1 , x 2 an abelian surface on x 3 , x 4 , x 5 IV-(3) 3 f = diag(1, 1, ω, ω, ω 2 , ω 2 )degree 3 monomials on x 0 , x 1 27 points

3 ,

 3 then by (8), we have j = 3e 0 mod p m , and hence e 0 = e 1 = • • • = e l-1 , contradicting to the definition of a cycle. Then we work out the classification case by case. The result is summarized in Theorem 0.1. Case 0. When p 2, 3, 5, 7 or 11. If we have a cycle of length l ≥ 3, since in Lemma 3.2, (-2) l -1 is Family 0 in Theorem 0.1. Case I. When p = 11.

  2e 0 + e 1 = 2e 1 + e 2 = 2e 2 + e 3 = 2e 3 + e 4 = 2e 4 + e 5 = 2e 5 + e 0 = j mod 7 m .

e 0 ; e 1 =

 1 e 0 + r • 7 m-1 ; e 2 = e 0 -r • 7 m-1 ; e 3 = e 0 + 3r • 7 m-1 ; e 4 = e 0 + 2r • 7 m-1 ; e 5 = e 0 -3r • 7 m-1 ; mod 7 m

e 0 ; e 1 = 5 m

 15 e 0 + r • 5 m-1 ; e 2 = e 0 -r • 5 m-1 ; e 3 = e 0 -2r • 5 m-1 ; mod Since e 0 e 1 , r cannot be 0. Since 2-cycle does not exists, for i = 4, 5, either e i takes the same value as e 0 , • • • , e 3 , or it is a 1-cycle, i.e. 3e i = j. In any case, we can write        e 4 = e 0 + ar • 5 m-1 ; e 5 = e 0 + br • 5 m-1 , mod 5 m where a, b ∈ Z/5Z. Taking into account of the first equation in (7), we obtain a + b = 4 mod 5.

  two 3-cycles (thus e i 's are distinct, so m ≥ 2):        2e 0 + e 1 = 2e 1 + e 2 = 2e 2 + e 0 = j 2e 3 + e 4 = 2e 4 + e 5 = 2e 5 + e 3 = j mod 3 m ,

  ) in Theorem 0.1. If j = -3, it reduces to the j = 3 case by replace ζ by ζ -1 , thus already included in Family IV-(5) of the theorem.

3 , i 1 ,

 31 • • • , i 5 = 0, 1 or 2 with sum i 1 + • • • + i 5 divisible by 3, and σ ∈ A 6 is a permutation of {0, 1, • • • , 5} with even sign.Then X is• not in Family I, II, IV-(4), IV-(5) or V-(3) simply because X does not admit automorphisms of order 11, 7, 9 or 8;• in Family III, since [x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] → [x 1 , x 2 , x 3 , x 4 , x 0 , x 5 ]is an order 5 automorphism which induces a symplectic automorphism on its Fano variety of lines. The eigenvalues of the corresponding permutation matrix is 1, 1, ζ, ζ 2 , ζ 3 , ζ 4 , thus it is exactly the automorphism in the list (up to a linear automorphism of P 5 ). • in IV-(1), IV-(2), IV-(3) obviously;

In the scheme-theoretic language, F(X) is defined to be the zero locus of s T ∈ H 0 Gr(P 1 , P 5 ), Sym

S ∨ , where S is the universal tautological subbundle on the Grassmannian, and s T is the section induced by T using the morphism of vector bundles Sym 3 H 0 (P[START_REF] Boissière | Automorphismes naturels de l'espace de Douady de points sur une surface[END_REF] , O(1)) ⊗ O → Sym 3 S ∨ on Gr(P 1 , P[START_REF] Boissière | Automorphismes naturels de l'espace de Douady de points sur une surface[END_REF] ).

could only be -3, 5, -11, 21, all of which are prime to p, this will lead to a contradiction. Therefore we only have cycles of length 1. As p 3, 3 -1 j mod p m is well-defined, hence we have e 0 = e 1 = • • • = e 5 . As a result, f is the identity action of P[START_REF] Boissière | Automorphismes naturels de l'espace de Douady de points sur une surface[END_REF] , which[START_REF]Antisymplectic involutions of holomorphic symplectic manifolds[END_REF] Here we use the terminology 'cycle' in the sens of graph theory: it means a loop in a oriented graph with no arrow repeated.The length of a cycle will refer to the number of arrows appearing in it.

on the parity of j: If j is even, then the equation for α i 's becomes α 4 +α 5 = 0 mod 2. This is Family V-(1) in Theorem 0.1, whose generic smoothness is easy to verify: x 2 4 x 0 + x 2 5 x 1 + x 4 x 5 x 2 + x 3 0 + x 3 1 + x 3 2 + x 3 3 is smooth. This family of cubic fourfolds has been studied in [START_REF] Camere | Symplectic involutions of holomorphic symplectic four-folds[END_REF]. If j is odd, then a basis for B is given by

x 5 , x 2 5 x 4 . However, any cubic fourfold in this family is singular along the intersection of two quadrics in the projective 3-planes (x 4 = x 5 = 0).

• (e 0 , • • • , e 5 ) = (0, 0, 0, 4, 2, 2) or (0, 4, 4, 4, -2, -2). They both correspond to the following automorphism of order n = 4:

We thus reduce to n = 4, and (e 0 , • • • , e 5 ) = (0, 0, 0, 2, 1, 1). Therefore 2 j = 0 mod 4, and thus j = 0, 2 mod 4. It also splits into two different cases depending on the value of j. If j = 0 mod 4, the equation ( 7) for α i 's becomes 2α 3 + α 4 + α 5 = 0 mod 4. We easily obtain a basis for B: degree 3 monomials on x 0 , x 1 , x 2 ; x 0 x 2 3 , x 1 x 2 3 , x 2 x 2 3 , x 3 x 4 x 5 , x 3 x 2 4 , x 3 x 2 5 . Unfortunately, any cubic fourfold in this family is singular on two points on the line defined by (x 0 = x 1 = x 2 = x 3 = 0). If j = 2 mod 4, the equation is 2α 3 + α 4 + α 5 = 2 mod 4. The following consists of a basis for B:

The general member is singular along a conic in the projective plane (x 3 = x 4 = x 5 = 0).

• (e 0 , • • • , e 5 ) = (0, 0, 0, 4, -2, -2) or (0, 4, 4, 4, 2, 2). They both correspond to the same f , which becomes the automorphism of the previous case if we replace i by -i. • (e 0 , • • • , e 5 ) = (0, 4, 2, 2, 2, -2) or (0, 4, 2, -2, -2, -2). They both correspond to the following automorphism of order n = 4:

We thus reduce to n = 4 and (e 0 , • • • , e 5 ) = (0, 0, 0, 2, 1, -1). Hence 2 j = 2 mod 4 gives two cases j = 1, -1 mod 4. If j = 1 mod 4, the equation for α i 's becomes 2α 3 + α 4 -α 5 = 1 mod 4. The basis for B is:

Each cubic fourfold in this family is singular along a conic curve in the plane defined by (x 3 = x 4 = x 5 = 0). If j = -1 mod 4, the equation for α i 's becomes 2α 3 + α 4 -α 5 = -1 mod 4, all the solutions are exactly the ones when j = 1 with x 4 and x 5 interchanged. LIE FU of this example is a little bit involved, we give the details here. Let T

To find the singular locus of (T = 0), we need to solve the system of equations

Thanks to the last two equations, we have four cases (x 2 = x 3 = 0), (x 2 = x 4 = 0), (x 5 = x 3 = 0), (x 5 = x 4 = 0). In the first three cases, it is easy to deduce that every variable is zero. In the last case, the system of equations simplifies to:

It is still easy to deduce that every variable is non-zero. And then from the last two equations, we find x 0 = ±2x 1 and x 2 = ∓x 3 . Putting these into the first two equations, we get contradictions. As a consequence, (T = 0) is smooth.

The classification is complete and the result is summarized in Theorem 0.1.

Remarks 3.3.

We have some explanations to make concerning the usage of our list.

• In the fifth column of the table in Theorem 0.1, we give a basis for the compactified parameter space B, which contains of course singular members. To pick out the smooth ones (i.e. to determine the non-empty open dense subset B), we have to apply usual method of Jacobian criterion. • Strictly speaking, the moduli space of cubic fourfolds is the geometric quotient M := P H 0 (P 5 , O(3)) // PGL 6 , and each B we have given in the theorem is a sub-projective space of P H 0 (P 5 , O(3)) , whose image in M is (a component of) the 'moduli space' of cubic fourfolds admitting a 'symplectic' automorphism of certain primary order.

• For an automorphism f of a given order n, say n = 2 r 2 3 r 3 5 r 5 7 r 7 11 r 11 , where r 2 = 0, 1, 2 or 3; r 3 = 0,1 or 2 and r 5 , r 7 , r 11 = 0 or 1. Then f = f 2 f 3 f 5 f 7 f 11 where f p is an automorphisms of order p r p commuting with each other. Thus they can be diagonalised simultaneously. Therefore to classify automorphisms of a given order, it suffices to intersect the corresponding families B's in the list, after independent scaling and permutation of coordinates, inside the complete linear system P H 0 (P 5 , O(3)) . Of course it may end up with an empty family or a family consisting of only singular members.

Example 3.4. We investigate the example of Fermat cubic fourfold X = (x 3 0 + x 3 1 + x 3 2 + x 3 3 + x 3 4 + x 3 5 = 0). We know that (cf. [START_REF] Shioda | Arithmetic and geometry of Fermat curves[END_REF], [START_REF] Kontogeorgis | Automorphisms of Fermat-like varieties[END_REF]) its automorphism group is Aut(X) = (Z/3Z) 5 ⋊ S 6 , which is generated by


Kummer

Fu

BEAUVILLE-VOISIN CONJECTURE FOR GENERALIZED

Inspired by their results on the Chow rings of projective K3 surfaces, Beauville and Voisin made the following conjecture: given a projective hyperkähler manifold, for any algebraic cycle which is a polynomial with rational coefficients of Chern classes of the tangent bundle and line bundles, it is rationally equivalent to zero if and only if it is numerically equivalent to zero. In this paper, we prove the Beauville-Voisin conjecture for generalized Kummer varieties.

Introduction

In [START_REF] Beauville | On the Chow ring of a K3 surface[END_REF], Beauville and Voisin observe the following property of the Chow rings of projective K3 surfaces.

Theorem 1.1 (Beauville-Voisin). Let S be a projective K3 surface. Then (i) There is a well defined 0-cycle o ∈ CH 0 (S ), which is represented by any point on any rational curve on S . It is called the canonical cycle. (ii) For any two divisors D, D , the intersection product D • D is proportional to the canonical cycle o in CH 0 (S ). (iii) c 2 (T S ) = 24o ∈ CH 0 (S ). In particular, for any algebraic cycle which is a polynomial on Chern classes of the tangent bundle T S and of line bundles on S , it is rationally equivalent to zero if and only if it is numerically equivalent to zero.

As is pointed out in their paper, the above result is surprising because CH 0 (S ) is very huge ('infinite dimensional' in the sense of Mumford [START_REF] Mumford | Rational equivalence of 0-cycles on surfaces[END_REF], cf. [START_REF] Voisin | Hodge theory and complex algebraic geometry[END_REF]Chapter 10]). In a subsequent paper [START_REF]On the splitting of the Bloch-Beilinson filtration[END_REF], Beauville proposed a conjectural explanation for Theorem 1.1 to put it into a larger picture. To explain his idea, let us firstly recall the following notion generalizing K3 surfaces to higher dimensions. See for example [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF], [START_REF] Huybrechts | Compact hyper-Kähler manifolds: basic results[END_REF], or [START_REF] Gross | Calabi-Yau manifolds and related geometries[END_REF] for a more detailed treatment. Definition 1.2 (cf. [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]). A smooth projective complex variety X is called hyperkähler or irreducible holomorphic symplectic, if it is simply connected and H 2,0 (X) is 1-dimensional and generated by a holomorphic 2-form which is non-degenerate at each point of X. In particular, a hyperkähler variety has trivial canoncial bundle.

Examples 1.3. Let us give some basic examples of projective hyperkähler manifolds:

• (Beauville [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]) Let S be a projective K3 surface and n ∈ N, then S [n] , which is the Hilbert scheme of subschemes of dimension 0 and length n, is hyperkähler of dimension 2n. • (Beauville [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]) Let A be an abelian surface and n ∈ N. Let s : A [n+1] → A be the natural morphism defined by the composition of the Hilbert-Chow morphism A [n+1] → A (n+1) and the summation A (n+1) → A using the group law of A. It is clear that s is an isotrivial fibration. Then a fibre K n := s -1 (O A ) is hyperkähler of dimension 2n, called generalized Kummer variety. The name is justified by the fact that K 1 is exactly the Kummer K3 surface associated to A. • (Beauville-Donagi [START_REF] Beauville | La variété des droites d'une hypersurface cubique de dimension 4[END_REF]) Let X ⊂ P 5 be a smooth cubic fourfold, then its Fano variety of lines F(X) := l ∈ Gr P 1 , P 5 | l ⊂ X is hyperkähler of dimension 4.

1

As an attempt to understand Theorem 1.1 in a broader framework, Beauville gives the point of view in [START_REF]On the splitting of the Bloch-Beilinson filtration[END_REF] that we can regard this result as a 'splitting property' of the conjectural Bloch-Beilinson-Murre filtration on Chow groups (see [START_REF] André | Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses[END_REF], [START_REF] Jannsen | Motivic sheaves and filtrations on Chow groups[END_REF]) for certain varieties with trivial canonical bundle. He suggests to verify the following down-to-earth consequence of this conjectural splitting of the conjectural filtration on Chow groups of hyperkähler varieties. As a first evidence, the special cases when X = S [2] or S [3] for a projective K3 surface S are verified in his paper loc.cit. Conjecture 1.4 (Beauville). Let X be a projective hyperkähler manifold, and z ∈ CH(X) Q be a polynomial with Q-coefficients of the first Chern classes of line bundles on X. Then z is homologically trivial if and only if z is (rationally equivalent to) zero.

Voisin pursues the work of Beauville and makes in [START_REF]On the Chow ring of certain algebraic hyper-Kähler manifolds[END_REF] the following stronger version of Conjecture 1.4, by involving also the Chern classes of the tangent bundle: Conjecture 1.5 (Beauville-Voisin). Let X be a projective hyperkähler manifold, and z ∈ CH(X) Q be a polynomial with Q-coefficients of the first Chern classes of line bundles on X and the Chern classes of the tangent bundle of X. Then z is numerically trivial if and only if z is (rationally equivalent to) zero.

Here we replaced 'homologically trivial' in the original statement in Voisin's paper [START_REF]On the Chow ring of certain algebraic hyper-Kähler manifolds[END_REF] by 'numerically trivial'. But according to the standard conjecture [START_REF] Steven | The standard conjectures, Motives[END_REF], the homological equivalence and the numerical equivalence are expected to coincide. We prefer to state the Beauville-Voisin conjecture in the above slightly stronger form since our proof for generalized Kummer varieties also works in this generality.

In [START_REF]On the Chow ring of certain algebraic hyper-Kähler manifolds[END_REF], Voisin proves Conjecture 1.5 for the Fano varieties of lines of cubic fourfolds, and for S [n] if S is a projective K3 surface and n ≤ 2b 2,tr + 4, where b 2,tr is the second Betti number of S minus its Picard number. We remark that here we indeed can replace the homological equivalence by the numerical equivalence since the standard conjecture in these two cases has been verified by Charles and Markman [START_REF] Franc | The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces[END_REF].

The main result of this paper is to prove the Beauville-Voisin conjecture 1.5 for generalized Kummer varieties.

Theorem 1.6. Let A be an abelian surface, n ≥ 1 be a natural number. Denote by K n the generalized Kummer variety associated to A (cf. Examples 1.3). Consider any algebraic cycle z ∈ CH(K n ) Q which is a polynomial with rational coefficients of the first Chern classes of line bundles on K n and the Chern classes of the tangent bundle of K n , then z is numerically trivial if and only if z is (rationally equivalent to) zero.

There are two key ingredients in the proof of the above theorem: on the one hand, as in [START_REF]On the Chow ring of certain algebraic hyper-Kähler manifolds[END_REF], the result of De Cataldo-Migliorini [START_REF] Andrea | The Chow groups and the motive of the Hilbert scheme of points on a surface[END_REF] recalled in Section 2 relates the Chow groups of A [n] to the Chow groups of various products of A. On the other hand, a recent result on algebraic cycles on abelian varieties due to Moonen [START_REF] Moonen | On the Chow motive of an abelian scheme with non-trivial endomorphisms[END_REF] and O'Sullivan [START_REF] Peter | Algebraic cycles on an abelian variety[END_REF], which is explained in Section 3, allows us to upgrade a relation modulo numerical equivalence to a relation modulo rational equivalence.

Convention: Throughout this paper, we work over the field of complex numbers. All Chow groups are with rational coefficients CH := CH ⊗Q. If A is an abelian variety, we denote by O A its origin and Pic s (A) its group of symmetric line bundles. For any smooth projective surface S , we denote by S [n] the Hilbert scheme of subschemes of length n, which is a 2n-dimensional smooth projective variety by [START_REF] Fogarty | Algebraic families on an algebraic surface[END_REF]. Finally, for an algebraic variety X, the big diagonal ∆ i j in a self-product X n is the subvariety (

x 1 , • • • , x n ) ∈ X n | x i = x j .

De Cataldo-Migliorini's result

As mentioned above, a crucial ingredient for the proof of Theorem 1.6 will be the following result due to De Cataldo and Migliorini. We state their result in the form adapted to our purpose.

Let S be a projective surface, n ∈ N + and P(n) be the set of partitions of n. For any such partition µ = ( µ 1 , • • • , µ l ), we denote by lµ := l its length. Define S µ := S lµ = S × • • • × S lµ , and also a natural morphism from it to the symmetric product:

S µ → S (n) (x 1 , • • • , x l ) → µ 1 x 1 + • • • + µ l x l . Now define Eµ := S [n] × S (n) S µ
red to be the reduced incidence variety inside S [n] × S µ . Then Eµ can be viewed as a correspondence from S [n] to S µ , and we will write t Eµ for the transpose correspondence, namely the correspondence from S µ to S [n] defined by the same subvariety Eµ in the product.

Let µ = ( µ 1 , • • • , µ l ) = 1 a 1 2 a 2 • • • n a n be a partition of n, we define mµ := (-1) n-l l j=1 µ j and cµ := 1 mµ 1 a 1 !•••a n ! .
Theorem 2.1 (De Cataldo-Migliorini [START_REF] Andrea | The Chow groups and the motive of the Hilbert scheme of points on a surface[END_REF]). Let S be a projective surface, n ∈ N + . For each µ ∈ P(n), let Eµ and t Eµ be the correspondences defined above. Then the sum of the compositions

µ ∈P(n) cµ t Eµ • Eµ = ∆ S [n]
is the identity correspondence of S [n] , modulo rational equivalence. In particular,

µ ∈P(n) cµ E * µ • Eµ * = id CH(S [n]
) : CH(S [n] ) → CH(S [n] ).

Return to the case where S = A is an abelian surface. We view A [n+1] as a variety over A by the natural summation morphism s : A [n+1] → A. Similarly, for each µ ∈ P(n + 1) of length l, A µ also admits a natural morphism to A, namely, the weighted sum:

sµ : A µ → A (x 1 , • • • , x l ) → µ 1 x 1 + • • • + µ l x l .
By definition, the correspondences Eµ, t Eµ are compatible with morphisms s and sµ to A, i.e. the following diagram commutes:

Eµ | | πµ A [n+1] s " " A µ sµ Ã .
We point out that the three morphisms to A are all isotrivial fibrations: they become products after the base change A •n+1 ---→ A given by multiplication by n + 1. Now let us take their fibres over the origin of A, or equivalently, apply the base change i : Spec(C) = O A → A to the above commutative diagram, we obtain the following correspondence, where K n := s -1 (O A ) is the generalized Kummer variety, Bµ is the possibly non-connected abelian variety Bµ := ker sµ : A µ → A , and

Γµ := π -1 µ (O A ). Γµ x x & & K n & & Bµ x x O A = Spec(C) .
In the sequel, we sometimes view Eµ simply as an algebraic cycle in CH A [n+1] × A µ and also by definition

Γµ = i ! (Eµ) ∈ CH K n × B µ
, where i ! is the refined Gysin map defined in [START_REF] Fulton | Intersection theory[END_REF]Chapter 6]. We need the following standard fact in intersection theory. Lemma 2.2. For any γ ∈ CH A [n+1] , we have

Γµ * γ| K n = (Eµ * (γ)) | Bµ in CH (Bµ) .
Similarly, for any β ∈ CH A µ , we have

Γ * µ β| Bµ = E * µ (β) | K n in CH (K n ) .
Proof. All squares are cartesian in the following commutative diagram.

K n × Bµ q / / p w ) ) Bµ w * * K n / / w ) ) O A w i * * A [n+1] × A µ p q / / A µ sµ A [n+1] s / / A. Now for any γ ∈ CH A [n+1] , we have Γµ * γ| K n = Γµ * i ! (γ) (by [14, Theorem 6.2(c)], as s is isotrivial) = q * p * i ! (γ) • i ! (Eµ) = q * i ! p * (γ) • i ! (Eµ) (by [14, Theorem 6.2(b)]) = q * i ! p * (γ) • Eµ = i ! q * p * (γ) • Eµ (by [14, Theorem 6.2(a)]) = i ! (Eµ * (γ)) = (Eµ * (γ)) | Bµ (by [14, Theorem 6.2(c)], as sµ is isotrivial)
The proof of the second equality is completely analogous. Theorem 2.1 together with Lemma 2.2 implies the following Corollary 2.3. For each µ ∈ P(n + 1), let Γµ be the correspondences between K n and Bµ defined above. Then for any γ ∈ CH A [n+1] , we have

µ ∈P(n+1) cµΓ * µ • Γµ * (γ| K n ) = γ| K n in CH(K n ),
where for a partition µ

= ( µ 1 , • • • , µ l ) = 1 a 1 2 a 2 • • • (n + 1) a n+1 ∈ P(n + 1)
, the constant cµ is defined as

1 (-1) n+1-l l j=1 µ j • 1 a 1 !•••a n+1 ! .
For later use, we now describe Bµ.

Let d := gcd ( µ 1 , • • • , µ l )
, then Bµ has d 4 isomorphic connected components. We denote by B 0 µ the identity component, which is a connected abelian variety; and the other components are its torsion translations. More precisely, define the weighted sum homomorphism

sµ : Z ⊕lµ → Z (m 1 , • • • , m l ) → µ 1 m 1 + • • • + µ l m l ,
whose image is clearly dZ. Let U be the kernel of sµ, which is a free abelian group of rank lµ -1. Define the reduced weighted sum sµ :

Z ⊕lµ → Z (m 1 , • • • , m l ) → µ 1 d m 1 + • • • + µ l d m l .
Then we have a short exact sequence of free abelian groups

(1) 0 → U → Z ⊕lµ sµ -→ Z → 0.
By tensoring with A, we obtain a short exact sequence of abelian varieties

(2) 0 → B 0 µ → A µ sµ -→ A → 0.
Since the short exact sequence (1) splits, so does the short exact sequence (2): B 0 µ is a direct summand of A µ , thus we can choose a projection pµ : A µ B 0 µ such that pµ • iµ = id B 0 µ , where iµ : B 0 µ → A µ is the natural inclusion.

Denoting A[d] for the set of d-torsion points of A, we have

Bµ = t∈A[d] B t µ ,
where

B t µ := (x 1 , • • • , x l ) ∈ A µ | l i=1 µ i d x i = t ∈
A . Now we specify the way that we view B t µ as a torsion translation of

B 0 µ . Since d is the greatest common divisor of µ 1 , • • • , µ l , it divides n + 1. We choose t ∈ A[n + 1] such that n+1 d • t = t in A.
Then the torsion translation on A by t will induce some 'torsion translation automorphism'

τ t := (t , • • • , t ) on A [n+1] τ t : A [n+1] → A [n+1]
z → z + t , (e.g. when z is a reduced subscheme of length n + 1 given by (x 1 , • • • , x n+1 ) with x j 's pairwise distinct, it is mapped to z + t := (x 1 + t , • • • , x n+1 + t )); as well as on A µ τ t :

A µ → A µ (x 1 , • • • , x l ) → (x 1 + t , • • • , x l + t ).
These actions are compatible: we have the following commutative diagram with actions:

τ t Eµ x x & & τ t A [n+1] s ' ' A µ τ t sµ x x id A
Moreover, the action of τ t on A µ translates B 0 µ isomorphically to B t µ .

Result of Moonen and O'Sullivan

In this section, A is an abelian variety of dimension g. For any m ∈ Z, let m be the endomorphism of A defined by the multiplication by m. To motivate the result of Moonen and O'Sullivan, let us firstly recall the Beauville conjectures for algebraic cycles on abelian varieties. In [START_REF] Beauville | Quelques remarques sur la transformation de Fourier dans l'anneau de Chow d'une variété abélienne[END_REF] and [START_REF]Sur l'anneau de Chow d'une variété abélienne[END_REF], Beauville investigates the Fourier transformation between the Chow rings of A and its dual abelian variety  and establishes the following: Theorem 3.1 (Beauville decomposition). Let A be a g-dimensional abelian variety. (i) For any 0 ≤ i ≤ g, there exists a direct-sum decomposition

CH i (A) = i s=i-g CH i (s) (A),
where

CH i (s) (A) := z ∈ CH i (A) | m * z = m 2i-s z, ∀m ∈ Z . (ii)
This decomposition is functorial: Let B be another abelian variety of dimension (g + c) and f : A → B be a homomorphism of abelian varieties. Then for any i,

f * CH i (s) (B) ⊂ CH i (s) (A); f * CH i (s) (A) ⊂ CH i+c (s) (B). (iii)
The intersection product respects the grading:

CH i (s) (A) • CH j (t) (A) ⊂ CH i+ j (s+t) (A).
In the spirit of Bloch-Beilinson-Murre conjecture (cf. [START_REF] André | Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses[END_REF], [START_REF] Jannsen | Motivic sheaves and filtrations on Chow groups[END_REF]), Beauville makes in [START_REF]Sur l'anneau de Chow d'une variété abélienne[END_REF] the following conjectures, which roughly say that F j CH i (A) := ⊕ s≥ j CH i (s) (A) should give the desired conjectural Bloch-Beilinson-Murre filtration.

Conjecture 3.2 (Beauville conjectures). (i)

For any i and any s < 0, CH i (s) (A) = 0; (ii) For any i, the restriction of the cycle class map cl :

CH i (0) (A) → H 2i (A, Q) is injective; (iii) For any i, the restriction of the Abel-Jacobi map AJ : CH i (1) (A) → J 2i-1 (A) Q is injective.
Obviously, the Beauville conjectures hold for divisors, i.e. CH 1 (A)

= CH 1 (0) (A) ⊕ CH 1 (1) (A) where CH 1 (0) (A) = Pic s (A) Q NS (A) Q ; CH 1 (1) (A) = Pic 0 (A) Q = Â ⊗ Z Q.
In particular, the Q-subalgebra of CH Remark 3.4. The above result is implicit in O'Sullivan's paper [START_REF] Peter | Algebraic cycles on an abelian variety[END_REF]. In fact, he constructs the so-called symmetrically distinguished cycles CH * (A) sd , which is a Q-subalgebra of CH * (A) containing the first Chern classes of symmetric line bundles and mapping isomorphically by the numerical cycle class map to CH * (A), the Q-algebra of cycles modulo the numerical equivalence.

Proof of Theorem 1.6

Let us prove the main result. To fix the notation, we recall the following description of line bundles on K n (see [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]Proposition 8]). Let : A [n+1] → A (n+1) be the Hilbert-Chow morphism, which is a resolution of singularities ( [START_REF] Fogarty | Algebraic families on an algebraic surface[END_REF]). Proposition 4.1 (Beauville). We have an injective homomorphism

j : NS (A) Q → NS (K n ) Q c top 1 (L) → L| K n such that Pic(K n ) Q = NS (K n ) Q = j NS (A) Q ⊕ Q • δ| K n ,
where δ is the exceptional divisor of A [n+1] .

Here for a line bundle L on A, the S n+1 -invariant line bundle L • • • L on A × • • • × A descends to a line bundle L on the symmetric product A (n+1) and we define L := * (L ). Remark 4.2. As the notation in this proposition indicates, modifying the line bundle L on A inside its numerical equivalence class will not change the resulting line bundle j(L)

= L| K n ∈ NS (K n ) Q = Pic(K n ) Q .
We hence obtain the following Lemma 4.3. Given any polynomial z ∈ CH(K n ) in the Chern classes of T K n and the first Chern classes of line bundles on K n , as in the main theorem, then: (i) There exists γ ∈ CH(A [n+1] ) which is a polynomial of algebraic cycles of one of the three forms: c 1 L for some symmetric line bundle L ∈ Pic s (A) Q , δ, and c j T A [n+1] for some j ∈ N, such that

γ| K n = z in CH(K n ).
(ii) Moreover, for such γ, the automorphism τ t of A [n+1] constructed at the end of Section 2 satisfies

(τ t * (γ)) | K n = γ| K n = z in CH(K n ). Proof. (i) Note that c j T K n = c j T A [n+1] | K n , since T A is trivial. Part (i) thus follows from Proposition 4.1 because c top 1 : Pic s (A) Q -→ NS (A) Q is an isomorphism (see [4, Page 649]). (ii) It is clear that τ t * (δ) = δ and τ t * (T A [n+1] ) = T A [n+1]
. On the other hand, the pushforward of L by a torsion translation on A has the same numerical class as L and hence by Remark 4.2, τ t * ( L) | K n = L| K n as line bundles. Therefore modifying γ by the automorphism τ t does not change its restriction to K n , although it might change the cycle γ itself.

Recall that d = gcd( µ 1 , • • • , µ l ) and for any d-torsion point t of A, the automorphism τ t constructed at the end of Section 2 translates B 0 µ to B t µ , therefore (4) implies that

τ t * (β)| B t µ = 0 for any t ∈ A[d].
However, τ t * (β) = τ t * (Eµ * (γ)) = Eµ * (τ t * (γ)) by the compatibility of the actions of τ t on A [n+1] and on A µ , as explained in Section 2.

We thus obtain that for any t ∈ A[d],

Γµ * (z) | B t µ = Γµ * τ t * γ| K n | B t µ = (Eµ * (τ t * γ)) | B t µ = τ t * (β)| B t µ = 0.
Here the first equality comes from Lemma 4.3(ii), see also Remark 4.2; the second equality uses Lemma 2.2. Since Bµ is the disjoint union of all B t µ for all t ∈ A[d], we have Γµ * (z) = 0 for any µ ∈ P(n + 1).

Using De Cataldo-Migliorini's result (rather Corollary 2.3), we have for z = γ| K n as before,

z = µ ∈P(n+1) cµ Γ * µ • Γµ * (z) = 0.
The proof of Theorem 1.6 is complete if one admits Proposition 4.4.

The proof of Proposition 4.4

The proof of Proposition 4.4 is quite technical but analogous to that of [START_REF]On the Chow ring of certain algebraic hyper-Kähler manifolds[END_REF]Proposition 2.4]. For the convenience of readers, we give in this section a more or less self-contained proof closely following [START_REF]On the Chow ring of certain algebraic hyper-Kähler manifolds[END_REF], emphasizing the differences from the case in [START_REF]On the Chow ring of certain algebraic hyper-Kähler manifolds[END_REF]. The author thanks Claire Voisin for allowing him to reproduce her arguments. For simplicity, we switch from n + 1 to n. Let A still be an abelian surface.

There are two natural vector bundles on A [n] . The first one is the tangent bundle T n := T A [n] , the second one is the rank n vector bundle O n := pr 1 * O U n , where U n ⊂ A [n] × A is the universal subscheme and pr 1 :

A [n] × A → A [n] is the first projection. As c 1 (O n ) = -1
2 δ, we can generalize Proposition 4.4 by proving it for any γ a polynomial of c 1 L for some L ∈ Pic s (A) Q , c i (O n ) for some i ∈ N, and c j (T n ) for some j ∈ N.

For any L ∈ Pic s (A), by the construction of Eµ

⊂ A [n] × A µ , the restriction pr * 1 L | Eµ is the pull back of the line bundle L ⊗ µ 1 • • • L ⊗ µ l on A µ .
Hence by projection formula, we only need to prove the following Proposition 5.1. For γ ∈ CH(A [n] ) a polynomial with rational coefficients of cycles of the forms:

• c i (O n ) for some i ∈ N; • c j (T n ) for some j ∈ N, the algebraic cycle β = Eµ * (γ) ∈ CH A µ is a polynomial with rational coefficients in the big diagonals ∆ i j of A µ = A lµ for 1 ≤ i j ≤ lµ.
To show Proposition 5.1, we actually prove the more general Proposition 5.2 below (note that Proposition 5.1 corresponds to the special case m = 0), which allows us to do induction on n. Let us introduce some notation first: for any m ∈ N, let Eµ ,m be the correspondence between A [n] × A m and A µ × A m defined by Eµ ,m := Eµ × ∆ A m . Let I n be the ideal sheaf of the universal subscheme U n ⊂ A [n] × A. For any 1 ≤ i j ≤ m, we denote by pr 0 :

A [n] ×A m → A [n] , resp. pr i : A [n] ×A m → A, resp. pr 0i : A [n] ×A m → A [n]
×A, resp. pr i j : A [n] × A m → A × A the projection onto the factor A [n] , resp. the i-th factor of A m , resp. the product of the factor A [n] and the i-th factor of A m , resp. the product of the i-th and j-th factors of A m . structure sheaf, T 1 = T A is trivial and I 1 = I ∆ A is the ideal sheaf of the diagonal, whose Chern classes are either zero or ∆ A (by Lemma 5.3), Proposition 5.2 is verified in this case. Now assuming the statement holds for n -1, let us prove it for n. In the rest of the proof, a partition of n means a grouping of the set {1, 2, • • • , n} rather than just a decreasing sequence of natural numbers with sum n as before. More precisely, a partition µ of length l is a sequence of mutually exclusive subsets

µ 1 , • • • , µ l ∈ 2 {1,••• ,n} such that l j=1 µ j = {1, • • • , n}.
Thus we can naturally identify A µ := A lµ with the diagonal (x 1 , . . . , x n ) ∈ A n | x i = x j if i, j ∈ µ k for some k ⊂ A n . Consider the reduced fibre product (A µ × A (n) A [n-1,n] ) red , which has lµ irreducible components dominating A µ , depending on the choice of the residue point. Let us pick one component, for example, the one where over a general point (x 1 , • • • , x n ) ∈ A µ , the residue point is x n . Let µ be the partition of {1, 2, • • • , n -1} given by µ i := µ i \{n} for all i. Let us call this irreducible component Eµ , µ . Set theoretically,

Eµ , µ = ((x 1 , • • • , x n ), z ⊂ z) ∈ A µ × A [n-1,n] | [z ] = x 1 + • • • + x n-1 , [z] = x 1 + • • • + x n ; Eµ = ((x 1 , • • • , x n ), z) ∈ A µ × A [n] | [z] = x 1 + • • • + x n ; Eµ = ((x 1 , • • • , x n-1 ), z ) ∈ A µ × A [n-1] | [z ] = x 1 + • • • + x n-1 ,
where [-] means the Hilbert-Chow morphism.

We have the following commutative diagram with natural morphisms:

(6) A µ × A A µ ι o o Eµ × A gµ ,1 O O fµ ,1 Eµ , µ χ o o χ / / p Eµ gµ O O fµ A [n-1] × A A [n-1,n] σ=(φ,ρ) o o ψ / / A [n]
Here and in the sequel, for any morphism h and any m ∈ N, we denote by h m the morphism h × id A m . In the above diagram, fµ, gµ, fµ , gµ are the natural projections; χ = (id A µ , ψ) :

((x 1 , • • • , x n ), z ⊂ z) → ((x 1 , • • • , x n ), z), χ = (pr A µ , σ) : ((x 1 , • • • , x n ), z ⊂ z) → ((x 1 , • • • , x n-1 ), z , x n ) both are of degree 1; and finally ι : (x 1 , • • • , x n ) → ((x 1 , • • • , x n-1 ), x n
) is either an isomorphism or a diagonal embedding depending on whether n is the only one element in the subset of partition where n belongs to.

Here comes the key setting for the induction process. For any m ∈ N, we make a product of the above diagram with A m and replace any morphism h by h

m := h × id A m : (7) A µ × A m+1 A µ × A m ι m o o Eµ × A m+1 gµ ,m+1 O O fµ ,m+1 Eµ , µ × A m χ m o o χ m / / p m Eµ × A m gµ ,m O O fµ ,m A [n-1] × A m+1 A [n-1,n] × A m σ m o o ψ m / / A [n] × A m
Given γ ∈ CH(A [n] × A m ) a polynomial expression as in Proposition 5.2, we want to prove that gµ ,m * f * µ ,m γ ∈ CH(A µ × A m ) is a polynomial of big diagonals of A lµ+m . Since ι m is either an isomorphism or a diagonal embedding, it suffices to prove the same thing for ι m * • gµ ,m * f * µ ,m γ ∈ CH(A µ × A m+1 ). However, [25, Page 626 (2.13)]).

ι m * • gµ ,m * f * µ ,m γ = ι m * • gµ ,m * • χ m * • χ * m • f * µ ,m (γ) (since χ m is of degree 1) = gµ ,m+1 * • χ m * • χ * m • f * µ ,m (γ) = gµ ,m+1 * • χ m * • p * m • ψ * m (γ) = gµ ,m+1 * • f * µ ,m+1 • σ m * • ψ * m (γ) (by
Using the induction hypothesis (since µ is a partition of n -1), we find that to finish the proof, it is enough to verify

Proposition 5.4. If γ ∈ CH(A [n] × A m
) is a polynomial expression in the cycles of the following forms:

• pr * 0 c j (O n ) for some j ∈ N; • pr * 0 c j (T n ) for some j ∈ N; • pr * 0i c j (I n ) for some 1 ≤ i ≤ m and j ∈ N; • pr * i j (∆ A ) for some 1 ≤ i j ≤ m.

then

σ m * • ψ * m (γ) ∈ CH(A [n-1] × A m+1
) is a polynomial in cycles of these four forms with n replaced by n -1 and m replaced by m + 1. This will follow essentially from the formulae below established in [START_REF] Ellingsrud | On the cobordism class of the Hilbert scheme of a surface[END_REF]. We adopt the notation in Diagram ( 5), ( 6), [START_REF] Beauville | On the Chow ring of a K3 surface[END_REF] and the definition of the line bundle L after Diagram [START_REF]On the splitting of the Bloch-Beilinson filtration[END_REF]. In our case of abelian surface, the formulae are simplified thanks to the fact that T A is trivial. an equality in the Grothendieck group K 0 (A [n-1,n] × A):

(iii) ψ ! 1 I n = φ ! 1 I n-1 -(L O A ) • ρ ! 1 (O ∆ A );
and an equality in the Chow group CH(A [n-1] × A):

(iv) σ * c 1 (L) i = (-1) i c i (-I n-1 ).

Return to the proof of Proposition 5.4. Taking the Chern classes of both sides of (i), (ii), (iii) in Theorem 5. where we also use pr 0 to denote the projection A [n-1,n] × A m → A [n-1,n] , etc.

When apply σ m * to a polynomial in cycles of the above five types, using the projection formula for the birational morphism σ m and Theorem 5.5(iv), we conclude that σ m * • ψ * m (γ) is of the desired form. This finishes the proof of Proposition 5.4 thus completes the proof of Proposition 4.4.

Theorem 5 . 5 (

 55 [START_REF] Ellingsrud | On the cobordism class of the Hilbert scheme of a surface[END_REF], Proposition 2.3, Lemma 2.1, in the proof of Proposition 3.1 and Lemma 1.1). We have the following equalities in the Grothendieck group K 0 (A[n-1,n] ):(i) ψ ! T n = φ ! T n-1 + L • σ ! I ∨ n-1 + 1; (ii) ψ ! O n = φ ! O n-1 + L;

  5, we get formulae for pull-backs by ψ or ψ 1 of the Chern classes of T n , O n , I n in terms of polynomial expressions of the first Chern class of L and the pull-backs by φ, ρ, σ of the Chern classes of T n-1 , O n-1 , I n-1 and O ∆ A . Therefore by the calculations in Lemma 5.3 and the fact that σ = (φ, ρ), we obtain that ψ * m (γ) ∈ CH A [n-1,n] × A m is a polynomial of cycles of the following five forms:• σ * m • pr * 0 c j (T n-1 ) for some j ∈ N; • pr * 0 (c 1 (L)); • σ * m • pr * 0i c j (I n-1 ) for some 1 ≤ i ≤ m + 1 and j ∈ N; • σ * m • pr * 0 c j (O n-1) for some j ∈ N; • σ * m • pr * i j (∆ A ) for some 1 ≤ i j ≤ m + 1,

  Voisin raised the natural question whether the cycle class map cl is injective on this subalgebra. Recently, Moonen [20, Corollary 8.4] and O'Sullivan [23, Theorem Page 2-3] have given a positive answer to Voisin's question: Theorem 3.3 (Moonen, O'Sullivan).Let A be an abelian variety. Let P ∈ CH * (A) be a polynomial with rational coefficients in the first Chern classes of symmetric line bundles on A, then P is numerically equivalent to zero if and only if P is (rationally equivalent to) zero.

* (A) generated by symmetric line bundles on A is contained in CH * (0) (A) (by Theorem 3.1(iii)). As a special case of Beauville's Conjecture 3.2(ii),

Acknowledgements

I would like to express my gratitude to Claire Voisin for her excellent mini-course at the colloquium GRIFGA as well as for the helpful discussions afterwards, to Kieran O'Grady for raising the question at the colloquium which is the main subject of this paper. I would like to thank Giuseppe Ancona for explaining me the result of O'Sullivan, to thank Ulrike Greiner for pointing out to me a gap in a first version and helping me to fix it. Finally, I want to thank Zhi Jiang for his careful reading of the preliminary version of this paper and the referees for their helpful suggestions which improved the paper a lot.

LIE FU

Let us start the proof of Theorem 1.6. We will use ≡ to denote the numerical equivalence. Given z ∈ CH(K n ) a polynomial of the Chern classes of T K n and line bundles on K n as in the main theorem 1.6. By Lemma 4.3(i), we can write z = γ| K n for γ ∈ CH(A [n+1] ) a polynomial of c 1 L for some L ∈ Pic s (A) Q , δ, and c j T A [n+1] for some j ∈ N.

Assuming z ≡ 0, we want to prove that z = 0. Adopting the previous notation, then for any µ ∈ P(n + 1) we have by Lemma 2.2

Define β := Eµ * (γ) ∈ CH A µ , the above equality says that β| Bµ ≡ 0, in particular, [START_REF]Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF] β| B 0 µ ≡ 0.

To describe β, we need the following proposition, which is the analogue of the corresponding result [25, Proposition 2.4] due to Voisin, and we will give its proof in the next section. Proposition 4.4. For γ ∈ CH(A [n+1] ) as above (i.e. a polynomial of cycles of the forms: c 1 L for some L ∈ Pic s (A) Q , δ, and c j T A [n+1] for some j ∈ N), the algebraic cycle β = Eµ * (γ) ∈ CH A µ is a polynomial with rational coefficients in cycles of the two forms:

• pr * i (L) for some symmetric line bundle L on A and

See the next section for its proof.

Corollary 4.5. With the same notation, β is a polynomial with rational coefficients in algebraic cycles of the form φ * (L), for some homomorphism of abelian varieties φ : A µ → A and some L ∈ Pic s (A) Q .

Proof. It is enough to remark that the big diagonal ∆ i j is nothing but the pull-back of O A ∈ CH 0 (A) via the homomorphism

Let us continue the proof of Theorem 1.6. Let B 0 µ be the identity component of Bµ, iµ : B 0 µ → A µ and pµ : A µ B 0 µ be the inclusion and the splitting constructed in Section 2. By assumption, we have equation

On the other hand, since iµ • pµ : A µ → A µ is an endomorphism of A µ , Corollary 4.5 implies that the numerically trivial cycle p * µ i * µ (β) is also a polynomial of cycles of the form φ * (L), for some homomorphism of abelian varieties φ : A µ → A and some L ∈ Pic s (A) Q . As a result, p * µ i * µ (β) is in the subalgebra of CH A µ generated by the first Chern classes of symmetric line bundles of A µ .

Therefore by the result of Moonen and O'Sullivan (Theorem 3.3),

LIE FU Proposition 5.2. For γ ∈ CH(A [n] × A m ) a polynomial with rational coefficients of cycles of the forms:

• pr * 0 c j (O n ) for some j ∈ N; • pr * 0 c j (T n ) for some j ∈ N; • pr * 0i c j (I n ) for some 1 ≤ i ≤ m and j ∈ N; • pr * i j (∆ A ) for some 1 ≤ i j ≤ m, the algebraic cycle Eµ ,m * (γ) ∈ CH A lµ+m is a polynomial with rational coefficients in the big diagonals ∆ i j of A lµ+m , for 1 ≤ i j ≤ lµ + m.

The main tool to prove this proposition is the so-called nested Hilbert schemes, which we briefly recall here (cf. [START_REF] Nakajima | Lectures on Hilbert schemes of points on surfaces[END_REF]). By definition, the nested Hilbert scheme is the incidence variety

where z ⊂ z means z is a closed subscheme of z. It admits natural projections to A [n-1] and A [n] , and also a natural morphism to A which associates the residue point to such a pair of subschemes (z ⊂ z). The situation is summarized by the following diagram:

(5)

A

We collect here some basic properties of the nested Hilbert scheme (cf. [START_REF] Ellingsrud | An intersection number for the punctual Hilbert scheme of a surface[END_REF], [START_REF] Lehn | Lectures on Hilbert schemes, Algebraic structures and moduli spaces[END_REF], [START_REF] Nakajima | Lectures on Hilbert schemes of points on surfaces[END_REF]):

• The nested Hilbert scheme A [n-1,n] is irreducible and smooth of dimension 2n (cf. [START_REF] Cheah | The cohomology of smooth nested Hilbert schemes of points[END_REF]).

• The natural morphism σ := (φ, ρ) : ,n] , where E is the exceptional divisor of the blow up. • The natural morphism σ = (φ, ρ) : A [n-1,n] → A [n-1] × A is also identified with the projection

Then L is identified with O P(In-1) (1) .

• The morphism ψ is generically finite of degree n.

• The natural morphism (ψ, ρ) :

where ω U n is the relative dualising sheaf (supported on U n ) of the universal subscheme U n ⊂ A [n] ×A.

Before we return to the proof of Proposition 5.2, we do the following calculation:

Proof. We apply the Grothendieck-Riemann-Roch formula to the diagonal embedding A → A × A, we get (since td (T A ) = td (T A×A ) = 1): ch(O ∆ A ) = ∆ A ∈ CH 2 (A × A), and the calculation of Chern classes follows.

Proof of Proposition 5.2. We do induction on n. When n = 0, there is nothing to prove. When n = 1, the only possible µ = (1), hence Eµ ,m is the identity correspondence of A m+1 . Since O 1 is the