1 Introduction
Given a smooth projective variety Y over
$\mathbb {C}$
, let
$\operatorname {A}^i(Y):=\operatorname {CH}^i(Y)_{\mathbb {Q}}$
denote the Chow groups of Y (i.e., the groups of codimension i algebraic cycles on Y with
$\mathbb {Q}$
-coefficients, modulo rational equivalence [Reference Fulton17, Reference Murre, Nagel and Peters32, Reference Voisin42]). The intersection product defines a ring structure on
$\operatorname {A}^\ast (Y)=\bigoplus _i \operatorname {A}^i(Y)$
, the Chow ring of Y. In the case of K3 surfaces, this ring structure has a remarkable property:
Theorem 1.1 (Beauville–Voisin [Reference Beauville and Voisin4])
Let S be a projective K3 surface. The
$\mathbb {Q}$
-subalgebra
$$ \begin{align*} R^\ast(S):= \bigl\langle \operatorname{A}^1(S), c_j(S) \bigr\rangle\ \ \ \subset\ \operatorname{A}^\ast(S) \end{align*} $$
injects into cohomology under the cycle class map.
A fancy way of rephrasing this result is as follows: for any variety Y, let
$$ \begin{align*} \operatorname{N}^i(Y):= \operatorname{A}^i(Y)/\operatorname{A}^i_{hom}(Y)\end{align*} $$
denote the quotient, where
$\operatorname {A}^i_{hom}(Y)\subset \operatorname {A}^i(Y)$
denotes the homologically trivial cycles. Then for K3 surfaces S (which have
$H^1(S,\mathbb {Q})=0$
and so
$A^1(S)$
injects into cohomology), Theorem 1.1 says that the
$\mathbb {Q}$
-algebra epimorphism
$$ \begin{align*} \operatorname{A}^\ast(S)\ \twoheadrightarrow\ \operatorname{N}^\ast(S)\end{align*} $$
admits a section, whose image contains the Chern classes of S—that is, S has the section property, in the language of [Reference Fu and Vial16].
It is then natural to ask which other varieties have the section property. An interesting partial answer is given in [Reference Fu and Vial16], by extending O’Sullivan’s theory of distinguished cycles from abelian varieties to varieties with motive of abelian type: if a variety Y verifies condition
$(\star)$
of loc. cit., then all powers of Y have the section property (cf. Section 2.3 below). One could say that varieties verifying the condition
$(\star)$
form a kind of “meilleur des mondes possibles,” a world in which Chow motives and their multiplicative behaviour are well-understood. Unfortunately, inhabitants of this meilleur des mondes are rather scarce; some examples of varieties verifying condition
$(\star)$
are given in [Reference Fu and Vial16, Reference Laterveer and Vial28].
The main result of the present paper exhibits special Calabi–Yau varieties of any dimension for which the Chow ring is just as well-behaved as that of K3 surfaces:
Theorem (=Theorem 3.1) Let X be a hyperelliptic Calabi–Yau variety of dimension
$n\ge 2$
. The
$\mathbb {Q}$
-subalgebra
$$ \begin{align*} R^\ast(X):= \Bigl\langle \operatorname{A}^1({X}), \, \operatorname{A}^i({X})\cdot \operatorname{A}^j({X}),\, c_k(X)\Bigr\rangle\ \ \ \subset\ \operatorname{A}^\ast(X)\ \ \ \ \ (i,j>0) \end{align*} $$
generated by divisors, Chern classes and by cycles that are intersections of two cycles of positive codimension injects into cohomology. In particular, the image of the intersection product
$$ \begin{align*} \operatorname{A}^i(X)\otimes \operatorname{A}^{j}(X)\ \ \to\ \operatorname{A}^{i+j}(X)\ \ \ \ (i,j>0) \end{align*} $$
injects into cohomology.
Here, a hyperelliptic Calabi–Yau variety is defined as follows: given
$2n+2$
hyperplanes in general position in
$\mathbb {P}^n$
, the double cover of
$\mathbb {P}^n$
branched along the union of hyperplanes admits a crepant resolution that is Calabi–Yau. We say that the resulting Calabi–Yau n-fold is hyperelliptic if the hyperplanes osculate a rational normal curve (cf. Section 2.2 below). These Calabi–Yau varieties have been studied in [Reference Gerkmann, Sheng, van Straten and Zuo18, Reference Matsumoto and Terasoma30, Reference Terasoma38].
The behavior exhibited by Theorem 3.1 is remarkable, in the sense that for general Calabi–Yau varieties Y one does not expect that the image of the intersection product
$$ \begin{align*} \operatorname{A}^i(Y)\otimes \operatorname{A}^{j}(Y)\ \ \to\ \operatorname{A}^{i+j}(Y)\ \ \ \ (i,j>0) \end{align*} $$
injects into cohomology; one only expects this for
$i+j=\dim Y$
(and this last expectation is known for complete intersection Calabi–Yau varieties [Reference Fu15], but wide open in general).
In proving Theorem 3.1, we rely on the “meilleur des mondes” formalism of [Reference Fu and Vial16]. We actually prove that a certain blow-up of X verifies condition
$(\star)$
of loc. cit.; the result for X is then a consequence of the nice behavior of the formalism. This reasoning is very similar to that of [Reference Laterveer and Vial28].
As a by-product of the argument, we also obtain some new cases where Voisin’s conjecture [Reference Voisin and Maruyama43] is verified:
Theorem (=Theorem 4.2) Let X be a hyperelliptic Calabi–Yau n-fold. Any two zero-cycles
$a,a^\prime \in \operatorname {A}^n_{hom}(X)$
satisfy
$$ \begin{align*} a\times a^\prime = (-1)^n \, a^\prime\times a\ \ \ \text{in}\ \operatorname{A}^{2n}(X\times X).\end{align*} $$
(Here,
$a\times a^\prime $
denotes the exterior product
$(p_1)^\ast (a)\cdot (p_2)^\ast (a^\prime)\in \operatorname {A}^{2n}(X\times X)$
, where
$p_j$
is projection to the jth factor.)
Another by-product concerns a conjecture of Voevodsky [Reference Voevodsky40]:
Theorem (=Theorem 4.7) Let X be a hyperelliptic Calabi–Yau variety of odd dimension. Then homological equivalence and smash-equivalence coincide for all algebraic cycles on X.
The aim of this paper is twofold: on the one hand, we want to promote the “meilleur des mondes” formalism of [Reference Fu and Vial16] (and encourage others to find new instances where this formalism can be applied); on the other hand, we want to raise interest for questions concerning the multiplicative structure of the Chow ring of varieties (and to this end, we have included some open questions concerning other Calabi–Yau varieties, cf. section 5).
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 are with rational coefficients: we will denote by
$\operatorname {A}_j(Y)$
the Chow group of j-dimensional cycles on Y with
$\mathbb {Q}$
-coefficients; for Y smooth of dimension n the notations
$\operatorname {A}_j(Y)$
and
$\operatorname {A}^{n-j}(Y)$
are used interchangeably. The notations
$\operatorname {A}^j_{hom}(Y)$
and
$\operatorname {A}^j_{AJ}(Y)$
will be used to indicate the subgroup of homologically trivial (resp. Abel–Jacobi trivial) cycles. For a morphism
$f\colon X\to Y$
, we will write
$\Gamma _f\in \operatorname {A}_\ast (X\times Y)$
for the graph of f.
The covariant category of Chow motives (i.e., pure motives with respect to rational equivalence with
$\mathbb {Q}$
-coefficients as in [Reference André2]) will be denoted
$\mathcal M_{\mathrm{rat}}$
.
2 Preliminaries
2.1 Intersection theory on quotient varieties
Lemma 2.1 Let M be a quotient variety, i.e.,
$M=M^\prime /G$
where
$M^\prime $
is a smooth quasi-projective variety and
$G\subset \operatorname {\textrm {Aut}}(M^\prime)$
is a finite group. Then
$\operatorname {A}^\ast (M):=\oplus _i \operatorname {A}_{\dim M-i}(M)$
is a commutative graded ring, with the usual functorial properties.
Proof According to [Reference Fulton17, Example 17.4.10], the natural map
$$ \begin{align*} \operatorname{A}^i(M)\ \to\ \operatorname{A}_{\dim M-i}(M) \end{align*} $$
from operational Chow cohomology (with
$\mathbb {Q}$
-coefficients) to the usual Chow groups (with
$\mathbb {Q}$
-coefficients) is an isomorphism. The lemma follows from the good formal properties of operational Chow cohomology.▪
Remark 2.2 In particular, Lemma 2.1 implies that the formalism of correspondences and pure motives (with
$\mathbb {Q}$
-coefficients) makes sense for projective quotient varieties.
2.2 Hyperelliptic Calabi–Yau varieties
It is a well-known fact that hyperplane arrangements give rise to Calabi–Yau varieties:
Proposition 2.3 Let
$H_1,\ldots ,H_{2n+2}$
be hyperplanes in
$\mathbb {P}^n$
that are in general position (i.e.,
$\dim H_{i_1}\cap \cdots \cap H_{i_j} = n-j$
for each subset
$\{i_1,\ldots ,i_j\}\subset \{1,\ldots ,2n+2\}$
). Let
$\bar {X}\to \mathbb {P}^n$
be the double cover ramified along
$\cup _{i=1}^{2n+2} H_i$
. Then
$\bar {X}$
is a quotient variety, and there exists a resolution of singularities
$f\colon X\to \bar {X}$
such that X is a Calabi–Yau variety. The morphism f is a sequence of blow-ups with smooth centers
$Z_i$
that have trivial Chow groups (i.e.,
$\operatorname {A}^\ast _{hom}(Z_i)=0$
).
Proof As explained in [Reference Gerkmann, Sheng, van Straten and Zuo18], the arrangement
$\{H_i\}$
is determined by an
$(n+1)\times (2n+2)$
-matrix
$(b_{ij})$
, where
$b_{ij}\in \mathbb {C}$
. Let
$Y\subset \mathbb {P}^{2n+1}$
be the complete intersection of quadrics
$$ \begin{align*} \begin{cases} b_{00} x_0^2+ b_{01}x_1^2+\cdots \ \cdots \ \cdots+ b_{0,2n+1}x_{2n+1}^2&=0,\\ \ \ \ \vdots\\ \ \ \ \vdots\\ b_{n0} x_0^2+b_{n1} x_1^2+\cdots \cdots \cdots + b_{n,2n+1} x_{2n+1}^2&=0. \end{cases}\end{align*} $$
The nonsingularity of Y is equivalent to the
$H_i$
being in general position [Reference Terasoma37, Proposition 3.1.2]. There is an isomorphism
$$ \begin{align*} \bar{X}\cong Y/G \end{align*} $$
for some finite group G (this is proven for
$n=3$
in [Reference Gerkmann, Sheng, van Straten and Zuo18, Proposition 2.5]; the argument works for general n), and so
$\bar {X}$
is a quotient variety.
A crepant resolution
$X\to \bar {X}$
is constructed in [Reference Cynk and Hulek7, Section 5.1] (an alternative construction is given in [Reference Matsumoto and Terasoma30, Proposition 4.2]). For later use, we give a precise description of the resolution algorithm. The resolution
$X\to \bar {X}$
is constructed as a cartesian diagram
$$ \begin{align*} \begin{array}[c]{cccccccc} X=:X_{m}\ \ \ \ & \xrightarrow{}\ \ \ & \cdots & \cdots & \xrightarrow{}& X_1 & \xrightarrow{}& \ \ X_0:= \bar{X} \\ &&&&&&&\\ \ \ \ \ \ \ \ \ \ \ \downarrow{\scriptstyle \pi_{m}} \ \ \ &&&&& \ \ \ \ \downarrow{\scriptstyle \pi_1} && \downarrow{\scriptstyle \pi_0}\ \ \ \ \ \ \\ &&&&&&&\\ \ \ \ \ \ \ \ P_{m}\ \ \ \ & \xrightarrow{r_{m-1}}\ \ \ & \cdots & \cdots & \xrightarrow{}& P_1 & \xrightarrow{r_0}& \ \ P_0:= \mathbb{P}^n, \end{array}\end{align*} $$
where
$r_0$
is a blow-up with center the codimension 2 intersection
$Q_0:=H_1\cap H_2$
, and each
$r_j$
is a blow-up with center
$Q_j\subset P_j$
, where
$Q_j$
is the strict transform of an intersection
$H_{i_1}\cap H_{i_2}$
. (This description is perhaps not immediately apparent from reading [Reference Cynk and Hulek7, Section 5.1], but this becomes crystal clear from the reinterpretation of [Reference Cynk and Hulek7] given in [Reference Ingalls and Logan19]: the arrangement
$\{H_i\}$
is splayed, in the sense of loc. cit. (cf. [Reference Ingalls and Logan19, Lemmas 3.24 and 3.25]), and hence the resolution algorithm [Reference Ingalls and Logan19, Algorithm 1.5] consists of blowing-up all pairwise intersections
$H_{i_1}\cap H_{i_2}$
, in arbitrary order.)▪
Remark 2.4 In case
$n=2$
, X as in Proposition 2.3 is a K3 surface of the type studied in [Reference Matsumoto, Sasaki and Yoshida29, Reference Paranjape34, Reference Yoshida44]. In case
$n=3$
, Calabi–Yau varieties X as in Proposition 2.3 are special cases of so-called “double octics”; these special cases have been intensively studied, particularly their modular properties [Reference Cynk and Kocel–Cynk8
–
Reference Cynk and van Straten12, Reference Gerkmann, Sheng, van Straten and Zuo18, Reference Meyer
31
, Reference Terasoma38].
In order to define hyperelliptic Calabi–Yau varieties, we consider a special case of the above construction:
Proposition 2.5 Let
$p_1,\ldots ,p_{2n+2}\in \mathbb {P}^1$
be distinct points, and let
$$ \begin{align*} H_i:= \gamma\bigl( p_i\times (\mathbb{P}^1)^{n-1}\bigr)\ \ \subset\ \mathbb{P}^n ,\end{align*} $$
where
$\gamma \colon (\mathbb {P}^1)^n\to \mathbb {P}^n$
is the natural map
$$ \begin{align*} \gamma\colon\ \ (\mathbb{P}^1)^n\to \operatorname{\textrm{Sym}}^n (\mathbb{P}^1)\cong \mathbb{P} H^0(\mathbb{P}^1,\mathcal O_{\mathbb{P}^1}(n))\cong \mathbb{P}^n .\end{align*} $$
Then
$H_1,\ldots ,H_{2n+2}$
is a hyperplane arrangement in general position, and hence gives rise to a Calabi–Yau n-fold X.
The hyperplanes
$H_1,\ldots ,H_{2n+2}$
are tangent to a rational normal curve of degree n. Conversely, any hyperplane arrangement osculating a rational normal curve of degree n arises in this way.
Proof The hyperplane
$H_i$
corresponds to degree n divisors on
$\mathbb {P}^1$
containing the point
$p_i$
. The intersection of hyperplanes
$H_{i_1}\cap \cdots \cap H_{i_r}$
then corresponds to degree n divisors on
$\mathbb {P}^1$
containing r distinct points; this has dimension equal to the expected dimension
$n-r$
.
The second statement (which is observed in [Reference Gerkmann, Sheng, van Straten and Zuo18, Remark 2.10]) follows from the fact that the image of the diagonal embedding of
$\mathbb {P}^1$
under
$\gamma $
is a rational normal curve tangent to the
$H_i$
.▪
Definition 2.6 Let
$\{H_i\}$
be a hyperplane arrangement as in Proposition 2.5, and let
$\bar {X}\to \mathbb {P}^n$
be the double cover branched along
$\cup _i H_i$
. The crepant resolution
$X\to \bar {X}$
constructed via the algorithm of Proposition 2.3 is called a hyperelliptic Calabi–Yau variety.
Remark 2.7 The appellation “hyperelliptic” comes from [Reference Gerkmann, Sheng, van Straten and Zuo18], where the moduli space of these Calabi–Yau varieties of dimension
$n=3$
is called the “hyperelliptic locus.” As observed in [Reference Gerkmann, Sheng, van Straten and Zuo18, Remark 2.10], the Calabi–Yau n-folds of Proposition 2.3 have moduli dimension
$n^2$
, while the hyperelliptic Calabi–Yau n-folds of Proposition 2.5 have moduli dimension
$2n-1$
.
In case
$n=2$
, this is the well-known fact that K3 surfaces coming from double planes branched along six lines form a four-dimensional family, while imposing that the six lines are tangent to a conic one obtains exactly the quartic Kummer K3 surfaces (which form a three-dimensional family). The moduli space of these surfaces is studied in [Reference Matsumoto, Sasaki and Yoshida29].
The next result provides justification for Definition 2.6:
Proposition 2.8 Let
$\bar {X}\to \mathbb {P}^n$
be a double cover branched along a hyperplane arrangement as in Proposition 2.5, and let
$X\to \bar {X}$
be the crepant resolution coming from Proposition 2.3 (i.e., X is a hyperelliptic Calabi–Yau n-fold).
(i) There is an isomorphism
$\bar {X}\cong C^n/G$
, where C is a hyperelliptic curve of genus n, and
$G\subset \operatorname {\textrm {Aut}}(C^n)$
is a finite group of automorphisms; in particular,
$\bar {X}$
has only quotient singularities.
(ii) There exist a motive
$M\in \mathcal M_{\mathrm{rat}}$
and isomorphisms of Chow motives
$$ \begin{align*} h(\bar{X}) &\cong M\oplus \bigoplus \mathbb{1}(\ast),\\ h({X}) &\cong M\oplus \bigoplus \mathbb{1}(\ast)\ \ \text{in}\ \mathcal M_{\mathrm{rat}},\end{align*} $$
where
$H^j(M,\mathbb {Q})=0$
for
$j\not =n$
.
Proof (i) In case
$n=3$
, this is contained in [Reference Gerkmann, Sheng, van Straten and Zuo18, Section 2.3]; the same argument works for arbitrary n, as we now explain. Let
$q\colon C\to \mathbb {P}^1$
be the hyperelliptic curve branched along the points
$p_1,\ldots , p_{2n+2}\in \mathbb {P}^1$
. The morphism
$$ \begin{align*} h\colon\ \ C^n\ \xrightarrow{q^n}\ (\mathbb{P}^1)^n\ \xrightarrow{\gamma}\ \mathbb{P}^n \end{align*} $$
is a Galois covering of degree
$2^n\cdot n!$
, with Galois group
$G_2\cong \langle \iota _1,\ldots ,\iota _n\rangle \rtimes \mathfrak S_n$
(here the
$\iota _j$
denotes the hyperelliptic involution of the jth copy of C). Let
$G\subset G_2$
be the index 2 subgroup
$G:=N\rtimes \mathfrak S_n$
, where N is the kernel of the sum homomorphism
$(\mathbb {Z}/2\mathbb {Z})^n\to \mathbb {Z}/2\mathbb {Z}$
. There is a cartesian diagram
$$ \begin{align*} \begin{array}[c]{ccc} C^n &\xrightarrow{p} & C^n/G\\ &&\\ \ \ \downarrow{\scriptstyle q^n}&& \ \downarrow{\scriptstyle \pi}\\ &&\\ (\mathbb{P}^1)^n &\xrightarrow{\gamma}& \mathbb{P}^n.\end{array}\end{align*} $$
The Galois group of
$\pi $
is isomorphic to
$G_2/G$
, and is generated by the image of
$\iota _1$
. Hence, the ramification locus of
$\pi $
is the image of the fixed locus
$L_1\subset C^n$
of
$\iota _1$
under
$\pi \circ p$
. This is the same as the image of
$L_1$
under
$\gamma \circ q^n$
. Since
$$ \begin{align*} q^n(L_1)=\bigcup_{i=1}^{2n+2} p_i\times (\mathbb{P}^1)^{n-1},\end{align*} $$
it follows that the ramification locus of
$\pi $
consists of the union of the hyperplanes
$$ \begin{align*} H_i:= \gamma\bigl( p_i\times (\mathbb{P}^1)^{n-1}\bigr)\ \ \subset\ \mathbb{P}^n.\end{align*} $$
As
$C^n/G$
and
$\bar {X}$
are double covers of
$\mathbb {P}^n$
with the same ramification locus, this proves (i).
As for (ii), note that
$\bar {X}$
(being a double cover of
$\mathbb {P}^n$
branched along a degree
$2n+2$
divisor) is isomorphic to an ample hypersurface in weighted projective space
$\mathbb {P}(1^{n+1},n+1)$
. By weak Lefschetz, plus the fact that
$\bar {X}$
is a quotient variety and hence satisfies Poincaré duality with
$\mathbb {Q}$
-coefficients (cf. for instance [Reference Dolgachev and Carrell14, Section 4.2.2]), it follows that
$$ \begin{align*} H^j(\bar{X},\mathbb{Q}) =\begin{cases} \mathbb{Q}&\ \ \text{if}\ j\not=n\ \text{even},\\ 0 &\ \ \text{if}\ j\not=n\ \text{odd}. \end{cases}\end{align*} $$
Let
$M\subset h(\bar {X})$
be the submotive defined by the projector
$$ \begin{align*} \Delta_{\bar{X}}-\sum_{j=0}^n {1\over d}\, h^j\times h^{n-j} \ \ \ \in\ \operatorname{A}^n(\bar{X}\times\bar{X}) \end{align*} $$
(where d is the degree of
$\bar {X}$
). This gives a decomposition of
$h(\bar {X})$
as requested.
The resolution
$X\to \bar {X}$
is done by blowing up subvarieties with trivial Chow groups, and so the blow-up formula gives an isomorphism
$$ \begin{align*} h(X)\cong h(\bar{X})\oplus \bigoplus\mathbb{1}(\ast)\ \ \ \text{in}\ \mathcal M_{\mathrm{rat}}.\end{align*} $$
This gives the requested decomposition of
$h(X)$
.▪
Remark 2.9 It seems likely that in Proposition 2.8(ii), one actually has an isomorphism
$$ \begin{align*} M\cong \operatorname{\textrm{Sym}}^n h^1(C)\end{align*} $$
(this is stated for
$n=3$
in [Reference Terasoma38, Equation (1.3)] and [Reference Gerkmann, Sheng, van Straten and Zuo18, Proposition 2.9]Footnote
1
).
2.3 The section property and distinguished cycles
The following notion was introduced by O’Sullivan [Reference O’Sullivan33].
Definition 2.10 (Symmetrically distinguished cycles on abelian varieties [Reference O’Sullivan33])
Let B be an abelian variety and
$\alpha \in \operatorname {A}^*(B)$
. For each integer
$m\geq 0$
, denote by
$V_{m}(\alpha)$
the
$\mathbb {Q}$
-vector subspace of
$\operatorname {A}^*(B^{m})$
generated by elements of the form
$$ \begin{align*}p_{*}(\alpha^{r_{1}}\times \alpha^{r_{2}}\times \cdots\times \alpha^{r_{n}}),\end{align*} $$
where
$n\leq m$
,
$r_{j}\geq 0 $
are integers, and
$p : B^{n}\to B^{m}$
is a closed immersion with each component
$B^{n}\to B$
being either a projection or the composite of a projection with
$[-1]: B\to B$
. Then
$\alpha $
is symmetrically distinguished if for every m the restriction of the projection
$\operatorname {A}^*(B^{m})\to \operatorname {N}^*(B^{m})$
to
$V_{m}(\alpha)$
is injective.
The main result of [Reference O’Sullivan33] is:
Theorem 2.11 (O’Sullivan [Reference O’Sullivan33])
Let B be an abelian variety. Then
$\operatorname {DA}^*(B)$
, the symmetrically distinguished cycles in
$\operatorname {A}^*(B)$
, form a graded
$\mathbb {Q}$
-subalgebra that contains symmetric divisors and that is stable under pull-backs and push-forwards along homomorphisms of abelian varieties. Moreover the composition
$$ \begin{align*}\operatorname{DA}^*(B)\hookrightarrow \operatorname{A}^*(B)\twoheadrightarrow \operatorname{N}^*(B)\end{align*} $$
is an isomorphism of
$\mathbb {Q}$
-algebras.
Let X be a smooth projective variety such that its Chow motive
$h(X)$
belongs to the strictly full and thick subcategory of Chow motives generated by the motives of abelian varieties. We say that X has motive of abelian type. A marking for X is an isomorphism of Chow motives
$$ \begin{align*} \phi\colon\ h(X)\stackrel{\cong}{\longrightarrow} M\ \ \text{in}\ \mathcal M_{\mathrm{rat}}\ \end{align*} $$
with M a direct summand of a Chow motive of the form
$\oplus _{i} h(B_{i})(n_{i})$
cut out by an idempotent matrix
$P \in \textrm {End}(\oplus _i h(B_i)(n_i))$
whose entries are symmetrically distinguished cycles, where
$B_{i}$
is an abelian variety and
$n_{i}$
is an integer (the Tate twist). We refer to [Reference Fu and Vial16, Definition 3.1] for the precise definition.
Given a marking
$\phi : h(X)\stackrel {\simeq }{\longrightarrow } M$
, we define the subgroup of distinguished cycles of X, denoted
$\operatorname {DA}^*_{\phi }(X)$
, to be the pre-image of
$\operatorname {DA}^*(M):= P_*\bigoplus _i \operatorname {DA}^{*-n_i}(B_i)$
via the induced isomorphism
$\phi _{*}:\operatorname {A}^*(X)\stackrel {\simeq }{\longrightarrow } \operatorname {A}^*(M)$
.
Given another smooth projective variety Y with a marking
$\psi : h(Y) \to N$
, the tensor product
$\phi \otimes \psi : h(X\times Y) \to M\otimes N$
naturally defines a marking for
$X\times Y$
. A morphism
$f: X\to Y$
is said to be a distinguished morphism if its graph is distinguished with respect to the product marking
$\phi \otimes \psi $
.
The composition
$$ \begin{align*}\operatorname{DA}^*_{\phi}(X)\hookrightarrow \operatorname{A}^*(X)\twoheadrightarrow \operatorname{N}^*(X)\end{align*} $$
is clearly bijective. In other words,
$\phi $
provides a section (as graded vector spaces) of the natural projection
$\operatorname {A}^*(X)\twoheadrightarrow \ \operatorname {N}^*(X)$
. In [Reference Fu and Vial16], sufficient conditions on the marking
$\phi $
are given such that
$\operatorname {DA}^*_{\phi }(X)$
defines a
$\mathbb {Q}$
-subalgebra of
$\operatorname {A}^*(X)$
:
Definition 2.12 (Definition 3.7 in [Reference Fu and Vial16])
We say that the marking
$\phi : h(X)\stackrel {\simeq }{\longrightarrow } M$
satisfies the condition
$(\star)$
if the following two conditions are satisfied:
-
(⋆Mult) the small diagonal
$\delta _{X}$
belongs to
$\operatorname {DA}^*_{\phi ^{\otimes 3}}(X^{3})$
; that is, under the induced isomorphism
$\phi ^{\otimes 3}_{*}: \operatorname {A}^*(X^{3})\stackrel {\simeq }{\longrightarrow } \operatorname {A}^*(M^{\otimes 3})$
, the image of
$\delta _{X}$
is symmetrically distinguished, i.e., in
$\operatorname {DA}^*(M^{\otimes 3})$
. -
(⋆Chern) all Chern classes
$c_{i}(X)$
belong to
$\operatorname {DA}^*_{\phi }(X)$
.
If in addition X is equipped with the action of a finite group G, we say that the marking
$\phi : h(X)\stackrel {\simeq }{\longrightarrow } M$
satisfies
$(\star _G)$
if:
-
(⋆ G ) the graph
$g_X$
of
$g: X\to X$
belongs to
$\operatorname {DA}^*_{\phi ^{\otimes 2}}(X^{2})$
for all
$g\in G$
.
The raison d’être for condition
$(\star)$
is its relation to the section property, as mentioned in the introduction:
Proposition 2.13 (Proposition 3.12 in [Reference Fu and Vial16])
If the marking
$\phi : h(X)\stackrel {\simeq }{\longrightarrow } M$
satisfies the condition
$(\star)$
, then there is a section, as graded
$\mathbb {Q}$
-algebras, for the natural surjective morphism
$\operatorname {A}^*(X)\to \operatorname {N}^*(X)$
such that all Chern classes of X are in the image of this section.
In other words, assuming
$(\star)$
we have a graded
$\mathbb {Q}$
-subalgebra
$\operatorname {DA}_\phi ^*(X)$
of the Chow ring
$\operatorname {A}^*(X)$
, which contains all the Chern classes of X and is mapped isomorphically to
$\operatorname {N}^*(X)$
. Elements of
$\operatorname {DA}_\phi ^*(X)$
are called distinguished cycles.
The raison d’être for condition
$(\star _G)$
is that it allows to easily treat quotients:
Proposition 2.14 Let Y be a smooth projective variety verifying
$(\star)$
and
$(\star _G)$
, for some finite group
$G\subset \operatorname {\textrm {Aut}}(Y)$
. Then
$X/G$
has a marking such that
$X/G$
verifies condition
$(\star _{\textrm {Mult}})$
, and the quotient morphism
$p\colon X\to X/G$
is distinguished. If p is étale then
$X/G$
verifies condition
$(\star)$
.
Proof This is [Reference Fu and Vial16, Proposition 4.12].▪
We refer to [Reference Fu and Vial16] for examples of varieties satisfying
$(\star)$
; for our purposes here, let us mention that these include abelian varieties, varieties with trivial Chow groups,Footnote
2
and hyperelliptic curves:
Proposition 2.15 Let C be a hyperelliptic curve equipped with the action of the group
$H\cong \mathbb {Z}/2\mathbb {Z}$
generated by the hyperelliptic involution. Then C has a marking that satisfies the conditions
$(\star)$
and
$(\star _H)$
, with the additional property that if P is a fixed point of H, then the embedding
$P\hookrightarrow C$
is distinguished.
Proof This is [Reference Laterveer and Vial28, Proposition 3.3(i)].▪
The property
$(\star)$
has great flexibility: as shown in [Reference Fu and Vial16, Section 4], this property is stable under product, projectivization of vector bundles, and (under certain conditions) blow-ups. The relevant result for blow-ups is as follows:
Proposition 2.16 [Reference Fu and Vial16]
Let X be a smooth projective variety and let
$i:Y\hookrightarrow X$
be a closed smooth subvariety. Let
$\tilde X$
be the blow-up of X along Y and let E be the exceptional divisor, so that we have a cartesian diagram
If we have markings satisfying the condition
$(\star)$
for X and Y such that
$i:Y\hookrightarrow X$
is distinguished, then E and
$\tilde X$
have natural markings that satisfy
$(\star)$
and are such that the morphisms
$i,j,\tau $
and p are all distinguished.
If, in addition, X is equipped with the action of a finite group G such that
$G\cdot Y = Y$
and such that the markings of X and Y satisfy
$(\star _G)$
, then the natural markings of E and
$\tilde X$
also satisfy
$(\star _G)$
.
Proof This is the content of [Reference Fu and Vial16, Propositions 4.5 and 4.8].▪
Let us also recall the following, which will come in useful in the proof of our main result (Theorem 3.1):
Proposition 2.17 Let X be a smooth projective variety of dimension
$n\geq 2$
with a marking
$\phi $
satisfying the condition
$(\star)$
of Definition 2.12. Assume that the cohomology of X is spanned by algebraic classes in degree
$\neq n$
. Then the graded
$\mathbb {Q}$
-subalgebra
$R^*(X) \subset \operatorname {A}^*(X)$
generated by divisors, Chern classes and by cycles that are the intersection of two cycles in X of positive codimension (is contained in
$\operatorname {DA}_\phi ^*(X)$
and hence) injects into
$\operatorname {N}^*(X)$
.
Proof This is [Reference Laterveer and Vial28, Proposition 2.10].▪
3 Main result
This section contains the proof of the main result, which is as follows:
Theorem 3.1 Let X be a hyperelliptic Calabi–Yau n-fold.
(i) There exists a sequence of blow-ups
$\widetilde {X}\to X$
such that
$\widetilde {X}$
verifies condition
$(\star)$
, and hence for each
$m\in \mathbb {N}$
the
$\mathbb {Q}$
-algebra epimorphism
$\operatorname {A}^\ast (\widetilde {X}^m)\to \operatorname {N}^\ast (\widetilde {X}^m)$
admits a section, whose image contains the Chern classes of
$\widetilde {X}^m$
.
(ii) Let
$n\ge 2$
. The
$\mathbb {Q}$
-subalgebra
$R^\ast (X)\subset \operatorname {A}^\ast (X)$
generated by divisors, Chern classes and by cycles that are intersections of two cycles of positive codimension injects into cohomology. In particular, the image of the intersection product
$$ \begin{align*} \operatorname{A}^i(X)\otimes \operatorname{A}^{j}(X)\ \ \to\ \operatorname{A}^{i+j}(X)\ \ \ \ (i,j>0) \end{align*} $$
injects into cohomology.
Proof To construct
$\widetilde {X}$
, we proceed as follows: starting from the singular double cover
$\pi \colon \bar {X}\to \mathbb {P}^n$
, we first blow-up all
$0$
-dimensional loci
$\pi ^{-1}(H_{j_1}\cap \cdots \cap H_{j_n})$
(where the
$j_i$
are pairwise distinct), then we blow-up all strict transforms of one-dimensional loci
$\pi ^{-1}(H_{j_1}\cap \cdots \cap H_{j_{n-1}})$
(where the
$j_i$
are pairwise distinct), and so on (ending with codimension 2 loci).
This resolution process can be encoded in a cartesian diagram
$$ \begin{align} \begin{array}[c]{cccccccc} \ \ \ \ \ \ Y_{n-1} &\xrightarrow{t_{n-2}}\ \ \ & \cdots & \cdots & \xrightarrow{}& Y_1 & \xrightarrow{t_0}& Y_0:= C^n \\ &&&&&&&\\ \ \ \ \ \ \ \ \ \ \ \downarrow{\scriptstyle p_{n-1}} \ \ \ &&&&& \ \ \ \ \downarrow{\scriptstyle p_1} && \downarrow{\scriptstyle p_0}\ \ \ \ \ \ \\ &&&&&&&\\ \widetilde{X}=:X_{n-1}\ \ \ \ & \xrightarrow{s_{n-2}}\ \ \ & \cdots & \cdots & \xrightarrow{}& X_1 & \xrightarrow{s_0}& \ \ X_0:= \bar{X} \\ &&&&&&&\\ \ \ \ \ \ \ \ \ \ \ \downarrow{\scriptstyle \pi_{n-1}} \ \ \ &&&&& \ \ \ \ \downarrow{\scriptstyle \pi_1} && \downarrow{\scriptstyle \pi_0}\ \ \ \ \ \ \\ &&&&&&&\\ \ \ \ \ \ \ \ P_{n-1}\ \ \ \ & \xrightarrow{r_{n-2}}\ \ \ & \cdots & \cdots & \xrightarrow{}& P_1 & \xrightarrow{r_0}& \ \ P_0:= \mathbb{P}^n . \end{array}\end{align} $$
Here,
$r_0$
is the blow-up with center
$\cup H_{j_1}\cap \cdots \cap H_{j_n}$
, and each
$r_i$
is the blow-up with center
$Q_i$
being the union of strict transforms of i-dimensional intersections
$H_{j_1}\cap \cdots \cap H_{j_{n-i}}$
(by construction, these strict transforms form a disjoint union). The arrows
$s_i$
and
$t_i$
are induced by
$r_i$
. Concretely, this means that
$t_0$
is the blow-up with center
$$ \begin{align*} Z_0:= {\displaystyle\bigcup_{\sigma\in\mathfrak S_n}}\ \bigcup_{1\le j_1,\ldots,j_n\le 2n+2}\sigma\Bigl( q_{j_1}\times\cdots\times q_{j_n} \Bigr)\ \ \ \subset\ C^n,\end{align*} $$
(where
$q_1,\ldots ,q_{2n+2}\in C$
are the Weierstrass points). Likewise, each
$t_i$
is a blow-up with center
$$ \begin{align*} Z_i:= {\displaystyle\bigcup_{\sigma\in\mathfrak S_n}}\ \bigcup_{1\le j_1,\ldots,j_{n-i}\le 2n+2}\overline{\sigma\bigl( q_{j_1}\times \cdots\times q_{j_{n-i}} \bigr)} \ \ \ \subset\ Y_{i}\ \end{align*} $$
(where
$\overline {a}$
means strict transform of a). By construction,
$Z_i$
is a disjoint union of smooth irreducible components of dimension i.
The arrows
$\pi _i$
are double covers. The arrow
$p_0$
is the quotient morphism for the action of
$G_0:=N\rtimes \mathfrak S_n$
, and the composition
$\pi _0\circ p_0$
is the quotient morphism for the action of
$H_0\cong (\mathbb {Z}/2\mathbb {Z})^n \rtimes \mathfrak S_n$
. Each arrow
$p_i$
(resp. each composition
$\pi _i\circ p_i$
) is the quotient morphism for the action of the finite group
$G_i$
(resp.
$H_i$
) on
$Y_i$
obtained by lifting the action of
$G_{i-1}$
(resp.
$H_{i-1}$
).
The idea is to prove property
$(\star)$
for
$\widetilde {X}$
inductively, moving from right to left in diagram (1). The induction base is
$Y_0$
:
Lemma 3.2 Let
$G_0\subset H_0\subset \operatorname {\textrm {Aut}}(C^n)$
be as above. The variety
$Y_0:=C^n$
verifies conditions
$(\star)$
and
$(\star _{G_0})$
and
$(\star _{H_0})$
.
Proof (of Lemma 3.2.) The self-product
$C^n$
verifies condition
$(\star)$
because hyperelliptic curves verify
$(\star)$
(Proposition 2.15), and
$(\star)$
is stable under products [Reference Fu and Vial16, Proposition 4.1]. To check condition
$(\star _{H_0})$
(which implies condition
$(\star _{H_0})$
), it suffices to check that the graph of g is distinguished for any
$g\in (\mathbb {Z}/2\mathbb {Z})^n$
and for any
$g\in \mathfrak S_n$
. For
$g\in N$
this follows from the fact that the graph of the hyperelliptic involution is distinguished (Proposition 2.15), plus the compatibility of group actions and products [Reference Fu and Vial16, Proposition 4.1]. For
$g\in \mathfrak S_n$
, this follows from [Reference Fu and Vial16, Remark 4.2].▪
The induction step is as follows:
Lemma 3.3 Assume
$Y_i$
verifies conditions
$(\star)$
and
$(\star _{G_i})$
and
$(\star _{H_i})$
. Then
$Y_{i+1}$
verifies conditions
$(\star)$
. and
$(\star _{G_{i+1}})$
and
$(\star _{H_{i+1}})$
.
Proof (of Lemma 3.3.) This is an application of the general blow-up result Proposition 2.16. Let us check that all hypotheses of Proposition 2.16 are met with. The variety
$Y_i$
and the center
$Z_i$
of the blow-up verify
$(\star)$
: for
$Y_i$
this is by assumption; for
$Z_i$
this is true by induction, because
$Z_i$
is
$C^{i}$
blown-up along certain explicit centers (i.e.,
$Z_i$
is of the form
$Y_i$
with a smaller value of n). To see that the embedding
$\iota _i\colon Z_i\hookrightarrow Y_i$
is distinguished, we note that its graph
$\Gamma _{\iota _i}$
is the pullback of the graph of the embedding
$\tau _i\colon Q_i\hookrightarrow P_i$
(here
$Q_i$
and
$P_i$
are as in the proof of Proposition 2.3). The embedding
$\tau _i$
is distinguished (indeed, one sees inductively that
$P_i$
and
$Q_i$
have trivial Chow groups, and so
$\operatorname {A}^\ast (Q_i\times P_i)=\operatorname {DA}^\ast (Q_i\times P_i)$
), and the assumption
$(\star _{H_i})$
implies that the quotient morphism
$\pi _i\circ p_i$
is distinguished [Reference Fu and Vial16, Proposition 4.12]. It follows that
$$ \begin{align*} \Gamma_{\iota_i}= (\pi_i\circ p_i \times \pi_i\circ p_i)^\ast \Gamma_{\tau_i}\ \ \ \in\ \operatorname{DA}^\ast(Z_i\times Y_i).\end{align*} $$
All hypotheses of Proposition 2.16 being verified, this proves that
$Y_{i+1}$
verifies condition
$(\star)$
.
As for the group action, this follows from the second part of Proposition 2.16. Both
$Y_i$
and the center
$Z_i$
verify condition
$(\star _{H_i})$
(and a fortiori
$(\star _{G_i})$
): for
$Y_i$
this is by assumption, for
$Z_i$
this is true by induction, because
$Z_i$
is
$C^{i}$
blown-up along certain explicit centers. The second part of Proposition 2.16 then guarantees that condition
$(\star _{H_i})$
(and a fortiori
$(\star _{G_i})$
) carries over to
$Y_{i+1}$
.▪
The induction set up by Lemmas 3.2 and 3.3 yields that
$Y_{n-1}$
verifies conditions
$(\star)$
and
$(\star _{G_{n-1}})$
. Using Proposition 2.14, this implies that
$\widetilde {X}= Y_{n-1}/G_{n-1}$
verifies condition
$(\star _{\textrm {Mult}})$
. Let us now check condition
$(\star _{\textrm {Chern}})$
for
$\widetilde {X}$
. For this, we view
$\widetilde {X}$
as the double cover
$$ \begin{align*}\pi\colon \widetilde{X}\to \widetilde{P}:=P_{n-1}\end{align*} $$
branched along the smooth divisor
$D\subset \widetilde {P}$
(obtained as strict transform of the hyperplane arrangement
$\cup _{j=1}^{2n+2} H_j\subset \mathbb {P}^n$
under the blow-ups
$r_i$
). The Chern classes of
$\widetilde {X}$
can be expressed in terms of the Chern classes of
$\widetilde {P}$
and the Chern classes of
$\mathcal O_{\widetilde {X}}(E)$
and
$\mathcal O_{\widetilde {X}}(2E)$
, where we write
$E\subset \widetilde {X}$
for the isomorphic pre-image of D in
$\widetilde {X}$
(cf. [Reference Fu and Vial16, Proof of Proposition 4.12]). But there is equality
$$ \begin{align*} E= d\, (p_{n-1})_\ast (p_{n-1})^\ast\pi^\ast(D)\ \ \ \text{in}\ \operatorname{A}^1(\widetilde{X})\ \ \ \ (d\in\mathbb{Q}),\end{align*} $$
and so (since
$D\in \operatorname {A}^\ast (\widetilde {P})=\operatorname {DA}^\ast (\widetilde {P})$
and
$\pi $
and
$p_{n-1}$
are distinguished) one has
$E\in \operatorname {DA}^\ast (\widetilde {X})$
. Likewise, one has
$$ \begin{align*} \pi^\ast c_j(\widetilde{P}) = d\, (p_{n-1})_\ast (p_{n-1})^\ast\pi^\ast c_j(\widetilde{P})\ \ \ \text{in}\ \operatorname{A}^\ast(\widetilde{X}) \ \ \ \ (d\in\mathbb{Q}),\end{align*} $$
and so also
$\pi ^\ast c_j(\widetilde {P})\in \operatorname {DA}^\ast (\widetilde {X})$
. We conclude that
$c_j(\widetilde {X})\in \operatorname {DA}^\ast (\widetilde {X})$
, i.e.,
$\widetilde {X}$
verifies condition
$(\star _{\textrm {Chern}})$
.
Since the Calabi–Yau variety X was obtained from
$\bar {X}$
by only blowing-up the codimension 2 loci
$H_{i_1}\cap H_{i_2}$
, there is a factorization
$$ \begin{align*} \widetilde{X}\ \xrightarrow{f}\ X\ \xrightarrow{}\ \bar{X},\end{align*} $$
where both arrows are sequences of blow-ups with smooth centers. This proves (i).
We record the following:
Lemma 3.4 Let
$\phi _i$
denote the marking for
$X_i$
constructed above. Let
$ S_i\subset X_i$
denote the center of the blow-up morphism
$s_i\colon X_{i+1}\to X_i$
, and let
$\xi _i\colon S_i\to X_i$
denote the inclusion morphism. The graphs of
$s_i$
and
$\xi _i$
are distinguished (with respect to the markings
$\phi _i$
).
Proof (of Lemma 3.4.) There is a cartesian diagram
$$ \begin{align*} \begin{array}[c]{ccc} Y_{i+1} & \xrightarrow{t_i}& Y_i \\ &&\\ \ \ \ \ \ \ \downarrow{\scriptstyle p_{i+1}} \ \ \ && \ \ \ \ \downarrow{\scriptstyle p_i} \\ &&\\ X_{i+1} & \xrightarrow{s_{i}} & \ \ X_i . \end{array} \end{align*} $$
The graph of
$s_i\circ p_{i+1}$
equals the graph of
$p_i\circ t_i$
, which is distinguished by the construction above. Since the graph of
$p_{i+1}$
is also distinguished (this is property
$(\ast _{G_{i+1}})$
), it follows that
$$ \begin{align*} \Gamma_{s_i}= {1\over \deg p_{i+1}}\ {}^t \Gamma_{p_{i+1}} \circ \Gamma_{p_{i+1}} \circ \Gamma_{s_i} \end{align*} $$
is distinguished.
The argument for
$\xi _i$
is similar, using the cartesian diagram
$$ \begin{align*} \begin{array}[c]{ccc} Z_{i} & \xrightarrow{\iota_i}& Y_i \\ &&\\ \ \ \ \ \ \ \downarrow{} \ \ \ && \ \ \ \ \downarrow{\scriptstyle p_i} \\ &&\\ S_{i} & \xrightarrow{\xi_{i}} & \ \ X_i . \end{array} \end{align*} $$
▪
(ii) Let us observe that
$f\colon \widetilde {X}\to X$
is a sequence of blow-ups with smooth centers having trivial Chow groups, and so
$$ \begin{align} h(\widetilde{X})\cong h(X)\oplus \bigoplus\mathbb{1}(\ast)\ \ \ \text{in}\ \mathcal M_{\mathrm{rat}}.\end{align} $$
Using Proposition 2.8(ii), we find that there is a decomposition
$$ \begin{align} h(\widetilde{X})\cong M\oplus \bigoplus\mathbb{1}(\ast)\ \ \ \text{in}\ \mathcal M_{\mathrm{rat}},\end{align} $$
where M has cohomology concentrated in degree n. It then follows from Proposition 2.17 that the
$\mathbb {Q}$
-subalgebra
$$ \begin{align*} R^\ast(\widetilde{X}):= \Bigl\langle \operatorname{A}^1(\widetilde{X}), \, \operatorname{A}^i(\widetilde{X})\cdot \operatorname{A}^j(\widetilde{X}), \, c_k(\widetilde{X})\Bigr\rangle\ \ \ \subset\ \operatorname{A}^\ast(\widetilde{X})\ \ \ \ \ (i,j>0) \end{align*} $$
is contained in
$\operatorname {DA}^\ast (\widetilde {X})$
and hence injects into cohomology. Using pullback along the morphism
$f\colon \widetilde {X}\to X$
, this implies at once that the
$\mathbb {Q}$
-subalgebra
$$ \begin{align*} \Bigl\langle \operatorname{A}^1({X}), \, \operatorname{A}^i({X})\cdot \operatorname{A}^j({X})\Bigr\rangle\ \ \ \subset\ \operatorname{A}^\ast({X})\ \ \ \ \ (i,j>0) \end{align*} $$
also injects into cohomology. However, to get a statement that includes the Chern classes of X some extra care is needed. Remark that one has
$$ \begin{align*} c_k(\widetilde{X}) = f^\ast c_k(X) + R_k\ \ \ \text{in}\ \operatorname{A}^k(\widetilde{X}),\end{align*} $$
where
$R_k$
is in the second part of the decomposition
$$ \begin{align} \operatorname{A}^k(\widetilde{X})\cong \operatorname{A}^k(X)\oplus \mathbb{Q}^s \end{align} $$
induced by (2). Lemma 3.4 implies that the marking for
$\widetilde {X}$
constructed above is induced (via the blow-up result Proposition 2.16) from a marking for
$\bar {X}$
, and so each trivial motive
$\mathbb {1}(\ast)$
in (3) is marked by a trivial motive
$\mathbb {1}(\ast)$
. A fortiori, the same is true for (2). This implies that the summand
$\mathbb {Q}^s$
in (4) is in
$\operatorname {DA}^\ast (\widetilde {X})$
, and so
$R_k\in \operatorname {DA}^\ast (\widetilde {X})$
.Footnote
3
But then one also has
$$ \begin{align*} f^\ast c_k(X)= c_k(\widetilde{X})-R_k\ \ \ \in\ \operatorname{DA}^\ast(\widetilde{X}),\end{align*} $$
and hence
$$ \begin{align*} f^\ast \Bigl\langle \operatorname{A}^1({X}), \, \operatorname{A}^i({X})\cdot \operatorname{A}^j({X}),\, c_k(X)\Bigr\rangle\ \ \ \subset\ \operatorname{DA}^\ast(\widetilde{X})\ \ \ \ \ (i,j>0) .\end{align*} $$
Since
$\operatorname {DA}^\ast (\widetilde {X})$
injects into cohomology (under the cycle class map), we conclude that the
$\mathbb {Q}$
-algebra
$$ \begin{align*} R^\ast(X):= \Bigl\langle \operatorname{A}^1({X}), \, \operatorname{A}^i({X})\cdot \operatorname{A}^j({X}),\, c_k(X)\Bigr\rangle\ \ \ \ \ (i,j>0) \end{align*} $$
also injects into cohomology (under the cycle class map).▪
Remark 3.5 To prove our main result (Theorem 3.1(ii), we show that for any hyperelliptic Calabi–Yau n-fold X there exists a surjection
$\widetilde {X}\to X$
such that
$\widetilde {X}$
verifies condition
$(\star)$
. This leaves open the question whether X itself verifies condition
$(\star)$
(which would provide an easier proof of the injectivity of Theorem 3.1(ii)).
Note that the hyperelliptic Calabi–Yau variety X is a group quotient of some blow-up of
$C^n$
, with centers having trivial Chow groups. However, these centers do not behave well with respect to the group action (an irreducible codimension 2 center is not invariant under the action of
$\mathfrak S_n$
), making it problematic to pass from the level of
$Y_j$
to the level of the quotient
$X_j$
. For this reason we prove our main result in a slightly “round-about” way, blowing up some more in order to make the irreducible codimension 2 centers disjoint.
4 Further consequences
4.1 Voisin’s conjecture
Voisin [Reference Voisin and Maruyama43] has formulated the following intriguing conjecture, which is a particular instance of the Bloch–Beilinson conjectures.
Conjecture 4.1 (Voisin [Reference Voisin and Maruyama43])
Let X be a smooth projective variety of dimension n, with
$h^{n,0}(X) = 1$
and
$h^{j,0}(X)=0$
for
$0<j<n$
. Then any two zero-cycles
$a,a^\prime \in \operatorname {A}^n_{hom}(X)$
satisfy
$$ \begin{align*} a\times a^\prime = (-1)^n a^\prime\times a\ \ \ \text{in}\ \operatorname{A}^{2n}(X\times X).\end{align*} $$
(Here,
$a\times a^\prime $
is the exterior product
$(p_1)^\ast (a)\cdot (p_2)^\ast (a^\prime)\in \operatorname {A}^{2n}(X\times X)$
, where
$p_j$
is projection to the jth factor.)
For background and motivation for Conjecture 4.1, cf. [Reference Voisin42, Section 4.3.5.2]. Conjecture 4.1 has been proven in some scattered special cases [Reference Bini, Laterveer and Pacienza5, Reference Burek6, Reference Laterveer23–Reference Laterveer and Vial28, Reference Voisin and Maruyama43], but is still wide open for a general K3 surface.
We now prove Voisin’s conjecture for the Calabi–Yau varieties under consideration in the present paper:
Proposition 4.2 Let X be a hyperelliptic Calabi–Yau variety of dimension n. Then Conjecture 4.1 is true for X: any
$a,a^\prime \in \operatorname {A}^n_{hom}(X)$
satisfy
$$ \begin{align*} a\times a^\prime = (-1)^n \, a^\prime\times a\ \ \ \text{in}\ \operatorname{A}^{2n}(X\times X).\end{align*} $$
Proof According to Proposition 2.8, we have a decomposition
$$ \begin{align*} h(X) = M\oplus \bigoplus {\mathbb{1}}(\ast)\ \ \ \text{in}\ \mathcal M_{\mathrm{rat}}, \end{align*} $$
with
$H^j(M)=0$
for
$j\not = n$
, and M isomorphic to a direct summand of
$h(C^n/G)$
, where C is a hyperelliptic curve. Since the symmetric group
$\mathfrak S_n$
is contained in G, the motive M is actually isomorphic to a direct summand of
$h(C^{(n)})$
. By Kimura finite-dimensionality [Reference Kimura22], M is then isomorphic to a direct summand of the motive
$$ \begin{align*} h^n(C^{(n)}):= (C^{(n)},\pi^n_{C^{(n)}},0)\ \ \ \in\ \mathcal M_{\mathrm{rat}},\end{align*} $$
where
$\pi ^n_{C^{(n)}}$
is the Chow–Künneth projector defined by the choice of a point on C, i.e.,
$\pi ^n_{C^{(n)}}$
corresponds to
$$ \begin{align*} \sum_{i_1+\cdots+i_n=n}\ \ \sum_{\sigma\in\mathfrak S_n} \sigma\Bigl( \pi_C^{i_1}\times\cdots \times \pi_C^{i_n} \Bigr) \ \ \in\ \operatorname{A}^n(C^n\times C^n)^{\mathfrak S_n}\end{align*} $$
under the natural isomorphism
$\operatorname {A}^\ast (C^{(n)})\cong \operatorname {A}^\ast (C^n)^{\mathfrak S_n}$
, and the
$\pi _C^{i_j}$
are Chow–Künneth projectors of C.
Let
$\mathcal M^\circ $
denote the category of birational motives [Reference Kahn and Sujatha21]. There is a functor
$$ \begin{align*} \operatorname{\textrm{Bir}}\colon\ \mathcal M_{\mathrm{rat}}^{\textrm{eff}}\ \to\ \mathcal M^\circ,\end{align*} $$
sending an effective motive
$(X,p,0)$
to
$(X,p\vert _{\operatorname {A}^n(X_{\mathbb {C}(X)})})$
. This functor has the property that
$\operatorname {A}_0(M)=\operatorname {A}_0(\operatorname {\textrm {Bir}}(M))$
for any
$M\in \mathcal M_{\mathrm{rat}}^{\textrm {eff}}$
. Looking at the image of
$ h^n(C^{(n)})$
under
$\operatorname {\textrm {Bir}}$
, one sees that all summands of
$\pi ^n_{C^{(n)}}$
where some
$i_j$
is
$0$
restrict to zero, i.e., one has
$$ \begin{align*} \operatorname{\textrm{Bir}} \bigl( h^n(C^{(n)})\bigr)=\operatorname{\textrm{Bir}} \bigl(\operatorname{\textrm{Sym}}^n h^1(C)\bigr)\ \ \ \text{in}\ \mathcal M^\circ.\end{align*} $$
Writing
$J:=\operatorname {\textrm {Jac}}(C)$
for the Jacobian of C and recalling that there is an isomorphism
$h^1(C)\cong h^1(J)$
, it follows that there are isomorphisms of birational motives
$$ \begin{align*} \operatorname{\textrm{Bir}} \bigl( h^n(C^{(n)})\bigr)\cong\operatorname{\textrm{Bir}} \bigl(\operatorname{\textrm{Sym}}^n h^1(C)\bigr)\cong \operatorname{\textrm{Bir}} \bigl(\operatorname{\textrm{Sym}}^n h^1(J)\bigr)\cong \operatorname{\textrm{Bir}}\bigl( h^n(J)\bigr) \ \ \ \text{in}\ \mathcal M^\circ,\end{align*} $$
where
$h^\ast (J)$
refers to the Deninger–Murre Chow–Künneth decomposition for abelian schemes [Reference Deninger and Murre13] (for the properties of Chow motives of abelian varieties that we use here, cf. [Reference Scholl and Jannsen35, Section 5]). In particular, taking Chow groups we find a split injection
$$ \begin{align*} \Gamma_\ast\colon\ \ \operatorname{A}_0(M)\ \hookrightarrow\ \operatorname{A}_0 ( h^n(C^{(n)})&=\operatorname{A}_0\bigl( \operatorname{\textrm{Bir}} ( h^n(C^{(n)})) \bigr)\\ &= \operatorname{A}_0\bigl(\operatorname{\textrm{Bir}}( h^n(J))\bigr)= \operatorname{A}_0(h^n(J))=\operatorname{A}^n_{(n)}(J),\end{align*} $$
where
$\operatorname {A}^\ast _{(\ast)}(J)$
refers to Beauville’s eigenspace decomposition [Reference Beauville3]. By the same argument, there is also a split injection
$$ \begin{align*} (\Gamma\times\Gamma)_\ast\colon\ \ \operatorname{A}_0(M\otimes M)\ \hookrightarrow\ \operatorname{A}_0(J\times J) . \end{align*} $$
This fits into a commutative diagram
$$ \begin{align*} \begin{array}[c]{ccc} \operatorname{A}^n(M)\otimes \operatorname{A}^n(M) & \xrightarrow{(\Gamma_\ast,\Gamma_\ast)}& \operatorname{A}^n_{(n)}(J)\otimes \operatorname{A}^n_{(n)}(J)\\ &&\\ \ \ \downarrow{\scriptstyle \Phi}&& \ \ \downarrow{\scriptstyle \Phi}\\ &&\\ \operatorname{A}^{2n}(M\otimes M) & \xrightarrow{(\Gamma\times\Gamma)_\ast}&\ \operatorname{A}^{2n}_{}(J\times J), \end{array} \end{align*} $$
where
$\Phi $
sends
$(a,a^\prime)$
to
$a\times a^\prime - (-1)^n \, a^\prime \times a$
. We are thus reduced to a general statement about zero-cycles on abelian varieties:
Proposition 4.3 (Voisin [Reference Voisin42])
Let B be an abelian variety of dimension n, and
$a,a^\prime \in \operatorname {A}^n_{(n)}(B)$
. Then
$$ \begin{align*}a\times a^\prime=(-1)^n \, a^\prime\times a\ \ \ \text{in}\ \operatorname{A}^{2n}(B\times B) .\end{align*} $$
Proof This is [Reference Voisin42, Example 4.40]. A generalization (and an alternative proof) is given in [Reference Vial39, Theorem 4.1].▪
This concludes the proof of the theorem.▪
4.2 Voevodsky’s conjecture
This subsection contains an application of our results to Voevodsky’s conjecture on smash-equivalence.
Definition 4.4 (Voevodsky [Reference Voevodsky40])
Let X be a smooth projective variety. A cycle
$a\in \operatorname {A}^i(X)$
is called smash-nilpotent if there exists
$m\in \mathbb {N}$
such that
$$ \begin{align*} \begin{array}[c]{ccc} a^m:= &\underbrace{a\times\cdots\times a}_{(m\text{ times})}&=0\ \ \text{in}\ \operatorname{A}^{mi}(X\times\cdots\times X)_{}. \end{array}\end{align*} $$
Two cycles
$a,a^\prime $
are called smash-equivalent if their difference
$a-a^\prime $
is smash-nilpotent. We will write
$\operatorname {A}^i_\otimes (X)\subseteq \operatorname {A}^i(X)$
for the subgroup of smash-nilpotent cycles.
Conjecture 4.5 (Voevodsky [Reference Voevodsky40])
Let X be a smooth projective variety. Then
$$ \begin{align*} \operatorname{A}^i_{hom}(X)\ \subseteq\ \operatorname{A}^i_\otimes(X)\ \ \ \text{for all }i.\end{align*} $$
Remark 4.6 It is known [Reference André, Kimura and O’Sullivan1, Théorème 3.33] that Conjecture 4.5 for all smooth projective varieties implies (and is strictly stronger than) Kimura’s conjecture “all smooth projective varieties have finite-dimensional motive” [Reference Kimura22].
Let us now verify Voevodsky’s conjecture for hyperelliptic Calabi–Yau varieties of odd dimension:
Theorem 4.7 Let X be a hyperelliptic Calabi–Yau variety. Assume that
$n:=\dim X$
is odd. Then
$$ \begin{align*} \operatorname{A}^i_{hom}(X)\ \subseteq\ \operatorname{A}^i_\otimes(X)\ \ \ \text{for all }i.\end{align*} $$
Proof According to Proposition 2.8, we have a decomposition
$$ \begin{align*} h(X) = M\oplus \bigoplus {\mathbb{1}}(\ast)\ \ \ \text{in}\ \mathcal M_{\mathrm{rat}}, \end{align*} $$
with
$H^j(M)=0$
for
$j\not = n$
, and M isomorphic to a direct summand of
$h(C^n)$
. By Kimura finite-dimensionality, M is isomorphic to a direct summand of the motive
$(C^n,\pi ^n,0)$
, where
$\pi ^n$
is any Chow–Künneth projector on the degree-n cohomology. But the Chow motive
$(C^n,\pi ^n,0)$
is oddly finite-dimensional (in the sense of [Reference Kimura22]). Hence, together with the fact that
$\operatorname {A}^{i}_{hom}({X})= \operatorname {A}^i_{hom}(M)$
, the theorem is implied by the fact that
$$ \begin{align*} \operatorname{A}^\ast_{}(M) \ \subseteq\ \operatorname{A}^\ast_\otimes(M)\end{align*} $$
for any oddly finite-dimensional Chow motive M (this is due to Kimura [Reference Kimura22, Proposition 6.1], and is also used in [Reference Kahn and Sebastian20]).▪
5 Some open questions
Question 5.1 Easy examples of Calabi–Yau varieties Y are given by smooth complete intersections of
$n+1$
quadrics in
$\mathbb {P}^{2n+1}$
. An interesting special case is when Y is defined by equations of the form
$$ \begin{align*} \begin{cases} x_0^2+x_1^2+\cdots \ \cdots \ \cdots+ x_{2n+1}^2&=0,\\[3pt] \lambda_0 x_0^2+\lambda_1 x_1^2+\cdots + \lambda_{2n+1}x_{2n+1}^2&=0,\\[3pt] \lambda_0^2 x_0^2+\lambda_1^2 x_1^2+\cdots + \lambda_{2n+1}^2 x_{2n+1}^2&=0,\\[3pt] \ \ \ \vdots\\ \ \ \ \vdots\\ \lambda_0^n x_0^2+\lambda_1^n x_1^2+\cdots + \lambda_{2n+1}^n x_{2n+1}^2&=0, \end{cases}\end{align*} $$
where
$\lambda _0,\ldots ,\lambda _{2n+1}\in \mathbb {C}$
are distinct numbers. Such varieties Y are isomorphic to a quotient
$D^n/G$
, where D is a curve and
$G\subset \operatorname {\textrm {Aut}}(D^n)$
a finite group [Reference Terasoma37, Theorem 2.4.2]. Moreover, Y is related to the double cover
$\bar {X}$
of Theorem 3.1: one has
$\bar {X}\cong Y/H$
for some finite group H [Reference Terasoma37, Proposition 2.4.4], [Reference Gerkmann, Sheng, van Straten and Zuo18, Section 2.2].
Can one prove condition
$(\star)$
for Y ? The problem is that the curve D is not hyperelliptic; D is a finite étale cover of a hyperelliptic curve, and it is not clear whether D verifies condition
$(\star)$
(this is closely related to the fact that as far as I am aware the only curves known to have a multiplicative Chow–Künneth decomposition, in the sense of [Reference Shen and Vial36], are hyperelliptic curves).
Question 5.2 Let Y be a Calabi–Yau variety as in Proposition 2.3, i.e., arising from a general arrangement of hyperplanes (without the condition that the hyperplanes osculate a rational normal curve). Is it still true that the image of intersection product
$$ \begin{align*} \operatorname{A}^i(Y)\otimes \operatorname{A}^{j}(Y)\ \ \to\ \operatorname{A}^{i+j}(Y)\ \ \ \ (i,j>0) \end{align*} $$
injects into cohomology? The problem is that for these Y, it is not even known that they have motive of abelian type, so that one cannot benefit from the formalism of [Reference Fu and Vial16].
Acknowledgment
Thanks to the referee for very helpful comments. Thanks to Kai and Len for enjoying Buurman en Buurman as much as I do.
