On the motive of Ito-Miura-Okawa-Ueda Calabi-Yau threefolds

TL;DR

This paper proves that two Calabi-Yau threefolds, previously known to be L-equivalent and derived equivalent but not stably birational, also share the same Chow motive, deepening understanding of their geometric relations.

Contribution

It establishes that the pair of Calabi-Yau threefolds have isomorphic Chow motives, completing the picture of their equivalences.

Findings

X and Y are isomorphic in Chow motives.

X and Y are L-equivalent and derived equivalent.

X and Y are not stably birational.

Abstract

Ito-Miura-Okawa-Ueda have constructed a pair of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, but not stably birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.

Equations

$$([X]-[Y])[\mathbb{A}^{1}]=0\ \ \ \hbox{in}\ K_{0}(\hbox{Var}(k))\$$

$$h(X)\cong h(Y)in{\mathcal{M}}_{\mathrm{rat}}.$$

$$[X]\ne [Y]in{K}_0(Var(k)),$$

$$([X]-[Y])[{\mathbb{A}}^1]=0in{K}_0(Var(k)).$$

$$D^b(X)\cong D^b(Y).$$

$$H^3(X,\mathbb{Z})\cong H^3(Y,\mathbb{Z}).$$

$$h(X)\cong h(Y)in{\mathcal{M}}_{\mathrm{rat}}.$$

$$\begin{array}{rl}h(X)& =\text{mathds}1\oplus \text{mathds}1(1)\oplus h^3(X)\oplus \text{mathds}1(2)\oplus \text{mathds}1(3),\\ h(Y)& =\text{mathds}1\oplus \text{mathds}1(1)\oplus h^3(Y)\oplus \text{mathds}1(2)\oplus \text{mathds}1(3)in{\mathcal{M}}_{\mathrm{rat}},\end{array}$$

$${\pi }_X^{2i}:=c_i{H}^{3-i}\times H\in A^3(X\times X),0\le i\le 3,$$

$$h^3(X)\cong h^3(Y)in{\mathcal{M}}_{\mathrm{rat}}.$$

$$A^{\ast }(h^3(X))=A_{hom}^{\ast }(X),$$

$${\Gamma }_{\ast }:A_{hom}^j(X_K)\stackrel{\cong }{\to }{A}_{hom}^j(Y_K)\forall j,$$

$$A^j(h^3(X{)}_K)\stackrel{\cong }{\to }{A}^j(h^3(Y{)}_K)\forall j,$$

$$\begin{array}[c]{ccccccccc}&&D&\xrightarrow{i}&M&\xleftarrow{j}&E&&\\ &&&&&&&&\\ &{\scriptstyle p}\swarrow&&{\scriptstyle\pi_{M}}\swarrow&\downarrow&\ \ \ \searrow{\scriptstyle\rho_{M}}&&\ \ \searrow{\scriptstyle q}&\\ &&&&&&&&\\ X&\hookrightarrow&Q&\xleftarrow{\pi}&F&\xrightarrow{\rho}&G&\hookleftarrow&Y\\ \end{array}$$

$$A_{hom}^i(Q)=A_{hom}^i(G)=0\forall i.$$

$$\begin{array}{rl}{i}_{\ast }{p}^{\ast }:A_{hom}^i(X)& \stackrel{\cong }{\to }{A}_{hom}^{i+1}(M),\\ j_{\ast }{q}^{\ast }:A_{hom}^i(Y)& \stackrel{\cong }{\to }{A}_{hom}^{i+1}(M).\end{array}$$

$$\begin{array}{rl}& A_{hom}^i(X)\stackrel{i_{\ast }{p}^{\ast }}{\to }{A}_{hom}^{i+1}(M)\stackrel{-p_{\ast }{i}^{\ast }}{\to }{A}_{hom}^i(X),\\ & A_{hom}^i(Y)\stackrel{j_{\ast }{q}^{\ast }}{\to }{A}_{hom}^{i+1}(M)\stackrel{-q_{\ast }{j}^{\ast }}{\to }{A}_{hom}^i(Y),\\ & A_{hom}^{i+1}(M)\stackrel{-p_{\ast }{i}^{\ast }}{\to }{A}_{hom}^i(X)\stackrel{i_{\ast }{p}^{\ast }}{\to }{A}_{hom}^{i+1}(M),\\ & A_{hom}^{i+1}(M)\stackrel{-q_{\ast }{j}^{\ast }}{\to }{A}_{hom}^i(Y)\stackrel{j_{\ast }{q}^{\ast }}{\to }{A}_{hom}^{i+1}(M),\end{array}$$

$$\Gamma :={\Gamma }_q\circ {}^t{\Gamma }_j\circ {\Gamma }_i\circ {}^t{\Gamma }_{p}in{A}^3(X\times Y).$$

$$\begin{array}{rl}& {\Gamma }^{\ast }{\Gamma }_{\ast }=id:A_{hom}^i(X)\to A_{hom}^i(X),\\ & {\Gamma }_{\ast }{\Gamma }^{\ast }=id:A_{hom}^i(Y)\to A_{hom}^i(Y)\end{array}$$

$${\Gamma }_{\ast }:A_{hom}^i(X)\to A_{hom}^i(Y)\forall i.$$

$${\Gamma }_{\ast }:A_{hom}^i(X_K)\to A_{hom}^i(Y_K)\forall i.$$

Full text

1. Introduction

Let $\hbox{Var}(k)$ denote the category of algebraic varieties over a field $k$. The Grothendieck ring $K_{0}(\hbox{Var}(k))$ encodes fundamental properties of the birational geometry of varieties. The intricacy of the ring $K_{0}(\hbox{Var}(k))$ is highlighted by the result of Borisov [2], showing that the class of the affine line $[\mathbb{A}^{1}]$ is a zero–divisor in $K_{0}(\hbox{Var}(k))$. Inspired by [2], Ito–Miura–Okawa–Ueda [6] exhibit a pair of Calabi–Yau threefolds $X,Y$ that are not stably birational (and so $[X]\not=[Y]$ in the Grothendieck ring), but

[TABLE]

(i.e., $X$ and $Y$ are “L-equivalent”, a notion studied in [8]).

As shown by Kuznetsov [7], the threefolds $X,Y$ of [6] are derived equivalent. According to a conjecture of Orlov [10, Conjecture 1], derived equivalent smooth projective varieties should have isomorphic Chow motives. The aim of this tiny note is to check that such is indeed the case for the threefolds $X,Y$:

Theorem **** (=theorem 3.1).

Let $X,Y$ be the two Calabi–Yau threefolds of [6]. Then

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An immediate corollary is that if $k$ is a finite field, then $X$ and $Y$ share the same zeta function (corollary 4.1).

**Conventions **.

In this note, the word variety will refer to a reduced irreducible scheme of finite type over a field $k$. For a smooth variety $X$, we will denote by $A^{j}(X)$ the Chow group of codimension $j$ cycles on $X$ with $\mathbb{Q}$-coefficients.

The notation $A^{j}_{hom}(X)$ will be used to indicate the subgroups of homologically trivial cycles. For a morphism between smooth varieties $f\colon X\to Y$, we will write $\Gamma_{f}\in A^{\ast}(X\times Y)$ for the graph of $f$, and ${}^{t}\Gamma_{f}\in A^{\ast}(Y\times X)$ for the transpose correspondence.

The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [12], [9]) will be denoted $\mathcal{M}_{\rm rat}$.

2. The Calabi–Yau threefolds

Theorem 2.1** (Ito–Miura–Okawa–Ueda [6]).**

Let $k$ be an algebraically closed field of characteristic [math]. There exist two Calabi–Yau threefolds $X,Y$ over $k$ such that

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but

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Theorem 2.2** (Kuznetsov [7]).**

Let $k$ be any field. The threefolds $X,Y$ over $k$ constructed as in [6] are derived equivalent: there is an isomorphism between the bounded derived categories of coherent sheaves

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In particular, if $k=\mathbb{C}$ then there is an isomorphism of polarized Hodge structures

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Proof.

The derived equivalence is [7, Theorem 5]. The isomorphism of Hodge structures is a corollary of the derived equivalence, in view of [11, Proposition 2.1 and Remark 2.3]. ∎

Remark 2.3**.**

The construction of the threefolds $X,Y$ in [6] works over any field $k$. However, the proof that $[X]\not=[Y]$ uses the MRC fibration and is (a priori) restricted to characteristic [math]. The argument of [7], on the other hand, has no characteristic [math] assumption.

3. Main result

Theorem 3.1**.**

Let $k$ be any field, and let $X,Y$ be the two Calabi–Yau threefolds over $k$ constructed as in [6]. Then

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Proof.

First, to simplify matters, let us slightly cut down the motives of $X$ and $Y$. It is known [6] that $X$ and $Y$ have Picard number $1$. A routine argument gives a decomposition of the Chow motives

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where $\mathds{1}$ is the motive of the point $\hbox{Spec}(k)$. (The gist of this “routine argument” is as follows: let $H\in A^{1}(X)$ be a hyperplane section. Then

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defines an orthogonal set of projectors lifting the Künneth components, for appropriate $c_{i}\in\mathbb{Q}$. One can then define $\pi^{3}_{X}=\Delta_{X}-\sum_{i}\pi^{2i}_{X}\in A^{3}(X\times X)$, and $h^{j}(X)=(X,\pi^{i}_{X},0)\in\mathcal{M}_{\rm rat}$, and ditto for $Y$.)

To prove the theorem, it will thus suffice to prove an isomorphism of motives

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We observe that the above decomposition (plus the fact that $H^{\ast}(h^{3}(X))=H^{3}(X)$ is odd–dimensional) implies equality

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and similarly for $Y$.

The rest of the proof will consist in finding a correspondence $\Gamma\in A^{3}(X\times Y)$ inducing isomorphisms

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for all field extensions $K\supset k$. By the above observation, this means that $\Gamma$ induces isomorphisms

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which (as is well-known, cf. for instance [5, Lemma 1.1]) ensures that $\Gamma$ induces the required isomorphism of Chow motives (1).

To find the correspondence $\Gamma$, we need look no further than the construction of the threefolds $X,Y$. As explained in [6] and [7], the threefolds $X,Y$ are related via a diagram

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Here $Q$ is a smooth $5$-dimensional quadric, and $G$ is a smooth intersection $G=\hbox{Gr}(2,V)\cap\mathbb{P}(W)$ of a Grassmannian and a linear subspace. The morphisms $\pi$ and $\rho$ are $\mathbb{P}^{1}$-fibrations. The morphisms $\pi_{M}$ and $\rho_{M}$ are the blow-ups with center the threefold $X$, resp. the threefold $Y$. The varieties $D,E$ are the exceptional divisors of the blow-ups.

Lemma 3.2**.**

Let $Q$ and $G$ be as above. We have

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Proof.

It is well-known that a $5$-dimensional quadric $Q$ has trivial Chow groups. (Indeed, [3, Corollary 2.3] gives that $A^{i}_{hom}(Q)=0$ for $i\geq 3$. The Bloch–Srinivas argument [1], combined with the fact that $H^{3}(Q)=0$, then implies that $A^{2}_{hom}(Q)=0$.)

As $\pi\colon F\to Q$ is a $\mathbb{P}^{1}$-fibration, it follows that the variety $F$ has trivial Chow groups. But $\rho\colon F\to G$ is a $\mathbb{P}^{1}$-fibration, and so $G$ also has trivial Chow groups. ∎

The blow-up formula, combined with lemma 3.2, gives isomorphisms

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What’s more, the inverse isomorphisms are induced by a correspondence: the compositions

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are all equal to the identity [13, Theorem 5.3].

This suggests how to find a correspondence $\Gamma$ doing the job. Let us define

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Then we have (by the above) that

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for all $i$, and so there are isomorphisms

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Given a field extension $K\supset k$, the threefolds $X_{K},Y_{K}$ are related via a blow-up diagram as above, and so the same reasoning as above shows that there are isomorphisms

[TABLE]

We have now established that $\Gamma$ verifies (2), which clinches the proof.

∎

4. A corollary

Corollary 4.1**.**

Let $k$ be a finite field, and let $X,Y$ be the Calabi–Yau threefolds over $k$ constructed as in [6]. Then $X$ and $Y$ have the same zeta function.

Proof.

The zeta function can be expressed (via the Lefschetz fixed point theorem) in terms of the action of Frobenius on $\ell$-adic étale cohomology, hence depends only on the motive. ∎

Remark 4.2**.**

Corollary 4.1 can also be deduced from [4], where it is proven that derived equivalent varieties of dimension $3$ have the same zeta function. The above proof (avoiding recourse to [7] and [4]) is more straightforward.

**Acknowledgements **.

This note was written at the Schiltigheim Math Research Institute. Thanks to the dedicated staff, who provide excellent working conditions.