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\begin{document}
\title{Potential density of rational points on the variety \\
of lines of a cubic fourfold}
\author{Ekaterina Amerik, Claire Voisin} \maketitle \setcounter{section}{-1}
\section{Introduction}
\setcounter{equation}{0}
Recall that for a variety $X$ defined over a field $k$,    rational points
are said to be potentially dense in $X$ if for some finite extension $k'$ of $k$,
$X(k')$ is Zariski dense in $X$.
For example, this is the case if $X$ is a rational
or a unirational variety.

If $k$ is a number field, a conjecture of Lang and Vojta, proved in dimension
$1$ by Faltings, but still open even in dimension two,
predicts that varieties of general type  never satisfy potential density over $k$.
On the contrary, it is generally expected that when the canonical bundle $K_X$ is
negative (i.e. $X$ is a Fano variety) or trivial, rational points are
always potentially dense in $X$ (see \cite{C} for a
conjectural geometric characterization
of potentially dense varieties).

In the Fano case, there are many examples
confirming this. First of all, many Fano varieties are unirational
(in dimension $\leq 2$, all of them are even rational over a finite extension of the definition field).
Furthermore, Fano threefolds such as a three-dimensional quartic
or a ``double Veronese cone" (a hypersurface of degree $6$ in $\p(1,1,1,2,3)$)
are birational to elliptic fibrations over $\p^2$, which makes
it possible to prove  potential density (see \cite{HarT}).

If the canonical bundle is trivial, the known examples are much
less convincing. It is well-known that rational points on
abelian varieties are potentially dense. But the simply-connected
case remains largely unsolved.


A number of  results concerning potential density of rational points on
$K3$ surfaces defined either over a number field or a function field
appeared in the last years.
Bogomolov and  Tschinkel \cite{Bo-Tschi}  proved potential
density of rational points for $K3$ surfaces  admitting an elliptic pencil, or an infinite
automorphisms group. Remark that these $K3$ surfaces are rather special;
in particular,
their geometric Picard number is never equal to $1$, whereas it is equal to $1$ for a
general projective $K3$.

The case
of $K3$ surfaces defined over a (complex) function field has been solved
for certain types of families
by Hassett and Tschinkel \cite{Hatschi}. They prove in particular the
existence
of $K3$ surfaces defined over $\mathbb{C}(t)$,
 whose geometric Picard group is  equal to $\mathbb{Z}$,
and which satisfy potential density. Their method can likely be adapted to
produce examples
over $\overline{\mathbb{Q}}(t)$.

However, no example of a $K3$ surface defined over a number field, and
with geometric Picard group equal to $\mathbb{Z}$ is known to satisfy potential density.


In this paper, we consider certain four-dimensional analogues of
$K3$
surfaces, namely,
Fano
varieties of lines
$F$ of cubic $4$-folds $X$.
These are $4$-dimensional varieties with trivial canonical bundles, and which
even possess a non-degenerate holomorphic $2$-form $\omega$ which will play an important role
in our study. Their similarity with
$K3$ surfaces is shown by the work of Beauville and Donagi
\cite{be-do}, where they prove that these varieties are deformations of
the second punctual Hilbert scheme of a
$K3$ surface. However,
for very  general $X$, the geometric Picard group of $F$ is equal to $\mathbb{Z}$, by an easy deformation
argument.
By a result due to Terasoma \cite{terasoma}, this property holds true even for many $X$'s defined over
$\mathbb{Q}$.
We prove the following result:

\begin{theo} \label{main} For many cubic $4$-folds $X$ defined over a number field,
such that the corresponding
variety of lines $F$ (which is defined over the same number field)  has cyclic
geometric Picard group,     $F$ satisfies the potential density
property.
\end{theo}
Here, the term ``many'' can be made precise and exactly depends on the
structure of the  set of points in a certain  moduli space satisfying Terasoma's density
condition (see \cite{terasoma}). In lemma \ref{conditionslemm}, we give precise conditions under which
the conclusions of Theorem \ref{main} hold.

\begin{rema}{\rm An interesting problem would be to write down the explicit equation
of some cubic fourfold as in Theorem \ref{main}, (cf. \cite{vL} for
some equations of K3 surfaces over the rationals, with cyclic geometric
Picard group
and infinitely many
rational points). Our existence result does not lead to such an explicit construction.}
\end{rema}


Let us describe the strategy of our proof. We will use
two geometric ingredients.
 The first one is the rational self-map $\phi:F\dashrightarrow F$
 constructed in
\cite{voisinturin} (cf section \ref{preli}); in \cite{am-cam},
this map was already used to prove the potential density
of rational points for such a variety $F$ over an uncountable field
(say, a function field of a complex curve, as in \cite{Hatschi}), assuming
its geometric  Picard group is
$\mathbb{Z}$.

The proof in \cite{am-cam} shows in fact that for a general complex point
$t\in F(\mathbb{C})$, its orbit
$\{\phi^l(t),\,l\in \mathbb{N}\}$ is Zariski dense in $F$.
In the case where $F$ is defined over
a number field $k$, this proof does not say anything about points of $F$ over
 finite extensions of $k$,
because it is based on discarding certain ``bad" countable unions of
proper subvarieties, and it seems difficult to get any better control
over those subsets. As they could contain all points
defined over a number field,  one cannot conclude from \cite{am-cam}
that there is a point $t\in F(\bar{\mathbb{Q}})$ with Zariski
dense orbit. Neither do we prove such a statement, and our strategy uses in fact the following
supplementary ingredient:
the variety  $F$ is covered by  a $2$-dimensional
 family of surfaces $(\Sigma_b)_{b\in B}$, which are lagrangian with
respect to the non-degenerate holomorphic 2-form $\omega$ and birationally
equivalent
to  abelian surfaces (cf \cite{voisincdm} and section \ref{preli}).

Of course, for any surface
as above defined over a number field, rational points are potentially dense in it.
Furthermore $\phi$ is defined over the same number field as $F$.  It
thus suffices
to prove that for one such surface
$\Sigma_b$, the countably many surfaces
$\phi^l(\Sigma_b)$ are Zariski dense in $F$.

This is done in two steps, and our proof shows that this density result holds once a
non-vanishing condition (see Proposition \ref{plusfort})
for a certain $l$-adic Abel-Jacobi cycle class holds for the corresponding
$\Sigma_b$. This non-vanishing is satisfied for many
$\overline{\mathbb{Q}}$-points in the moduli
space $B$ of these surfaces, by an argument which is taken from \cite{ggp} (and attributed there
to Bloch and Nori).

In section \ref{sec2}, we prove that for many $\Sigma_b$'s defined over a number field, the Zariski closure
of the union of the surfaces
$\phi^l(\Sigma_b)$ contains at least a divisor (that is, $\Sigma_b$ is not
preperiodic), and this already necessitates this non-vanishing
condition.

In the last section, we want to exclude the possibility
that the union  of the surfaces
$\phi^l(\Sigma_b)$ has a Zariski closure
which is of dimension $3$. We thus have
to study divisors in $F$ invariant under a power of $\phi$ and containing
a Zariski dense union of lagrangian surfaces birationally equivalent to abelian surfaces.
We use the restriction of the holomorphic symplectic  form of $F$ to such divisors
to classify them (cf section \ref{secclassi}) into two types (see corollary \ref{twofibrat}).
The conclusion then comes in two steps, the first  one
being  geometric, the second one being arithmetic
 :
we exclude one type by geometric considerations in section  \ref{secclassigeo} and
we show in section
 \ref{conclusion} that the  remaining case
can be excluded using the non-vanishing of the Abel-Jacobi invariant as above.


\section{Preliminaries \label{preli}}
\setcounter{equation}{0}

Let  $X\subset \mathbb{P}^5$ be a smooth cubic fourfold,
and $F$  the variety of lines of $X$; let us denote by $P\subset F\times X$
 the universal incidence correspondence, that is,
 $p:P\rightarrow F$ is the universal family of lines parameterized by $F$ and $q:P\rightarrow X$
  is the natural
 inclusion in  $X\subset\mathbb{P}^5$ on each of these lines. Recall from \cite{be-do} that
 the Abel-Jacobi map
 $$p_*q^*:H^4(X_\mathbb{C},\mathbb{Q})\rightarrow H^2(F_\mathbb{C},\mathbb{Q})$$
 is an isomorphism of rational Hodge structures (of bidegree $(-1,-1)$),
 which sends $h^2,\,h:=c_1(\mathcal{O}_X(1))$,
 to $c_1(\mathcal{L})$, where $\mathcal{L}$ is the restriction to $F$ of the Pl\"ucker line bundle.
 (In the whole paper, we will denote by $X_\mathbb{C}$ the complex manifold associated
 to a variety defined over a subfield of $\mathbb{C}$, and
 $H^*(X_\mathbb{C},R)$ will then denote the Betti cohomology of this complex manifold
 with coefficients in  $R$.)
The number $h^{3,1}(X)$ is equal to $1$, as shown by Griffiths theory, and we thus have
$h^{2,0}(F)=1$. Let $\omega=q_*p^*\alpha$ be a generator of $H^{2,0}(F)=H^0(F,\Omega_F^2)$,
where $\alpha$ generates $H^{3,1}(X)$. Then
$\omega$ is a non-degenerate $2$-form by \cite{be-do}.

On the other hand, it was proved by Terasoma
\cite{terasoma}  that many cubic fourfolds $X$ defined over
number fields satisfy the property that the Hodge classes in $H^4(X_\mathbb{C},\mathbb{Q})$
 are rational multiples
of $h^2$. (Terasoma formulates his result in terms of the non-existence
of non-zero primitive algebraic cycle classes, but this is equivalent
to the non-existence
of non-zero primitive Hodge classes,
as the Hodge conjecture is known to be true for cubic $4$-folds,
see \cite{zu}, \cite{comu}.) We deduce from this together with the fact that $H^2(F_\mathbb{C},\mathbb{Z})$
has no torsion and the class
$c_1(\mathcal{L})$ is not divisible
in $H^2(F_\mathbb{C},\mathbb{Z})$, that the corresponding variety $F$, which is also defined
over a number field, satisfies $Pic\,F_\mathbb{C}=\mathbb{Z}$,
with generator
 $\mathcal{L}$. Equivalently, the geometric Picard group of $F$ is equal
 to $\mathbb{Z}\mathcal{L}$.


For each smooth hyperplane section $Y\subset X$, the variety of lines contained in $Y$ is a
smooth surface $\Sigma_Y$ of class $c:=c_2(\mathcal{E})$, where $\mathcal{E}$ is the restriction
to $F$ of the universal rank $2$ bundle on the Grassmannian. Furthermore, these surfaces
are isotropic with respect to $\omega$. Indeed, denoting by
$$p_Y:P_Y\rightarrow \Sigma_Y,\,\,q_Y:P_Y\rightarrow Y$$
the incidence variety for $Y$, we have the following equality:
$$\omega_{\mid \Sigma_Y}=p_{Y*}\circ q_Y^*(\alpha_{\mid Y})\,\,{\rm in}\,\,H^{2,0}(\Sigma_Y).$$
But by Lefschetz theorem on hyperplane sections, we have $H^{3,1}(Y)=0$, and hence
$\alpha_{\mid Y}=0$. Thus $\omega_{\mid \Sigma_Y}=0$.

Recall next the observation made in \cite{voisincdm} : consider  first the
case where $Y$ has one ordinary
double point $y\in Y$. Then from \cite{cl-gr}, we know that
the normalization of $\Sigma_Y$ is isomorphic to
the symmetric product $C^{(2)}_x$, where $C_x$ is the curve of lines in $Y$ passing through $y$.
This curve is the complete intersection of a cubic and a quadric in $\mathbb{P}^3$.
The map $C^{(2)}_x\rightarrow \Sigma_Y$ sends a pair of lines $l_1,\,l_2$ through $y$  to the
residual line $l$ of the intersection  $Q\cap Y$ of the plane $Q:=<l_1,l_2>$
generated by  $l_1$ and $l_2$ ($Q\cap Y$ is a plane cubic curve which  already contains the two lines
$l_1,\,l_2$). The
 inverse of this map associates to a generic line $l$ in $Y$ not passing through $y$
the pair of lines which forms the residual intersection of the plane $<l,y>$ with $Y$
(this is indeed a  plane conic singular at $y$).

The curve $C_x$ is smooth and has genus $4$.

Assume now that $Y$ has exactly two more double points $y'$ and $y''$, (so
 $y$,
$y'$, $y''$ are not on the same line, as otherwise this line would be contained in
$Sing\,Y$).  Then
the curve $C_x$ also acquires exactly two double points, corresponding to the
lines $l',\,l''$ joining $y$ respectively
to $y'$ and $y''$, which  are obviously contained in $Y$ (we refer to \cite{cl-gr}, or
to the proof of lemma
\ref{ledoublecubic} below for the geometry of nodal cubic threefolds).

Thus the normalization of $C_x$ has genus $2$, and its second
symmetric product is birational to an abelian surface.
\begin{rema} {\rm This abelian surface is isomorphic to the intermediate Jacobian of the desingularization
$\widetilde{Y}$
of $Y$ obtained by blowing-up the nodes. It is also isomorphic via the Abel-Jacobi map to the
group $CH_1(\widetilde{Y})_{hom}$ (cf \cite{bloch}, \cite{cl-gr}).}
\end{rema}

Imposing three double points to $Y$ imposes three conditions to the corresponding hyperplane
of $\mathbb{P}^5$. Thus we get a $2$-dimensional family
$(\Sigma_b)_{b\in B}$ of such (highly singular) surfaces in $F$, and we will
denote by $(A_b)_{b\in B}$ the corresponding family of abelian surfaces, which are birationally
 equivalent
to them. Thus $A_b$ can be defined as $Alb\,\Sigma'_b$ for some desingularization
$\Sigma'_b$ of $\Sigma_b$.

Note that if $X$ is defined over a number field, so are $B$ and the family of
abelian  surfaces
$(A_b)_{b\in B}$.

We will need the following lemma:
\begin{lemm} \label{ledoublecubic}A generic  genus $2$ curve appears as the normalization
of the family of lines through a singular point of some cubic threefold with three
non-collinear double points.
\end{lemm}
{\bf Proof.} A generic genus $2$ curve $C$ is the normalization of a complete intersection
$C'$ of a smooth quadric and a cubic surface in $\mathbb{P}^3$, where $C'$
 admits exactly $2$ double points.
Indeed, it suffices to choose two generic line bundles of degree $3$ on $C$,
which will provide two maps from $C$ to $\mathbb{P}^1$. Then for a generic choice,
the induced map from $C$ to $\mathbb{P}^1\times \mathbb{P}^1$ has for image a nodal curve
$C'$.
This curve is in the linear system $\mid\mathcal{O}(3,3)\mid$ on
$\mathbb{P}^1\times \mathbb{P}^1$, which equivalently means that $C'$ is a complete intersection
$Q\cap S$, where $S$ has degree $3$ in $\mathbb{P}^3$ and $Q\cong \mathbb{P}^1\times \mathbb{P}^1$
is a smooth quadric. As the arithmetic genus of $C'$ is $4$, $C'$ has $2$ double points.

Now, following \cite{cl-gr}, let $Z$ be the blow-up of $\mathbb{P}^3$ along $C'$, and consider
the morphism from $Z$ to
$\mathbb{P}^4$ given by the cubic equations vanishing on $C'$: if $q,\,t$ are given choices
of quadratic, resp. cubic generating equations for the ideal of $C'$, this morphism is given by the linear
system of cubic polynomial on $\mathbb{P}^3$ vanishing along $C'$:
$$X_0q,\ldots, X_3q,t.$$

Denoting by $Y_0,\ldots,Y_4$ the homogeneous coordinates on $\mathbb{P}^4$ used above,
the image of $Z$ under this morphism is the cubic threefold
$Y$ which has
for equation
$$t(Y_0,\ldots,Y_3)=Y_4q(Y_0,\ldots, Y_3).$$
It is immediate to verify that $Y$ has exactly $3$ ordinary   double points, one of them
being $o:=(0,\ldots,0,1)$, the others satisfying $Y_4=0$ and corresponding to the double points of
$C'$.
The curve of lines in $Y$ though $o$ is sent isomorphically
to $C'$ by the projection from $o$.


\cqfd

Our arguments in the next sections   will apply to those $X$ satisfying the
conclusions
of the following lemma. These $X$'s will thus satisfy the conclusion of theorem \ref{main}.
\begin{lemm} \label{conditionslemm}For many $X$'s defined over
 over a number field,  the following two properties
are satisfied:

- The corresponding variety of lines $F$ defined over the same number field has geometric Picard group
equal to $\mathbb{Z}$.

-  For  some (and in fact many) closed points $b\in B$ defined over a number field,
 the abelian surfaces
$A_b$
have geometric N\'eron-Severi group equal to $\mathbb{Z}$.
\end{lemm}
{\bf Proof.} The proof goes exactly as in Terasoma \cite{terasoma}.
Consider the family $\mathcal{M}$ parameterizing pairs
$(A,X)$ where $A$ is a principally polarized abelian variety which is the
intermediate Jacobian of the desingularization of a hyperplane section $Y$ of $X$ with
exactly three independent
nodes (with its canonical polarization).
$\mathcal{M}$ maps via the first projection
to the moduli space of principally polarized abelian varieties
$\mathcal{A}_2$, and via the second projection, to the
open set  $U\subset \mathbb{P}(H^0(\mathbb{P}^5,\mathcal{O}_{\mathbb{P}^5}(3)))$ parameterizing
smooth cubic $4$-folds.
Lemma \ref{ledoublecubic} shows that
both maps are dominating.
Replacing
$\mathcal{M}$ and $\mathcal{A}_2$ by  finite covers $\mathcal{M}'$, $\mathcal{A}_2'$
if necessary, we may assume there
are universal families
$$\mathcal{T}\rightarrow \mathcal{A}_2',\,\,\mathcal{X}\rightarrow U,\,\mathcal{T}
\times_{\mathcal{M}'}\mathcal{X}_{\mathcal{M}'}\rightarrow\mathcal{M}'
$$
(where $\mathcal{X}_{\mathcal{M}'}$ is the pull-back of $\mathcal{X}$ to 
$\mathcal{M}'$).
Note furthermore that all families are defined over $\mathbb{Q}$.
We fix compatible base-points $o,\,o_\mathcal{A},\,o_U$ for $\mathcal{M}',\,\mathcal{A}_2',\,U$.

For the general polarized abelian complex surface $T$,
one has $NS(T)=\mathbb{Z}\lambda$,
which is equivalent to the condition that the only Hodge classes in
$H^2(T,\mathbb{Q})$ are multiples of the polarization
$\lambda\in H^2(A,\mathbb{Q})$.

This is  a consequence of the stronger result saying that the monodromy
group
$$Im\,\rho_\mathcal{A}:\pi_1(\mathcal{A}'_{2,\mathbb{C}},o_\mathcal{A})\rightarrow Aut\,
H^2(T_{o_\mathcal{A},\mathbb{C}},\mathbb{Q})$$
does not act via a finite group on any non-zero class in $\lambda^\perp$.

As the image of $\pi_1(\mathcal{M}'_\mathbb{C},o)$ in
$\pi_1(\mathcal{A}'_{2,\mathbb{C}},o_\mathcal{A})$ has finite index, the same holds for
$$Im\,\rho'_A:\pi_1(\mathcal{M}'_\mathbb{C},o)\rightarrow Aut\,
H^2(T_{o,\mathbb{C}},\mathbb{Q}).$$
Similarly, there is no non-zero class in
$H^4(X_{o_U,\mathbb{C}},\mathbb{Q})_{prim}$ on which the monodromy group
$$Im\,\rho_U:\pi_1(U_\mathbb{C},o_U)\rightarrow Aut\,
H^4(X_{o_U,\mathbb{C}},\mathbb{Q})_{prim}$$
acts via a finite group, which implies as above  the Noether-Lefschetz
theorem, saying that for general $X$, there is no non-trivial  Hodge class
in $H^4(X,\mathbb{Q})_{prim}$.
It follows that
there is no non-zero class in
$H^4(X_{o_U,\mathbb{C}},\mathbb{Q})_{prim}$ on which the monodromy group
$$Im\,\rho'_U:\pi_1(\mathcal{M}'_\mathbb{C},o_U)\rightarrow Aut\,
H^4(X_{o_U,\mathbb{C}},\mathbb{Q})_{prim}$$
acts via a finite group. Of course, we have the same conclusions
for the monodromy acting on cohomology with $\mathbb{Q}_l$ coefficients:
$$\rho'_A:\pi_1(\mathcal{M}'_\mathbb{C},o)\rightarrow Aut\,
H^2(T_{o,\mathbb{C}},\mathbb{Q}_l),\,\rho'_U:\pi_1(\mathcal{M}'_\mathbb{C},o_U)\rightarrow Aut\,
H^4(X_{o_U,\mathbb{C}},\mathbb{Q}_l)_{prim}.$$
Finally, we use the density theorem of Terasoma \cite{terasoma}, which says that
 for infinitely many points $o$ of $\mathcal{M}$ defined over a number field
$k_o$, the image of the
Galois group $Gal\,(\overline{k_o}/k_o)$ acting
on
$$H^2_{et}(T_{o,\overline{k_o}},\mathbb{Q}_l)
\oplus
H^4_{et}(X_{{o},\overline{k_o}},\mathbb{Q}_l)_{prim}$$
$$\cong
 H^2(T_{{o},\mathbb{C}},\mathbb{Q}_l)
\oplus  H^4(X_{{o},\mathbb{C}},\mathbb{Q}_l)_{prim}$$
is equal to the algebraic  monodromy group, that is, to the
image of
$\pi_{1,alg}(\mathcal{M}')$ acting on
$$H^2_{et}(\mathcal{T}_{\overline{k(\mathcal{M}')}},\mathbb{Q}_l)
\oplus
H^4_{et}(\mathcal{X}_{\overline{k(\mathcal{M}')}},\mathbb{Q}_l).$$


As the algebraic monodromy group contains the classical monodromy group
$Im\,(\rho'_A,\rho'_U)$,
it thus follows that for such a point $o$, there is no non-zero cycle class
in $H^2_{et}(T_{o,\overline{k_o}},\mathbb{Q}_l)$
which is not proportional to
$\lambda$,
or in $H^4_{et}(X_{o,\overline{k_o}},\mathbb{Q}_l)_{prim}$, because on cycle classes,
the Galois group acts via a
finite group. As cycle classes in $H^4_{et}(X_{o,\overline{k_o}},\mathbb{Q}_l)$
 correspond bijectively via $p_*q^*$ to $(Pic\,F_{o,\overline{k_o}})
 \otimes\mathbb{Q}_l$, the proof is complete.
\cqfd

Fix $X$ satisfying the
conclusion of the above lemma, and
let us come back  to the  $2$-dimensional variety $B$ parameterizing very singular
 surfaces $\Sigma_b$ which are birationally equivalent to
 abelian surfaces $A_b$ (thus  $B$ is the fiber
 of the map $\mathcal{M}\rightarrow U$ introduced above). We shall take a desingularisation
of the total space of the family $(\Sigma_{b})_{b\in B}$. Eventually after
replacing $B$ by a Zariski open subset, this gives us a new family
parameterized  by $B$, with smooth generic fiber. The fibers
will be denoted by $\Sigma'_b$. We have then $A_b=Alb\,\Sigma'_b$.

Let $k(B)$ be the function field of $B$.
Consider  the following supplementary
assumption on  the point $b\in B$ defined over a number field
$k(b)\subset \mathbb{C}$. We will assume first  $b\in B_{reg}$, the Zariski open subset
of $B$ over which the family  $(\Sigma'_{b})_{b\in B}$ is a smooth family of surfaces. Then there is a family of Albanese varieties $A_b$ over $B_{reg}$,
so that we have on one hand 
the algebraic monodromy acting on the {\'e}tale cohomology of the generic  geometric fiber
$$\pi_{1,alg}(B_{reg})\rightarrow Aut\,(H^1_{et}(A_{\overline{k(B)}},\mathbb{Q}_l)\oplus H^2_{et}(A_{\overline{k(B)}},\mathbb{Q}_l)_{prim})\cong $$
$$\cong Aut\,(H^1_{et}(A_{b,\overline{k(b)}},\mathbb{Q}_l)\oplus 
H^2_{et}(A_{b,\overline{k(b)}},\mathbb{Q}_l)_{prim}) $$
and on the other hand the Galois group action
acting on the {\'e}tale cohomology of the closed geometric fiber
$$Gal\,(\overline{k(b)}/k(b))\rightarrow Aut\,(H^1_{et}(A_{b,\overline{k(b)}},\mathbb{Q}_l)\oplus H^2_{et}(A_{b,\overline{k(b)}},\mathbb{Q}_l)_{prim}).$$



By Terasoma \cite{terasoma}, the image of the Galois group representation is 
equal to the algebraic
monodromy group for many closed
points $b$ of $B$ defined over a number field. When this is satisfied, the
abelian surface $A_b=Alb\,\Sigma'_b$ is geometrically simple, and in fact has 
geometric
N\'eron-Severi group equal to $\mathbb{Z}$. Indeed, by our choice of $X$,
we know by lemma \ref{conditionslemm} that this is true for the abelian surface corresponding to a
sufficiently general point of
$B(\mathbb{C})$, and arguing as in the proof
of lemma \ref{conditionslemm}, this implies the same results for those fibers $A_b$
satisfying this condition.

The action on the first cohomology is considered because later it shall
be used in the proof of Proposition 2.1, which is the key ingredient for our
main theorem. Note that $H^1_{et}(A_{b,\overline{k(b)}},\mathbb{Q}_l)=
H^1_{et}(\Sigma'_{b,\overline{k(b)}},\mathbb{Q}_l),$ since $A_b=Alb\,\Sigma'_b$.

To conclude this preliminary section, recall from
\cite{voisinturin} that for any cubic $4$-fold $X$, the Fano variety
of lines $F$ carries a rational self-map
$$\phi:F\dashrightarrow F$$
which can be described as follows:

If $t\in F$ parameterizes a line $\Delta_t\subset X$, and $t$ is generic, then there
exists a unique plane $Q_t\subset \mathbb{P}^5$ containing
$\Delta_t$  which is everywhere tangent to $X$ along $\Delta_t$.
As $X$ is not swept out by planes, $\Delta_t$ is not contained in any plane
contained in $X$, and thus $Q_t\cap X=2\Delta_t+\Delta_{t'}$ for some point
$t'=:\phi(t)\in F$.

In the generic case, $X$ does not contain any plane, and thus the indeterminacy locus
of $\phi$
consists of those points $t\in F$ for which the plane $Q_t$ is not unique,
which means equivalently that
there is a $\mathbb{P}^3\subset \mathbb{P}^5$ everywhere tangent to $X$ along $\Delta_t$.

One can show that the indeterminacy locus of $\phi$ is then a smooth
surface of general type, and compute its invariants (see for example 
\cite{amerik}). The indeterminacy of $\phi$ is resolved by a single
blow-up along this surface.


An obvious but crucial property of $\phi$ that we will use constantly in the paper is the following
(cf \cite{voisinturin}) :
\begin{lemm}\label{propertyphi}
 For  $t\in F$ away from the indeterminacy locus of $\phi$, one has the equality
$$2\Delta_t+\Delta_{\phi(t)}=h^3\,\,{\rm in}\,\,CH^3(X);$$
in particular, the class of $2\Delta_t+\Delta_{\phi(t)}$ in $CH^3(X)$ does
not depend on $t$.

\end{lemm}


One first consequence is the following (see \cite{voisinturin}):
\begin{coro}  One has $\phi^*\omega=-2\omega$.
\end{coro}
Indeed, recall the incidence correspondence
$P\subset F\times X$. Then the line $\Delta_t$ has by definition its class in $CH^3(X)$ equal to
$P_*(t)$. Thus lemma \ref{propertyphi} can be restated by saying that
\begin{eqnarray}
\label{crucialpropertyphi} P_*(\phi(t))=-2P_*t+h^3\,\,{\rm in}\,\,CH^3(X)
\end{eqnarray}
for generic $t\in F$.
Mumford's theorem conveniently generalized (cf \cite{voisinbook}, Proposition
10.24) then tells us
that we have
$$\phi^*(P^*\alpha)=-2P^*\alpha,$$
for $\alpha\in H^{3,1}(X)$ as above.
As we have $P^*\alpha=\omega$, this gives the result.
\cqfd
\begin{rema}{\rm We use here a very easy special instance of Mumford's theorem. Indeed,
we actually have (away from a codimension $2$ subset) an equality of effective  $1$-cycles
$$P_*(\phi(t))+2P_*t=Q_t\cap X,$$
from which one deduces that the $(2,0)$-form $\phi^*(P^*\alpha)+2P^*\alpha$ on $F$
is the restriction to $F$ of a $(2,0)$-form on the Grassmannian of $2$-planes in $\mathbb{P}^5$, hence is $0$.}
\end{rema}
Another corollary which will be used is the following:
\begin{coro}\label{degphi}\cite{voisinturin} We have
\begin{eqnarray}\label{eqdegphi} deg\,\phi=16.
\end{eqnarray}
\end{coro}
{\bf Proof.} Indeed, denoting by $\tau:\widetilde{F}\rightarrow F$ the
blow-up along the indeterminacy locus,
and by $\tilde{\phi}:\widetilde{F}\rightarrow F$ the resulting desingularization of the rational map
$\phi$,  it follows from
the previous corollary that $\tilde{\phi}^*\omega^2=4\tau^*\omega^2$ so that $$\tilde{\phi}^*
(\omega^2\wedge\overline{\omega}^2)
=16\tau^*(\omega^2\wedge\overline{\omega}^2).$$
Integrating over $\widetilde{F}$ gives the result, as $\omega^2\wedge\overline{\omega}^2$
is  a volume form on $F$.
\cqfd
\section{Non-preperiodicity \label{sec2}}
\setcounter{equation}{0}
As mentioned in the introduction,
 Theorem  \ref{main} will be obtained as a consequence of the following
 fact :
 there exist  (many) $X$'s defined over a number field $k$, and
(many)  surfaces $\Sigma_b$ as in the previous section, also defined over a number field,
such that
the surfaces $\phi^l(\Sigma_b),\,l\in \mathbb{N},$ are Zariski dense in $X$.

Our goal in this section is to show that for many surfaces $\Sigma_b$, the 
set of distinct iterations $\phi^l(\Sigma_b)$ is infinite. This is proved in proposition
\ref{step1}.
Note that  $\phi^l$ is always generically defined along
$\Sigma_b$. Indeed, we may assume by induction on
$l$ that $\phi^{l-1}$ is generically defined along
$\Sigma_b$. Then   $\phi^{l-1}(\Sigma_b)$ must be a surface, as proved in \cite{amerik}.
If $\phi^l$ were not
generically defined along
$\Sigma_b$, then $\phi^{l-1}(\Sigma_b)$ would be contained in the indeterminacy locus of $\phi$.
But this indeterminacy locus is a surface of general type (\cite{amerik}),
hence cannot be dominated by
a surface which is birationally equivalent to an abelian surface.
So it does make sense to speak of $\phi^l(\Sigma_b)$, and each 
$\phi^l(\Sigma_b)$
is indeed a surface.

The strategy is as follows. Recall from lemma \ref{degphi} that we have $deg\,\phi=16$ and thus $deg\,\phi^l=16^l$.
What we will show, using proposition \ref{subprop1}, is the fact that if a  surface $\Sigma_b$ satisfies
$$\phi^l(\phi^k(\Sigma_b))=\phi^k(\Sigma_b)$$ for some $l>0$, and if 
furthermore a certain Terasoma-type condition holds for the point $b$, then
$$\phi^l:\phi^k(\Sigma_b)\rightarrow \phi^k(\Sigma_b)$$
has to be of degree $16^l$. This shall imply the following equality of degree $4$ cohomology classes
$${\phi^l}^*([\phi^k(\Sigma_b)])=[\phi^k(\Sigma_b)],$$
contradicting the results of \cite{amerik}, which show that $1$ is not an eigenvalue of
${\phi^l}^*$ acting on $H^4(F,\mathbb{Q})$.





An important step in the proof 
is the following result
concerning the restriction to the surfaces $\Sigma'_b$
of the incidence
correspondence $P$ between $F$ and $X$ :
\begin{prop} \label{subprop1} 1) The morphism
$$P_{b*}:CH_0(\Sigma'_{b,\mathbb{C}})\to CH_1(X_{\C})$$ induced by the pull-back to
$\Sigma'_b\rightarrow\Sigma_b\subset F$ of the incidence correspondence
$P\subset F\times X$ satisfies:
 ${P_{b*}}_{\mid CH_0(\Sigma'_{b,\mathbb{C}})_{hom}}$ factors through
the Albanese map $$CH_0(\Sigma'_{b,\mathbb{C}})_{hom}\stackrel{alb}{\rightarrow}
Alb\,\Sigma'_{b,\mathbb{C}}\cong A_{b,\mathbb{C}}.$$

2) For many closed points $b\in B$ defined over a number field, the  group morphism induced by $P_{b,*}$
$$P_{b,alb}: A_{b,\mathbb{C}}=Alb\,\Sigma'_{b,\mathbb{C}}\rightarrow CH_1(X_\mathbb{C})$$
is non-zero.
\end{prop}

The closed points of the second part of the proposition are exactly
the points satisfying a certain Terasoma-type condition, which shall 
be
stated in the end of this section.

\begin{rema} {\rm It is essential here to consider the Chow groups of the
corresponding complex algebraic variety, because conjecturally
$CH_1(X)_{hom}$ is trivial for $X$ defined over a number field. Also,
$CH_1(X_\mathbb{C})$ has no torsion, and thus the second part of the 
proposition is equivalent to saying that the group morphism $P_{b,alb}$ is
not of torsion.}
\end{rema}

We postpone the proof of proposition \ref{subprop1} and prove now the following proposition, which is the main result
of this section.
\begin{prop}\label{step1} If $b$ satisfies the conclusion of Proposition 2.1, 
2), the
Zariski closure of the union of surfaces $\phi^l(\Sigma_b),\,l\in \mathbb{N}$,
contains at least a divisor.
\end{prop}


As mentioned above,  because we already know
that the $\phi^{l}(\Sigma_b)$ are surfaces, the statement
 is equivalent to the fact that there are actually
infinitely many distinct surfaces $\phi^l(\Sigma_b),\,l\in \mathbb{N}$.
Equivalently, we have to prove that we do not have
$\phi^{l'}(\Sigma_b)=\phi^{k}(\Sigma_b)$
for some $l',\,k,\,l'>k$.
This is equivalent to saying that for any $l>0,\,k\geq 0$
\begin{eqnarray}
\label{equationperiodic}\phi^{l}(\phi^{k}(\Sigma_b))\neq \phi^{k}(\Sigma_b),
\end{eqnarray}
or that no surface $\phi^{k}(\Sigma_b)$, $k\in \Z_{\geq 0}$, is periodic for $\phi$.
In current terminology, one says that
$\Sigma_b$ is not preperiodic for $\phi$.


   The proof of proposition \ref{step1}  will proceed by contradiction. So let us  assume that
  $\Sigma_{b,k}=\phi^{l}(\Sigma_{b,k})$ for some $l>1$, where
  $\Sigma_{b,k}:=\phi^k(\Sigma_b)$.
Let us choose a desingularization
$\Sigma'_{b,k}$ of $\Sigma_{b,k}$. We first prove:
  \begin{lemm}\label{alksigmabk} The Albanese map
  $$alb: \Sigma'_{b,k}\rightarrow Alb\,{\Sigma'_{b,k}}$$
  is a birational map.
\end{lemm}
{\bf Proof.} First observe that $\Sigma'_{b,k}$ is rationally dominated via the rational
map $\phi^k$ by
the abelian surface $A_b=Alb\,{\Sigma'_b}$. On the other hand,
let
  $$\phi^k_*:A_{b,\mathbb{C}}=Alb\,\Sigma'_{b,\mathbb{C}}\rightarrow Alb\,{\Sigma'_{b,k,\mathbb{C}}}$$
  be  induced by the rational map $\phi^k$ between $\Sigma'_b$ and
  $\Sigma'_{b,k}$. We claim that $\phi^k_*$ is an isogeny between
$Alb\,{\Sigma'_{b,k}}$ and  $Alb\,{\Sigma'_{b}}$.

Indeed, this follows from the following three facts:
We know from proposition \ref{subprop1}, 1)  that for $\Sigma'_b$ (hence
 also for $\Sigma'_{b,k}$ by property (\ref{crucialpropertyphi})), the
 restriction $P_{b*}:CH_0(\Sigma'_{b,\mathbb{C}})_{hom}\rightarrow CH_1(X_\mathbb{C})$ (resp.
 $P_{b,k,*}:CH_0(\Sigma'_{b,k,\mathbb{C}})_{hom}\rightarrow CH_1(X_\mathbb{C})$) factors
 through $Alb\,\Sigma'_{b,\mathbb{C}}$ (resp. $Alb\,\Sigma'_{b,k,\mathbb{C}}$).  Let
 $$P_{b,k,alb}:Alb\,{\Sigma'_{b,k,\mathbb{C}}}\rightarrow CH_1(X_\mathbb{C}),\,\,
 P_{b,alb}: A_{b,\mathbb{C}}\rightarrow CH_1(X_\mathbb{C})$$
be induced respectively by the maps $P_{b,k,*}$ and $P_{b*}$.

 By formula (\ref{crucialpropertyphi}), factoring everything through the Albanese
 maps,  we get
\begin{eqnarray}\label{memeformule}P_{b,k,alb}\circ \phi^k_*=(-2)^kP_{b,alb}:A_{b,\mathbb{C}}
\rightarrow CH_1(X_\mathbb{C}).
\end{eqnarray}


On the other hand,
we know by Proposition \ref{subprop1}, 2) that $P_{b, alb}: A_{b,\mathbb{C}}
\rightarrow CH_1(X_\mathbb{C})$ is non-zero.
As $P_{b,*}$ is a group morphism which is induced by a correspondence, its kernel is
a countable union of translates of an abelian subvariety of $A_{b,\mathbb{C}}$. Thus it must
be countable
because we know  that $A_b$ is a geometrically simple abelian variety
and that $P_{b,*}$ is non-zero.
Formula (\ref{memeformule}) then shows that the surjective morphism
$\phi^k_*$ has  countable kernel, hence that it is an isogeny.

In conclusion, the surface $\Sigma'_{b,k}$ sits between two abelian surfaces:
$$ A_b\dashrightarrow \Sigma'_{b,k}\stackrel{alb}{\rightarrow} Alb\,\Sigma'_{b,k},$$
where the composed map is an isogeny. It follows immediately that
$alb: \Sigma'_{b,k}{\rightarrow} Alb\,{\Sigma'_{b,k}}$ is birational.
\cqfd
As a consequence of this lemma and formula (\ref{memeformule}),
we get now:
\begin{coro}\label{corointer} The rational map
$$\phi^l:\Sigma'_{b,k}\dashrightarrow \Sigma'_{b,k}$$
has degree $(-2)^{4l}=(16)^l$.
\end{coro}
{\bf Proof.}  Indeed,
formula (\ref{memeformule}) applied to the maps $\phi^k$ and $\phi^{k+l}$ tells us that
\begin{eqnarray}
\label{above15juillet}
P_{b,k,alb}\circ\phi^l_*=(-2)^lP_{b,k,alb}: Alb\,{\Sigma'_{b,k,\mathbb{C}}}\rightarrow CH_1(X_\mathbb{C}).
\end{eqnarray}
Here $\phi^l_*:Alb\,{\Sigma'_{b,k,\mathbb{C}}}\rightarrow Alb\,{\Sigma'_{b,k,\mathbb{C}}}$
 is induced by the rational self-map
$\phi^l:{\Sigma'_{b,k}}\rightarrow {\Sigma'_{b,k}}$.
Furthermore,  we know that
$Ker\,P_{b,k,alb}$ is countable.
 From  formula (\ref{above15juillet}), we
 thus  deduce that $\phi^l_*$ must be multiplication by $(-2)^l$, hence must have degree
 $(-2)^{4l}$.
As  the  surface
 $\Sigma'_{b,k}$ is birationally equivalent to its Albanese variety by lemma
 \ref{alksigmabk}, the same result holds for
 $\phi^l:\Sigma'_{b,k}\rightarrow \Sigma'_{b,k}$.

\cqfd
{\bf Proof of Proposition \ref{step1}}.
We know that $\phi^l$ is generically well-defined along $\Sigma'_{b,k}$.
Take the standard desingularization
$$\widetilde{\phi}:\widetilde{F}\rightarrow F,\,
\pi: \widetilde{F}\rightarrow F$$ obtained by  blowing-up the indeterminacy locus
of $\phi$. It is proved in \cite{amerik} that
$\widetilde{\phi}$ does not contract divisors and can only contract a
surface which is already contracted by $\pi$ (in fact even this is impossible,
but needs some extra argument). It follows that for
the natural  desingularization
$\widetilde{\phi^l}:\widetilde{\widetilde{F}}\rightarrow F$,
$\pi':\widetilde{\widetilde{F}}\rightarrow F$ of $\phi^l$
obtained by taking
the graph, the following holds:
For any  subvariety $\Sigma\subset\widetilde{\widetilde{F}} $
which is contracted under
$\widetilde{\phi^l}$,  the image of
$\Sigma$ under $\pi'$ is at most $1$-dimensional.
We apply this to any component $\Sigma$ of
$\widetilde{\phi^l}^{-1}(\Sigma_{b,k})$.  We conclude that
if either $dim\,\Sigma\geq 3$ or $\Sigma$ does not dominate $\Sigma_{b,k}$ via
$\widetilde{\phi^l}$, $\pi'(\Sigma)$ has dimension at most $1$.
On the other hand,
$\pi'(\widetilde{\phi^l}^{-1}(\Sigma_{b,k}))$ contains $\Sigma_{b,k}$ by periodicity, and we
have just seen that the restriction of $\phi^l$ to $\Sigma_{b,k}$  is
already of degree $(16)^l=deg\,\widetilde{\phi^l}$. It thus follows that   the only
$2$-dimensional component of
$\widetilde{\phi^l}^{-1}(\Sigma_{b,k})$ dominating $\Sigma_{b,k}$ is
sent birationally by $\pi'$ to
$\Sigma_{b,k}$, while the non-dominating components or the three dimensional
component are contracted to curves or points in
$F$ by $\pi'$.

 This implies  that we have the equality
of $2$-dimensional cycles:
$$(\phi^l)^{*}(\Sigma_{b,k})=\Sigma_{b,k}$$ and thus
$$(\phi^l)^{*}([\Sigma_{b,k}])=[\Sigma_{b,k}],$$
where $[\,\cdot\,]$ denotes the cohomology class of a cycle.
This contradicts the computation of the eigenvalues of $(\phi^l)^*$ on $H^4(F_\mathbb{C},\mathbb{Q})$
performed in \cite{amerik}.

The proof is now complete, assuming Proposition \ref{subprop1}.
\cqfd
{\bf Proof of Proposition \ref{subprop1}.}
Statement 1) follows from the fact that the surface $\Sigma_b$ is the surface of lines of a
nodal cubic
threefold $Y_b$, for which we know that the Abel-Jacobi map on a desingularization
$\widetilde{Y}_b$ induces an isomorphism:
$$CH_1(\widetilde{Y}_{b,\mathbb{C}})_{hom}\stackrel{\Phi_{\widetilde{Y}_b}}{\cong}
J^3(\widetilde{Y}_{b,\mathbb{C}}).$$
The correspondence from $\Sigma'_b$ to $X$ thus factors to a correspondence from
$\Sigma'_b$ to $\widetilde{Y}_b$, which, when restricted to
$CH_0(\Sigma'_{b,\mathbb{C}})_{hom}$ has to factor through $Alb\,\Sigma'_{b,\mathbb{C}}=J^3(\widetilde{Y}_{b,\mathbb{C}})$.

For the proof of statement 2), we will follow the argument of
\cite{ggp}. In fact this will be   as in \cite{ggp}
a consequence  of  an even stronger
statement (Proposition \ref{plusfort}), the proof of which is attributed
there to Bloch and Nori, and  which we will also need
in section \ref{conclusion}.

Let us first introduce a corrected codimension $3$ cycle
$$P'\in CH^3( F\times X)_\mathbb{Q}$$
which has the property that its cohomology class satisfies
\begin{eqnarray}
\label{formPprime}[P']\in H^2(F_\mathbb{C},\mathbb{Q})\otimes H^4(X_\mathbb{C},\mathbb{Q})_{prim},
\end{eqnarray}
and which differs from the incidence cycle $P$ by a combination of cycles of the form
$\Gamma_i\times h^i$, $\Gamma_i\in CH^{3-i}(F)$.
This cycle is obtained by considering
the  cycle $j_*P$ on $F\times \mathbb{P}^5$, where $j$ is the inclusion of $X$
in $\mathbb{P}^5$ (or of $F\times X$ in $X\times \mathbb{P}^5$). Denoting by
$H\in CH^1(\mathbb{P}^5)$ the class of $\mathcal{O}_{\mathbb{P}^5}(1)$, this cycle
$j_*P\in CH^4(F\times
\mathbb{P}^5)$ can be written as $\sum_{i=1}^{i=4}pr_1^*\Gamma_i\cdot pr_2^*H^{i}$, with $\Gamma_i\in
CH^{4-i}(F)$ (cf \cite{voisinbook}, Theorem 9.25).
The fact that the sum is only over $i\geq1$ is due to the fact that
$H^5 j_*P=0$.
Let $$P':=P-\frac{1}{3}(\sum_{i=0}^{i=3}pr_1^*\Gamma_{i+1}\cdot pr_2^*h^{i})\in CH^3(F\times X)_\mathbb{Q},$$
where now the projectors are those of $F\times X$ to its factors.
Then $j_*P'=0$ because $j_*h^i=3H^{i+1}$, and property
(\ref{formPprime})  follows by Lefschetz theorem on hyperplane sections, which tells that
$Ker\,(j_*:H^*(X_\mathbb{C},\mathbb{Q})\rightarrow H^{*+2}(\mathbb{P}^5_\mathbb{C},\mathbb{Q}))$ is zero for
$*\not=4$.



We can see, using Poincar{\'e} duality on $X$, the vector space
$H^2(F_\mathbb{C},\mathbb{Q})\otimes H^4(X_\mathbb{C},\mathbb{Q})_{prim}$ as
$$Hom\,(H^4(X_\mathbb{C},\mathbb{Q})_{prim},H^2(F_\mathbb{C},\mathbb{Q})).$$ Thus,
for any morphism $f: T\rightarrow F$, the cycle
$f^*P'$ is cohomologous to $0$ if and only if the map
$$f^*\circ P^*:H^4(X_\mathbb{C},\mathbb{Q})_{prim}\rightarrow H^2(T_\mathbb{C},\mathbb{Q})$$
is $0$.  (Note that, by definition of $P'$, $P^*={P'}^*$ on $H^4(X_\mathbb{C},\mathbb{Q})_{prim}$.)

Assume now  that this last condition is satisfied. (This is true in particular if
$T$ is one of the surfaces $\Sigma'_b$, by the same argument
as in section \ref{preli}, where we prove that the surfaces $\Sigma_b$ are Lagrangian with respect to
$\omega$.)

Then the cycle $f^*P'$ is cohomologous to $0$. If furthermore
$T$ and $f$ are defined over a number field $k$, then by the comparison theorems
between {\'e}tale $l$-adic and Betti cohomology for algebraically closed fields of characteristic $0$,
the cycle $f^*P'$ has trivial  $l$-adic {\'e}tale cohomology class in
$H^6_{et}(T_{\overline{k}}\times_{\overline{k}} X_{\overline{k}},\mathbb{Q}_l(3))$.
On the other hand, the cycle
$f^*P'$ defined over  $k$ has a {\it continuous {\'e}tale} (cf \cite{jannsen}, \cite{raskind})
cohomology class
$$[f^*P']\in
H^6(T_{k}\times_{{k}} X_{{k}},\mathbb{Q}_l(3)).$$

(Continuous \'etale cohomology is first defined for \'etale coefficients
sheaves like $\mathbb{Z}_l$, which are inverse limits of finite
coefficients sheaves. Continuous \'etale cohomology with coefficients in a
$\mathbb{Q}_l$-module is then defined by tensoring with $\mathbb{Q}_l$.)


Following \cite{raskind}, we omit in the sequel both ``continuous'' and ``\'etale'' in the notation
for these cycle classes.

The Hochschild-Serre spectral sequence for continuous {\'e}tale cohomology then gives us
the $l$-adic Abel-Jacobi invariant
$$\gamma_{f^*P'}\in H^1(k, H^5
(T_{\overline{k}}\times_{\overline{k}} X_{\overline{k}},\mathbb{Q}_l(3))),$$
which lies in fact (because the cycle $P'$ is deduced from $P$ by
a Chow-K\"unneth decomposition) in
the subspace
$H^1(k, H^1(T_{\overline{k}},\mathbb{Q}_l)
\otimes_{\mathbb{Q}_l} H^4(X_{\overline{k}},\mathbb{Q}_l)(3))$.

(Here $H^1(k,\cdot)$ denotes the continuous \'etale cohomology of $Spec\,k$.)

Coming back to the case where $T$ is one of our surfaces $\Sigma'_b$, with $b\in B$ a point
defined over a number field $k(b)$, let us denote by
$P'_b\in CH^3(\Sigma'_b\times X)_\mathbb{Q}$ the cycle $\tau_b^*P'$, where
$\tau_b:\Sigma'_b\times X\rightarrow F\times X$
is the natural map.
Using
 the Bloch-Srinivas lemma (cf \cite{blS}
or \cite{voisinbook}, Corollary 10.20) and the fact that there is no
non-zero algebraic cycle class in
$H^4_{et}(X_{\overline{k(b)}}, \Q_l(2))_{prim}$, we get as in \cite{ggp} the following
lemma:
\begin{lemm} \label{intermediate} If the $l$-adic Abel-Jacobi invariant
$$\gamma_{P'_b}\in H^1(k(b), H^1(\Sigma'_{\overline{k(b)}},\mathbb{Q}_l)
\otimes_{\mathbb{Q}_l} H^4(X_{\overline{k(b)}},\mathbb{Q}_l)(3))$$
is non zero, then the morphism
$$P_{b,alb}: Alb\,\Sigma'_{b,\mathbb{C}}=A_{b,\mathbb{C}}\rightarrow CH_1(X_\mathbb{C})$$
is non zero.
\end{lemm}

Using this lemma, the statement 2) of proposition \ref{subprop1} is a consequence of the following
proposition:

\begin{prop} \label{plusfort} For many closed points $b\in B$ defined over a number field, the
$l$-adic Abel-Jacobi invariant
$$\gamma_{P'_b}\in H^1(k(b)), H^1(\Sigma'_{\overline{k(b)}},\mathbb{Q}_l)
\otimes_{\mathbb{Q}_l} H^4(X_{\overline{k(b)}},\mathbb{Q}_l)(3))$$
is non-zero.
\end{prop}
{\bf Proof.}
The proof of this statement goes as in \cite{ggp}, once we notice the following facts:
We have a $2$-dimensional family of surfaces which sweep out $F$ :
$$\mathcal{S}_B:=\cup_{b\in B}\Sigma_b'\stackrel{s}{\rightarrow} F$$
 where $s$ is dominating. For a generic curve $B'\subset B$
defined over a number field, the family of surfaces
$$\mathcal{S}_{B'}:=\cup_{b\in B'}\Sigma_b'\stackrel{s}{\rightarrow} F$$
covers an hypersurface in $F$, and it thus follows that
the pull-back map by $s$:
$$ H^{2,0}(F)\rightarrow H^{2,0}(\mathcal{S}_{B'})$$
is non-zero because the $(2,0)$-form $\omega$ is non-degenerate, hence cannot vanish
on a divisor of $F$.

More precisely, for any Zariski open set $B'_0\subset B'$, the cohomology class
$[s^*\omega]$ does not vanish in the Betti cohomology space
$H^2(\mathcal{S}_{B'_0,\mathbb{C}}, \mathbb{C})$, because it is of type $(2,0)$.
We will take for $B'_0$ the locus where the fiber of our family of surfaces
$\pi:\mathcal{S}_{B'}\rightarrow B'$ is smooth.

Hence we conclude similarly that the Betti
cohomology class of the cycle
$$s^*P'_{\mid \mathcal{S}_{B'_0}}\in CH^3(\mathcal{S}_{B'_0}\times X)$$
does not vanish in
$$ H^6(\mathcal{S}_{B'_0,\mathbb{C}}\times X_\mathbb{C},\mathbb{Q}).$$
However, we know already that the cycle $s^*P'$ restricted to the fibers
$\Sigma'_b\times X$ of $\pi$ has vanishing cohomology class. Hence
$[s^*P'_{\mid \mathcal{S}_{B'_0}}]$ lies at least in the $L^1$ level of the Leray filtration
associated to
$$\pi'=\pi\circ pr_1:\mathcal{S}_{B'_0,\mathbb{C}}\times X_\mathbb{C}\rightarrow B'_{0,\mathbb{C}},$$
 and gives an element of
$$H^1(B'_{0,\mathbb{C}},R^5\pi'_{\mathbb{C}*}\mathbb{Q})$$
and in fact more precisely (using the fact that $P'$ was obtained
from $P$ by applying a Chow-K\"unneth projector on
$X$) in
$$H^1(B'_{0,\mathbb{C}},R^1\pi_{\mathbb{C}*}\mathbb{Q}\otimes H^4(X_\mathbb{C},\mathbb{Q})_{prim}).$$
This last class is non-zero because the Leray spectral sequence has only two terms, as
the affine curve $B'_0$ has the homotopy type of a CW-complex of dimension $1$.


We can now make the same construction in $l$-adic {\'e}tale cohomology for
the corresponding algebraic varieties defined over $\overline{\mathbb{Q}}$.
Using the comparison theorems between Betti cohomology of the complex manifold and {\'e}tale cohomology
of the algebraic variety, we conclude that
the {\'e}tale cohomology class
$$[p^*P'_{\mid \mathcal{S}_{B'_0}}]_{et}\in H^6_{et}
(\mathcal{S}_{B'_0,\overline{\mathbb{Q}}}\times_{\overline{\mathbb{Q}}} X_{\overline{\mathbb{Q}}},
\mathbb{Q}_l(3))$$
is non-zero, lies in the $L^1$ level of the Leray filtration for {\'e}tale
cohomology, and has a non-zero image in
$$H^1_{et}(B'_{0,\overline{\mathbb{Q}}},R^1\pi_{et*}\mathbb{Q}_l\otimes_{\mathbb{Q}_l}
 H^4(X_{\overline{\mathbb{Q}}},\mathbb{Q}_l)_{prim}(3)).$$

This provides a non-trivial extension $E$ of the {\'e}tale local system
$R^1\pi_{et*}\mathbb{Q}_l\otimes_{\mathbb{Q}_l} H^4(X_{\overline{\mathbb{Q}}},
\mathbb{Q}_l(3))$
on $B'_0$  by the trivial local system. As in \cite{ggp}, let us choose $b\in
B'_0(\overline{\mathbb{Q}})$
satisfying the Terasoma condition (that is, the coincidence of the images
of the monodromy and Galois representations) for $E$ . Then we conclude as in 
\cite{ggp}, Lemma 7, that the
corresponding
extension of $Gal\,(\overline{k(b)}/k(b))$-modules is also non-trivial, and it
follows
that the class
 $$\gamma_{P'_b}\in H^1(k(b), H^1(\Sigma'_{\overline{k(b)}},\mathbb{Q}_l)
\otimes_{\mathbb{Q}_l} H^4(X_{\overline{k(b)}},\mathbb{Q}_l)(3))$$
is non-zero.



\cqfd
\section{Study of invariant divisors}
\setcounter{equation}{0}
We have now established that the Zariski closure of the union of surfaces
$\phi^l(\Sigma_b)$ contains at least a divisor.
Our goal in this section is to exclude  the case where
the Zariski closure of the union of the surfaces
$\Sigma_{b,i}$ is of dimension $3$.
Observe now that if this closure
is not equal to $F$,  the union of its divisorial components   is invariant under $\phi$,
hence each of its
irreducible components must be invariant under some power $\phi^l$.




Reasoning by contradiction,
we will denote in the sequel by $\tau:D\rightarrow F$ a desingularization of a
$3$-dimensional  irreducible
component of the Zariski closure of the union of the surfaces $\Sigma_{b,i}$, where $b$ is fixed and is supposed to
satisfy the conclusion of Proposition \ref{step1}.
As this Zariski closure has only finitely many $3$-dimensional components and is stable under
$\phi$, it follows that each of its irreducible components is stable under some power
$\phi^l$ of $\phi$.

We will denote by
 $\Sigma_{b,n_i}$ the surfaces $\Sigma_{b,k}$ contained
in $\tau(D)$ and  $\Sigma'_{b,n_i}$ their proper transforms under $\tau$.

We shall also denote by $\omega_D$ the holomorphic $2$-form
$\tau^*\omega$. Notice that $\omega_D$ is a non-zero form, because $\omega$ is non-degenerate.
Thus at a point where $\tau $ is of maximal rank $3$, $\omega_D$ cannot vanish.

\subsection{A Jouanolou type argument \label{secclassi}}

The kernel of $\omega_D$ at the generic point is one-dimensional. This
provides a line bundle $L$ and an injective map
$L\to T_D$ with torsion-free cokernel, generically defined as
$Ker\,\omega_D:T_D\rightarrow\Omega_D$. As the surfaces $\Sigma'_{b,n_i}$ are
isotropic for $\omega_D$, the image of $L$ must be tangent to any
$\Sigma'_{b,i}$, or at least to those not contained in the singular set of
$\tau(D)$. Recall that the  surfaces $\Sigma'_{b,n_i}$ are Zariski dense
in $D$. In particular there are infinitely many such surfaces which are
distinct and not contained in the singular set of
$\tau(D)$.

 We start by proving an analogue of
Jouanolou's theorem (cf \cite{J}) for this situation:

\begin{prop}\label{jouan} There exists a rational map
$f: D\dasharrow \p^1$ such that $L$ is tangent to the fibers of $f$.
\end{prop}
{\bf Proof:} For convenience of the reader, we recall at the same time
the construction
of Jouanolou. For large $N$, the classes of the surfaces $\Sigma'_{b,n_i},\,i\leq N$ are not linearly
independent in $NS(D)$. Therefore one can find integers $m_i$ such that
the line bundle associated to the divisor $M=\sum_{i=i_0}^Nm_i\Sigma'_{b,n_i}$
has zero Chern class in $H^1(D, \Omega_D)$. Here, $i_0$ is chosen in such way that
 $\omega_D$ does not vanish on $\Sigma'_{b,n_i}$ for $i\geq i_0$.
Let $U_{\alpha}$ be an open
covering of $D$, and $f_{\alpha}$ be a meromorphic function defining $M$ on $U_{\alpha}$. The vanishing
of the first Chern class of the cocycle $g_{\alpha \beta}\in
H^1(D, {\cal O}^*_D)$ means that on the intersections
$U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}$
$$\frac{df_{\alpha}}{f_{\alpha}}-\frac{df_{\beta}}{f_{\beta}}=
\sigma_{\alpha}-\sigma_{\beta}$$
for some $\sigma_{\alpha}\in \Omega^1_{U_{\alpha}}$,
$\sigma_{\beta}\in \Omega^1_{U_{\beta}}$. Therefore, the
$\frac{df_{\alpha}}{f_{\alpha}}-\sigma_{\alpha}$ patch together into a
meromorphic $1$-form $\psi_M$ with logarithmic singularities
along the components of the support of
$M$, defined up to a holomorphic $1$-form, that is, we have
a map
$$\psi: Div^0_{\Sigma}(D)\otimes \C \to H^0(D, \Omega^1_D
\otimes {\cal M}^*_D)/H^0(D,\Omega^1_D),$$
where ${\cal M}^*_D$ denotes the sheaf of meromorphic functions and
where $Div^0_{\Sigma}(D)$ is the kernel of
the cycle class map, restricted to the subspace generated by the $\Sigma'_{b,n_i},\,i\geq i_0$.
 Moreover,
on $U_{\alpha \beta}$, $d\sigma_{\alpha}=d\sigma_{\beta}$, so there is
a well-defined holomorphic 2-form $d\sigma$. It follows from the
degeneration of ``Hodge to de Rham" spectral sequence that $d\sigma$
is actually zero (cf. \cite{J}). Therefore $\psi_M$ is closed.

Moreover, the map $\psi$ is injective (this is \cite{J}, Lemma 2.8).


The meromorphic $1$-form $\psi_M$ induces a meromorphic section of $L^*$.
Since outside of the singularities of the foliation, that is, outside of an
analytic subset of codimension at least two in $D$, $L$ is tangent to $\Sigma'_{b,n_i}$
along $\Sigma'_{b,n_i}$ for $i\geq i_0$,
this section is actually holomorphic, because $\psi_M$ has logarithmic
singularities along the smooth part of the  $\Sigma'_{b,n_i}$'s appearing in the support of $M$.
So the map $\psi$ induces a map
$$\chi:  Div^0_{\Sigma}(D)\otimes \C \to H^0(D,L^*)/V,$$
where $V$ is the image in $H^0(D,L^*)$ of the space of global holomorphic 1-forms on $D$.
The kernel of $\chi$ is clearly infinite-dimensional.

Let $M$ and $M'$
be two non-proportional elements
of  $Ker\,\chi\subset Div^0_{\Sigma}(D)\otimes \C$.
Then we can choose the meromorphic forms $\psi_M$ and $\psi_{M'}$ in such a way that
$$\psi_{M\mid L}=0,\,\,\psi_{M'\mid L}=0.$$
It then follows that $L$ is contained in the kernel
of the meromorphic $2$-form $\psi_M\wedge \psi_{M'}$. But this property is also satisfied
by our  original 2-form
$\omega_D$. Therefore $\psi_M\wedge \psi_{M'}=f\omega_D$ for some meromorphic function
$f$ on $D$.
We meet now two cases:

1) Assume that for all $M$ and $M'$, $f$ is a constant function on $D$.
In this case, the meromorphic form $\psi_M\wedge \psi_{M'}$ is nonsingular.
On the other hand, the meromorphic $1$-form $\psi_{M'}$ has logarithmic singularities
along the surfaces $\Sigma'_{b,n_j}$ appearing in $M'$. The absence of singularities
in $\psi_M\wedge \psi_{M'}$ then implies that the restriction of $\psi_M$
to the surfaces $\Sigma'_{b,n_j}$ appearing in $M'$ and not in $M$ is zero.
But as $M'$ is arbitrary, it then follows that there are infinitely
many surfaces $\Sigma'_{b,n_i}$ which are integral surfaces for the $1$-form
$\psi_M$. Jouanolou's theorem
\cite{J} then tells that there exists a non-constant
meromorphic  function from
 $D$ to $\mathbb{P}^1$,  whose fibers are leaves of $\psi_M$. As $\psi_M$ vanishes on $L$,
 we are done in this case.

2) In the second case,  we have for some
$M'$ a non-constant rational map $f: D\dasharrow \p^1$.
Finally, differentiating
the equality $\psi_M\wedge \psi_{M'}=f\omega_D$,
an using the fact  that all our forms are closed, we get $df\wedge\omega_D=0$, so $df$ vanishes on
$Ker\,\omega_D=L$.

\cqfd

\begin{coro}\label{twofibrat}
We have two possibilities: either

(1) $D$ is rationally fibered in (birationally) abelian varieties,
and $\Sigma'_{b,n_i}$
are fibers for all but finitely many $i$'s; or

(2) $D$ is rationally fibered in curves over a surface $T$, the fibers are
integral curves of the foliation $L$, and $\Sigma'_{b,n_i}$
project onto curves in $T$. Furthermore the $2$-form $\omega_D$ is pulled-back from a
$2$-form on $T$.
\end{coro}
Here,``rationally fibered" means that the fibration map is only rational,
not necessarily regular. We can, of course, arrange for it to be regular
by blowing-up $D$.

\medskip

\noindent{\bf Proof:} Consider the Stein factorisation
$g: D\dasharrow C$ of the map $f$. If all but finitely many
$\Sigma'_{b,n_i}$'s are fibers, we are done; if not, consider the general
fiber $D_c$. On $D_c$, we still have the foliation $L$ from the
proposition \ref{jouan}. This foliation has infinitely many integral curves, which
are components of $\Sigma'_{b,n_i}\cap D_c$, where $i$ is such that
$\Sigma'_{b,n_i}$ is not a fiber. So, by the original
Jouanolou's theorem, there must be a fibration
$h_c:D_c\dasharrow Z_c$ over a curve, whose fibers are generically connected
and tangent to
$L$. A countability argument for the Chow varieties
of curves in $D$  shows that we can assume that
the   fibers of $h_c$, $c\in C$, form a family of algebraic curves covering $D$ and consisting of
integral leaves of $L$.
It is then immediate that these curves provide a fibration of $D$ to a surface
$T$, which we may assume to have connected fibers.

As $L$ is tangent to these fibers, and to the surfaces $\Sigma'_{b,n_i}$, it follows that the
$\Sigma'_{b,n_i}$'s
 are contracted to curves in $T$. Finally, as the fibration has connected fibers, whose tangent space
 is generically the kernel of $\omega_D$, it follows that $\omega_D$ comes from a
 form on $T$.

\cqfd



\subsection{Case where $D$ is fibered into abelian surfaces \label{secclassigeo}}

In this subsection, we will exclude case 1) of Corollary \ref{twofibrat}. We argue again by contradiction,
and assume that we have a divisor $D$, invariant under some power $\phi^l$
of $\phi$, and
admitting a fibration over a curve, whose fibers are surfaces birationally equivalent to abelian surfaces.
We know furthermore that infinitely many fibers are surfaces $\Sigma'_{b,n_i}$, which
are not periodic under $\phi^l$.

The starting point is the following

\begin{prop}\label{kod}
The Kodaira dimension of $D$ is zero.
\end{prop}
{\bf Proof:} $D$ is (rationally) fibered over a curve in varieties
of Kodaira dimension zero. By the universal property of the Iitaka
fibration, $\kappa(D)$ is thus at most one.

To exclude the case where $\kappa(D)=1$, we use  the result of \cite{NZ}, saying that
a rational self-map $\psi$ of any variety $X$ induces an automorphism
of finite order on the base $B$ of the Iitaka fibration (note that the fact
that $\psi$ induces a polarization-preserving automorphism of $B$ is clear
because the pull-back $\psi^*$
is an automorphism of the vector space of pluricanonical forms; the non-trivial
statement is the finiteness of the order of the induced automorphism  on $B$. Also, we do not need the full strength of
Theorem A of \cite{NZ}; the crucial Proposition 2.3 of {\it loc. cit.} for an abelian fibration
suffices, and this is shown by a monodromy argument).

 We now apply this to $X=D$ and $\psi=\phi^l$. If $\kappa(D)=1$,
then the map $f:D\dasharrow \p^1$ constructed in \ref{jouan} is the Iitaka
fibration, and the finiteness of the order of the induced map on $\p^1$ implies
that the $\Sigma_{b,n_i}$ (which are fibers of $f$) are periodic, whereas we know that this is not the case.

Suppose now that $\kappa(D)=-\infty$. Then $D$ is uniruled. Consider the
maximal rationally connected  fibration $r:D\dasharrow Q$. As the fibers are rationally connected, every
holomorphic form on $D$ must be pulled-back from $Q$. Since $D$ carries a non-zero
holomorphic 2-form $\omega_D$, $Q$ is a surface, and $\omega_D=r^*\eta$
for some $0\not=\eta\in H^{2,0}(Q)$.
The surfaces $\Sigma'_{b,n_i}$ cannot  project onto curves on $Q$, since
they are not uniruled. Therefore they dominate $Q$ via $r$. But as they are
isotropic for $\omega_D$, we get
$$\omega_{D\mid \Sigma'_{b,n_i}}=0=r^*\eta_{\mid \Sigma'_{b,n_i}},$$ hence
$\eta=0$, which is  a contradiction.


\cqfd

If $D$ had numerically trivial canonical class, the natural idea to continue
would be to use the Bogomolov-Beauville decomposition; it would imply,
because of the existence of a holomorphic 2-form, that up to a finite
covering, $D$ is either an abelian threefold or a product of an elliptic
curve and a $K3$-surface. In our situation,
we can take the minimal model of $D$, but it can have terminal singularities,
and at this moment, no analogue of Bogomolov-Beauville decomposition is known
for terminal varieties. Nevertheless, as observed already by Campana and 
Peternell
\cite{CP}, it exists for threefolds with
a holomorphic $2$-form.


The following proposition is proved in \cite{CP}; we give a somewhat more
detailed argument for reader's convenience.

\begin{prop}\label{mmp} (Theorem 3.2 of \cite{CP})
For any smooth projective threefold $D$ such that $\kappa(D)=0$ and
$h^{2,0}(D)\neq 0$,
there exists a dominant rational map $\mu:Y\dasharrow D$, where $Y$ is either
an abelian threefold, or a product $E\times S$ of a projective K3 surface $S$
and an elliptic curve $E$.

\end{prop}


{\bf Proof.} By minimal model theory, there is a birational map
$D\dasharrow D^0$, such that $D^0$ has at most canonical singularities,
the Weil divisor $K_{D^0}$ is $\Q$-Cartier and $mK_{D^0}=0$ for
some $m>0$. Take the canonical cover $D'=Spec(\oplus_{i=0}^{m-1}{\cal O}
(iK_{D^0}))$ (where ${\cal O}(iK_{D^0})$ are sheaves associated to Weil
divisors; see \cite{R}, appendix to section 1, for a discussion of the
definition of the pluricanonical sheaves etc. in this case). It is well-known
(cf \cite{R}) that it is \'etale in codimension 2 (wherever
$K_{D^0}$ is Cartier), the singularities of $D'$
 are canonical and that $K_{D'}=0$. Let $Y$ be a resolution of $D'$.

Since canonical singularities are rational by \cite{E},
 that is, there are no higher direct images of the structure sheaf,
we have $h^2({\cal O}_{D^0})=h^2({\cal O}_{D})>0$. Therefore
$h^2({\cal O}_{D'})\geq h^2({\cal O}_{D^0})>0$. Since canonical class
of $D'$ is trivial, we obtain by Serre duality that $h^1({\cal O}_{D'})>0$,
and so, by rationality of the singularities, $h^1({\cal O}_{Y})>0$. 

Thus the Albanese map of $Y$ is non-trivial. It factors through $D'$
and
gives a fibration
$\alpha:D'\to Alb\,D':= Alb\,Y$ (\cite{Kaw}). Now the results of
\cite{Kaw}, section 8, say that this becomes a product after a finite \'etale
covering. Thus if the dimension of fibers of $\alpha$ is zero or
one,
$D$ is dominated by an abelian threefold; if this dimension is two,
then either the fibers are birational to a $K3$ surface $S$ and $D$
is dominated by a product of $S$ and an elliptic curve, or,
repeating the
process (i.e. considering
the Albanese map) for the fiber of $\alpha$, we get again that $D$
is dominated by an abelian threefold.

\cqfd

Let us denote by $H^2_0(F, \Q)$ the orthogonal of
$c_1(\mathcal{L})$ with respect to the Beauville-Bogomolov form. This is just
the  image of the map
$p_*q^*:H^4(X,\mathbb{Q})_{prim}\rightarrow H^2(F, \Q)$ (cf section \ref{preli}).

\begin{lemm}\label{hodge}
For $F$ such that $Pic(F)=\Z$, $H^2_0(F, \Q)=H^2_{tr}(F, \Q)$ is a
simple
Hodge
structure of rank 22.
\end{lemm}
{\bf Proof:} The simplicity follows from $h^{2,0}(F)=1$ by a standard argument:
let $V$ be a non-trivial Hodge
substructure of $H^2_0(F, \Q)$. If $V$ is of level zero, that is,
$V\otimes \C\subset H^{1,1}(F)$, then there are integral $(1,1)$ classes
in $H^2_0(F, \C)$, contradicting $Pic(F)=\Z$. If $V$ is not of level zero,
$V\otimes \C$ contains $H^{2,0}(F)$, so the orthogonal of $V$ is of level
zero, hence must be trivial, and we conclude that $V=H^2_0(F, \Q)$.

The rank is computed in \cite{be-do}: indeed, it is proved there
that $F$ is deformation equivalent to the $Hilb^2$ of a K3 surface,
so that  $b_2(F)=23$; and since $Pic(F)=\Z$, the $b_2^{tr}(F)$ is  only one less.


\cqfd

In the above lemma and in what follows,
we denote by $b_2^{tr}(Y)$ the difference
between $b_2(Y)$ and the Picard number $\rho(Y)$, where $Y$ is a projective
manifold.

\begin{lemm}\label{betti}
Let $Y,\,Z$ be smooth projective of the same dimension. If there exists
a dominant rational map $f:Y\dasharrow Z$, then $b_2^{tr}(Z)\leq b_2^{tr}(Y)$.
\end{lemm}
{\bf Proof:} Let $p:Y'\to Y$ be a resolution of $f$, so that $p$ is a
composition
of blow-ups with smooth centers and $q:Y'\to Z$ is a generically finite
morphism, $q=f\circ p$. Since $q_*q^*=deg(q)\cdot Id$ on  cohomology groups,
$q^*:H^2(Z,\C)\to H^2(Y',\C)$
is injective and takes transcendental cohomology to transcendental cohomology. So
$b_2^{tr}(Z)\leq b_2^{tr}(Y')$. Since $p$ is a composition of blow-ups,
$b_2^{tr}(Y')=b_2^{tr}(Y)$.

\cqfd

We now conclude as follows. From propositions
\ref{kod}  and \ref{mmp}, our divisor
$D$ is rationally dominated either by an abelian threefold, or by a
product of an elliptic curve and a K3-surface. Consider
the morphism of Hodge structures
$$\tau^*: H^2_{tr}(F, \C)\to H^2(D, \C)$$
As  $\omega$ restricts
non-trivially to $D$,  this morphism is non-zero. By irreducibility
of the Hodge structure on $H^2_{tr}(F, \Q)$, it is injective and its image
is contained in  the transcendental part of $H^2(D,\mathbb{Q})$,
so $b_2^{tr}(D)$ is at least
$22$ (lemma \ref{hodge}).
This is clearly greater than $b_2^{tr}$ of a three-dimensional torus.
Finally, let $E$ be an elliptic curve and $S$ a projective K3-surface; we have
$$H^2(E\times S,\Q)=H^2(E,\Q)\otimes H^0(S,\Q)\oplus H^0(E,\Q)
\otimes H^2(S,\Q);$$
this is of dimension $23$ and has at least a $2$-dimensional subspace of
algebraic classes, so that $b_2^{tr}(E\times S)$ is again strictly less
than $22$, contradicting lemma \ref{betti}.

We thus proved  that the case of a fibration in abelian surfaces is impossible.

\subsection{Case where $D$ is fibered over a surface \label{conclusion}}
In this section we consider the remaining situation, where $D$ admits a rational fibration
 over a surface $T$, in such a way that the $2$-form $\omega_D$ is pulled-back from
 a $2$-form on $T$
 and the
 countably many surfaces $\Sigma'_{b,n_i}\subset D$ contained in $D$
  are inverse images of curves
 $T_i\subset T$:
 $$\Sigma'_{b,n_i}=\pi^{-1}(T_i).$$
Note that the curves $T_i$ are then Zariski dense in $T$.
Our goal is to show that this last case cannot occur, if the point $b\in B$ satisfies the Terasoma-type condition from the previous section. This will be 
done in several steps.

 We claim  first that this  fibration
  $\pi:D\dashrightarrow T$   is preserved by $\phi^l$.  Indeed, the fibration
 is determined by the
  data of the  $2$-form
 $\omega_D$ up to a scalar :  the fibers of $\pi$ are the integral leaves of the
 vector field defined generically on $D$ as the kernel of $\omega_D$. As we have
 $(\phi^l)^*\omega_D=(-2)^l\omega_D$, the claim follows.

   As $D$ is defined only up to bimeromorphic
 transformations, we may assume $\pi$ is actually a morphism. We will denote by
 $\psi:T\dashrightarrow T$ the rational map induced by $\phi^l$ on the basis of the fibration.


 \begin{lemm}\label{utile1} The generic fiber of $\pi$ is a curve of general type.
 \end{lemm}
{\bf Proof.} As the $T_i$ are Zariski dense in $T$, it suffices to prove the same result for the fibration
$\pi:\Sigma'_{b,n_i}\rightarrow T_i$.

Recall now that the surfaces $\Sigma'_{b,j}$ are birationally equivalent to an abelian surface
$A_b$,
and that we proved in section \ref{preli} that we could assume $A_b$ to be
geometrically simple.
We then simply use the fact that curves in  a simple abelian surface are of general type.
\cqfd
Let $g\geq 2$ be the genus of the
 generic fiber of $\pi$. Let $m:T\dashrightarrow \mathcal{M}_g$ the rational map  associated to the fibration
 $\pi$ and let $Z$ be its image. By definition, the fibration becomes
  isotrivial when restricted to the fibers of $m$.

 As $\phi^l$ acts on $D$ preserving the fibration $\pi$, and the fibers of $\pi$ are of general type,
 $\phi^l$ induces for generic $t\in T$ an isomorphism
 $$C_t\cong C_{\psi(t)},\,t\in T,$$
 where $C_t:=\pi^{-1}(t)$. This isomorphism and the definition of $m$ shows that
 $m\circ \psi=m$.
\begin{lemm}\label{utile2} $Z$ has dimension $1$, and the restriction of $m$ to
$T_i\subset T$ maps each $T_i$  onto $Z$.
 \end{lemm}
 {\bf Proof.}
As the rational self-map $\psi$ is not of finite order on $T$ (because the $T_i$ are of the form
$\psi^k(T_1)$ and are Zariski dense in $T$), the equality
$m\circ \psi=m$ implies that $m$ is not generically finite-to-one.

Thus either $m$ is constant or its image $Z$ has dimension $1$.
We will show
that the restriction of $m$ to $T_i$
cannot be constant, which implies the full statement
of lemma \ref{utile2}.
Recall that we have $ \pi^{-1}(T_i)=\Sigma'_{b,n_i}$, which is birationally equivalent to an abelian surface
$A_b$.
If $m$ were constant on $T_i$, then the rational fibration
$A_b\dashrightarrow T_i$ would be isotrivial with fiber $C$ of general type. This does not exist
on any abelian surface $A$, because otherwise, $A$ would be dominated by a product
$C\times C'$, which would send the curves $C\times c'$
to the fibers of the fibration. But a rational map
$$\alpha:C\times C'\dashrightarrow A$$
is necessarily a morphism which is a ``sum morphism'', that is of the form
$$\alpha(c,c')=\alpha_1(c)+\alpha_2(c').$$
Thus the fibers of this isotrivial fibration would be translates
in $A$ (in the directions given by
$\alpha_2(C')$)  of a given
curve of general type. But two fibers must intersect because
they are of general type, and they can intersect only along the $0$-dimensional
  indeterminacy locus of the considered fibration on $A$. Their intersection
  is thus stable under translation by $\alpha_2(C')$, which is a contradiction.



 \cqfd
 \begin{coro} \label{corocorr}The fibers $T_z:=m^{-1}(z),\,z\in Z$ intersect the curves $T_i$.
 \end{coro}
 Indeed, this is equivalent to the surjectivity of the restricted map
 $m:T_i\rightarrow Z$.
 \cqfd

 Let now $z\in Z$ be a point defined over a number field and consider the curve $T_z=m^{-1}(z)\subset T$.
   The surface $\pi^{-1}(T_z)$ admits an isotrivial fibration
 with fiber $C_z$ defined over a number field $k$, and thus is dominated by
 a product $C_z\times_k T_z'$, where $T_z'$ is a finite cover of $T_z$.

For each point $y\in T'_z$ defined over a number field $k(y)$, the curve
$D_y:=\pi^{-1}(y)$ is isomorphic to
$C_z$ and the restricted cycle
$$P'_{y}:=P'_{\mid D_y\times X}\subset D_{y}\times X_{k(y)}\cong C_{z,k(y)}\times X_{k(y)}$$ is
 homogically trivial in the sense that its {\'e}tale cohomology class
 in $H^6_{et}(D_{y,\overline{k}}\times X_{\overline{k}},\mathbb{Q}_l(3))$ is zero.
 Thus it admits an Abel-Jacobi $l$-adic invariant
 $$\gamma_y\in
 H^1(k(y),H^5(D_{y,\overline{k(y)}}\times X_{\overline{k(y)}},\mathbb{Q}_l(3)))$$
$$ =H^1(k(y),H^5(C_{z,\overline{k(y)}}\times X_{\overline{k(y)}},\mathbb{Q}_l(3)))$$
 which is obtained as in the previous section by considering the continuous {\'e}tale cycle class of
 $P'_y$ in $H^6(D_y\times X_{k(y)},\mathbb{Q}_l(3))$
 and applying the Hochschild-Serre spectral sequence to it.

We choose a point $z\in Z$ defined over a number field
and which is sufficiently general (so as to avoid singular fibers).
 We will prove the following two propositions:
 \begin{prop} \label{prop1fin} The class $\gamma_y$ does not depend on $y\in T'_z(\overline{\mathbb{Q}})$.
 \end{prop}
This makes sense, as for $y,\,y'\in T'_z(\overline{\mathbb{Q}})$, one may choose a common definition field
$k$ for $y$ and $y'$, on which the isomorphism
$D_y\cong D_{y'}\cong C_z$ is also defined. Then
we compare both classes in
$H^1(k,H^5(C_{z,\overline{k}}\times X_{\overline{k}},\mathbb{Q}_l)(3))$.

\vspace{0,5cm}

{\bf Proof of Proposition \ref{prop1fin}.} Indeed, consider the surface
$\pi^{-1}(T_z)$ which admits as a rational finite cover $C_z\times_k T'_z$. Let us denote
by $r:C_z\times_k T'_z\dashrightarrow F$ the natural rational map.
We claim that $(r,Id)^*P'\in CH^3(C_z\times _kT'_z\times X)_\mathbb{Q}$ is a cycle homologous to $0$,
which means that its  {\'e}tale cohomology class on
$C_{z,\overline{k}}\times _{\overline{k}}T'_{z,\overline{k}}
\times _{\overline{k}}X_{\overline{k}}$ or equivalently its Betti cohomology class
on $C_{z,\mathbb{C}}\times T'_{z,\mathbb{C}}
\times X_{\mathbb{C}}$ vanishes.
Indeed, we know that the $2$-form $\omega=P^{'*}\alpha$ on $F$ has the property that
its restriction $\omega_D$ to $D$ is of the form $\pi^*\omega_T$ for some holomorphic
$2$-form on $T$. Of course $\omega_T$ vanishes on the curve $T_z$.
It follows that $\omega_D$ vanishes on the surface $\pi^{-1}(T_z)$.
Let us work in the complex setting: by lemma
\ref{hodge},  the Hodge structure on $H^4(X,\mathbb{Q})_{prim}$
is simple.
As the morphism
$$r^*\circ P^{'*}:H^4(X_\mathbb{C},\mathbb{Q})_{prim}\rightarrow H^2(C_{z,\mathbb{C}}\times
T'_{z,\mathbb{C}},\mathbb{Q})$$
is not injective, because its complexification vanishes on the $(3,1)$-part,
it must be $0$, which proves the claim.

Having this, we conclude that the cycle
$(r,Id)^*P'\in CH^3(C_z\times _kT'_z\times X)$ has an $l$-adic Abel-Jacobi invariant
$$\gamma\in
H^1(k,H^5(C_{z,\overline{k}}\times_{\overline{k}}
 T'_{z,\overline{k}}\times _{\overline{k}} X_{\overline{k}},\mathbb{Q}_l(3)))$$
 which in fact lies in
 $$H^1(k,H^1(C_{z,\overline{k}}\times_{\overline{k}}
 T'_{z,\overline{k}},\mathbb{Q}_l)\otimes _{\mathbb{Q}_l}H^4( X_{\overline{k}},\mathbb{Q}_l)(3))$$
 because $P'$ is obtained from $P$  by applying a Chow-K\"unneth projection on $X$.

We have $$H^1_{et}(C_{z,\overline{k}}\times_{\overline{k}}
 T'_{z,\overline{k}},\mathbb{Q}_l)\cong H^1_{et}(C_{z,\overline{k}},\mathbb{Q}_l)\oplus H^1_{et}(
 T'_{z,\overline{k}},\mathbb{Q}_l),$$
 and thus the first projection of $\gamma$ gives us a class
 $$\gamma_z\in
 H^1(k,H^1(C_{z,\overline{k}},\mathbb{Q}_l)
 \otimes _{\mathbb{Q}_l}H^4( X_{\overline{k}},\mathbb{Q}_l)(3))$$
 which  by definition restricts to
 $\gamma_y$, for any $y\in T'_z$.
\cqfd

\begin{prop} \label{prop2fin}  The class
 $\gamma_y,\,y\in T'_z$ is non-zero, for many
closed points $b\in B$ defined over a number field.
 \end{prop}
{\bf Proof.} The fiber $T_z$ meets the countably many curves $T_i$, by corollary
\ref{corocorr}.
Let $y\in T_1\cap T_z$ be defined over a number field $k$. The class
$\gamma_y$ is equal to the class $\gamma_{z}$ by Proposition \ref{prop1fin}.
Now, note that, as $y\in T_1$, the fiber $D_{y}$ is a curve in the surface
$\Sigma'_{b,n_1}=\pi^{-1}(T_1)$. This surface is birationally equivalent
to a simple abelian surface, and thus we have an injective
morphism of {\'e}tale cohomology groups:
$$ H^1_{et}(\Sigma'_{b,n_1,\overline{k}},\mathbb{Q}_l)\rightarrow H^1_{et}(D_{y,\overline{k}},
\mathbb{Q}_l),$$
where $k$ is a common field of definition of $T_1$ and $y$, and $\overline{k}$ is
its Galois closure.
This restriction map is $Gal\,(\overline{k}/k)$-equivariant and it makes
the left hand side into a direct summand of right hand side, using the existence
of a non-degenerate $Gal\,(\overline{k}/k)$-invariant
intersection pairing on the right hand side.
It follows that there is an induced injection
$$H^1(k, H^1(\Sigma'_{b,n_1,\overline{k}},\mathbb{Q}_l)
\otimes_{\mathbb{Q}_l}H^4(X_{\overline{k}},\mathbb{Q}_l)(3))$$
$$
\hookrightarrow
H^1(k, H^1(D_{y,\overline{k}},\mathbb{Q}_l)
\otimes_{\mathbb{Q}_l}H^4(X_{\overline{k}},\mathbb{Q}_l)(3)),$$
which obviously sends the Abel-Jacobi $l$-adic invariant of the cycle
$P'_{\Sigma'_{b,n_1}}$ to the
$l$-adic invariant of the cycle $P'_{y}$, that is $\gamma_{y}$.
We already proved in proposition \ref{plusfort} that the first one is non-zero,
because it
is equal to $(-2)^{n_1}\gamma_{P'_b}$ by property (\ref{crucialpropertyphi}), hence
$\gamma_{y}\not=0$.
\cqfd

Let us now explain how to use these propositions
to get a contradiction, which will exclude the case considered in this section.


The fiber $T_z$ meets the countably many curves $T_i$, as we proved in the proof of Lemma
\ref{utile2}. This fiber is acted on by $\psi$, as proved in lemma \ref{utile2}.
Let $y\in T_1\cap T_z$ be defined over a number field $k$.


By proposition \ref{prop1fin}, its
$l$-adic Abel-Jacobi invariant
$$\gamma_{y}\in H^1(k,H^5(D_{y,\overline{k}}\times X_{\overline{k}},\mathbb{Q}_l(3)))$$
 must be equal
to the $l$-adic Abel-Jacobi invariant
$$\gamma_{y'}\in H^1(k,H^5(D_{y',\overline{k}}\times
_{\overline{k}} X_{\overline{k}},\mathbb{Q}_l(3)))$$
of the cycle
$P'_{y'}\subset D_{y'}\times X$, where $y'=\psi(y)\in \psi(T_1)\cap T_z$,
and $D_{y}$ and $D_{y'}$ are identified via the trivialisation
of the fibration of $\pi$ over $T_z$. Up to replacing the power $l$ by an higher power,
we may assume that this identification is given by
$\phi^l:D_{y}\cong C_{y'}$ because the automorphisms group of these curves is finite.
Thus we get
\begin{eqnarray}\label{autre1}{\phi^l}^*\gamma_{y'}=\gamma_{y},
\end{eqnarray}
where we see $\phi^l$ as a map from $D_{y}$ to $D_{y'}$.

On the other hand, we
 get  from
(\ref{crucialpropertyphi}) and  Bloch-Srinivas decomposition theorem
 (cf \cite{blS} and
\cite{voisinbook}, Corollary 10.20)  the equality
$${\phi^l}^*(P'_{y'})=(-2)^lP'_y+\Gamma\,\,{\rm in}\,\, CH^3(D_{y}\times X)_\mathbb{Q},$$
 where $\Gamma\subset D_{y}\times X$ is a ``vertical'' cycle, supported over
a divisor of $D_{y}$.
As vertical cycles have trivial $l$-adic Abel-Jacobi invariant in the
considered group, this  implies  that
\begin{eqnarray}\label{autre2}{\phi^l}^*\gamma_{y'}=(-2)^l\gamma_{y}.
\end{eqnarray}
As
$$0\not=\gamma_{y}\,\,{\rm in}\,\,
 H^1(k,H^5(D_{y,\overline{k}}\times_{\overline{k}}
 X_{\overline{k}},\mathbb{Q}_l(3)))$$
by proposition \ref{prop2fin},
(\ref{autre1}) and (\ref{autre2}) provide a contradiction.
\cqfd




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