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\newcommand{\xra}[1]{\xrightarrow{#1}}   % indexed right arrow 

\newcommand{\zp}{\mathbb{Z}/p}      % integers mod p

\newcommand{\LL}{\mathbb{L}}   % Lazard ring

\newcommand{\DD}{\mathcal{D}}      % Dynkin diagram

\newcommand{\Om}{\Omega}           % cobordism

\newcommand{\bk}{{\overline{k}}}  % algebraic closure of k
\newcommand{\bK}{{\overline{K}}}  % algebraic closure of K
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\newcommand{\pl}{\mathbb{P}}      % projective line

\newcommand{\res}{\mathrm{res}}     % restriction

\DeclareMathOperator{\Spec}{\mathrm{Spec}}  % Spectrum

\DeclareMathOperator{\CH}{\mathrm{CH}}          % Chow group
\DeclareMathOperator{\Ch}{\mathrm{Ch}}          % Chow group mod p
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\title{Special correspondences and
Chow traces of Landweber-Novikov operations}

\author{K.~Zainoulline}

\begin{document}

\maketitle

\section{Introduction}



In paper \cite{Vi07} A.~Vishik using the techniques of symmetric
operations in algebraic cobordism
proved that changing the base field
by the function field of a smooth projective quadric 
doesn't change the property of being rational 
for cycles of small codimension. 
This fact which he calls  Main Tool Lemma
plays the crucial role in his construction of fields
with $u$-invariant $2^r+1$. 

In the present paper we prove
M.T.L. for the class of varieties introduced by M.~Rost
in the context of the proof of Bloch-Kato conjecture. 
Namely, for varieties 
which possess a special correspondence (see \cite[Definition~5.1]{Ro06}).
As in the Vishik's proof the main technical tools are 
algebraic cobordism of M.~Levine and F.~Morel \cite{LM07}, 
generalized Rost degree formula and 
divisibility of Chow traces of certain Landweber-Novikov operations.
Therefore, we always assume 
that our base field $k$ has characteristic $0$.

We use the following notation.
By $\bk$ we denote the algebraic closure of $k$ and
by $\SX$ the respective base change $X\times_k \bk$ of a variety $X$.
By $K$ we denote the function field of $X$.
Given a prime $p$ by $\ChO^*(X)$ 
we denote the Chow ring of $X$ modulo $p$-torsion 
taken with $\zp$-coefficients, i.e.,
$\ChO^*(X)=\CHO^*(X)/p$, where 
$\CHO^*(X)=\CH^*(X)/(\text{\it p-tors})$.
We say that a cycle from $\ChO^*(\SX)$ is {\it defined over $k$}
if it belongs to the image of the restriction map 
$\ChO^*(X)\to\ChO^*(\SX)$.


The following notion will be central in this paper

\begin{dfn} 
Let $X$ be a smooth proper irreducible variety over a field $k$ 
of dimension $n$ and $p$ be a prime. 
Assume that $X$ has no zero-cycles of degree coprime to $p$.
Let $d$ be an integer $0\le d\le n$.
We say $X$ is a {\it $d$-splitting variety mod $p$} if 
for any smooth quasi-projective variety $Y$ over $k$ 
and all integers $m$ satisfying 
$0\le m < d$, the following condition holds
\begin{equation}\label{meq}
\text{A cycle } y\in \ChO^m(\SY)\text{ is defined over }k 
\Longleftrightarrow
y\times_\bk \bK\text{ is defined over }K 
\end{equation}
\end{dfn}


\begin{ex} 
According to the results of A.~Vishik
\begin{enumerate}
\item[(a)]
any smooth projective quadric $Q$ of dimension $n$
is a $[\tfrac{n+1}{2}]$-splitting variety mod $2$ 
(see \cite[Cor.~3.5.(1)]{Vi07}); 
\item[(b)] a quadric $Q$ is a $n$-splitting variety (mod $2$)
if and only if $Q$ possesses a Rost projector
(see \cite[p.~373]{Vi07}).
\end{enumerate}
\end{ex}


The main result of the paper is the following

\begin{thm}\label{mthm} Let $X$ be a smooth proper irreducible
variety over a field of characteristik $0$.
Assume that $X$ has no zero-cycles of degree coprime to $p$.
If $X$ possesses a special correspondence in the sense of Rost, 
then $X$ is a $\frac{n}{p-1}$-splitting variety 
and the value $\frac{n}{p-1}$ is optimal.
\end{thm}

As an application of the techniques used
in the proof of \ref{mthm}, we provide
a complete list of $d$-splitting 
projective homogeneous varieties of type $F_4$.

To do this we introduce the following notation.
Let $G$ be a simple linear algebraic group over $k$.
We say a projective homogeneous $G$-variety $X$ 
is of type $\DD$, 
if the group $\SG$ has root system of type
$\DD$. Moreover, if 
$\SX$ is the variety of parabolic subgroups of $\SG$
defined by the subset of simple roots $S$ of $\DD$, then
we say that $X$ is of type $\DD/P_S$.
In this notation $P_\DD$ defines a Borel subgroup
and $P_i$, $i\in\DD$, defines a maximal parabolic subgroup. 


\begin{cor}
Let $X$ be a projective homogeneous variety of type $F_4$ and
$p$ be one of its torsion primes ($2$ or $3$).
Assume that $X$ has no zero-cycles of degree coprime to $p$. Then depending
on $p$ we have
\begin{itemize}
\item[{\footnotesize $p=2$:}] If $X$ is of type $F_4/P_4$, 
then $X$ is a $(\dim X)$-splitting variety.
For all other types $X$ is a $3$-splitting variety and this value is 
optimal.
\item[{\footnotesize $p=3$:}] 
$X$ is always a $4$-splitting variety and this value is optimal.
\end{itemize}
\end{cor}





\section{Mod-$p$ operations}
In the present section we introduce certain operations 
$$
\phi^{q(t)}_p \colon \Om^* \to \ChO^*\text{ parametrized by }
q(t)\in\ChO^*[[t]]
$$
from the ring of algebraic cobordism $\Om^*$ of Levine-Morel 
to the Chow group modulo $p$-torsion $\ChO^*$, where $p$ is a given prime.
All the facts used here
can be found in \cite{LM07}, \cite{Me03} and \cite{Vi06}. 

Let $\CHO^*$ and $\ChO^*$
denote the respective Chow groups modulo $p$-torsion.
Consider the commutative diagram
$$
\xymatrix{
\Om^m \ar[r]^-{S_{LN}^{k(p)}}\ar[d]^{pr} & 
\Om^{m+(p-1)k}\ar[d]^{pr} \\
\CHO^m \ar[d] & \CHO^{m+(p-1)k} \ar[d] \\
\ChO^m\ar[r]^-{S^k} & \ChO^{m+(p-1)k},
}
$$
where $S_{LN}^{k(p)}$ 
is the Landweber-Novikov operation corresponding
to the partition 
$k(p)=\scriptsize{\underbrace{(p-1,p-1,\ldots,p-1)}_{k-times}}$,
$pr\colon \Om^*\to \CH^* \to \CHO^*$ is the natural surjection and
$S^k$ is the reduced $p$-th power operation.
By the properties of reduced power operations, 
$S^k=0$ if $k>m$.

Define (cf. \cite[Prop.~2.1]{Vi07}) 
$$
\phi^{t^{(p-1)a}}_p=\frac{pr\circ S_{LN}^{k(p)}}{p},\text{ where } a=k-m.
$$
By commutativity of the diagram $\phi^{t^{(p-1)a}}_p$ 
is well-defined.
If $r$ is not divisible by $(p-1)$, then
$pr\circ S_{LN}^{k(r)}=0$ and we set $\phi^{t^r}_p=0$.
Hence, we have constructed an operation $\phi^{t^r}_p$, $r>0$, 
which maps $\Om^m$ to $\ChO^{r+pm}$.


Finally, given a power series $q(t)\in \ChO^*[[t]]$ we define 
$$
\phi^{q(t)}_p=\sum_{r\ge 0} q_r\phi^{t^r}_p, \text{ where } 
q(t)=\sum_{r\ge 0} q_rt^r.
$$
By definition the operations $\phi^{q(t)}_p$ 
are additive and respect pull-backs.


Let $E$ be a vector bundle over a smooth variety $U$,
Consider
its total Chern class
$c(E)=\prod_j (t+\xi_j)$, 
where $\xi_j$ are the roots.
Define $c(E)^{(p)}(t)=\prod_j(t^{p-1}+\xi_j^{p-1})$.
Let $E=T_U$ be the tangent bundle of rank $l$.
Set $b_l(U)=\deg c(-T_U)^{(p)}$. 
According to \cite[\S6]{Me03}
this number is always divisible by $p$. Moreover, it is trivial if
$l$ is not divisible by $(p-1)$.
We define the Rost number $\eta_p$ as
$$
\eta_p(U)=\frac{b_l(U)}{p} \in \zp.
$$

We will extensively use the following property of the operation $\phi_p$
which follows from the multiplicativity of Landweber-Novikov operations

\begin{lem}(cf. \cite[Prop.2.3]{Vi07})\label{RDF} 
Let $U$ be a smooth projective variety of positive dimension $l$ 
and $[U]$ be its class in the Lazard ring $\LL$.
Let $\beta\in \Om^{j}(X)$.
Then
$$
\phi^{t^r}_p([U]\cdot\beta)=\eta_p(U)\cdot S^k(pr(\beta)), \text{ where }
r=(p-1)(k-j)+lp>0.
$$
Observe that $\phi^{t^r}_p([U]\cdot\beta)=0$ 
if $r$ is not divisible by $(p-1)$.
\end{lem}






\section{Construction of a cycle defined over $k$}

In the present section we prove Theorem~\ref{mthm}.

\paragraph{I.}
We proceed following the proof of \cite[Thm.~3.1]{Vi07}.
Let $Y$ be a smooth projective
variety over $k$.
Let $y\in\ChO^m(\SY)$ be
such that $y_{\bK}$ is defined over $K$. We want to show that
$y$ is defined over $k$ for all $m<d$.
Let $\Om^*$ be the algebraic cobordism of Levine-Morel. There is
a natural surjection $pr\colon\Om^*\to \ChO^*$.
Consider the commutative diagram
$$
\xymatrix{
\omega \ar@{|->}[d]& \Om^m(X\times Y) \ar[d]_-{\res}\ar@{>>}[r]^-{pr}& 
\ChO^m(X\times Y)\ar@{>>}[r]^-{p^*}\ar[d]^-{\res} & \ChO^m(Y_{K})\ar[d]^-{\res} & u \ar@{|->}[d]\\
\bar\omega &\Om^m(\SX\times \SY) \ar@{>>}[r]^-{pr} & \ChO^m(\SX\times\SY)\ar@{>>}[r]^-{\bar p^*} & \ChO^m(Y_\bK) & y_{\bK}
}
$$
where $p\colon \Spec(K) \times Y\to X\times Y$ and the pull-back $p^*$
is surjective because of the localization
sequence for Chow groups.
By the hypothesis there exists a preimage $u$ of $y_{\bK}$
by means of $\res$. By surjectivity of $pr$ and $p^*$, there exists
a preimage $\omega$ of $u$.
Let $\bar\omega=\res(\omega)$.


\paragraph{II.}
Let $X$ be a variety which possesses a special correspondence.
By the results of \cite{Ro06} this implies that
\begin{itemize}
\item[(a)] $n=p^s-1$, 
\item[(b)] $\eta_p(\SX)\neq 0 \mod p$
\item[(c)] the Chow motive of $X$ contains an indecomposable summand
$M$ which over $\bk$ splits as a direct sum of Tate motives
shifted by the multiples of $d=\tfrac{p^s-1}{p-1}$
$$
M_\bk\simeq \bigoplus_{i=0}^{p-1} \zp\{di\}.
$$
\end{itemize}


Let $\pi$ be an idempotent defining $M$. We can choose $\omega$ in such a way
that the realization $\rho=\pi_*(\bar\omega)\in \Om^m(\SX\times\SY)$ 
will have the following form (cf.~\cite[p.368]{Vi07})
\begin{equation}\label{mexp}
\rho=x_n\times y_n + \sum_i 
x_i\times y_i + x_0\times y_0,
\end{equation}
where the sum in the middle is taken over all $i\in\{d,2d,\ldots,(p-2)d\}$,
$x_i\in \Om_i(\SX)$, 
$y_i\in\Om^{m-n+i}(\SY)$  are certain cobordism classes,
$x_0$ is the class of a rational point,
$x_n=\pi_*(1)$ is the class given by a proper birational morphism
$\hat X \to \SX$ and $pr(y_n)=y$ (cf.~\cite[Lemma~3.2]{Vi07}).



\paragraph{III.}
Consider the cycle $\phi^{t^{r}}_p(p_{Y*}(\rho))\in \ChO^*(\SY)$ 
defined over $k$.
It has codimension $m$, if $r=(dp-m)(p-1)>0$.

\begin{lem}\label{phip} $\phi^{t^r}_p(p_{Y*}(\rho))=
\eta_p(\SX)\cdot y+\phi^{t^r}_p(y_0)$.
\end{lem}

\begin{proof}  Applying Lemma~\ref{RDF} to each summand $x_i\times y_i$
in the middle of \eqref{mexp} we obtain
$$
\phi^{t^{r}}_p(p_{Y*}(x_i\times y_i))=\eta_p(x_i)\cdot 
S^{(n-i)/(p-1)}(pr(y_i)).
$$
Since $pr(y_i)\in \ChO^{m-n+i}(\SY)$ and 
$m-n+i\le m-n+(p-2)d=m-d<0$, it must be trivial.
Hence, only the very right and left summands of \eqref{mexp} survive.

By Rost Degree Formula \cite[Cor.~13.8]{LM07} we have
\begin{equation}
x_n=1 +\sum_{u_Z\in \LL_{>0}} u_Z\cdot [Z\to \SX].
\end{equation}
Since $\pi$ is an idempotent, 
$\pi_*\circ\pi_*(1)=\pi_*(1)$.
The latter implies that 
$$\pi_*(\sum u_Z\cdot [Z\to \SX])=
\sum u_Z\cdot \pi_*([Z\to\SX])=0.$$
Hence, applying $\eta_p$ we get
$$
0=\sum_{u_Z\in \LL_{>0}}
\eta_p\Big(u_Z\cdot \pi_*([Z\to\SX])\Big)=
u_{pt}+\sum_{u_Z\in\LL_{<n}} u_Z\cdot \pi_*([Z\to\SX]).
$$
Since $\eta_p(u_Z\cdot \pi_*([Z\to\SX]))$ is divisible by $p$ 
if $u_Z\in \LL_{<n}$, we obtain that $\eta_p(u_{pt})=0 \mod p$.

Therefore, for the very left summand of \eqref{mexp} we get
$$
\phi^{t^{r}}_p(p_{Y*}(x_n\times y_n))=\eta_p\Big([\SX]+
\sum_{u_Z\in\LL_{<n}} 
u_Z\cdot [Z]+u_{pt}\Big)\cdot pr(y_n)=\eta_p(\SX)\cdot y.
$$
Observe that $\eta_p(u_Z\cdot [Z])$ is divisible by $p$ if 
$u_Z\in\LL_{<n}$.
\end{proof}


Consider the cycle $p_{Y*}(\phi^{t^{r'}}_p(\rho))\in\ChO^*(\SY)$ 
defined over $k$.
It has codimension $m$, if $r'=(d-m)(p-1)>0$.

\begin{lem} \label{pphi}
$p_{Y*}(\phi^{t^{r'}}_p(\rho))=\phi^{t^r}(y_0)$.
\end{lem}

\begin{proof} By definition
$p_{Y*}(\phi^{t^{r'}}_p(x_i\times y_i))=p_{Y*}(\tfrac{1}{p}\cdot pr( 
S_{LN}^{d(p)}(x_i\times y_i)))$.
By multiplicativity of $S_{LN}^*$ the latter can be written as
$$
p_{Y*}(\tfrac{1}{p}\cdot pr( 
S_{LN}^{d(p)}(x_i\times y_i)))=p_{Y*}(\tfrac{1}{p}\cdot
\sum_{\alpha+\beta=n} pr(S_{LN}^\alpha(x_i))\times pr(S_{LN}^\beta(y_i)))=
$$
$$
=\sum_{\alpha +\beta =d(p)}\tfrac{1}{p}\cdot \deg\big(pr(S_{LN}^\alpha(x_i))\big)\cdot pr(S_{LN}^\beta(y_i))=
$$
Since $m<d$, $y_i$ has negative codimension for all $i<n$, therefore, 
$pr(S_{LN}^\beta(y_i))$ is divisible by $p$ for all $i<n$. 

On the other hand, we have $S^*=S_*\cdot c(-T_{\SX})^{(p)}$,
where $S_*$ (resp. $S^*$) are the (co-)homological operations.
Hence, 
$$
\deg\big(pr(S_{LN}^\alpha(x_i))\big)=
\deg\big(S_*(pr(x_i))\cdot c(-T_{\SX})^{(p)}\big).
$$
Since all Chern classes of the tangent bundle of $\SX$
are defined over $k$, and $X$ possesses a special correspondence,
according to \cite[Lemma~9.3]{Ro06} we obtain that
$\deg\big(S_*(pr(x_i))\cdot c(-T_{\SX})^{(p)}\big)$ is congruent
to $\deg(S_*(pr(x_i)))$ mod $p$.
Since $S_*$ respect push-forwards,  $\deg(S_*(pr(x_i)))$ is trivial mod $p$
for all $i>0$.
Hence, $\deg\big(pr(S_{LN}^\alpha(x_i))\big)$ is divisible by $p$ for all 
$i>0$.

Combining all together we get
$$
p_{Y*}(\phi^{t^{r'}}_p(\rho))=
\sum_{|\alpha|=n}\tfrac{1}{p}\deg\big(pr(S_{LN}^{\alpha}(x_n))\big) y+ \phi^{t^r}_p(y_0)=\phi^{t^r}_p(y_0),
$$
since $S_{LN}^{\alpha}(x_n)=0$. 
\end{proof}


By Lemmas~\ref{phip} and \ref{pphi} 
the following cycle is defined over $k$
$$
\phi^{t^r}_p(p_{Y*}(\rho))-p_{Y_*}(\phi^{t^{r'}}_p(\rho))=\eta_p(\SX)\cdot y.
$$
Since $\eta_p(\SX)\neq 0\mod p$, the cycle $y$ is defined over $k$, therefore,
$X$ is a $d$-splitting variety.

To see that $d=\tfrac{n}{p-1}$ is an optimal value 
take $Y=X$ and consider the cycle $y\in\ChO^d(\SX)$ 
defining the Tate motive $\zp\{n-d\}$ 
in the decomposition of the Rost motive $M$
over $\bk$. Since $M$ splits over $K$, $y_{\bK}$ is defined over $K$.
By condition \eqref{meq} it implies that $y$ is defined over $k$, i.e.,
$M$ splits over $k$ which contradicts to the indecomposability of $M$.
The theorem is proven.
 



\section{$F_4$-varieties}

In the present section we apply our methods to describe
all $d$-splitting varieties of type $F_4$.

\paragraph{I.}
Let $X$ be a smooth proper irreducible variety over a field $k$.
As in the beginning of the proof of \ref{mthm} given a cycle $y\in\ChO^m(\SY)$
we construct a cobordism class $\bar\omega\in\Om^m(\SX\times\SY)$ defined
over $k$.

Assume that the motive of $X$ contains a motive $M=(X,\pi)$ 
such that the idempotent $\pi$ can be written as
$$
\pi_\bk=\gamma\times \gamma^* + \sum_i x_i\times x_i^*,
$$
where $\gamma\in\Om_{(p-1)d}(\SX)$ is defined over $k$, 
$\gamma^*$ denotes its Poincare dual, i.e., $pr(\gamma\cdot\gamma^*)=pt$ and
$x_i\in \Om_i(\SX)$, $i\in\{0,d,2d,\ldots,(p-2)d\}$.
 
Then the realization 
$\rho=\pi_*(p_X^*(\gamma)\cdot\bar\omega)\in\Om^{m+n-g}(\SX\times\SY)$,
$g=(p-1)d$, 
can be written as (cf.~\eqref{mexp})
$$
\rho=\gamma \times y_g + \sum_i x_i \times y_i+ x_0\times y_0,
$$
where $x_i\in \Om_i(\SX)$, $y_i\in\Om^{m-g+i}(\SY)$,
$i\in\{d,2d,\ldots,(p-2)d\}$.

As in the proof of Lemma~\eqref{phip} we obtain that 
for $r=(dp-m)(p-1)>0$ and any $m<d$ the following cycle in $\ChO^m(\SY)$
is defined over $k$
$$
\phi^{t^r}_p(p_{Y*}(\rho))=\eta_p(\gamma)\cdot y+
\phi^{t^r}_p(y_0).
$$

The transposed cycle $\pi^t$ defines a direct summand $M^t=(X,\pi^t)$
of the motive of $X$ 
(the one which over $\bk$ contains the generic point of $X$).
The realization $\rho'=\pi^t_*(\bar\omega)\in\Om^m(\SX\times\SY)$
can be written as
$$
\rho'=\sum_i x^{(i)}\times y^{(i)} + \gamma^*\times y^{(g)},
$$
where $x^{(i)}\in\Om^i(\SX)$,
$i\in\{0,d,2d,\ldots,(p-2)d\}$, $y^{(g)}=y_0$.

Consider the cycle $\delta=p_{Y*}(p_X^*(\gamma)\cdot\phi^{t^{r'}}_p(\rho'))$.
It is defined over $k$ and for $r'=(d-m)(p-1)>0$ belongs to $\ChO^m(\SY)$.
Assume that $\delta=\phi^{t^r}_p(y_0)$, then subtracting it from 
$\phi^{t^r}_p(p_{Y*}(\rho))$ we obtain the cycle $\eta_p(\gamma)\cdot y$.
Hence, to prove that $X$ is a $d$-splitting variety it is enough
to show that $\eta_p(\gamma)\neq 0\mod p$ and $\delta=\phi^{t^r}_p(y_0)$.

\paragraph{II.}
Observe that for $p=2$ we have $\delta=\phi^{t^r}_p(y_0)$
by simple dimension reasons. Indeed, in this case the cycle $\rho'$
consists only of two terms 
$$
\rho'=x^{(0)}\times y^{(0)}+\gamma^*\times y^{(g)},
$$ 
where the first summand vanishes in $\delta$, 
since $S^{\alpha}_{LN}(x^{(0)})=0$ if
$|\alpha|>0$ and the second summand gives the required cycle $\phi^{t^r}_p(y_0)$.

For $p=3$ the cycle $\rho'$ consists of three terms
$$
\rho'=x^{(0)}\times y^{(0)}+x^{(d)}\times y^{(d)}+ \gamma^*\times y^{(g)},
$$
where again the first summand vanishes, the last gives $\phi^{t^r}(y_0)$
and the middle gives
\begin{equation}\label{mids}
\sum_{\alpha+\beta=d}\deg\big(pr(\gamma\cdot S^{\alpha}_{LN}(x^{(d)}))\big)
\cdot 
\tfrac{1}{p}pr(S^{\beta}_{LN}(y^{(d)})),\text{ where }
|\alpha|=\tfrac{p-2}{p-1}d.
\end{equation}

\paragraph{III.}
Given an $F_4$-variety $X$ we provide a cycle $\gamma$ satisfying the
conditions above as follows

\subparagraph{$p=2$:} If $X$ is generically split, then
we may assume that $X$ is of type $F_4/P_1$.
By the main result of \cite{PSZ} the cycle 
$$
\gamma=H\cdot c_{4}(T_{\SX})\cdot c_{7}(T_{\SX}),
$$ 
where
$H$ is a generator of the Picard group of $\SX$ defined over $k$,
is the generic point of a Rost motive sitting inside
the motive of $X$.
Observe that $[\gamma]=[Q_3]$ is represented by the class
of a $3$-dimensional quadric. 
Hence, $\eta_2(Q_3)=1\mod 2$
and $X$ is a $d$-splitting variety with $d=3$.

If $X$ is not generically split, i.e., $X$ is of the type $F_4/P_4$,
then $M(X)$ contains a Rost motive with $\gamma=[X]$, i.e.,
$X$ is a variety which possesses a special correspondence.
Indeed, one can show that
$X$ is a norm variety corresponding to the cohomological
invariant $f_5$.
Hence, $X$ is a $n$-splitting variety (here $n=15$).

\subparagraph{$p=3$} 
In this case all $F_4$-varieties are generically split.
Hence, by the results of \cite{PSZ} the motive of $X$ contains 
a generalized Rost motive
with the generic point given by the cycle 
$$
\gamma=H^7.
$$ 
The direct computation
using the Adjunction formula shows that 
$\eta_3([H^7])\neq 0 \mod 3$. Moreover, in this case
the expression \eqref{mids} will have the following form 
$$
\deg\big(\gamma S^2(pr(x^{(4)}))\big)\cdot \phi^{t^{12-2m}}_p(y^{(4)}).
$$
which is equal to zero since $S^2(pr(x^{(4)}))$ is equal to zero.
Hence, $X$ is a $d$-splitting variety
with $d=4$.


\begin{rem} Let $X$ be a $d$-splitting geometrically cellular
variety. As an immediate consequence of the definition
of a canonical $p$-dimension (see \cite{KM06}) we obtain 
the following inequality
$$
cd_p(X)\ge d.
$$
In the case of a variety of type $F_4/P_4$ it gives
$cd_2(X)=\dim X=15$.
\end{rem}

\paragraph{Acknowledgements} I am very grateful to Alexander Vishik and Fabien
Morel for various discussions on the subject of this paper.
This work was partially supported by SFB 701 Bielefeld and
INTAS 05-1000008-8118.

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