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\controldates{8-SEP-1998,8-SEP-1998,8-SEP-1998,8-SEP-1998}
 
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\begin{document}

\title{The Nash conjecture for threefolds}

%    Information for the author
\author{J\'anos Koll\'ar}
%    Address of record for the research reported here
\address{University of Utah, Salt Lake City, UT 84112}
\email{kollar@math.utah.edu} 


%    General info
\subjclass{Primary 14P25}


\issueinfo{4}{10}{}{1998}
\dateposted{September 15, 1998}
\pagespan{63}{73}
\PII{S 1079-6762(98)00049-3}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}

\date{July 17, 1998}

\commby{Robert Lazarsfeld}


%\keywords{}

%\noindent {\it AMS 1991 subject classifications}:  

\begin{abstract}
Nash conjectured in 1952 that every compact
differentiable manifold can be realized as the set of real
points of a real algebraic variety which is birational
to projective space. This paper announces the negative
solution of this conjecture in dimension 3.
The proof shows that in fact very few 3-manifolds
can be realized this way. 
\end{abstract}

\maketitle


%\tableofcontents


\section{Introduction}



In real algebraic   geometry, one of the main directions of
investigation  is the   topological study of the set of  real solutions
of algebraic equations. The first general theorem was proved in
\cite{Nash52} and later improved by \cite{Tognoli}. 
Their result says that every compact differentiable manifold is
algebraic:

\begin{thm}[\cite{Nash52,Tognoli}]\label{nash.thm}
  Let $M^n$ be a 
compact differentiable manifold. Then there are
real polynomials $f_i(x_1,\dots,x_N)$ such that their common zero set
%$$
\[
V(f_1,\dots,f_s)({\mathbb R}):=\{ {\bold x}\in {\mathbb R}^N: f_i({\bold x})=0\
 \forall
i\}\subset
{\mathbb R}^N
%$$
\]
is diffeomorphic to $M^n$.
\end{thm}

Nash also considered further refinements of this result. To state
these,   we need to fix basic concepts, since the terminology in real
algebraic geometry is, unfortunately, not at all uniform.



\begin{defn}\label{what.is.rav?}
 By a \textit{real algebraic variety} I mean a 
variety given by real equations, as defined in most algebraic
geometry books (see, for instance, 
\cite{Shafarevich72,Hartshorne77}). This is  different from the
definition  frequently used in real algebraic geometry  which
essentially considers only the germ of $X$ along its real points
(cf.\  \cite{BCR87}). 


If $X$ is a real algebraic
variety, then      $X({\mathbb R})$ denotes the set of real
points of $X$   as a topological space
and $X({\mathbb C})$ denotes the set of complex points as a complex space. 
For all practical purposes we can identify $X$ with the pair
($X({\mathbb C})$, complex conjugation).
\end{defn}

The varieties $X=V(f_1,\dots,f_s)$ constructed by 
Nash do not seem to have any special properties. Nash considered the
question if (\ref{nash.thm}) remains true for real algebraic varieties
which are rational: 


\begin{defn} Let $X, Y$ be real algebraic varieties. We say that
$X$ and $Y$ are \textit{birational} if there is a birational map
$\phi:X\map Y$ which is defined over ${\mathbb R}$.  If $X$ is birational to
$\bp^n$, then 
$X$ is    called
\textit{rational over ${\mathbb R} $} or just \textit{rational}. (Note 
that in many papers
$X$ is called rational if
$X({\mathbb C})$ is birational to $\bp^n$.)
\end{defn}

\begin{conj}[{\cite[p. 421]{Nash52}}]  Let $M^n$ be a 
compact differentiable manifold. Then there 
is a smooth real algebraic variety  $X^n$ such that $X$
is birational to $\bp^n$ and
$X({\mathbb R})$ 
is diffeomorphic to $M^n$.
\end{conj}


Unbeknownst to Nash, this question has been settled
for surfaces much earlier.

\begin{thm}[\cite{Comessatti14}]\label{comm.thm}
  Let $S$ be a 
smooth real algebraic surface. Assume that $S$ is birational to $\bp^2$
and $S({\mathbb R})$ is orientable.

Then 
$S({\mathbb R})$ 
is either a sphere or  a torus.
\end{thm}

In higher dimensions the conjecture of Nash remained open, and some
partial results seemed to support the hope that it may hold
in dimensions 3 and up. \cite{ben-mar} showed that for every
3-manifold $M^3$ there
is a {\em singular} real algebraic variety  $X^3$ such that $X$
is birational to $\bp^3$ and
$X({\mathbb R})$ 
is homeomorphic to $M^3$. \cite{akb-king} and \cite{mikh}
showed that a weaker variant, the so-called ``topological Nash
conjecture" is true.
%\medskip

The aim of this paper is to announce a solution to
the Nash conjecture in dimension 3. The result says that
the Nash conjecture again fails completely. (The relevant  basic
concepts of 3-manifold topology are recalled in (\ref{lens.defn}).
See \cite{Hempel76,Rolfsen76,Scott83} for more details.)


\begin{mainthm}\label{main.thm}
 Let $X$ be a smooth, projective,  real algebraic
3-fold. Assume that $X$ is birational to $\bp^3$
and that $X({\mathbb R})$ is orientable. Then $X({\mathbb R})$ is
diffeomorphic  to  a 3-manifold
%$$
\[
M\# a{\mathbb R}\bp^3\# b (S^1\times S^2)\qtq{for some $a,b\geq 0$,}
%$$
\]
where $M$ is one of the following:
\begin{enumerate}
\item connected sum of lens spaces,
\item Seifert fibered,
\item $S^1\times S^1$-bundle over $S^1$ or a $\bz_2$-quotient of such.
\item finitely many other possibilities.
\end{enumerate}
\end{mainthm}

The proof  establishes a tight connection between certain 
algebraic properties of $X({\mathbb C})$ and 
  geometric structures of $X({\mathbb R})$.  In some cases
 such a relationship has not been proved, and this accounts for the
finitely many unknown cases. I believe, however, that
there are no exceptions:

\begin{conj}\label{main.thm.conj} The cases \eqref{main.thm}, 3--4
do not occur.
\end{conj} 

\begin{rem}\label{main.thm.compl}
 The assumption that 
$X$ be birational to $\bp^3$ can be weakened considerably. Namely, 
(\ref{main.thm}) also holds if we assume the following equivalent
conditions
\begin{enumerate}
\item $X({\mathbb C})$ is \textit{uniruled} (that is, covered by rational
curves),
\item $X$ has \textit{Kodaira dimension $-\infty$}
(that is, $H^0(X,\mathcal{O} _X(mK_{X}))=0$ for every $m\geq 1$).
\end{enumerate}
In this case $X({\mathbb R})$ may have several connected components
and each  satisfies the conclusions of 
(\ref{main.thm}).
I do not know what happens if some components of $X({\mathbb R})$ are orientable
and some are not.
\end{rem}



The orientability of $X({\mathbb R})$ is not a crucial point for
(\ref{main.thm}). The proof proceeds by a  reduction argument.
At each step we either get a nice description or we exhibit
a special surface in $X({\mathbb R})$. As far as I can
tell, it is only an accident that all  these special surfaces
imply nonorientability. They also imply that $X({\mathbb R})$ is not hyperbolic,
and we obtain the following. (Again I conjecture that there are no
 exceptions.)


\begin{thm}\label{main.hyp.thm}
There are only finitely many hyperbolic $3$-manifolds
(orientable or not)
among the  $X({\mathbb R})$ where
 $X$ is a smooth, projective, real algebraic
$3$-fold such that $X({\mathbb C})$ is  uniruled.
\end{thm}

A similar result was obtained in all dimensions by Viterbo, using  
stronger conditions on rational curves.


\begin{thm}[\cite{viterbo}]\label{vit.hyp.thm}
Let $X$ be a smooth, projective, real algebraic variety
of dimension $n\geq 3$. 
Assume that $H_2(X({\mathbb C}),\bz)\cong \bz$ and that 
$X({\mathbb C})$ is covered by rational curves $C_{\lambda}$ such that
$[C_{\lambda}]\in H_2(X({\mathbb C}),\bz)$ is a generator. 

Then $X({\mathbb R})$ does not carry any metric with negative sectional
curvature.
\end{thm}

\begin{defn}\label{lens.defn}{\ } For relatively prime
$04}:$] \quad $x^2+y^2z+ayt^r+h_{\geq s}(z,t)$,\quad 
($a\in {\mathbb R}$, $r\geq 3$, $s\geq 4$). 
\item[$cE_6:$] \quad $x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq
4}(z,t)$,\quad 
($h_4\neq 0$).
\item[$cE_7:$] \quad $x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq
5}(z,t)$,\quad 
($g_3\neq 0$).
\item[$cE_8:$] \quad $x^2+y^3+yg_{\geq 4}(z,t)+h_{\geq
5}(z,t)$,\quad 
($h_5\neq 0$).
\end{enumerate}} 
\end{thm}

\begin{exmp}\label{top.norm}
 If $0\in X$ is an isolated $3$-dimensional singularity,
then $X({\mathbb R})$ is a cone  over a surface near $0$. Consider for instance
the singularities $X_f:=(x^2+ y^2+f(z,t)=0)$. Assume that $f(z,t)$ is
negative on $m$  connected domains    near the
origin. Then $X_f({\mathbb R})$ is a point for $m=0$, a cone over  a torus for
$m=1$ and a cone over  $m$ disjoint spheres for
$m>1$.  

If $M$ is a $3$-complex which is locally always like a cone over disjoint
spheres, then its \textit{topological normalization} $\overline M$
is a manifold and $M$ is obtained from $\overline M$ by pinching
together finite collections of points.
\end{exmp}




Quotient singularities frequently have unexpected real forms.
Consider for instance the $\bz_4$-action on ${\mathbb C}^2$ given by
$(x,y)\mapsto (ix,-iy)$. There is an isomorphism 
%$$
\[
{\mathbb C}^2/\bz_4\cong
(uv-w^4=0)\subset {\mathbb C}^3 \qtq{given by} u,v,w\mapsto x^4,y^4,xy.
%$$
\]
The singularity $(uv-w^4=0)$ has 3 different  real forms:
%$$
\[
uv-w^4=0,\quad u^2+v^2-w^4=0 \qtq{and} u^2+v^2+w^4=0.
%$$
\]
For $3$-dimensional terminal singularities this does not happen.
As Reid pointed out, this is closely related to the fact that
the canonical divisor class generates the torsion subgroup of the local
Picard group. We obtain the following classification.
($\mbox{Index}=n$  means that we take quotient by $\bz_n$
and $\mbox{weights}=(a,b,c,d)$ means that the action is
$(x,y,z,t)\mapsto (\epsilon^ax,\epsilon^by,\epsilon^cz,\epsilon^dt)$
where $\epsilon$ is a primitive $n$th root of unity.
Note that in each case the action itself is not defined over ${\mathbb R}$ but
the ring of invariants  has a ${\mathbb C}$-basis consisting of
monomials, which gives  a real structure.)

\begin{thm}[{\cite[3.4]{rat1}}]\label{hind.term.thm}
 Let  $0\in X$ be a $3$-fold terminal
nonhypersurface singularity over
${\mathbb R}$. Then $0\in X$ is isomorphic over ${\mathbb R}$ to a singularity
described by the following list:
%$$
\[
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{|c|l|c|l|l|}
\hline name &\qquad equation  & index  & \quad weights &
condition\\
\hline $cA/2$& $x^2\pm y^2+f(z,t)$ & $2$ & $(1,1,1,0)$&\\
\hline $cA/n$& $xy+f(z,t)$ & $n\geq 3$ & $(r,-r,1,0)$&
$(n,r)=1$\\
\hline $cAx/2$&  $ x^2\pm y^2 +f_{\geq 4}(z,t)$  & $2$ & $(0,1,1,1)$&\\
\hline $cAx/4$ & $ x^2\pm y^2 +f_{\geq 2}(z,t)$ & $4$ & $(1,3,1,2)$&
\\
\hline $cD/2$&$x^2+f_{\geq 3}(y,z,t)$ & $2$ &$(1,0,1,1)$&\\
\hline $cD/3$&$x^2+f_{\geq 3}(y,z,t)$ &$3$ &$(0,2,1,1)$&
$f_3(1,0,0)\neq 0$\\
\hline $cE/2$&$x^2+y^3+f_{\geq 4}(y,z,t)$ &$2$
&$(1,0,1,1)$&\\
\hline
\end{tabular}
%$$
\]
\end{thm} 


\section{The minimal model program over ${\mathbb R}$}


In order to understand how $X({\mathbb R})$ is obtained from $X^*({\mathbb R})$
it is necessary to study the intermediate steps of the MMP.
These steps are, unfortunately, not  well understood even over ${\mathbb C}$.
Thus we change point of view somewhat, and  try to describe how a
given $Y$ can be the target of one step  $Y_1\to Y$.
Again we run into problems, and the answer is not known even
if $Y$ is smooth.

We thus try to describe only the ``simplest" steps
$Y_1\to Y$. To do this, we need to develop a measure of how
to compare the exceptional divisors of different birational maps
$Y_i\to Y$. 

\subsection{The hierarchy of exceptional divisors}
Let $S$ be a smooth surface and $f_1:S_1\to S$ the blowup of a smooth
point with exceptional curve $E_1\subset S_1$.
Next blow up a point on $E_1$ to obtain $f_2:S_2\to S_1$
with exceptional curve  $E_2$. The composite $f_1\circ f_2:S_2\to S$
has two exceptional curves which I denote by $E_1$ and $E_2$ by a
slight abuse of notation. One can easily compute that the Jacobian of 
$f_1\circ f_2$ vanishes along $E_1$ to order 1 and along $E_2$ to order
2. If we perform further blowups, we obtain curves with
higher and higher order vanishing of the Jacobian along them.

Thus the order of vanishing of the Jacobian establishes a hierarchy of 
all exceptional curves, with  the smallest order of vanishing
corresponding to curves
that appear after just one blowup.

A similar argument applies to blowups of smooth varieties in any
dimension. Terminal singularities are essentially defined to
make a similar hierarchy possible, but there are two   problems.
First, the correct analog of the order of vanishing
of the Jacobian  of $f:Z\to Y$  along an exceptional divisor
$E\subset Z$  is only a positive rational
number. It is  called the \textit{disrepancy} and it is denoted by
$a(E,Y)$ (see \cite[3.3]{rat2} or \cite[2.25]{KoMo98} for more details).
Second, and this is  more serious,
if we have $f_2:Y_2\to Y_1$ and $f_1:Y_1\to Y$ with exceptional divisors
$E_2$ and
$E_1$, then we can only say that 
%$$
\[
a(E_2, Y)\geq \frac{1}{m}(a(E_1,Y)+1)
%$$
\]
where $m$ is the index of $Y_1$ at $f_2(E_2)$. 
If  $Y_1$ has index 1 at $f_2(E_2)$, then we are in good shape, but not
in general.

\subsection{Gateways}
Assume now that everything is over ${\mathbb R}$ and that $Y_1$ has index 1
along $Y_1({\mathbb R})$. If we know that $E_2({\mathbb R})$ is 
Zariski dense in $E_2({\mathbb C})$,
then $f_2(E_2)$ is a real point of  $Y_1$. 

This observation leads us to study those steps $X_i\to X_{i+1}$ of the
MMP over
${\mathbb R}$ such that $K_{X_i}$ is Cartier along $X_i({\mathbb R})$, 
but this fails for
$X_{i+1}$. These are called \textit{gateway contractions} in \cite{rat2},
since they are the gateways through which the MMP can leave the
category of ``nice" varieties.

A lengthy case by case analysis produces a list of such gateways
\cite[8.2]{rat2}. A   study of each of these gateways is
relatively straightforward, and  in most cases
there is a 
surface of nonnegative Euler
characteristic in $X_i({\mathbb R})$.
The remaining cases are rather mild and
we obtain the following.


\begin{thm}[{\cite[1.13]{rat2}}]\label{int.mmp.steps}
 Let $X$ be a smooth, projective, real algebraic $3$-fold such that 
$X({\mathbb R})$   is orientable.
Let $f_i:X_i\map X_{i+1}$ be any of the intermediate steps of the MMP
over
${\mathbb R}$ starting with $X$.   Then
the topological normalization $\overline{X_i({\mathbb R})}$ \eqref{top.norm}
is a $3$-manifold and the
following is a complete
list of possibilities for $f_i$:
\begin{enumerate}

\item (${\mathbb R}$-trivial) $f_i$ is an isomorphism in a (Zariski)
neighborhood of the set of real points.

\item (${\mathbb R}$-small) $f_i:X_i({\mathbb R})\to X_{i+1}({\mathbb R})$ 
collapses a $1$-complex
to   points and there are small perturbations $\tilde f_i$ of
$f_i$ such  that 
$\tilde f_i: \overline{X_i({\mathbb R})}\to \overline{X_{i+1}({\mathbb R})}$
is a homeomorphism.

\item (smooth point blowup)  $f_i$ is the inverse of the blowup of a
smooth point $P\in X_{i+1}({\mathbb R})$. 


\item (singular point blowup) $f_i$ is the inverse of a  weighted  
blowup of a singular point $P\in X_{i+1}({\mathbb R})$.  Up to real analytic
equivalence near $P$, there are two cases:


\begin{enumerate}
\item   $X_{i+1}\cong (x^2+y^2+g_{\geq 2m}(z,t)=0)$ where
$g_{2m}(z,t)\neq 0$,
$m\geq 1$ and $X_i\cong B_{(m,m,1,1)}X_{i+1}$.


\item 
 $X_{i+1}\cong (x^2+y^2+g_{\geq 2m+1}(z,t)=0)$ where  $m\geq 1$,
$z^{2m+1}\in g$ and 
$z^it^j\not\in g$ for  $2i+j< 4m+2$;
$X_i\cong B_{(2m+1,2m+1,2,1)}X_{i+1}$. 
\end{enumerate}
\end{enumerate}
\end{thm}

By repeatedly applying (\ref{int.mmp.steps}) to each step of
the MMP, we can compare $X^*({\mathbb R})$ and $X({\mathbb R})$.
This   shows that it is sufficient to
prove (\ref{main.thm}) for $X^*$.


\begin{thm}[{\cite[1.2]{rat2}}]\label{int.orient.thm}
 Let $X$ be a smooth, projective, real algebraic $3$-fold and $X^*$
the result of the MMP over ${\mathbb R}$. Assume that  $X({\mathbb R})$ is orientable.

Then   $K_{X^*}$ is Cartier along $X^*({\mathbb R})$, $\overline{X^*({\mathbb R})}$ 
is a topological $3$-manifold and
 $X({\mathbb R})$  can be obtained from
$\overline{X^*({\mathbb R})}$ by repeated application of the following
operations: 
\begin{enumerate}
\setcounter{enumi}{-1}
\item throwing away all isolated points of $\overline{X^*({\mathbb R})}$,
\item taking connected sums of connected components,
\item taking connected sum with $S^1\times S^2$,
\item taking connected sum with ${\mathbb R}\bp^3$.
\end{enumerate}
\end{thm}

The operations
\eqref{int.orient.thm}, 2--3 appear even if $X^*$ is smooth.

\begin{exmp}\label{connsum.exmp}
 Let $X$ be a smooth $3$-fold over ${\mathbb R}$ and $0\in X({\mathbb R})$ a
real point. Set $Y=B_0X$. Then
$Y({\mathbb R})\sim X({\mathbb R})\ \#\  {\mathbb R}\bp^3$. 

 Let $X$ be a
smooth $3$-fold over ${\mathbb R}$ and 
$D\subset X$  a real curve which has a unique real point
$\{0\}= D({\mathbb R})$. Assume furthermore that  near $0$ the curve is given by
equations
$(z=x^2+y^2=0)$. Let
$Y_1$ be the blowup of $D$ in $X$. $Y_1$ has a unique singular point;
let
$Y$ denote the result of blowing it up. It is not hard to see that $Y$
is smooth and 
$Y({\mathbb R})\sim X({\mathbb R})\ \#\  (S^1\times S^2)$. 
\end{exmp}




\section{Conic and Del Pezzo fibrations}


If $X^*$ is either a conic or a Del Pezzo fibration, then 
we obtain a map of $X^*({\mathbb R})$ to a 2- or 1-dimensional
space. 
One
can try to use this map to get a geometric description
of $X^*({\mathbb R})$. It turns out that this is indeed possible, but the
two cases require  different methods.
We start with the conic bundle case.


Let $Y$ be a real algebraic $3$-fold and $f:Y\to S$ a conic fibration.
Then $Y({\mathbb R})\to S({\mathbb R})$ is a map of a $3$-complex to a 
$2$-complex
such that every fiber has dimension 1. Moreover,  
$Y({\mathbb R})\to S({\mathbb R})$ is a circle bundle over the complement of a
$1$-complex. It seems quite promising that $Y({\mathbb R})\to S({\mathbb R})$
is a Seifert fibration. Nonetheless, one has to be more careful, as the
following example shows.


\begin{exmp} It is known that every $3$-manifold $M$ can be written
as a degree 3 branched covering of $S^3$, branched along a
knot $C\subset S^3$ (cf.\ \cite[10.G]{Rolfsen76}). Let $S^3\to S^2$ be
a  Hopf fibration which is in general position with respect to $C$. The
composite  can be factored as $M\to F\to S^2$ where $F$ is a  2-complex
and $M\to F$ has connected fibers.
$M\to F$ has all the properties enumerated above, but it is not a
Seifert fibration in general.
\end{exmp}

From this we see that in order to understand the topology of a conic
fibration, one has to study the special fibers in detail.

Let 
$Y$ be a real projective $3$-fold   with
terminal singularities such that $K_Y$ is Cartier along $Y({\mathbb R})$ and
$\overline{Y({\mathbb R})}$ is an orientable   $3$-manifold.
Let $f:Y\to S$ be a  rational curve fibration
such that $-K_Y$ is
$f$-ample.
An elementary topological argument shows that one can factor
$\overline{Y({\mathbb R})}\to S({\mathbb R})$ as
%$$
\[
\overline{Y({\mathbb R})}\stackrel{\tilde f}{\to} F
\to S({\mathbb R})
%$$
\]
where $F$ is a 2-manifold with boundary, $\tilde f$ is surjective and
has connected fibers. 
It is not hard to see that $\tilde f$ is an $S^1$-bundle over $\inter
F$, except at finitely many points $P_1,\dots,P_s$. At the boundary
points of
$F$  one can choose local coordinates such that $\tilde f$ is given as
$(x,y,z)\mapsto (x^2+y^2,z)$.
It remains to  establish a connection
between the algebraic  $f$ and the topological $\tilde f$ near the
special points $P_i$.

\begin{thm}[{\cite[1.9]{rat3}}]\label{intro.local.seif.fib}
Notation as above.
For each $P_i$ there is   $m=m(P_i)\geq 2$ such that
\begin{enumerate}
\item  $\tilde f: \overline{Y({\mathbb R})}\to F$  has a
Seifert fiber of multiplicty
$m$ above $P_i$, and
\item  near $P_i$,  $f$ is real analytically isomorphic to
%$$
\[
\left(\bp^2_{x:y:z}\times \ab^2_{s,t}\right)/\bz_m\supset
(x^2+y^2-z^2=0)/\bz_m
\to \ab^2_{s,t}/\bz_m,
%$$
\]
where $\bz_m$ acts by rotation with angle $2\pi/m$ on $(s,t)$, it fixes
$z$ and acts by rotation with angle $2a\pi/m$ on $(x,y)$  for some
$(a,m)=1$.
\end{enumerate}
\end{thm}


If $F$ has no boundary, then $\tilde f: \overline{Y({\mathbb R})}\to F$ is a 
Seifert fibration. If $F$ does have boundary, then it is easy to see that
$\overline{Y({\mathbb R})}$ is a connected sum of lens spaces.
This proves (\ref{main.thm}) in the conic fibration case.
%\medskip



Next we study Del Pezzo fibrations. Here we have
 a morphism $f:Y\to C$ such that  $Y$ has only isolated singularities, 
$-K_{Y}$ is $f$-ample,  $K_{Y}$  is Cartier at all real points, 
every fiber of $f$ is irreducible (over ${\mathbb R}$) and  $\overline{Y({\mathbb R})}$ 
  is a $3$-manifold.


As in the conic fibration case, the essential point  is the analysis of
the singular fibers. 


Let $A\sim S^1$ be a connected component of $C({\mathbb R})$ and  
$p_1,\dots,p_s\in A$   the points (in cyclic order) over which $f$ is
not smooth. For each $i$ pick a point $q_i\in (p_i,p_{i+1})$. Then
$Y_{q_i}:=f^{-1}(q_i)$ is a smooth Del Pezzo surface  and $Y_{q_i}({\mathbb R})$
is orientable. Thus by (\ref{comm.thm})
$Y_{q_i}({\mathbb R})$ is either $S^1\times S^1$ or a disjoint union of copies
of $S^2$. Gluing $3$-manifolds along such surfaces is  a relatively
simple operation, thus one can expect to  get a good description of
$Y({\mathbb R})$ by describing the pieces
$Z_i:=(f^{-1}[q_{i-1},q_i])({\mathbb R})$ for every $i$.

$f:Z_i\to [q_{i-1},q_i]$ is a function whose only   critical value is
$p_i$. Thus $Z_i$ can be viewed as a regular neighborhood of the
critical level set $(f^{-1}(p_i))({\mathbb R})$. 

The complex projective surface $S_i:= f^{-1}(p_i)$ is a ``singular Del
Pezzo" surface which appears as a degeneration of smooth Del Pezzo
surfaces. Quite a lot is known about such surfaces (see, for instance,
\cite{Keel-Mc,Manetti1,Manetti2,Reid94}). Unfortunately,  a complete
classification of such singular Del Pezzo surfaces is not   feasible
because of the combinatorial complexity of the problem. 

It is nonetheless possible to use the methods of these authors
to develop a rough topological description of the $S_i({\mathbb R})$. 
As in  \cite{rat4}, the end result is that  $\overline{Y({\mathbb R})}$
 is glued together from the following
pieces:
\begin{enumerate}
\item $S^1\times S^2$ minus open balls,
\item  lens space minus open balls,
\item solid torus minus open balls,
\item interval bundle over a torus or a Klein bottle.
\end{enumerate}

It is not hard to see that all such manifolds belong to 
one of the cases (\ref{main.thm}.1--3). 
This proves (\ref{main.thm}) in the Del Pezzo fibration case.


\section{Speculations}


\subsection{Fano threefolds}
In order to complete the proof of (\ref{main.thm}), we still have to
deal with the case when $X^*$ is a Fano $3$-fold.
There is a complete list of   smooth Fano $3$-folds
\cite{Iskovskikh80}. It would be interesting to
 determine the possible topological types of smooth
real Fano $3$-folds. Unfortunately this may not be easy. For instance
it is not known if $Y({\mathbb R})$ can be hyperbolic for a smooth degree 4
hypersuface $Y\subset
\bp^4$. (The case of degree 4 surfaces $S\subset \bp^3$ was treated in
\cite{Kharlamov76}.)



In general it is known that there are only finitely many families of
singular Fano varieties in dimension 3
\cite{Kawamata92}, but there is no explicit list. 
From this we conclude that as $Y$ runs through all singular Fano
$3$-folds, we obtain only finitely many topological spaces $Y({\mathbb R})$ 
up to
homeomophism. The methods of \cite{koll93} can be used to derive an
explicit upper bound. (I have not computed the resulting bound; it is
probably something like $10^{10^{10}}$.)

This concludes the  proof of (\ref{main.thm}). 

Many Fano $3$-folds are birational to conic  or  Del Pezzo fibrations,
and this could be used to understand the topology of their real part.
In other cases there is frequently a fibration  whose general fibers
are elliptic curves. It should be possible to develop a topological
theory of real elliptic fibrations. Unfortunately I do not know a
general result which asserts that such fibrations always exist.


\subsection{Higher-dimensional Nash conjecture}
It is natural to consider the Nash conjecture in higher dimensions
as well. The minimal model program is still conjectural in dimensions
above 3 and it is very unlikely that terminal singularities or the
steps of the program will ever be completely described. 
Nonetheless, (\ref{main.hyp.thm}) and (\ref{vit.hyp.thm})
suggest that the following may be true:


\begin{conj}\label{hyp.conj}
Let $X$ be a smooth, projective, real algebraic variety
of dimension $n\geq 3$. 
Assume that $X({\mathbb C})$ is uniruled.
Then $X({\mathbb R})$ does not carry any metric with negative sectional
curvature.
\end{conj}

One can even go further and pose the following question, which is open
even in dimension 3.

\begin{ques}\label{hyp.conj2}
Let $X$ be a smooth, projective, real algebraic variety
of dimension $n\geq 3$. 
Assume that $X({\mathbb R})$ does   carry a  metric with negative sectional
curvature.
Is it true that $X$ is of general type?
\end{ques}

\subsection{The nonprojective Nash conjecture}
For the minimal model program to work it is essential to
have projective varieties. It would be interesting to know
what happens for proper but nonprojective varieties. 
There are reasons to believe that the answer may be quite
different from
the projective case.

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%\noindent University of Utah, Salt Lake City UT 84112 

%\begin{verbatim}kollar@math.utah.edu\end{verbatim}


\end{document}