On the Chow groups of some hyperk\"ahler fourfolds with a non-symplectic involution

TL;DR

This paper verifies predictions about how a non-symplectic involution acts on the Chow groups of certain hyperk"ahler fourfolds, specifically Fano varieties of lines on cubic fourfolds, impacting their Chow rings.

Contribution

It confirms the Bloch-Beilinson conjecture predictions for specific hyperk"ahler fourfolds with non-symplectic involutions, providing new insights into their Chow groups.

Findings

Verification of Bloch-Beilinson conjecture predictions for specific hyperk"ahler fourfolds

Analysis of the action of involution on Chow groups

Implications for the Chow ring of the quotient variety

Abstract

This note concerns hyperk\"ahler fourfolds $X$ having a non-symplectic involution $\iota$. The Bloch-Beilinson conjectures predict the way $\iota$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient $X/\iota$.

Equations

$$\begin{array}{rl}{\iota }^{\ast }=-\mathrm{id}:& H^{2i,0}(X)\to H^{2i,0}(X)foriodd,\\ {\iota }^{\ast }=\mathrm{id}:& H^{2i,0}(X)\to H^{2i,0}(X)forieven.\end{array}$$

$$A^i(X)=\underset{j}{⨁}{A}_{(j)}^i(X),$$

$$\begin{array}{rl}{\iota }^{\ast }=(-1{)}^i\mathrm{id}:& A_{(2i)}^{2i}(X)\to A^{2i}(X),\\ {\iota }^{\ast }=(-1{)}^i\mathrm{id}:& A_{(2i)}^{2m}(X)\to A^{2m}(X).\end{array}$$

$$(X_0{)}^2\ell (X_1,\dots ,X_5)+g(X_1,\dots ,X_5)=0.$$

$$\begin{array}{rl}{\mathbb{P}}^5(\mathbb{C})& \to {\mathbb{P}}^5(\mathbb{C}),\\ [X_0,X_1,\dots ,X_5]& \mapsto [-X_0,X_1,\dots ,X_5].\end{array}$$

$$\begin{array}{rl}{\iota }^{\ast }=-\mathrm{id}:& A_{(2)}^i(X)\to A_{(2)}^i(X)fori=2,4;\\ {\iota }^{\ast }=\mathrm{id}:& A_{(j)}^4(X)\to A_{(j)}^4(X)forj=0,4.\end{array}$$

$$A^2(Z)\otimes A^2(Z)\to A^4(Z)$$

$$A^2(Z)\otimes A^1(Z)\to A^3(Z)$$

$$A_{hom}^2(Z)\stackrel{??}{=}0.$$

$${\Delta }_X={\pi }_0+{\pi }_1+\cdots +{\pi }_{2n}in{A}^n(X\times X),$$

$$\Delta_{X}^{sm}:=\bigl{\{}(x,x,x)\ |\ x\in X\bigr{\}}\ \subset\ X\times X\times X\ .$$

$${\pi }_k^X\circ {\Delta }_X^{sm}\circ ({\pi }_i^X\times {\pi }_j^X)=0in{A}^{2n}(X\times X\times X)for\; alli+j\ne k.$$

$$\Bigl{(}\pi^{X}_{k}\circ\Delta_{X}^{sm}\circ(\pi^{X}_{i}\times\pi^{X}_{j})\Bigr{)}{}_{\ast}(a\times b)=0\ \ \ \hbox{for\ all\ }a,b\in\ A^{\ast}(X)\ .$$

$${\Delta }_X^{sm}:h(X)\otimes h(X)\to h(X)in{\mathcal{M}}_{\mathrm{rat}}.$$

$$h(X)=\underset{i=0}{\overset{2n}{⨁}}{h}^i(X)in{\mathcal{M}}_{\mathrm{rat}}.$$

$$h^i(X)\otimes h^j(X)\to h(X)\otimes h(X)\stackrel{{\Delta }_X^{sm}}{\to }h(X)in{\mathcal{M}}_{\mathrm{rat}}$$

$$A_{(j)}^i(X):=({\pi }_{2i-j}^X{)}_{\ast }{A}^i(X),$$

$$A_{(j)}^i(X)\stackrel{??}{=}0forj<0,A_{(0)}^i(X)\cap A_{hom}^i(X)\stackrel{??}{=}0;$$

$$({\pi }_{2i-j}^X{)}_{\ast }{A}^i(X)=A_{(j)}^i(X),$$

$$A_{(j)}^i(X)=0forj<0and\; forj>i.$$

$$A_{(0)}^2(X)\cdot A_{(0)}^2(X)\stackrel{??}{\subset }{A}_{(0)}^4(X)$$

$$\cdot \ell :A_{(2)}^2(X)\stackrel{\cong }{\to }{A}_{(2)}^4(X).$$

$$A_{(2)}^2(X)\otimes A_{(2)}^2(X)\twoheadrightarrow A_{(4)}^4(X).$$

$$[X_0,X_1,\dots ,X_5]\mapsto [-X_0,X_1,\dots ,X_5].$$

$$(X_0{)}^2\ell (X_1,\dots ,X_5)+g(X_1,\dots ,X_5)=0.$$

$$H^2(X)\cong H^4(Y).$$

$$\sum _{i=0}^5(-1{)}^i{X}_i\frac{d{X}_0\wedge \dots \wedge d\widehat{X_i}\wedge \dots \wedge d{X}_5}{f^2},$$

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

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

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

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 of $X$ (i.e. the groups of codimension $i$ algebraic cycles on $X$ with $\mathbb{Q}$–coefficients, modulo rational equivalence). As explained for instance in [12] or [29], the Bloch–Beilinson conjectures form a powerful and coherent heuristic guide, allowing to make concrete predictions about Chow groups. In this note, we focus on one particular instance of such a prediction, concerning non–symplectic involutions on hyperkähler varieties.

Let $X$ be a hyperkähler variety (i.e., a projective irreducible holomorphic symplectic manifold, cf. [1], [2]), and suppose $X$ has an anti–symplectic involution $\iota$. The action of $\iota$ on the subring $H^{\ast,0}(X)$ is well–understood: we have

[TABLE]

The action of $\iota$ on the Chow ring $A^{\ast}(X)$ is more mysterious. To state the conjectural behaviour, we will now assume the Chow ring of $X$ has a bigraded ring structure $A^{\ast}_{(\ast)}(X)$, where each $A^{i}(X)$ splits into pieces

[TABLE]

and the piece $A^{i}_{(j)}(X)$ is isomorphic to the graded $\hbox{Gr}^{j}_{F}A^{i}(X)$ for the Bloch–Beilinson filtration that conjecturally exists for all smooth projective varieties. (It is expected such a bigrading $A^{\ast}_{(\ast)}(-)$ exists for all hyperkähler varieties [4].)

Since the pieces $A^{i}_{(i)}(X)$ and $A^{\dim X}_{i}(X)$ should only depend on the subring $H^{\ast,0}(X)$, we arrive at the following conjecture:

Conjecture 1.1**.**

Let $X$ be a hyperkähler variety of dimension $2m$, and let $\iota\in\operatorname{Aut}(X)$ be a non–symplectic involution. Then

[TABLE]

This conjecture is studied, and proven in some particular cases, in [13], [15], [14], [16]. The aim of this note is to provide some more examples where conjecture 1.1 is verified, by considering Fano varieties of lines on cubic fourfolds. The main result is as follows:

Theorem **** (=theorem 3.4).

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

[TABLE]

Let $X=F(Y)$ be the Fano variety of lines in $Y$. Let $\iota\in\operatorname{Aut}(X)$ be the anti–symplectic involution induced by

[TABLE]

Then

[TABLE]

The notation $A^{\ast}_{(\ast)}(X)$ in theorem 3.4 refers to the Fourier decomposition of the Chow ring of $X$ constructed by Shen–Vial [21]. (We mention in passing that for $X$ as in theorem 3.4, it is unfortunately not yet known whether $A^{\ast}_{(\ast)}(X)$ is a bigraded ring, cf. remark 2.6 below.)

To prove theorem 3.4, we employ the fact that the family of cubics under consideration is sufficiently large for the method of “spread” developed by Voisin [26], [27] to apply. It is worth mentioning that the action of polarized symplectic automorphisms on Chow groups of Fano varieties of cubic fourfolds has already been treated by L. Fu [9], similarly using the method of “spread”.

Theorem 3.4 has some rather striking consequences for the Chow ring of the quotient (this quotient is a slightly singular Calabi–Yau fourfold):

Corollary **** (=corollary 4.1).

Let $(X,\iota)$ be as in theorem 3.4. Let $Z:=X/\iota$ be the quotient. Then the image of the intersection product map

[TABLE]

has dimension $1$.

This means that for any $2$–cycles $b,c\in A^{2}(Z)$, the [math]–cycle $b\cdot c$ is rationally trivial if and only if it has degree [math]. This is similar to results for Calabi–Yau complete intersections obtained in [25], [8].

Corollary **** (=corollary 4.4).

Let $(X,\iota)$ be as in theorem 3.4. Let $Z:=X/\iota$ be the quotient. Then the image of the intersection product map

[TABLE]

is a finite–dimensional $\mathbb{Q}$–vector space.

Corollaries 4.1 and 4.4 provide some (admittedly very meagre) support in favour of the conjecture that

[TABLE]

(The Bloch–Beilinson conjectures would imply that $A^{2}_{AJ}(M)=0$ for any Calabi–Yau variety $M$ of dimension $>2$. As far as I am aware, there is not a single Calabi–Yau variety $M$ for which this is known to be true.)

**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 [20], [18]) will be denoted $\mathcal{M}_{\rm rat}$.

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

2. Preliminaries

2.1. MCK decomposition

Definition 2.1** (Murre [17]).**

Let $X$ be a smooth projective variety of dimension $n$. We say that $X$ has a CK decomposition if there exists a decomposition of the diagonal

[TABLE]

such that the $\pi_{i}$ are mutually orthogonal idempotents and $(\pi_{i})_{\ast}H^{\ast}(X)=H^{i}(X)$.

(NB: “CK decomposition” is shorthand for “Chow–Künneth decomposition”.)

Remark 2.2**.**

*The existence of a CK decomposition for any smooth projective variety is part of Murre’s conjectures [17], [12]. *

Definition 2.3** (Shen–Vial [21]).**

Let $X$ be a smooth projective variety of dimension $n$. Let $\Delta_{X}^{sm}\in A^{2n}(X\times X\times X)$ be the class of the small diagonal

[TABLE]

An MCK decomposition is a CK decomposition $\{\pi^{X}_{i}\}$ of $X$ that is multiplicative, i.e. it satisfies

[TABLE]

(NB: “MCK decomposition” is shorthand for “multiplicative Chow–Künneth decomposition”.)

A weak MCK decomposition is a CK decomposition $\{\pi^{X}_{i}\}$ of $X$ that satisfies

[TABLE]

Remark 2.4**.**

The small diagonal (seen as a correspondence from $X\times X$ to $X$) induces the multiplication morphism

[TABLE]

Suppose $X$ has a CK decomposition

[TABLE]

By definition, this decomposition is multiplicative if for any $i,j$ the composition

[TABLE]

factors through $h^{i+j}(X)$.

If $X$ has a weak MCK decomposition, then setting

[TABLE]

one obtains a bigraded ring structure on the Chow ring: that is, the intersection product sends $A^{i}_{(j)}(X)\otimes A^{i^{\prime}}_{(j^{\prime})}(X)$ to $A^{i+i^{\prime}}_{(j+j^{\prime})}(X)$.

It is expected (but not proven !) that for any $X$ with a weak MCK decomposition, one has

[TABLE]

this is related to Murre’s conjectures B and D, that have been formulated for any CK decomposition [17].

The property of having an MCK decomposition is severely restrictive, and is closely related to Beauville’s “(weak) splitting property” [4]. For more ample discussion, and examples of varieties with an MCK decomposition, we refer to [21, Section 8], as well as [24], [22], [10].

In what follows, we will make use of the following:

Theorem 2.5** (Shen–Vial [21]).**

Let $Y\subset\mathbb{P}^{5}(\mathbb{C})$ be a smooth cubic fourfold, and let $X:=F(Y)$ be the Fano variety of lines in $Y$. There exists a CK decomposition $\{\pi^{X}_{i}\}$ for $X$, and

[TABLE]

where the right–hand side denotes the splitting of the Chow groups defined in terms of the Fourier transform as in [21, Theorem 2]. Moreover, we have

[TABLE]

In case $Y$ is very general, the Fourier decomposition $A^{\ast}_{(\ast)}(X)$ forms a bigraded ring, and hence $\{\pi^{X}_{i}\}$ is a weak MCK decomposition.

Proof.

(A remark on notation: what we denote $A^{i}_{(j)}(X)$ is denoted $CH^{i}(X)_{j}$ in [21].)

The existence of a CK decomposition $\{\pi^{X}_{i}\}$ is [21, Theorem 3.3], combined with the results in [21, Section 3] to ensure that the hypotheses of [21, Theorem 3.3] are satisfied. According to [21, Theorem 3.3], the given CK decomposition agrees with the Fourier decomposition of the Chow groups. The “moreover” part is because the $\{\pi^{X}_{i}\}$ are shown to satisfy Murre’s conjecture B [21, Theorem 3.3].

The statement for very general cubics is [21, Theorem 3]. ∎

Remark 2.6**.**

Unfortunately, it is not yet known that the Fourier decomposition of [21] induces a bigraded ring structure on the Chow ring for all Fano varieties of smooth cubic fourfolds. For one thing, it has not yet been proven that

[TABLE]

for the Fano variety of a given (not necessarily very general) cubic fourfold (cf. [21, Section 22.3] for discussion).

To prove that $A^{\ast}_{(\ast)}()$ is a bigraded ring for all Fano varieties of smooth cubic fourfolds, it would suffice to construct an MCK decomposition for the Fano variety of the very general cubic fourfold.

2.2. A multiplicative result

Let $X$ be the Fano variety of lines on a smooth cubic fourfold. As we have seen (theorem 2.5), the Chow ring of $X$ splits into pieces $A^{i}_{(j)}(X)$. The work [21] contains a thorough analysis of the multiplicative behaviour of these pieces. Here are the relevant results we will be needing:

Theorem 2.7** (Shen–Vial [21]).**

Let $Y\subset\mathbb{P}^{5}(\mathbb{C})$ be a smooth cubic fourfold, and let $X:=F(Y)$ be the Fano variety of lines in $Y$.

(i) There exists $\ell\in A^{2}_{(0)}(X)$ such that intersecting with $\ell$ induces an isomorphism

[TABLE]

(ii) Intersection product induces a surjection

[TABLE]

Proof.

Statement (i) is [21, Theorem 4]. Statement (ii) is [21, Proposition 20.3]. ∎

2.3. The involution

Lemma 2.8**.**

Let $\iota_{\mathbb{P}}\in\operatorname{Aut}(\mathbb{P}^{5}(\mathbb{C}))$ be the involution defined as

[TABLE]

The cubic fourfolds invariant under $\iota_{\mathbb{P}}$ are exactly those defined by an equation

[TABLE]

Let $Y\subset\mathbb{P}^{5}(\mathbb{C})$ be a smooth cubic invariant under $\iota_{\mathbb{P}}$, and let $\iota_{Y}\in\operatorname{Aut}(Y)$ be the involution induced by $\iota_{\mathbb{P}}$. Let $X=F(Y)$ be the Fano variety of lines in $Y$, and let $\iota\in\operatorname{Aut}(X)$ be the involution induced by $\iota_{Y}$. The involution $\iota$ is anti–symplectic.

Proof.

The only thing that needs explaining is the last phrase; this is proven in [6, Section 7]. The idea is that there is an isomorphism of Hodge structures, compatible with the involution

[TABLE]

The action of $\iota_{Y}$ on $H^{3,1}(Y)$ is minus the identity, because $H^{3,1}(Y)$ is generated by the meromorphic form

[TABLE]

where $f$ is an equation for $Y$. ∎

2.4. Spread

Lemma 2.9** (Voisin [26], [27]).**

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 [26, Proof of Proposition 3.13] or [27, 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 2.10** (Voisin [27]).**

Assumptions as in lemma 2.9. 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 [27, Proposition 1.6]. ∎

The following is an equivariant version of proposition 2.10:

Proposition 2.11** (Voisin [27]).**

Let $M$ and $L$ be as in proposition 2.10. 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 [27], but it is contained in the proof of [27, 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 [27, 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 [27, 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 [27, 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

This section contains the proof of the main result of this note, theorem 3.4. The proof is split in two parts. In the first part, we prove a statement (theorem 3.1) about the action of the involution on $1$–cycles on the cubic $Y$. The proof is based on the technique of “spread” of cycles in a family, as developed by Voisin [26], [27], [28], [29] (more precisely, the results recalled in subsection 2.4).

In the second part, we deduce from this our main result, theorem 3.4. This second part builds on the structural results of Shen–Vial [21] (notably the results recalled in subsections 2.1 and 2.2).

3.1. First part

Theorem 3.1**.**

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

[TABLE]

Let $\iota_{Y}\in\operatorname{Aut}(Y)$ be the involution of lemma 2.8. Then

[TABLE]

Proof.

We have seen (proof of lemma 2.8) that

[TABLE]

Let $H^{4}_{tr}(Y)$ denote the orthogonal complement (under the cup–product pairing) of $N^{2}H^{4}(Y)$ (which coincides with $H^{2,2}(Y,\mathbb{Q})$ since the Hodge conjecture is true for $Y$). Since $H^{4}_{tr}(Y)\subset H^{4}(Y)$ is the smallest Hodge substructure containing $H^{3,1}(Y)$, we must also have

[TABLE]

This implies that there is a decomposition

[TABLE]

where $\gamma\in A^{4}(Y\times Y)$ is a “completely decomposed” cycle, i.e.

[TABLE]

and $\gamma_{2i}$ has support on $V_{i}\times W_{i}\subset Y\times Y$ with $\dim V_{i}=i$ and $\dim W_{i}=4-i$. (Indeed, the cycle $\gamma$ is obtained by considering

[TABLE]

where $\pi_{i}$ denotes the Künneth component. For $i\not=4$, the claimed support condition is obviously satisfied since it is satisfied by $\pi_{i}$. For $i=4$, one uses (1) to see that $\gamma_{4}$ is supported on $N^{2}H^{4}(Y)\otimes N^{2}H^{4}(Y)\subset H^{8}(Y\times Y)$.)

We now consider things family–wise. Let

[TABLE]

denote the universal family of all smooth cubic fourfolds defined by an equation where $X_{0}$ occurs only in even degree. Let $Y_{b}\subset\mathbb{P}^{5}(\mathbb{C})$ denote the fibre over $b\in B$.

The involution $\iota_{\mathbb{P}}$ defines by restriction an involution $\iota_{\mathcal{Y}}\in\operatorname{Aut}(\mathcal{Y})$. Let $\Delta_{\mathcal{Y}}\in A^{4}(\mathcal{Y}\times_{B}\mathcal{Y})$ denote the relative diagonal. Obviously the argument leading to the decomposition (2) applies to each fibre $Y_{b}$. This means that for each $b\in B$, there exists a completely decomposed cycle $\gamma_{b}\in A^{4}(Y_{b}\times Y_{b})$ such that

[TABLE]

Applying the “spread” result [26, Proposition 3.7], we can find a “completely decomposed” relative correspondence $\gamma\in A^{4}(\mathcal{Y}\times_{B}\mathcal{Y})$ such that

[TABLE]

(By this, we mean the following: there exist subvarieties $\mathcal{V}_{i},\mathcal{W}_{i}\subset\mathcal{Y}$ for $i=0,2,4,6,8$ with

[TABLE]

and such that the cycle $\gamma$ is supported on

[TABLE]

Actually, for $i\not=4$ this is obvious since the $\pi_{i},i\not=4$ obviously exist relatively. The recourse to [26, Proposition 3.7] can thus be limited to $i=4$.)

That is, the relative correspondence

[TABLE]

is fibrewise homologically trivial:

[TABLE]

At this point, we note that the family $\mathcal{Y}\to B$ is large enough to verify the hypotheses of proposition 2.11; this will be proven in lemma 3.2 below. Applying proposition 2.11 to the relative correspondence $\Gamma$, we find that there exists $\delta\in A^{4}(\mathbb{P}^{5}\times\mathbb{P}^{5})$ such that

[TABLE]

But

[TABLE]

(indeed, the action factors over $A^{4}_{hom}(\mathbb{P}^{5})$ which is [math]). Also, we have

[TABLE]

(indeed, for general $b\in B$ the restriction $\gamma|_{Y_{b}\times Y_{b}}$ is a completely decomposed cycle; such cycles do not act on $A^{3}_{hom}$ for dimension reasons).

By definition of $\Gamma$, this means that

[TABLE]

This proves theorem 3.1 for general $b\in B$. To extend to all $b\in B$, one can reason as in [9, Lemma 3.1].

It only remains to check the hypotheses of Voisin’s result are satisfied:

Lemma 3.2**.**

Let $\mathcal{X}\to B$ be the family of smooth cubic fourfolds as in theorem 3.4, i.e.

[TABLE]

is the open subset parametrizing smooth $G$–invariant cubics, and $G=\{id,\iota_{\mathbb{P}}\}\subset\operatorname{Aut}(\mathbb{P}^{5})$ as above. This set–up verifies the hypotheses of proposition 2.11.

Proof.

Let us first prove hypothesis (i) of proposition 2.11 is satisfied.

To this end, we consider the quotient morphism

[TABLE]

where $\mathbb{P}(2,1^{5})$ denotes a weighted projective space.

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 sections coming from $P$, i.e. there is an isomorphism

[TABLE]

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

[TABLE]

Then

[TABLE]

and so (using lemma 3.3 below) there exists \sigma\in\mathbb{P}H^{0}\bigl{(}P,\mathcal{O}_{P}(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}.

To establish hypothesis (ii) of proposition 2.11, we proceed by contradiction. Let us suppose hypothesis (ii) is not met with, i.e. there exists a smooth cubic $Y_{b}$ as in theorem 3.4, 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}(Y_{b}\times Y_{b})\bigr{)}. Looking at the action on $H^{3,1}(Y_{b})$, we find that necessarily $c=d$ (indeed, $\delta$ does not act on $H^{3,1}(Y_{b})$, and $\iota_{Y_{b}}$ acts as minus the identity on $H^{3,1}(Y_{b})$). That is, we would have a relation

[TABLE]

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

[TABLE]

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

[TABLE]

Consider now the Fano variety of lines $X=F(Y_{b})$ with the involution $\iota$. Using the Beauville–Donagi isomorphism [3], one obtains that also

[TABLE]

and so the trace of $(\iota)^{\ast}$ on $\hbox{Gr}^{1}_{F}H^{2}(X,\mathbb{C})$ would be $-19$. However, this contradicts the fact that (as noted in [5, 3.6]) the trace is actually $-7$, and so hypothesis (ii) must be satisfied.

Lemma 3.3**.**

Let $P=\mathbb{P}(2,1^{5})$. 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 [7, 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. ∎

∎

∎

3.2. Second part

Theorem 3.4**.**

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

[TABLE]

Let $X=F(Y)$ be the Fano variety of lines in $Y$. Let $\iota\in\operatorname{Aut}(X)$ be the anti–symplectic involution of lemma 2.8. Then

[TABLE]

Proof.

First, we note that

[TABLE]

where $I\subset X\times X$ is the incidence correspondence [21, Proof of Proposition 21.10]. On the other hand,

[TABLE]

where $X\leftarrow P\to Y$ denotes the universal family of lines on $Y$ [21, Lemma 17.2]. Hence,

[TABLE]

But $P_{\ast}\colon A^{4}_{hom}(X)\to A^{3}_{hom}(Y)$ is surjective [19], and so

[TABLE]

It is readily checked that the diagram

[TABLE]

is commutative (this is because the involution extends to an involution on $P$). Using this diagram, theorem 3.1 implies that $\iota$ acts as minus the identity on $({}^{t}P)_{\ast}A^{3}_{hom}(Y)=A^{2}_{(2)}(X)$.

Since intersection product induces a surjection

[TABLE]

(theorem 2.7(ii)), it follows that $\iota$ acts as the identity on $A^{4}_{(4)}(X)$.

Next, we want to exploit the fact that there is an isomorphism

[TABLE]

(theorem 2.7(i)). Since $\iota^{\ast}(\ell)=\ell$ (proposition 3.5 below), this implies that $\iota$ acts as minus the identity on $A^{4}_{(2)}(X)$.

Proposition 3.5**.**

Let $X$ be the variety of lines on a smooth cubic fourfold $Y\subset\mathbb{P}^{5}(\mathbb{C})$, and let $\iota\in\operatorname{Aut}(X)$ be an involution induced by an involution $\iota_{Y}\in\operatorname{Aut}(Y)$. Let $\ell\in A^{2}(X)$ be the class of theorem 2.7(i). Then

[TABLE]

Proof.

We give two proofs of this fact. The first proof has the benefit of brevity; the second proof will be useful in proving another result (lemma 3.6 below).

First proof: It is known that

[TABLE]

(where the right–hand side denotes the second Chern class of the tangent bundle $T_{X}$ of $X$) [21, Equation (108)]. Since

[TABLE]

this proves the proposition.

Second proof: Shen–Vial define the class $L\in A^{2}(X\times X)$ (lifting the Beauville-Bogomolov class $\mathfrak{B}\in H^{4}(X\times X)$) as

[TABLE]

[21, Equation (107)]. Here $I$ is the incidence correspondence, and

[TABLE]

where $\mathcal{E}_{2}$ is the rank $2$ vector bundle coming from the tautological bundle on the Grassmannian, and $p_{i}\colon X\times X\to X$ denote the two projections.

Clearly one has

[TABLE]

In view of the definition of $L$, it follows that

[TABLE]

Using Lieberman’s lemma [23, Lemma 3.3], plus the fact that ${}^{t}\Gamma_{\iota}=\Gamma_{\iota}$, this means there is a commutativity relation

[TABLE]

The class $\ell$ is defined as $\ell:=(i_{\Delta})^{\ast}(L)\in A^{2}(X)$. We now find that

[TABLE]

Here the second equality is by virtue of the commutative diagram

[TABLE]

The third equality is again Lieberman’s lemma, plus the fact that ${}^{t}\Gamma_{\iota}=\Gamma_{\iota}$. The last equality is (3). ∎

It only remains to prove theorem 3.4 is true for $(i,j)=(4,0)$. This follows from the fact that $A^{4}_{(0)}(X)$ is generated by $\ell^{2}$ [21], plus the fact that $\ell$ is $\iota$–invariant (proposition 3.5). Theorem 3.4 is now proven. ∎

For later use, we remark that the argument of proposition 3.5 also proves the following compatibility statement:

Lemma 3.6**.**

Let $X$ be the variety of lines on a smooth cubic fourfold $Y\subset\mathbb{P}^{5}(\mathbb{C})$, and let $\iota\in\operatorname{Aut}(X)$ be an involution induced by an involution $\iota_{Y}\in\operatorname{Aut}(Y)$. Then

[TABLE]

Proof.

Given the Shen–Vial class $L\in A^{2}(X\times X)$ as above, consider (as in [21, Section 2]) the eigenspaces

[TABLE]

We observe that equality (3) implies

[TABLE]

Using Lieberman’s lemma, this is equivalent to the commutativity relation

[TABLE]

This commutativity relation implies that the involution respects the eigenspaces:

[TABLE]

As shown in [21, Part 3], $X$ verifies the hypotheses of [21, Theorem 2.1]. As such, [21, Theorem 2.1] applies to $X$, which means that each $A^{i}_{(j)}(X)$ is a sum of eigenspaces $\Lambda^{i}_{\lambda}$ except for $(i,j)=(4,0)$ and $(i,j)=(4,2)$, for which one has

[TABLE]

Property (4), plus the fact that $\iota^{\ast}A^{\ast}_{hom}(X)\subset A^{\ast}_{hom}(X)$, thus implies that

[TABLE]

The remaining case $(i,j)=(4,0)$ was already treated above: $A^{4}_{(0)}(X)$ is generated by $\ell^{2}$ and $\iota^{\ast}(\ell)=\ell$. ∎

4. Corollaries

In this last section, we consider the quotient $Z:=X/\iota$, for $(X,\iota)$ as in theorem 3.4. The variety $Z$ is a slightly singular Calabi–Yau variety. As is well–known, Chow groups with $\mathbb{Q}$–coefficients of quotient varieties such as $Z$ still have a ring structure [11, Examples 8.3.12 and 17.4.10]. For this reason, we will write $A^{i}(Z)$ for the Chow group of codimension $i$ cycles on $Z$ (just as in the smooth case).

Corollary 4.1**.**

Let $(X,\iota)$ be as in theorem 3.4. Let $Z:=X/\iota$ be the quotient. Then the image of the intersection product map

[TABLE]

has dimension $1$.

Proof.

We first establish a lemma:

Lemma 4.2**.**

Let $(X,\iota)$ be as in theorem 3.4. Then

[TABLE]

Proof.

Let $c\in A^{2}(X)^{\iota}$, and suppose

[TABLE]

where $c_{j}\in A^{2}_{(j)}(X)$. Since $c$ is $\iota$–invariant, we also have

[TABLE]

(where we have used theorem 3.4 to conclude that $\iota^{\ast}(c_{2})=-c_{2}$. But $\iota^{\ast}(c_{0})\in A^{2}_{(0)}(X)$ (lemma 3.6), and so (by unicity of the decomposition $c=c_{0}+c_{2}$) we must have

[TABLE]

∎

Let $p\colon X\to Z$ be the quotient morphism. Lemma 4.2 says that

[TABLE]

It follows that

[TABLE]

Here for the last inclusion we have used [21, Proposition 22.8].

On the other hand, we have

[TABLE]

and so (by combining with the above inclusion) we find that

[TABLE]

Applying lemma 4.3 below, it follows that

[TABLE]

Since $\dim A^{4}_{(0)}(X)=1$ and $p^{\ast}$ is injective, this proves

[TABLE]

Since p^{\ast}\operatorname{Im}\bigl{(}A^{2}(Z)\otimes A^{2}(Z)\to A^{4}(Z)\bigr{)} contains the non–trivial element $\ell^{2}$, the dimension must be equal to $1$.

Lemma 4.3**.**

Let $(X,\iota)$ be as in theorem 3.4. Then

[TABLE]

Proof.

This is similar to the proof of lemma 4.2. Let

[TABLE]

(where $b_{j}\in A^{4}_{(j)}(X)$), and suppose $b$ is $\iota$–invariant. In view of theorem 3.4, this means that

[TABLE]

But $\iota^{\ast}(b_{0})\in A^{4}_{(0)}(X)$ (lemma 3.6), and so (because of unicity of the decomposition $b=b_{0}+b_{2}$) we must have

[TABLE]

This proves the inclusion “$\subset$”. The opposite inclusion holds since $\ell^{2}$ generates $A^{4}_{(0)}(X)$ and is $\iota$–invariant. ∎

∎

One can also say something about $1$–cycles on the quotient:

Corollary 4.4**.**

Let $(X,\iota)$ be as in theorem 3.4. Let $Z:=X/\iota$ be the quotient. Then the image of the intersection product map

[TABLE]

is a finite–dimensional $\mathbb{Q}$–vector space.

Proof.

As above, let $p\colon X\to Z$ denote the quotient morphism. We have seen (lemma 4.2) that

[TABLE]

and so

[TABLE]

Let $N^{2}H^{4}(X)\subset H^{4}(X)$ denote the $\mathbb{Q}$–vector space of cycle classes. There is an exact sequence

[TABLE]

Let $b_{1},\ldots,b_{r}$ be (non–canonical) lifts of a basis of the finite–dimensional $\mathbb{Q}$–vector space $N^{2}H^{4}(X)$ to $A^{2}_{(0)}(X)$, and let $B\subset A^{2}_{(0)}(X)$ be the finite–dimensional sub–vector space spanned by the $b_{i}$.

It is known that the intersection product map

[TABLE]

is the zero–map [21, Proposition 22.4]. It follows that

[TABLE]

which is finite–dimensional.

Since $p^{\ast}$ is injective, this proves the corollary. ∎

Remark 4.5**.**

Let $X$ and $Z$ be as in corollary 4.4. The statement obtained in corollary 4.4 is less than optimal, in the following sense. It should be the case that the cycle class map induces an injection

[TABLE]

Indeed, as we have seen $p^{\ast}A^{2}(Z)\subset A^{2}_{(0)}(X)$. Now if we knew that

[TABLE]

then one could conclude that (5) is true.

Unfortunately, the inclusion (6) is known only for Fano varieties of very general cubic fourfolds. The problem is thus (once again !) the absence of an MCK decomposition for Fano varieties of arbitrary smooth cubic fourfolds.

5. Closing remarks

Remark 5.1**.**

The family of hyperkähler fourfolds as in theorem 3.4 occurs as an example in [5, 3.6]. Applying [5, Theorem 2], one finds that the pair $(X,\iota)$ (where $X$ and $\iota$ are as in theorem 3.4) deforms in a $14$–dimensional family. The cubic fourfolds $Y$ as in theorem 3.4 form a $14$–dimensional family, and so any deformation of $X$ that is not a Fano variety of lines “loses” the anti–symplectic involution.

Remark 5.2**.**

Let $(X,\iota)$ be as in theorem 3.4. As explained in [5, 3.6], the fixed–locus of $\iota$ is the disjoint union $S\cup T$, where $S$ is a cubic surface and $T$ is the Fano variety of lines on the smooth cubic threefold $g(X_{1},\ldots,X_{5})=0$. The surfaces $S$ and $T$ are Lagrangian subvarieties of $X$ [5]. Lemma 4.2 implies that $S$ and $T$ in $A^{2}_{(0)}(X)$. (NB: for $T$, this also follows from the fact that

[TABLE]

as it is known that $c\in A^{2}_{(0)}(X)$ [21, Theorem 21.9(iii)].)

This raises the following question:

(i) Is it true that intersecting with $S$ resp. $T$ induces the zero–map

[TABLE]

The answer is “yes” in both cases (for $T$ because $T=c$ in $A^{2}(X)$, and for $S$ because $S$ is rational, hence a constant cycle subvariety).

A stronger version of question (i) is

(i′) Is it true that the restriction map

[TABLE]

(resp. $A^{2}_{hom}(X)\to A^{2}(S)$) is the zero–map ? The answer is trivially “yes” for $S$, but “I don’t know” for $T$.

Remark 5.3**.**

Let $Y$ be a smooth cubic fourfold, and $X=F(Y)$ the Fano variety of lines on $X$. Let us say that an automorphism $\sigma\in\operatorname{Aut}(X)$ is induced if it comes from an automorphism of $Y$. (An automorphism of $X$ is induced if and only if it is polarized, cf. [9, Proposition 6.1].)

Apart from the family considered in theorem 3.4, there is only one other family of Fano varieties of lines on cubic fourfolds with an induced anti–symplectic involution. This is the third case of the table given in [6, Section 7]. It would be interesting to prove theorem 3.4 for this family as well.

I do not know examples of non–induced involutions of Fano varieties of lines on cubic fourfolds.

Also, it would be interesting to consider non–symplectic automorphisms of order $>2$ of Fano varieties of lines on cubic fourfolds.

**Acknowledgements **.

Many thanks to Kai for reading “Pluk van de Petteflat” with me.