Hyper-K\"ahler Fourfolds Fibered by Elliptic Products
Ljudmila Kamenova
\noindent Every fibration of a projective hyper-K\"ahler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We classify those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypotheses. Unlike the genus two case, we prove there are infinitely many deformation families for hyper-K\"ahler fourfolds fibered by elliptic products. %We consider hyper-K\"ahler manifolds of complex %dimension $4$ which are fibrations. It is known that the %fibers are abelian varieties and the base is $\mathbb P^2$. %We assume that the general fiber is isomorphic to a product of %two elliptic curves. We prove that such a hyper-K\"ahler %manifold is deformation equivalent to a Hilbert scheme of two points on a %K3 surface.
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\begin{document}


\title{Hyper-K\"ahler Fourfolds Fibered by Elliptic Products}
\author{Ljudmila Kamenova}
\email{kamenova@math.sunysb.edu}
\address{Department of Mathematics, SUNY Stony Brook}
\date{} \footnotetext{Research partially supported by NSF 
Grants DMS-0111298 and DMS-0805594}


\begin{abstract}
\noindent
Every fibration of a projective hyper-K\"ahler fourfold has fibers which 
are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus 
two curve, these have been classified by Markushevich. We classify 
those cases where the Abelian surface is a product of two elliptic curves, 
under some mild genericity hypotheses. Unlike the genus two case, 
we prove there are infinitely many deformation families for 
hyper-K\"ahler fourfolds fibered by elliptic products. 

%We consider hyper-K\"ahler manifolds of complex 
%dimension $4$ which are fibrations. It is known that the 
%fibers are abelian varieties and the base is $\mathbb P^2$. 
%We assume that the general fiber is isomorphic to a product of 
%two elliptic curves. We prove that such a hyper-K\"ahler 
%manifold is deformation equivalent to a Hilbert scheme of two points on a 
%K3 surface. 
\end{abstract}

\maketitle

\section{Introduction} 

Among projective complex manifolds with zero first Chern class, hyper-K\"ahler 
manifolds play a distinguished role, generalizing the class of K3
surfaces.  In fact the hyper-K\"ahler manifolds of (complex) dimension
2 are precisely the K3 surfaces.
In higher dimensions, dimension 4 and higher, few hyper-K\"ahler 
manifolds are known: two infinite classes introduced by Beauville 
(\cite{Beauville1}) and two exceptional cases discovered by O'Grady 
(\cite{og1} and \cite{og2}).  Nonetheless, even in
the first case of hyper-K\"ahler fourfolds, we are still far from a
classification or even a proof that hyper-K\"ahler fourfolds form a
bounded family.

Just as elliptically fibered K3 surfaces are more amenable to study,
so also fibered hyper-K\"ahler manifolds are better
understood. 
By works of Matsushita and Hwang, (\cite{m} and \cite{hw}), every 
holomorphic fibration whose total space is a $2n$-dimensional 
hyper-K\"ahler has base manifold $\mathbb{P}^n$ and has general fiber
an $n$-dimensional Abelian variety, up to a finite (unramified)
cover.  A hyper-K\"ahler manifold fibered over $\mathbb{P}^n$ with
general fiber an Abelian variety is called an \emph{Abelian
  fibration}.  Every Abelian fibered manifold is a ``Tate-Shafarevich
twist'' of an Abelian fibration with a section, and the set of these
twists coming from a given Abelian fibration
is classified by a well-studied Tate-Shafarevich group.  Thus from now
on we assume that the Abelian fibration admits a section. 

The principal case is when the fibers are \emph{principally polarized}
Abelian varieties.  For hyper-K\"ahler fourfolds, the fiber will be a
principally polarized Abelian surface.  There are two types of such
surfaces, depending on whether or not the polarizing divisor is
irreducible or reducible, i.e., whether the Abelian surface is the
Jacobian of a genus $2$ curve or a product of elliptic curves.
%% Our ultimate goal is to relate Abelian fibered hyper-K\"ahler manifolds 
%% to Beauville's standard series of examples. 
%% Matsushita proved that the 
%% fibers of such a fibration are Lagrangian with respect to the 
%% holomorphic symplectic form. 
%% Therefore, the Abelian fibration can be 
%% considered as an algebraic completely integrable Hamiltonian system. 
Markushevich classified the first case, when the Abelian surface is
the Jacobian of a genus $2$ curve.

\begin{thm}\cite[Markushevich]{markushevich96} 
Every irreducible holomorphic symplectic fourfold $X$ which 
is fibered by Jacobians of genus two curves and such that the fibration 
$X \rightarrow \mathbb P^2$ admits a section is birational 
to ${\it Hilb}^2 (S)$, for a K3 surface $S$ which is a 
double cover of $\mathbb P^2$ branched over a plane sextic. 
\end{thm}

Our goal is to classify those Abelian fibrations where the generic
fiber is a product of two elliptic curves, an \emph{elliptic
  product}.  But unlike Markushevich's theorem, there are infinitely
many deformation classes of projective hyper-K\"ahler fourfolds 
together with a
fibration by elliptic products.  Moreover, these hyper-K\"ahler
fourfolds can become ``wild'' when the fibration is allowed to become
pathologically degenerate.  In fact we conjecture that these
pathological degenerations never occur, at least for a generic member
of a deformation family.  But, at the moment, we add our genericity
conditions as hypotheses: the conditions 
are quite natural and are satisfied in every case we know of.

In order to explain the genericity conditions, we first introduce some
notation.  

As we shall see, every hyper-K\"ahler fibration by elliptic products 
$p: X \rightarrow \mathbb{P}^2$ arises from a degree-2 branched cover 
$f: \bar V \rightarrow \mathbb P^2$, where $\bar V$ is a normal 
variety, and a family of elliptic curves 
$\pi : {\mathcal E} \rightarrow \bar V$. Let $D$ be the branch locus 
of $f$, $\tilde D = f^{-1}(D)$ and let $G$ be the discriminant locus
of $\pi$. These are our \emph{genericity conditions}: 
\begin{enumerate}
\item Over every generic point of $G$, the fiber of $\pi$ is
  irreducible and semistable. 
\item The image $f(G)$ contains no singular point of $D$.  In fact it
  suffices to assume that $f(G)$ contains no intersection point of two
  irreducible components of $D$ which are lines.
\end{enumerate}

Here is the main theorem. 

\begin{thm}
Let $p: X \rightarrow \mathbb P^2$ be a projective 
hyper-K\"ahler fibration whose general 
fiber is a product of two elliptic curves and which satisfies the
genericity conditions above.
Assume that the fibration admits a 
section $\tau$. 
Then $X$ is birational to $Hilb^2(S)$ for a projective, 
elliptic K3 surface $S$ with a section.
\end{thm}

Notice that for every projective, elliptic K3 surface $S$ with a
section, $\text{Hilb}^2(S)$ is a projective hyper-K\"ahler fourfold fibered
by elliptic products and which admits a section.
There are countably many families of projective, elliptic K3 surfaces
$S$, hence countably many families of elliptic product hyper-K\"ahler
fourfolds.  And for the general member of each of these families, the
genericity conditions are satisfied.  As mentioned, we conjecture that
every sufficiently general elliptic product hyper-K\"ahler fourfold
satisfies the genericity conditions.  All such are deformation
equivalent to one of the countably many families arising from
projective, elliptic K3 surfaces.


\section{Preliminaries}

First we define our main objects of study, {\it irreducible holomorphic 
symplectic manifolds} or {\it irreducible hyper-K\"ahler manifolds}. 

\begin{defn}
A compact complex K\"ahler manifold $X$ is called {\it irreducible 
holomorphic symplectic}  
if it is simply connected and if $H^0(X, \Omega_X^2)$ is spanned by an 
everywhere non-degenerate 2-form  $\omega$.
\end{defn}


Any holomorphic two-form $\sigma$ induces a homomorphism ${\mathcal T}_X 
\to \Omega_X$. The two-form is {\it everywhere non-degenerate} 
if ${\mathcal T}_X \to \Omega_X$ is bijective. 
The last condition in the definition implies that 
$h^{2,0}(X)=h^{0,2}(X)=1$ and $K_X \cong {\mathcal O}_X$, i.e., 
$c_1(X)=0$. \\


\begin{defn}
A compact connected $4n$-dimensional Riemannian manifold $(M,g)$ is called
{\it irreducible hyper-K\"ahler} if its holonomy is ${\rm Sp}(n)$.
\end{defn}

As Huybrechts notes \cite{h}, irreducible holomorphic symplectic manifolds 
with a fixed K\"ahler class are equivalent to 
irreducible hyper-K\"ahler manifolds. 
In the rest of the paper, we refer to irreducible hyper-K\"ahler 
manifolds just as hyper-K\"ahler manifolds for simplicity. \\ 

\begin{defn}
An {\it Abelian fibration} on a $2n$-dimensional hyper-K\"ahler manifold $X$ 
is the structure of a fibration over ${\mathbb P}^n$ whose generic fibre is 
a smooth abelian variety of dimension $n$. 
\end{defn}

This is a higher dimensional analogue of elliptic
fibrations on K3 surfaces. Any fibration structure of a projective 
hyper-Kahler manifold is an abelian fibration due to the following 
theorems by Matsushita~\cite{m} and Hwang~\cite{hw}:


\begin{thm}
\label{matsushita} (Matsushita, \cite{m})
For a projective holomorphic symplectic manifold $X$, let 
$f : X \rightarrow B$ be a proper surjective 
morphism such that the generic fibre $F$ is connected. Assume that $B$ 
is smooth and $0<{\mathrm{dim}}B<{\mathrm{dim}}X$. Then 

(1) $F$ is an abelian variety up to a finite unramified cover,

(2) $B$ is $n$-dimensional and has the same Hodge numbers as ${\mathbb P}^n$, 

(3) the fibration is Lagrangian with respect to the holomorphic
symplectic form.

\end{thm}


\begin{thm} (Hwang, \cite{hw})
In the setting of Matsushita's theorem, $B$ is biholomorphic to 
${\mathbb P}^n$.
\end{thm}

%In particular, if $X$ is 4-dimensional, we can use the
%Castelnuovo-Enriques classification of surfaces to deduce that the
%generic fibre is an abelian surface and the base is ${\mathbb P}^2$. \\


Fogarty gives the following description of the Hilbert schemes of  
complex surfaces. 

\begin{thm} \label{fog} (Fogarty, \cite{f})
For a nonsingular surface $X$ and $n \in \mathbb N$ \\
(1) ${\it Hilb}^n (X)$ is non-singular of dimension $2n$ and \\
(2) $\pi : {\it Hilb}^n (X) \rightarrow S^n X$ is a resolution of 
singularities, where $S^n X$ is the n-th symmetric product of $X$.
\end{thm}



% \section{Basic Tools}



\section{Proof of the Main Theorem}

In this section we prove our main theorem. 

\begin{proof}
Let us be more precise about the hypothesis that the general 
singular fiber is irreducible and semistable. Precisely, there exists a closed 
codimension two subscheme $N$ of $\mathbb P^2$ such that the fiber of $p$ 
over every point outside of $N$ is either smooth or irreducible and 
semi-stable. \\

Take the open subset $U \subset \mathbb P^2$ over which the fibers of $p$ 
are smooth. $U$ is algebraic. 
% Indeed, $p$ is a proper morphism, the singular 
% locus of the morphism is closed and the image of a Zariski closed subset of 
% $X$ is Zariski closed subset in $\mathbb P^2$. Therefore, the complement of 
% the image of the singular locus, which is $U$, is an algebraic open set. \\
The fibers over $U$ are of the form $E^1_t \times E^2_t$. 
We can form the fibration $Y \rightarrow U$ with fibers of the 
form $E^1_t \cup E^2_t$, where the two elliptic curves are glued along the 
section $\tau$. Take the normalization 
${\tilde Y} \rightarrow Y \rightarrow U$. Since U is 
algebraic, we can apply Stein factorization. 
The morphism ${\tilde Y} \rightarrow U$ factors through a 
smooth proper morphism with connected fibers and an \'etale morphism of 
degree 2: ${\tilde Y} \rightarrow V \rightarrow U$. \\

According to Grothendieck's lemma ( \cite{g}, Section 6.3.), there is a 
unique normal variety $\bar V$, a finite 
degree-2 morphism $f : \bar V \rightarrow \mathbb P^2$ and a fiber diagram:
$$\begin{array}{ccc}
V & \stackrel{2:1}{\longrightarrow} & U \\ 
\curvearrowright  &               & \curvearrowright \\
\bar V & \stackrel{2:1}{\longrightarrow} & \mathbb P^2 \\
\end{array}$$ \\

% Indeed, let's cover $\mathbb P^2 = \bigcup E_i$ with open affine sets. Then 
% we have the following maps between the rings of functions: 
% $\mathbb C (E_i) \rightarrow \mathbb C (U) \rightarrow \mathbb C (V).$ 
% Take the integral closure of $\mathbb C (E_i)$ inside the fraction field of 
% $\mathbb C (V)$. It corresponds to $\bar E_i$. We can glue all of them 
% together to get a variety $\bar V$ (\cite{har}) which is normal by 
% construction. $\bar V$ is unique, because it is such that for every normal 
% $T$ and dominant morphism $T \rightarrow \mathbb P^2$, the set of lifts: 
% $T \dashrightarrow \bar V \rightarrow \mathbb P^2$ is naturally in bijection 
% with the set of factorizations: 
% $\mathbb C (\mathbb P^2) \rightarrow \mathbb C (\bar V) \dashrightarrow 
% \mathbb C (T)$. \\


Since the general fiber of $X$ is semi-stable, 
the fibration 
${\tilde Y} \rightarrow V$ extends to a minimal family of elliptic 
curves $\pi : {\mathcal E} \rightarrow \bar V$ with a general fiber being 
semi-stable. The singular fibers are from Kodaira's list of 
degenerations of elliptic curves. In codimension one there are no multiple 
fibers. In codimension two there might be multiple fibers, the 
fiber dimension can jump or $\mathcal E$ might not be even defined. 
However, we are interested in codimension one.
Notice that $\mathcal E$ is a N\'eron model \cite{neron} since the fibers are 
abelian varieties.
There is an induced section $\sigma$ of the fibration. \\

Since the map $f : \bar V \rightarrow \mathbb P^2$ is 2:1, there is an 
involution 
$i$ acting on $\bar V$ which interchanges the sheets of the fibers. 
The involution is well defined on $V$ and from the fiber diagram above 
it is well defined on $\bar V$ as well, because with $i^{-1} V$ we can 
construct a similar fiber diagram since the maps to $\mathbb P^2$ are the same. 
Therefore there will be an involution on $\bar V$ compatible with the 
involution on $V$. 
% Using the maps 
% $\tau_{\bar V}$ and $\tau_{\bar V} \circ i$, we get the cofiber diagram: 
% $$\bar V \rightrightarrows {\cal E} \rightarrow Y \rightarrow \mathbb P^2$$
% which defines $Y$ in a universal way. 
We recall the data from the Introduction:  
denote the branched locus of $f$ by $D$ and $f^{-1}(D) = \tilde D$. 
Let $G$ be the discriminant locus of $\pi : {\mathcal E} \rightarrow \bar V$.
Note that the intersection $G \cap \tilde D$ consists of finitely many 
points. Indeed, if it wasn't true, then $f(G)$ and $D$ would have a 
whole component in common.
The fibers of $p$ above this component would be very 
degenerate (they will be products of two degenerate elliptic curves). 
According to the genericity conditions, in codimension one the fibers of the 
original fibration have at worst simple normal crossing singularities, so this 
cannot happen.  \\

The section $\sigma$ induces a section ($\sigma$, $\sigma \circ i$) of the map:
$$pr_{\bar V} : {\mathcal E} \times_{\bar V} i^\ast {\mathcal E} \rightarrow 
\bar V$$
and the involution $i : \bar V \circlearrowleft $ induces an involution 
$i_{\mathcal E}$ on ${\mathcal E} \times_{\bar V} i^\ast {\mathcal E}$. \\

Consider the non-branched locus $\mathbb P^2 - D$ and its pre-image $X_0$ in 
$X$. Denote the induced fibration by 
$p_0 : X_0 \rightarrow \mathbb P^2 - D$. By construction,
\begin{equation} \label{0}
X_0 \cong {\mathcal E} \times_{\bar V} i^\ast {\mathcal E} \vert_{V} / 
i_{\mathcal E}
\end{equation}  \\

Let $L \subset \mathbb P^2$ be a line intersecting $D$ transversally at 
general points of $D$. Denote the pre-image of $L$ in $\bar V$ by $L_V$. 
The restricted map $f_V : L_V \rightarrow L$ is 2:1. We choose $L$ general 
so that $L_V$ doesn't intersect $\tilde D \cap G$, so every fiber of 
${\mathcal E}_{L_V} \times_{L_V} i^\ast {\mathcal E}_{L_V} \rightarrow L_V$
over a point of $L_V \cap \tilde D$ is smooth. 
After pulling back our construction to $L_V$, there a rational morphism:
$$L_V \times_{\mathbb P^2} X \dashrightarrow 
{\mathcal E}_{L_V} \times_{L_V} i^\ast {\mathcal E}_{L_V}$$
which by Weil's extension theorem \cite{neron} is regular on the smooth locus 
of $L_V \times_{\mathbb P^2} X \rightarrow L_V$ since the fibers are 
abelian varieties. \\

Every section of $p : X \rightarrow \mathbb P^2$ is contained in the 
smooth locus of $ X \rightarrow \mathbb P^2$, so it pulls-back to a 
curve in the smooth locus of 
$L_V \times_{\mathbb P^2} X \rightarrow L_V$ via the section $\tau f_V$. \\

Since ${\mathcal K}_X = {\mathcal O}_X$, the relative sheaf of 
$p: X \rightarrow \mathbb{P}^2$ is 
$\omega_{X / \mathbb P^2} = p^* {\mathcal O}_{\mathbb P^2}(3)$ and 
therefore $\tau^\ast \omega_{X / \mathbb P^2} = {\mathcal O}_{\mathbb P^2}(3)$. \\

Consider the relative tangent bundle with the natural isomorphism:
$$(\tau f_V)^\ast N_{\tau f_V (L_V) / L_V \times_{\mathbb P^2} X } = 
f_V^\ast (\tau^\ast T_{X / \mathbb P^2}) \stackrel{\sim}{\rightarrow} 
(\sigma, \sigma \circ i)^\ast T_{{\mathcal E} \times i^\ast {\mathcal E} / 
\bar V} 
\vert_{L_V} $$

For every point $p \in L_V \cap \tilde D$, the map is  $L_1 \oplus L_2 
\rightarrow L_1 \oplus L_2(p)$, where $L_1 \in T_{\Delta_{\mathcal E}}|_p$ 
and $L_2 \in N_{\Delta_{\mathcal E} / {\mathcal E} \times {\mathcal E}} 
\cong T_{\mathcal E}$, and $\Delta_{\mathcal E}$ is the diagonal. 
Outside the branched locus, the map is the identity.

From the above we get: 
$$f_V^\ast (\tau^\ast \Lambda^2 T_{X / \mathbb P^2}) 
\stackrel{\sim}{\rightarrow} 
(\sigma, \sigma \circ i)^\ast \Lambda^2 T_{{\mathcal E} \times i^\ast 
{\mathcal E} / \bar V} \vert_{L_V} ,$$
or equivalently,
$$f_V^\ast  {{\omega}_{X / \mathbb P^2}} \check{} 
\stackrel{\sim}{\rightarrow} 
(\sigma, \sigma \circ i)^\ast 
\omega_{{\mathcal E} \times i^\ast {\mathcal E} / {\bar V}} \check{}
 ~ \vert_{L_V} $$


Also, by the description of the map above, after twisting with the 
branched locus, we have:
\begin{equation} \label{e1}
\omega_{{\mathcal E} \times i^\ast {\mathcal E} / \bar V} \vert_{L_V} \cong 
f_V^\ast \omega_{X / \mathbb P^2} (-L_V \cap \tilde D) \cong 
f^\ast [ {\mathcal O}_{\mathbb P^2} (3) (-\frac{1}{2} D)] \vert_{L_V},
\end{equation} 
and the last isomorphism holds because $\tilde D \rightarrow D$ is 2:1. \\

However, 
$\omega_{{\mathcal E} \times i^\ast {\mathcal E} / \bar V} = 
\omega_{{\mathcal E} / \bar V} \otimes \omega_{i^\ast {\mathcal E} / \bar V}$. 
Denote by $\bar {\mathcal M}_{1,1}$ the coarse moduli space of marked elliptic 
curves. The singular locus of a family of elliptic curves maps to the 
boundary of $\bar {\mathcal M}_{1,1}$. If we denote the pull-back of the 
boundary of $\bar {\mathcal M}_{1,1}$ by $\delta$, 
then it is well known  that 
$\omega_{{\mathcal E} / \bar V} = \frac{\delta}{12}$  
(see for example \cite{hm}). \\ 

%and similarly, 
%$ \omega_{i^\ast {\mathcal E} / \bar V} = \frac{\delta}{12}$. \\

Therefore, 
\begin{equation} \label{e2}
(\sigma, \sigma \circ i)^\ast 
\omega_{{\mathcal E} \times i^\ast {\mathcal E} / \bar V} \cong 
{\mathcal O}_{\bar V} \Big ( \frac{G + i^{-1}G}{12} \Big ) = 
{\mathcal O}_{\bar V} \Big ( \frac{f^{-1}(f(G))}{12} \Big ) = 
f^\ast {\mathcal O}_{\mathbb P^2} \Big ( \frac{f(G)}{12} \Big ) 
\end{equation}

When we compare the isomorphisms (\ref{e1}) and (\ref{e2}), we get: 
$$f^\ast {\mathcal O}_{\mathbb P^2} \Big ( \frac{f(G)}{12} \Big ) \vert_{L_V} 
\cong 
f^\ast [ {\mathcal O}_{\mathbb P^2} (3) (-\frac{1}{2} D)] \vert_{L_V},$$
or equivalently,
$$f^\ast {\mathcal O}_{\mathbb P^2} \Big ( \frac{D}{2} + \frac{f(G)}{12} \Big )
\vert_{L_V} \cong 
f^\ast {\mathcal O}_{\mathbb P^2} (3) \vert_{L_V}$$

Comparing the degrees, we obtain the relation: 
$$\frac{1}{2} deg (D) + \frac{1}{12} deg f(G) = 3$$
The degrees of $D$ and $f(G)$ are positive even integers (otherwise we would 
have trivial fibrations), hence there are two possibilities: 
$(deg(D), deg(G)) = (2,24)$ or $(4,12)$. \\

{\bf Case 1:} $(deg(D), deg(f(G))) = (4,12)$ \\

First we consider the case when $D$ is smooth. 
Since $deg(D)=4$, $\bar V$ is a del Pezzo surface
($K_{\bar V} < 0$). We want to show that 
${\mathcal E}$ is rationally connected. Take two general points $p, q \in 
{\mathcal E}$. Then $f(\pi(p)), f(\pi(q))$ are two general points in 
$\mathbb P^2$. Then $f(\pi(p)) \in L_p$, where $L_p$ is a tangent line to 
$D$ at $f(\pi(p))$ and $f(\pi(q)) \in L_q$, where $L_q$ is also a tangent 
line. Let $L_p \cap L_q = \{ r \}$. \\

Take a tangent line $L$ to $D$ and pull it back to $\bar V$: 
$L_{\bar V} \doteq L \times_{\mathbb P^2} \bar V$. Its normalization 
is $\tilde L_{\bar V}$ and ${\mathcal E} \vert_{\tilde L_{\bar V}}$ is an 
elliptic fibration over $\mathbb P^1$ with 12 nodal fibers. 
The surface is rational, because it is deformation equivalent to $\mathbb P^2$ 
blown-up at 9 points in the base locus of a pencil of plane cubics 
(\cite{bpv}, section 5.12). \\

 We can lift the lines $L_p$ and $L_q$ 
to ${\mathcal E}$ and get ${\mathcal E} \vert_{\tilde L_{p, \bar V} }$ and 
 ${\mathcal E} \vert_{\tilde L_{q, \bar V} }$ which are rational surfaces. We 
can connect any two points on a rational surface with a rational curve. 
Connect $p$ to $\tilde r$ and $q$ to $\tilde r$, where 
$\tilde r \in (f\pi)^{-1} (r)$, so $p$ is rationally chain connected to $q$. \\

The point $\tilde r$ is a smooth point of ${\mathcal E}$. In characteristic 0, 
we can smooth the nodal rational curve if we have a general pair $p, q$ and 
if $\tilde r$ is smooth. Therefore, ${\mathcal E}$ is rationally connected. \\

Fix a section $s$ of $i^\ast {\mathcal E} \rightarrow \bar V$. Then we get a 
section $\tilde s$ of ${\mathcal E} \times_{\bar V} i^\ast {\mathcal E} 
\rightarrow {\mathcal E}$. Since we have a finite morphism 
${\mathcal E} \times_{\bar V} i^\ast {\mathcal E} \rightarrow X$, we get a 
finite morphism ${\mathcal E} \rightarrow X$. \\

The image of a finite morphism from a rationally connected variety 
to a hyper-K\"ahler manifold is of dimension at most $\frac{1}{2} dim(X)$,
because we have a $(2,0)$ form on $X$ and there are no holomorphic $(2,0)$ or 
$(1,0)$ forms on a rational variety. However, $dim({\mathcal E}) = 3$ and 
it is bigger than $\frac{1}{2} dim(X) = 2$ - a contradiction. \\

Now assume $D$ is singular. Consider the linear system of lines $l$ 
containing a fixed singular point $r$ of $D$. Consider the associated 
linear system $f^{-1}(l)$. These divisors will typically be all singular 
because $\bar V$ is singular at $f^{-1}(r)$. However, the linear system 
of strict transforms $\widetilde{f^{-1}(l)}$ on $Bl_{f^{-1}(r)}{\bar V}$ 
is a basepoint free pencil of divisors on the normal surface 
$Bl_{f^{-1}(r)}{\bar V}$. Thus, by Bertini's theorem, a general member 
of this pencil is smooth and intersects $G$ transversally. In particular, 
the surface ${\mathcal E} \vert_{\widetilde{f^{-1}(l)}}$ is smooth for 
such a member. \\

Let $L$ be any line passing through a singular point $r \in D$. 
Every component of $f^{-1}(L)$ is a rational curve. Without loss of 
generality we shall consider the irreducible case. The normalization 
$\widetilde{f^{-1}(L)}$ is isomorphic to $\mathbb P^1$. Therefore, 
${\mathcal E} \vert_{\widetilde{f^{-1}(L)}}$ is an elliptic surface 
over $\mathbb P^1$. Since there is the following relation of 
intersection numbers: $f^{-1}(L) \cdot G = L \cdot f(G) = 12$, 
it follows that ${\mathcal E} \vert_{\widetilde{f^{-1}(L)}}$ is a 
rational surface. \\

Consider the 1-parameter family $\mathcal F$ of such rational surfaces 
parametrized by $\mathbb P^1$ (since the line $L$ varies in $\mathbb P^1$). 
The 3-fold $\mathcal F$ is rationally connected inside a 4-dimensional 
hyper-K\"ahler manifold which is impossible. \\

We ruled out the first case completely and the only remaining case is: \\

{\bf Case 2:} $(deg(D), deg(f(G))) = (2,24)$ \\

First we consider the case when $D$ is smooth. 
Then $\bar V \cong \mathbb P^1 \times \mathbb P^1$. \\

Take a tangent line $L$ to the conic and pull it back to $\bar V$: 
$L_{\bar V} \doteq L \times_{\mathbb P^2} \bar V$. Its normalization 
$\tilde L_{\bar V}$ is reducible and consists of two copies of $\mathbb P^1$, 
say $\tilde L_{\bar V, 1}$ and $\tilde L_{\bar V, 2}$. \\


{\bf Case 2.1:} \\
${\mathcal E} \vert_{\tilde L_{\bar V, k} }$ is an elliptic fibration over 
$\mathbb P^1$ with 12 nodal fibers, $k = 1,2$. 
Then we repeat the same argument as in Case 1 in order to exclude this case. \\

{\bf Case 2.2:} \\
${\mathcal E} \vert_{\tilde L_{\bar V, 1} }$ is an elliptic fibration over 
$\mathbb P^1$ with 24 nodal fibers and ${\mathcal E} 
\vert_{\tilde L_{\bar V, 2}}$ 
is an elliptic fibration over $\mathbb P^1$ with no singular fibers. 
Then  ${\mathcal E} \vert_{\tilde L_{\bar V, 2}}$ is the trivial fibration and 
therefore, ${\mathcal E}$ is the pull back of an elliptic fibration on 
$\mathbb P^1$ through the projection on this factor. And, since the elliptic 
fibration on $\mathbb P^1$ has 24 nodal fibers, it is an elliptic K3 surface 
$S \rightarrow \mathbb P^1$. \\

After considering all the cases, we see that: 
$${\mathcal E} \times_{(\mathbb P^1 \times \mathbb P^1)} i^\ast {\mathcal E} 
 = (S \times \mathbb P^1) \times_{(\mathbb P^1 \times \mathbb P^1)}
( \mathbb P^1 \times S).$$
We want to prove that 
$(S \times \mathbb P^1) \times_{(\mathbb P^1 \times \mathbb P^1)}
( \mathbb P^1 \times S) \cong S \times S$. Indeed, we have the following 
commutative fiber diagram:

$$\begin{array}{ccc}
S \times S & \longrightarrow & \mathbb P^1 \times S \\ 
\downarrow  &  \searrow    & \downarrow \\
S \times \mathbb P^1 & \longrightarrow & \mathbb P^1 \times \mathbb P^1 \\
\end{array}$$

Therefore, 
$${\mathcal E} \times_{(\mathbb P^1 \times \mathbb P^1)} i^\ast {\mathcal E} 
 = (S \times \mathbb P^1) \times_{(\mathbb P^1 \times \mathbb P^1)}
( \mathbb P^1 \times S) \cong S \times S.$$


But $X$ is birational to the desingularization of 
${\mathcal E} \times_{(\mathbb P^1 \times \mathbb P^1)} i^\ast {\mathcal E} / 
{\tilde i}$ (by (\ref{0})) which is birational to 
$S \times S / {\mathbb Z_2}$. Therefore, $X$ is birational 
to the desingularization of $S \times S / {\mathbb Z_2}$ which is 
$\text{Hilb}^2(S)$
by Fogarty's theorem (Theorem \ref{fog}.) \\

Now consider the case when $D$ is singular. Since $\bar V$ is normal, 
$D$ cannot be a double line. Therefore, $D$ is the union of two lines and 
$\bar V$ is a singular quadric cone. \\ 

Let $L$ be a line in ${\mathbb P}^2$ passing through the node 
$r \in D$. Then $f^{-1}(L)$ is the union of two lines $L_1$ and $L_2$ 
each one of which is a line in the cone $\bar V$ passing through its 
vertex. On a quadric cone all lines are algebracially equivalent, and in 
particular $L_1$ and $L_2$ are algebraically equivalent. Since 
$deg(G)=24$, $L_1 \cdot G = L_2 \cdot G = 12$. 
Therefore, the surface ${\mathcal E} \vert_{L_i}$ is rational (for $i=1,2$). 
We constructed a rationally parametrized 1-parameter family of rational 
surfaces which is impossible to exist in a 4-dimensional hyper-K\"ahler 
manifold. With this we finish the proof of our main theorem. 

\end{proof}


{\bf Acknowledgments.} The author is thankful to Prof. G. Tian for suggesting 
the problem and for the helpful discussions. She also thanks the Institute 
of Advanced Study for the opportunity to be there during the 2006-07 
academic year. 


\begin{thebibliography}{99}

\bibitem{bpv} Barth, W.; Peters, C.; Van de Ven, A., 
        {\it Compact complex surfaces,} 
        EMG {\bf 3}, Springer-Verlag, Berlin, 1984.

\bibitem{Beauville1} Beauville, A., 
        {\it Vari\'et\'es K\"ahleriennes dont la
        premi\`ere classe de Chern est nulle,} 
        J. Diff. Geom. {\bf 18} (1983) 755 - 782. 

\bibitem{neron}  Bosch, S.; L\"utkebohmert, W.; Raynaud, M., 
        {\it N\'eron models,}
        Springer-Verlag, Berlin, 1990.

\bibitem{f} Fogarty, J.,
        {\it Algebraic families on an algebraic surface,}
        Amer. J. Math. {\bf 90} (1968) 511 - 521.

% \bibitem{Fujiki2} Fujiki, A., 
%        {\it On the de Rham Cohomology Group of a Compact
%        K\"ahler Symplectic Manifold, } 
%        Adv. Stud. Pure Math. {\bf 10} (1987) 105 - 165,

\bibitem{g} Grothendieck, A., 
        {\it \'El\'ements de G\'eom\'etrie Alg\'ebrique II,} 
        I.H.E.S. Publ. Math. {\bf 8}, 1961.

\bibitem{hm} Harris, J., Morrison, I.,
         {\it Moduli of curves,}
         G.T.M. {\bf 187}, Springer-Verlag, New York, 1998.

\bibitem{har}  Hartshorne, R.,
        {\it Algebraic geometry,}
        Springer-Verlag, New York, 1983.

\bibitem{h} Huybrechts, D., 
        {\it Compact hyper-K\"ahler manifolds: basic results,}
        Invent. Math. {\bf 135} (1999) 63 - 113.

\bibitem{hw} Hwang, J.-M., 
        {\it Base manifolds for fibrations of projective irreducible 
        symplectic manifolds,} 
        Invent. Math. {\bf 174} (2008) 625 - 644.

% \bibitem{kkmsd} Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.,
%        {\it Toroidal embeddings}, LNM {\bf 339}, Springer-Verlag (1973),

% \bibitem{k} Kulikov, V, {\it Degenerations of $K3$ surfaces and Enriques 
%        surfaces,} 
%        Izv. Akad. Nauk SSSR Ser. Mat. {\bf 41} (1977) 1008 - 1042,

\bibitem{markushevich96} Markushevich D.,
        {\it Lagrangian families of Jacobians of genus 2 curves},
        J. Math. Sci. {\bf 82} (1996) 3268-3284.


\bibitem{m} Matsushita, D, {\it On fibre space structures of a
         projective  irreducible 
         symplectic manifold,}  
         Topology  {\bf 38}  (1999) 79 - 83.

% \bibitem{n} Nakajima, H., 
%        {\it Lectures on Hilbert schemes of points on surfaces,}
%        AMS, Providence (1991),

\bibitem{og1} O'Grady, K., 
        {\it Desingularized moduli spaces of sheaves on a K3,} 
        J. Reine Angew. Math. {\bf 512} (1999) 49 - 117. 

\bibitem{og2} O'Grady, K., 
        {\it A new six-dimensional irreducible symplectic variety,} 
        J. Algebr. Geom. {\bf 12} (2003) 435 - 505. 

% \bibitem{book} Parshin, A; Shafarevich, I.R.(eds.),
%        {\it Algebraic geometry III: Complex algebraic varieties, algebraic 
%        curves and their Jacobians,} 
%        Encyclopaedia of Mathematical Sciences {\bf 36} Berlin: Springer 
%        (1998),

% \bibitem{p} Persson, U, 
%        {\it On degenerations of algebraic surfaces,} 
%        Mem. Amer. Math. Soc. {\bf 11} (1977) no. 189, 

% \bibitem{Salamon2} Salamon, S., 
%        {\it Riemannian Geometry and Holonomy Groups,}
%        Pitman Research Notes in Math. {\bf 201} 
%        Longman Scientific New York (1989),

% \bibitem{s} Sawon, J,
%         {\it Abelian fibred holomorphic symplectic manifolds,}
%         Turkish J. Math. {\bf 27} (2003) 197 - 230,

% \bibitem{Tian} Tian, G., 
%         {\it Smoothness of the universal deformation space of compact 
%         Calabi-Yau manifolds and its Petersson-Weil metric, }
%         In: Mathematical Aspects of String theory, 
%         World Scientific (1987) 629 - 646,

% \bibitem{ueno1} Ueno, K., 
%        {\it On fibre spaces of normally polarized abelian varieties of 
%        dimension $2$. I. Singular fibres of the first kind,}
%        J. Fac. Sci. Univ. Tokyo Sect. IA Math. {\bf 18}  (1971) 37 - 95,

% \bibitem{ueno2} Ueno, K., 
%        {\it On fibre spaces of normally polarized abelian varieties of 
%        dimension $2$. II. Singular fibres of the first kind,}
%        J. Fac. Sci. Univ. Tokyo Sect. IA Math. {\bf 19}  (1972) 163 - 199,

% \bibitem{Varouchas} Varouchas, J., 
%        {\it K\"ahler Spaces and Proper Open Morphisms,} 
%        Math. Ann. {\bf 283} (1989) 13 - 52.


\end{thebibliography}

\end{document}



1208.3778.extracted_bib
\begin{thebibliography}{99}

\bibitem{bpv} Barth, W.; Peters, C.; Van de Ven, A., 
        {\it Compact complex surfaces,} 
        EMG {\bf 3}, Springer-Verlag, Berlin, 1984.

\bibitem{Beauville1} Beauville, A., 
        {\it Vari\'et\'es K\"ahleriennes dont la
        premi\`ere classe de Chern est nulle,} 
        J. Diff. Geom. {\bf 18} (1983) 755 - 782. 

\bibitem{neron}  Bosch, S.; L\"utkebohmert, W.; Raynaud, M., 
        {\it N\'eron models,}
        Springer-Verlag, Berlin, 1990.

\bibitem{f} Fogarty, J.,
        {\it Algebraic families on an algebraic surface,}
        Amer. J. Math. {\bf 90} (1968) 511 - 521.

% \bibitem{Fujiki2} Fujiki, A., 
%        {\it On the de Rham Cohomology Group of a Compact
%        K\"ahler Symplectic Manifold, } 
%        Adv. Stud. Pure Math. {\bf 10} (1987) 105 - 165,

\bibitem{g} Grothendieck, A., 
        {\it \'El\'ements de G\'eom\'etrie Alg\'ebrique II,} 
        I.H.E.S. Publ. Math. {\bf 8}, 1961.

\bibitem{hm} Harris, J., Morrison, I.,
         {\it Moduli of curves,}
         G.T.M. {\bf 187}, Springer-Verlag, New York, 1998.

\bibitem{har}  Hartshorne, R.,
        {\it Algebraic geometry,}
        Springer-Verlag, New York, 1983.

\bibitem{h} Huybrechts, D., 
        {\it Compact hyper-K\"ahler manifolds: basic results,}
        Invent. Math. {\bf 135} (1999) 63 - 113.

\bibitem{hw} Hwang, J.-M., 
        {\it Base manifolds for fibrations of projective irreducible 
        symplectic manifolds,} 
        Invent. Math. {\bf 174} (2008) 625 - 644.

% \bibitem{kkmsd} Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.,
%        {\it Toroidal embeddings}, LNM {\bf 339}, Springer-Verlag (1973),

% \bibitem{k} Kulikov, V, {\it Degenerations of $K3$ surfaces and Enriques 
%        surfaces,} 
%        Izv. Akad. Nauk SSSR Ser. Mat. {\bf 41} (1977) 1008 - 1042,

\bibitem{markushevich96} Markushevich D.,
        {\it Lagrangian families of Jacobians of genus 2 curves},
        J. Math. Sci. {\bf 82} (1996) 3268-3284.


\bibitem{m} Matsushita, D, {\it On fibre space structures of a
         projective  irreducible 
         symplectic manifold,}  
         Topology  {\bf 38}  (1999) 79 - 83.

% \bibitem{n} Nakajima, H., 
%        {\it Lectures on Hilbert schemes of points on surfaces,}
%        AMS, Providence (1991),

\bibitem{og1} O'Grady, K., 
        {\it Desingularized moduli spaces of sheaves on a K3,} 
        J. Reine Angew. Math. {\bf 512} (1999) 49 - 117. 

\bibitem{og2} O'Grady, K., 
        {\it A new six-dimensional irreducible symplectic variety,} 
        J. Algebr. Geom. {\bf 12} (2003) 435 - 505. 

% \bibitem{book} Parshin, A; Shafarevich, I.R.(eds.),
%        {\it Algebraic geometry III: Complex algebraic varieties, algebraic 
%        curves and their Jacobians,} 
%        Encyclopaedia of Mathematical Sciences {\bf 36} Berlin: Springer 
%        (1998),

% \bibitem{p} Persson, U, 
%        {\it On degenerations of algebraic surfaces,} 
%        Mem. Amer. Math. Soc. {\bf 11} (1977) no. 189, 

% \bibitem{Salamon2} Salamon, S., 
%        {\it Riemannian Geometry and Holonomy Groups,}
%        Pitman Research Notes in Math. {\bf 201} 
%        Longman Scientific New York (1989),

% \bibitem{s} Sawon, J,
%         {\it Abelian fibred holomorphic symplectic manifolds,}
%         Turkish J. Math. {\bf 27} (2003) 197 - 230,

% \bibitem{Tian} Tian, G., 
%         {\it Smoothness of the universal deformation space of compact 
%         Calabi-Yau manifolds and its Petersson-Weil metric, }
%         In: Mathematical Aspects of String theory, 
%         World Scientific (1987) 629 - 646,

% \bibitem{ueno1} Ueno, K., 
%        {\it On fibre spaces of normally polarized abelian varieties of 
%        dimension $2$. I. Singular fibres of the first kind,}
%        J. Fac. Sci. Univ. Tokyo Sect. IA Math. {\bf 18}  (1971) 37 - 95,

% \bibitem{ueno2} Ueno, K., 
%        {\it On fibre spaces of normally polarized abelian varieties of 
%        dimension $2$. II. Singular fibres of the first kind,}
%        J. Fac. Sci. Univ. Tokyo Sect. IA Math. {\bf 19}  (1972) 163 - 199,

% \bibitem{Varouchas} Varouchas, J., 
%        {\it K\"ahler Spaces and Proper Open Morphisms,} 
%        Math. Ann. {\bf 283} (1989) 13 - 52.


\end{thebibliography}