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\begin{document}
\title{Rational fibrations of certain
  K-trivial varieties: english summary}
\author{Ekaterina Amerik and Fr\'ed\'eric Campana}
\date{\today}
\maketitle


In this paper we make several observations related to the search
for smooth projective varieties $X$ with an endomorphism $h$ of degree 
greater than
one. Such examples are very rare if we suppose that $h$ is regular, and 
they are less rare if $h$ is only a rational map. Still, certain examples
should be thought of as  ``not quite interesting'': for instance, any
surface of Kodaira dimension one has a rational endomorphism of high degree,
but such an endomorphism respects the elliptic fibration 
(the Iitaka fibration), that is, it sends its fibers to its fibers;
moreover it induces an automorphism of the base, which, as one can check,
is always of finite order. 
So in fact such endomorphisms ``come from'' endomorphisms
elliptic curves.

It is often difficult to decide whether a given variety has a rational
endomorphism of higher degree or not. For example, whereas some special
$K3$ surfaces (elliptic surfaces, Kummer surfaces) do, the answer is not
known for a generic $K3$ surface (say with Picard group $\Z$). That is why the 
following example from \cite{V} is very interesting:

Consider a smooth cubic hypersurface $V\subset\p^5$. Let $X$ be the variety
of lines on $V$; it is smooth, 4-dimensional, and $K_X=0$. In fact $X$ is
an irreducible hyperk\"ahler variety (\cite{BD}). For $V$ generic, $Pic(X)=\Z$.

It turns out that for a
general line $l\subset V$, there is a unique plane $P$ tangent to $V$ along
$l$. One has thus a rational endomorphism $h$ of $X$: set $h(l)=l'$, where
$V\cap P=l\cup l'$. Voisin proves that $deg(h)=16$.

It is a natural question whether this endomorphism is ``interesting''
in our sense; that is, is it true that it does not preserve a 
(meromorphic) fibration, at least for a general $X$? Our first result
is the following

\medskip

{\bf Theorem 2.1} {\it Let $X$ be a smooth projective variety such that
$K_X=0$, $Pic(X)=\Z$. Suppose that there is a meromorphic
fibration $g:X\dasharrow B$.
Then a general fiber $F$ of $g$ is of general type.}

\medskip

For the proof, one compares the Kodaira dimension of the (desingularized)
generic fiber with its
Iitaka dimension with respect to the inverse image of an ample line
bundle on $B$.

It is well-known that varieties of general type do not have
rational endomorphisms of higher degree. That is, at least in the generic case
(of cyclic Picard group), no power of
Voisin's endomorphism
can preserve a fibration.

As $X$ from Voisin's example is hyperk\"ahler, it is interesting
to ask what can one say about meromorphic fibrations of hyperk\"ahler
varieties (of dimension $2n$). Of course one has to suppose that the
fibers are not of general type; so one is naturally led (by Iitaka fibration)
to meromorphic fibrations in varieties of zero Kodaira dimension. 
Recall that in the holomorphic case we have 
Matsushita's results
saying that the fibers are lagrangian tori of dimension $n$, and the base
is ``very similar'' to $\p^n$. For $n=2$ (that is, for four-dimensional $X$),
we prove the following results showing
that the situation is similar to Matsushita's:

\medskip

{\bf Proposition 3.1} {\it Let $L$ be a line bundle defining $g$; then
$L^{4}=0$, that is, $L$ is an isotropic vector of the Beauville-Bogomolov
form.} 

\medskip



In particular, if the Beauville-Bogomolov form does not represent zero
on $Pic(X)$, then any rational fibration of $X$ is in varieties of
general type. Proposition 3.1 also implies immediately (Proposition 3.4)
that the general fiber of $g$ is not a curve.

\medskip

{\bf Proposition 3.3} {\it The proper transform of any $M\in Pic(B)$ is 
proportional to $L$.}

This is analogous to Matsushita's $Pic(B)=\Z$.

\medskip

The reason that our proofs are valid only in dimension 4 is that we use
the mininal model program for the fibers of $g$. Modulo MMP, the proofs
work in arbitrary dimension.
 
The crucial point in the proof of 3.1 is a theorem of Huybrechts saying
that any curve of negative Beauville-Bogomolov square lies in the
uniruled subvariety of $X$.
The proof of 3.3 is along Matsushita's lines.

For the following theorems we use the results of Wierzba
\cite{W} on symplectic flops in dimension 4, which in turn rely on the
work of Wierzba-Wisniewski. The results of Wierzba say that $L$ can be
rendered nef by a finite sequnce of Mukai flops (in fact there is a gap
in Wierzba's proof, but it concerns the termination of flops, which has
been proved for all 4-folds by Matsuki \cite{Mat}; so the result
is valid). We observe that $L$ nef
implies $g$ almost holomorphic, and deduce the

\medskip

{\bf Theorem 3.5}{\it The fibers of $g$ are lagrangian, birational to
abelian surfaces.}

\medskip

{\bf Theorem 3.6} {\it If $L$ is nef, $g$ has a holomorphic model; that is, 
there is a birational map $\phi: B\dasharrow B_0$ such that the composition
$\phi\circ g$ is holomorphic.}

\medskip

The main idea here is to modify $X$ and $B$ in such a way that one has
a well-defined map from $B$ to the Chow variety of $X$, mapping a point
$b\in B$ to the fiber $X_b$. The image of this map can then be taken for
$B_0$.

\medskip

The main result of the section 4 is the following 

\medskip

{\bf Theorem 4.1} {\it Let $X$ be an arbitrary compact K\"ahler variety, and
$f:X\dasharrow X$ a dominant meromorphic map. Then there is a meromorphic
fibration $g:X\dasharrow Y$, such that for $x$ sufficiently general, the
fiber of $g$ passing through $x$ is the Zariski closure of the iterated orbit
$\{x, f(x), \dots f^k(x), \dots \}$ of $x$.}

\medskip

Here ``sufficiently general'' means ``in the complement of a countable
union of proper subvarieties''.


This implies immediately 
that in Voisin's example, the iterated orbit of a generic
(complex!) point is Zariski-dense, at least for $X$ generic.
 We observe that there exist such an $X$
with $Pic(X)=\Z$ and defined over a number field; the natural question is,
can one prove that the iterated orbit of a generic $\bar \Q$-point is
Zariski-dense? (If yes, this would of course imply the potential density
of rational points on $X$). Unfortunately, our method does not apply
at all: while proving Theorem 4.1, we systematically discard countable unions
of proper subvarieties; we cannot do that over $\bar \Q$ because of its
countability.

In Section 5, we prove that if $g:X\rightarrow B$ is a meromorphic
fibration with $X$ compact hyperk\"ahler (not necessarily projective)
and $B$ K\"ahler (without the loss of generality, we may suppose
that $B$ is smooth), then all covariant holomorphic tensors on $B$ vanish.
If $dim(B)=2$, this implies the rationality of $B$, and if $dim(B)=3$,
its rational connexity.

Finally, in section 6, we analyse the behaviour of some natural
fibrations on $X$ (compact K\"ahler)  with respect to a dominant 
meromorphic self-map $f:X\dasharrow X$.
It turns out that the ``best-behaved'' fibration is the {\it core} 
$c:X\dasharrow C$, introduced by the second author in \cite{C3}. 
Namely the following theorem holds:

\medskip

{\bf Theorem } {\it In the situation as above, $f$ induces a rational
self-map $c_f$ of $C$, such that $fc=c_ff$ and $c_f^N=id_C$ for some
positive integer $N$. In other words, a power of $f$ commutes with $c$.} 

\medskip 

In particular, if $X$ admits an $f$ with a Zariski-dense iterated orbit, 
then $X$ must be special in the sense of \cite{C3}.

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\bibitem[M]{M} D. Matsushita, On fibre space structures of a
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http://www.dpmms.cam.ac.uk/~jw227/publications.html.

\end{thebibliography}{}
\end{document}
