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\begin{document}


\title{Fields of $u$-invariant $2^r+1$}
\author{A.Vishik}
%\institute{Institute for Information Transmission Problems,
%\email{vishik@mccme.ru}}
%\titlerunning{Fields of $u$-invariant $2^r+1$}
%\authorrunning{A.Vishik}

\maketitle

\begin{abstract}
In this article we provide the uniform construction of
fields with all known $u$-invariants. We also obtain the new values for
the $u$-invariant: $2^r+1$, for $r>3$. The main tools here are the
new discrete invariant of quadrics (so-called, {\it elementary discrete
invariant}), and the methods of \cite{GPQCG} (which permit to reduce
the questions of {\it rationality} of elements of the Chow ring over
the base field to that over bigger field - the generic point of a quadric). 
%\keywords{Quadrics, Grassmannians, Chow groups, Algebraic Cobordisms, 
%Landweber-Novikov operations, Steenrod operations}
\end{abstract}

\section{Introduction}
\label{1}

The $u$-invariant of a field is defined as the maximal dimension
of anisotropic quadratic form over it. For some time people were
interested in the set of possible values of this invariant.
Using elementary quadratic form theory it is easy to establish
that $u$-invariant can not take values $3$, $5$ and $7$. The
conjecture of Kaplansky (1953) suggested that the only possible values
are the powers of $2$ (such examples were produced). This conjecture
was disproved by A.Merkurjev in 1991, who constructed fields with all
even $u$-invariants. The next challenge was to find out if the odd
$u$-invariants are possible at all. The breakthrough here was made
by O.Izhboldin who in 1999 constructed a field of $u$-invariant $9$ - see
\cite{u9}.
Still the question of other possible values remained open.
In the current article I would like to provide certain uniform method
of constructing fields with various $u$-invariants. In particular,
we get fields with even $u$-invariants without using the {\it index
reduction formula} of Merkurjev, as well as fields of $u$-invariant $2^r+1$,
for all $r\geq 3$
(note, that O.Izhboldin conjectured the existence 
and had an idea of constructing fields with such $u$-invariants). 
In the case of the $u$-invariant $9$ our construction
is different from that of O.Izhboldin. I would say that it uses substantially
more coarse invariants (like {\it generic discrete invariant of quadrics}),
while the original construction used very subtle ones (like
the {\it cokernel on the unramified cohomology}, etc. ...). So, I want
to demonstrate that the $u$-invariant questions can be solved just with
the help of ``coarse'' invariants.
The costruction is based on the ideas from \cite{GPQCG}, which, in turn,
are based on {\it symmetric operations} (see \cite{so1},\cite{so2}).


{\bf Acknowledgements:}
I'm very grateful to V.Chernousov for the very usefull
discussions and to J.Minac and U.Rehmann for the very helpful suggestions.
This text was partially written while I was visiting
Indiana University, and I would like to express my gratitude to this
institution for the support and excellent working conditions. The support
of CRDF award RUM1-2661-MO-05, INTAS 05-1000008-8118, and RFBR grant 
06-01-72550 is gratefully acknowledged.


\section{Elementary discrete invariant}
\label{2}

Everywhere below we will assume that the base-field $k$ has characteristic $0$.
Although, many things work for odd characteristics as well, the use of
algebraic cobordisms of M.Levine-F.Morel will require such assumption.

To each quadratic form $q/k$ one can assign so-called {\it generic
discrete invariant} $GDI(Q)$ - see \cite{CGQG}, which is defined as the
collection of subrings
$$
GDI(Q,i):=image(\op{CH}^*(G(Q,i))/2\row\op{CH}^*(G(Q,i)|_{\kbar})/2),
$$
for all $0\leq i\leq d:=[\ddim(Q)/2]$,
where $G(Q,i)$ is the grassmannian of $i$-dimensional projective subspaces
on $Q$.

For $J\subset I\subset\{0,\ldots,d\}$ let us denote the natural projection
between partial flag varieties $F(Q,I)\row F(Q,J)$ as $\pi$ with subindex 
$I$ with $J$ 
underlined inside it. Consider natural correspondences:
$$
f_i:G(Q,i)\stackrel{\pi_{(0,\un{i})}}{\low}F(Q,0,i)\stackrel{\pi_{(\un{0},i)}}{\row}Q.
$$

In $\op{CH}^*(G(Q,i)|_{\kbar})$ we have special classes:
$\zzj{i-d}_j\in\op{CH}^j, \ddim(Q)-d-i\leq j\leq\ddim(Q)-i$, and
$\wwj{i-d}_j\in\op{CH}^j,0\leq j\leq d-i$, defined by:
$$
\zzj{i-d}_j:=(f_i)^*(l_{\ddim(Q)-i-j});\,\,\,\,
\wwj{i-d}_j:=(f_i)^*(h^{i+j}),
$$
where $l_s\in\op{CH}_s(Q|_{\kbar})$
is the class of a projective subspace of dimension $s$, and
$h^s\in\op{CH}^s(Q|_{\kbar})$ is the class of plane section of codimension
$s$.

Let us denote as $\zj{i-d}_j$ and $\wj{i-d}_j$ the same classes in $\op{CH}^*/2$.
We will call classes $\zj{i-d}_j$ {\it elementary}. Notice, that the classes
$\wj{i-d}_j$ always belong to $GDI(Q,i)$.




Let $Tav_i$ be the tautological $(i+1)$-dimensional vector bundle on $G(Q,i)$.

\begin{prop}
\label{chern}
For any $0\leq i\leq d$, and $\ddim(Q)-d-i\leq j\leq\ddim(Q)-i$,
$$
c_{\bullet}(-Tav_i)=\sum_{j=0}^{d-i}\wwj{i-d}_j+2\sum_{d-i<j\leq\ddim(Q)-i}
\zzj{i-d}_j.
$$
\end{prop}

\begin{proof}
Since $Tav_0=\co(-1)$ on $Q$, the statement is true for $i=0$.
Consider the correspondence
$$
f_i:G(Q,i)\stackrel{\pi_{(0,\un{i})}}{\low}F(Q,0,i)\stackrel{\pi_{(\un{0},i)}}{\row}Q.
$$
Notice, that $F(Q,0,i)$ is naturally identified with the projective bundle
$\pp_{G(Q,i)}(Tav_i)$, and the sheaf $\pi_{(\un{0},i)}^*(Tav_0)$ is naturally
identified with $\co(-1)$. Thus, 
$$
(\pi_{(0,\un{i})})_*(\pi_{(\un{0},i)})^*(c_{\bullet}(-Tav_0))=
(\pi_{(0,\un{i})})_*(c_{\bullet}(-\co(-1)))=c_{\bullet}(-Tav_i).
$$ 
\Qed
\end{proof}

\begin{remark} 
In particular, for $i=d$ we get another proof of 
\cite[Theorem 2.5(3)]{CGQG}.
\end{remark}


\begin{defi}
Define the elementary discrete invariant $EDI(Q)$ as the collection
of subsets $EDI(Q,i)$ consisting of those $j$ that
$\zj{i-d}_j\in GDI(Q,i)$.
\end{defi}

One can visualize $EDI(Q)$ as the coordinate $d\times d$ square,
where some integral nodes are marked, each row corresponds to
particular grassmannian, and the codimension of a ``node'' is
decreasing up and right. The lower row corresponds to a quadric
itself, and the upper one - to the last grassmannian. The
south-west corner is marked if and only if $Q$ is isotropic.

\begin{exa}
The $EDI(Q)$ for the $10$-dimensional excellent form looks as:
$$
\def\objectstyle{\scriptstyle}
\def\labelstyle{\scriptstyle}
\xymatrix @-1.3pc{
\bub &\bub &\bub &\bub &\circ \\
\bub &\bub &\circ &\bub &\circ \\
\bub &\bub &\circ &\circ &\circ \\
\circ &\bub &\circ &\circ &\circ \\
\circ &\circ &\circ &\circ &\circ
}
$$
\end{exa}


The following statement puts serious constraints on possible markings.

\begin{prop}
\label{uul}
Let $0\leq i<d$, and $j\in EDI(Q,i)$. Then $j,j-1\in EDI(Q,i+1)$.
\end{prop}

This can be visualized as:
$$
\def\objectstyle{\scriptstyle}
\def\labelstyle{\scriptstyle}
\xymatrix @-1.3pc{
\bub & \bub \\
& \bub\ar @{->} [u]\ar @{->} [ul]
}
$$

\begin{proof}

The Proposition easily follows from the next Lemma.
Let us temporarily denote $\pi_{(\un{i},i+1)}$ as $\alpha$, and
$\pi_{(i,\un{i+1})}$ as $\beta$.

\begin{lem}
\label{pi*}
\begin{equation*}
\begin{split}
&\alpha^*(\zzj{i-d}_j)=\beta^*(\zzj{i+1-d}_j)+
c_1(\co(1))\cdot
\beta^*(\zzj{i+1-d}_{j-1});\\
&\alpha^*(\wwj{i-d}_j)=\beta^*(\wwj{i+1-d}_j)+
c_1(\co(1))\cdot
\beta^*(\wwj{i+1-d}_{j-1}), 0\leq j<d-i;\\
&\alpha^*(\wwj{i-d}_{d-i})=2\beta^*(\zzj{i+1-d}_{d-i})+
c_1(\co(1))\cdot\beta^*(\wwj{i+1-d}_{d-i-1}),
\end{split}
\end{equation*}
where $\co(1)$ is the standard sheaf on the projective bundle
$$
F(Q,i,i+1)=\pp_{G(Q,i+1)}(Tav^{\vee}_{i+1}).
$$
\end{lem}

\begin{proof}
By definition,
$\zzj{i-d}_j,\wwj{i-d}_j$ have the form 
$(\pi_{(0,\un{i})})_*(\pi_{(\un{0},i)})^*(x)$, for certain
$x\in\op{CH}^*(Q|_{\kbar})$.
Since the square
$$
\begin{CD}
F(Q,i,i+1)&@<{\pi_{(0,\un{i},\un{i+1})}}<<&F(Q,0,i,i+1)\\
@V{\pi_{(\un{i},i+1)}}VV&&@VV{\pi_{(\un{0},\un{i},i+1)}}V\\
G(Q,i)&@<<{\pi_{(0,\un{i})}}<&F(Q,0,i)
\end{CD}
$$
is transversal cartesian, $(\pi_{(\un{i},i+1)})^*$ of these elements is equal to
\begin{equation*}
\begin{split}
(\pi_{(0,\un{i},\un{i+1})})_*(\pi_{(\un{0},\un{i},i+1)})^*(\pi_{(\un{0},i)})^*(x)=
(\pi_{(0,\un{i},\un{i+1})})_*(\pi_{(\un{0},i,\un{i+1})})^*(\pi_{(\un{0},i+1)})^*(x).
\end{split}
\end{equation*}
Variety $F(Q,0,i,i+1)$ is naturally a divisor $D$ on the
transversal product $F(Q,0,i+1)\times_{G(Q,i+1)}F(Q,i,i+1)$ with
$\co(D)=\pi_{(\un{0},i,i+1)}^*(\co(h))\otimes\pi_{(0,\un{i},\un{i+1})}^*(\co(1))$,
where $\co(h)$ is the sheaf given by the hyperplane section on
$Q$. Then
\begin{equation*}
\begin{split}
(\pi_{(0,\un{i},\un{i+1})})_*(\pi_{(\un{0},i,\un{i+1})})^*(\pi_{(\un{0},i+1)})^*(x)=&\\
c_1(\co(1))\cdot
(\pi_{(i,\un{i+1})})^*(\pi_{(0,\un{i+1})})_*(\pi_{(\un{0},i+1)})^*(x)+&\\
(\pi_{(i,\un{i+1})})^*(\pi_{(0,\un{i+1})})_*(\pi_{(\un{0},i+1)})^*(h\cdot x)&.
\end{split}
\end{equation*}
It remains to plug in the appropriate $x$.
\Qed
\end{proof}

Notice, that the projectors giving the decomposition:
$$
\op{CH}^*(\pp_{G(Q,i+1)}(Tav^{\vee}_{i+1}))/2=
\oplus_{l=0}^{i}c_1(\co(1))^l\cdot\op{CH}^*(G(Q,i+1))/2
$$
are defined over the base field. Thus, the element of this group is
defined over $k$ if and only if all of it's coordinates are. Since
the cycle $\zj{i-d}_j$ is defined over $k$, by Lemma \ref{pi*} the
cycles $\zj{i+1-d}_j$ and $\zj{i+1-d}_{j-1}$ are defined too. \Qed
\end{proof}



If we have a codimension $1$ subquadric $P$ of a quadric $Q$, the
$EDI$'s of them are related by the following.

\begin{prop}
\label{subform} Let $d_P:=[\ddim(P)/2]$ and $d_Q:=[\ddim(Q)/2]$.
Then
$$
\zj{i-d_Q}_j(Q)\,\text{is
defined}\,\Rightarrow\zj{i-d_P}_j(P)\,\text{is defined}\,;
$$
$$
\zj{i-d_P}_j(P)\,\text{is
defined}\,\Rightarrow\zj{i+1-d_Q}_j(Q)\,\text{is defined}\,.
$$
\end{prop}

\begin{proof}
Consider the natural embedding: $e:G(P,i)\row G(Q,i)$.
Then, it follows from the definition that $e^*(\zj{i-d_Q}_j)=
\zj{i-d_P}_j$. To prove the second statement just observe that
$Q$ is a codimension 1 subquadric in $P'$, where $p'=p\perp\hh$,
and $\zj{i-d_P}_j(P)$ is defined $\Rightarrow$ $\zj{i-d_{P'}}_j(P')$ is
defined.
\Qed
\end{proof}


Unfortunately, the Steenrod operations, in general, do not act on the
$EDI(Q,i)$,
since they do not preserve {\it elementary classes}. But they act in
the lower and the upper row: for the quadric itself, and for the
last grassmannian. 
Also, it follows from \cite[Main Theorem 5.8]{CGQG} that
$EDI(Q,d)$ carries the same information as $GDI(Q,d)$. The same is true about
$EDI(Q,0)$ and $GDI(Q,0)$ by the evident reasons.

The action of the Steenrod operations on the {\it elementary classes}
can be described as follows.

\begin{prop}
\label{steenrod}
Let $0\leq i\leq d$, and $\ddim(Q)-d-i\leq j\leq\ddim(Q)-i$, then
$$
S^m(\zj{i-d}_j)=\sum_{l=0}^{d-i}\binom{j-l}{m-l}
\zj{i-d}_{j+m-l}\cdot\wj{i-d}_l,
$$
where {\it elementary classes} of codimension more than $\ddim(Q)-i$ are
assumed to be $0$.
\end{prop}

\begin{proof}
We know that $S^{\bullet}(l_s)=(1+h)^{\ddim(Q)-s+1}l_s$. Since
$(\pi_{\un{0},i})^*(\co(1))$ is the sheaf $\co(1)$ on
$F(Q,0,i)=\pp_{G(Q,i)}(Tav_i)$, 
$$
S_{\bullet}(\pi_{(\un{0},i)})^*(l_{\ddim(Q)-i-j})=c_{\bullet}(-T_{F(Q,0,i)})\cdot
(1+H)^{i+j+1}\cdot (\pi_{(\un{0},i)})^*(l_{\ddim(Q)-i-j}),
$$
where $H=c_1(\co(1))$.
Since $S_{\bullet}$ commutes with the push-forward morphisms,
\begin{equation*}
\begin{split}
S^{\bullet}(\zj{i-d}_j)=S^{\bullet}(\pi_{(0,\un{i})})_*(\pi_{(\un{0},i)})^*
(l_{\ddim(Q)-i-j})=&\\
(\pi_{(0,\un{i})})_*(c_{\bullet}(-T_{fiber})\cdot(1+H)^{i+j+1}\cdot 
(\pi_{(\un{0},i)})^*(l_{\ddim(Q)-i-j})).&\end{split}
\end{equation*}
Since $c_{\bullet}(-T_{fiber})=c_{\bullet}(-Tav_i\otimes\co(1))$, 
in the light of Proposition \ref{chern},
$(\,mod\,2)$ this is equal to
$\sum_{l=0}^{d-i}\wj{i-d}_l(1+H)^{-i-1-l}$.
Thus,
\begin{equation*}
\begin{split}
S^{\bullet}(\zj{i-d}_j)=(\sum_{l=0}^{d-i}\wj{i-d}_l)(\pi_{(0,\un{i})})_*(\pi_{(\un{0},i)})^*
((1+h)^{j-l}l_{\ddim(Q)-i-j})=&\\
\sum_{r\geq 0}\sum_{l=0}^{d-i}\binom{j-l}{r-l}\zj{i-d}_{j+r-l}\wj{i-d}_{l}.&
\end{split}
\end{equation*}
\end{proof}
\Qed


\begin{remark}
In particular, for $i=d$ we get a new proof of 
\cite[Theorem 4.1]{CGQG}.
\end{remark}


The following fact is well-known (see, for example, \cite{HB}).
We will give an independent proof below.

\begin{prop}
\label{generate}
The ring $\op{CH}^*(G(Q,i)|_{\kbar})$ is generated by the classes
$$
\zzj{i-d}_j, \ddim(Q)-d-i\leq j\leq\ddim(Q)-i,\,\,\,\text{and}\,\,\,
\wwj{i-d}_j,0\leq j\leq d-i.
$$
\end{prop}

\begin{proof}
For $0\leq l\leq i$, let us denote the pull back of $\zzj{l-d}_j$ to
$F(Q,0,\ldots,i)$ by the same symbol. On this flag variety we have natural
line bundles $\cl_k:=Tav_k/Tav_{k-1}$. Let us denote $h_k:=c_1(\cl_k^{-1})$.

\begin{lem}
\label{Fgenerate}
Suppose that $Q|_E$ is completely split. Then
the ring $\op{CH}^*(F(Q,0,\ldots,i)|_E)$ is generated by
$\wwj{l-d}_j,0\leq l\leq i, 1\leq j\leq d-l$, and
$\zzj{l-d}_j,0\leq l\leq i, \ddim(Q)-d-l\leq j\leq\ddim(Q)-l$.
\end{lem}

\begin{proof}
Induction on $i$. For $i=0$ the statement is evident.

\begin{statement}
\label{yx}
Let $\pi:Y\row X$ be smooth morphism, $X$-smooth. For $x\in X^{(r)}$, $Y_x$ be
the fiber over the point $x$. Let $\zeta$ denote the generic point of $X$,
and $s_x:\op{CH}^*(Y_{\zeta})\row\op{CH}^*(Y_x)$ be the specialization map.
Let $N\subset\op{CH}^*(Y)$ be a subgroup.
Suppose:
\begin{itemize}
\item[$(a)$\ ] the map $N\row\op{CH}^*(Y|_{\zeta})$ is surjective;
\item[$(b)$\ ] all the maps $s_x$ are surjective.
\end{itemize}
Then $\op{CH}^*(Y)=N\cdot\pi^*(\op{CH}^*(X))$.
\end{statement}

\begin{proof}
On $\op{CH}^*(Y)$ we have decreasing filtration $F^{\bullet}$, where
$F^r$ consists of classes, having a representative with the
image under $\pi$ of codimension $\geq r$.
This gives the surjection:
$$
\oplus_r\oplus_{x\in X^{(r)}}\op{CH}^*(Y_x)\row
gr_{F^{\bullet}}\op{CH}^{r+*}(Y).
$$
Let $[x]\in\op{CH}^r(X)$ be the class represented by the closure of $x$.
Clearly, the image of $\pi^*([x])\cdot N$ covers the image of
$\op{CH}^*(Y_x)$ in $F^r/F^{r+1}$. 
\Qed
\end{proof}

Consider the projection
$$
\pi_{(\un{0},\ldots,\un{i-1},i)}:F(Q,0,\ldots,i-1,i)\row F(Q,0,\ldots,i-1).
$$
Let $Q_{\{i\},x}/E(x)$ be the fiber of this projection over the point $x$. It
is a completely split quadric of dimension $\ddim(Q)-2i$. Thus, the condition
$(b)$ of the Statement \ref{yx} is satisfied. Since 
$$
\zzj{i-d}_j|_{Q_{\{i\},\zeta}}=l_{\ddim(Q)-2i-j},\hspace{7mm}
\wwj{i-d}_j|_{Q_{\{i\},\zeta}}=h^j
$$ 
we can take $N$ additively generated
by  $\zzj{i-d}_j,\ddim(Q)-d-i\leq j\leq \ddim(Q)-2i$, and
$\wwj{i-d}_j,0\leq j\leq d-i$. Then the condition $(a)$ will be satisfied too.
The induction step follows. 
\Qed
\end{proof}

Lemma \ref{pi*} implies that the $\zzj{l-d}_j,\wwj{l-d}_j$, for $l<i$ are
expressible in terms of $\zzj{i-d}_k,\wwj{i-d}_k$ and $h_m,0\leq m\leq i$.
Let $A\subset\op{CH}^*(G(Q,i))$ be the subring generated by
$\zzj{i-d}_j,\wwj{i-d}_j$. Since $F(Q,0,\ldots,i)$ is a variety of complete
flags of subspaces of the vector bundle $Tav_i$ on $G(Q,i)$, the ring
$\op{CH}^*(F(Q,0,\ldots,i))$ is isomorphic to 
$$
\op{CH}^*(G(Q,i))[h_0,\ldots,h_i]/(\sigma_r(h)-c_r(Tav_i^{\vee}),1\leq r\leq i+1).
$$
But $c_r(Tav_i^{\vee})\in A$, by Proposition \ref{chern}. Since $A$ and
$h_m,0\leq m\leq i$ generate $\op{CH}^*(F(Q,0,\ldots,i))$, $A$ must coinside
with $\op{CH}^*(G(Q,i))$.
\Qed
\end{proof}


In particular, since the cycles $\wwj{i-d}_j$ are defined over $k$, we have:
\begin{cor}
\label{=<d-i} The graded part of $\op{CH}^*(G(i,Q)|_{\kbar})$ of
degree less or equal $(d-i)$ consists of classes which are defined over
$k$.
\end{cor}


Notice, that for $i=d$,
$\op{CH}^*(G(Q,d)|_{\kbar})/2$ is generated as a ring by
$\zj{0}_j$, and moreover, $GDI(Q,d)$ is always generated as a ring
by the subset of $\zj{0}_j$ contained in it - see \cite[Main
Theorem 5.8]{CGQG}.

















\section{Generic points of quadrics and Chow groups}
\label{3}

In this section I would like to remind the principal result of
\cite{GPQCG}. Let $Q$ be smooth projective quadric, $Y$ be smooth 
quasiprojective variety, and
$\ov{y}\in\op{CH}^m(Y|_{\kbar})/2$. This will be our main tool in
the construction of fields with various $u$-invariants.

\begin{thm}
\label{3.5} {\rm (\cite[Corollary 3.5]{GPQCG},\cite[Theorem 4.3]{so2}.)} \\
Suppose
$m<[\ddim(Q)+1/2]$. Then 
$$
\ov{y}|_{\ov{k(Q)}}\,\,\,\text{is defined over}\,\,\,k(Q)\,\,\,\Leftrightarrow\,\,\,
\ov{y}\,\,\,\text{is defined over}\,\,\,k.
$$
\end{thm}


\begin{exa}
\label{pfister} Let $\alpha=\{a_1,\ldots,a_n\}\in\K^M_n(k)/2$ be a
nonzero pure symbol, and $Q_{\alpha}$ be the respective
anisotropic Pfister quadric. Then in $EDI(Q_{\alpha})$ the marked
nodes will be exactly those ones which live above the main (N-W to
S-E) diagonal.
$$
\def\objectstyle{\scriptstyle}
\def\labelstyle{\scriptstyle}
\xymatrix @-1.3pc{
\circ &\bub &\bub &\dots &\bub &\bub \\
\circ &\circ &\bub &\dots &\bub &\bub \\
\circ &\circ &\circ &\dots &\bub &\bub \\
\dots &\dots &\dots &\dots &\dots &\dots \\
\circ &\circ &\circ &\dots &\circ &\bub \\
\circ &\circ &\circ &\dots &\circ &\circ }
$$
\end{exa}

Really, over own generic point $k(Q_{\alpha})$ quadric
$Q_{\alpha}$ becomes hyperbolic, and so, all the elementary cycles
are defined there. Then the cycles above the main diagonal got to
be defined already over the base field, since their codimension is
smaller than $[\ddim(Q_{\alpha})+1/2]$. On the other hand, the N-W
corner could not be defined over $k$, since otherwise all the
elementary cycles on the last grassmannian of $Q_{\alpha}$ would
be defined over $k$, but the product of all these cycles is the
class of a rational point on this grassmannian (see \cite{CGQG}).
Since $Q_{\alpha}$ is not hyperbolic over $k$, this is impossible.
The rest of the picture follows from Proposition \ref{uul}.

The proof of Theorem \ref{3.5} uses Algebraic Cobordisms of
M.Levine-F.Morel. Let me say few words about the latter.



\subsection{Algebraic Cobordisms}
In \cite{LM1,LM2,LMI,L} M.Levine and F.Morel have constructed the
universal oriented generalized cohomology theory $\om^*$ on the category
of smooth quasiprojective varieties over the field $k$ of
characteristic $0$, called {\it Algebraic Cobordism}.

For any smooth quasiprojective $X$ over $k$, the additive group
$\om^*(X)$ is generated by the classes $[v:V\row X]$ of
projective maps from smooth varieties $V$ subject to certain
relations, and the upper grading is the codimensional one. There
is natural morphism of theories $pr:\om^*\row\op{CH}^*$. The main
properties of $\om^*$ are:
\begin{itemize}
\item[$(1)$\ ] $\om^*(\op{Spec}(k))=\laz=\op{MU}(pt)$ - the
Lazard ring, and the isomorphism is given by the topological
realization functor; \item[$(2)$\ ]
$\op{CH}^*(X)=\om^*(X)/\laz^{<0}\cdot\om^*(X)$.
\end{itemize}

On $\om^*$ there is the action of the Landweber-Novikov
operations. Let
$R(\sigma_1,\sigma_2,\ldots)\in\laz[\sigma_1,\sigma_2,\ldots]$ be
some polynomial, where we assume $\ddeg(\sigma_i)=i$. Then
$\oln{R}:\om^*\row\om^{*+\ddeg(R)}$ is given by:
$$
\oln{R}([v:V\row X]):=v_*(R(c_1,c_2,\ldots)\cdot 1_V),
$$
where $c_j=c_j(-T_V+v^*T_X)$.

If $R=\sigma_i$, we will denote the respective operation simply as
$\oln{i}$. The following statement follows from the definition of
Steenrod and Landweber-Novikov operations - see P.Brosnan \cite{PB1},
A.Merkurjev \cite{M2}, and M.Levine \cite{Lst}

\begin{prop}
\label{ln-s} There is commutative
square:
$$ \begin{CD} \om^*(X)&@>{\oln{i}}>>&\om^{*+i}(X)\\
@VVV&&@VVV\\
\op{CH}^*(X)/2&@>>{S^i}>&\op{CH}^{*+i}(X)/2,
\end{CD}
$$
\end{prop}
where $S^i$ is the Steenrod operation (\cite{VoOp,PB1}).

In particular, $pr\circ\oln{i}(\om^m)\subset
2\cdot\op{CH}^{i+m}$, if $i>m$, and
$pr\circ(\oln{m}-\square)(\om^m)\in 2\cdot\op{CH}^{2m}$. This
implies that $(\,\text{modulo}\,2-\text{torsion}\,)$ we have well
defined maps $\frac{pr\circ\oln{i}}{2}$ and
$\frac{pr\circ(\oln{m}-\square)}{2}$. In reality, these maps can
be lifted to a well defined, so-called, {\it symmetric operations}
$\Phi^{t^{i-m}}:\om^m\row\om^{m+i}$ - see \cite{so2}. Since
over algebraically closed field all our varieties are cellular,
and thus, the Chow groups of them are torsion-free, we will not
need such subtleties, but we will keep the notation from
\cite{so2}, and denote our maps as $\phi^{t^{i-m}}$.



\subsection{Beyond the Theorem \ref{3.5}}

Below we will need to study the relation between the rationality
of $\ov{y}$ and $\ov{y}|_{\ov{k(Q)}}$ for $\op{codim}(\ov{y})$
slightly bigger than $[\ddim(Q)+1/2]$. The methods involved are
just the same as are employed for the proof of Theorem \ref{3.5}.

Let $Y$ be smooth quasiprojective variety, $Q$ be smooth
projective quadric. Let $v\in\op{CH}^*(Y\times Q)/2$ be some
element, and $w\in\om^*(Y\times Q)$ be it's arbitrary lifting.
Over $\kbar$, quadric $Q$ becomes a cellular variety with basis of
Chow groups and Cobordisms given by the set $\{l_i,h^i\}_{0\leq
i\leq [\ddim(Q)/2]}$ of projective subspaces and plane sections.
This implies that
$$
\op{CH}^*(Y\times
Q|_{\kbar})=\oplus_{i=0}^{[\ddim(Q)/2]}(\op{CH}^*(Y|_{\kbar})\cdot
l_i\oplus\op{CH}^*(Y|_{\kbar})\cdot h^i),\,\,\,\text{and}
$$
$$
\om^*(Y\times
Q|_{\kbar})=\oplus_{i=0}^{[\ddim(Q)/2]}(\om^*(Y|_{\kbar})\cdot
l_i\oplus\om^*(Y|_{\kbar})\cdot h^i)
$$
- see \cite[Section 2]{VY}.
In particular, 
$$
\ov{v}= \sum_{i=0}^{[\ddim(Q)/2]}(\ov{v}^i\cdot
h^i+\ov{v}_i\cdot l_i),\,\,\,\,\,\,\text{and}\,\,\,\,\,\,
\ov{w}=\sum_{i=0}^{[\ddim(Q)/2]}(\ov{w}^i\cdot h^i+\ov{w}_i\cdot
l_i).
$$

\begin{prop}
\label{4n+1} Let $Q$ be smooth projective quadric of dimension
$\geq 4n-1$, $Y$ be smooth quasiprojective variety, and
$v\in\op{CH}^{2n+1}(Y\times Q)/2$ be some element. Then the class
$$
\ov{v}^0+S^1(\ov{v}^1)+\ov{v}^1\cdot\ov{v}_{\ddim(Q)-2n}+\ov{v}^0\cdot\ov{v}_{\ddim(Q)-2n-1}
$$
in $\op{CH}^{2n+1}(Y|_{\kbar})/(2;2-torsion)$ is defined over $k$.
\end{prop}

\begin{cor}
\label{cor4n+1} Let $Q$ be smooth projective quadric of
dimension $\geq 4n-1$, $Y$ be smooth quasiprojective variety, and
$\ov{y}\in\op{CH}^{2n+1}(Y|_{\kbar})/2$ is defined over $k(Q)$.
Then, either
\begin{itemize}
\item[$(a)$\ ] $\zj{-[\ddim(Q)/2]}_{2n+1}(Q|_{k(Y)})$ is defined; or
\item[$(b)$\ ] for certain
$\ov{v}^1\in\op{CH}^{2n}(Y|_{\kbar})/2$, and for certain divisor
$\ov{v}_{\ddim(Q)-2n}\in\op{CH}^1(Y|_{\kbar})/2$, the element
$$
\ov{y}+S^1(\ov{v}^1)+\ov{v}^1\cdot\ov{v}_{\ddim(Q)-2n}
$$
in $\op{CH}^{2n+1}(Y|_{\kbar})/(2;2-torsion)$ is defined over $k$.
\end{itemize}
\end{cor}

\begin{proof}
Since $\ov{y}$ is defined over $k(Q)$, there is
$x\in\op{CH}^{2n+1}(Y|_{k(Q)})/2$ such that
$\ov{x}=\ov{y}|_{\ov{k(Q)}}$. Using the surjection
$\op{CH}^*(Y\times Q)\twoheadrightarrow\op{CH}^*(Y|_{k(Q)})$ lift
the $x$ to an element $v\in\op{CH}^{2n+1}(Y\times Q)/2$. Then
$\ov{v}=\sum_{i=0}^{[\ddim(Q)/2]}(\ov{v}^i\cdot h^i+\ov{v}_i\cdot
l_i)$, where $\ov{v}^0|_{\ov{k(Q)}}=\ov{y}|_{\ov{k(Q)}}$. But for
any extension of fields $F/{\kbar}$ (with smaller one
algebraically closed), the restriction morphism on Chow groups
(with any coefficients) is injective by the specialization
arguments. Thus, $\ov{v}^0=\ov{y}$. It remains to apply the
Proposition \ref{4n+1}, and observe that if
$\ov{v}_{\ddim(Q)-2n-1}\in\op{CH}^0(Y|_{\kbar})=\zz/2\cdot 1$ is
nonzero, then the cycle $l_{\ddim(Q)-2n-1}=\zj{-[\ddim(Q)/2]}_{2n+1}$
is defined over $k(Y)$. Really, this cycle is just equal to
$\ov{v}|_{\ov{k(Y)}}$. \Qed
\end{proof}

\begin{remark} 
One can get rid of factoring $(2-torsion)$ in the statements
above by using genuine {\it symmetric operations} 
(see \cite{so2}, cf. \cite{GPQCG})
instead of Landweber-Novikov operations. As was explained above, for our
purposes it is irrelevant.
\end{remark}

Before proving the Proposition let us study a bit some special
power series.
Denote as $\gamma(t)\in\zz/2[[t]]$ the power series $1+\sum_{i\geq
0}t^{2^i}$. Then $\gamma(t)$ satisfies the equation:
$$
\gamma^2-\gamma=t,
$$
and generates the quadratic extension of $\zz/2(t)$. In
particular, for any $m\geq 0$, $\gamma^m=a_m\gamma+b_m$ for
certain unique $a_m,b_m\in \zz/2(t)$. The following statement is
clear.

\begin{observation}
\label{obs}
\begin{itemize}
\item[$(1)$\ ] $a_{m+1}=a_m+b_m$,\ \ \ \ $b_{m+1}=ta_m$
\item[$(2)$\ ] $a_m$ and $b_m$ are polynomials in $t$ of degree
$\leq$ $[m-1/2]$ and $[m/2]$, respectively.
\end{itemize}
\end{observation}

For the power series $\beta(t)$ let us denote as $(\beta)_{\leq
l}$ the polynomial $\sum_{j=0}^l\beta_jt^j$, and as $(\beta)_{>l}$
- the remaining part $\beta-(\beta)_{\leq l}$.

\begin{lem}
\label{ambm}
$$
a_m=(\gamma^m)_{\leq [m/2]}=(\gamma^m)_{\leq [m-1/2]}
$$
\end{lem}

\begin{proof}
Let $m=2^k+m_1$, where $0\leq m_1<2^k$. Then
$\gamma^m=\gamma^{2^k}\cdot\gamma^{m_1}=(a_{m_1}\gamma+b_{m_1})+
O(t^{2^{k}})=
(1+\sum_{i=0}^{k-1}t^{2^i})a_{m_1}+b_{m_1}+O(t^{2^k})$. Observation
\ref{obs} implies that $(\gamma^m)_{\leq
[m/2]}=(1+\sum_{i=0}^{k-1}t^{2^i})a_{m_1}+b_{m_1}$. On the other
hand, $\gamma^{2^k}=\gamma+(\sum_{j=1}^{k-1}t^{2^j})$, thus
$\gamma^m$ is equal to
\begin{equation*}
\begin{split}
(\gamma+(\sum_{j=1}^{k-1}t^{2^j}))(a_{m_1}\gamma+b_{m_1})=&\\
a_{m_1}\gamma+a_{m_1}t+(\sum_{j=1}^{k-1}t^{2^j})a_{m_1}\gamma+b_{m_1}\gamma+
(\sum_{j=1}^{k-1}t^{2^j})b_{m_1}=&\\
((1+\sum_{i=0}^{k-1}t^{2^i})a_{m_1}+b_{m_1})\gamma+
(ta_{m_1}+(\sum_{j=1}^{k-1}t^{2^j})b_{m_1}).
\end{split}
\end{equation*}
Hence,
$a_m=((1+\sum_{i=0}^{k-1}t^{2^i})a_{m_1}+b_{m_1})=(\gamma^m)_{\leq
[m/2]}$. The second equality follows from Observation \ref{obs}(2)
\Qed
\end{proof}

Lemma \ref{ambm} implies that
$$
\gamma^m=(\gamma^m)_{\leq
[m-1/2]}\cdot\gamma+t(\gamma^{m-1})_{\leq [m-2/2]}.
$$

\begin{lem}
\label{um>}
$$
(\gamma^m)_{>[m/2]}=t^{m}(1+mt)+O(t^{m+2})
$$
\end{lem}

\begin{proof}
Use induction on $m$ and on the number of $1$'s in the binary
presentation of $m$. For $m=2^k$ the statement is clear. Let now
$m=2^k+m_1$, where $0<m_1<2^k$. We have: $(\gamma^m)_{>[m/2]}=
((\gamma^{m+1})_{\leq [m/2]})_{>{[m/2]}}+t((\gamma^m)_{\leq
[m-1/2]}\cdot\gamma^{-1})_{>([m/2]-1)}=t(a_m\cdot\gamma^{-1})_{>[m/2]-1}$.

$a_m=(\gamma^m)_{\leq [m/2]}=(\gamma^{2^k}\cdot\gamma^{m_1})_{\leq
[m/2]}=(\gamma^{m_1})_{\leq [m/2]}=(a_{m_1}\gamma+b_{m_1})_{\leq
[m/2]}$, and since degrees of $a_{m_1}$ and $b_{m_1}$ are no more
than $[m_1/2]$, this expression should be equal to $\gamma^{m_1}+
a_{m_1}t^{2^k}+O(t^{2^{k+1}})$. Then
\begin{equation*}
\begin{split}
a_m\cdot\gamma^{-1}=
\gamma^{m_1-1}+a_{m_1}\gamma^{-1}t^{2^k}+O(t^{2^{k+1}})=&\\
(a_{m_1-1}\gamma+b_{m_1-1})+
a_{m_1}\gamma^{-1}t^{2^k}+O(t^{2^{k+1}})=&\\
(a_{m_1-1}(1+\sum_{i=0}^{k-1}t^{2^i})+b_{m_1-1})+
t^{2^k}(a_{m_1-1}+a_{m_1}\gamma^{-1})+O(t^{2^{k+1}}).&
\end{split}
\end{equation*}
Since the degree of $a_{m_1-1}$ is no more than $[m_1/2]-1$, using
Observation \ref{a}, we get:
\begin{equation*}
\begin{split}
(a_m\cdot\gamma^{-1})_{>[m/2]-1}=t^{2^k}(a_{m_1-1}+a_{m_1}\gamma^{-1})+
O(t^{2^{k+1}})=&\\
t^{2^k}(\gamma^{m_1-2}+a_{m_1-1}\gamma^{-1})+O(t^{2^{k+1}})=&\\
t^{2^k}\gamma^{-1}(\gamma^{m_1-1}+(\gamma^{m_1-1})_{\leq
[m_1-1/2]})+O(t^{2^{k+1}}).&
\end{split}
\end{equation*}
Consequently,
$(\gamma^m)_{>[m/2]}=t^{2^k+1}(\gamma^{m_1-1})_{>[m_1-1/2]}\cdot\gamma^{-1}+
O(t^{2^{k+1}})$.
And, by the inductive assumption, this is equal to
$$
t^{2^k+1}(t^{m_1-1}(1+(m_1-1)t))\gamma^{-1}+O(t^{2^k+m_1+2})=
t^m(1+mt)+O(t^{m+2}).
$$
\Qed
\end{proof}


\begin{cor}
\label{amu-1}
$$
(a_m\cdot\gamma^{-1})_{>[m/2]-1}=t^{m-1}(1+mt)+O(t^{m+1})
$$
\end{cor}

Observe now that $\gamma^{-1}(t)=\sum_{i\geq 0}t^{2^i-1}$. Denote
as $\delta(t)$ the polynomial $a_{2n+1}(t)$. Then
\begin{equation}
\label{a}
\delta(t)\gamma^{-1}(t)=\alpha(t)+t^{2n}+t^{2n+1}+O(t^{2n+2}),
\end{equation}
where $\delta(t)$ and $\alpha(t)$ are polynomials of degree $\leq
n$. Observation \ref{obs}(1) shows that $\delta=1+t+\ldots$.








\begin{prof}{Proof of Proposition \ref{4n+1}}
The case of $\ddim(Q)\geq 4n-1$ can be reduced to that of
$\ddim(Q)=4n-1$ by considering arbitrary subquadric $Q'\subset Q$
of dimension $4n-1$, and restricting $v$ to $Y\times Q'$. So, we
will assume that $\ddim(Q)=4n-1$.

Let $Q_s\stackrel{e_s}{\row}Q$ be arbitrary smooth subquadric of
$Q$ of dimension $s$. Denote as $w(s)$ the class $(id\times
e_s)^*(w)\in\om^{2n+1}(Q_s\times Y)$. Then
$$
\ov{w(s)}=\sum_{0\leq i\leq\op{min}(2n-1,s)}\ov{w}^i\cdot
h^i+\sum_{4n-s-1\leq j\leq 2n-1}\ov{w}_j\cdot l_{j-4n+s+1}.
$$
Let $\pi_{Y,s}:Q_s\times Y\row Y$ be the natural projections.

Consider the element
\begin{equation*}
\begin{split}
u:=(\pi_{Y,2n+1})_*\phi^{t^0}(w(2n+1))+
\sum_{p=n+1}^{2n+1}\delta_{2n+1-p}\phi^{t^{2p-(2n+1)}}(\pi_{Y,p})_*(w(p))
\end{split}
\end{equation*}
in $\op{CH}^{2n+1}(Y)/2$, where $\delta_j$ are the coefficients of
the power series $\delta$ above. Let us compute $\ov{u}$. Since we
are computing modulo $2$-torsion, it is sufficient to compute
$2\ov{u}$, which is equal to the Chow-trace of
\begin{equation*}
\begin{split}
(\pi_{Y,2n+1})_*(\oln{2n+1}-\square)(\ov{w}(2n+1))+
\sum_{p=n+1}^{2n+1}\delta_{2n+1-p}\oln{p}(\pi_{Y,p})_*(\ov{w}(p)).
\end{split}
\end{equation*}

Using multiplicative properties of the Landweber-Novikov
operations:
$$
\oln{a}(x\cdot y)=\sum_{b+c=a}\oln{b}(x)\oln{c}(y),
$$
and Proposition \ref{ln-s}, we get $(\,modulo\,4)$:
\begin{equation*}
\begin{split}
pr(\pi_{Y,2n+1})_*\oln{2n+1}(\ov{w}(2n+1))=
\sum_{j=0}^{2n-1}\binom{j}{2n-j+1}\cdot 2\cdot
S^j(\ov{v}^j)+&\\
pr(\binom{2n+1}{1}\cdot\oln{2n}(\ov{w}_{2n-1})+\binom{2n+2}{0}\cdot
\oln{2n+1}(\ov{w}_{2n-2})).&
\end{split}
\end{equation*}
Codimension of $\ov{v}^j$ is $2n+1-j$, thus either
$\binom{j}{2n-j+1}$ is zero, or $S^j(\ov{v}^j)$ is, and our
expression is equal to
$pr(\oln{2n}(\ov{w}_{2n-1})+\oln{2n+1}(\ov{w}_{2n-2}))$. Also,
$(\,modulo\,4)$,
$$
pr(\pi_{Y,2n+1})_*\square(\ov{w}(2n+1))=2\cdot
pr(\ov{w}^0\ov{w}_{2n-2}+\ov{w}^1\ov{w}_{2n-1}).
$$

In the same way, $(\,modulo\,4)$,
\begin{equation*}
\begin{split}
pr\oln{p}(\pi_{Y,p})_*(\ov{w}(p))=
\sum_{j=0}^{min(2n-1,p)}\binom{-(p+2-j)}{p-j}\cdot
2\cdot S^j(\ov{v}^j)+& \\
pr(\sum_{i=0}^{p-2n}
\binom{-(i+1)}{i}\oln{p-i}(\ov{w}_{i+4n-1-p})).&
\end{split}
\end{equation*}

Observe, that the second sum is empty for $p<2n$, is equal
$(\,modulo\,4)$, to $pr\oln{2n}(\ov{w}_{2n-1})$ for $p=2n$, and to
$pr\oln{2n+1}(\ov{w}_{2n-2})$ for $p=2n+1$ (we used the fact that
$pr\oln{m}(x)$ is divisible by $2$, if $m>\op{codim}(x)$ - see
Proposition \ref{ln-s}).

Since the coefficient $\binom{-(l+2)}{l}$ is odd if and only if
$l=2^k-1$, for some $k$, the first sum is equal to:
$$
2\sum_{j=0}^{min(2n-1,p)}(\gamma^{-1})_{p-j}\cdot S^j(\ov{v}^j).
$$
Taking into account that $\delta(t)=1+t+\ldots$, we get:
\begin{equation*}
\begin{split}
pr\sum_{p=n+1}^{2n+1}\delta_{2n+1-p}\oln{p}(\pi_{Y,p})_*(\ov{w}(p))=&\\
2\sum_{p=n+1}^{2n+1}\sum_{j=0}^{min(2n-1,p)}\delta_{2n+1-p}(\gamma^{-1})_{p-j}\cdot
S^j(\ov{v}^j)+&\\
(pr\oln{2n}(\ov{w}_{2n-1})+pr\oln{2n+1}(\ov{w}_{2n-2}))=&\\
2\sum_{j=0}^{2n-1}(\delta\cdot\gamma^{-1})_{2n+1-j}S^j(\ov{v}^j)+
(pr\oln{2n}(\ov{w}_{2n-1})+pr\oln{2n+1}(\ov{w}_{2n-2}))=&\\
2(\ov{v}^0+S^1(\ov{v}^1))+(pr\oln{2n}(\ov{w}_{2n-1})+pr\oln{2n+1}(\ov{w}_{2n-2})),
\end{split}
\end{equation*}
in the light of Corollary \ref{amu-1}.

Putting things together (and again using the fact that
$pr\oln{m}(x)$ is divisible by $2$, if $m>\op{codim}(x)$), we
obtain:
\begin{equation*}
2\ov{u}=2(\ov{v}^0+S^1(\ov{v}^1)+
\ov{v}^1\cdot\ov{v}_{2n-1}+\ov{v}^0\cdot\ov{v}_{2n-2}).
\end{equation*}

Since $u$ is defined over the base-field $k$, the Proposition is
proven. \Qed
\end{prof}


There is another result which extends a bit Theorem \ref{3.5}.

\begin{prop} {\rm (\cite[Statement 3.8]{GPQCG})}
\label{3.8}
Let $Y$ be smooth quasiprojective variety, $Q$ smooth projective quadric over
$k$. Let $\ov{y}\in\op{CH}^m(Y|_{\kbar})/2$. Suppose
$\zj{0}_{[\ddim(Q)+1/2]}(Q)$ is defined. Then for $m\leq [\ddim(Q)+1/2]$,
$$
\ov{y}|_{\ov{k(Q)}}\,\,\text{is defined over}\,\,k(Q)\,\,\Leftrightarrow\,\,
\ov{y}\,\,\text{is defined over}\,\,k.
$$
\end{prop}


Proposition \ref{3.8} extends Theorem \ref{3.5} in the direction of the
following:

\begin{conj} {\rm (\cite[Conjecture 3.11]{GPQCG})}
\label{conj}
In the notations of Theorem \ref{3.5},
suppose $\zj{\ddim(Q)-l-d}_l(Q)$ is defined. Then for any $m\leq l$,
$$ 
\ov{y}|_{\ov{k(Q)}}\,\,\text{is defined over}\,\,k(Q)\,\,\Leftrightarrow\,\,
\ov{y}\,\,\text{is defined over}\,\,k.
$$
\end{conj}

This conjecture is known for $l=[\ddim(Q)+1/2],\ddim(Q)-1,\ddim(Q)$.

\subsection{Some auxilary facts}

For our purposes it will be important to be able (under certain conditions)
to get rid of the
last term in the formula from Proposition \ref{4n+1}.
For this we will need the following facts.

\begin{prop}
\label{ftz} Let $R$ be smooth quadric, $d_R:=[\ddim(R)/2]$, $0\leq
i\leq d_R$, and $f:
G(R,i)\stackrel{\alpha}{\low}F(R,0,i)\stackrel{\beta}{\row} R$ be
the natural correspondence. Let $\zj{i-d_R}_{\ddim(R)-i}$ is
defined. Let $t\in\op{CH}_{\ddim(R)-i}(G(R,i))/2$ be such that
$f_*(t)=1\in\op{CH}^0(R)/2$. Then
$f_*(t\cdot\zj{i-d_R}_{\ddim(R)-i})=l_i\in\op{CH}_i(R)/2$.
\end{prop}

\begin{proof}
Really, by the definition,
$\zj{i-d_R}_{\ddim(R)-i}=f^*(l_0)=\alpha_*\beta^*(l_0)$. By the
projection formula, 
$$
f_*(t\cdot\zj{i-d_R}_{\ddim(R)-i})=
\beta_*\alpha^*(t\cdot\zj{i-d_R}_{\ddim(R)-i})=
\beta_*\alpha^*\alpha_*(\alpha^*(t)\cdot\beta^*(l_0)).
$$ 
Again, by
the projection formula,
$\beta_*(\alpha^*(t)\cdot\beta^*(l_0))=l_0$. Thus,
$\alpha^*(t)\cdot\beta^*(l_0)$ is a zero-cycle of degree $1$ on
$F(R,0,i)$, and $\alpha_*(\alpha^*(t)\cdot\beta^*(l_0))$ is a zero
cycle of degree $1$ on $G(R,i)$. Proposition follows. \Qed
\end{proof}

Let $v\in\op{CH}^m(Y\times Q)/2$ be some element. Then 
$$
\ov{v}=\sum_{i=0}^{d}(\ov{v}^i\cdot h^i+\ov{v}_i\cdot l_i).
$$


\begin{statement}
\label{s0} 
Suppose $\zj{\ddim(Q)-m-d}_m(Q)$ is defined. 
Then for any $v$ as above,
there exists $u\in\op{CH}^m(Y\times Q)/2$ such that
$\ov{u}^0=\ov{v}^0$, and $\ov{u}_{\ddim(Q)-m}=0$.
\end{statement}

\begin{proof}
If $\ov{v}_{\ddim(Q)-m}=0$, there is nothing to prove.
Otherwise, the class $l_{\ddim(Q)-m}\in\op{CH}_{\ddim(Q)-m}(Q|_{k(Y)})/2$
is defined. Indeed, let 
$$
\rho_X:\op{CH}^*(Y\times X)/2\twoheadrightarrow
\op{CH}^*(X|_{k(Y)})/2
$$ 
be the natural restriction. Then
$\rho_Q(\ov{v})=l_{\ddim(Q)-m}$ plus
$\lambda\cdot h^m$, if $2m=\ddim(Q)$ (notice, that 
$\ov{v}_{\ddim(Q)-m}\in\op{CH}^0$).  
Anyway, this implies that
over $k(Y)$ variety $G(Q,\ddim(Q)-m)$ has a zero-cycle of degree $1$, and
thus, a rational point. Let $x\in\op{CH}_{\ddim(Y)}(G(Q,\ddim(Q)-m)\times
Y)/2$ be arbitrary lifting of the class of a point on
$G(Q,\ddim(Q)-m)|_{k(Y)}$ with respect to $\rho_{G(Q,\ddim(Q)-m)}$. Let
$$
f:G(Q,\ddim(Q)-m)\stackrel{\alpha}{\low}F(Q,0,\ddim(Q)-m)\stackrel{\beta}{\row}Q
$$
be the natural correspondence. Consider $u':=(f\times
id)_*(x)\in\op{CH}^{m}(Q\times Y)/2$. Proposition
\ref{ftz} implies that the (defined over $k$) cycle
$$
u":=\pi_Y^*(\pi_Y)_*((h^{\ddim(Q)-m}\times 1_Y)\cdot (f\times
id)_*(x\cdot\zj{\ddim(Q)-m-d}_m(Q)))
$$ 
satisfy:
$\ov{u"}^0=\ov{u'}^0$, and (evidently) $\ov{u"}_{\ddim(Q)-m}=0$. Since
$\ov{u'}_{\ddim(Q)-m}=1=\ov{v}_{\ddim(Q)-m}$, it remains to take:
$u:=v-u'+u"$. \Qed
\end{proof}


\section{Even $u$-invariants}
\label{4}

The fields of any given $u$-invariant were constructed by A.Merkurjev
in \cite{MeU} using his {\it index-reduction formula} for central
simple algebras. Namely, A.Merkurjev showed that over the generic
point of a quadric $Q$ the index of a central simple algebra $A$ can drop
at most by the factor 2, and the latter happens if and only if
$C_0(q)$ can be mapped to $A$. In particular, division algebra of index $2^t$
will stay divison over the field $k(Q)$, for any $q$ of dimension
$>2(t+1)$. So, if over a base field $k$ we have a form $p\in I^2$
of dimension $2t+2$ whose $C_0$ has index $2^t$, then it will have
the same index over $k(Q)$ for any form $q$ of bigger dimension, and
thus, $p|_{k(Q)}$ will stay anisotropic. It remains to construct
Merkurjev tower of fields making all forms of dimension $>2(t+1)$
isotropic.

Let me give another construction, which does not use {\it index reduction
formula}. Instead, I will use the class
$\zj{0}_{\ddim(P)-d}$ - the North-West corner.

Let $p$ be form of dimension $2t+2$ such that $\zj{0}_{t}(P)$ is not defined over
$k$. One can use
generic form - see \cite[Statement 3.6]{GPQCG}. Then, by Theorem
\ref{3.5}, for any form $q$ of dimension $>\ddim(p)$, $\zj{0}_{t}(P|_{k(Q)})$
is not defined as well. But then Proposition \ref{uul} implies
that $\zj{-d}_{\ddim(P)}(P|_{k(Q)})$ is not defined. That is,
$p|_{k(Q)}$ is anisotropic. It remains to construct Merkurjev
tower of fields.



\section{Odd $u$-invariants}
\label{5}

Let us analize a bit the above construction. Instead of working with
the cycle $\zj{-[\ddim(P)/2]}_{\ddim(P)}$ -
the class of a rational point
on a quadric $P$, we worked with the (smaller codimensional!) cycle
$\zj{0}_{[\ddim(P)+1/2]}$, and used the fact that rationality
of the former implies rationality of the latter (Proposition \ref{uul}).

Unfortunately, for odd-dimensional forms we can not use
the class $\zj{0}_{[\ddim(P)+1/2]}$. Really, if $p$ is any such form, then
for $q:=p\perp\la det_{\pm}(p)\ra$,
$\zj{0}_{[\ddim(P)+1/2]}(P|_{k(Q)})$ will be defined, since
rationality of this class is equivalent to the rationality
of $\zj{0}_{[\ddim(Q)+1/2]}(Q|_{k(Q)})$ - observe that
$$
G(Q,[\ddim(Q)/2])=G(P,[\ddim(P)/2])\coprod G(P,[\ddim(P)/2]),
$$ and the
rationality of the latter follows from the rationality of the class
$\zj{-[\ddim(Q)/2]}_{\ddim(Q)}(Q|_{k(Q)})$ (isotropity of $Q|_{k(Q)}$).
So, even if we start from the form, where our class is not defined,
over generic point of some bigger-dimensional form it will become
rational, and we can not control anisotropity of $P$.

But observe that the rationality of $\zj{-[\ddim(P)/2]}_{\ddim(P)}$
implies rationality not just of  $\zj{0}_{[\ddim(P)+1/2]}$, but
of all West edge $\zj{-s}_{[\ddim(P)+1/2]-s}$,
$0\leq s\leq [\ddim(P)/2]$. So, let us use these other cycles.


Let the form $p$ has dimension $2^r+1$. In this case, one can use
previous to the last grassmannian, and the class
$\zj{-1}_{2^{r-1}+1}$ on it.


\begin{thm}
\label{todd}
Let $\ddim(p)=2^r+1$, $r\geq 3$, and $EDI(P)$ looks as
$$
\def\objectstyle{\scriptstyle}
\def\labelstyle{\scriptstyle}
\xymatrix @-1.3pc{
? &\circ &\dots &\circ \\
\circ &\circ &\dots &\circ \\
\dots &\dots &\dots &\dots \\
\circ &\circ &\dots &\circ
}
$$
Let $\ddim(q)>\ddim(p)$. Then $EDI(P|_{k(Q)})$ has the same property.

In particular, $p|_{k(Q)}$ is anisotropic.
\end{thm}

\begin{cor}
\label{oddu}
For any $r\geq 3$ there is a field of $u$-invariant $2^r+1$.
\end{cor}

\begin{proof}
Start with {\it generic form} $p$ over
$k=k_0(x_1,\ldots,x_{2^r+1})$. Then $EDI(P)$ is empty. This
follows from Proposition \ref{uul} and \cite[Statement
3.6]{GPQCG}. Let $F$ be the limit of the Merkurjev tower of fields
which makes all forms of dimension $>\ddim(p)$ isotropic. Then it
follows from Theorem \ref{todd} that $p|_F$ is anisotropic. Thus,
$u(F)=2^r+1$. \Qed
\end{proof}

\begin{prof}{Proof of Theorem \ref{todd}}

Let $d:=[\ddim(P)/2]=2^{r-1}-1$. It follows from Theorem \ref{3.5}
that the cycles $\zj{0}_j(P|_{k(Q)})$, $1\leq j\leq d$ are not
defined. That is, we have $\circ$'s to the right of "$?$". In the
light of Proposition \ref{uul} it remains only to treat the case
of $\zj{-1}_{d+2}(P|_{k(Q)})$ (that is, the node just below the
"$?$").

Suppose this cycle is defined over $k(Q)$.  We clearly can assume
that $\ddim(q)=\ddim(p)+1=2^r+2$. Let us denote $G(P,d-1)$
temporarily as $Y$. We have $y\in\op{CH}^{d+2}(Y|_{k(Q)})/2$ such
that $\ov{y}=\zj{-1}_{d+2}\in\op{CH}^{d+2}(Y|_{\ov{k(Q)}})/2$. 
Let us lift it to $v\in\op{CH}^{d+2}(Y\times Q)/2$ via the natural projection
$\op{CH}^{d+2}(Y\times
Q)/2\stackrel{\rho_Y}{\twoheadrightarrow}\op{CH}^{d+2}(Y|_{k(Q)})/2$.

\begin{statement}
\label{s} There exists such $v\in\op{CH}^{d+2}(Y\times Q)/2$ that
$\ov{v}^0=\ov{y}$, and
$\ov{v}_d=0\in\op{CH}_{d}(Q|_{\ov{k(Y)}})/2$.
\end{statement}

\begin{proof}
Let $v$ be arbitrary lifting of $y$ with respect to $\rho_Y$. 
If $\ov{v}_d=0$, there is nothing to prove. Otherwise, let
us show that $\zj{-1}_{2^{r-1}+1}(Q)$ is defined over $k$.


Suppose $\ov{v}_d\neq 0$, then it is equal to $1\in\op{CH}^0(Y|_{\kbar})/2$.
Then $\ov{\rho_Q(v)}=l_d\in\op{CH}_{d}(Q|_{\ov{k(Y)}})/2=\zz/2\cdot l_d$. Thus,
$\zj{-[\ddim(Q)/2]}_{d+2}(Q|_{k(Y)})$ is defined. By Proposition
\ref{uul}, $\zj{-2}_3(Q|_{k(Y)})$ is defined too (it lives in the
same column above). We want to show that $\zj{-2}_3(Q)$ is defined.

Consider the two towers of fibrations:
\begin{equation*}
\begin{split}
&\op{Spec}(k)\low P\low \ldots\low F(P,0,1,\ldots,d-1);\\
&\op{Spec}(k)\low Q\low \ldots\low F(Q,0,1,\ldots,d-1),
\end{split}
\end{equation*}
with the generic fibers - quadrics $P=P_1,\ldots,P_d$, $Q=Q_1,\ldots,Q_d$
of dimension $2d+1,2d-1,\ldots,3$, and $2d+2,2d,\ldots,4$, respectively.
Let us denote $k_{a,b}:=k(F(P,0,\ldots,a-1)\times F(Q,0,\ldots,b-1))$.
Then 
$$
k_{a+1,b}=k_{a,b}(P_a)\,\,\,\text{ and }\,\,\, k_{a,b+1}=k_{a,b}(Q_b).
$$

Since we have embeddings of fields 
$$
k\subset
k(Y)=k(G(P,d-1))\subset k(F(P,0,\ldots,d-1))=k_{d,0},
$$
$\zj{-2}_3(Q|_{k_{d,0}})$ is defined. Then by Proposition \ref{uul}, 
$\zj{0}_1(Q|_{k_{d,0}})$ 
is defined. By Theorem \ref{3.5}, 
$\zj{0}_1(Q)$
and $\zj{0}_2(Q)=(\zj{0}_1(Q))^2=S^1(\zj{0}_1(Q))$ are defined.

It follows from Corollary \ref{=<d-i} that for arbitrary elements \\
$\ov{\alpha},\ov{\beta}\in\op{CH}^*(G(Q,d-1)|_{\ov{k_{a-1,0}}})/2$
of codimension $2$ and $1$, respectively, the class
$S^1(\ov{\alpha})+\ov{\alpha}\cdot\ov{\beta}$ is defined over any field,
where $Q$ is defined, in particular, over $k_{a-1,0}$.
It follows from Corollary \ref{cor4n+1} that
\begin{equation*}
\zj{-2}_3(Q|_{k_{a,0}})\,\,\,\text{is defined}\,\,\,\Rightarrow
\begin{cases}
\text{either}\,\,\, &\zj{-2}_3(Q|_{k_{a-1,0}})\,\,\,\text{is defined};\\
\text{or}\,\,\,&\zj{-[\ddim(P_a)/2]}_3(P_a|_{k_{a-1,d}})\,\,\,\text{is
  defined,}\\
 & \text{and}\,\,\,[\ddim(P_a)/2]\leq 3.
\end{cases}
\end{equation*}
Let us show that the second case is impossible.
Really, by Proposition \ref{uul},
$$
\zj{-[\ddim(P_a)/2]}_3(P_a|_{k_{a-1,d}})\,\,\,\text{is def.}\,\,\,\Rightarrow
\zj{0}_{3-[\ddim(P_a)/2]}(P_a|_{k_{a-1,d}})\,\,\,\text{is def.}
$$
Since $3-[\ddim(P_a)/2]\leq 2$, $\ddim(Q_b)\geq 4$, and $\zj{0}_2(Q_d)$ is defined,
by Proposition \ref{3.8} and Theorem \ref{3.5},
$$
\zj{0}_{3-[\ddim(P_a)/2]}(P_a|_{k_{a-1,b}})\,\,\,\text{is def.}\,\,\,
\Rightarrow
\zj{0}_{3-[\ddim(P_a)/2]}(P_a|_{k_{a-1,b-1}})\,\,\,\text{is def.}
$$
Then $\zj{0}_{3-[\ddim(P_a)/2]}(P_a/k_{a-1,0})$ is
defined, and $\zj{0}_{3-[\ddim(P_a)/2]}(P)$ is defined (by Theorem \ref{3.5}).
This contradicts to the conditions of our Theorem (here we are using 
the fact that $r\geq
3$). Thus,
$$
\zj{-2}_3(Q|_{k_{a,0}})\,\,\,\text{is defined}\,\,\,\Rightarrow
\zj{-2}_3(Q|_{k_{a-1,0}})\,\,\,\text{is defined}, 
$$
and, consequently, $\zj{-2}_3(Q)$ is defined.
By Proposition \ref{uul},
$\zj{-1}_3(Q)$ is defined too. Proposition \ref{steenrod} implies
that
$$
\zj{-1}_{2^{r-1}+1}(Q)=S^{2^{r-2}}S^{2^{r-3}}\dots
S^2(\zj{-1}_3(Q))
$$
is also defined over $k$. 
Since $\zj{-1}_{2^{r-1}+1}(Q)$ is defined, everything follows from
Statement \ref{s0}. \Qed
\end{proof}

Consider $v\in\op{CH}^{d+2}(Y\times Q)/2$ satisfying the
conditions of Statement \ref{s}. As above,
$\ov{v}=\sum_{i=0}^{2^{r-1}}(\ov{v}^i\cdot h^i+\ov{v}_i\cdot
l_i)$. Then, by Proposition \ref{4n+1}, the class
$\ov{v}^0+S^1(\ov{v}^1)+\ov{v}^1\ov{v}_{2^{r-1}}+\ov{v}^0\ov{v}_{2^{r-1}-1}$
is defined over $k$. But
$\ov{v}_{2^{r-1}-1}=0$. Thus on
$Y$ we have the class
$\ov{v}^0+S^1(\ov{v}^1)+\ov{v}^1\ov{v}_{2^{r-1}}$ defined over
$k$.

Now it is time to use the specific of $Y$ and $v$. Our $Y$ is a
grassmannian $G(P,d-1)$. In particular, it is a geometrically
cellular variety, and the map
$\op{CH}^*(Y|_{\kbar})/2\row\op{CH}^*(Y|_{\ov{k(Q)}})/2$ is an
isomorphism. Thus, $\ov{v}^0=\zj{-1}_{d+2}$. On the other hand,
$\ov{v}_{2^{r-1}}=\ov{v}_{d+1}$ belongs to
$\op{CH}^1(Y|_{\kbar})/2$, and so is equal either to $0$, or to
$\wj{-1}_1$. So, on $G(P,d-1)$ we have class either of the form
$\zj{-1}_{d+2}+S^1(\ov{v}^1)$, or of the form
$\zj{-1}_{d+2}+S^1(\ov{v}^1)+\ov{v}^1\wj{-1}_1$ defined over $k$.
The following Statement shows that such class should be nonzero.

\begin{statement}
\label{nonzero} Let $R$ be a smooth projective quadric of
dimension $4n-1$, $d=[\ddim(R)/2]=2n-1$. Then
$\zj{-1}_{\ddim(R)-d+1}(R)$ belongs to the image of neither of two maps:
$S^1,(S^1+\wj{-1}_1\cdot\phantom{x}):$
$$
\op{CH}^{\ddim(R)-d}(G(R,d-1))/2\row\op{CH}^{\ddim(R)-d+1}(G(R,d-1))/2
$$
\end{statement}

\begin{proof}
We can assume that $k=\kbar$. Consider the natural projections:
$$
G(R,d-1)\stackrel{\alpha}{\low}F(R,d-1,d)\stackrel{\beta}{\row}G(R,d).
$$
The map $\beta$ provides $F(R,d-1,d)$ with the structure of the
projective bundle $\pp_{G(R,d)}(Tav^{\vee}_d)$, and the Chern classes of
$Tav_d$ are divisible by $2$ - see Proposition \ref{chern} (and
\cite{CGQG}). Thus,
$\op{CH}^*(F(R,d-1,d))/2=\op{CH}^*(G(R,d))/2[h]/(h^{d+1})$, where
$h=c_1(\co(1))$. By Lemma \ref{pi*},
$$
\alpha^*(\zj{-1}_{\ddim(R)-d+1})=\beta^*(\zj{0}_{\ddim(R)-d})\cdot
h,\,\,\,\text{and}\,\,\, 
h=\alpha^*(\wj{-1}_1).
$$
The first fact now is simple, since
$$
S^1(\alpha^*(\zj{-1}_{\ddim(R)-d+1}))=S^1(\beta^*(\zj{0}_{\ddim(R)-d})\cdot
h)=\beta^*(\zj{0}_{\ddim(R)-d})\cdot h^2, 
$$
by Proposition
\ref{steenrod}, and the latter element is nonzero. Thus, even
$\alpha^*(\zj{-1}_{\ddim(R)-d+1})$ can not be in the image of
$S^1$, since $S^1\circ S^1=0$.

To prove the second fact, observe that
$$
\alpha^*(\zj{-1}_{\ddim(R)-d+1})=\beta^*(\zj{0}_{\ddim(R)-d})\cdot
h=(S^1+h\cdot\phantom{x})(\beta^*(\zj{0}_{\ddim(R)-d})).
$$ 
Let
$u\in\op{CH}^{\ddim(R)-d}(G(R,d-1))/2$ be such that
$(S^1+\wj{-1}_1\cdot\phantom{x})(u)=\zj{-1}_{\ddim(R)-d+1}$. Then
$(S^1+h\cdot\phantom{x})(\beta^*(\zj{0}_{\ddim(R)-d})-\alpha^*(u))=0$.
Since $d$ is odd, the differential $(S^1+h\cdot\phantom{x})$ acts
without cohomology on $\op{CH}^*(G(R,d))/2[h]/(h^{d+1})$.
Consequently,
$(\beta^*(\zj{0}_{\ddim(R)-d})-\alpha^*(u))=(S^1+h\cdot\phantom{x})(w)$,
for some $w\in\op{CH}^{\ddim(R)-d-1}(F(R,d-1,d))/2$. This implies
$$
\alpha_*\beta^*(\zj{0}_{\ddim(R)-d})=\alpha_*(S^1+h\cdot\phantom{x})(w),
$$
since $\alpha_*\alpha^*=0$. Notice that $\alpha:F(R,d-1,d)\row
G(R,d-1)$ is a conic bundle with relative tangent sheaf
$\alpha^*(\co(\wj{-1}_1))=\co(h)$. Thus,
$\alpha_*(S^1+h\cdot\phantom{x})(w)=S^1(\alpha_*(w))$, and
$\alpha^*\alpha_*(\beta^*(\zj{0}_{\ddim(R)-d}))=S^1(\alpha^*\alpha_*(w))$.
But
$\alpha^*\alpha_*(\beta^*(\zj{0}_{\ddim(R)-d}))=h^{\ddim(R)-d-1}=h^d$,
and this element is not in the image of $S^1$, as one can easily
see. The contradiction shows that $u$ as above does not exist.
\Qed
\end{proof}

It follows from the Statement \ref{nonzero} that in
$\op{CH}^{d+2}(G(P,d-1)|_{\kbar})/2$ we have nonzero class $t$
defined over $k$. Then $\alpha^*(t)\in\op{CH}^{d+2}(F(P,d-1,d))/2$
will be also nonzero class defined over $k$. But the subring of
$k$-rational classes in $\op{CH}^{*}(F(P,d-1,d)|_{\kbar})/2$ is
$GDI(P,d)[h]/(h^{d+1})$, and by the main result of \cite{CGQG},
$GDI(P,d)$ as a ring is generated by the elementary classes
$\zj{0}_j$ contained in it. By the conditions of our Theorem,
among such classes only $\zj{0}_{d+1}$ could be defined over $k$.
Then the degree $=(d+2)$ component of the subring of $k$-rational
classes in $\op{CH}^{*}(F(P,d-1,d))/2$ is contained in
$\zz/2\cdot(\beta^*(\zj{0}_{d+1})\cdot
h)=\zz/2\cdot\alpha^*(\zj{-1}_{d+2})$. Thus, if $t$ is nonzero, it
got to be $\zj{-1}_{d+2}$. But this class is not defined over $k$
by the condition of the Theorem. And the contradiction shows that
the class $\zj{-1}_{d+2}$ is not defined over $k(Q)$ as well. 
Theorem \ref{todd} is proven.
\Qed
\end{prof}






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\end{thebibliography}




\end{document}
