A remark on the Chow ring of K\"uchle fourfolds of type $d3$

TL;DR

This paper proves that K"uchle fourfolds of type d3 possess a multiplicative Chow-K"unneth decomposition, leading to new insights into their Chow ring structure.

Contribution

It establishes the existence of a multiplicative Chow-K"unneth decomposition for K"uchle fourfolds of type d3, a novel result in the study of their algebraic cycles.

Findings

Existence of multiplicative Chow-K"unneth decomposition for these fourfolds

Consequences for the structure of their Chow ring

Implications for algebraic cycle theory on K"uchle fourfolds

Abstract

We prove that K\"uchle fourfolds $X$ of type d3 have a multiplicative Chow-K\"unneth decomposition. We present some consequences for the Chow ring of $X$.

Equations

$$({\wedge }^2{\mathcal{U}}^{\vee }{)}^{\oplus 2}\oplus {\mathcal{O}}_G(1)\to G:=\mathrm{Gr}(5,V_{10}),$$

$$\begin{array}[c]{ccccccccccccc}&&&&&&1&&&&&&\\ &&&&&0&&0&&&&&\\ &&&&0&&5&&0&&&&\\ &&&0&&0&&0&&0&&&\\ &&0&&1&&26&&1&&0&&\\ &&&0&&0&&0&&0&&&\\ &&&&0&&5&&0&&&&\\ &&&&&0&&0&&&&&\\ &&&&&&1&&&&&&\\ \end{array}$$

$$K\ddot{u}chle\; fourfolds\; of\; type\; d3\leftrightsquigarrow K3\; surfaces$$

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

$${\Delta }_X={\pi }_X^0+{\pi }_X^1+\cdots +{\pi }_X^{2n}in{A}^n(X\times X),$$

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

$${\pi }_X^i={}^t{\pi }_X^{2n-i}in{A}^n(X\times X)\forall i.$$

$$A^i(X{)}_{(j)}\stackrel{??}{=}0forj<0,A^i(X{)}_{(0)}\cap A_{num}^i(X)\stackrel{??}{=}0.$$

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

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

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

$$\hbox{Im}\Bigl{(}A^{i}(X)_{(j)}\otimes A^{i^{\prime}}(X)_{(j^{\prime})}\xrightarrow{\cdot}A^{i+i^{\prime}}(X)\Bigr{)}\ \subseteq\ A^{i+i^{\prime}}(X)_{(j+j^{\prime})}\ .$$

$$\pi^{i}_{M\times N}:=\sum_{k+\ell=i}\pi^{k}_{M}\times\pi^{\ell}_{N}\ \ \ \in A^{\dim M+\dim N}\bigl{(}(M\times N)\times(M\times N)\bigr{)}$$

$$\bar{B}:=\mathbb{P}H^{0}\bigl{(}(\mathbb{P}^{1})^{4},\mathcal{O}(1,1,1,1)^{\oplus 2}\bigr{)}$$

$$\bar{\mathcal{S}}\to \bar{B}.$$

$$\Gamma {|}_{S_b}=0in{H}^{2i}(S_b)\forall b\in B.$$

$$\Gamma {|}_{S_b}=0in{A}^i(S_b)\forall b\in B.$$

$${\pi }_{\mathcal{S}}^0:=\frac{1}{24}(p_1{)}^{\ast }{c}_2(T_{\mathcal{S}/B}),{\pi }_{\mathcal{S}}^4:=\frac{1}{24}(p_2{)}^{\ast }{c}_2(T_{\mathcal{S}/B}),{\pi }_{\mathcal{S}}^2:={\Delta }_{\mathcal{S}}-{\pi }_{\mathcal{S}}^0-{\pi }_{\mathcal{S}}^4\in A^4(\mathcal{S}{\times }_B\mathcal{S})$$

$$({\pi }_{S_b}^2{)}_{\ast }{c}_2({\mathcal{N}}_{S_b/({\mathbb{P}}^1{)}^4})\stackrel{??}{=}0in{A}^2(S_b).$$

$$(\pi^{2}_{S_{b}})_{\ast}c_{2}(\mathcal{N}_{S_{b}/(\mathbb{P}^{1})^{4}})=\Bigl{(}(\pi^{2}_{\mathcal{S}})_{\ast}c_{2}(\mathcal{N}_{\mathcal{S}/((\mathbb{P}^{1})^{4}\times B)})\Bigr{)}|_{S_{b}}\ \ \ \hbox{in}\ A^{2}(S_{b})\$$

$$({\pi }_{S_b\times ({\mathbb{P}}^1{)}^4}^{\ell }{)}_{\ast }({\Gamma }_{{\iota }_b})\stackrel{??}{=}0in{A}^4(S_b\times ({\mathbb{P}}^1{)}^4)\forall \ell \ne 8,\forall b\in B.$$

$$(\pi^{\ell}_{S_{b}\times(\mathbb{P}^{1})^{4}})_{\ast}(\Gamma_{\iota_{b}})=\Bigl{(}(\sum_{j+k=\ell}\pi^{j}_{\mathcal{S}}\times\pi^{k}_{(\mathbb{P}^{1})^{4}})_{\ast}(\Gamma_{\iota})\Bigr{)}|_{S_{b}\times(\mathbb{P}^{1})^{4}}\ \ \ \hbox{in}\ A^{4}(S_{b}\times(\mathbb{P}^{1})^{4})\ .$$

$$\begin{array}[c]{ccc}A^{4}(\mathcal{S}\times(\mathbb{P}^{1})^{4})&\to&A^{4}(S_{b}\times(\mathbb{P}^{1})^{4})\\ &&\\ \ \ \ \downarrow{\cong}&&\ \ \ \downarrow{\cong}\\ &&\\ \bigoplus A^{\ast}(\mathcal{S})&\to&\bigoplus A^{\ast}(S_{b})\\ \end{array}$$

$$\begin{array}[c]{ccc}\bar{\mathcal{S}}&\xrightarrow{\pi}&(\mathbb{P}^{1})^{4}\\ \ \ \ \ \downarrow{\scriptstyle\phi}&&\\ \ \ \bar{B}&&\\ \end{array}$$

$$\bar{\Gamma }=\sum _{\ell =0}^2{\pi }^{\ast }(a_{\ell })\cdot {\xi }^{\ell }in{A}^2(\bar{\mathcal{S}}),$$

$${\varphi }^{\ast }(h)=\alpha \xi +{\pi }^{\ast }(h_1)in{A}^1(\bar{\mathcal{S}}),$$

$$\xi={1\over\alpha}\bigl{(}\phi^{\ast}(h)-\pi^{\ast}(h_{1})\bigr{)}\ \ \ \hbox{in}\ A^{1}(\bar{\mathcal{S}})\ .$$

$$\bar{\Gamma}|_{S_{b}}=a_{0}^{\prime}|_{S_{b}}\ \ \ \hbox{in}\ A^{2}(S_{b})\$$

$$A^2(({\mathbb{P}}^1{)}^4)=\underset{i+j+k+\ell =2}{⨁}{A}^i({\mathbb{P}}^1)\otimes A^j({\mathbb{P}}^1)\otimes A^k({\mathbb{P}}^1)\otimes A^{\ell }({\mathbb{P}}^1)$$

$$\bar{\Gamma }{|}_{S_b}=a_0^{\prime }{|}_{S_b}\in \mathbb{Q}{\mathfrak{o}}_{S_b}\subset A^2(S_b),$$

Full text

1. Introduction

Küchle [5] has classified Fano fourfolds that are obtained as sections of globally generated homogeneous vector bundles on Grassmannians. In Küchle’s list, a Fano fourfold of type d3 is defined as a smooth $4$–dimensional section of the vector bundle

[TABLE]

where $\mathcal{U}$ denotes the tautological bundle on the Grassmannian $G$ (of $5$–dimensional subspaces of a fixed $10$–dimensional complex vector space).

Let $X$ be a Küchle fourfold of type d3. The Hodge diamond of $X$ is

[TABLE]

This Hodge diamond looks like the one of a $K3$ surface. Interestingly, the relation

[TABLE]

goes further than a similarity of Hodge numbers: Kuznetsov [6, Corollary 3.6] has shown that Küchle fourfolds $X$ of type d3 are related to $K3$ surfaces on the level of derived categories (in the sense that the derived category of $X$ admits a semi–orthogonal decomposition, and the interesting part of this decomposition is isomorphic to the derived category of a $K3$ surface).

The main result of the present note is that Küchle fourfolds of type d3 behave like $K3$ surfaces from a Chow–theoretic point of view:

Theorem **** (=theorem 3.1).

Let $X$ be a Küchle fourfold of type d3. Then $X$ has a multiplicative Chow–Künneth decomposition (in the sense of [13]).

This is very easily proven, provided one uses Kuznetsov’s alternative description [6] of Küchle fourfolds of type d3. Theorem 3.1 has interesting consequences for the Chow ring $A^{\ast}(X)_{\mathbb{Q}}$:

Corollary **** (=corollary 4.1).

Let $X$ be a Küchle fourfold of type d3. Let $R^{3}(X)\subset A^{3}(X)_{\mathbb{Q}}$ be the subgroup generated by the Chern class $c_{3}(T_{X})$ and intersections $A^{1}(X)\cdot A^{2}(X)$ of divisors with $2$–cycles. The cycle class map induces an injection

[TABLE]

This is reminiscent of the famous result of Beauville–Voisin describing the Chow ring of a $K3$ surface [2]. More generally, there is a similar injectivity result for the Chow ring of certain self–products $X^{m}$ (corollary 4.1).

There are two other families of fourfolds on Küchle’s list which have a Hodge diamond of $K3$ type: the families of type c5 and c7. There is also the complete intersection eightfold in $\operatorname{Gr}(2,V_{8})$ of [3, Proposition 5.2] which has a Hodge diamond of $K3$ type. It would be interesting to establish a multiplicative Chow–Künneth decomposition for those varieties as well; I hope to return to this in the future.

**Conventions **.

In this note, the word variety will refer to a reduced irreducible scheme of finite type over $\mathbb{C}$. 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 $f$. The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [12], [10]) will be denoted $\mathcal{M}_{\rm rat}$.

We will write $H^{\ast}(X):=H^{\ast}(X,\mathbb{Q})$ for singular cohomology with $\mathbb{Q}$–coefficients.

2. Multiplicative Chow–Künneth decomposition

Definition 2.1** (Murre [9]).**

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

[TABLE]

such that the $\pi^{i}_{X}$ are mutually orthogonal idempotents and $(\pi^{i}_{X})_{\ast}H^{\ast}(X)=H^{i}(X)$. Given a CK decomposition for $X$, we set

[TABLE]

The CK decomposition is said to be self-dual if

[TABLE]

(Here ${}^{t}\pi$ denotes the transpose of a cycle $\pi$.)

(NB: “CK decomposition” is short-hand for “Chow–Künneth decomposition”.)

Remark 2.2**.**

The existence of a Chow–Künneth decomposition for any smooth projective variety is part of Murre’s conjectures [9], [10]. It is expected that for any $X$ with a CK decomposition, one has

[TABLE]

These are Murre’s conjectures B and D, respectively.

Definition 2.3** (Definition 8.1 in [13]).**

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

[TABLE]

A CK decomposition $\{\pi^{i}_{X}\}$ of $X$ is multiplicative if it satisfies

[TABLE]

In that case,

[TABLE]

defines a bigraded ring structure on the Chow ring ; that is, the intersection product has the property that

[TABLE]

(For brevity, we will write MCK decomposition for “multiplicative Chow–Künneth decomposition”.)

Remark 2.4**.**

The property of having an MCK decomposition is severely restrictive, and is closely related to Beauville’s “(weak) splitting property” [1]. For more ample discussion, and examples of varieties admitting a MCK decomposition, we refer to [13, Chapter 8], as well as [15], [14], [4], [8].

There are the following useful general results:

Proposition 2.5** (Shen–Vial [13]).**

Let $M,N$ be smooth projective varieties that have an MCK decomposition. Then the product $M\times N$ has an MCK decomposition.

Proof.

This is [13, Theorem 8.6], which shows more precisely that the product CK decomposition

[TABLE]

is multiplicative. ∎

Proposition 2.6** (Shen–Vial [14]).**

Let $M$ be a smooth projective variety, and let $f\colon\widetilde{M}\to M$ be the blow–up with center a smooth closed subvariety $N\subset M$. Assume that

1. (1)

$M$* and $N$ have a self–dual MCK decomposition;* 2. (2)

the Chern classes of the normal bundle $\mathcal{N}_{N/M}$ are in $A^{\ast}_{(0)}(N)$; 3. (3)

the graph of the inclusion morphism $N\to M$ is in $A^{\ast}_{(0)}(N\times M)$; 4. (4)

the Chern classes $c_{j}(T_{M})$ are in $A^{\ast}_{(0)}(M)$.

Then $\widetilde{M}$ has a self–dual MCK decomposition, the Chern classes $c_{j}(T_{\widetilde{M}})$ are in $A^{\ast}_{(0)}(\widetilde{M})$, and the graph $\Gamma_{f}$ is in $A^{\ast}_{(0)}(\widetilde{M}\times M)$.

Proof.

This is [14, Proposition 2.4]. ∎

3. Main result

Theorem 3.1**.**

Let $X$ be a Küchle fourfold of type d3. Then $X$ has a self–dual MCK decomposition. Moreover, the Chern classes $c_{j}(T_{X})$ are in $A^{\ast}_{(0)}(X)$.

Proof.

The argument relies on the following alternative description of $X$ (this is [6, Corollary 3.5]):

Theorem 3.2** (Kuznetsov [6]).**

Let $X$ be a Küchle fourfold of type d3. Then $X$ is isomorphic to the blow–up of $(\mathbb{P}^{1})^{4}$ in a $K3$ surface $S$ obtained as smooth intersection of two divisors of multidegree $(1,1,1,1)$.

(The argument of [6, Corollary 3.5] also shows that conversely, any blow–up of $(\mathbb{P}^{1})^{4}$ in a $K3$ surface $S$ of this type is a Küchle fourfold of type d3. We will not need this.)

We wish to apply the general result proposition 2.6 to $M=(\mathbb{P}^{1})^{4}$ and $N=S$. All we need to do is to check that the assumptions of proposition 2.6 are met with. Assumption (1) is OK, since varieties with trivial Chow groups and $K3$ surfaces have a self–dual MCK decomposition [13, Example 8.17]. Assumption (4) is trivially satisfied, since $A^{\ast}_{hom}((\mathbb{P}^{1})^{4})=0$ and so $A^{\ast}((\mathbb{P}^{1})^{4})=A^{\ast}_{(0)}((\mathbb{P}^{1})^{4})$.

To check assumptions (2) and (3), we consider things family–wise. That is, we write

[TABLE]

and we consider the universal complete intersection

[TABLE]

We write $B\subset\bar{B}$ for the Zariski open parametrizing smooth dimensionally transversal intersection, and $\mathcal{S}\to B$ for the base change (so the fibres $S_{b}$ of $\mathcal{S}\to B$ are exactly the $K3$ surfaces considered in theorem 3.2). We make the following claim:

Claim 3.3**.**

Let $\Gamma\in A^{i}(\mathcal{S})$ be such that

[TABLE]

Then also

[TABLE]

We argue that the claim implies that assumptions (2) and (3) are met with (and so proposition 2.6 can be applied to prove theorem 3.1). Indeed, let $p_{j}\colon\mathcal{S}\times_{B}\mathcal{S}\to\mathcal{S}$, $j=1,2$, denote the two projections. We observe that

[TABLE]

defines a “relative MCK decomposition”, in the sense that for any $b\in B$, the restriction $\pi^{i}_{\mathcal{S}}|_{S_{b}\times S_{b}}$ defines an MCK decomposition (which we denote $\pi^{i}_{S_{b}}$) for $S_{b}$.

To check that assumption (2) is satisfied, we need to check that for any $b\in B$ there is vanishing

[TABLE]

But we can write

[TABLE]

(for the formalism of relative correspondences, cf. [10, Chapter 8]), and besides we know that $(\pi^{2}_{S_{b}})_{\ast}c_{2}(\mathcal{N}_{S_{b}/(\mathbb{P}^{1})^{4}})$ is homologically trivial ($\pi^{2}_{S_{b}}$ acts as zero on $H^{4}(S_{b})$). Thus, the claim implies the necessary vanishing (1).

Assumption (3) is checked similarly. Let $\iota_{b}\colon S_{b}\to(\mathbb{P}^{1})^{4}$ and $\iota\colon\mathcal{S}\to(\mathbb{P}^{1})^{4}\times B$ denote the inclusion morphisms. To check assumption (3), we need to convince ourselves of the vanishing

[TABLE]

The fact that $\ell\not=8$ implies that $(\pi^{\ell}_{S_{b}\times(\mathbb{P}^{1})^{4}})_{\ast}(\Gamma_{\iota_{b}})$ is homologically trivial. Furthermore, we can write the cycle we are interested in as the restriction of a universal cycle:

[TABLE]

For any $b\in B$, there is a commutative diagram

[TABLE]

where horizontal arrows are restriction to a fibre, and where vertical arrows are isomorphisms by repeated application of the projective bundle formula for Chow groups. Claim 3.3 applied to the lower horizontal arrow shows the vanishing (2), and so assumption (3) holds.

It is left to prove the claim. Since $A^{i}_{hom}(S_{b})=0$ for $i\leq 1$, the only non–trivial case is $i=2$. Given $\Gamma\in A^{2}(\mathcal{S})$ as in the claim, let $\bar{\Gamma}\in A^{2}(\bar{\mathcal{S}})$ be a cycle restricting to $\Gamma$. We consider the two projections

[TABLE]

Since any point of $(\mathbb{P}^{1})^{4}$ imposes exactly one condition on $\bar{B}$, the morphism $\pi$ has the structure of a projective bundle. As such, any $\bar{\Gamma}\in A^{2}(\bar{\mathcal{S}})$ can be written

[TABLE]

where $a_{\ell}\in A^{2-\ell}((\mathbb{P}^{1})^{4})$ and $\xi\in A^{1}(\bar{\mathcal{S}})$ is the relative hyperplane class.

Let $h:=c_{1}(\mathcal{O}_{\bar{B}}(1))\in A^{1}(\bar{B})$. There is a relation

[TABLE]

where $\alpha\in\mathbb{Q}$ and $h_{1}\in A^{1}((\mathbb{P}^{1})^{4})$. As in [11, Proof of Lemma 1.1], one checks that $\alpha\not=0$ (if $\alpha$ were [math], we would have $\phi^{\ast}(h^{\dim\bar{B}})=\pi^{\ast}(h_{1}^{\dim\bar{B}})$, which is absurd since $\dim\bar{B}>4$ and so the right–hand side is [math]). Hence, there is a relation

[TABLE]

For any $b\in B$, the restriction of $\phi^{\ast}(h)$ to the fibre $S_{b}$ vanishes, and so it follows that

[TABLE]

for some $a_{0}^{\prime}\in A^{2}((\mathbb{P}^{1})^{4})$. But

[TABLE]

is generated by intersections of divisors, and so Beauville–Voisin’s result [2] implies that

[TABLE]

where ${\mathfrak{o}}_{S_{b}}\in A^{2}(S_{b})$ is the distinguished [math]–cycle of [2]. This proves the claim. ∎

4. A consequence

Corollary 4.1**.**

Let $X$ be a Küchle fourfold of type d3, and let $m\in\mathbb{N}$. Let $R^{\ast}(X^{m})\subset A^{\ast}(X^{m})$ be the $\mathbb{Q}$–subalgebra

[TABLE]

(Here $p_{i}\colon X^{m}\to X$ and $p_{ij}\colon X^{m}\to X^{2}$ denote projection to the $i$th factor, resp. to the $i$th and $j$th factor.)

The cycle class map induces injections

[TABLE]

in the following cases:

1. (1)

$m=1$* and $j$ arbitrary;* 2. (2)

$m=2$* and $j\geq 5$;* 3. (3)

$m=3$* and $j\geq 9$.*

Proof.

Theorem 3.1, in combination with proposition 2.5, ensures that $X^{m}$ has an MCK decomposition, and so $A^{\ast}(X^{m})$ has the structure of a bigraded ring under the intersection product. The corollary is now implied by the combination of the two following claims:

Claim 4.2**.**

There is inclusion

[TABLE]

Claim 4.3**.**

The cycle class map induces injections

[TABLE]

provided $m=1$, or $m=2$ and $j\geq 5$, or $m=3$ and $j\geq 9$.

To prove claim 4.2, we note that $A^{k}_{hom}(X)=0$ for $k\leq 2$, which readily implies the equality $A^{k}(X)=A^{k}_{(0)}(X)$ for $k\leq 2$. The fact that $c_{3}(T_{X})$ is in $A^{3}_{(0)}(X)$ is part of theorem 3.1. The fact that $\Delta_{X}\in A^{4}_{(0)}(X\times X)$ is a general fact for any $X$ with a self–dual MCK decomposition [14, Lemma 1.4]. Since the projections $p_{i}$ and $p_{ij}$ are pure of grade [math] [14, Corollary 1.6], and $A^{\ast}_{(0)}(X^{m})$ is a ring under the intersection product, this proves claim 4.2.

To prove claim 4.3, we observe that Manin’s blow–up formula [12, Theorem 2.8] gives an isomorphism of motives

[TABLE]

Moreover, in view of proposition 2.6 (cf. also [14, Proposition 2.4]), the correspondence inducing this isomorphism is of pure grade [math].

In particular, for any $m\in\mathbb{N}$ we have isomorphisms of Chow groups

[TABLE]

and this isomorphism respects the $A^{\ast}_{(0)}()$ parts. Claim 4.3 now follows from the fact that for any surface $S$ with an MCK decomposition, and any $m\in\mathbb{N}$, the cycle class map induces injections

[TABLE]

(this is noted in [15, Introduction], cf. also [7, Proof of Lemma 2.20]).

∎

**Acknowledgements **.

Thanks to my colleagues Kai–kun and Len–kun at the Schiltigheim Centre for Pure Mathematics.