On the Chow groups of certain cubic fourfolds

TL;DR

This paper investigates the Chow groups of a special family of smooth cubic fourfolds with symplectic involutions, establishing a relation with associated K3 surfaces and proving finite-dimensionality of their motives.

Contribution

It introduces a new family of cubic fourfolds with symplectic involutions and relates their Chow motives to those of K3 surfaces, showing finite-dimensionality.

Findings

Relation between cubic fourfolds and K3 surfaces on Chow motives

Finite-dimensionality of motives for certain fourfolds

Symplectic involution induces fixed K3 surface

Abstract

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold $X$ in the family has an involution such that the induced involution on the Fano variety $F$ of lines in $X$ is symplectic and has a $K3$ surface $S$ in the fixed locus. The main result establishes a relation between $X$ and $S$ on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.

Equations

$$f(x_{0},x_{1},x_{2},x_{3})+(x_{4})^{2}\ell_{1}(x_{0},\ldots,x_{3})+(x_{5})^{2}\ell_{2}(x_{0},\ldots,x_{3})+x_{4}x_{5}\ell_{3}(x_{0},\ldots,x_{3})=0\$$

$${\Gamma }_{\ast }:A_{hom}^3(X)\hookrightarrow A_{hom}^2(S).$$

$${\Gamma }_{\ast }:H^{3,1}(X)\stackrel{\cong }{\to }{H}^{2,0}(S).$$

$$\dim H^4(X)\cap H^{2,2}(X,\mathbb{C})\ge 20.$$

$$L^{n-i}:H^i(X)\to H^{2n-i}(X)$$

$$(C_i{)}_{\ast }:H^{2n-i}(X)\to H^i(X)$$

$${\tilde{N}}^j{H}^i(X)\subset N^j{H}^i(X)$$

$${\Delta }_X=\sum _{i,j}{\pi }_{i,j}^{X}in{H}^{2n}(X\times X),$$

$$\pi :\mathcal{X}\to B$$

$$p:\widetilde{\mathcal{X}{\times }_B\mathcal{X}}\to \mathcal{X}{\times }_B\mathcal{X}$$

$$V:=\Bigl{\{}\bigl{(}(x,y,z),\sigma\bigr{)}\ |\ \sigma|_{z}=0\Bigr{\}}\ \ \subset\ \widetilde{M\times M}\times|L|\ .$$

$$H^n(X_b{)}_{prim}\ne 0andR{|}_{\widetilde{X_b\times X_b}}=0\in H^{2n}(\widetilde{X_b\times X_b}).$$

$$(p_{b})_{\ast}\bigl{(}R|_{\widetilde{X_{b}\times X_{b}}}\bigr{)}=\gamma|_{X_{b}\times X_{b}}\ \ \in A^{n}({X_{b}\times X_{b}})$$

$$\sum _{g\in G}{c}_g{\Gamma }_g+\gamma =0in{H}^{2n}(X_b\times X_b),$$

$$R{|}_{X_b\times X_b}=0\in H^{2n}(X_b\times X_b)\forall b\in B.$$

$$R{|}_{X_b\times X_b}=\gamma {|}_{X_b\times X_b}\in A^n(X_b\times X_b)\forall b\in B.$$

$$V:=\Bigl{\{}\bigl{(}(x,y,z),\sigma\bigr{)}\ |\ \sigma|_{z}=0\Bigr{\}}\ \ \subset\ \widetilde{M\times M}\times|L|^{G}\ .$$

$$R{|}_{X_b\times X_b}=R_0{|}_{X_b\times X_b}+\sum _{g\in G}{\lambda }_g{\Gamma }_{g}in{A}^n(X_b\times X_b),$$

$$f(x_{0},x_{1},x_{2},x_{3})+(x_{4})^{2}\ell_{1}(x_{0},\ldots,x_{3})+(x_{5})^{2}\ell_{2}(x_{0},\ldots,x_{3})+x_{4}x_{5}\ell_{3}(x_{0},\ldots,x_{3})=0\$$

$${\Gamma }_{\ast }:A_{hom}^3(X)\hookrightarrow A_{hom}^2(S).$$

$$\begin{array}{rl}\iota :{\mathbb{P}}^5& \to {\mathbb{P}}^5,\\ [x_0:x_1:x_2:x_3:x_4:x_5]& \mapsto [x_0:x_1:x_2:x_3:-x_4:-x_5].\end{array}$$

$${\iota }_X:X\to X$$

$${\iota }_F:F\to F$$

$${\tau }^{\ast }:H^{2,0}(F)\stackrel{\cong }{\to }{H}^{2,0}(S).$$

$${\tau }^{\ast }:H_{tr}^2(F)\stackrel{\cong }{\to }{H}_{tr}^2(S),$$

$$(\Gamma_{BD})_{\ast}\colon\ \ H^{4}(X)\ \xrightarrow{\cong}\ H^{2}(F)\$$

$$\Gamma :={}^t{\Gamma }_{\tau }\circ {\Gamma }_{BD}\in A^3(X\times S).$$

$${\Gamma }_{\ast }:H^4(X)/N^2\stackrel{\cong }{\to }{H}_{tr}^2(F)\to H_{tr}^2(S).$$

$$\Gamma :(X,{\pi }_{4,1}^X,0)\stackrel{\cong }{\to }(S,{\pi }_{2,tr}^S,0)in{\mathcal{M}}_{\mathrm{hom}}.$$

$$(\Psi \circ \Gamma {)}_{\ast }=\mathrm{id}:H^4(X)/N^2\to H^4(X)/N^2,$$

Full text

1. Introduction

For a smooth projective variety $X$ over $\mathbb{C}$, let $A^{i}(X):=CH^{i}(X)_{\mathbb{Q}}$ denote the Chow groups (i.e. the groups of codimension $i$ algebraic cycles on $X$ with $\mathbb{Q}$–coefficients, modulo rational equivalence). Let $A^{i}_{hom}(X)$ denote the subgroup of homologically trivial cycles.

When $X\subset\mathbb{P}^{5}(\mathbb{C})$ is a smooth cubic fourfold, we have $A^{i}_{hom}(X)=0$ for $i\not=3$, but $A^{3}_{hom}(X)\not=0$ (this is related to the fact that $H^{3,1}(X)\not=0$). The main result of this note shows that for a certain family of cubic fourfolds, the group $A^{3}_{hom}(X)$ is not larger than the Chow group of [math]–cycles on a $K3$ surface:

Theorem **** (=theorem 3.1).

Let $X\subset\mathbb{P}^{5}(\mathbb{C})$ be a smooth cubic fourfold defined by an equation

[TABLE]

(here $f$ has degree $3$ and $\ell_{1},\ell_{2},\ell_{3}$ are linear forms). There exists a $K3$ surface $S$ and a correspondence $\Gamma\in A^{3}(X\times S)$ inducing a split injection

[TABLE]

In a nutshell, the argument proving theorem 3.1 is as follows: cubics $X$ as in theorem 3.1 have an involution inducing a symplectic involution $\iota_{F}$ of the Fano variety of lines $F=F(X)$. The fixed locus of $\iota_{F}$ contains a $K3$ surface $S$. The inclusion $S\subset F$ being symplectic, there is a (correspondence–induced) isomorphism

[TABLE]

Because the cubics $X$ as in theorem 3.1 form a large family, and the correspondence $\Gamma$ exists for the whole family, one can apply Voisin’s method of “spread” [33], [34], [35], [36] to this isomorphism, and obtain a statement on the level of rational equivalence which proves theorem 3.1.

As an application of theorem 3.1, we obtain some new examples of cubics with finite–dimensional motive (in the sense of Kimura/O’Sullivan [19], [1], [17]):

Corollary **** (=corollary 4.1).

Let $X$ be as in theorem 3.1, and assume

[TABLE]

Then $X$ has finite–dimensional motive.

For $X$ as in corollary 4.1, one can also prove finite–dimensionality for the Fano varieties of lines on $X$ (remark 4.2). This gives new examples of hyperkähler fourfolds with finite–dimensional motive.

**Conventions **.

In this article, the word variety will refer to a reduced irreducible scheme of finite type over $\mathbb{C}$. A subvariety is a (possibly reducible) reduced subscheme which is equidimensional.

All Chow groups will be with rational coefficients*: we will denote by $A_{j}(X)$ the Chow group of $j$–dimensional cycles on $X$ with $\mathbb{Q}$–coefficients; for $X$ smooth of dimension $n$ the notations $A_{j}(X)$ and $A^{n-j}(X)$ are used interchangeably.*

The notations $A^{j}_{hom}(X)$, $A^{j}_{AJ}(X)$ will be used to indicate the subgroups of homologically trivial, resp. Abel–Jacobi trivial cycles. For a morphism $f\colon X\to Y$, we will write $\Gamma_{f}\in A_{\ast}(X\times Y)$ for the graph of $f$. The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [29], [26]) will be denoted $\mathcal{M}_{\rm rat}$.

We will write $H^{j}(X)$ to indicate singular cohomology $H^{j}(X,\mathbb{Q})$.

Given a group $G\subset\operatorname{Aut}(X)$ of automorphisms of $X$, we will write $A^{j}(X)^{G}$ (and $H^{j}(X)^{G}$) for the subgroup of $A^{j}(X)$ (resp. $H^{j}(X)$) invariant under $G$.

2. Preliminaries

2.1. Refined Künneth decomposition

Definition 2.1**.**

Let $X$ be a smooth projective variety, and $h\in\hbox{Pic}(X)$ an ample class. The hard Lefschetz theorem asserts that the map

[TABLE]

obtained by cupping with $h^{n-i}$ is an isomorphism, for any $i<n$. One of the standard conjectures, often denoted $B(X)$, asserts that the inverse isomorphism is algebraic: we say that $B(X)$ holds if for any $i<n$, there exists a correspondence $\mathbb{C}_{i}\in A^{i}(X\times X)$ such that

[TABLE]

is an inverse to $L^{n-i}$.

Remark 2.2**.**

For more on the standard conjectures, cf. [20], [21]. In this note, we will be using the following two facts: Any smooth hypersurface $X\subset\mathbb{P}^{m}(\mathbb{C})$ verifies $B(X)$ [20], [21]. For any smooth cubic fourfold $X\subset\mathbb{P}^{5}(\mathbb{C})$, the Fano variety of lines $F:=F(X)$ verifies $B(F)$ (this follows from [9, Theorem 1.1], or alternatively from [23, Corollary 6]).

Remark 2.3**.**

Let $N^{\ast}H^{\ast}$ denote the coniveau filtration on cohomology [6]. Vial [31] has introduced a variant filtration $\widetilde{N}^{\ast}H^{\ast}$, called the niveau filtration. There is an inclusion

[TABLE]

for any $X$ and all $i,j$. Conjecturally, this is always an equality (this would follow from the standard conjecture $B$). If $B(X)$ holds and $j\geq{i-1\over 2}$, this inclusion is an equality [31].

Theorem 2.4** (Vial [31]).**

Let $X$ be a smooth projective variety of dimension $n\leq 5$. Assume $B(X)$ holds. There exists a decomposition of the diagonal

[TABLE]

where the $\pi_{i,j}$’s are mutually orthogonal idempotents. The correspondence $\pi_{i,j}$ acts on $H^{\ast}(X)$ as a projector on $\hbox{Gr}^{j}_{\widetilde{N}}H^{i}(X)$. Moreover, $\pi_{i,j}$ can be chosen to factor over a variety of dimension $i-2j$ (i.e., for each $\pi_{i,j}$ there exists a smooth projective variety $Z_{i,j}$ of dimension $i-2j$, and correspondences $\Gamma_{i,j}\in A^{n-j}(Z_{i,j}\times X),\Psi_{i,j}\in A^{i-j}(X\times Z_{i,j})$ such that $\pi_{i,j}=\Gamma_{i,j}\circ\Psi_{i,j}$ in $H^{2n}(X\times X)$).

Proof.

This is a special case of [31, Theorem 1]. Indeed, as mentioned in loc. cit., varieties $X$ of dimension $\leq 5$ such that $B(X)$ holds verify condition (*) of loc. cit. ∎

Remark 2.5**.**

If $X$ is a surface, $\pi^{X}_{2,0}$ is the homological realization of the projector $\pi^{X}_{2,tr}$ constructed on the level of Chow motives in [18].

2.2. Spread

Lemma 2.6** (Voisin [33], [34]).**

Let $M$ be a smooth projective variety of dimension $n+1$, and $L$ a very ample line bundle on $M$. Let

[TABLE]

denote a family of hypersurfaces, where $B\subset|L|$ is a Zariski open. Let

[TABLE]

denote the blow–up of the relative diagonal. Then $\widetilde{\mathcal{X}\times_{B}\mathcal{X}}$ is Zariski open in $V$, where $V$ is a projective bundle over $\widetilde{M\times M}$, the blow–up of $M\times M$ along the diagonal.

Proof.

This is [33, Proof of Proposition 3.13] or [34, Lemma 1.3]. The idea is to define $V$ as

[TABLE]

The very ampleness assumption ensures $V\to\widetilde{M\times M}$ is a projective bundle. ∎

This is used in the following key proposition:

Proposition 2.7** (Voisin [34]).**

Assumptions as in lemma 2.6. Assume moreover $M$ has trivial Chow groups. Let $R\in A^{n}(V)$. Suppose that for all $b\in B$ one has

[TABLE]

Then there exists $\gamma\in A^{n}(M\times M)$ such that

[TABLE]

for all $b\in B$. (Here $p_{b}$ denotes the restriction of $p$ to $\widetilde{X_{b}\times X_{b}}$, which is the blow–up of $X_{b}\times X_{b}$ along the diagonal.)

Proof.

This is [34, Proposition 1.6]. ∎

The following is an equivariant version of proposition 2.7:

Proposition 2.8** (Voisin [34]).**

Let $M$ and $L$ be as in proposition 2.7. Let $G\subset\operatorname{Aut}(M)$ be a finite group. Assume the following:

(i) The linear system |L|^{G}:=\mathbb{P}\bigl{(}H^{0}(M,L)^{G}\bigr{)} has no base–points, and the locus of points in $\widetilde{M\times M}$ parametrizing triples $(x,y,z)$ such that the length $2$ subscheme $z$ imposes only one condition on $|L|^{G}$ is contained in the union of (proper transforms of) graphs of non–trivial elements of $G$, plus some loci of codimension $>n+1$.

(ii) Let $B\subset|L|^{G}$ be the open parametrizing smooth hypersurfaces, and let $X_{b}\subset M$ be a hypersurface for $b\in B$ general. There is no non–trivial relation

[TABLE]

where $\gamma$ is a cycle in \operatorname{Im}\bigl{(}A^{n}(M\times M)\to A^{n}(X_{b}\times X_{b})\bigr{)}.

Let $R\in A^{n}(\mathcal{X}\times_{B}\mathcal{X})$ be such that

[TABLE]

Then there exists $\gamma\in A^{n}(M\times M)$ such that

[TABLE]

Proof.

This is not stated verbatim in [34], but it is contained in the proof of [34, Proposition 3.1 and Theorem 3.3]. We briefly review the argument. One considers

[TABLE]

The problem is that this is no longer a projective bundle over $\widetilde{M\times M}$. However, as explained in the proof of [34, Theorem 3.3], hypothesis (i) ensures that one can obtain a projective bundle after blowing up the graphs $\Gamma_{g},g\in G$ plus some loci of codimension $>n+1$. Let $M^{\prime}\to\widetilde{M\times M}$ denote the result of these blow–ups, and let $V^{\prime}\to M^{\prime}$ denote the projective bundle obtained by base–changing.

Analyzing the situation as in [34, Proof of Theorem 3.3], one obtains

[TABLE]

where $R_{0}\in A^{n}(M\times M)$ and $\lambda_{g}\in\mathbb{Q}$ (this is [34, Equation (15)]). By assumption, $R|_{X_{b}\times X_{b}}$ is homologically trivial. Using hypothesis (ii), this implies that all $\lambda_{g}$ have to be [math]. ∎

3. Main result

Theorem 3.1**.**

Let $X\subset\mathbb{P}^{5}(\mathbb{C})$ be a smooth cubic fourfold defined by an equation

[TABLE]

(here $f$ has degree $3$ and $\ell_{1},\ell_{2},\ell_{3}$ are linear forms). There exists a $K3$ surface $S$ and a correspondence $\Gamma\in A^{3}(X\times S)$ inducing a split injection

[TABLE]

Proof.

Let us consider the involution

[TABLE]

The family of cubic fourfolds $X$ as in theorem 3.1 is exactly the family of smooth cubic fourfolds invariant under $\iota$ (this was observed in [8, Section 7], and also in [14], where this family appears as “Family V-(1)” in the classification table of [14, Theorem 0.1]). Let us denote by

[TABLE]

the involution of $X$ induced by $\iota$.

Let $F:=F(X)$ denote the Fano variety parametrizing lines contained in $X$. The variety $F$ is a hyperkähler variety [4]. The involution

[TABLE]

induced by $\iota_{X}$ is symplectic [8, Section 7], [14, Theorem 0.1]. The fixed locus of $\iota_{F}$ consists of $28$ isolated points and a $K3$ surface $S\subset F$ [8, Section 7], [14, Section 4]. The involution $\iota_{F}$ being symplectic, the surface $S\subset F$ is a symplectic subvariety, i.e. the inclusion $\tau\colon S\to F$ induces an isomorphism

[TABLE]

As is readily seen, this implies there is also an isomorphism

[TABLE]

where $H^{2}_{tr}()\subset H^{2}()$ denotes the smallest Hodge–substructure containing $H^{2,0}()$. Let $\Gamma_{BD}$ be the correspondence inducing the Beauville–Donagi isomorphism

[TABLE]

[4]. (That is, let $P\subset X\times F$ denote the incidence variety, with morphisms $p\colon P\to F$, $q\colon P\to X$. Then $\Gamma_{BD}:=\Gamma_{p}\circ{}^{t}\Gamma_{q}\in A^{5}(X\times F)$.)

Let us define a correspondence

[TABLE]

Combining isomorphisms (1) and (2), we obtain an isomorphism

[TABLE]

A bit more formally, this implies there is an isomorphism of homological motives

[TABLE]

Here, $\pi^{X}_{4,1}=\pi^{X}_{4}-\pi^{X}_{4,2}$ is a projector on $H^{4}(X)/N^{2}$; this exists thanks to theorem 2.4. The projector $\pi^{S}_{2,tr}$ is the projector on $H^{2}_{tr}(S)$ constructed in [18]. Let $\Psi\in A^{3}(S\times X)$ be a correspondence inducing an inverse to the isomorphism (3). This means that we have

[TABLE]

which means that there is a homological equivalence of cycles

[TABLE]

where $\gamma_{1}\in A^{4}(X\times X)$ is some cycle supported on $V\times V\subset X\times X$, where $V\subset X$ is a codimension $2$ closed subvariety (this is because $\gamma_{1}$ is supported on the support of $\pi^{X}_{4,2}$, which is supported on $V\times V$ as indicated, by theorem 2.4).

As $X\subset\mathbb{P}^{5}$ is a hypersurface, the only interesting Künneth component is $\pi^{X}_{4}$. That is, we can write

[TABLE]

where $\gamma_{2}$ is a “completely decomposed” cycle, i.e. a cycle with support on $\cup_{i}V_{i}\times W_{i}\subset X\times X$, where $\dim V_{i}+\dim W_{i}=4$. Plugging this in equation (4), we obtain a homological equivalence of cycles

[TABLE]

where $\gamma$ is a “completely decomposed” cycle in the above sense.

We now proceed to upgrade the homological equivalence (5) to a rational equivalence. This can be done thanks to the work of Voisin on the Bloch/Hodge equivalence [33], [34], using the technique of “spread” of algebraic cycles in good families.

Following the approach of [33], [34], we put the above construction in family. We define

[TABLE]

to be the family of all smooth cubic fourfolds given by an equation as in theorem 3.1. (That is, we let $G\subset\operatorname{Aut}(\mathbb{P}^{5})$ be the order $2$ group generated by the involution $u$, and we define

[TABLE]

as the open subset parametrizing smooth $G$–invariant cubics.) We will write $X_{b}:=\pi^{-1}(b)$ for the fibre over $b\in B$. We also define families

[TABLE]

of Fano varieties of lines, resp. of $K3$ surfaces. (That is, $\mathcal{S}\subset\mathcal{F}$ is the fixed locus of the involution of $\mathcal{F}$ induced by $\iota$.) We will write $F_{b}$ and $S_{b}$ for the fibre over $b\in B$.

The correspondence $\Gamma$ constructed above readily extends to this relative setting:

Lemma 3.2**.**

There exists a relative correspondence $\Gamma\in A^{3}(\mathcal{X}\times_{B}\mathcal{S})$, such that for all $b\in B$, the restriction

[TABLE]

induces the isomorphism

[TABLE]

as in (3).

Proof.

Let $\mathcal{P}\subset\mathcal{X}\times_{B}\mathcal{F}$ denote the incidence variety, with projections $p\colon\mathcal{P}\to\mathcal{F}$, $q\colon\mathcal{P}\to\mathcal{X}$. Let $\tau$ denote the inclusion morphism $\mathcal{S}\to\mathcal{F}$. We define

[TABLE]

(For composition of relative correspondences in the setting of smooth quasi–projective families that are smooth over a base $B$, cf. [10], [15], [27], [12], [26, 8.1.2].) ∎

The correspondences $\Psi$ and $\gamma$ also extend to the relative setting:

Lemma 3.3**.**

There exist subvarieties $\mathcal{V}_{i},\mathcal{W}_{i}\subset\mathcal{X}$ with $\hbox{codim}(\mathcal{V}_{i})+\hbox{codim}(\mathcal{W}_{i})=4$, and relative correspondences

[TABLE]

where $\gamma$ is supported on $\cup_{i}\mathcal{V}_{i}\times_{B}\mathcal{W}_{i}$, and such that for all $b\in B$, the restrictions

[TABLE]

verify the equality

[TABLE]

as in (5).

Proof.

The statement is different, but this is really the same Hilbert schemes argument as [33, Proposition 3.7], [35, Proposition 4.25].

Let $\Gamma\in A^{3}(\mathcal{X}\times_{B}\mathcal{S})$ be the relative correspondence of lemma 3.2, and let $\Delta_{\mathcal{X}}\in A^{4}(\mathcal{X}\times_{B}\mathcal{X})$ be the relative diagonal. By what we have said above, for each $b\in B$ there exist subvarieties $V_{b,i},W_{b,i}\subset X_{b}$ (with $\dim(V_{b,i})+\dim(W_{b,i})=4$), and a cycle $\gamma_{b}$ supported on

[TABLE]

and a cycle $\Psi_{b}\in A^{3}(S_{b}\times X_{b})$, such that there is equality

[TABLE]

The point is that the data of all the $(b,V_{b,i},W_{b,i},\gamma_{b},\Psi_{b})$ that are solutions of the equality (6) can be encoded by a countable number of algebraic varieties $p_{j}\colon M_{j}\to B$, with universal objects

[TABLE]

(where $\mathcal{V}_{i,j},\mathcal{W}_{i,j}\subset\mathcal{X}_{M_{j}}$, and $\gamma_{j}$ is a cycle supported on $\cup_{i}\mathcal{V}_{i,j}\times_{M_{j}}\mathcal{W}_{i,j}$, and $\Psi_{j}\in A^{3}(\mathcal{S}\times_{M_{j}}\mathcal{X})$), with the property that for $m\in M_{j}$ and $b=p_{j}(m)\in B$, we have

[TABLE]

By what we have said above, the union of the $M_{j}$ dominate $B$. Since there is a countable number of $M_{j}$, one of the $M_{j}$ (say $M_{0}$) must dominate $B$. Taking hyperplane sections, we may assume $M_{0}\to B$ is generically finite (say of degree $d$). Projecting the cycles $\gamma_{0}$ and $\Psi_{0}$ to $\mathcal{X}\times_{B}\mathcal{X}$, resp. to $\mathcal{S}\times_{B}\mathcal{X}$, and then dividing by $d$, we have obtained cycles $\gamma$ and $\Psi$ as requested. ∎

Lemma 3.3 can be succinctly restated as follows: the relative correspondence

[TABLE]

has the property that for all $b\in B$, the restriction is homologically trivial:

[TABLE]

Applying theorem 2.8 to $R$ (this is possible in view of proposition 3.4 below), we find that

[TABLE]

where $\delta$ is some cycle

[TABLE]

Since $A^{\ast}_{hom}(\mathbb{P}^{5}\times\mathbb{P}^{5})=0$, we have

[TABLE]

For $b\in B$ general, the fibre $X_{b}\times X_{b}$ will be in general position with respect to the $\mathcal{V}_{i}$ and $\mathcal{W}_{i}$ and so

[TABLE]

which ensures that

[TABLE]

Plugging in the definition of $R$ into the rational equivalence (7), this means that

[TABLE]

which proves theorem 3.1 for $b\in B$ general.

To prove theorem 3.1 for any given $b_{0}\in B$, we note that the above construction can also be made locally around the point $b_{0}$: in the construction of lemma 3.3, we throw away all the data $M_{j}$ for which the subvarieties $\mathcal{V}_{i,j},\mathcal{W}_{i,j}$ are not all in general position with respect to $X_{b_{0}}\times X_{b_{0}}$. The union of the remaining $M_{j}$ will dominate an open $B^{\prime}\subset B$ containing $b_{0}$, and so the above proof works for the cubic $X_{b_{0}}$.

To end the proof, it remains to verify the hypotheses of theorem 2.8 (which we applied above) are met with. This is the content of the following:

Proposition 3.4**.**

Let $\mathcal{X}\to B$ be the family of smooth cubic fourfolds as in theorem 3.1, i.e.

[TABLE]

is the open subset parametrizing smooth $G$–invariant cubics, and $G=\{id,\iota\}\subset\operatorname{Aut}(\mathbb{P}^{5})$ as above. This set–up verifies the hypotheses of proposition 2.8.

Proof.

Let us first prove hypothesis (i) of proposition 2.8 is satisfied.

To this end, we consider the tower of morphisms

[TABLE]

where $\mathbb{P}(1^{4},2^{2})=\mathbb{P}^{5}/(\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z})$ denotes a weighted projective space. Let us write $\iota_{4},\iota_{5}$ for the involutions of $\mathbb{P}^{5}$

[TABLE]

(We note that $\iota=\iota_{4}\circ\iota_{5}$, and $P=\mathbb{P}^{5}/<\iota_{4},\iota_{5}>$.)

The sections in \bigl{(}\mathbb{P}H^{0}\bigl{(}\mathbb{P}^{5},\mathcal{O}_{\mathbb{P}^{5}}(3)\bigr{)}\bigr{)}^{G} are in bijection with \mathbb{P}H^{0}\bigl{(}P^{\prime},\mathcal{O}_{P^{\prime}}(3)\bigr{)}, and so there is an inclusion

[TABLE]

Let us now assume $x,y\in\mathbb{P}^{5}$ are two points such that

[TABLE]

Then

[TABLE]

and so (using lemma 3.5 below) there exists \sigma\in\mathbb{P}H^{0}\bigl{(}P^{\prime},\mathcal{O}_{P^{\prime}}(3)\bigr{)} containing $p(x)$ but not $p(y)$. The pullback $p^{\ast}(\sigma)$ contains $x$ but not $y$, and so these points $(x,y)$ impose $2$ independent conditions on \bigl{(}\mathbb{P}H^{0}\bigl{(}\mathbb{P}^{5},\mathcal{O}_{\mathbb{P}^{5}}(3)\bigr{)}\bigr{)}^{G}.

It only remains to check that a generic element $(x,y)\in\Gamma_{\iota_{4}}\cup\Gamma_{\iota_{5}}$ also imposes $2$ independent conditions. Let us assume $(x,y)$ is generic on $\Gamma_{4}$ (the argument for $\Gamma_{5}$ is only notationally different). Let us write $x=[a_{0}:a_{1}:\ldots:a_{5}]$. By genericity, we may assume all $a_{i}$ are $\not=0$ (intersections of $\Gamma_{4}$ with a coordinate hyperplane have codimension $>n+1$ and so need not be considered for hypothesis (i) of proposition 2.8). We can thus write

[TABLE]

The cubic

[TABLE]

is $G$–invariant and contains $x$ while avoiding $y$. This proves hypothesis (i) is satisfied.

To establish hypothesis (ii) of proposition 2.8, we proceed by contradiction. Let us suppose hypothesis (ii) is not met with, i.e. there exists a smooth cubic $X_{b}$ as in theorem 3.1, and a non–trivial relation

[TABLE]

where $c,d\in\mathbb{Q}^{\ast}$ and \delta\in\operatorname{Im}\bigl{(}A^{4}(\mathbb{P}^{5}\times\mathbb{P}^{5})\to A^{4}(X_{b}\times X_{b})\bigr{)}. Looking at the action on $H^{3,1}(X_{b})$, we find that necessarily $c=-d$ (indeed, $\delta$ does not act on $H^{3,1}(X_{b})$, and $\iota$ acts as the identity on $H^{3,1}(X_{b})$). That is, we would have a relation

[TABLE]

Looking at the action on $H^{2,2}(X_{b})$, we find that

[TABLE]

Since there is a codimension $2$ linear subspace in $\mathbb{P}^{5}$ fixed by $\iota$, it follows that actually

[TABLE]

Consider now the Fano variety of lines $F=F(X_{b})$ with the involution $\iota_{F}$. Using the Beauville–Donagi isomorphism [4], one obtains that also

[TABLE]

As $\dim\hbox{Gr}^{1}_{F}H^{2}(F,\mathbb{C})=21$, this would imply that the trace of $(\iota_{F})^{\ast}$ on $\hbox{Gr}^{1}_{F}H^{2}(F,\mathbb{C})$ is $21$. However, this contradicts proposition 3.6 below, and so hypothesis (ii) must be satisfied.

Lemma 3.5**.**

Let $P=\mathbb{P}(1^{4},2^{2})$. Let $r,s\in P$ and $r\not=s$. Then there exists \sigma\in\mathbb{P}H^{0}\bigl{(}P,\mathcal{O}_{P}(3)\bigr{)} containing $r$ but avoiding $s$.

Proof.

It follows from Delorme’s work [11, Proposition 2.3(iii)] that the locally free sheaf $\mathcal{O}_{P}(2)$ is very ample. This means there exists \sigma^{\prime}\in\mathbb{P}H^{0}\bigl{(}P,\mathcal{O}_{P}(2)\bigr{)} containing $r$ but avoiding $s$. Taking the union of $\sigma^{\prime}$ with a hyperplane avoiding $s$, one obtains $\sigma$ as required. ∎

Proposition 3.6** (Camere [8]).**

Let $X_{b}\subset\mathbb{P}^{5}$ be a cubic as in theorem 3.1, and let $\iota_{X_{b}}$ be the involution as above. Let $F=F(X_{b})$ be the Fano variety of lines, and let $\iota_{F}$ be the involution of $F$ induced by $\iota_{X_{b}}$. The trace of $(\iota_{F})^{\ast}$ on the $21$–dimensional vector space $\hbox{Gr}^{1}_{F}H^{2}(F,\mathbb{C})$ is $5$.

Proof.

This follows from [8, Theorem 5]. ∎

∎

∎

Remark 3.7**.**

Let $X$ and $S$ be as in theorem 3.1. One expects there is actually an isomorphism

[TABLE]

I am unsure whether the argument of theorem 3.1 can also be used to prove surjectivity.

Remark 3.8**.**

To find the $K3$ surface $S$ of theorem 3.1, we have used the existence of the symplectic involution $\iota_{F}$ on the Fano variety $F=F(X)$ of lines on the cubic fourfold $X$, for which $S\subset F$ is in the fixed locus. One could ask if there exist cubic fourfolds $X$ other than those of theorem 3.1, such that the Fano variety $F(X)$ has a symplectic automorphism with a $2$–dimensional component in the fixed locus.

However, if one restricts to polarized symplectic automorphisms of $F(X)$, there are only $2$ families with a surface in the fixed locus: the family of theorem 3.1, and a family with an abelian surface in the fixed locus. This follows from the classification obtained by L. Fu in [14, Theorem 0.1] (the first family is labelled “Family V-(1)”, and the second family is labelled “Family IV-(2)” in loc. cit.).

The second family (with an abelian surface in the fixed locus) is studied from the point of view of algebraic cycles in [24].

Remark 3.9**.**

Let $X$ and $F$ be as in theorem 3.1. We mention in passing that the automorphisms $\iota$ and $\iota_{F}$ of $X$ resp. of $F$ act as the identity on $A^{3}(X)$, resp. on $A^{4}(F)$ (for $X$, this follows immediately from theorem 3.1).

This is proven more generally for any polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold [13, Theorems 0.5 and 0.6] (for a slightly different take on this, cf. [30, Theorem 5.3]). The argument of [13] is (just like the argument proving theorem 3.1) based on the idea of spread of algebraic cycles in a family, inspired by [33], [34].

4. Finite–dimensionality

Corollary 4.1**.**

Let $X\subset\mathbb{P}^{5}(\mathbb{C})$ be a smooth cubic fourfold defined by an equation

[TABLE]

Assume

[TABLE]

Then $X$ has finite–dimensional motive.

Proof.

It follows from (the proof of) theorem 3.1 there is an inclusion as direct summand

[TABLE]

where $S$ is a $K3$ surface. We have also seen (in the proof of theorem 3.1) there is an isomorphism

[TABLE]

Since the Hodge conjecture is known for $X$ (because $X$ is Fano), there is equality

[TABLE]

Thus, the hypothesis on the dimension of the space of Hodge classes implies that

[TABLE]

and so

[TABLE]

This implies the Picard number $\rho(S)$ is at least $19$, and so $S$ has finite–dimensional motive [28]. In view of inclusion (8), this concludes the proof. ∎

Remark 4.2**.**

Let $X$ be a cubic as in corollary 4.1. Applying [22], it follows that the Fano variety of lines $F:=F(X)$ also has finite–dimensional motive.

**Acknowledgements **.

Thanks to all participants of the Strasbourg 2014/2015 “groupe de travail” based on the monograph [35] for a very pleasant atmosphere. Many thanks to Kai and Len and Yoyo for stimulating discussions not related to this work.