About Chow groups of certain hyperk\"ahler varieties with non-symplectic automorphisms

TL;DR

This paper investigates the action of non-symplectic automorphisms on the Chow groups of hyperk"ahler varieties, providing partial proofs of conjectures and implications for quotient varieties.

Contribution

It proves the predicted behavior of certain Chow group pieces under automorphisms for Hilbert schemes of K3 surfaces, supporting Beauville's and Bloch-Beilinson's conjectures.

Findings

Confirmed invariance of specific Chow group pieces under automorphisms.

Established that certain Chow groups contain no non-trivial invariant cycles.

Derived consequences for the Chow ring of quotient varieties.

Abstract

Let $X$ be a hyperk\"ahler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The Bloch-Beilinson conjectures predict how $G$ should act on these pieces of the Chow groups: certain pieces should be invariant under $G$, while certain other pieces should not contain any non-trivial $G$-invariant cycle. We can prove this for two pieces of the Chow groups when $X$ is the Hilbert scheme of a $K3$ surface and $G$ consists of natural automorphisms. This has consequences for the Chow ring of the quotient $X/G$.

Equations

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

$$\begin{array}{rl}& A_{(j)}^{2m}(X)\cap A^{2m}(X{)}^G=\{\begin{array}{ll}0& ifj<2k;\\ A_{(j)}^{2m}(X)& ifj=2k;\end{array}\\ & A_{(i)}^i(X)\cap A^i(X{)}^G=\{\begin{array}{ll}0& ifi<2k;\\ A_{(i)}^i(X)& ifi=2k;\end{array}\end{array}$$

$$A_{(2)}^i(X)\cap A^i(X{)}^G=0fori\in \{2,2m\}.$$

$$A^{i_1}(Y)\otimes A^{i_2}(Y)\otimes \cdots \otimes A^{i_r}(Y)\to A^{2m-1}(Y),$$

$$A^3(Y)\otimes A^{i_1}(Y)\otimes \cdots \otimes A^{i_r}(Y)\to A^{2m}(Y),$$

$$Y=X/G,$$

$$A^i(Y)\to A_{n-i}(Y)$$

$$h(Y)\cong h(X{)}^G:=(X,{\Delta }_X^G,0)in{\mathcal{M}}_{\mathrm{rat}},$$

$${\Delta }_X^G:=\frac{1}{|G|}\sum _{g\in G}{\Gamma }_g\in A^n(X\times X).$$

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

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

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

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

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

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

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

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

$$A_{(j)}^i(X)=0for\; alljodd\; and\; for\; allj>i.$$

$$\underset{\mu \in \mathfrak{B}(m)}{⨁}({}^t{\hat{\Gamma }}_{\mu }{)}_{\ast }:A^i(X)\stackrel{\cong }{\to }\underset{\mu \in \mathfrak{B}(m)}{⨁}{A}^{i+l(\mu )-m}(S^{(\mu )}),$$

$${\Pi }_i^X:=\sum _{\mu \in \mathfrak{B}(m)}\frac{1}{m_{\mu }}{\hat{\Gamma }}_{\mu }\circ {\Pi }_{i-2m+2l(\mu )}^{S^{(\mu )}}\circ {}^t{\hat{\Gamma }}_{\mu },$$

$$({\Pi }_{\ell }^X{)}_{\ast }{A}^i(X)=0for\; all\ell <i.$$

$$\underset{\mu \in \mathfrak{B}(m)}{⨁}({}^t{\hat{\Gamma }}_{\mu }{)}_{\ast }:A_{(j)}^i(X)\stackrel{\cong }{\to }\underset{\mu \in \mathfrak{B}(m)}{⨁}{A}_{(j)}^{i+l(\mu )-m}(S^{(\mu )}).$$

$$\underset{\mu \in \mathfrak{B}(m)}{⨁}({}^t{\Gamma }_{\mu }{)}_{\ast }:A_{(j)}^i(X)\hookrightarrow \underset{\mu \in \mathfrak{B}(m)}{⨁}{A}_{(j)}^{i+l(\mu )-m}(S^{\mu }).$$

$$\begin{array}{rl}& {\Delta }_X\in A_{(0)}^n(X\times X),\\ & {\Delta }_X^{sm}\in A_{(0)}^{2n}(X\times X\times X).\end{array}$$

$$\begin{array}{rl}{\Delta }_X& =\sum _{i=0}^{2n}{\Pi }_i^X=\sum _{i=0}^{2n}{\Pi }_i^X\circ {\Pi }_i^X\\ & =\sum _{i=0}^{2n}({}^t{\Pi }_i^X\times {\Pi }_i^X{)}_{\ast }{\Delta }_X\\ & =\sum _{i=0}^{2n}({\Pi }_{2n-i}^X\times {\Pi }_i^X{)}_{\ast }{\Delta }_X\\ & =({\Pi }_{2n}^{X\times X}{)}_{\ast }{\Delta }_X\in A_{(0)}^n(X\times X).\end{array}$$

$${\Theta }_1,\dots ,{\Theta }_m\in A^{2m}(S^m\times S),{\Xi }_1,\dots ,{\Xi }_m\in A^2(S\times S^m)$$

$$\begin{array}{rl}{A}_{(2)}^{2m}(S^m)\stackrel{(({\Theta }_1{)}_{\ast },\dots ,({\Theta }_m{)}_{\ast })}{\to }{A}^2(S)\oplus \cdots \oplus A^2(S)& \\ \stackrel{({\Xi }_1{)}_{\ast }+\dots +({\Xi }_m{)}_{\ast }}{\to }{A}^{2m}(S^m)& \end{array}$$

$${\Pi }_0^S={\mathfrak{o}}_S\times S,{\Pi }_4^S=S\times {\mathfrak{o}}_S,{\Pi }_2^S={\Delta }_S-{\pi }_0^S-{\pi }_4^S.$$

$$p_{i,j}:S^{2m}\to S^2(1\le i<j\le 2m)$$

$$p_i:S^m\to S(1\le i\le m)$$

Full text

1 Introduction

Let $X$ be a hyperkähler variety of dimension $2m$ (i.e., a projective irreducible holomorphic symplectic manifold, cf. Beau0 , Beau1 ). Let $G\subset\operatorname{Aut}(X)$ be a finite cyclic group of order $k$ consisting of non–symplectic automorphisms. We will be interested in the action of $G$ on the Chow groups $A^{\ast}(X)$. (Here, $A^{i}(X):=CH^{i}(X)_{\mathbb{Q}}$ denotes the Chow group of codimension $i$ algebraic cycles modulo rational equivalence with $\mathbb{Q}$–coefficients.) Let us suppose $X$ has a multiplicative Chow–Künneth decomposition, in the sense of SV . This implies the Chow ring of $X$ is a bigraded ring $A^{\ast}_{(\ast)}(X)$, where each Chow group splits as

[TABLE]

and the piece $A^{i}_{(j)}(X)$ is expected to be isomorphic to the graded $\operatorname{Gr}^{j}_{F}A^{i}(X)$ for the conjectural Bloch–Beilinson filtration $F^{\ast}$ on Chow groups. (Conjecturally, all hyperkähler varieties have a multiplicative Chow–Künneth decomposition; this is related to Beauville’s conjectural splitting property Beau3 . The existence of a multiplicative Chow–Künneth decomposition has been established for Hilbert schemes of $K3$ surfaces SV , V6 , and for generalized Kummer varieties FTV .)

Since $H^{2k,0}(X)=H^{2,0}(X)^{\otimes k}$, the group $G$ acts as the identity on $H^{2k,0}(X)$. For $i<2k$, we have that $\sum_{g\in G}g^{\ast}$ acts as [math] on $H^{i,0}(X)$. The Bloch–Beilinson conjectures J2 , combined with the expected isomorphism $A^{i}_{(j)}(X)\cong\operatorname{Gr}^{j}_{F}A^{i}(X)$, thus imply the following conjecture:

Conjecture 1

Let $X$ be a hyperkähler variety of dimension $2m$, and let $G\subset\operatorname{Aut}(X)$ be a finite cyclic group of order $k$ of non–symplectic automorphisms. Then

[TABLE]

(Here $A^{i}(X)^{G}\subset A^{i}(X)$ denotes the $G$–invariant part of the Chow group $A^{i}(X)$.)

The main result in this note is a partial verification of conjecture 1 for a certain class of hyperkähler varieties and a certain class of automorphisms.

Theorem **** (=theorem 3.1)

Let $S$ be a projective $K3$ surface, and let $X=S^{[m]}$ be the Hilbert scheme of length $m$ subschemes. Let $G\subset\operatorname{Aut}(X)$ be a subgroup of order $k$ of natural non–symplectic automorphisms. Then

[TABLE]

A natural automorphism of $X$ is an automorphism induced by an automorphism of $S$. Theorem 3.1 applies to Hilbert schemes of any $K3$ surface $S$ having a finite order non–symplectic automorphism. Such $K3$ surfaces have been intensively studied, and there are lots of examples known Nik1 , Nik2 , Kon , Vor , LSY , Schu , Tak , AS , AST , AST2 , GP . It would be interesting to prove theorem 3.1 also for non–symplectic automorphisms that are non–natural; this seems considerably more difficult (cf. BlochHK4 for one special case where theorem 3.1 is proven for a non–natural involution).

Theorem 3.1 has interesting consequences for the Chow ring of the quotient:

Corollary **** (=corollaries 1 and 2)

Let $X$ and $G$ be as in theorem 3.1, and let $Y:=X/G$.

(i) Let $a\in A^{2m-1}(Y)$ be a $1$–cycle which is in the image of the intersection product map

[TABLE]

where all $i_{j}$ are $\leq 2$. Then $a$ is rationally trivial if and only if $a$ is homologically trivial.

(ii) Let $a\in A^{2m}(Y)$ be a [math]–cycle which is in the image of the intersection product map

[TABLE]

where all $i_{j}$ are $\leq 2$. Then $a$ is rationally trivial if and only if $a$ is homologically trivial.

These corollaries illustrate the following expectation: for certain special varieties with a multiplicative Chow–Künneth decomposition, the subring $A^{\ast}_{(0)}\subset A^{\ast}$ on which the cycle class map is injective should be larger than for hyperkähler varieties. Indeed, for a quotient $Y=X/G$ where $X$ is hyperkähler and $G\subset\operatorname{Aut}(X)$ is a finite order group of non–symplectic automorphisms, one expects that codimension $2$ cycles lie in $A^{\ast}_{(0)}(Y)$.

Results similar in spirit have been obtained for certain other hyperkähler varieties and their Calabi–Yau quotients in EPW , LSYmoi , BlochHK4 .

**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 Sc , MNP ) 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 Preliminary

2.1 Quotient varieties

Definition 1

A projective quotient variety is a variety

[TABLE]

where $X$ is a smooth projective variety and $G\subset\operatorname{Aut}(X)$ is a finite group.

Proposition 1 (Fulton F )

Let $Y$ be a projective quotient variety of dimension $n$. Let $A^{\ast}(Y)$ denote the operational Chow cohomology ring. The natural map

[TABLE]

is an isomorphism for all $i$.

Proof

This is (F, , Example 17.4.10).

Remark 1

It follows from proposition 1 that the formalism of correspondences goes through unchanged for projective quotient varieties (this is also noted in (F, , Example 16.1.13)). We can thus consider motives $(Y,p,0)\in\mathcal{M}_{\rm rat}$, where $Y$ is a projective quotient variety and $p\in A^{n}(Y\times Y)$ is a projector. For a projective quotient variety $Y=X/G$, one readily proves (using Manin’s identity principle) that there is an isomorphism

[TABLE]

where $\Delta^{G}_{X}$ denotes the idempotent

[TABLE]

(NB: $\Delta^{G}_{X}$ is a projector on the $G$–invariant part of the Chow groups $A^{\ast}(X)^{G}$.)

2.2 MCK decomposition

Definition 2 (Murre Mur )

Let $X$ be a projective quotient 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 $(\pi_{i})_{\ast}H^{\ast}(X)=H^{i}(X)$. A CK decomposition is self–dual if $\pi_{i}={}^{t}\pi_{2n-i}$ in $A^{n}(X\times X)$ for all $i$ (here ${}^{t}\pi$ denotes the transpose of $\pi$).

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

Remark 2

The existence of a CK decomposition for any smooth projective variety is part of Murre’s conjectures Mur , J2 .

Definition 3 (Shen–Vial SV )

Let $X$ be a projective quotient 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 $\{\pi_{i}\}$ of $X$ that is multiplicative, i.e. it satisfies

[TABLE]

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

Remark 3

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 $h^{i+j}(X)$. It follows that if $X$ has an 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 an MCK decomposition, one has

[TABLE]

this is related to Murre’s conjectures B and D Mur .

The property of having an MCK decomposition is severely restrictive, and is closely related to Beauville’s “(weak) splitting property” Beau3 . For more ample discussion, and examples of varieties with an MCK decomposition, we refer to (SV, , Section 8), as well as V6 , SV2 , FTV , EPW .

Theorem 2.1 (Vial V6 )

Let $S$ be an algebraic $K3$ surface, and let $X=S^{[m]}$ be the Hilbert scheme of length $m$ subschemes of $S$. Then $X$ has a self–dual MCK decomposition. One has

[TABLE]

Proof

This is (V6, , Theorem 1). For later use, we briefly review the construction. First, one takes an MCK decomposition $\{\Pi^{S}_{i}\}$ for $S$ (this exists, thanks to SV ). Taking products, this induces an MCK decomposition $\{\Pi^{S^{r}}\}$ for $S^{r}$, $r\in\mathbb{N}$. This product MCK decomposition is invariant under the action of the symmetric group $\mathfrak{S}_{r}$, and hence it induces an MCK decomposition $\{\Pi^{S^{(r)}}\}$ for the symmetric products $S^{(r)}$, $r\in\mathbb{N}$. There is the isomorphism of de Cataldo–Migliorini CM

[TABLE]

where $\mathfrak{B}(m)$ is the set of partitions of $m$, $l(\mu)$ is the length of the partition $\mu$, and $S^{(\mu)}=S^{l(\mu)}/{\mathfrak{S}_{l(\mu)}}$, and ${}^{t}\hat{\Gamma}_{\mu}$ is a correspondence in $A^{m+l(\mu)}(S^{[m]}\times S^{(\mu)})$. Using this isomorphism, Vial defines (V6, , Equation (4)) a natural CK decomposition for $X$, by setting

[TABLE]

where the $m_{\mu}$ are rational numbers coming from the de Cataldo–Migliorini isomorphism. The $\{\Pi^{X}_{i}\}$ of definition (1) are proven to be an MCK decomposition.

The self–duality of the $\{\Pi^{X}_{i}\}$ is apparent from definition (1). The fact that $A^{i}_{(j)}(X)$ vanishes for $j$ odd is because $\Pi^{X}_{j}=0$ for $j$ odd. The vanishing for $j>i$ follows from the fact that by construction, the projector $\Pi^{X}_{\ell}$ is supported on $V\times X$ with $\dim V=\ell$; this implies (for reasons of dimension) that

[TABLE]

Remark 4

It follows from definition (1) that the de Cataldo–Migliorini isomorphism is compatible with the bigrading of the Chow ring, in the sense that there are induced isomorphisms

[TABLE]

In particular, there are split injections

[TABLE]

(Here, the right–hand side refers to the product MCK decomposition of $S^{\mu}$.)

Lemma 1 (Shen–Vial)

Let $X$ be a projective quotient variety of dimension $n$, and suppose $X$ has a self–dual MCK decomposition. Then

[TABLE]

Proof

The first statement follows from (SV2, , Lemma 1.4) when $X$ is smooth. The same argument works for projective quotient varieties; the point is just that

[TABLE]

(Here, the second line follows from Lieberman’s lemma (V3, , Lemma 3.3), and the last line is the fact that the product of $2$ MCK decompositions is MCK (SV, , Theorem 8.6).)

The second statement is proven for smooth $X$ in (SV, , Proposition 8.4); the same argument works for projective quotient varieties.

2.3 MCK for products

Proposition 2

Let $S$ be a $K3$ surface. There exist correspondences

[TABLE]

such that the composition

[TABLE]

is the identity.

Proof

By construction SV , the MCK decomposition for $S$ is given by

[TABLE]

Here $\mathfrak{o}_{S}\in A^{2}(S)$ denotes the “distinguished point” of BV (any point lying on a rational curve in $S$ equals $\mathfrak{o}_{S}$ in $A^{2}(S)$). Let

[TABLE]

denote projection to the $i$-th and $j$-th factor, and let

[TABLE]

denote projection to the $i$–th factor.

We now claim that there is equality

[TABLE]

Indeed, using Lieberman’s lemma (V3, , Lemma 3.3), we find that

[TABLE]

Let us now (by way of example) consider the first summand of the right–hand–side of (2). For brevity, let

[TABLE]

denote the projection on the first $m$ and last $m$ factors. Writing out the definition of composition of correspondences, we find that

[TABLE]

(Here, we use the notation $(C)_{(i,j)}$ to indicate that the cycle $C$ lies in the $i$th and $j$th factor, and likewise for $(D)_{(i,j,k)}$.)

Doing the same for the other summands in (2), one convinces oneself that both sides of (2) are equal to the product Chow–Künneth component

[TABLE]

thus proving the claim.

Let us now define

[TABLE]

where $1\leq i\leq m$. It follows from equation (2) that there is equality

[TABLE]

Taking $(i,j)=(2m,2)$, this proves the proposition.

The following is a version of proposition 2 for the group $A^{2}_{(2)}(S^{m})$:

Proposition 3

Let $S$ be a $K3$ surface. There exist correspondences

[TABLE]

such that the composition

[TABLE]

is the identity.

Proof

By construction, the product MCK decomposition $\{\Pi_{i}^{S^{m}}\}$ satisfies

[TABLE]

Hence, the transpose of equation (3) gives the equality

[TABLE]

Taking $(i,j)=(2,2)$, this proves the proposition.

2.4 Birational invariance

Proposition 4 (RießRie , Vial V6 )

Let $X$ and $X^{\prime}$ be birational hyperkähler varieties. Assume $X$ has an MCK decomposition. Then also $X^{\prime}$ has an MCK decomposition, and there are natural isomorphisms

[TABLE]

Proof

As noted by Vial (V6, , Introduction), this is a consequence of Rieß’s result that $X$ and $X^{\prime}$ have isomorphic Chow motive (as algebras in the category of Chow motives). For more details, cf. (SV, , Section 6) or (BlochHK4, , Lemma 2.8).

2.5 A commutativity lemma

Lemma 2

Let $S$ be an algebraic $K3$ surface, and let $\{\Pi_{i}^{S}\}$ be the MCK decomposition as above. Let $h\in\operatorname{Aut}(S)$. Then

[TABLE]

Proof

It suffices to prove this for $i=0$. Indeed, by definition of $\{\Pi_{i}^{S}\}$ we have

[TABLE]

Supposing the lemma holds for $i=0$, by taking transpose correspondences we get an equality

[TABLE]

Composing on both sides with $\Gamma_{h}$, we get

[TABLE]

Next, since obviously the diagonal $\Delta_{S}$ commutes with $\Gamma_{h}$, we also get

[TABLE]

It remains to prove the lemma for $i=0$. The projector $\Pi_{0}^{S}$ is defined as

[TABLE]

where $\mathfrak{o}_{S}\in A^{2}(S)$ is the “distinguished point” of BV . Let $x\in S$ be a point lying on a rational curve. Then $h^{\ast}(\mathfrak{o}_{S})=h^{-1}(x)$ is again a point lying on a rational curve, and so

[TABLE]

Using Lieberman’s lemma (V3, , Lemma 3.3), we find that

[TABLE]

whereas obviously

[TABLE]

This proves the $i=0$ case of the lemma.

The following lemmas establish some corollaries of lemma 2:

Lemma 3

Let $S$ be an algebraic $K3$ surface, and $G_{S}\subset\operatorname{Aut}(S)$ a group of finite order $k$. For any $r\in\mathbb{N}$, let $\{\Pi_{i}^{S^{r}}\}$ denote the product MCK decomposition of $S^{r}$ induced by the MCK decomposition of $S$ as above. Let

[TABLE]

Then

[TABLE]

is an idempotent, for any $i$.

Proof

It suffices to prove the commutativity statement. (Indeed, since both $\Delta^{G}_{S^{r}}$ and $\Pi_{i}^{S^{r}}$ are idempotent, the idempotence of their composition follows immediately from the stated commutativity relation.) To prove the commutativity statement, we will prove more precisely that for any $h\in\operatorname{Aut}(S)$ we have equality

[TABLE]

This can be seen as follows: we have

[TABLE]

Here, the first and last lines are the definition of the product MCK decomposition for $S^{r}$; the second and fourth line are just regrouping, and the third line is lemma 2.

Lemma 4

Let $S$ be an algebraic $K3$ surface, and $G_{S}\subset\operatorname{Aut}(S)$ a group of finite order $k$. For any $r\in\mathbb{N}$, let $X=S^{[r]}$ and let $G\subset\operatorname{Aut}(X)$ be the group of natural automorphisms induced by $G_{S}$. Let $\{\Pi_{i}^{X}\}$ be the MCK decomposition of theorem 2.1. Let $\Delta^{G}_{X}$ denote the correspondence

[TABLE]

Then

[TABLE]

is an idempotent, for any $i$.

Proof

Again, it suffices to prove the commutativity statement. This can be done as follows: for any $g\in G$, we can write $g=h^{[r]}$ where $h\in\operatorname{Aut}(S)$. Then we have

[TABLE]

Here, the first line follows from the definition of $\Pi_{i}^{X}$ (definition (1)). The second line is just regrouping, the third line is by construction of natural automorphisms of $X$, the fourth line is equality (4) above, and the fifth line is again by construction of natural automorphisms.

Remark 5

In view of (SV2, , Lemma 1.4) the commutativity property (5) is equivalent to the following: for any natural automorphism $g$ of $X$, the graph $\Gamma_{g}\in A^{2r}(X\times X)$ is “of pure grade [math]”, i.e. $\Gamma_{g}\in A^{2r}_{(0)}(X\times X)$.

Lemma 5

Let $S$ be an algebraic $K3$ surface, and $X=S^{[m]}$ the Hilbert scheme of length $m$ subschemes. Let $G\subset\operatorname{Aut}(X)$ a group of finite order $k$ of natural automorphisms. Then the quotient $Y:=X/G$ has a self–dual MCK decomposition.

Proof

Let $p\colon X\to Y$ denote the quotient morphism. One defines

[TABLE]

where $\{\Pi^{X}_{j}\}$ is the self–dual MCK decomposition of theorem 2.1. This defines a self–dual CK decomposition $\{\Pi_{j}^{Y}\}$, since

[TABLE]

(Here, in the third line we have used lemma 2.)

It remains to check this CK decomposition is multiplicative. To this end, let $i,j,k$ be integers with $k\not=i+j$. We note that

[TABLE]

Here, the first equality is by definition of the $\Pi^{Y}_{i}$, the second equality is lemma 6 below, the third equality follows from lemma 4, and the fourth equality is the fact that the $\{\Pi^{X}_{i}\}$ are an MCK decomposition for $X$.

Lemma 6

There is equality

[TABLE]

Proof

The second equality is just the definition of $\Delta^{G}_{X}$. As to the first equality, we first note that

[TABLE]

This implies that

[TABLE]

But ${}^{t}\Gamma_{p}\circ\Gamma_{p}=\sum_{g\in G}\Gamma_{g}$, and thus

[TABLE]

as claimed.

There is also the following commutativity relation:

Lemma 7

Let

[TABLE]

be as in propositions 2 and 3. Let $h\in\operatorname{Aut}(S)$. The diagrams

[TABLE]

and

[TABLE]

are commutative.

Proof

First, we observe that $h^{\times m}$ and $h$ preserve the bigrading in view of (4), so the diagrams make sense. Next, we recall (proposition 2) that ${}^{t}\Xi_{i}$ is defined on $A^{2}_{(2)}(S^{m})$ as projection on the $i$th factor (which also preserves the bigrading, cf. (SV2, , Corollary 1.6)). The commutativity of the first diagram now follows from the commutativity of

[TABLE]

As for the second diagram: $\Xi_{i}$ acts on $A^{2}_{(2)}(S)$ as $(p_{i})^{\ast}$. Since we can write $h_{\ast}=(h^{-1})^{\ast}$, the second diagram is also commutative.

2.6 Natural automorphisms of Hilbert schemes

Definition 4 (Boissière Bo )

Let $S$ be a surface, and let $X=S^{[m]}$ denote the Hilbert scheme of length $m$ subschemes. An automorphism $\psi\in\operatorname{Aut}(S)$ induces an automorphism $\psi^{[m]}$ of $X$. This determines a homomorphism

[TABLE]

which is injective Bo . The image of this homomorphism is called the group of natural automorphisms of $X$.

Remark 6

It is known (BoSa, , Theorem 1) that an automorphism of a Hilbert scheme is natural if and only if it fixes the exceptional divisor of the Hilbert–Chow morphism. To find examples of non–natural automorphisms of a Hilbert scheme $X$, Boissière and Sarti introduce the notion of index of an automorphism of $X$. For Hilbert schemes of a generic algebraic $K3$ surface, the index of an automorphism is $1$ if and only if the automorphism is natural (BoSa, , section 4).

3 Main result

This section contains the proof of the main result of this note, theorem 3.1.

Definition 5

Let $S$ be a $K3$ surface, and let $h\in\operatorname{Aut}(S)$ be an automorphism of order $k$. We say that $h$ is non–symplectic if

[TABLE]

where $\nu$ is a primitive $k$–th root of unity.

(NB: this is sometimes referred to as a “purely non–symplectic automorphism”.)

Theorem 3.1

Let $S$ be a projective $K3$ surface, and let $X=S^{[m]}$ be the Hilbert scheme of length $m$ subschemes. Let $G\subset\operatorname{Aut}(X)$ be a subgroup of order $k$ of natural non–symplectic automorphisms. Then

[TABLE]

Proof

Let us start with the case $i=2$, i.e. codimension $2$ cycles. To prove the required vanishing

[TABLE]

is equivalent to showing that

[TABLE]

where $\Pi_{2}^{X}$ is part of an MCK decomposition for $X$.

As we have seen (remark 4, plus the obvious fact that $A^{1}_{(2)}(S^{m-1})=0$), there is a commutative diagram

[TABLE]

where horizontal arrows are split injective. Here, the correspondence $\Delta^{G}_{S^{m}}$ is defined as

[TABLE]

and the diagram commutes because of the construction of natural automorphisms of $X$.

To prove (6), we are thus reduced to proving that

[TABLE]

where $\Pi_{2}^{S^{m}}$ is part of an MCK decomposition $\{\Pi_{i}^{S^{m}}\}$ for $S^{m}$. We will suppose $\{\Pi_{i}^{S^{m}}\}$ is the product MCK decomposition used in the proof of theorem 2.1.

We state a lemma:

Lemma 8

The surface $R:=S/G_{S}$ has

[TABLE]

Equivalently, for any MCK decomposition $\{\Pi^{S}_{j}\}$ one has

[TABLE]

Proof

The quotient variety $R$ has geometric genus [math]. Since quotient singularities are rational singularities, there exists a resolution $Y\to R$ with $p_{g}(Y)=0$. Since $Y$ is not of general type, Bloch’s conjecture is known to hold for $Y$ BKL , i.e. $A^{2}_{hom}(Y)=0$. This implies that also $A^{2}_{hom}(R)=0$.

Armed with this lemma, we can prove the vanishing (7): There is a commutative diagram

[TABLE]

The commutativity of this diagram is lemma 7. Horizontal arrows are injections thanks to proposition 3. Since the right vertical arrow is the zero map (lemma 8), the left vertical arrow is also the zero map; this proves the vanishing (7).

The statement for $i=2m$ is proven similarly: in view of remark 4, there is a commutative diagram

[TABLE]

where horizontal arrows are split injective. It thus suffices to prove the right vertical arrow is the zero map.

Thanks to proposition 2 and lemma 7, there is a commutative diagram

[TABLE]

where horizontal arrows are surjections. Combined with lemma 8, this settles the $i=2m$ case.

Remark 7

Let $X$ and $G$ be as in theorem 3.1. Let $X^{\prime}$ be a hyperkähler variety birational to $X$, and let $G^{\prime}$ be the group of birational self–maps of $X^{\prime}$ induced by $G$. Applying proposition 4, it follows from theorem 3.1 that also

[TABLE]

4 Some corollaries

Corollary 1

Let $X$ and $G$ be as in theorem 3.1, and let $Y:=X/G$ be the quotient. For any $r\in\mathbb{N}$, let

[TABLE]

be the subalgebra generated by (pullbacks of) $A^{1}(Y)$ and $A^{2}(Y)$ and $\Delta_{Y}$, $\Delta_{Y}^{sm}$. Then the cycle class map induces maps

[TABLE]

that are injective for $i\geq 2mr-1$.

Proof

First, it follows from lemma 5 that $Y$, and hence $Y^{r}$, has a self–dual MCK decomposition. Consequently, the Chow ring $A^{\ast}(Y^{r})$ is a bigraded ring. Theorem 3.1 (plus the obvious fact that $A^{1}_{hom}(Y)=0$) implies that

[TABLE]

Lemma 1 ensures that

[TABLE]

Since pullbacks for projections of type $Y^{r}\to Y^{s},s<r$, preserve the bigrading (this follows from (SV2, , Corollary 1.6), or alternatively can be checked directly), this implies that

[TABLE]

The corollary now follows from the fact that

[TABLE]

is injective (this is true for any $i$ and $j$), and the fact that

[TABLE]

(as noted in (V6, , Introduction)).

Corollary 2

Let $X$ and $G$ be as in theorem 3.1, and let $Y:=X/G$ be the quotient. Let $a\in A^{2m}(Y)$ be a [math]–cycle which is in the image of the intersection product map

[TABLE]

with all $i_{j}\leq 2$ (and $i_{1}+\cdots+i_{s}=2m-3$). Then $a$ is rationally trivial if and only if $\deg(a)=0$.

Proof

The point is that

[TABLE]

(theorem 3.1), and so

[TABLE]

But we know that $A^{2m}_{(r)}(Y)=0$ for $r<0$ (this is a general fact for any variety with an MCK decomposition), and we have seen that $A^{2m}_{(2)}(Y)=0$ (theorem 3.1), and so

[TABLE]

Remark 8

Results similar to corollaries 1 and 2 have been obtained for [math]–cycles on certain Calabi–Yau varieties. If $Y$ is a Calabi–Yau variety (of dimension $n$) that is a generic complete intersection in projective space, it is known that the image of the intersection product

[TABLE]

is of dimension $1$, and hence injects into cohomology V13 , LFu .

Going beyond the Calabi–Yau case, there is also a result of L. Fu for generic hypersurfaces $Y$ of general type. Here, the image of the intersection product

[TABLE]

is again of dimension $1$, provided $m$ is large enough relative to the degree of $Y$ (LFu, , Theorem 2.13). This is very similar to the behaviour of the Chow ring exhibited in corollary 2.

Acknowledgements.

Thanks to Len, Kai and Yasuyo for numerous pleasant coffee breaks.