A remark on Beauville's splitting property

TL;DR

This paper discusses a conjecture related to the injectivity of a subring of algebraic cycles on hyperk"ahler varieties into cohomology, providing evidence for its validity.

Contribution

It proposes a new conjecture extending Beauville's splitting property to algebraic cycles modulo algebraic equivalence and offers supporting evidence.

Findings

Evidence supporting the conjecture is presented.

The subring containing divisors and codimension 2 cycles likely injects into cohomology.

The conjecture extends Beauville's original splitting property.

Abstract

Let $X$ be a hyperk\"ahler variety. Beauville has conjectured that a certain subring of the Chow ring of $X$ should inject into cohomology. This note proposes a similar conjecture for the ring of algebraic cycles on $X$ modulo algebraic equivalence: a certain subring (containing divisors and codimension $2$ cycles) should inject into cohomology. We present some evidence for this conjecture.

Equations

$$\sum _i{D}_i\cdot D_i^{\prime }=0in{A}^2(S)\iff \sum _i[D_i]\cup [D_i^{\prime }]=0in{H}^4(S,\mathbb{Q}).$$

$$A^i(X)=\underset{j=0}{\overset{i}{⨁}}{A}_{(j)}^i(X)for\; alli.$$

$$A_{hom}^i(X)=\underset{j\ge 1}{⨁}{A}_{(j)}^i(X).$$

$$D^{i}(X)\ \hookrightarrow\ H^{2i}(X,\mathbb{Q})\$$

$$B^2(X)=B_{(0)}^2(X).$$

$$E^i(X)\hookrightarrow H^{2i}(X,\mathbb{Q})\forall i.$$

$$E^3(X)\hookrightarrow H^6(X,\mathbb{Q}).$$

$$E^{2m-1}(X)\hookrightarrow H^{4m-2}(X,\mathbb{Q}).$$

$$E^i(X)\hookrightarrow H^{2i}(X)fori\ge 3.$$

$$A_{(2)}^2(X)=L_{\ast }{A}_{(2)}^4(X).$$

$$A_{(2)}^2(X)\subset A_{alg}^2(X).$$

$$B^{2}(X)=A^{2}(X)/A^{2}_{alg}(X)=\bigl{(}A^{2}_{(0)}(X)\oplus A^{2}_{(2)}(X)\bigr{)}/A^{2}_{alg}(X)=A^{2}_{(0)}(X)/A^{2}_{alg}(X)=B^{2}_{(0)}(X)\ .$$

$$\frac{25}{2}a=\mathcal{F}\circ \mathcal{F}(a)=0in{A}^3(X).$$

$$A^1(X{)}_{prim}:=P_{\ast }(A^2(Y{)}_{prim})\subset A^1(X),$$

$$b=\sum _{k=1}^r{a}_k\cdot d_k\in B^3(X),$$

$$A_{(0)}^2(X)\cdot A^1(X{)}_{prim}\subset A_{(0)}^3(X)$$

$$E^{i}(X)\ \hookrightarrow\ H^{2i}(X)\$$

$$B^2(X)=\underset{j\le 0}{⨁}{B}_{(j)}^2(X).$$

$$B_{(1)}^2(X)=({\Pi }_3^X{)}_{\ast }{B}^2(X),$$

$$B_{(1)}^2(X)=({\Pi }_3^X{)}_{\ast }{B}_{hom}^2(X).$$

$${\gamma }_{\ast }{B}_{hom}^2(X)=0.$$

$$({\Pi }_3^X{)}_{\ast }{B}_{hom}^2(X)=0,$$

$${\pi }_{\ast }{B}_{hom}^i(X)=0\Rightarrow {\Pi }_{\ast }{B}_{hom}^i(X)=0.$$

$$\Pi -\pi \in A_{hom}^n(X\times X).$$

$$(\Pi -\pi {)}^{\circ N}=0in{A}^n(X\times X).$$

$$\Pi ={\Pi }^{\circ N}=P_1+P_2+\cdots +P_{m}in{A}^n(X\times X{)}_{\mathbb{Q}},$$

$$E^{\ast }(X)\subset \underset{j\le 0}{⨁}{B}_{(j)}^{\ast }(X).$$

$$B_{(j)}^{2m-1}(X)=0forj<0,and{B}_{(0)}^{2m-1}(X)\cap B_{hom}^{2m-1}(X)=0.$$

$$B_{(j)}^{2m-1}(X)=({\Pi }_{4m-2-j}^X{)}_{\ast }{B}^{2m-1}(X),$$

$${\gamma }_{\ast }{B}_{hom}^{2m-1}(X)=0.$$

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 group of codimension $i$ algebraic cycles modulo rational equivalence with $\mathbb{Q}$–coefficients. Intersection product defines a ring structure on $A^{\ast}(X)=\oplus_{i}A^{i}(X)$. In the case of $K3$ surfaces, this ring structure has a remarkable property:

Theorem 1.1** (Beauville–Voisin [6]).**

Let $S$ be a $K3$ surface. Let $D_{i},D_{i}^{\prime}\in A^{1}(S)$ be a finite number of divisors. Then

[TABLE]

In the wake of this result (combined with results concerning the Chow ring of abelian varieties [4]), Beauville has asked which varieties have behaviour similar to theorem 1.1. This is the problem of determining which varieties verify the “splitting property” of [5]. We briefly state this problem here as follows:

Problem 1.2** (Beauville [5]).**

Find a class $\mathcal{C}$ of varieties (containing $K3$ surfaces, abelian varieties and hyperkähler varieties), such that for any $X\in\mathcal{C}$, the Chow ring of $X$ admits a multiplicative bigrading $A^{\ast}_{(\ast)}(X)$, with

[TABLE]

This bigrading should split the conjectural Bloch–Beilinson filtration, in particular

[TABLE]

This question is hard to answer in practice, since we do not have the Bloch–Beilinson filtration at our disposal. However, as noted by Beauville, the class $\mathcal{C}$ has some nice properties that can be tested in practice. In particular, the conjecture that hyperkähler varieties are in $\mathcal{C}$ leads to the so–called weak splitting property conjecture, which is the following falsifiable statement:

Conjecture 1.3** (Beauville [5], Voisin [18]).**

Let $X$ be a hyperkähler variety, and let $D^{\ast}(X)\subset A^{\ast}(X)$ be the $\mathbb{Q}$–subalgebra generated by divisors and Chern classes. The cycle class map induces an injection

[TABLE]

for all $i$.

(cf. [18], [19], [20], [21], [8], [14], [23], [7] for extensions and partial results concerning conjecture 1.3.)

An interesting novel approach to problem 1.2 (as well as a reinterpretation of theorem 1.1) is provided by the concept of multiplicative Chow–Künneth decomposition, giving rise to unconditional constructions of a bigraded ring structure on the Chow ring of certain varieties [15], [17], [16], [9]. (The bigrading constructed in these works should be seen as a candidate for the (only ideally existing) bigrading evoked in problem 1.2; in particular, it is not known whether property (1) holds for these candidates.)

This note does not directly address problem 1.2 or conjecture 1.3. Instead, our aim is to propose a modified version of conjecture 1.3. The modification consists in considering the groups $B^{\ast}(X)$ of cycles with $\mathbb{Q}$–coefficients modulo algebraic equivalence. For any $X\in\mathcal{C}$ (in particular, for a hyperkähler variety), the conjectural bigrading $A^{\ast}_{(\ast)}(X)$ is expected to be of motivic origin (i.e., induced by a Chow–Künneth decomposition). As such, one expects the bigrading to pass to algebraic equivalence and induce a bigrading $B^{\ast}_{(\ast)}(X)$. Now, it has been conjectured that (for any smooth projective variety) the deepest level $F^{i}A^{i}(X)$ of the conjectural Bloch–Beilinson filtration should be algebraically trivial [10], and so $B^{2}_{(2)}(X)=0$. For a hyperkähler variety, one expects that also $B^{2}_{(1)}(X)=0$ (this is clear when $X$ is of $K3^{[n]}$ type; for general hyperkähler varieties, one can reason as in the proof of proposition 3.2 below), and so conjecturally

[TABLE]

This leads to the following variant of conjecture 1.3:

Conjecture 1.4**.**

Let $X$ be a hyperkähler variety. Let $E^{\ast}(X)\subset B^{\ast}(X)$ be the $\mathbb{Q}$–subalgebra generated by $B^{1}(X)$, $B^{2}(X)$ and the Chern classes. The cycle class map induces injections

[TABLE]

Here is some evidence we have found for conjecture 1.4:

Theorem **** (=theorem 2.1).

Let $X$ be either

(i) a Hilbert scheme $X=S^{[2]}$, where $S$ is a projective $K3$ surface, or

(ii) a Fano variety of lines $X=F(Y)$, where $Y\subset\mathbb{P}^{5}(\mathbb{C})$ is a very general cubic fourfold.

The cycle class map induces an injection

[TABLE]

Theorem **** (=theorem 3.1).

Let $X=K_{m}(A)$ be a generalized Kummer variety of dimension $2m$. The cycle class map induces an injection

[TABLE]

Our evidence is, alas, restricted to $1$–cycles. The reason for this restriction is that in proving theorems 2.1 and 3.1, we rely on the bigrading of the Chow ring of $X$ constructed unconditionally in [15] resp. [9]. In both cases, it is not known whether the bigrading satisfies property (1) for all $i$ (this is only known for $i\geq\dim X-1$).

**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 groups of cycles will be with rational coefficients*: we will denote by $A_{j}(X)$ the Chow group of $j$–dimensional algebraic cycles on $X$ with $\mathbb{Q}$–coefficients; for $X$ smooth of dimension $n$ we will write $A^{i}(X):=A_{n-i}(X)$. Likewise, we will write $B_{j}(X)$ for the group of $j$–dimensional algebraic cycles on $X$ with $\mathbb{Q}$–coefficients, modulo algebraic equivalence, and $B^{i}(X):=B_{n-i}(X)$ for $X$ smooth.*

The notations $A^{i}_{hom}(X)$, $A^{i}_{alg}(X)$ will be used to indicate the subgroups of homologically trivial, resp. algebraically trivial cycles. Likewise, we write $B^{i}_{hom}(X)$ for what is commonly known as the Griffiths group of $X$.

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

2. Some hyperkähler fourfolds

Theorem 2.1**.**

Let $X$ be either

(i) a Hilbert scheme $X=S^{[2]}$, where $S$ is a projective $K3$ surface, or

(ii) a Fano variety of lines $X=F(Y)$, where $Y\subset\mathbb{P}^{5}(\mathbb{C})$ is a very general cubic fourfold.

Let $E^{\ast}(X)\subset B^{\ast}(X)$ be the $\mathbb{Q}$–subalgebra generated by $B^{1}(X)$, $B^{2}(X)$ and the Chern classes. Then the cycle class map induces an injection

[TABLE]

Proof.

Since algebraic and homological equivalence coincide for [math]–cycles, the $i=4$ case is trivially true. The interesting part of the statement is thus only the injectivity of $E^{3}(X)\to H^{6}(X)$.

In both cases (i) and (ii), there exists a bigraded ring structure $A^{\ast}_{(\ast)}(X)$ induced by the Fourier transform constructed in [15]. In both cases, the bigrading is also described by the action of a Chow–Künneth decomposition, and therefore the ring $B^{\ast}(X)$ inherits a bigrading $B^{\ast}_{(\ast)}(X)$. The Chern classes of $X$ are in $A^{\ast}_{(0)}(X)$ (in case (i), this is [17, Theorem 2]; in case (ii), this follows from the fact that the Chern classes are polynomials in the classes labelled $l\in A^{1}(X),c\in A^{2}(X)$ in [15] (coming from the tautological bundle on the Grassmannian), and it is known that $c\in A^{2}_{(0)}(X)$ [15, Theorem 21.9(iii)]).

The theorem now follows from the following claim:

Claim 2.2**.**

One has $B^{2}(X)=B^{2}_{(0)}(X)$.

Indeed: the claim, combined with the above remarks, implies that $E^{\ast}(X)\subset B^{\ast}_{(0)}(X)$. But we know (lemma 2.3 below) that $B^{3}_{(0)}(X)\to H^{6}(X)$ is injective, and so theorem 2.1 is proven.

The claim follows from the fact, proven by Shen–Vial [15, Theorems 2.2 and 2.4], that there exists a correspondence $L\in A^{2}(X\times X)$ with the property that

[TABLE]

Indeed, any [math]–cycle $a\in A^{4}_{(2)}(X)$ is (homologically trivial hence) algebraically trivial. As algebraic equivalence is an adequate equivalence relation, it follows that $L_{\ast}(a)$ is algebraically trivial and so

[TABLE]

This proves the claim:

[TABLE]

It only remains to prove the following lemma:

Lemma 2.3** (Shen–Vial [15]).**

Let $X$ be either

(i) a Hilbert scheme $X=S^{[2]}$, where $S$ is a projective $K3$ surface, or

(ii) a Fano variety of lines $X=F(Y)$, where $Y\subset\mathbb{P}^{5}(\mathbb{C})$ is any smooth cubic fourfold.

Then $A^{3}_{(0)}(X)\cap A^{3}_{hom}(X)=0$.

Proof.

This is contained in [15]. A quick way of proving the lemma is as follows: let $\mathcal{F}$ be the Fourier transform of [15]. We have that $a\in A^{3}(X)$ is in $A^{3}_{(0)}(X)$ if and only if $\mathcal{F}(a)\in A^{1}_{(0)}(X)=A^{1}(X)$ [15, Theorem 2]. Suppose $a\in A^{3}_{(0)}(X)$ is homologically trivial. Then also $\mathcal{F}(a)\in A^{1}(X)$ is homologically trivial, hence $\mathcal{F}(a)=0$ in $A^{1}(X)$. But then, using [15, Theorem 2.4], we find that

[TABLE]

∎

∎

In theorem 2.1(ii), we restricted to very general cubic fourfolds. The reason is that for the Fano variety $X$ of lines on any given smooth cubic fourfold, it is not yet known that the Fourier decomposition $A^{\ast}_{(\ast)}(X)$ is a bigraded ring structure (cf. [15, Remark 22.9]). If we abandon the hypothesis “very general”, we can obtain a weaker statement:

Definition 2.4**.**

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

[TABLE]

where $P\in A^{3}(Y\times X)$ is the universal family of lines, and $A^{2}(Y)_{prim}:=\{c\in A^{2}(Y)\ |\ [c]\in H^{4}(Y)_{prim}\}$. We set $B^{1}(X)_{prim}=A^{1}(X)_{prim}$.

Proposition 2.5**.**

Let $Y\subset\mathbb{P}^{5}(\mathbb{C})$ be any smooth cubic fourfold, and let $X=F(Y)$ be the Fano variety of lines in $Y$. Let $b\in B^{3}(X)$ be a cycle of the form

[TABLE]

where $a_{k}\in B^{2}(X)$ and $d_{k}\in B^{1}(X)_{prim}$. Then $b$ is algebraically trivial if and only if $b$ is homologically trivial.

Proof.

Claim 2.2 still applies to $X$, and so the $a_{k}$ are in $B^{2}_{(0)}(X)$. As such, they can be lifted to $\bar{a_{k}}\in A^{2}_{(0)}(X)$. One knows that

[TABLE]

[15, Proposition 22.7], and thus $\bar{a_{k}}\cdot d_{k}\in A^{3}_{(0)}(X)$. It follows that $b\in B^{3}_{(0)}(X)$, and one concludes using lemma 2.3. ∎

3. Generalized Kummer varieties

Theorem 3.1**.**

Let $A$ be an abelian surface, and let $X=K_{m}(A)$ be a generalized Kummer variety of dimension $2m$. Let $E^{\ast}(X)\subset B^{\ast}(X)$ be the $\mathbb{Q}$–subalgebra generated by $B^{1}(X)$, $B^{2}(X)$ and the Chern classes. The cycle class map induces injections

[TABLE]

for $i\geq 2m-1$.

Proof.

Thanks to [9, Theorem 7.9], there exists a multiplicative Chow–Künneth decomposition for $X$ and so the Chow ring has a bigrading $A^{\ast}_{(\ast)}(X)$. Moreover, the Chern classes of $X$ are in $A^{\ast}_{(0)}(X)$ [9, Proposition 7.13]. Let $B^{\ast}_{(\ast)}(X)$ denote the induced bigrading modulo algebraic equivalence.

Proposition 3.2**.**

We have

[TABLE]

Proof.

One knows that $B^{i}_{(i)}(X)=0$ for all $i>0$ [13, Corollary 18]. It remains to check that $B^{2}_{(1)}(X)=0$. By definition, we have

[TABLE]

where $\{\Pi^{X}_{j}\}$ is the multiplicative Chow–Künneth decomposition furnished by [9]. Since $B^{2}_{(1)}(X)\subset B^{2}_{hom}(X)$, and $\Pi^{X}_{3}$ is idempotent, we also have

[TABLE]

Next, we observe that (as $X$ is hyperkähler) $H^{3}(X,\mathcal{O}_{X})=0$. Since the generalized Hodge conjecture is known to hold for self–products of abelian surfaces [1, 7.2.2], [2, 8.1(2)], and generalized Kummer varieties are motivated by abelian surfaces in the sense of [3], the generalized Hodge conjecture is true for generalized Kummer varieties (for the usual Hodge conjecture, this was noted in [22, Theorem 3.3]). In particular, $H^{3}(X)$ is supported on a divisor $D\subset X$, and $H^{2n-3}(X)$ is supported on a $2$–dimensional subvariety $S\subset X$. Using the Lefschetz $(1,1)$ theorem, one can find a cycle $\gamma\in A^{2m}(X\times X)$ representing the Künneth component $\pi^{X}_{3}$ and supported on $S\times D$. For dimension reasons, we have

[TABLE]

(Indeed, the action of $\gamma$ on $B^{2}_{hom}(X)$ factors over $B^{2}_{hom}(\widetilde{S})=0$, where $\widetilde{S}\to S$ denotes a desingularization.) Applying lemma 3.3 below, this implies that also

[TABLE]

and we are done in view of (2).

Here, we have used the following lemma. (The lemma applies to our set–up, because generalized Kummer varieties have finite–dimensional motive [22], [9].)

Lemma 3.3**.**

Let $X$ be a smooth projective variety of dimension $n$, and assume $X$ has finite–dimensional motive. Let $\Pi$ and $\pi\in A^{n}(X\times X)$ be such that $\Pi$ is idempotent and $\Pi=\pi$ in $H^{2n}(X\times X)$. Then

[TABLE]

Proof.

We have

[TABLE]

From Kimura’s nilpotence theorem [12], it follows that there exists $N\in\mathbb{N}$ such that

[TABLE]

Developing this expression, we obtain

[TABLE]

where each $P_{j}$ is a composition of correspondences containing at least one copy of $\pi$. But then (by hypothesis) the right–hand side acts as zero on $B^{i}_{hom}(X)$, and hence so does the left–hand side. ∎

This ends the proof of proposition 3.2. ∎

Proposition 3.2, combined with the fact that the Chern classes are in $B^{\ast}_{(0)}(X)$, implies that there is an inclusion

[TABLE]

Theorem 3.1 follows from this inclusion, combined with the following lemma:

Lemma 3.4**.**

Let $X$ be a generalized Kummer variety of dimension $2m$. Then

[TABLE]

To prove the lemma, we note that by definition,

[TABLE]

where $\{\Pi^{X}_{k}\}$ is a multiplicative Chow–Künneth decomposition [9]. Inspecting the construction in [9], one finds that $\Pi^{X}_{4m-1}=0$ and $\Pi^{X}_{4m}$ is of the form $X\times x$, where $x\in X$. This proves the first statement.

As for the second statement of the lemma, we observe that there exists a cycle $\gamma\in A^{2m}(X\times X)$ representing the Künneth component $\pi^{X}_{4m-2}$ and supported on $X\times S$, where $S\subset X$ is a smooth surface (this is a general fact, for any variety $X$ verifying the Lefschetz standard conjecture $B(X)$, cf. [11, Theorem 7.7.4]). For dimension reasons, we have

[TABLE]

(Indeed, the action of $\gamma$ on $B^{2m-1}_{hom}(X)$ factors over $B^{1}_{hom}({S})=0$). By lemma 3.3, this implies that

[TABLE]

On the other hand, $\Pi^{X}_{4m-2}$ is a projector on $B^{2m-1}_{(0)}(X)$, and so

[TABLE]

∎

**Acknowledgements **.

Thanks to all participants of the Strasbourg 2014/2015 “groupe de travail” based on the monograph [20] for a stimulating atmosphere. Thanks to Yoyo, Kai and Len for wonderful Christmas holidays. Thanks to the referee for highly pertinent remarks.