On the Chow ring of certain hypersurfaces in a Grassmannian

TL;DR

This paper investigates the Chow ring structure of Plücker hyperplane sections of a Grassmannian, showing the main Chow group is generated by specific sub-Grassmannians and a subring injects into cohomology.

Contribution

It establishes the generators of the Chow group for these hypersurfaces and proves an injection of a subring into cohomology, inspired by analogies with cubic fourfolds.

Findings

The main Chow group is generated by Grassmannians of type Gr(3,W_6) contained in X.

A certain subring of the Chow ring injects into cohomology.

Provides new insights into the algebraic cycles of these hypersurfaces.

Abstract

This small note is about Pl\"ucker hyperplane sections $X$ of the Grassmannian $\operatorname{Gr}(3,V_{10})$. Inspired by the analogy with cubic fourfolds, we prove that the only non-trivial Chow group of $X$ is generated by Grassmannians of type $\operatorname{Gr}(3,W_{6})$ contained in $X$. We also prove that a certain subring of the Chow ring of $X$ (containing all intersections of positive-codimensional subvarieties) injects into cohomology.

Equations

$$X\in |\mathcal{L}|$$

$$\begin{array}[c]{ccccccccc}&&&&1&&&&\\ &&&&2&&&&\\ &&&&3&&&&\\ &&&&\ast&&&&\\ &&&&\vdots&&&&\\ &&&&\ast&&&&\\ 0&\dots\dots&0&1&30&1&0&\dots\dots&0\\ &&&&\ast&&&&\\ &&&&\vdots&&&&\\ &&&&\ast&&&&\\ &&&&3&&&&\\ &&&&2&&&&\\ &&&&1&&&&\\ \end{array}$$

$$Pl\ddot{u}cker\; hypersurfaces\; in\mathrm{Gr}(3,V_{10})\leftrightsquigarrow cubic\; fourfolds$$

$$A^{i}_{hom}(X)_{\mathbb{Q}}=0\ \ \ \forall\ i\not=11\$$

$$R^{11}(X)\hookrightarrow H^{22}(X,\mathbb{Q}).$$

$$A_{hom}^i(X)=0\forall i>11.$$

$$\hbox{Niveau}\bigl{(}A^{\ast}(X)\bigr{)}\leq 2\$$

$$A_{AJ}^i(X)=0\forall i\ne 11.$$

$$A_{hom}^i(X)=0\forall i\ne 11.$$

$$\mathcal{X}\to B$$

$$X_{\sigma }\subset \mathrm{Gr}(3,V_{10})\subset \mathbb{P}({\wedge }^3{V}_{10})$$

$$Y_{\sigma}:=\bigl{\{}W_{6}\in\operatorname{Gr}(6,V_{10})\ \hbox{such\ that}\ \sigma|_{W_{6}}=0\bigr{\}}\ \ \subset\ \operatorname{Gr}(6,V_{10})$$

$$(\Gamma {|}_{X_{\sigma }\times X_{\sigma }}{)}_{\ast }{H}^{11,9}(X_{\sigma })=0for\; very\; general\sigma \in B.$$

$$(\Gamma {|}_{X_{\sigma }\times X_{\sigma }}{)}_{\ast }{A}_{hom}^{11}(X_{\sigma })=0for\; all\sigma \in B.$$

$$\Gamma {|}_{X_{\sigma }\times X_{\sigma }}=\sum _{i=1}^s{V}_{\sigma }^i\times W_{\sigma }^{i}in{H}^{40}(X_{\sigma }\times X_{\sigma }).$$

$$\bigl{(}\Gamma-\delta)|_{X_{\sigma}\times X_{\sigma}}=0\ \ \ \hbox{in}\ H^{40}(X_{\sigma}\times X_{\sigma})\ ,\ \ \ \hbox{for\ very\ general\ }\sigma\in B\ .$$

$$R:=\Gamma -\delta \in A^{20}(\mathcal{X}{\times }_B\mathcal{X}).$$

$$I:=\bigl{\{}(\sigma,y)\in\bar{B}\times\widetilde{{M}\times{M}}\ |\ s|_{\mu(y)}=0\bigr{\}}\ .$$

$$f:\widetilde{\mathcal{X}{\times }_B\mathcal{X}}\to \mathcal{X}{\times }_B\mathcal{X},$$

$$\bar{R}{|}_{\widetilde{\mathcal{X}{\times }_B\mathcal{X}}}=f^{\ast }(R)in{A}^n(\widetilde{\mathcal{X}{\times }_B\mathcal{X}}).$$

$$\bar{R}|_{\widetilde{X_{\sigma}\times X_{\sigma}}}=\bigl{(}f^{\ast}(R)\bigr{)}|_{\widetilde{X_{\sigma}\times X_{\sigma}}}=(f_{\sigma})^{\ast}(R|_{{X_{\sigma}\times X_{\sigma}}})=0\ \ \ \hbox{in}\ H^{40}(\widetilde{X_{\sigma}\times X_{\sigma}})\ ,$$

$$f_{\sigma }:\widetilde{X_{\sigma }\times X_{\sigma }}\to X_{\sigma }\times X_{\sigma }$$

$$R{|}_{X_{\sigma }\times X_{\sigma }}=(f_{\sigma }{)}_{\ast }(\bar{R}{|}_{\widetilde{X_{\sigma }\times X_{\sigma }}})=\gamma {|}_{X_{\sigma }\times X_{\sigma }}in{A}^{20}(X_{\sigma }\times X_{\sigma })\forall \sigma \in B.$$

$$\bigl{(}(\iota\times\iota)^{\ast}(\gamma)\bigr{)}{}_{\ast}(a)=\iota^{\ast}\gamma_{\ast}\iota_{\ast}(a)\ \ \ \hbox{in}\ A^{11}_{hom}(X_{\sigma})\ .$$

$$\bigl{(}R|_{X_{\sigma}\times X_{\sigma}}\bigr{)}{}_{\ast}=0\colon\ \ A^{11}_{hom}(X_{\sigma})\ \to\ A^{11}_{hom}(X_{\sigma})\ \ \ \forall\sigma\in B\ .$$

$$\delta {|}_{X_{\sigma }\times X_{\sigma }}\in A^{20}(X_{\sigma }\times X_{\sigma })$$

$$\bigl{(}\Gamma|_{X_{\sigma}\times X_{\sigma}}\bigr{)}{}_{\ast}=\bigl{(}(R+\delta){}|_{X_{\sigma}\times X_{\sigma}}\bigr{)}{}_{\ast}=0\colon\ \ A^{11}_{hom}(X_{\sigma})\ \to\ A^{11}_{hom}(X_{\sigma})\ \ \ \forall\sigma\in B\ .$$

$$G_{\sigma}:=\Bigl{\{}(W_{3},W_{6})\ \in\operatorname{Gr}(3,V_{10})\times\operatorname{Gr}(6,V_{10})\ \big{|}\ W_{3}\subset W_{6}\ ,\ \sigma|_{W_{6}}=0\ \Bigr{\}}$$

$$\begin{array}[c]{ccc}G_{\sigma}&\xrightarrow{p_{\sigma}}&X_{\sigma}\\ \ \ \ \ \downarrow{\scriptstyle q_{\sigma}}&&\\ \ \ Y_{\sigma}\ .&&\\ \end{array}$$

$$\begin{array}[c]{ccc}\mathcal{G}&\xrightarrow{p}&\mathcal{X}\\ \ \ \ \ \downarrow{\scriptstyle q}&&\\ \ \ \mathcal{Y}\ .&&\\ \end{array}$$

Full text

1. Introduction

Let $\mathcal{L}$ be the Plücker polarization on the complex Grassmannian $\operatorname{Gr}(3,V_{10})$, and let

[TABLE]

be a smooth hypersurface in the linear system of $\mathcal{L}$. The Hodge diamond of the $20$-dimensional variety $X$ is

[TABLE]

(Here $\ast$ indicates some unspecified number, and all empty entries are [math]. The Hodge numbers of the vanishing cohomology can be found in [4, Theorem 1.1]; alternatively, they can be computed using [7, Theorem 1.1].)

This looks much like the Hodge diamond of a cubic fourfold. To further this analogy, Debarre and Voisin [4] have constructed, for a general such hypersurface $X$, a hyperkähler fourfold $Y$ that is associated (via an Abel–Jacobi isomorphism) to $X$. Just as in the famous Beauville–Donagi construction starting from a cubic fourfold [2], the hyperkähler fourfolds $Y$ form a $20$-dimensional family, deformation equivalent to the Hilbert square of a K3 surface. The analogy

[TABLE]

also exists on the level of derived categories [8, Section 4.4].

In this note we will be interested in the Chow ring $A^{\ast}(X)_{\mathbb{Q}}$ of the hypersurface $X$. Using her celebrated method of spread of algebraic cycles in families, Voisin [18, Theorem 2.4] (cf. also the proof of theorem 2.1 below) has already proven a form of the Bloch conjecture for $X$: one has vanishing

[TABLE]

(where $A^{i}_{hom}(X)_{\mathbb{Q}}$ is defined as the kernel of the cycle class map to singular cohomology). This is the analogue of the well-known fact that the only non-trivial Chow group of a cubic fourfold is the Chow group of $1$-cycles.

We complete Voisin’s result, by describing the only non-trivial Chow group of $X$:

Theorem **** (=theorem 2.1).

Let $\mathcal{L}$ be the Plücker polarization on $\operatorname{Gr}(3,V_{10})$. Let $X\in|\mathcal{L}|$ be a smooth hypersurface for which the associated hyperkähler fourfold $Y$ is smooth. Then $A^{11}_{hom}(X)_{\mathbb{Q}}$ is generated by Grassmannians $\operatorname{Gr}(3,W_{6})$ contained in $X$.

This is the analogue of the well-known fact that for a cubic fourfold $V\subset\mathbb{P}^{5}(\mathbb{C})$, the Chow group $A^{3}(V)$ is generated by lines [12]. Theorem 2.1 is readily proven using the spread method of [17], [18], [19]; as such, theorem 2.1 could naturally have been included in [18].

The second result of this note concerns the ring structure of the Chow ring of $X$, given by the intersection product:

Theorem **** (=theorem 3.1).

Let $\mathcal{L}$ be the Plücker polarization on $\operatorname{Gr}(3,V_{10})$, and let $X\in|\mathcal{L}|$ be a smooth hypersurface. Let $R^{11}(X)\subset A^{11}(X)_{\mathbb{Q}}$ be the subgroup containing intersections of two cycles of positive codimension, the Chern class $c_{11}(T_{X})$ and the image of the restriction map $A^{11}(\operatorname{Gr}(3,V_{10}))_{\mathbb{Q}}\to A^{11}(X)_{\mathbb{Q}}$. The cycle class map induces an injection

[TABLE]

This is reminiscent of the famous result about the Chow ring of a K3 surface [2]. It is also an analogue of the fact that for a cubic fourfold $V$, the subgroup $A^{2}(V)_{\mathbb{Q}}\cdot A^{1}(V)_{\mathbb{Q}}\subset A^{3}(V)_{\mathbb{Q}}$ is one-dimensional. Theorem 3.1 suggests that the hypersurfaces $X$ might have a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial [14]. This seems difficult to establish, however (cf. remark 3.6).

**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 notations $A^{j}_{hom}(X)$, $A^{i}_{AJ}(X)$ will be used to indicate the subgroups of homologically trivial (resp. Abel–Jacobi 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.

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

2. Generators for $A^{11}$

Theorem 2.1**.**

Let $\mathcal{L}$ be the Plücker polarization on $\operatorname{Gr}(3,V_{10})$. Let $X\in|\mathcal{L}|$ be a smooth hypersurface for which there is an associated smooth hyperkähler fourfold $Y$. Then $A^{11}_{hom}(X)$ is generated by the classes of Grassmannians $\operatorname{Gr}(3,W_{6})\subset X$ (where $W_{6}\subset V_{10}$ is a six-dimensional vector space).

Proof.

As mentioned in the introduction, Voisin [18, Theorem 2.4] has proven that

[TABLE]

Using the Bloch–Srinivas “decomposition of the diagonal” method [3], [19, Chapter 3] (in the precise form of [9, Theorem 1.7]), this implies that

[TABLE]

in the language of [9], and also (using [9, Remark 1.8.1])

[TABLE]

But all intermediate Jacobians of $X$ are trivial (there is no odd-degree cohomology), and so

[TABLE]

That is, the $20$-dimensional variety $X$ motivically looks like a surface, and so in particular the Hodge conjecture is true for $X$ [9, Proposition 2.4].

Let

[TABLE]

denote the universal family of smooth hypersurfaces in the linear system $|\mathcal{L}|$. The base $B$ is the Zariski open in $\mathbb{P}(\wedge^{3}V_{10}^{\ast})$ parametrizing $3$-forms $\sigma$ such that the corresponding hyperplane section

[TABLE]

is smooth.

Let $B^{\prime}\subset B$ be the Zariski open such that the fibre $X_{\sigma}$ has an associated hyperkähler fourfold $Y_{\sigma}$, in the sense of [4]. That is, $B^{\prime}$ parametrizes $3$-forms $\sigma$ such that both $X_{\sigma}$ and

[TABLE]

are smooth of the expected dimension.

We rely on the spread result of Voisin’s, in the following form:

Theorem 2.2** (Voisin [18]).**

Let $\Gamma\in A^{20}(\mathcal{X}\times_{B}\mathcal{X})$ be a relative correspondence with the property that

[TABLE]

Then

[TABLE]

(For basics on the formalism of relative correspondences, cf. [10, Section 8.1].) Since theorem 2.2 is not stated precisely in this form in [18], we briefly indicate the proof:

Proof.

(of theorem 2.2) The assumption on $\Gamma$ (plus the shape of the Hodge diamond of $X_{\sigma}$, and the truth of the Hodge conjecture for $X_{\sigma}$, as shown above) implies that for the very general $\sigma\in B$ there exist closed subvarieties $V_{\sigma}^{i},W_{\sigma}^{i}\subset X_{\sigma}$ with $\dim V_{\sigma}^{i}+\dim W_{\sigma}^{i}=20$, and such that

[TABLE]

One has the Noether–Lefschetz property that $H^{20}(X_{\sigma},\mathbb{Q})_{van}\cap F^{10}=0$ for very general $X_{\sigma}$ (this is because the Picard number of the very general Debarre–Voisin hyperkähler fourfold is $1$). This implies that all the subvarieties $V_{\sigma}^{i},W_{\sigma}^{i}$ are obtained by restriction from subvarieties of $\operatorname{Gr}(3,V_{10})$, hence they exist universally. (Instead of evoking Noether–Lefschetz, one could also apply Voisin’s Hilbert scheme argument [17, Proposition 3.7] to obtain that the $V_{\sigma}^{i},W_{\sigma}^{i}$ exist universally). That is, there exist closed subvarieties $\mathcal{V}^{i},\mathcal{W}^{i}\subset\mathcal{X}$ with $\hbox{codim}\mathcal{V}^{i}+\hbox{codim}\mathcal{W}^{i}=20$, and a cycle $\delta$ supported on $\cup\mathcal{V}^{i}\times_{B}\mathcal{W}^{i}$, such that

[TABLE]

We now define a relative correspondence

[TABLE]

For brevity, from now on let us write $M:=\operatorname{Gr}(3,V_{10})$. Since $M$ has trivial Chow groups (this is true for all Grassmannians, and more generally for linear varieties, cf. [15, Theorem 3]), and the hypersurfaces $X_{\sigma}$ have non-zero primitive cohomology (indeed $h^{11,9}(X_{\sigma})=1$), we are in the set–up of [18]. As in loc. cit., we consider the blow-up $\widetilde{{M}\times{M}}$ of ${M}\times{M}$ along the diagonal, and the quotient morphism $\mu\colon\widetilde{{M}\times{M}}\to M^{[2]}$ to the Hilbert scheme of length $2$ subschemes. Let $\bar{B}:=\mathbb{P}H^{0}({M},\mathcal{L})$ and as in [18, Lemma 1.3], introduce the incidence variety

[TABLE]

Since $\mathcal{L}$ is very ample on ${M}$, $I$ has the structure of a projective bundle over $\widetilde{M\times M}$.

Next, let us consider

[TABLE]

the blow-up along the relative diagonal $\Delta_{\mathcal{X}}$. There is an open inclusion $\widetilde{\mathcal{X}\times_{B}\mathcal{X}}\subset I$. Hence, given our relative correspondence $R\in A^{n}(\mathcal{X}\times_{B}\mathcal{X})$ as above, there exists a (non-canonical) cycle $\bar{R}\in A^{n}(I)$ such that

[TABLE]

Hence, we have

[TABLE]

for $\sigma\in B$ very general, by assumption on $R$. (Here, as one might guess, the notation

[TABLE]

indicates the blow-up along the diagonal $\Delta_{X_{\sigma}}$.)

We now apply [18, Proposition 1.6] to the cycle $\bar{R}$. The result is that there exists a cycle $\gamma\in A^{20}(M\times M)$ such that there is a rational equivalence

[TABLE]

We know that the restriction of $\gamma$ acts as zero on $A^{11}_{hom}(X_{\sigma})$. (Indeed, let $\iota\colon X_{\sigma}\to M$ denote the inclusion, and let $a\in A^{11}_{hom}(X_{\sigma})$. With the aid of Lieberman’s lemma [16, Lemma 3.3], one finds that

[TABLE]

But $\iota_{\ast}(a)\in A^{12}_{hom}(M)=0$).

Thus, it follows that

[TABLE]

For any given $\sigma\in B$, one can construct the subvarieties $\mathcal{V}^{i},\mathcal{W}^{i}\subset\mathcal{X}$ in the above argument in such a way that they are in general position with respect to the fibre $X_{\sigma}$. This implies that the restriction

[TABLE]

is a completely decomposed cycle, i.e. a cycle supported on a union of subvarieties $V^{\sigma}_{j}\times W^{\sigma}_{j}\subset X_{\sigma}\times X_{\sigma}$ with $\hbox{codim}(V_{j}^{\sigma})+\hbox{codim}(W_{j}^{\sigma})=20$. But completely decomposed cycles do not act on $A^{\ast}_{hom}()$ [3], and so

[TABLE]

This ends the proof of theorem 2.2. ∎

Let us now pick up the thread of the proof of theorem 2.1. As in [4, Section 2], for any $3$-form $\sigma\in B^{\prime}$ let

[TABLE]

denote the incidence variety, with projections

[TABLE]

The fibres of $q_{\sigma}$ are $9$-dimensional Grassmannians $\operatorname{Gr}(3,W_{6})$.

Let $\mathcal{Y}\to B^{\prime}$ denote the universal family of Debarre–Voisin fourfolds (i.e., $\mathcal{Y}\subset\operatorname{Gr}(6,V_{10})\times B^{\prime}$ is the subvariety of pairs $(W_{6},\sigma)$ such that $\sigma|_{W_{6}}=0$), and let $\mathcal{G}\to B^{\prime}$ be the relative version of $G_{\sigma}$, with projections

[TABLE]

We will rely on an Abel–Jacobi type result from [4], concerning the vanishing cohomology defined as

[TABLE]

Lemma 2.3**.**

Let $\sigma\in B^{\prime}$ be very general. Then there is an isomorphism

[TABLE]

The inverse isomorphism is given by

[TABLE]

(Here $\mu\in\mathbb{Q}$ is some non-zero number independent of $\sigma$, and $g\in A^{1}(Y_{\sigma})$ is the Plücker polarization.)

Proof.

The first part (i.e. the fact that $(q_{\sigma})_{\ast}(p_{\sigma})^{\ast}$ is an isomorphism on the vanishing cohomology) is [4, Theorem 2.2 and Corollary 2.7]. For the second part, we observe that the dual map (with respect to cup product)

[TABLE]

is also an isomorphism. In particular, using hard Lefschetz, this means that the composition

[TABLE]

is non-zero (and actually an isomorphism). Hence, the assignment

[TABLE]

defines a polarization on $H^{2}(Y_{\sigma},\mathbb{Q})_{van}$. Here, $<\alpha,\beta>_{Y_{\sigma}}$ is the Beauville–Bogomolov form. However, as explained in [18, Proof of Lemma 2.2], for very general $\sigma$ the Hodge structure on $H^{2}(Y_{\sigma},\mathbb{Q})_{van}$ is simple, and admits a unique polarization up to a coefficient. That is, there exists a non-zero number $\mu\in\mathbb{Q}$ such that

[TABLE]

The Beauville–Bogomolov form being non-degenerate, this proves that

[TABLE]

Reasoning likewise starting from $H^{20}(X_{\sigma},\mathbb{Q})_{van}$ (now using the cup product instead of the Beauville–Bogomolov form), we find that the other composition is also the identity.

Finally, the fact that the constant $\mu$ is the same for all fibres $X_{\sigma}$ is because the map in cohomology $H^{2}(Y_{\sigma},\mathbb{Q})_{van}\to H^{20}(X_{\sigma},\mathbb{Q})_{van}$ is locally constant in the family. ∎

Let us define the relative correspondence

[TABLE]

where $\Gamma_{g^{2}}\in A^{6}(\mathcal{Y}\times_{B^{\prime}}\mathcal{Y})$ is the correspondence acting fibrewise as intersection with two Plücker hyperplanes. Lemma 2.3 implies that

[TABLE]

That is, the relative correspondence $\Gamma$ satisfies the assumption of theorem 2.2. Thanks to theorem 2.2, we thus conclude that

[TABLE]

Unraveling the definition of $\Gamma$, this means in particular that there is a surjection

[TABLE]

As we have seen, for any point $y\in Y_{\sigma}$ the fibre $(q_{\sigma})^{-1}(y)$ is a $9$-dimensional Grassmannian $\operatorname{Gr}(3,W_{6})$ such that the $3$-form $\sigma$ vanishes on $W_{6}$. Such a Grassmannian is contained in the hypersurface $X_{\sigma}$, and so

[TABLE]

The theorem is proven. ∎

Remark 2.4**.**

The above argument actually shows that

[TABLE]

is a non-zero multiple of the identity, for any $\sigma\in B^{\prime}$. This is very much reminiscent of cubic fourfolds and their Fano varieties of lines [1], [14]. Inspired by this analogy, it is tempting to ask the following: can one somehow prove that

[TABLE]

is the same as the subgroup of [math]-cycles supported on a uniruled divisor ?

3. An injectivity result

Theorem 3.1**.**

Let $\mathcal{L}$ be the Plücker polarization on $\operatorname{Gr}(3,V_{10})$, and let $X\in|\mathcal{L}|$ be a smooth hypersurface. Let $R^{11}(X)\subset A^{11}(X)_{\mathbb{Q}}$ be the subgroup containing intersections of two cycles of positive codimension, the Chern class $c_{11}(T_{X})$ and the image of the restriction map $A^{11}(\operatorname{Gr}(3,V_{10}))\to A^{11}(X)$. The cycle class map induces an injection

[TABLE]

In order to prove theorem 3.1, we first establish a “generalized Franchetta conjecture” type of statement (for more on the generalized Franchetta conjecture, cf. [11], [13], [6]):

Theorem 3.2**.**

Let $\mathcal{X}\to B$ denote the universal family of Plücker hyperplanes in $\operatorname{Gr}(3,V_{10})$ (as in section 2). Let $\Psi\in A^{11}(\mathcal{X})$ be such that

[TABLE]

Then

[TABLE]

Proof.

This is a two-step argument:

Claim 3.3**.**

There is equality

[TABLE]

Claim 3.4**.**

Restriction of the cycle class map induces an injection

[TABLE]

Clearly, the combination of these two claims proves theorem 3.2. To prove claim 3.3, let $\bar{B}:=\mathbb{P}H^{0}(\operatorname{Gr}(3,V_{10}),\mathcal{L})$ and let

[TABLE]

denote the universal hyperplane (including the singular hyperplanes). The morphism $\pi$ is a projective bundle, and so any $\Psi\in A^{11}(\bar{\mathcal{X}})$ can be written

[TABLE]

where $a_{\ell}\in A^{11-\ell}(\operatorname{Gr}(3,V_{10}))$ and $h:=c_{1}(\mathcal{O}_{\bar{B}}(1))\in A^{1}(\bar{B})$. For any $\sigma\in B$, the restriction of $\phi^{\ast}(h)$ to the fibre $X_{\sigma}$ vanishes, and so

[TABLE]

which establishes claim 3.3.

Let us prove claim 3.4. For any given $\sigma\in B$, let $\iota\colon X_{\sigma}\to\operatorname{Gr}(3,V_{10})$ denote the inclusion morphism. We know that

[TABLE]

equals multiplication by the ample class $c_{1}(\mathcal{L})\in A^{1}(\operatorname{Gr}(3,V_{10}))$. Now let

[TABLE]

be such that the restriction $\iota^{\ast}(b)\in A^{11}(X_{\sigma})$ is homologically trivial. Then we have that also

[TABLE]

To conclude that $b=0$, it suffices to show that

[TABLE]

is injective (and hence, by hard Lefschetz, an isomorphism). By hard Lefschetz, this is equivalent to showing that

[TABLE]

is surjective (hence an isomorphism).

According to [5, Theorem 5.26], the Chow ring of the Grassmannian is of the form

[TABLE]

where $c_{j}\in A^{j}(\operatorname{Gr}(3,V_{10}))$ are Chern classes of the universal subbundle, and $I$ is a certain complete intersection ideal generated by the $3$ relations

[TABLE]

in degree $8,9,10$. With the aid of the relations in $I$, we find that

[TABLE]

is $10$-dimensional (the classes $c_{1}^{2}c_{2}^{4},c_{2}^{5}$ are eliminated thanks to the relation in degree $8$ containing $c_{2}^{4}$; the class $c_{1}c_{3}^{3}$ is eliminated thanks to the relation in degree $9$; the class $c_{2}^{2}c_{3}^{2}$ is eliminated thanks to the relation in degree $10$). Inspecting this description of $A^{10}(\operatorname{Gr}(3,V_{10}))$, we observe that the inclusion

[TABLE]

is an equality. Since $c_{1}$ is proportional to $c_{1}(\mathcal{L})$, this proves claim 3.4. ∎

It remains to prove theorem 3.1:

Proof.

(of theorem 3.1) Clearly, the Chern class is universally defined: for any $\sigma\in B$, we have

[TABLE]

Also, the image

[TABLE]

consists of universally defined cycles. (For a given $a\in A^{11}(\operatorname{Gr}(3,V_{10}))$, the relative cycle

[TABLE]

does the job.)

Likewise, for any $j<10$ the fact that $A^{j}_{hom}(X_{\sigma})=0$, combined with weak Lefschetz in cohomology, implies that

[TABLE]

and so $A^{j}(X_{\sigma})$ consists of universally defined cycles for $j<10$. In particular, all intersections

[TABLE]

consist of universally defined cycles.

It remains to make sense of intersections

[TABLE]

To this end, we note that $A^{1}(X_{\sigma})$ is $1$-dimensional, generated by the restriction $g$ of the Plücker line bundle $\mathcal{L}$. Let $\iota\colon X_{\sigma}\to\operatorname{Gr}(3,V_{10})$ denote the inclusion. The normal bundle formula implies that

[TABLE]

It follows that

[TABLE]

also consists of universally defined cycles.

In conclusion, we have shown that $R^{11}(X_{\sigma})$ consists of universally defined cycles, and so theorem 3.1 is a corollary of theorem 3.2. ∎

Remark 3.5**.**

There are more cycle classes that can be put in the subgroup $R^{11}(X)$ of theorem 3.1. For instance, let $Y_{\sigma}$ be the hyperkähler fourfold associated to $X=X_{\sigma}$, and assume $Y_{\sigma}$ is smooth. Then (as we have seen above) the class

[TABLE]

is universally defined, hence it can be added to the subgroup $R^{11}(X)$ of theorem 3.1.

Remark 3.6**.**

Theorem 3.1 is an indication that perhaps the hypersurfaces $X\subset\operatorname{Gr}(3,V_{10})$ have a multiplicative Chow–Künneth decomposition, in the sense of [14, Chapter 8]. Unfortunately, establishing this seems difficult; one would need something like theorem 3.2 for

[TABLE]

**Acknowledgements **.

Thanks to my mythical colleague Gilberto Kiwi for inspiring conversations, and thanks to a zealous referee for constructive comments.