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\begin{document}
 


\begin{center}
{\Large
\bf
Uniruled varieties with splitting tangent bundle
}
\vspace{0.7ex}

Andreas H\"oring
\end{center}

\section{Introduction}

The goal of this paper is to give a partial answer to the following conjecture, initially asked by A. Beauville:

\begin{conjecture}
\label{beauvilleconj}
Let $X$ be a compact K\"ahler manifold such that $T_X = V_1 \oplus V_2$ where $V_1$ and $V_2$ are vector bundles.
Let \holom{\mu}{\tilde{X}}{X} be the universal covering of $X$. Then $\tilde{X} \simeq Y_1 \times Y_2$, where
$\dim Y_j = \rk V_j$. If moreover $V_j$ is integrable, then $\mu^* V_j = p_{Y_j}^* T_{Y_j}$ (up to an appropriate
isomorphism of $\tilde{X}$).
\end{conjecture}

The integrability of the vector bundles $V_j$ is not true in general (see example \ref{beauvilleexample}),
but the decomposition of the universal covering $\tilde{X}$ still holds in these examples. The conjecture
has been studied before by Beauville \cite{Bea00}, Druel \cite{Dru00}, Campana-Peternell \cite{CP02}
and very recently by Brunella-Pereira-Touzet. Their paper contains most of the preceeding
results, its main result is the

\begin{theo} \cite{BPT04}
\label{brunella}
Let $M$ be a compact connected K\"ahler manifold. Suppose that its tangent bundle $T_M$ splits as $D \oplus L$,
where $D \subset T_M$ is a subbundle of rank $\dim M - 1$ and $L \subset T_M$ is a subbundle of rank 1. Then:
\begin{enumerate}
 \item If $D$ is not integrable, then $L$ is tangent to the fibres of a $\PP^1$-bundle.
 \item If $D$ is integrable then \holom{\mu}{\tilde{M}}{M}, the universal covering of $M$, splits as $\tilde{N} \times E$,
 where $E$ is  a connected curve. Moreover $\mu^* D = p_N^* T_N$ and $\mu^* L = p_E^* T_E$. 
 \end{enumerate}
\end{theo}

The theorem establishes a surprising link between the existence of rational curves along the foliation $L$ and
the integrability of the complement $D$. We obtain a similar statement in the projective case:

\begin{theo}
\label{nonuniruledintegrable}
Let $X$ be a projective manifold with splitting tangent bundle $T_X=V_1 \oplus V_2$.
If $X$ is not uniruled, then $V_j$ is integrable for $j=1,2$.
\end{theo} 

In this paper we will then concentrate on projective 
uniruled varieties. By the theorem above this is
the class of varieties where the integrability of the direct factors $V_j$ fails in general, but we will obtain
several integrability results in special cases.

The global strategy is to construct a fibre space structure on $X$ which is
related to the decomposition $T_X = V_1 \oplus V_2$ and deduce properties of the universal covering of $X$ from this 
fibre space structure. The main reason why this works (at least in some cases) is the classical Ehresmann theorem.

\begin{theo} \cite{Ehr50}
\label{ehresmann}
Let \holom{\phi}{X}{Y} be a submersion of complex manifolds
such that
\[
\xymatrix{
0 \ar[r] & \phi^{*} \Omega_Y  \ar[r]  & \Omega_X \ar[r]  & \Omega_{X/Y} \ar[r] & 0
}
\]
splits. This exhibits a 
connection $E$ on the fibration, i.e. a subbundle $E \subset T_X$ such that $T_X = E \oplus T_{X/Y}$. 
If the connection is integrable, then \holom{\phi}{X}{Y} is an analytic fibre bundle
with typical fibre $F$ .
In that case there is a representation
\holom{\rho}{\pi_1(Y)}{Aut(F)} so that $\pi_1(Y)$ acts on $F$ via $\rho$.

Moreover $X$ is the fibre bundle over $Y$ with fibre $F$ given by $(\tilde{Y} \times F)/\pi_1(Y)$. Finally the splitting 
$\tilde{X} = \tilde{Y} \times \tilde{F}$ satisfies $\mu^* E = p_{\tilde{Y}}^* T_{\tilde{Y}}$ and 
$\mu^* T_{X/Y} = p_{\tilde{F}}^* T_{\tilde{F}}$, where \holom{\mu}{\tilde{X}}{X} is the universal covering.
\end{theo}

The general structure of the paper is as follows: in the next section we recall some well-known results about
foliations and rational curves. The third section gives details about the link between uniruledness and
the integrability of the direct factors, in particular we give the short proof of theorem \ref{nonuniruledintegrable}.
The fourth section treats rationally connected manifolds, which are known to
be simply connected, where we obtain a rather
satisfactory answer to the conjecture if we assume integrability of one direct factor (theorem \ref{bogomolovbetter}). 
In particular we obtain the 

\begin{theo}
\label{rationalsurfaces}
Let $X$ be a rationally connected projective manifold
such that $T_X = \oplus_{j=1}^n V_j$ where $\rk V_j=2$ for all $j$.
Then $X$ is isomorphic to a product of rationally connected surfaces $S_j$ such that $V_j = p_j^* T_{S_j}$
where \holom{p_j}{X \simeq \Pi_i S_i}{S_j}.
\end{theo}

Using appropriate families of rational curves on $X$,
we will obtain in section 5 various results on the structure of elementary Mori contractions of fibre type.
Section 6 provides an analogue of these considerations for birational contractions in dimension 4, which for obvious reasons do not generalise to higher dimension.
Combining the results from section 5 and 6, we obtain

\begin{theo}
\label{dim4uniruled}
Let $X$ be a projective uniruled fourfold such that $T_X = V_1 \oplus V_2$ where $\rk V_j=2$ for $j=1,2$.
If $V_1$ and $V_2$ are integrable, conjecture \ref{beauvilleconj} holds.
\end{theo}

{\bf Acknowledgements:}

This paper is a first result of the author's binational Ph.D. thesis supervised by L. Bonavero (Institut Fourier Grenoble) and
T. Peternell (Universit\"at Bayreuth) which is partially supported by the DFG-Schwerpunkt \lq Globale Methoden in der Komplexen
Geometrie\rq. I want to thank St\'ephane Druel for his careful proofreading of a rather awkward first version of this
paper.

\section{Notations and basic material}
\label{basics}

We work over the complex field $\C$. For standard definitions in complex algebraic geometry we refer to  \cite{Ha77} or \cite{Kau83}, 
for positivity notions of vector bundles we follow the definitions from \cite{Laz04II}. For $X$ a compact K\"ahler manifold, we denote $\chow{X}$ the cycle space as defined in \cite{EMS7}[Ch.VIII].

A fibration is a surjective holomorphic map \holom{\phi}{X}{Y} from a complex manifold to a normal variety $Y$ 
such that the general fibre is connected and $\dim X > \dim Y$. 
We say that a fibration is almost smooth if the reduction of every fibre is a
smooth variety.

Let \holom{\phi}{X}{Y} be a morphism between complex manifolds. The tangent map is a generically surjective
sheaf homomorphism \holom{d\phi}{T_X}{\phi^* T_Y}. For a subsheaf $\Ssheaf \subset T_X$, we will 
use the slightly abusive notation 
\[
d \phi (\Ssheaf) := \phi_* (d\phi(\Ssheaf)) \subset T_Y.
\]




\subsection{Rational curves and splitting tangent bundle}
\label{rationalcurves}

\begin{defn}
\label{normedrational}
Let $X$ be a complex manifold and $C \subset X$ a rational curve with normalization \holom{f}{\PP^1}{C}. 
Then $C$ is pseudo-standard if
\[
f^* T_X = \Osheaf (2) \oplus \bigoplus_{i=2}^{\dim X} \Osheaf (a_i) \qquad a_i \leq 1 \quad \forall i>1
\]
\end{defn}

\begin{defn}
Let $X$ be a projective manifold. $X$ is uniruled if through every point $x \in X$ there exists a rational 
curve. $X$ is rationally connected if for every pair $(x_1,x_2) \in X \times X$ there exists a rational
curve that contains $x_1$ and $x_2$.
\end{defn}

\begin{theo}
\cite{Ko96}[IV.2.10]
\label{uniruledcovering}
Let $X$ be a smooth uniruled variety. Then there exists a pseudo-standard rational curve $C \subset X$ such that 
\[
f^* T_X = \Osheaf (2) \oplus \Osheaf (1)^{\oplus a}  \oplus \Osheaf ^{\oplus b}.
\] 
\nopagebreak
Such a $C$ is called a minimal free curve.  
\end{theo}


\begin{theo}
\cite{Ko96}
\label{connectedcovering}
Let $X$ be a smooth rationally connected variety. Then there exists a pseudo-standard rational curve $C \subset X$ such that 
\[
f^* T_X = \bigoplus \Osheaf (a_i) \qquad a_i > 0 \quad \forall i.
\] 
\nopagebreak
Such a $C$ is called a very free rational curve.  
\end{theo}

The following lemma is a cornerstone of our approach. The main ingredient of its proof is a vanishing result
by Baum-Bott \cite{BB70}, which implies that if $T_X=V_1 \oplus V_2$, then 
\[
c_k(V_j) \in H^k(X, \wedge^k V_j^*) \subset H^k(X, \wedge^k \Omega_X) 
\] 
for $j=1,2$. 

\begin{lemma}
\cite{CP02}[Lemma 1.3.]
\label{CP02Lemma13}
Suppose $T_X = \oplus_j V_j$. Let $C \subset X$ be a pseudo-standard rational curve \holom{f}{\PP^1}{C}. Note
\[
f^* V_j = \oplus_{i=1}^{\rk V_j} \Osheaf (a_i^j) \qquad a_i^j \in \Z \quad \forall i=1, \ldots, \rk V_j.
\] 
After possible renumeration 
we may assume, that the factor $\Osheaf(2)$ appears in the decomposition of $f^* V_1$, i.e. 
\[
\forall j>1 \qquad f^* V_j = \oplus \Osheaf (a_i^j) \qquad a_i^j \leq 1 \qquad \forall\ i=1, \ldots, \rk V_j.
\] 
Then $det V_j \cdot C = \sum_{i=1}^{\rk V_j} a_i^j = 0$.
If $f^* T_X$ is nef, i.e. all $a_i^j \geq 0$, then 
\[
f^* V_j = \Osheaf^{\oplus \rk V_j} \qquad \forall\ j>1.
\]
\end{lemma}



We will use the standard terminology from Mori theory. A Mori contraction of a projective manifold $X$ is
a morphism with connected fibres \holom{\phi}{X}{Y} on a normal variety $Y$ such that the anticanonical bundle $-K_X$ is $\phi$-ample.
We say that the contraction is elementary if there exists a rational curve $C \subset X$ such that a curve
$C' \subset X$ is contracted by $\phi$ if and only if we have an equality in $N_1(X)$
\[
[C'] = \lambda \ [C] \qquad \lambda \in \Q^+
\]
By $N_1(X)$ we denote the $\Q$-vector space of 1-cycles on $X$ modulo numerical equivalence (cf. \cite{Deb01}).
The contraction is said to be of fibre type if $\dim Y < \dim X$, otherwise it is birational. If
the contraction is birational and the exceptional locus $\phi$-locus $E$ is irreducible, 
let $k:=\dim E$ and $l$ be the dimension
of $\phi(E)$. The birational contraction is then said to be of type $(k,l)$. 

\begin{prop}
\cite{CP02}[Prop.1.8.]
\label{CP02Prop18}
Let $X$ be a projective manifold with $T_X = \oplus_j V_j$. Let \holom{\phi}{X}{Y} be an elementary 
extremal contraction.

1. If $L_j = det V_j$ is not $\phi$-trivial, then $\dim F \leq \rk V_j =: r_j $ for every $\phi$-fibre $F$.

2. $\dim F \leq  max_j (r_j)$

3. Suppose some fibre of $\phi$ contains a pseudo-standard rational curve (e.g. $\dim Y < \dim X$ or $\phi$ has a scheme-theoretic
smooth fibre). 
Then one of the $L_j$ is $\phi$-ample and the remaining ones are $\phi$-trivial. 

\end{prop}

{\bf Easy, but important remark:}

In the situation of the proposition, let $\holom{\psi}{\PP^1}{C}$ be a pseudo-standard rational curve and suppose that $L_1$
is $\phi$-ample. From the proof of \ref{CP02Lemma13} we see that the decomposition of $\psi^* V_1$ contains the 
unique factor $\Osheaf(2)$. We will show the slighly stronger statement $T_{\PP^1} \subset \psi^* V_1$. 

Clearly $\psi^* V_j = \oplus_{i=1}^{\rk V_j} \Osheaf (a_i^j)$ for
$j>1$ does not
contain a direct factor $\Osheaf(2)$.
But this implies that the projection of $T_{\PP^1}$ along $\psi^* V_1$ on $(\oplus_{j>1} \psi^* V_j)$ is zero, 
since
\[
H^0(\PP^1, \Osheaf(-2) \otimes (\oplus_{j>1} \psi^* V_j))=0.
\]  
This shows the inclusion. 

We conclude this section with a standard argument that we will use frequently
in the following. Let \holom{\phi}{X}{Y} be a morphism from a projective manifold $X$
to a normal variety $Y$ and $Z$ be fibre such that its scheme-theoretic structure is reduced. 
Let $F$ be an irreducible component of $Z$ that is smooth. Then the canonical morphism
\[
\Isheaf_Z/\Isheaf_Z^2 \otimes \Osheaf_F \rightarrow \Isheaf_F/\Isheaf_F^2
\]
is generically surjective. Since $\Isheaf_Z/\Isheaf_Z^2$ is globally generated, 
$\Isheaf_F/\Isheaf_F^2=N_{F/X}^*$ is generically generated.
Let \holom{f}{\PP^1}{F} be a rational curve
such that $f^* T_F$ is nef. Since the deformation of the rational curve cover $F$, 
the restriction of the bundle $N_{F/X}^*$ to a general deformation of $f$ is generically generated. 
So up to replacing \holom{f}{\PP^1}{F} by such a general deformation, we may suppose that
$f^* N_{F/X}^*$ is generically generated, hence nef.
So the exact sequence
\[
0 \rightarrow f^* T_F  \rightarrow f^* T_X \rightarrow f^* N_{F/X} \rightarrow  0
\]
splits by the $h^1$-criterion ($f^* (N_{F/X}^* \otimes T_F$) is nef). In particular 
\begin{equation}
\label{fibresplitting}
f^* T_X = f^* T_F \oplus f^* N_{F/X}
\end{equation}

\subsection{Foliations}
\label{foliations}

In this section we introduce the terminology that we will need in the following and state some well-known results
about holomorphic foliations. 

Let $X$ be a complex compact manifold. By definition a subbundle $V \subset T_X$ is integrable if it is closed under Lie brackets.
Equivalently if $E:= (T_X/V)^*$, then 
\[
dE \subset E \wedge \Omega_X^1.
\]
We recall that the Lie brackets
\[
[.,.] \ : \ V  \times V \ \rightarrow \ T_X
\]
is a bilinear antisymmetric mapping that is not $\Osheaf_X$-linear but 
induces an $\Osheaf_X$-linear map $\wedge^2 V \rightarrow (T_X / V)$, that is zero if and only if $V$ is integrable.
In particular 
\[
H^0(X, \Homsheaf( \wedge^2 V, (T_X / V)) = 0
\] 
implies that $V$ is integrable. In general we will show this vanishing property using a covering family $S$ (cf. \cite{Ca04} for the terminology of covering families) of
subvarieties $(Z_s)_{s \in S}$ of $X$ such that a general member of the family satisfies
\[
H^0(Z_s, \Homsheaf( \wedge^2 V, (T_X / V)|_{Z_s}) = 0.
\] 
Since the general members of the family cover a nonempty Zariski open subset of the variety $X$ and $\Homsheaf(\wedge^2 V, (T_X/V)))$ is a vector
bundle, we obtain the vanishing on $X$.

By the Frobenius theorem an integrable subbundle $V$ of $T_X$ induces a foliation on $X$, i.e. for every $x \in X$ there exists
an analytic neighborhood $U$ and a submersion \holom{q}{U}{W} such that $T_{U/W}=V|_U$. This shows that $V$ can be realised
as the tangent bundle of locally closed subsets of $X$. The foliation induces an equivalence relation
on $X$, two points being equivalent if and only if they can be connected by chains of smooth curves $C_i$
such that $T_{C_i} \subset V|_{C_i}$. An equivalence class is then called a leaf of the foliation. The equivalence relation induces a quotient map \holom{\phi}{X}{X/V} on
the so-called leaf space $X/V$ such that the fibres are the leaves of the foliation. This map can
always be defined set-theoretically, but in general $X/V$ is not Hausdorff.

We will say that the general leaf of a foliation is compact if there exists a non-empty saturated Zariski open subset
$X^* \subset X$ such that every leaf contained in $X^*$ is compact.


\begin{prop}
\label{generalleafcompact}
Let $X$ be a compact K\"ahler manifold, and let $V \subset T_X$ be an integrable subbundle. Assume that the general $V$-leaf
is compact, then every $V$-leaf of the foliation is compact and the projection on the leaf space \holom{\phi}{X}{Y:=X/V}
is an almost smooth holomorphic map. 
\end{prop}

{\bf Proof:}

The compactness of the leaves follows from the global stability theorem on K\"ahler manifolds (cf. \cite{Pe01}
for a short proof), in this case Holmann \cite{Ho80} has shown in a more general setting that the leaf space admits the structure of an analytic space such that the projection is almost smooth. In our case
we can see this in a more elementary way: since the leaves are compact subvarieties of $X$ varying in a 
holomorphic way, there exists a proper variety $Y \subset \chow{X}$ such that a general cycle parametrized by $Y$
is a $V$-leaf. If we note $\Gamma \subset Y \times X$ the graph of this family of cycles, 
it is easy to see that the
projection \holom{p_X}{\Gamma}{X} is an isomorphism and all cycles parametrized by $Y$ correspond to $V$-leaves. 
The projection \holom{p_Y}{\Gamma \simeq X}{Y} then provides the quotient map.
$\blacksquare$

\begin{cor}
\label{quotientsubmersion}
Let $X$ be a compact K\"ahler manifold, and let $V \subset T_X$ be an integrable subbundle such that one leaf
is compact and rationally connected. Then there exists a submersion \holom{g}{X}{Y} on the smooth variety $Y$ 
such that $T_{X/Y}=V$.
\end{cor}

{\bf Remark:}

Since \holom{g}{X}{Y} is a submersion, $Y$ is a compact 
K\"ahler manifold. If furthermore $X$ is projective, then $Y$
is projective.

{\bf Proof:}
\nopagebreak

A rationally connected leaf is simply connected, so by Reeb's
local stability theorem (cf. \cite{Cam85}[IV, �3, Thm.3]), the general $V$-leaf
is compact. The proposition \ref{generalleafcompact} yields an almost
smooth map \holom{g}{X}{Y} on the leaf space $Y:=X/V$. We will
first establish that the reduction of every fibre is rationally
connected.

Let $y \in Y$ be an arbitrary point, then by \cite{Mol88}[Prop.3.7.]
there exists a neighborhood $U \simeq \D^{k}/G$
with $G$ a finite group and $k:=\dim Y$. More precisely $G$ is the image of a representation
of the fundamental group $\pi_1(\fibre{g}{y}_{red})$.  Denote now
$\holom{q}{\D^k}{U}$ the quotient map and $X_U$ the normalisation of $\fibre{g}{U} \times_U \D^k$. 
Let \holom{g'}{X_U}{\D^k} be the induced map and \holom{q'}{X_U}{\fibre{g}{U}} the induced \'etale
covering. Then $g'$ is a smooth map and the $g'$-fibres are the
$q'^* V$-leaves.

A $V$-leaf is rationally connected if and only if the
corresponding $q'^* V$-leaf is rationally connected. Furthermore
one $g'$-fibre is rationally connected if and only if all
$g'$-fibres are rationally connected (deformation invariance of
rational connectedness, cf. \cite{Ko96}). Now by hypothesis there
exists one rationally connected $V$-leaf, so by connectedness of
$Y$ all $V$-leaves are rationally connected. Since rationally
connected varieties are simply connected, it now follows that $G$ is trivial, so $Y$ is smooth.

Let us now show that $g$ is a smooth map. 
For $y \in Y$ and $x \in \fibre{g}{y}$ arbitrary, there exists an analytic neighborhood $U \subset X$ such that 
$U$ admits
a product structure relative to the foliation, i.e. a submersion \holom{q}{U}{W} such that $T_{U/W}=V|_U$. 
Since the fibres of $q$ and $g|_U$ coincide topologically,  
we have a factorisation \holom{h}{W}{g(U)} that is a ramified covering with ramification locus exactly the $y' \in g(U)$ such
that the scheme-theoretic fibre \fibre{g}{y'} is multiple. By the purity of branch theorem applied to $h$, we see that $g$ is either without 
multiple fibres or has multiple fibres in codimension 1. 

Suppose that there are multiple fibres and note $H \subset Y$ the branch locus. Let $y \in H$ be a point such
that $H$ is smooth at $y$. Then for a general smooth curve $C$ intersecting $H$ transversally in $Y$,
the preimage  $Z:=\fibre{g}{C}$ is smooth.
So we obtain an almost smooth, but not smooth map \holom{g'}{Z}{C}
on a smooth curve $C$. If $F$ is a multiple fibre, then $\Osheaf_{F_{red}}(F_{red})$ restricted to $F_{red}$ is a nontrival torsion line
bundle (cf. \cite{BPV84} or \cite{Bea92}). This contradicts the fact that $F_{red}$ is simply connected.
$\blacksquare$



\begin{cor}
\label{quotientcodim0}
Let $X$ be a compact K\"ahler manifold and $V \subset T_X$ a subbundle. Suppose that
there exists a fibration \holom{\phi}{X}{Y} such that $T_{X/Y}=V$ holds generically. Then $V$ is integrable
with compact leaves. The almost smooth map \holom{\phi'}{X}{Y'} from the proposition provides a factorisation
of $\phi$, i.e. there exists a birational morphism \holom{g}{Y'}{Y} such that $g \circ \phi' = \phi$. 
In particular if $\phi$ is equidimensional, it is almost smooth. $\blacksquare$
\end{cor}



\section{An integrability result}
\label{integrability}

The integrability of the direct factors $V_j$ will play an important role in the following sections. 
In this paragraph we show that if we suppose $X$ to be projective and non-uniruled, the integrability is immediate. In particular the theory of holomorphic foliations can be used to study the conjecture \ref{beauvilleconj}.

\begin{theo} \cite{CP04}[Thm.1.5]
Let $X$ be a projective manifold with splitting tangent bundle $T_X=V_1 \oplus V_2$.
Note $L_j:= \det V_j^*$ for $j=1,2$. If $X$ is not uniruled, then $L_j$ is pseudoeffective.
\end{theo}

\begin{theo} \cite{De02}
Let $X$ be a compact K\"ahler manifold. Assume that there exists a pseudo-effective line bundle $L$ on $X$ and a nonzero
holomorphic section $\theta \in H^0(X, \Omega_X^p \otimes L^{-1})$, where $0 \leq p \leq n = \dim X$. Let
$S_\theta$ be the coherent subsheaf of germs of vector fields $\zeta$ in the tangent sheaf $T_X$, such that
the contraction $i_\zeta  \theta$ vanishes. Then $S_\theta$ is integrable, namely $[S_\theta,S_\theta] \subset S_\theta$,
and $L$ has flat curvature along the leaves of the (possibly singular) foliation defined by $S_\theta$. 
\end{theo}

{\bf Proof of theorem \ref{nonuniruledintegrable}:}

By the theorem of Campana-Peternell, $L:=\det V_1^*$ is pseudoeffective.
Trivially, we have
\[
\wedge^{\rk V_1} \Omega_X = 
\oplus_{i=0}^{\rk V_1} \wedge^{\rk V_1 - i} V_1^* \otimes \wedge^i V_2^*
\]
Hence
\[
\det V_1 \otimes \wedge^{\rk V_1} \Omega_X = 
\Osheaf_X \oplus \oplus_{i=1}^{\rk V_1} \wedge^{\rk V_1 - i} V_1^* \otimes \wedge^i V_2^*
\]
Let $\theta \in H^0(X, L^{-1} \otimes \wedge^{\rk V_1} \Omega_X)$ be a section corresponding to 
the direct factor $\Osheaf_X$, then Demailly's theorem shows that the
coherent subsheaf $\Ssheaf_\theta$ 
is integrable. Since we may interpret $\theta$ as a section of $H^0(X, \det V_1^* \otimes L^{-1})$, 
we will show by a local calculus that generically $\Ssheaf_\theta=V_2$. 
Since integrability of a vector bundle is a generic property this is sufficient.
The statement for $V_1$ follows completely analogous.

Let $x \in X$ be a general point, i.e. $\theta_x \neq 0$. Then in an analytic neighborhood of $x$ we 
choose local generators
$v_1, \ldots, v_{\rk V_1}$ of $V_1$ and $v_{\rk V_1 + 1}, \ldots, v_{\dim X}$ of $V_2$. A germ of a vector field
$\zeta$ through $x$ can then be written as
\[
\zeta = \sum_{j=1}^{\dim X} f_j v_j
\]
where $f_j$ are holomorphic functions. Furthermore locally
\[
\theta = g v_1^* \wedge \ldots \wedge v_{\rk V_1}^* \otimes e
\]
where $g$ holomorphic and $g(x) \neq 0, v_j^*$ the dual basis of $v_j$ and $e$ a local generator 
of the line bundle $L^{-1}$.
Then
\begin{eqnarray*}
i_\zeta \theta 
&=& g v_1^* \wedge \ldots \wedge v_{\rk V_1}^* \otimes e (\sum_{j=1}^{\dim X} f_j v_j)
\\
&=&
\sum_{j=1}^{\rk V_1} (g f_j)  v_1^* \wedge \ldots v_{j-1}^* \wedge v_{j+1}^* \wedge \ldots v_{\rk V_1}^* \otimes e
\end{eqnarray*}
This equals zero iff $f_j=0$ for $j=1, \ldots, \rk V_1$.
$\blacksquare$

{\bf Remark:}

The integrability lemma is optimal, in the sense that there is the following counterexample due to A. Beauville to the integrability in the uniruled case. 

\begin{exa} 
\label{beauvilleexample}
Let $A$ be an abelian surface and  $u_1, u_2$ be non-zero vector fields on $A$. Let $z_1, z_2$ 
be non-zero vector fields on $\PP^1$ such that $[z_1,z_2] \neq 0$.
Then $v_1:=p_A^*(u_1) + p_{P_1}^* (z_1)$ and $v_2:=p_A^*(u_2) + p_{P_1}^* (z_2)$ are everywhere nonzero vector fields
on $X:=A \times \PP^1$. The subbundle $V:= \Osheaf_X v_1 \oplus \Osheaf_X v_2 \subset T_X$
is not integrable and $T_X = V \oplus p_{\PP^1}^* T_{\PP^1}$. 
\end{exa}









\section{Rationally connected varieties}

The goal of this section is to show theorem \ref{rationalsurfaces}. The strategy is to show the integrability
of the direct factors of $T_X$ and to apply a difficult result by Bogomolov and McQuillan (theorem \ref{bogomolov}).

\begin{lemma}
\label{foliationrationally}
Let $X$ be a projective rationally connected manifold such that $T_X=V_1 \oplus V_2$ and $V_1 \subset T_X$ is
an integrable subbundle. 
Then there exists a submersion \holom{\phi}{X}{Y} such that $V_1=T_{X/Y}$.
\end{lemma}

{\bf Proof:}

Since $X$ is rationally connected, there exists a (very free) 
rational curve  \holom{f}{\PP^1}{C} on $X$ such that
\[
f^* V_1 \oplus f^* V_2 = f^*(T_X) = \oplus_i \Osheaf(a_i) 
\qquad a_i>0 \quad \forall \quad i=1, \ldots ,\dim X
\]

In particular $f^*(V_1)$ is an ample vector bundle. 
By the theorem of Bogomolov-McQuillan below this 
implies that the $V_1$-leaves passing through a point $x \in C$ are rationally
connected subvarieties of $X$. Now apply corollary \ref{quotientsubmersion}.
$\blacksquare$ 

\begin{theo} (\cite{BM01}[Thm. 0.1.a)], \cite{KST05}[Thm.1.1])
\label{bogomolov}
Let $X$ be a projective manifold, and let $V \subset T_X$ be an integrable subbundle. Let \holom{f}{C}{X} be
a curve on $X$ such that $f^* V$ is ample. Then all $V$-leaves passing through $f(C)$ are rationally 
connected submanifolds of $X$.
\end{theo}


\begin{lemma}
\label{connectionrationally}
Let $X$ be a compact complex manifold that admits a submersion \holom{\phi}{X}{Y} on a  variety $Y$.
Suppose furthermore that $\phi$ admits a connection, i.e. a vector bundle
$V \subset T_X$ such that $T_X =V \oplus T_{X/Y}$. If $Y$ is rationally connected of dimension at least 3 or a uniruled surface,
then  $V$ is integrable. If furthermore $Y$ is simply connected, then $X \simeq Y \times F$ where $F$ is a general fibre. 
\end{lemma}

{\bf Remark:}

If $Y$ is a smooth curve, then the connection is always integrable, so there is nothing to show. If $Y$ is
a non-uniruled surface, there are counterexamples to the integrability of the connection (cf. section \ref{integrability}).

{\bf Proof:}

We will construct a covering family of curves on $X$ such that the general member $C'$ is a smooth rational curve
\holom{f}{\PP^1}{C'} that satisfies
\begin{eqnarray*}
f^* T_{X/Y} &=& \Osheaf^{\oplus \dim X - \dim Y}
\\
f^* \wedge^2 V &=& \oplus_i \Osheaf (a_i) \qquad a_i > 0
\end{eqnarray*}
Then 
\[
f^* \Homsheaf(\wedge^2 V, (T_X/V)) \simeq (\oplus_{i} \Osheaf (-a_i))^{\oplus \dim X - \dim Y}
\]
So clearly
\[
H^0 (C', \Homsheaf(\wedge^2 V, (T_X/V)))= H^0 (\PP^	1, f^* \Homsheaf(\wedge^2 V, (T_X/V))) = 0
\]
This shows the integrability of $V$ (see subsection \ref{foliations}).
We continue with the construction of the family:

if $Y$ is rationally connected of dimension at least 3, there
exists a covering family of curves on $Y$ such that the general member is a very free
{\it smooth} rational curve $\holom{f}{\PP^1}{C}$ (cf. \cite{Deb01}[Ex.4.8.5]). By definition
\[
f^* T_Y = \oplus_j \Osheaf (b_j) \qquad b_j > 0.
\]
If $Y$ is 
a uniruled surface, then its relative minimal model is a ruled surface or $\PP^2$. In the first case
the lines of the ruling give a family such that the general member is a smooth rational curve
satisfying $f^* T_Y = \Osheaf (2) \oplus \Osheaf $. In the second case just take lines in $\PP^2$, so
the general member satisfies $f^* T_Y = \Osheaf (2) \oplus \Osheaf (1)$. In all cases, we obtain
$ f^* \wedge^2 T_Y = \oplus_i \Osheaf (a_i)$ with all $a_i$ strictly positive.

Fix now a general member $\holom{f}{\PP^1}{C}$ of the covering family. Since C is smooth and $\phi$ is a 
submersion, 
the pull-back $Z:= X \times_Y \PP^1$ is smooth and admits a submersion
\holom{\tilde{\phi}}{Z}{\PP^1}.
Denote \holom{\mu}{Z}{X} the natural projection, then $\mu^* T_{X/Y}= T_{Z/\PP^1}$
and $T_Z \subset \mu^* T_X$ is a subbundle.
We then consider the sequence of sheaf homomorphisms on $Z$
\[
T_{Z/\PP^1} \hookrightarrow T_Z \hookrightarrow \mu^* T_X \rightarrow \mu^* T_{X/Y}.
\]
The first two maps are the embeddings, while the last one is the projection along $\mu^* V$.
By what precedes it is clear that $T_{Z/\PP^1}$ maps isomorphically on $\mu^* T_{X/Y}$,
so the map $T_Z \rightarrow \mu^* T_{X/Y}$ has maximal rank in every point.
It follows that $L:=(\mu^* V \cap T_Z)= \ker(T_Z \rightarrow \mu^* T_{X/Y})$ is a rank 1 subbundle of $T_Z$ such that
\[
T_Z := L \oplus T_{Z/\PP^1}
\]
Since $L$ has rank 1 it is integrable. This shows that \holom{\tilde{\phi}}{Z}{\PP^1} admits an 
integrable connection, so by the Ehresmann theorem $Z \simeq \PP^1 \times F$ where $F$ is a fibre.
It follows that for $a \in F$, we obtain a rational curve \holom{\mu'}{\PP^1 \times \{a\}}{C'}  
on $X$ such that $\mu'^* T_{X/Y} = \Osheaf^{\oplus \dim X - \dim Y}$
and $\mu'^* \wedge^2 V = \oplus_i \Osheaf (a_i)$. Since the rational curves we used cover a dense open subset on $Y$, the constructed curves cover a dense open subset on $X$. 

Since $\phi$ admits an integrable connection, it follows from the Ehresmann theorem that 
the structure of an analytic bundle
on $X$ arises as a representation of the fundamental group of $Y$. 
If $Y$ is simply connected, this 
implies $X \simeq Y \times F$,
where $F$ is a $\phi$-fibre.
$\blacksquare$

From the two lemmas above, we obtain immediately

\begin{theo}
\label{bogomolovbetter}
Let $X$ be a rationally connected manifold such that $T_X = V_1 \oplus V_2$. 
If one of the direct factors is integrable, then 
$X$ is isomorphic to a product $Z_1 \times Z_2$ such that
$V_j = p_{Z_j}^* T_{Z_j}$ for $j=1,2$.
\end{theo}

The integrability condition is probably not necessary. The following lemma shows that it is indeed superflous for a splitting
in bundles of low rank.

\begin{lemma}
\label{uniruledintegrable}
Let $X$ be a uniruled projective manifold
such that $T_X = \oplus_{j=1}^n V_j$ where $\rk V_j=2$ for all $j$. Then one of the direct factors is integrable.
\end{lemma}

{\bf Proof:}

Since $X$ is uniruled, by theorem \ref{uniruledcovering} there
exists a covering family $S$ of rational curves such that a general member \holom{f}{\PP^1}{C} 
is a  minimal free rational curve
\[
f^* T_X = \oplus_j f^* V_j = \Osheaf (2) \oplus \Osheaf (1)^{\oplus a} \oplus \Osheaf^{\oplus b}.
\]

Suppose wlog that the factor $\Osheaf(2)$ is part of the decomposition of $f^* V_1$. 
This implies that $f^* V_1 = \Osheaf (2) \oplus \Osheaf (1)$ or
$f^* V_1 = \Osheaf (2) \oplus \Osheaf$. 
Clearly $f^* \wedge^2 V_1 = \Osheaf (b)$ with $b \geq 2$ and $f^* (T_X/V_1)= \Osheaf(1)^{\oplus a'} \oplus \Osheaf^{\oplus b'}$. So 
\[
H^0(\PP^1, \Homsheaf(f^* \wedge^2 V_1, f^* (T_X/V_1)))=0.
\] 
We have seen in subsection \ref{foliations} that this implies the integrability of $V_1$. 
$\blacksquare$

\vspace{1ex}

{\bf Proof of theorem \ref{rationalsurfaces}:} 

By the lemma above one of the direct factors $V_j$ is integrable, so by theorem \ref{bogomolovbetter} we have
\[
X \simeq S_j \times Y \qquad V_j = p_{S_j}^* T_{S_j} \qquad  \oplus_{i \neq j} V_i = p_{Y}^* T_{Y}.
\]
The theorem follows by induction on the dimension. $\blacksquare$



\section{Fibre space structures on uniruled varieties}

We use the decomposition of the 
tangent bundle to give a description of Mori contractions of fibre type. These structure results
provide the main ingredient to the proof of theorem \ref{dim4uniruled} in the next section.



\begin{lemma}
\label{ungenericpositionmorifibrespace}

Let $X$ be a projective manifold with $T_X = \oplus_j V_j$. Let \holom{\phi}{X}{Y} be an elementary
extremal contraction of fibre type.
Then after possible renumeration, we have $T_{X/Y} \subset V_1$ generically, that is there exists a 
non-empty Zariski
open subset $Y^*$ of $Y$ such that this holds on \fibre{\phi}{Y^*}. 
\end{lemma}

{\bf Proof:}

The general fibre $F$ is a Fano manifold,
so by theorem \ref{connectedcovering} there exists
a covering family such that the general member \holom{f}{\PP^1}{F} is a  very free rational curve, that is
\[
f^* T_F = \oplus_i \Osheaf (a_i) \qquad a_i > 0.
\]
We use equation \ref{fibresplitting} (cf. the remark after proposition \ref{CP02Prop18}) to obtain
\[
f^* T_X = f^* (T_F) \oplus \Osheaf^ {\oplus \dim Y} = \oplus_i \Osheaf (a_i) \oplus \Osheaf^ {\oplus \dim Y}.
\]
By proposition \ref{CP02Prop18} we may assume after possible renumeration that 
$\det V_1$ is $\phi$-ample and $\det V_j$ is $\phi$-trivial for $j \geq 2$.
This implies
$V_j |_C = \Osheaf_C^{\oplus \rk V_j}$ for $j \geq 2$. 
It is then clear that the canonically defined map \holom{\delta}{T_F}{\oplus_{j>1} V_j|_F} (we project along $V_1|_F$) is zero  
which is equivalent to $T_F \subset V_1|_F$. 
$\blacksquare$

\begin{prop}
\label{moriequality}
Let $X$ be a projective manifold with $T_X = V_1 \oplus V_2$, let \holom{\phi}{X}{Y} be an elementary contraction of fibre type on
a normal variety $Y$. 
Then by \ref{ungenericpositionmorifibrespace} up to renumeration
$T_{X/Y} \subset V_1$
holds generically. Assume $\dim X - \dim Y = \rk V_1$, 
then $\phi$ is smooth and $T_{X/Y}= V_1$. 
Moreover $\phi$ is an analytic bundle with general
fibre $F$ a Fano manifold with Picard number 1. 
\end{prop}

{\bf Proof:}

By proposition \ref{CP02Prop18} all fibres have dimension $\rk V_1$. Since the general fibre is a Fano manifold and
a $V_1$-leaf, corollary \ref{quotientsubmersion} and corollary \ref{quotientcodim0}  
show that $\phi$ is smooth. The vector bundle $V_2$ provides a connection, so
by lemma \ref{connectionbundle} we know that $\phi$ is a fibre bundle.
A well-known spectral sequence argument shows that $b_2(X)=b_2(F)+b_2(Y)$. 
Since $\phi$ is elementary this implies $b_2(F)=1$.
$\blacksquare$

\begin{lemma}
\label{connectionbundle}
Let $X$ be a complex manifold that admits a proper submersion \holom{\phi}{X}{Y} on a  variety $Y$.
Suppose furthermore that $\phi$ admits a connection, i.e. a vector bundle
$V \subset T_X$ such that $T_X =V \oplus T_{X/Y}$. Then $\phi$ is an analytic fibre bundle.
\end{lemma}

{\bf Proof:}

In general $V$ will not be integrable,
but if $C \subset Y$ is a smooth curve, then the restriction \holom{\phi|_{\fibre{\phi}{C}}}{\fibre{\phi}{C}}{C} is
a smooth map over a curve and $V \cap T_{\fibre{\phi}{C}}$ is a rank 1 bundle that provides a connection
(cf. the proof of lemma \ref{connectionrationally} for details). Since the connection has rank 1 it is integrable,
so $\phi|_{\fibre{\phi}{C}}$ is an analytic bundle. In particular its fibres are isomorphic. Since we can connect
any two points in $Y$ by a chain of smooth curves, this shows that all fibres are isomorphic complex manifolds. 
By the Grauert-Fischer theorem \cite{FG65}
this shows that $\phi$ is a fibre bundle. $\blacksquare$


\begin{cor}
\label{moricurve}
Let $X$ be a projective manifold with $T_X = V_1 \oplus V_2$. Let \holom{\phi}{X}{C} be an elementary contraction on
a smooth curve $C$. Then $\phi$ is smooth and after possible renumeration $T_{X/C}= V_1$. 
The morphism $\phi$ induces the structure of a fibre bundle on $X$ such that
$X = (\tilde{C} \times F)/ \pi_1(C)$. 
\end{cor}

{\bf Proof:}
The first statement is clear from \ref{ungenericpositionmorifibrespace} and \ref{moriequality}.
The bundle $V_2$ has rank 1, so provides
an integrable connection on $\phi$. We conclude with the Ehresmann theorem \ref{ehresmann}.
$\blacksquare$

{\bf Remark:}

The corollary is a generalization of \cite{CP02}[Thm.2.9.] which gives the same statement for an elementary contraction of a threefold on a curve. 
We show a technical lemma which seems to be standard in foliation theory, but for which we did not find a proof in the literature.


\begin{lemma}
\label{closedcondition}
Let $X$ be a compact K\"ahler manifold and $V \subset T_X$ an integrable subbundle.
Let $Y \subset \chow{X}$ be an irreducible subvariety such that the cycles parametrized by $Y$ are connected, and let 
$\Gamma \subset Y \times X$ be the graph over $Y$. Denote $X_y$ the fibre of the
projection \holom{p_Y}{\Gamma}{Y}. If a general fibre $X_y$ is contained 
in a $V$-leaf, this holds for all fibres.
\end{lemma}

{\bf Proof:}

Fix a cycle $X_y$ corresponding to a point $y \in Y$. It is sufficient to show that every irreducible component  of
the smooth locus $X_y'$ of $(X_y)_{red}$ is contained in a unique $V$-leaf. As soon as we know this, the closure
of every irreducible component of $X_y'$  is contained in a $V$-leaf. Since $X_y$ is connected and the $V$-leaves are disjoint, all irreducible components of $X_y'$ must lie in one and the same leaf.

Let $X_{y_n}$ be a sequence of general $p_{Y}$-fibres 
such that $y_n$ converges to $y$. 
Then by hypothesis every
$X_{y_n}$ is contained in a unique $V$-leaf and the 
closed subsets $X_{y_n} \subset X$ converge to $X_y$
for some Hausdorff topology induced by a metric on $X$ (cf. \cite{SGAN82}).
 
Fix a closed point $x \in X_y$, let $U$ be a product neighborhood for
the foliation $V$, i.e. let \holom{q}{U}{W} be a submersion such that $T_{U/W}=V|_U$ and $q(x)=0$.
As seen above, we may suppose $(X_y)_{red}$ to be smooth in $x$, so it is locally irreducible and
in particular we may suppose $(X_y)_{red} \cap U$ to be irreducible. 

The cycles $X_{y_n}$ are analytic and $U \cap X_{y_n}$ may have up to $\deg_x X_y$ irreducible components
(cf. \cite{Ca04}). Since $(X_y)_{red} \cap U$ is irreducible, we can choose irreducible components
$X_{y_n}' \subset U \cap X_{y_n}$ such that the limit of the sequence $X_{y_n}'$ for the Hausdorff topology is 
still $(X_y)_{red} \cap U$. 
 
Let $x_n \in X_{y_n}'$ be a sequence such that $x_n$ converges to $x$. Then by continuity $q(x_n)$ converges to
$q(x)=0$, hence $q(X_{y_n}') = q(x_n)$ converges to 0 (here we use that $X_{y_n}$ is contained in a $V$-leaf). 
Since every point of $X_y \cap U$ is in the limit of $X_{y_n}'$, this shows (again by continuity) that
$X_y \cap U$ is contained in the central fibre \fibre{q}{0}.
This shows the assertion locally and we conclude by connectedness.
$\blacksquare$



\begin{lemma}
\label{codim1flattening}
Let $X$ be a projective manifold with $T_X = V_1 \oplus V_2$, 
let \holom{\phi}{X}{Y} be an elementary contraction of fibre type on
a normal variety $Y$. 
Then by \ref{ungenericpositionmorifibrespace} up to renumeration
$T_{X/Y} \subset V_1$
holds generically. Assume $\dim X - \dim Y + 1 = \rk V_1$, 
then $\phi$ is equidimensional. 
\end{lemma}

{\bf Proof:}

Let us first note that $V_1$ is integrable: using the same family of rational curves \holom{f}{\PP^1}{F} 
as in the proof of lemma \ref{ungenericpositionmorifibrespace}, we have
\[
f^* \wedge^2 V_1 = \oplus_{i<j} \Osheaf(a_i + a_j) \oplus \oplus_i \Osheaf(a_i).
\]
Every direct summand is strictly positive, since $f^* V_2=f^* (T_X/V_1)$ is trivial, we obtain 
\[
H^0(\PP^1, \Homsheaf(f^* \wedge^2 V_1, f^* (T_X/V_1)))=0.
\]
We have seen in subsection \ref{foliations} that this implies the integrability of $V_1$. 

Let $Y^* \subset Y$ be the flat $\phi$-locus which we consider as embedded in $\chow{X}$ and
note $\bar{Y}$ its closure. 
Let $\Gamma$ denote the graph over $\bar{Y}$ and \holom{p_{\bar{Y}}}{\Gamma}{\bar{Y}}
and \holom{p_X}{\Gamma}{X} the projections. Furthermore note $X_y$ be the scheme-theoretic fibres of
 $p_{\bar{Y}}$.
Since every $p_{\bar{Y}}$-fibre is contracted by $\phi$ (this depends only on the homology class), 
a standard rigidity result \cite{Deb01}[Lemma1.15] implies
the existence of a factorization \holom{g}{\bar{Y}}{Y}, so that 
we obtain a commutative diagram
\[
\xymatrix{
\Gamma \ar[d]_{p_{\bar{Y}}} \ar[r]^{p_X} & X \ar[d]_\phi
\\
\bar{Y} \ar[r]^g & Y
}
\]

Our first goal is to show that $p_X$ is bijective, since $X$ is smooth and $\Gamma$ reduced this implies that
$p_X$ is an isomorphism. Assume this for the moment, then
 we can identify $\Gamma$ to $X$ using the isomorphism $p_X$.
The factorization map $g$ is birational and the 
exceptional locus projects on the points where $\phi$ is not equidimensional. 
So if $\phi$ is not equidimensional there exists an irreducible curve $C' \subset \bar{Y}$ that is contracted by $g$
($Y$ is normal, so we apply Zariski's lemma).
Let $C \subset X$ be an irreducible curve such that $p_{\bar{Y}}(C)=C'$. 
Then $C$ is contracted by $\phi$, since $\phi$ is
elementary the homology class $[C]$ is equivalent to some multiple of $[\tilde{C}]$, 
where $\tilde{C}$ is contained in a general $\phi$-fibre. This implies that $(p_{\bar{Y}})_*[\tilde{C}]$ is zero,
so $(p_{\bar{Y}})_* [C]= k [C']$ is zero. We have reached a contradiction.

In order to show that $p_X$ is bijective we first show
that the intersection of any $V_1$-leaf and a $\phi$-fibre is a subvariety of dimension $k:=\dim X - \dim Y$.

Step 1: The intersection does not have dimension $k+1$. 

Suppose the contrary, i.e. let $V_1^x$ be a $V_1$-leaf passing through 
a point $x$, such that $\fibre{\phi}{\phi(x)}$ contains $V_1^x$ (as a set).
Since every irreducible component of the fibre has dimension at most $k+1= \rk V_1$, this implies that the leaf $V_1^x$
is an irreducible component $F \subset \fibre{\phi}{\phi(x)}$. This shows that $F_{red}$ is smooth and $V_1^x$ is compact. Since 
$T_{F_{red}}=V_1|_{F_{red}}$ and $\det V_1$ is $\phi$-ample, we see that $V_1^x$ is a Fano variety. 
Then lemma \ref{quotientsubmersion} shows that there
exists a submersion \holom{f}{X}{Z} such that all fibres are $V_1$-leaves. 
Since $\phi$ contracts $F$, we can apply the rigidity lemma \cite{Mum74} to obtain a factorisation 
\holom{g}{Z}{Y}. 
But this is impossible, since $\dim X - k - 1 = \dim Z < \dim Y = \dim X - k$.

Step 2: The intersection has dimension at least $k$.

Let $x \in X$ be an arbitrary point, then there exists an $X_y$ passing through $x$. 
The reduction of $X_y$ is contained in the fibre \fibre{\phi}{\phi(x)} and
the technical lemma \ref{closedcondition} shows that it is contained in the leaf $V_1^x$ . 
The scheme $X_y$ has dimension $k$.


Step 3: $p_X$ is bijective.

For an arbitary $x \in X$, there exists exactly one $\phi$-fibre $F_x$  and one $V_1$-leaf 
$V_1^x$ passing through $x$. 
Step 1 and 2 show that for $z \in \fibre{p_X}{x}$ and $z \in X_y$, we have $X_y \subset F_x \cap V_1^x$.
Since $F_x \cap V_1^x$ has dimension equal to $X_y$, 
there exist at most finitely many $X_y$ passing through $x$, in particular $p_X$ is finite.
Since $p_X$ is birational and $X$ is smooth, the fibres are connected, so $p_X$ is bijective.
$\blacksquare$

{\bf Remark:}

The conditions in the lemma seem to be much too strong, but there are some 
technical difficulties to overcome to proof it in a more general setting. 
If $\phi$ is 
not elementary, there is no reason why it should be equidimensional. Consider a product $Y \times Z$
where $\holom{\psi}{Y}{C}$ is a ruled surface and \holom{\sigma}{Z}{Z'} is the blow-up of a smooth surface $Z'$.
Then the non-elementary contraction \holom{\psi \times \sigma}{Y \times Z}{C \times Z'} is not equidimensional.




\begin{cor}
\label{morisurface}
Let $X$ be a projective manifold with $T_X = V_1 \oplus V_2$. Let \holom{\phi}{X}{S} be an 
elementary extremal contraction on
a normal surface $S$. 
By \ref{ungenericpositionmorifibrespace} we have $T_{X/S} \subset V_1$ generically, 
so we distinguish two cases:

1. $\rk V_1 = \dim X - 2$. Then $T_{X/S}= V_1$, $\phi$ is an analytic bundle with general
fibre $F$ a Fano manifold with Picard number 1. In this case $V_2$ is not necessarily integrable.

2. $\rk V_1 = \dim X - 1$. Then $V_1$ and $V_2$ are integrable and $\phi$ is a flat map.
\end{cor}


{\bf Proof:}
In the first case, just apply proposition \ref{moriequality}. In the second case, 
lemma \ref{codim1flattening} yields the equidimensionality. Since $S$ is a normal surface and $\phi$ elementary, we 
know that $S$ is smooth (cf. e.g. \cite{AW96}). Hence equidimensionality implies flatness.
$\blacksquare$

\begin{prop}
\label{fibretype1}
Suppose that $X$ is a projective manifold with splitting tangent bundle $T_X = V_1 \oplus V_2$ 
and admits an elementary Mori contraction \holom{\phi}{X}{Y} such that
$\dim Y = \dim X - 1$. 
By \ref{ungenericpositionmorifibrespace} we have $T_{X/S} \subset V_1$ generically, 
suppose furthermore $\rk V_1 \leq 2$. 
Then $\phi$ is a $\PP^1$-bundle or a conic bundle. If $F$ is a singular fibre, then $F$  is isomorphic
to $C_1 \cup C_2$ where $C_j$ are smooth rational curves meeting transversally.
\end{prop}
\newpage

{\bf Proof:}

By lemma \ref{ungenericpositionmorifibrespace}, we have generically $T_{X/Y} \subset V_j$ for some $j$, say $j=1$.
If $V_1$ has rank 1, then it is clearly integrable and its general leaves are the general $\phi$-fibres. So
as in the proof of proposition \ref{moriequality} we see that this induces on $X$ a structure of a $\PP^1$-bundle.
This structure generically coincides with the fibration $\phi$ so they are the same.
The interesting case is where $V_1$ has rank 2, here lemma \ref{codim1flattening} implies that $\phi$ is equidimensional.
By Ando's well-known result (\cite{An85}), we know that $\phi$ is a  $\PP^1$-bundle or a conic bundle. 

For the structure of the singular fibres we use the classification
by Ando, 
so we just have to show that there exists no fibre $F$ such that $F \simeq \PP^1$ and the scheme-theoretic
structure of $F$ is that of a double line. If $F$ is such a fibre, then the foliation $V_1$ induces a germ of an
analytic surface $S \subset X$ that contains $F_{red}$ and projects via $\phi$ 
on a germ of an analytic curve $C \subset Y$.
We CLAIM that $C$ is smooth at $\phi(F)$: then the restriction of $N_{F/S}$ to $F_{red}$ is
a non-trivial torsion line bundle. This contradicts the fact that $F_{red}$ is simply connected.

Proof of the CLAIM:
 
\holom{\phi}{X}{Y} is a conic bundle, so $Y$ is smooth (\cite{An85}[Thm.3.1]). The subvariety $D \subset Y$
such that the fibre is a double line has codimension at least 2 
(otherwise we could take $\dim Y -1$ general hyperplane sections in $Y$ and apply our
argument above to the induced
morphism \holom{\phi|_{\fibre{\phi}{C}}}{\fibre{\phi}{C}}{C}). So there exists a Zariski open subset $X^* \subset X$
such that $X \setminus X^*$ has codimension at least 2 and every fibre of $\phi|_{X^*}$ has one point where
$d \phi$ is injective. By lemma \ref{pushdownlemma}, this shows that $\phi$ induces a splitting of $T_Y$ such that
\[
T_Y = (d \phi (V_1))^{**} \oplus (d \phi (V_2))^{**} 
\]
In particular $L_1:=(d \phi V_1)^{**}$ is a line bundle. If we consider the analytic germ of an analytic curve 
$C \subset Y$, then by construction $T_{C}=L_1|_C$ generically. But this implies $T_C=L_1|_C$ in every point, so
$C$ is smooth.
$\blacksquare$

{\bf Remark:}

Since elementary contractions \holom{\phi}{S}{C} from a projective surface on a curve are always 
$\PP^1$-bundles, one might expect that in the situation above there are no singular fibres at all.
The following counterexample, for which we thank M. Brunella, shows that this is not true, thereby
correcting and completing \cite{CP02}[Thm.2.8]:
 
Let $S' := \PP^1 \times \PP^1$, we identify $\PP^1 = \C \cup \infty$ 
and put coordinates $(z,w)$ on $S'$. Then the following map is an
involution on $S'':=S' \setminus \{ (0,0) , (\infty,\infty) \}$.
\[
\holom{\phi'}{S''}{S''} \quad (z,w) \mapsto (z,\frac{z}{w})
\]
If $C_z:=z \times \PP^1$ is a fibre of the projection $pr_1$ on the first factor,
then $\phi'|_{C_z}$ is the involution on $\PP^1$ with fixed points $\sqrt{z}$ and
$- \sqrt{z}$.
Blow up $S'$ in $(0,0)$ and $(\infty,\infty)$ to resolve the indeterminacies of
$\phi'$. If we note $\holom{\mu}{S}{S'}$ the blow-up, then $\phi'$ lifts to
an involution $\phi$ of $S$. The total transform of $0 \times \PP^1$ has two
irreducible components $E_1$ and $E_2$ such that $\phi(E_1)=E_2$ and $\phi(E_2)=E_1$
(an analogous statement holds for the total transform of $\infty \times \PP^1$).

Let $T:=\C/(\Z \oplus i \Z)$ be an elliptic curve and $\holom{\psi}{T}{T} \quad t \mapsto t+\frac{1}{2}$. Then
$T' \simeq T/\{id_T,\psi\}$ is an elliptic curve. 
Define $X:= (S \times T)/ \{id_S \times id_T, \phi \times \psi\}$, then $X$ is a smooth
projective variety with splitting tangent bundle. The map $(pr_1 \circ \mu) \times id_T$ induces
a morphism 
\[
\holom{f}{X}{\PP^1 \times T'}
\]
We will show that $f$ is an elementary Mori contraction that is not a $\PP^1$ - bundle. 
The only property that is not obvious is that $f$ is elementary. 
Let  $F_0' = E_1' + E_2'$ be a singular fibre and let $F_0 = E_1 + E_2$ be its lifting to $S \times T$.
Then $F_0 \subset S \times t$ for some $t \in T$ and
clearly 
\[
E_1 = (\phi \times \psi) (\tilde{E})
\] 
where $\tilde{E}$ is the copy of $E_2$ in $S \times (t+\frac{1}{2})$.
Since $\tilde{E}$ is a deformation of $E_2$ in $S \times T$, this shows that  
$E_2'$ is a deformation
of $E_1'$ in $X$, hence $E_1'=E_2'$ in $N_1(X)$. 
The special fibre $F_0$ is homologous to a general fibre $F$, so
\[
F = F_0 = E_1' + E_2' = 2 E_1'
\] 
in $N_1(X)$. This shows that $f$ is the elementary contraction of the ray generated by $E_1'$.

\begin{lemma}
\label{pushdownlemma}
Let $X$ be a complex manifold with splitting tangent bundle $T_X = V_1 \oplus V_2$.
Suppose that there exists a morphism 
\holom{f}{X}{Y} on a manifold $Y$ such that on a general fibre F,
we have $T_{F} = (V_1 \cap T_F) \oplus (V_2 \cap T_F)$. 
Suppose furthermore that there exists a subvariety $Z \subset Y$ of codimension at least 2, such that for 
$y \in Y^*:=(Y \setminus Z)$, the following condition
is satisfied:
\[
 \exists \ x \in \fibre{f}{y} \ \mbox{s.t.} \ \rk df_x = \dim X - \dim Y.
\]
Then $W_j:= (df(V_j))^{**}$ is a subbundle of $T_Y$ such that $T_Y=W_1 \oplus W_2$.
\end{lemma}

{\bf Proof:}

Suppose first that $f$ is smooth. Then the tangent map \holom{df}{T_X}{f^* T_Y} is surjective.
Its restriction to the subbundles $V_j$, denoted \holom{q_j}{V_j}{f^* T_Y} is a morphism of sheaves.
Since $T_{X/Y}= \ker df$, we have $\ker q_j= (T_{X/Y} \cap V_j)$, so the 
rank of $q_j$ in every point is $\rk V_j - \rk (T_{F} \cap V_j|_F)$. By hypothesis this implies
$\rk q_1 + \rk q_2= \dim Y$. Since $df=q_1 \oplus q_2$ and $df$ has rank equal to $\dim Y$ in every
point, this implies that $q_j$ is a morphism of vector bundles and $\im(q_1) \oplus \im(q_2) = f^* T_Y$.
We verify that this induces a splitting of $T_Y$: 
for every fibre $F$, we have 
\[
\im(q_1)|_F \oplus \im(q_2)|_F = f^* T_Y|_F=  \simeq \Osheaf^{\oplus \dim Y},
\] 
so it is elementary to see that $\im(q_j)|_F$ is trivial. This shows $\im(q_j)=f^* E_j$ where $E_j$ is a vector
bundle on $Y$, so the splitting pushes down to $Y$.

We show how this implies the general case:
The set 
\[
X^*:= \{ x \in \fibre{f}{y} \ \mbox{s.t.} \ \rk df_x = \dim X - \dim Y, \ y \in Y^* \} \subset \fibre{f}{Y^*}
\]
is a non-empty Zariski open subset of $X$. By the first part, 
the differential of the smooth map \holom{f|_{X^*}}{X^*}{Y^*} induces
a splitting on $T_{Y^*}=S_1 \oplus S_2$. Note
\holom{i}{Y^*}{Y} the canonical embedding and define $W_j:= i_* S_j$. Then
\[
T_Y = W_1 \oplus W_2,
\]
so the upper semicontinous functions $\phi_j(y)= \dim (W_j \otimes \C(y))$ are constant.
This implies that the $W_j$ are locally free.
$\blacksquare$



\section{Birational geometry}
\label{birational}

This section completes the proof of theorem \ref{dim4uniruled}. The main step is a 
discussion of elementary contractions of birational type which for general fourfolds are very difficult. 
Our case is much simpler, we will see that concerning its birational behaviour, the fourfold $X$ is
similar to a product of surfaces.

\begin{prop}
\label{moribirational}
Let $X$ be projective fourfold such that $T_X = V_1 \oplus V_2$ where $\rk V_j=2$
and let \holom{\phi}{X}{Y} be a contraction of an 
extremal ray such that $\dim X = \dim Y$ (birational Mori contraction).
Then $\phi$ is the blow-up of a smooth $2$-dimensional subvariety of the manifold $Y$.
\end{prop}

{\bf Proof:}

Let $E$ be the exceptional locus of the contraction
and
$F$ an irreducible component of a non-trivial fibre, then by \ref{CP02Prop18} we have $\dim F \leq 2$. In particular the contraction
can't be of type $(3,0)$.
Contractions of type $(3,1)$ have been classified by Takagi \cite{Ta99}, the contraction 
\holom{\phi|_E}{E}{\phi(E)} is a $\PP^2$-bundle or quadric bundle, so 
the general fibre is reduced and isomorphic to $\PP^2$, $\PP^1 \times \PP^1$, or a quadric cone.
If $\phi$ is of type $(3,2)$, then a fibre of dimension 2 is isolated
(in the sense of \cite{AW96}), so $F$ is reduced and 
either $\PP^2$,  $\PP^2 \cup \PP^2$, $\PP^1 \times \PP^1$, or a quadric cone \cite{AW96}[Thm.4.6].
Elementary fourfold contractions such that the exceptional locus is not an irreducible divisor 
have been classified by Kawamata \cite{Ka89},
the non-trivial fibres are reduced and isomorphic to $\PP^2$.

We will exclude case by case the existence of 2-dimensional fibres. The strategy is to
choose appropriately rational curves \holom{f}{\PP^1}{F} such that $f^* T_F$ is nef
so that we can use equation \ref{fibresplitting} to compute $f^* T_X$


Case 1: $F \simeq \PP^2$ or $F \simeq \PP^2 \cup \PP^2$.

If $F \simeq \PP^2$, let \holom{f}{\PP^1}{F} be a line $C=f(\PP^1)$, then 
\[
f^* T_X = \Osheaf (2) \oplus \Osheaf (1) \oplus \Osheaf (-a) \oplus \Osheaf (-b)
\]
where $a \geq 0$, $b \geq 0$ and $a+b>0$ (otherwise the deformations of $C$ would cover $X$).
Since the curve is pseudo-standard, we have up to renumeration  $\Osheaf (2) \subset f^* V_1$.
 Since $C$ is a line in projective space it deforms keeping a point fixed.
This shows that $\Osheaf(2) \subset f^* V_1$ implies $T_{\PP^2}=V_1|_{\PP^2}$. In particular 
$f^* V_1 = \Osheaf (2) \oplus \Osheaf (1)$, so $f^* V_2 = \Osheaf (-a) \oplus \Osheaf(-b)$. This implies that
$f^* \det V_2 = \Osheaf(-a-b)$ is not $\phi$-trivial. This contradicts \ref{CP02Prop18}.

The argument before formula \ref{fibresplitting} shows that the same argument works in the case
$F \simeq \PP^2 \cup \PP^2$.


Case 2: $F \simeq \PP^1 \times \PP^1$.

Choose a ruling line \holom{f}{\PP^1}{F} with image $C=f(\PP^1)$, then
\[
f^* T_X = \Osheaf (2) \oplus \Osheaf \oplus 
\Osheaf (-a) \oplus \Osheaf (-b)
\]
where $a \geq 0$ and $b \geq 0$ and $a+b>0$. Up to renumeration, this implies $\Osheaf(2) \subset f^* V_1$. 
Hence $\det V_1$ is $\phi$-ample and $\det V_2$ is $\phi$-trivial.
Now choose a line \holom{f'}{\PP^1}{F} from
the second ruling that
is transversal to the first one, then
\[
f'^* T_X = \Osheaf (2) \oplus \Osheaf \oplus 
\Osheaf (-a) \oplus \Osheaf (-b)
\]
with coefficients as above.
Then again $\Osheaf(2) \subset f'^* V_1$, since otherwise $f'^* c_1(V_1) = 0$ (cf. the proof of \cite{CP02}[Lemma1.3]), which would contradict
the $\phi$-ampleness of $\det V_1$. Since the lines are transversal, we obtain
\[
\Osheaf(2) \oplus \Osheaf = f^* T_F = f^* V_1.
\]
As in the first case we obtain $f^* V_2 = \Osheaf (-a) \oplus \Osheaf(-b)$. Clearly
$f^* \det V_2 = \Osheaf(-a-b)$ is not trivial, a contradiction. 

Case 3 : $F$ is a quadric cone.

Let \holom{f}{\PP^1}{F} be a line passing through the vertex $x$ of the cone $F$. Then Campana and Peternell
have shown that $C=f(\PP^1)$ is pseudo-standard \cite{CP02}[Theorem3.6], so $\Osheaf(2) \subset f^* V_1$ by the remark after \ref{CP02Prop18}.
In particular $(T_C)_x \subset (V_1)_x$ for every such line. Since $x$ is a singularity of $F$ and the 
$T_C$ generate the Zariski tangent space of $F$ in $x$, they generate a subspace of dimension at least 3 in 
$(V_1)_x$. This contradicts $\rk V_1=2$.


We resume: $\phi$ is a birational contraction of divisorial type such that all non-trivial
fibres are of pure dimension 1. So the lemma is a consequence of a well-known result
by Ando \cite{An85}[Thm.2.1].
$\blacksquare$


\begin{prop}
\label{moribirationalcovering}
Let $X$ be a projective manifold such that $T_X = \oplus V_j$ where
$rk V_j \leq 2$. Let \holom{\phi}{X}{Y} be the blow-up of a  
of a smooth $(n-2)$-dimensional subvariety of $Y$.
Then the tangent bundle of $Y$ splits in $T_Y = \oplus_j (d\phi(V_j))^{**}$.

If the universal covering of $Y$ splits in $\tilde{Y} = Y_1 \times \ldots \times Y_k$  such that $\mu^* V_j' = T_{V_j}$, then the analogous statement holds for $X$.
\end{prop}

{\bf Proof:}


The splitting of the tangent bundle of $T_Y$ is a direct consequence of lemma \ref{pushdownlemma}.
Let \holom{\mu}{\tilde{Y}}{Y} be the universal covering map and
suppose that $\tilde{Y} = Y_1 \times \ldots \times Y_k$  and $\mu^* V_j' = p_{Y_j}^* T_{Y_j}$ .
Note that this implies the integrability of $V_j'$ for all $j$.
Since $\pi_1(X)=\pi_1(Y)$, the pull-back $\tilde{X} = X \times_Y \tilde{Y}$
is the universal covering of $X$ so that we obtain a commutative diagram
\[
\xymatrix{
\tilde{X} \ar[r]^{\tilde{\mu}} \ar[d]^{\tilde{f}} & X  \ar[d]^f
\\
\tilde{Y}  \ar[r]^\mu & Y 
}
\]
By the commutativity of the diagram, we have 
\[
\tilde{\mu}^* V_j = \tilde{f}^* \circ \mu^* V_j' = \tilde{f}^* p_{Y_j}^* T_{Y_j}.
\]
Since $\tilde{f}$ is birational, this shows that $\tilde{\mu}^* V_j$ is integrable for all $j$.
The morphism $\tilde{f}$ is the blow-up of $\tilde{Y}$ along $\fibre{\mu}{\phi(E)}$.
Since the contracted fibres of $\tilde{f}$ are contained in the
leaves of $\tilde{\mu}^* V_1$, the morphism \holom{pr_{Y_2 \times Y_k} \circ \tilde{f}}{X}{Y_2 \times Y_k}
is a submersion with integrable connection $\tilde{\mu}^* V_1$. By the Ehresmann theorem \ref{ehresmann}
we obtain $\tilde{X} \simeq F \times Y_2 \times \ldots \times Y_k$ with $F$ the general fibre,
but clearly $F \simeq Bl_{y_1}(Y_1)$ for some $y_1 \in Y_1$.
$\blacksquare$

\begin{prop}
\label{mmp}
Let $X$ be a projective fourfold such that $T_X = V_1 \oplus V_2$ where $\rk V_j=2$ for $j=1,2$.
Then $X'$ admits a relative minimal model, i.e. a birational morphism 
\holom{\phi}{X}{X'} on a smooth variety $X'$ such that $T_{X'} = d\phi(V_1)^{**} \oplus d\phi(V_2)^{**}$
and either
\begin{itemize}
\item $K_{X'}$ is nef
\item $X'$ admits an elementary contraction of fibre type. 
\end{itemize}
If conjecture \ref{beauvilleconj} holds for $X'$, then it holds for $X$.
\end{prop}

{\bf Proof:} 

The two lemmas above show that the decomposition of $T_X$ and conjecture  \ref{beauvilleconj} are stable
under birational contractions. By the general minimal model program it is clear that after finitely many steps
we obtain either a variety with nef canonical bundle or an elementary contraction of fibre type. $\blacksquare$

{\bf Proof of theorem \ref{dim4uniruled}:}

By proposition \ref{mmp} we may suppose that $X$ admits an elementary contraction of fibre type \holom{\phi}{X}{Y}.
By lemma \ref{ungenericpositionmorifibrespace} it is clear that the general fibre $F$ has $\dim F=2$ or $\dim F=1$
and up to renumeration $T_F \subset V_1|_F$.

Case 1.
If $\dim F=2$, then $\dim F= \rk V_1$. By proposition \ref{moriequality} the morphism $\phi$ is an analytic bundle
with connection $V_2$. Since $V_2$ is integrable, we conclude with the Ehresmann theorem (see also the remark below).

Case 2.
If $\dim F=1$, then proposition \ref{fibretype1} shows that $\phi$ is flat without multiple fibres and
$T_Y= d\phi(V_1)^{**} \oplus d\phi(V_2)^{**}=: L \oplus V$. By theorem \ref{brunella} applied to $Y$, there are two
cases:

a) $Y$ is a $\PP^1$-bundle \holom{\psi}{Y}{Z} such that $T_{Y/Z}=L$. Then \holom{\psi \circ \phi}{X}{Z} is a submersion
with integrable connection $V_2$ and we conclude with the Ehresmann theorem.

b) $V$ is integrable, so the universal covering \holom{\nu}{\tilde{Y}}{Y} satisfies $\tilde{Y} \simeq S \times C$
such that $\nu^* V = p_S^* T_S$ and $\nu^* L = p_C^* T_C$. Furthermore we have a commutative diagram
\[
\xymatrix{
X' \ar[r]^{\mu'} \ar[d]^{\tilde{\phi}} & X  \ar[d]^\phi
\\
\tilde{Y}  \ar[r]^\nu & Y 
}
\]
where \holom{\mu'}{X':= X \times_Y \tilde{Y}}{X} is \'etale. Note that $\tilde{\phi}$ does not have any multiple fibres, so the morphism 
\holom{q:=p_S \circ \tilde{\phi}}{X'}{S} is a submersion such that $\mu'^* V_1 = T_{X'/S}$. The bundle $\mu'^* V_2$ provides an integrable connection on $q$, so we conclude as above. $\blacksquare$

{\bf Remark:}

One should note that we used the integrability hypothesis on the direct factors $V_j$ 
only to apply the Ehresmann theorem,
while it was not necessary to establish the preceeding structure results. Furthermore Campana and Peternell
have given an affirmative answer to conjecture \ref{beauvilleconj} for a $\PP^n$-bundle \holom{\phi}{X}{Y} with a (not necessarily integrable) connection, so the only case where we can not conclude without the
integrability hypothesis is case 2. We resume our considerations in this direction with a supplement to the Ehresmann theorem.

\begin{lemma}
Let \holom{\phi}{X}{Y} be a proper submersion of complex manifolds
that admits a connection $E$, i.e. a subbundle $E \subset T_X$ such that $T_X = E \oplus T_{X/Y}$. 
Then $\phi$ induces the structure of an analytic fibre bundle on $X$. 
If $X$ is projective and non-uniruled, the connection is integrable.
\end{lemma}

{\bf Proof:} Apply lemma \ref{connectionbundle} and theorem \ref{nonuniruledintegrable}. $\blacksquare$

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\end{thebibliography}






{\it Andreas H\"oring,  UFR de Math\'ematiques, 
Universit\'e de Grenoble 1, UMR 5582 du CNRS, BP 74, 38402 Saint Martin d'H\`eres, FRANCE

E-mail address:  ahoering@fourier.ujf-grenoble.fr }

\end{document}


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  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
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\newblock Rationally connected foliations after bogomolov and mcquillan.
\newblock {\em arXiv preprint}, 0505222, 2005.

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\newblock {\em Positivity in algebraic geometry. {II}}, volume~49 of {\em
  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
  Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd
  Series. A Series of Modern Surveys in Mathematics]}.
\newblock Springer-Verlag, Berlin, 2004.
\newblock Positivity for vector bundles, and multiplier ideals.

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\newblock {\em Riemannian foliations}, volume~73 of {\em Progress in
  Mathematics}.
\newblock Birkh\"auser Boston Inc., Boston, MA, 1988.
\newblock Translated from the French by Grant Cairns, With appendices by
  Cairns, Y. Carri\`ere, \'E. Ghys, E. Salem and V. Sergiescu.

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\newblock {\em Abelian varieties}.
\newblock Tata Institute of Fundamental Research Studies in Mathematics, No. 5.
  Published for the Tata Institute of Fundamental Research, Bombay, 1970.

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\newblock Global stability for holomorphic foliations on {K}aehler manifolds.
\newblock {\em Qual. Theory Dyn. Syst.}, 2(2):381--384, 2001.

\bibitem[Tak99]{Ta99}
Hiromichi Takagi.
\newblock Classification of extremal contractions from smooth fourfolds of
  {$(3,1)$}-type.
\newblock {\em Proc. Amer. Math. Soc.}, 127(2):315--321, 1999.

\end{thebibliography}