# On the Chow groups of hypersurfaces in symplectic Grassmannians

Robert Laterveer  Institut de Recherche Mathématique Avancée, CNRS –
Université de Strasbourg, 7 Rue René Descartes, 67084 Strasbourg CEDEX,
FRANCE<EMAIL_ADDRESS>
###### Abstract.

Let $Y$ be a Plücker hypersurface in a symplectic Grassmannian
$I_{1}\operatorname{Gr}(3,n)$ or a bisymplectic Grassmannian
$I_{2}\operatorname{Gr}(3,n)$. We show that many Chow groups of $Y$ inject
into cohomology.

###### Key words and phrases:

Algebraic cycles, Chow groups, motive, Bloch-Beilinson conjectures

###### 2010 Mathematics Subject Classification:

Primary 14C15, 14C25, 14C30.

##  1\. Introduction

Given a smooth projective variety $Y$ over $\mathbb{C}$, let
$A_{i}(Y):=CH_{i}(Y)_{\mathbb{Q}}$ denote the Chow groups of $Y$ (i.e. the
groups of $i$-dimensional algebraic cycles on $Y$ with
$\mathbb{Q}$-coefficients, modulo rational equivalence). Let
$A_{i}^{hom}(Y)\subset A_{i}(Y)$ denote the subgroup of homologically trivial
cycles.

The famous Bloch–Beilinson conjectures [8], [25] predict that the Hodge level
of the cohomology of $Y$ should have an influence on the size of the Chow
groups of $Y$. For surfaces, this is the notorious Bloch conjecture, which is
still an open problem. For hypersurfaces in projective space, the precise
prediction is as follows:

######  Conjecture 1.1.

Let $Y\subset\mathbb{P}^{n}$ be a smooth hypersurface of degree $d$. Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq{n\over d}-1\ .$

Conjecture 1.1 is still open; partial results have been obtained in [16],
[23], [19], [5], [6].

In [14], I considered a version of Conjecture 1.1 for Plücker hyperplane
sections of Grassmannians. In this note, we look at the case of Plücker
hyperplane sections of symplectic Grassmannians. Recall that inside the
Grassmannian $\operatorname{Gr}(3,n)$ (of $3$-dimensional subspaces of an
$n$-dimensional vector space), the symplectic Grassmannian
$I_{1}\operatorname{Gr}(3,n)\subset\operatorname{Gr}(3,n)$ parametrizes
subspaces that are isotropic with respect to some fixed skew-symmetric 2-form.
The precise prediction (cf. subsection 3.2 below) is as follows:

######  Conjecture 1.2.

Let

$Y:=I_{1}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-4\ .$

The main result of this note is a partial verification of Conjecture 1.2:

######  Theorem  (=Theorem 3.4).

Let

$Y:=I_{1}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-5\ .$

Moreover, in case $n\leq 10$ or $n=12$ we have

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-4\ .$

To prove Theorem 3.4, we rely on the recent notion of projections among
(symplectic) Grassmannians [2]. Combined with the Chow-theoretic Cayley trick
[9], this reduces Theorem 3.4 to understanding the Chow groups of a hyperplane
section in an ordinary Grassmannian $\operatorname{Gr}(3,n+1)$. This last
problem was handled in [14].

As a consequence of Theorem 3.4, some instances of the generalized Hodge
conjecture are verified:

######  Corollary  (=Corollary 4.1).

Let $Y$ be as in Theorem 3.4, and assume $n\leq 10$ or $n=12$. Then $H^{\dim
Y}(Y,\mathbb{Q})$ is supported on a subvariety of codimension $n-3$.

Other consequences are as follows:

######  Corollary  (=Corollary 4.2).

Let

$Y:=I_{1}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding).

(i) If $n\leq 8$, then $Y$ has finite-dimensional motive (in the sense of
[11]).

(ii) If $n\leq 9$, then $Y$ has trivial Griffiths groups (and so Voevodsky’s
smash conjecture is true for $Y$).

(iii) If $n\leq 10$, the Hodge conjecture is true for $Y$.

Applying the same method, we can also say something about hypersurfaces in
bisymplectic Grassmannians. (Recall that the bisymplectic Grassmannian
$I_{2}\operatorname{Gr}(3,n)\subset\operatorname{Gr}(3,n)$ is the locus of
3-spaces that are isotropic with respect to two fixed generic skew-forms.)

######  Theorem  (=Theorem 3.5).

Assume $n$ is even, and let

$Y:=I_{2}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-7\ .$

Moreover, in case $n\leq 10$ we have

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-6\ .$

The hyperplane sections $I_{1}\operatorname{Gr}(3,9)\cap H$ and
$I_{2}\operatorname{Gr}(3,8)\cap H$ are of particular interest: they are Fano
varieties of K3 type, and they are related to hyperplane sections
$\operatorname{Gr}(3,10)\cap H$ and hence to Debarre–Voisin hyperkähler
fourfolds, cf. [2, Section 4].

######  Conventions .

In this note, the word variety will refer to a reduced irreducible scheme of
finite type over $\mathbb{C}$. A subvariety is a (possibly reducible) reduced
subscheme which is equidimensional.

All Chow groups will be with rational coefficients: we denote by
$A_{j}(Y):=CH_{j}(Y)_{\mathbb{Q}}$ the Chow group of $j$-dimensional cycles on
$Y$ with $\mathbb{Q}$-coefficients; for $Y$ smooth of dimension $n$ the
notations $A_{j}(Y)$ and $A^{n-j}(Y)$ are used interchangeably. The notations
$A^{j}_{hom}(Y)$ and $A^{j}_{AJ}(X)$ will be used to indicate the subgroup of
homologically trivial (resp. Abel–Jacobi trivial) cycles.

The contravariant category of Chow motives (i.e., pure motives with respect to
rational equivalence as in [20], [17]) will be denoted $\mathcal{M}_{\rm
rat}$.

##  2\. Preliminaries

###  2.1. Cayley’s trick and Chow groups

######  Theorem 2.1 (Jiang [9]).

Let $E\to U$ be a vector bundle of rank $r\geq 2$ over a projective variety
$U$, and let $S:=s^{-1}(0)\subset U$ be the zero locus of a regular section
$s\in H^{0}(U,E)$ such that $S$ is smooth of dimension $\dim
U-\operatorname{rank}E$. Let $X:=w^{-1}(0)\subset\mathbb{P}(E)$ be the zero
locus of the regular section $w\in
H^{0}(\mathbb{P}(E),\mathcal{O}_{\mathbb{P}(E)}(1))$ that corresponds to $s$
under the natural isomorphism $H^{0}(U,E)\cong
H^{0}(\mathbb{P}(E),\mathcal{O}_{\mathbb{P}(E)}(1))$. There is an isomorphism
of integral Chow motives

$h(X)\cong h(S)(1-r)\oplus\bigoplus_{i=0}^{r-2}h(U)(-i)\ \ \ \hbox{in}\
\mathcal{M}^{\mathbb{Z}}_{\rm rat}\ .$

###### Proof.

This is [9, Theorem 3.1]. Both the isomorphism and its inverse are explicitly
described. ∎

######  Remark 2.2.

In the set-up of Theorem 2.1, a cohomological relation between $X$, $S$ and
$U$ was established in [12, Prop. 4.3] (cf. also [7, section 3.7], as well as
[2, Proposition 46] for a generalization). A relation on the level of derived
categories was established in [18, Theorem 2.10] (cf. also [10, Theorem 2.4]
and [2, Proposition 47]).

###  2.2. Linear sections of $\operatorname{Gr}(2,n)$

######  Proposition 2.3.

Let

$Y:=\operatorname{Gr}(2,n)\cap H_{1}\cap\cdots\cap H_{s}\ \ \ \subset\
\mathbb{P}^{{n\choose 2}-1}$

be a smooth dimensionally transverse intersection with $s$ hyperplanes (with
respect to the Plücker embedding). Assume $s\leq 2$. Then

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

###### Proof.

This uses a geometric construction that can be found in [4].

Let $P\subset\mathbb{P}(V_{n})$ be a fixed hyperplane, and consider (as in [4,
Section 2.3]) the rational map

$\operatorname{Gr}(2,V_{n})\ \dashrightarrow\ P$

sending a line in $\mathbb{P}(V_{n})$ to its intersection with $P$. This map
is resolved by blowing up a subvariety
$\sigma_{11}(P)\cong\operatorname{Gr}(2,n-1)$, resulting in a morphism

$\Gamma\colon\ \ \widetilde{\operatorname{Gr}}\ \to\ P$

(where $\widetilde{\operatorname{Gr}}\to\operatorname{Gr}(2,V_{n})$ denotes
the blow-up with center $\sigma_{11}(P)$).

Let $\widetilde{Y}\to Y$ be the blow-up of $Y$ with center $\sigma_{11}(P)\cap
Y$, and let us consider the morphism

$\Gamma_{Y}\colon\ \ \widetilde{Y}\ \to\ P\ ,$

obtained by restricting $\Gamma$.

In case $s=1$ and $P$ is generic with respect to $Y$, the morphism
$\Gamma_{Y}$ is a $\mathbb{P}^{n-3}$-fibration over $P$. It follows that
$\widetilde{Y}$, and hence $Y$, has trivial Chow groups.

In case $s=2$, and $P$ chosen generically with respect to $Y$, the morphism
$\Gamma_{Y}$ is generically a $\mathbb{P}^{n-4}$-fibration over $P$, and there
are finitely many points in $P$ where the fiber is $\mathbb{P}^{n-3}$.
Applying Theorem 2.1, this implies that $\widetilde{Y}$, and hence also $Y$,
has trivial Chow groups. ∎

###  2.3. Hyperplane sections of $\operatorname{Gr}(3,n)$

######  Theorem 2.4.

Let

$Y:=\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose 3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-3\ .$

Moreover, in case $n\leq 11$ or $n=13$ we have

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-2\ .$

###### Proof.

This is [14, Theorems 3.1 and 3.2], which uses the notion of jumps between
Grassmannians as developed in [2]. ∎

##  3\. Main results

###  3.1. Projections

As in [2], let $I_{r}\operatorname{Gr}(k,n)\subset\operatorname{Gr}(k,n)$
parametrize linear subspaces that are isotropic with respect to $r$ fixed
generic skew-forms. One has

$\dim I_{r}\operatorname{Gr}(k,n)=k(n-k)-r{k\choose 2}\ .$

For example, $I_{r}\operatorname{Gr}(2,n)$ is just the intersection of
$\operatorname{Gr}(2,n)$ with $r$ Plücker hyperplanes. The case
$I_{2}\operatorname{Gr}(k,n)$ is studied in detail in [1].

To relate hyperplane sections of different symplectic Grassmannians,
Bernardara–Fatighenti–Manivel [2] have developed a theory of projections. The
starting point is a rational map

$\pi\colon\operatorname{Gr}(k,n+1)\ \dashrightarrow\ \operatorname{Gr}(k,n)\
,$

determined by the choice of a line in the $n+1$-dimensional vector space. If
$Y^{\prime}$ is a hyperplane section of $I_{r}\operatorname{Gr}(k,n+1)$, one
can restrict $\pi$ to $Y^{\prime}$. A detailed analysis of the case $k=3$
yields the following:

######  Theorem 3.1 ([2]).

Assume $n$ is even and $r\leq 1$, or $n$ is odd and $r=0$. Let

$Y^{\prime}:=I_{r}\operatorname{Gr}(3,n+1)\cap H$

be a smooth hyperplane section. There exists a commutative diagram

$\begin{array}[c]{ccccccc}E&\hookrightarrow&\widetilde{Y}^{\prime}&\hookleftarrow&F&&\\\
&&&&&&\\\ \downarrow&&\ \ \downarrow{\scriptstyle\sigma}&\
\searrow{\scriptstyle\tau}&&\ \searrow{\scriptstyle q}&\\\ &&&&&&\\\
Z^{\prime}&\hookrightarrow&Y^{\prime}&\stackrel{{\scriptstyle\pi^{\prime}}}{{\dashrightarrow}}&I_{r}\operatorname{Gr}(3,n)&\hookleftarrow&Y\\\
\end{array}$

where $Y:=I_{r+1}\operatorname{Gr}(3,n)\cap H$ is a smooth hyperplane section.
The morphism $\sigma$ is the blow-up with center $Z^{\prime}\cong
I_{r+1}\operatorname{Gr}(2,n)$. The morphism $q$ is a
$\mathbb{P}^{3}$-fibration, while $\tau$ is a $\mathbb{P}^{2}$-fibration over
the complement of $Y$.

###### Proof.

This is contained in [2, Section 3.2] (NB: note that our $n$ is $n-1$ in loc.
cit.). As explained in loc. cit., the assumptions on $n$ and $r$ guarantee
that the target $I_{r}\operatorname{Gr}(3,n)$ and the hyperplane section $Y$
are generic and hence smooth. ∎

###  3.2. Motivating the conjecture

As a consequence of Theorem 3.1, one can compute the Hodge level of hyperplane
sections $Y$ of symplectic Grassmannians: surprisingly, it turns out that (at
least for $n>8$) $Y$ is of “Calabi–Yau type”:

######  Theorem 3.2 ([2]).

Let

$Y:=I_{1}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
k}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Assume
$n>8$. Then $Y$ has Hodge coniveau $n-3$. More precisely, the Hodge numbers
verify

$h^{p,\dim Y-p}(Y)=\begin{cases}1&\hbox{if}\ p=n-3\ ,\\\ 0&\hbox{if}\ p<n-3\
.\\\ \end{cases}$

###### Proof.

This is implicit in [2], as we now explain. Let the set-up be as in Theorem
3.1. Then [2, Proposition 6] relates $Y$ and $Y^{\prime}$ on the level of
cohomology: one has an isomorphism of Hodge structures

(1)
$\begin{split}H^{j-6}(Y,\mathbb{Q})(-3)\oplus\bigoplus_{i=0}^{2}H^{j-2i}(I_{r}\operatorname{Gr}(3,n),\mathbb{Q})(-i)\
\xrightarrow{\cong}&\ H^{j}(Y^{\prime},\mathbb{Q})\oplus\\\
&\bigoplus_{i=1}^{c-1}H^{j-2i}(I_{r+1}\operatorname{Gr}(2,n),\mathbb{Q})(-i)\\\
\end{split}$

(where $c$ denotes the codimension of $Z^{\prime}$ in $Y^{\prime}$). Setting
$r=0$ and combining with [2, Theorem 3] (which gives the Hodge numbers of
$Y^{\prime}$), plus the fact that $\operatorname{Gr}(3,n)$ and
$I_{1}\operatorname{Gr}(2,n)$ have algebraic cohomology, this gives the
required Hodge numbers of $Y$. ∎

Theorem 3.2 motivates Conjecture 1.2. Indeed, the generalized Bloch conjecture
[25, Conjecture 1.10] predicts that any variety $Y$ with Hodge coniveau $\geq
c$ has

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i<c\ .$

Note that at least for $n>8$, the bound of Conjecture 1.2 is optimal: assuming
$A_{i}^{hom}(Y)=0$ for $j\leq n-3$ and applying the Bloch–Srinivas argument
[3], one would get the vanishing $h^{n-3,\dim Y-n+3}(Y)=0$, contradicting
Theorem 3.2.

###  3.3. A relation of motives

The cohomological relation (1) between $Y$ and $Y^{\prime}$ also exists as
(and actually is implied by) a relation on the level of the Grothendieck ring
of varieties [2, Proposition 4], and on the level of derived categories [2,
Proposition 5]. To complete the picture, we now lift the relation (1) to the
level of Chow motives:

######  Proposition 3.3.

Let notation and assumptions be as in Theorem 3.1. There is an isomorphism of
integral Chow motives

$\begin{split}h(Y)(-3)\oplus\bigoplus_{i=0}^{2}h(I_{r}\operatorname{Gr}(3,n))(-i)\
\xrightarrow{\cong}\
h(Y^{\prime})\oplus\bigoplus_{i=1}^{c-1}h(I_{r+1}\operatorname{Gr}(2,n))(-i)&\\\
\ \ \ \hbox{in}\ \mathcal{M}^{\mathbb{Z}}_{\rm rat}&\ .\\\ \end{split}$

(Here $c$ denotes the codimension of $Z^{\prime}$ in $Y^{\prime}$.)

###### Proof.

The idea is to express the motive of $\widetilde{Y}^{\prime}$ in two different
ways:

The blow-up formula expresses $h(\widetilde{Y}^{\prime})$ in terms of
$h(Y^{\prime})$; this gives the right-hand side of the relation.

Looking at [2, Section 3.2], one finds that $\widetilde{Y}^{\prime}$ is the
total space of a projectivization $\mathbb{P}(E)$ where $E$ is the vector
bundle $E:=\mathcal{O}\oplus\mathcal{U}^{\ast}$ on
$I_{r}\operatorname{Gr}(3,n)$ (in the notation of loc. cit.), and $Y$ is given
by a section of $E$. That is, we are in the setting of Cayley’s trick, and so
Theorem 2.1 expresses $h(\widetilde{Y}^{\prime})$ in terms of $h(Y)$; this
gives the left-hand side of the relation. ∎

###  3.4. Hyperplane sections of $I_{1}\operatorname{Gr}(3,n)$

######  Theorem 3.4.

Let

$Y:=I_{1}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-5\ .$

Moreover, in case $n\leq 10$ or $n=12$ we have

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-4\ .$

###### Proof.

A generic hyperplane section $Y$ is attained by the construction of Theorem
3.1 with $r=0$, i.e. there is a smooth hyperplane section

$Y^{\prime}:=\operatorname{Gr}(3,n+1)\cap H\ ,$

related to $Y$ via the projection of Theorem 3.1. In this case, Proposition
3.3 implies that there is an injection of Chow groups

$A_{i}^{hom}(Y)\ \hookrightarrow\ A^{hom}_{i+3}(Y^{\prime})\oplus\bigoplus
A_{\ast}^{hom}(I_{1}\operatorname{Gr}(2,n))\ .$

The symplectic Grassmannian $I_{1}\operatorname{Gr}(2,n)$ is nothing but a
Plücker hyperplane section of $\operatorname{Gr}(2,n)$, and so Proposition 2.3
gives the vanishing

$A_{\ast}^{hom}(I_{1}\operatorname{Gr}(2,n))=0\ .$

The variety $Y^{\prime}$ is a hyperplane section of
$\operatorname{Gr}(3,n+1)$, and so Theorem 2.4 gives the vanishing

$A_{i+3}^{hom}(Y^{\prime})=0\ \ \ \forall\ i\leq n-5\ ,$

with the additional vanishing for $i=n-4$ for small $n$. This proves the
theorem for generic sections $Y$.

A standard spread argument allows to extend to all smooth hyperplane sections:
Let $\mathcal{Y}\to B$ denote the universal family of all smooth hyperplane
sections of $\operatorname{Gr}(k,n)$, and let $B^{\circ}\subset B$ denote the
Zariski open subset parametrizing smooth $Y$ verifying the set-up of Theorem
3.1. Doing the Bloch–Srinivas argument [3] (cf. also [13]), the above implies
that for each $b\in B^{\circ}$ one has a decomposition of the diagonal

(2) $\Delta_{Y_{b}}=\gamma_{b}+\delta_{b}\ \ \ \hbox{in}\ A^{\dim
Y_{b}}(Y_{b}\times Y_{b})\ $

where $\gamma_{b}$ is completely decomposed (i.e. $\gamma_{b}\in
A^{\ast}(Y_{b})\otimes A^{\ast}(Y_{b})$) and $\delta_{b}$ is supported on
$Y_{b}\times W_{b}$ with $\hbox{codim}\,W_{b}=n-2$ (and
$\hbox{codim}\,W_{b}=n-1$ for small $n$). Using the Hilbert schemes argument
of [24, Proposition 3.7] (cf. also [15, Proposition A.1] for the precise form
used here), the $\gamma_{b},\delta_{b},W_{b}$ exist relatively, i.e. one can
find a cycle
$\gamma\in(p_{1})^{\ast}A^{\ast}(\mathcal{Y})\cdot(p_{2})^{\ast}A^{\ast}(\mathcal{Y})$,
a subvariety $\mathcal{W}\subset\mathcal{Y}$ of codimension $n-2$, and a cycle
$\delta$ supported on $\mathcal{Y}\times_{B^{\circ}}\mathcal{W}$ such that

$\Delta_{\mathcal{Y}}|_{b}=\gamma|_{b}+\delta|_{b}\ \ \ \hbox{in}\ A^{\dim
Y_{b}}(Y_{b}\times Y_{b})\ \ \ \forall\ b\in B^{\circ}\ .\ $

Let $\bar{\gamma},\bar{\delta}\in A^{\dim
Y_{b}}(\mathcal{Y}\times_{B}\mathcal{Y})$ be cycles that restrict to $\gamma$
resp. $\delta$. The spread lemma [25, Lemma 3.2] implies that

$\Delta_{\mathcal{Y}}|_{b}=\bar{\gamma}|_{b}+\bar{\delta}|_{b}\ \ \ \hbox{in}\
A^{\dim Y_{b}}(Y_{b}\times Y_{b})\ \ \ \forall\ b\in B\ .\ $

Given any $b_{1}\in B\setminus B^{\circ}$, using the moving lemma, one can
find representatives for $\bar{\gamma}$ and $\bar{\delta}$ in general position
with respect to the fiber $Y_{b_{1}}\times Y_{b_{1}}$. Restricting to the
fiber, this implies that the diagonal of $Y_{b_{1}}$ has a decomposition as in
(2). Letting the decomposition (2) act on Chow groups, this shows that

$A_{i}^{hom}(Y_{b})=0\ \ \ \forall\ i\leq n-5\ ,\ \ \ \forall\ b\in B\ $

(with the additional vanishing for $i=n-4$ for small $n$). ∎

###  3.5. Hyperplane sections of $I_{2}\operatorname{Gr}(3,n)$

######  Theorem 3.5.

Assume $n$ is even, and let

$Y:=I_{2}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-7\ .$

Moreover, in case $n\leq 10$ we have

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-6\ .$

###### Proof.

For $n$ even, a generic hyperplane section $Y$ is attained by the construction
of Theorem 3.1 with $r=1$, i.e. there is a smooth hyperplane section

$Y^{\prime}:=I_{1}\operatorname{Gr}(3,n+1)\cap H\ ,$

related to $Y$ via the projection of Theorem 3.1. In this case, Proposition
3.3 implies that there is an injection of Chow groups

$A_{i}^{hom}(Y)\ \hookrightarrow\ A^{hom}_{i+3}(Y^{\prime})\oplus\bigoplus
A_{\ast}^{hom}(I_{2}\operatorname{Gr}(2,n))\ .$

The bisymplectic Grassmannian $I_{2}\operatorname{Gr}(2,n)$ is nothing but an
intersection $\operatorname{Gr}(2,n)\cap H_{1}\cap H_{2}$ (where the $H_{j}$
are Plücker hyperplanes), and so Proposition 2.3 gives the vanishing

$A_{\ast}^{hom}(I_{2}\operatorname{Gr}(2,n))=0\ .$

The variety $Y^{\prime}$ is a hyperplane section of
$I_{1}\operatorname{Gr}(3,n+1)$, and so Theorem 3.4 gives the vanishing

$A_{i+3}^{hom}(Y^{\prime})=0\ \ \ \forall\ i\leq n-7\ ,$

with the additional vanishing for $i=n-6$ for small $n$. This proves the
theorem for generic sections $Y$.

The extension to all smooth hyperplane sections $Y$ is done just as in the
proof of Theorem 3.4. ∎

##  4\. Some consequences

######  Corollary 4.1.

(i) Let $Y$ be as in Theorem 3.4 and $n\leq 10$ or $n=12$, or as in Theorem
3.5 and $n\leq 10$. Then $H^{\dim Y}(Y,\mathbb{Q})$ is supported on a
subvariety of codimension $n-3$.

(ii) Let $Y$ be as in Theorem 3.5 and $n\leq 10$. Then $H^{\dim
Y}(Y,\mathbb{Q})$ is supported on a subvariety of codimension $n-5$.

###### Proof.

This follows in standard fashion from the Bloch–Srinivas argument [3]. Let us
treat (i) (the argument for (ii) is the same). The vanishing

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-4\ $

(Theorem 3.4) is equivalent to the decomposition

$\Delta_{Y}=\gamma+\delta\ \ \ \hbox{in}\ A^{\dim Y}(Y\times Y)\ ,$

where $\gamma$ is a completely decomposed cycle (i.e. $\gamma\in
A^{\ast}(Y)\otimes A^{\ast}(Y)$), and $\delta$ has support on $Y\times W$ with
$W\subset Y$ of codimension $n-3$ (to see this equivalence, one can look for
instance at [13, Theorem 1.7]). Let $H^{\dim Y}_{tr}(Y,\mathbb{Q})$ denote the
transcendental cohomology (i.e. the complement of the algebraic part under the
cup product pairing). The cycle $\gamma$ does not act on $H^{\dim
Y}_{tr}(Y,\mathbb{Q})$. The action of $\delta$ on $H^{\dim
Y}_{tr}(Y,\mathbb{Q})$ factors over $W$, and so

$H^{\dim Y}_{tr}(Y,\mathbb{Q})\ \ \subset\ H^{\dim Y}_{W}(Y,\mathbb{Q})\ .$

Since the algebraic part of $H^{\dim Y}(Y,\mathbb{Q})$ is (by definition)
supported in codimension $\dim Y/2$, this settles the corollary. ∎

######  Corollary 4.2.

Let

$Y:=I_{1}\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose
3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding).

(i) If $n\leq 8$, then $Y$ has finite-dimensional motive (in the sense of
[11]).

(ii) If $n\leq 9$, then $Y$ has trivial Griffiths groups (and so Voevodsky’s
smash conjecture [22] is true for $Y$, i.e. numerical equivalence and smash-
equivalence coincide on $Y$).

(iii) If $n\leq 10$, the Hodge conjecture is true for $Y$.

###### Proof.

This is similar to the argument of Corollary 4.1.

(i) The vanishing

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-4\ $

(Theorem 3.4) is equivalent to the decomposition of the diagonal

$\Delta_{Y}=\gamma+\delta\ \ \ \hbox{in}\ A^{\dim Y}(Y\times Y)\ ,$

where $\gamma$ is a completely decomposed cycle, and $\delta$ has support on
$Y\times W$ with $W\subset Y$ of codimension $n-3$ (cf. [3] or [13]). The
dimension of $Y$ is $3n-13$, and so (looking at the action of the diagonal)
one finds that

$A^{\ast}_{AJ}(Y)=0\ $

as long as $n\leq 8$. This implies Kimura finite-dimensionality of $Y$ [21,
Theorem 4].

(ii) The vanishing of Theorem 3.4 implies that

$\hbox{Niveau}(A_{\ast}(Y))\leq 2$

(in the sense of [13]), i.e. the motive of $Y$ factors over a surface. Since
surfaces have trivial Griffiths groups, the conclusion follows.

(iii) The vanishing of Theorem 3.4 implies that

$\hbox{Niveau}(A_{\ast}(Y))\leq 3$

(in the sense of [13]), i.e. the motive of $Y$ factors over a threefold. Since
threefolds verify the Hodge conjecture, the conclusion follows. ∎

We leave it to the zealous reader to formulate and prove a version of
Corollary 4.2 for bisymplectic Grassmannians.

######  Acknowledgments .

Thanks to Kai and Len for enjoying Kuifje movies. Thanks to the referee for
constructive comments that helped to improve the presentation.

## References

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