% December 22, 1992
% Algebraic approximations of holomorphic maps
% from Stein domains to projective manifolds
% J.P. Demailly, B. Shiffman
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%BEGIN TEXT
\centerline{\bftwelve Algebraic approximations of holomorphic maps}
\vskip4pt
\centerline{\bftwelve from Stein domains to projective manifolds}
 
\vskip5pt
{\noindent\hangindent0.6cm\hangafter-1
{\bf Jean-Pierre Demailly$\,$\footnote{${}^\star$}{\eightpoint\rm
Research partially supported by Institut Universitaire de France}
\hfill Bernard Shiffman$\,$\footnote{${}^{\star\star}$}{\eightpoint\rm
Research partially supported by National Science Foundation Grant No.\
DMS-9204037}\kern0.6cm
 
}}
 
{\noindent\hangindent0.6cm\hangafter-4{\sl
\llap{${}^\star~$}Universit\'e de Grenoble I\hfill
${}^{\star\star}$ The Johns Hopkins University\kern0.6cm\break
Institut Fourier, BP 74\hfill
Department of Mathematics\kern0.6cm\break
U.R.A. 188 du C.N.R.S.\hfill
Baltimore, MD 21218, U.S.A.\kern0.6cm\break
38402 Saint-Martin d'H\`eres, France
 
}}

\vskip12pt\noindent
{\bf Abstract}\pointir{\sl It is shown that every holomorphic map $f$
from a Stein manifold $\Omega$ into a projective algebraic manifold $X$
is a locally uniform limit of maps $f_\nu$ defined over an exhausting
sequence of relatively compact open sets $\Omega_\nu$ in~$\Omega$, such
that the images $f_\nu(\Omega_\nu)$ are contained in algebraic
subvarieties of $X$ of dimension equal to~$\dim f_\nu(\Omega_\nu)$; if
$\Omega$ is a Runge open set in an affine algebraic manifold, then the
$f_\nu$ can be chosen to be Nash algebraic maps. The main point is a
reduction to the case where $f(\Omega)$ is contained in an affine
Zariski open subset of~$X$. In the case where $\Omega$ is the unit
disc, these results can be used to show that the Kobayashi-Royden
infinitesimal metric of a quasi-projective algebraic manifold~$Z$ is
computable solely in terms of the closed algebraic curves in~$Z$.  More
generally, when $\Omega$ is the unit ball in $\cb^p$, the approximation
theorem shows that the $p$-dimensional Eisenman metric of a
quasi-projective algebraic manifold can be computed in terms of the
Eisenman volumes of its $p$-dimensional algebraic subvarieties.}
\vskip8pt
 
\noindent{\bf Table of contents}\vskip3pt
{\eightpoint\noindent
1. Introduction\dotfill p. 2\break
2. Existence of holomorphic and Nash algebraic retractions\dotfill p. 6\break
3. Nash algebraic approximations of disc mappings and hyperbolicity\dotfill
   p. 8\break
4. Complete pluripolar sets and $L^2$ approximation theorem\dotfill
   p. 14\break
5. Algebraic approximation on Runge domains
   of affine algebraic manifolds\dotfill p. 20\break
6. Exhaustion of Stein manifolds by Runge
   domains of affine algebraic manifolds\dotfill p. 24\break
References\dotfill p. 28}\vskip8pt
\noindent
{\bf Key words:}~ Stein manifold, Runge domain, projective algebraic
manifold, \hbox{affine algebraic manifold}, Zariski open set, algebraic
curve, \hbox{holomorphic map}, Nash algebraic mapping, holomorphic vector
bundle, algebraic approximation, \hbox{Nash algebraic retraction},
plurisubharmonic function, complete pluripolar~set, \hbox{H\"ormander's
$L^2$-estimates for~$\ov\partial$}, Kobayashi-Royden infinitesimal metric,
hyperbolic space, Eisenman metric, strongly $p$-measure
hyperbolic space.
 
\vskip5pt
\noindent
{\bf A.M.S. Classification 1985:} 32E30, 32H20, 14C30
 
 
\titre 1. Introduction|
The present work proceeds from the general philosophy that analytic
objects should be approximable by algebraic objects under suitable
restrictions. Such questions have been extensively studied in the case
of holomorphic functions of several complex variables and can be traced
back to the Oka-Weil approximation theorem (see [We35] and~[Oka37]).
Our goal is to show that algebraic approximation is always possible in
the case of nonsingular objects such as locally free sheaves and
holomorphic maps from Stein domains to projective manifolds. Of course,
to avoid difficulties arising from the possible occurrence of infinite
topology or geometry near the boundary, it is necessary to shrink the
objects and to restrict to relatively compact domains. Since we deal
with algebraic approximation, a central notion is that of Runge open
set: By definition, an open set $\Omega$ in a Stein manifold $Y$ is
said to be a {\sl Runge open set} if $\Omega$ is Stein and if the
restriction map $\oc(Y)\to\oc(\Omega)$ has dense range. It is well
known that $\Omega$ is a Runge open set if and only if the holomorphic
hull with respect to $\oc(Y)$ of any compact subset $K\subset\Omega$ is
contained in~$\Omega$. If~$Y$ is an affine algebraic manifold, a Stein
open set $\Omega$ is Runge if and only if the polynomial functions
on~$Y$ are dense in~$\oc(\Omega)$.
 
Our first result given below concerns approximations of holomorphic
mappings between algebraic manifolds. If $Y$, $Z$ are quasi-projective
algebraic mani\-folds, a mapping $f:\Omega\to Z$ defined on an open subset
$\Omega\subset Y$ is said to be {\sl (complex) Nash algebraic} if $f$
is holomorphic and the graph
$$\Gamma_f:=\big\{(y,f(y)\in\Omega\times Z: y\in\Omega\big\}$$
is contained in an algebraic subvariety $G$ of $Y\times Z$ of dimension
equal to~$\dim Y$. If~$f$ is Nash algebraic, then the image $f(\Omega)$
is contained in an algebraic subvariety $A$ of $Z$ with
$\dim A=\dim f(\Omega)\le\dim Y$. (Take $A=\pr_2(G)\subset Z$,
after eliminating any unnecessary components in~$G$.)
 
\than Theorem 1.1|Let $S$ be an affine algebraic manifold.
Let $\Omega$ be a Runge domain in~$S$, and $(\Omega_\nu)$ an increasing
sequence of relatively compact domains $\Omega_\nu\compact\Omega$ with
$\Omega=\bigcup\Omega_\nu$. For every holomorphic map
$$f:\Omega\to X$$
into a projective algebraic manifold~$X$, there is a sequence of
Nash algebraic maps \hbox{$f_\nu:\Omega_\nu\to X$} such that
$f=\lim_{\nu\to+\infty}f_\nu$ uniformly on every compact subset of~$\Omega$.
Moreover, if for each $\nu$ we are given a positive integer $m_\nu$
and a finite sequence of points $(t_{j,\nu})$ in~$\Omega_\nu$, then $f_\nu$
can be taken to have the same $m_\nu$-jet as~$f$ at each of the points
$(t_{j,\nu})$. \fintha
 
Recall that the $k$-jet of a holomorphic mapping $f:Y\to Z$ at a point
$a\in Y$ is the equivalence class of holomorphic mappings $g:Y\to Z$
such that $g(a)=f(a)$ and the derivatives of $g$ of order~$\leq k$ agree with
those of $f$.
 
In the case where $f$ in Theorem~1.1 is an embedding with the property that
$f^\star L$ is trivial for some ample line bundle $L$ on $X$, we can require
in addition that each $f_\nu$ is an embedding with image contained in an
affine Zariski open set of the form $X\ssm h_\nu^{-1}(0)$, where
$h_\nu\in H^0(X,L^{\otimes\mu(\nu)})$; see Proposition 5.1.
 
One of our main applications is the study of the Kobayashi-Royden
infinite\-simal metric on algebraic manifolds ([Ko67], [Ko70], [Ro71]).
We use throughout the following notation (which provides the
definition of the Kobayashi-Royden metric):
 
\smallskip \noindent
$\Delta_R(a) = \{t\in \cb:|t-a|<R\}$,~ $\Delta_R = \Delta_R(0)$,~ $\Delta =
\Delta_1$.\hfil\break
$T_Y={}$ the holomorphic tangent bundle of a complex
manifold $Y$.\hfil\break
$N_S=N_{S/Y}={}$ the normal bundle $T_{Y|S}/T_S$ of a submanifold
$S\subset Y$.\hfil\break
$\kappa_Y(v)={}$ the Kobayashi-Royden infinitesimal length of
a vector $v\in T_Y$, i.e.
$$\kappa_Y(v)=\inf\Big\{\lambda>0\,:\,\exists
f:\Delta\to Y~ \hbox{\rm holomorphic with}~
f_\star\Big(\lambda{\partial\over\partial\zeta}{}_{|0}\Big)=v\Big\}.$$
In applying Theorem 1.1 to compute $\kappa_Z$ for a quasi-projective
manifold~$Z$, we only need to find algebraic approximations of holomorphic
maps~$\Delta\to Z$. In that case, a rather elementary proof can be given,
avoiding the heavy analytic machinery of the general case. The details are
given in~\S$\,$3. A straightforward consequence is:
 
\than Corollary 1.2|Let $Z$ be a quasi-projective algebraic manifold.
Then the Kobayashi-Royden infinitesimal pseudometric $\kappa_Z$ is given
by $$\kappa_Z(v)=\inf_C\kappa_{\wt C}(v),~~~~\forall v\in T_Z\,,$$
where $C$ runs over all $($possibly singular$)$ closed algebraic curves
in $Z$ tangent to~$v$, and $\wt C$ is the normalization of~$C$.
\fintha
 
In the case where $\wt C$ is hyperbolic (e.g., is compact and has
genus $\geq 2$), the Kobayashi-Royden metric $\smash{\kappa_{\wt C}}$ just
coincides with the Poincar\'e metric induced by the universal covering
$\Delta\to\smash{\wt C}\,$; thus in some sense the computation of
$\kappa_Z$ is reduced to a problem which is more algebraic in nature.
Of particular interest is the case where $Z$ is projective; then the
computation becomes one of determining the genus and Poincar\'e metric of
curves in each component of the Hilbert scheme. This observation strongly
suggests that it should be possible to characterize Kobayashi hyperbolicity
of projective manifolds in purely algebraic terms; this would follow for
instance from S.~Lang's conjecture [La86], [La87,~IV.5.7] that a projective
manifold is hyperbolic if and only if it has no rational curves and no
nontrivial images of abelian varieties; see also [De93] for further results
on this question.
 
Eisenman [Ei70] has also introduced an invariant pseudometric on the
decomposable $p$-vectors of an arbitrary complex space~$Y$. Let $B^p$ be
the unit ball of $\cb^p$. At each nonsingular point of~$Y$, the Eisenman
$p$-metric is defined by $$E^p_Y(v)=\inf\Big\{\lambda>0:\,\exists
f:B^p\to Y~ \hbox{\rm holomorphic
with}~f_\star\Big(\lambda{\partial\over\partial z_1}
\wedge\ldots\wedge{\partial\over\partial z_p} {}_{|0}\Big)=v\Big\}$$ for all
decomposable tensors $v\in\Lambda^pT_Y$. The space $Y$ is said to be {\sl
strongly $p$-measure hyperbolic} if $E^p_Y(v)>0$ whenever~$v\ne 0\,$; it is
said to be {\sl strongly measure hyperbolic} if the Eisenman volume form
$E^n_Y$ ($n=\dim Y$) is everywhere positive. An argument similar to the one
used in Corollary~1.2 gives
 
\than Corollary 1.3|The Eisenman $p$-metric of a quasi-projective
algebraic manifold $Z$ can be computed in terms of the Eisenman volumes of
its $p$-dimensional algebraic subvarieties. More precisely
$$E^p_Z(v)=\inf_YE^p_Y(v)$$ for all decomposable tensors
$v\in\Lambda^pT_Z$, where $Y$ runs over all $($possibly singular$)$
$p$-dimensional closed algebraic subvarieties of~$Z$ tangent to~$v$. $($A
$p$-dimensional subvariety $Y$ is said to be tangent to $v$ if for a
desingularization $\wt Y\buildo{\pi}\over\longrightarrow Y$ there is a tensor
$\wt v\in\Lambda^pT_{\wt Y}$ with $v=f_\star\wt v\,.)$ \fintha
 
If $Y$ is compact and $K_{\wt Y}$ is ample, then a form of the maximum
principle tells us that the Eisenman volume $E^p_{\wt Y}$ is bounded from
below by the volume form of the K\"ahler-Einstein metric (with curvature
$-1$) on ${\wt Y}$ (see [Yau78], [NO90,~\S 2.5]). It would be very
interesting to know when these two intrinsic volumes are actually equal.  In
connection to this question, one would like to know whether the infimum
appearing in Corollary~1.3 can be restricted to subvarieties $Y$ with
$\wt Y$ satisfying this property, in virtue of some density argument. This
would make the Eisenman $p$-metric theory completely parallel to the case of
the Kobayashi metric $\kappa_Z=E^1_Z$.
 
Let us now provide a few words about the proof of Theorem~1.1.  The theorem
is obtained by a reduction to the case where the image of $f$ is contained in
an affine Zariski open set. This can be done by creating essential
singularities on the boundary of $f(\Omega_\nu)$, in such a way that the
closure $f(\ov\Omega_\nu)$ becomes complete pluripolar (in the sense of
plurisubharmonic function theory). The existence of an ample divisor avoiding
$f(\Omega_\nu)$ is then obtained by means of an approximation theorem based
on H\"ormander's $L^2$ estimates for~$\ov\partial$ (see~\S$\,$4). Once
$f(\Omega_\nu)$ is known to be contained in an affine Zariski open
set~$Y\subset X$, the method is to embed $Y$ in an affine space~$\cb^N$ and
to approximate the components of $f$ by polynomials. Of course, a polynomial
approximation of $f$ need not have its image contained in~$Y$, so we need a
further Nash algebraic retraction onto $Y$ to complete the argument. The
existence of Nash algebraic retractions is standard and briefly discussed
in~\S$\,$2.
 
In order to extend Theorem~1.1 to mappings from arbitrary Stein manifolds
(Theorem~1.6), we need the following result of Stout [St84] on the
exhaustion of Stein manifolds by Runge domains in affine algebraic manifolds;
this result has been refined into a relative version by
Tancredi-Tognoli~[TT89].
 
\than Theorem 1.4 {\rm(E.L.~Stout)}|Let $S$ be a Stein
manifold. For every Runge open set $\Omega\compact S$, there exists an
affine algebraic manifold $Y$ such that $\Omega$ is biholomorphic to a
Runge open set in~$Y$.
\fintha
 
We must point out at this stage that Theorem~1.4 completely breaks
down for singular Stein spaces. In fact, there are examples of germs of
analytic sets with non-isolated singularities which are not
algebraizable, i.e., which are not biholomorphic to a germ of an algebraic
set; Whitney [Wh65] has even given an example which is not
$C^1$-diffeomorphic to an algebraic germ. The technique used by Stout
and Tancredi-Tognoli to prove Theorem~1.4 consists of a generalization
to the complex case of the methods introduced by Nash~[Nh52].
In the present work, we give a completely different proof of Theorem~1.4
based on some of the arguments used in the proof of the approximation
theorem~1.1.  Our proof starts with the observation that every Stein domain
$\Omega\compact S$ can be properly embedded in an affine algebraic manifold
in such a way that the normal bundle is trivial. This key step requires the
following intimately related approximation result for holomorphic vector
bundles over Stein manifolds. (A similar result has also been announced by
Tancredi and Tognoli~[TT92], Theorem~4.1.)
 
\than Theorem 1.5|Let $S$ be a Stein manifold, and let $E\to S$ be
a holomorphic vector bundle. For every Runge open set $\Omega\compact S$,
there exists a projective algebraic manifold~$Z$, an ample divisor
$D$ in~$Z$ and an algebraic vector bundle $\wt E\to Z$ with the
following properties:
\smallskip
\item{\rm(i)} $\Omega$ is biholomorphic to a Runge domain
in the affine Zariski open set $Z\ssm D\,;$
\vskip0pt
\item{\rm(ii)} the restriction $\wt E_{|\Omega}$ is isomorphic
to~$E_{|\Omega}$.
\fintha
 
Theorem~1.5 is proved by observing that there is a holomorphic map from $S$
into a Grassmannian such that the given vector bundle $E$ on $S$
is isomorphic to the pull-back of the universal quotient vector bundle. The
desired result then follows from Theorem~1.4 applied to $\Omega\compact S$
and from the existence of Nash algebraic approximations of the map from
$\Omega$ to the Grassmannian (see~\S\S$\,$5,~6).
 
A consequence of Theorem~1.1 and Stout's Theorem~1.4 is the
following approximation theorem for holomorphic maps defined on Stein
manifolds with values in projective algebraic manifolds:
 
\than Theorem 1.6|Let $S$ be a Stein manifold and let $f:S\to X$ be a
holomorphic map into a projective algebraic manifold~$X$.
Fix an increasing sequence $(\Omega_\nu)$ of relatively compact domains
$\Omega_\nu\compact S$ with $S=\bigcup\Omega_\nu$. Then
there is a sequence of holomorphic maps \hbox{$f_\nu:\Omega_\nu\to X$} such
that $f=\lim_{\nu\to+\infty}f_\nu$ uniformly on every compact subset of~$S$,
and such that the images $f_\nu(\Omega_\nu)$ are contained in algebraic
subvarieties $A_\nu$ of~$X$ with $\dim A_\nu=\dim f_\nu(\Omega_\nu)$.
Moreover, if for each $\nu$ we are given a positive integer $m_\nu$ and a
finite sequence of points $(t_{j,\nu})$ in~$\Omega_\nu$, then $f_\nu$ can be
taken to have the same $m_\nu$-jet as~$f$ at each of the points $(t_{j,\nu})$.
\smallskip
\fintha
 
A major part of the results obtained in the present work were obtained
during a stay of the second author at Institut Fourier, Universit\'e de
Grenoble~I. \vfill\eject
 
\titre 2. Existence of holomorphic and Nash algebraic retractions|
 
Let $S$ be an $n$-dimensional Stein manifold. The Bishop-Narasimhan
embedding theorem ([Bi61], [Na60]) implies that $S$ can be embedded as
a closed submanifold of a complex vector space~$\cb^N$ with e.g.
$N=2n+1\,$; by Eliashberg and Gromov [EG92], it is in fact enough to take
$N=[(3n+3)/2]$, and this value is optimal for even~$n$. Such optimal
values will not be needed here. We first recall a well-known and elementary
lemma about the existence of holomorphic retractions for Stein submanifolds.
 
\than Lemma 2.1|Let $S$ be a Stein $($not necessarily closed$)$ submanifold
of a complex manifold~$Y$ and let $n=\dim Y$.
\smallskip
\item{\rm(i)} There exist a neighborhood $U$ of the zero section in the
normal bundle $N_S$ of~$S$, a neighborhood $V$ of $S$ in $Y$ and a
biholomorphism $\psi:U\to V$, such that $\psi$ maps the zero section of~$N_S$
identically onto~$S$ with normal derivative $(d_n\psi)_{|S}=\Id_{N_S}$.
Moreover, if $Y$ is affine algebraic and $S$ is a closed algebraic
submanifold, then $\psi$ can be taken to be Nash algebraic.
\smallskip
\item{\rm(ii)} Let $\pi:N_S\to S$ be the natural projection. Then
$\rho=\pi\circ\psi^{-1}:V\to S$ is a holomorphic retraction, i.e. a
holomorphic map such that $\rho_{|S}=\Id_S$. If $Y$ and $S$ are algebraic,
then $\rho$ is Nash algebraic $($provided that $\psi$ is Nash algebraic$)$.
\smallskip
\fintha
 
\proof We first outline the proof of the well-known case where $Y=\cb^n$.
(Details can be found in [GR65,~p.~257].) As $S$ is Stein, by Cartan's
theorem~B the normal bundle sequence $$0\la T_S\la T_{\cb^n|S}\la N_S\la 0$$
splits; i.e., there is a global morphism
$\sigma:N_S\to T_{\cb^n|S}$ which is
mapped to~$\Id_{N_S}$ by composition with the projection onto~$N_S$. Then
the map $\psi:N_S\la\cb^n$ given by
$$\psi(z,\zeta)=z+\sigma(z)\cdot\zeta\;,~~~~\zeta\in N_{S,z}\;,$$ coincides
with $\Id_S$ on the zero section of $N_S$ (in particular $\psi$ has tangential
derivative $(d_t\psi)_{|S}=\Id_{T_S}$), and $\psi$ has normal derivative
$d_n\psi=\Id_{N_S}$ along~$S$ because $\sigma$ projects onto~$\Id_{N_S}$.
Hence $d\psi$ is invertible at all points of the zero section of~$N_S$. By
the implicit function theorem, $\psi$ defines a biholomorphism from a
neighborhood $U$ of the zero section of $N_S$ onto a neighborhood $V$ of~$S$
in~$\cb^n$. Then $\psi_{|U}:U\to V$ is the required biholomorphism. When $S$
is affine algebraic, $N_S$ can be realized as an affine algebraic manifold
such that $\pi:N_S\to S$ is an algebraic map. The splitting morphism $\sigma$
can also be taken to be algebraic. Then $\Gamma_\psi$ is an open subset of a
$n$-dimensional subvariety $G\subset N_S\times Y$. The assertion that
$\rho=\pi\circ\psi^{-1}$ is a retraction is clear, and in the algebraic case
$\Gamma_\rho=(\Id_Y\times\pi)(\Gamma_\psi^{-1})$ is contained in the
$n$-dimensional algebraic variety $(\Id_Y\times\pi)(G)$.
 
In the case of a general complex manifold~$Y$, it is enough to consider
the case where $Y$ is Stein, since by [Siu76] every Stein subvariety $S$
in a complex space $Y$ has arbitrarily small Stein neighborhoods.
(See also [De90] for a simpler proof; this property is anyhow very easy to
check in the case of nonsingular subvarieties.) Thus we can
suppose $Y$ to be a closed $n$-dimensional submanifold of
some~$\cb^p$, e.g. with $p=2n+1$. Observe that the normal bundle
$N_{S/Y}$ of $S$ in $Y$ is a subbundle of $N_{S/\cb^p}$. By the first case
applied to the pairs $S\subset\cb^p$, $Y\subset\cb^p$, we get a map
$\psi':N_{S/\cb^p}\longrightarrow \cb^p$ inducing a biholomorphism
$\psi':U'\to V'$ and a retraction $\rho'':V''\to Y$, where $U'$ is a
neighborhood of the zero section of~$N_{S/\cb^p}$, $V'$~a
neighborhood of $S$ and $V''$ a neighborhood of~$Y$. We define $\psi$ to
be the composition
$$\psi:N_{S/Y}\lhra N_{S/\cb^p}\buildo{\displaystyle \psi'}\over
{\relbar\joinrel\la}\cb^p\buildo{\displaystyle\rho''}\over
{\relbar\joinrel\la}Y.$$
Of course, $\psi$ is not defined on the whole of~$N_{S/Y}$, but only on
$N_{S/Y}\cap(\psi')^{-1}(V'')$. Clearly $\psi_{|S}=\Id_S$, and the
property $(d_n\psi)_{|S}=\Id_{N_{S/Y}}$ follows from the equalities
$(d_n\psi')_{|S}=\Id_{N_{S/\cb^p}}$ and $(d\rho'')_{|T_Y}=\Id_{T_Y}$.
It follows as before that $\psi$ induces a biholomorphism $U\to V$ from
a neighborhood of the zero section of $N_{S/Y}$ onto a neighborhood $V$
of~$S$. If $Y$ and $S$ are algebraic, then we can take $\psi'$ to be
algebraic and $\Gamma_{\rho''}$ to be contained in an algebraic
$p$-dimensional subvariety $G''$ of $\cb^p\times Y$. Then $\Gamma_\psi$
is contained in some $n$-dimensional component of the algebraic set
$$\big(N_{S/Y}\times Y\big)\cap(\psi'\times\Id_Y)^{-1}(G'').$$
Therefore $\psi$ is Nash algebraic.\finpr
 
\than Lemma 2.2|Let $S$ be a Stein submanifold of a complex
manifold~$Y$,  and let $E\to Y$ be a holomorphic vector bundle. Fix
a holomorphic retraction \hbox{$\rho:V\to S$} of a neighborhood $V$
of $S$ onto~$S$. Then there is a neighborhood $V'\subset V$ of~$S$ such
that $E$ is isomorphic to $\rho^\star(E_{|S})$ on~$V'$. In particular,
if the restriction $E_{|S}$ is trivial, then $E$ is trivial on a
neighborhood of~$S$.\fintha
 
\proof By [Siu76] we can find a Stein neighborhood $V_0\subset V$. As
$V_0$ is Stein, the identity map from $E_{|S}$ to $(\rho^\star E)_{|S}$
can be extended to a section of $\Hom(E,\rho^\star E)$ over~$V_0$. By
continuity, this homomorphism is an isomorphism on a sufficiently small
neighborhood $V'\subset V_0$ of~$S$.\finpr\vskip7pt
 
Finally, we need a simple lemma about the existence of sections of
holomorphic vector bundles on Stein manifolds.
 
\than Lemma 2.3|Let $E$ be a holomorphic vector bundle of rank $r$
on a Stein space~$S$. Then there are finitely many holomorphic sections
$g_1\ld g_N$ in $H^0(X,E)$ such that $g_1(x)\ld g_N(x)$ generate every
fibre $E_x$, $x\in X$.
\fintha
 
\proof By induction on~$\dim S$. Cartan's theorem B implies that there
are holomorphic sections with prescribed values at any point or on any
discrete sequence of points. Select a point $x_j$ on each irreducible
component of $S$ and $r$ sections $g_1\ld g_r$ of $E$ which are
linearly independent at each point~$x_j$ (these points form a discrete
sequence). Then $g_1\ld g_r$ generate $E$ except possibly on some
analytic subset $S'$ with $\dim S'\leq\dim S-1$. By the induction
hypothesis, we find finitely many sections $g'_{r+1}\ld g'_N$
generating $E_{|S'}$. Then, taking $g_j$ to be a holomorphic
extension of $g'_j$ to~$S$, for $r<j\leq N$, we see that $g_1\ld g_N$ generate
$E$ over~$S$. (This obvious proof yields $N=r(\dim S+1)\,$; in fact, a
slightly more careful induction argument shows that
$N=r+\dim S$ is enough.)\finpr
 
\vbox{
\than Corollary 2.4|Let $E$ be a holomorphic vector bundle over a Stein
space~$S$. Then there is a holomorphic map $\varphi:S\to G$ into
a Grassmannian such that $E$ is isomorphic to the pull-back
$\varphi^\star Q$ of the universal quotient vector bundle $Q$ on~$G$.
\fintha}
 
\proof With the sections $g_j$ of Lemma~2.3, we define a holomorphic map
$$\varphi:S\to G(N,r),~~~~x\mapsto
V_x=\big\{(\lambda_1\ld\lambda_N)\in\cb^N\,:\,\sum\lambda_jg_j(x)=0\big\}$$
into the Grassmannian $G(N,r)$ of subspaces of codimension $r$ in~$\cb^N$.
Then the generators $g_j(x)$ define a canonical isomorphism between $E_x$
and $\cb^N/V_x$, which is precisely the fibre at the point $\varphi(x)=V_x$
of the universal quotient vector bundle~$Q$ on $G(N,r)$ (see, for
example, [GA74,~Ch.~V]). Hence $E\simeq\varphi^\star Q$.\finpr
\vskip12pt
 
\titre 3. Nash algebraic approximations of disc mappings and hyperbolicity|
 
In this section, we give a simple proof of the approximation theorem in the
case of holomorphic maps from the unit disc to projective manifolds. This
case is also of independent interest in the study of the Kobayashi-Royden
infinitesimal metric of projective varieties. We denote by
$J_a^k(f)$ the $k$-jet of a holomorphic map $f:\Delta \rightarrow X$ at a
point $a\in\Delta$.
 
\than Theorem 3.1|Let $f:\Delta \rightarrow X$ be a
holomorphic map into a projective algebraic manifold $X$.  Then for any
$R<1$ and any positive integer $k$, there exists a sequence of Nash
algebraic maps $f_\nu:\Delta_R \rightarrow X$, $\nu=1,2,\ldots,$ such that
$J^k_0(f_\nu) = J^k_0(f)$ for all $\nu$, and $f_\nu\rightarrow f$ uniformly.
\fintha
 
It is important to observe that in general $f_\nu$ can only be extended
to $\cb$ as a branched map $\cb\to X\,$; otherwise, if the extension were
univalent, then the graph $\Gamma_{f_\nu}$ would extend to a
rational curve $G_\nu$, and $\pr_2(G_\nu)\subset X$ would
also be rational, whereas $X$ need not have any rational curve at all. The
following corollary is a generalization of~1.2.
 
\than Corollary 3.2|Let $\Omega$ be an open subset of a projective
algebraic manifold $X$ and let $v\in T_\Omega$.  Then $\kappa_\Omega(v)$ is
the infimum of the set of $r\geq 0$ such that there exist a
closed algebraic curve $A\subset X$, a normalization $\gamma:C\rightarrow
A\cap \Omega$, and a vector $u \in T_C$ with $\kappa_C(u)=r$ and
$\gamma_\star u=v$. \fintha
 
\noindent {\sl Proof of Corollary~3.2 (assuming Theorem~3.1)}\pointir
Let $r_0$ be the infimum given above.  By the distance-decreasing
property of holomorphic maps, $\kappa_X (v)\leq r_0$.  We must verify
the reverse inequality.  Let $\varepsilon >0$ be arbitrary.  Choose a
holomorphic map $g:\Delta\rightarrow \Omega$ and $\lambda\in
[0,\kappa_X(v)+\varepsilon]$ such that $g_\star(\lambda e_0)=v$,
where $e_0={\partial\over\partial\zeta}{}_{|0}$. By Theorem~3.1,
there exists an algebraic map $f:\Delta _{1-\varepsilon}\rightarrow
\Omega$ such that $f_\star(e_0)=g_\star (e_0)$. Let $\wt f:
\Delta\rightarrow\Omega$ be given by $\wt f(\zeta)=f((1-\varepsilon)\zeta)$
and let $\wt\lambda=\lambda/(1-\varepsilon)$. Then
$\wt f_\star(\wt\lambda e_0)=v$. Since $\wt f$ is algebraic, there exists
an algebraic curve $A\subset X$ such that $\wt f(\Delta )\subset A$. Let
$\gamma:C\rightarrow A\cap\Omega$ be the normalization of~$A\cap\Omega$.
We then have a commutative diagram
$$\eqalign{
\Delta&\buildo{{\displaystyle \wh f}}\over{\longrightarrow}C\cr
\wt f&\searrow\quad\downarrow\gamma\cr
&\quad A\cap \Omega\,.\cr}$$
Let $u = \wh f_\star (\wt \lambda e_0)$. We have $\gamma_\star u=v$. Thus
$$r_0\leq\kappa_C(u) \leq\wt\lambda  = {\lambda \over 1-\varepsilon}\le
{\kappa_\Omega(v)+\varepsilon\over 1-\varepsilon}\,.\eqno(3.1)$$
Since $\varepsilon>0$ is
arbitrary, we conclude that $r_0\leq\kappa_\Omega(v)$.\finpr\vskip7pt
 
The proof of Theorem~3.1 is based on the following elementary lemmas:
 
\than Lemma 3.3|Suppose $f_\nu\in\oc(\Delta )$ for
$\nu = 1,2,\ldots$ such that $f_\nu\rightarrow 0$ uniformly on $\Delta $
and $f_\nu^{(j)}(0)=0$ for $0\leq j\leq s-1$.  Then
$f_\nu(z)/z^s\rightarrow 0$ uniformly for $z\in \Delta\ssm \{0\}$.
\fintha
 
\proof  By the maximum modulus principle.\finpr
 
\than Lemma 3.4|Let $\varphi:\rb^+\rightarrow \rb^+$ be a positive
semi-algebraic function.  Then there exist $\varepsilon >0$, $t_0 >0$,
$\lambda \in \nb$ such that $\varphi(t) \geq \varepsilon t^\lambda $ for
$0<t<t_0$.\fintha
 
\proof Proposition 2.6.1 of [BCR87] with $f(x) =1/\varphi(1/x)$.  (A {\sl
semi-algebraic function} is a real-valued function whose graph is a
semi-algebraic set; see [BCR87] for the definitions.) \finpr
 
\than Lemma 3.5|Let $g,h$ be continuous semi-algebraic
functions on a compact real-algebraic manifold $M$ and let $U$ be an open
semi-algebraic set such that $U\supset g^{-1} (0) \ssm h^{-1}(0)$.  Then
there exist $\lambda \in \nb$ and $c>0$ such that
$$U\supset \big\{z \in M:|g(z)|<c|h(z)|^\lambda \big\}\,.$$
\fintha
 
\rem Remark|$M$ can be replaced by any compact semi-algebraic space.\bigskip
 
\proof For $t>0$ let $\varphi(t)$ denote the supremum of the
set of $r\in [0,1]$ such that $$U\supset \big\{z\in M:|h(z)|\ge
t,\,\,\,|g(z)|\leq r\big\}\,.$$ By the Tarski-Seidenberg principle [Se54],
[BCR87], $\varphi:\rb^+\rightarrow [0,1]$ is a semi-algebraic function.  By
definition, $\varphi$ is monotonically non-decreasing.  We now verify that
$\varphi(t)>0$ for all $t>0$:  Suppose on the contrary that $\varphi(t_0)=0$
for some $t_0>0$.  Let
$$E_\nu=\big\{z\in M\ssm U:|h(z)|\geq t_0,~|g(z)|\le
{1\over \nu} \big\}\neq \emptyset$$
for $\nu=1,2,\ldots$.  However
$$\bigcap^\infty_{\nu=1} E_\nu \subset\big(M \ssm U\big) \cap
\big(g^{-1}(0) \ssm h^{-1}(0)\big)=\emptyset$$
which contradicts the compactness of $M$.
 
Thus $\varphi:\rb^+\rightarrow{}]0,1]$.  Therefore, by Lemma~3.4 there
exists $\varepsilon$, $t_0>0$, $\lambda\in\nb$ such that
$$\varphi(t)\geq\varepsilon t^\lambda~~~~\hbox{\rm for}~~0<t<t_0\,.$$
Since $\varphi$ is monotonic, we can choose $c>0$ such that
$$\varphi(t)\geq ct^\lambda~~~~
\hbox{\rm for}~~0<t\leq \sup_{z\in M}|h(z)|\,.$$
Thus
$$\eqalign{
 U&\supset\big\{z\in M:h(z)\neq 0,~|g(z)|<\varphi
(|h(z)|)\big\}\cr
  &\supset\big\{z\in M:|g(z)|<c |h(t)|^\lambda\big\}\,.\cr}$$
\finpr
 
\noindent {\sl Proof of Theorem 3.1}\pointir Let $f,R,k$ be as in
Theorem~3.1. Suppose $X \subset\pb^{n+m}$, where $\dim X=n$.  Choose
homogeneous coordinates $z_0,\ldots,z_{n+m}$ in $\pb^{n+m}$ such that
$f(0) \not\in \{z_0=0\}$. Write $H_0=\{z_0=0\}$ and identify
$\pb^{n+m}\ssm H_0$ with $\cb^{n+m}$ using the coordinates
$z_1/z_0,\ldots,z_{n+m}/z_0$, and let $X_0=X\ssm H_0\subset\cb^{n+m}$.
We let
$$f^{-1} (H_0)\cap\ov\Delta_R=\{t_1,\ldots,t_\ell\}~~~~
\hbox{\rm (with the $t_j$ distinct).}$$
Assume without loss of generality that
$f^{-1}(H_0)\cap \partial \Delta_R=\emptyset$ (increase $R$ if necessary).
By the proof of Lemma~2.1, the normal bundle sequence
$$0\la T_{X_0}\la T_{\cb^{n+m}|X_0}\la N_{S_0}\la 0$$
has an algebraic splitting
$$\sigma=(\pi_0,\sigma_0):N_{X_0}\longrightarrow T_{\cb^{n+m}|X_0}=
X_0\times\cb^{n+m}\;,$$
where $\pi_0:N_{X_0}
\rightarrow X_0$ is the vector-bundle projection. Then the map
$$\psi:N_{X_0}\to\cb^{n+m},~~~~\psi(y)=\pi_0(y)+\sigma_0(y)\;,$$
induces a biholomorphism from a neighborhood of the zero section of $N_{X_0}$
onto a neighborhood of~$X_0$. Our first goal is to describe more precisely
these neighborhoods.
 
We compactify the fibres of the normal bundle $N_X$ to obtain an
algebraic manifold $Y\supset N_X$ and an algebraic fibre bundle
$Y\buildo{\pi_1}\over\longrightarrow X$ with fibre $\pb^m$ and group
$GL(m,\cb)\subset PGL(m+1,\cb)$.  Note that $\pi_{1|N_{X_0}}=\pi_0$. We
regard $\psi$ as a meromorphic map $\psi:Y\merto\pb^{n+m}$. Let $\wt
Y=\Gamma_{\psi}\subset Y\times\pb^{n+m}$ and let $\wt\psi: \wt
Y\rightarrow \pb^{n+m}$ be the projection.  Identifying $X$ with the
zero-section in $N_X$, we regard $X\subset N_X\subset Y$.  Let
$\wt X$ denote the diagonal in $X\times X$.  We note that
$\wt X\subset\wt Y$ and that we have the commutative diagrams
$$\eqalignno{
\wt X&\lhra\wt Y\qquad\qquad\qquad\wt X\lhra\wt Y\cr
\hfill{}\approx\big\downarrow&\phantom{\lhra}~\,\big\downarrow
\qquad\qquad\quad\hfil{}\,\!\approx\big\downarrow\phantom{\lhra}
~~\big\downarrow\wt\psi&(3.2)\cr
X&\lhra Y\qquad\qquad\qquad X\lhra\pb^{n+m}\cr}$$
or in other words, $\wt X$ is the proper transform of $X$ under the
blow-up $\wt Y\rightarrow Y$, and $\psi_{|X}=\,\hbox{\rm Id}_X$.
For $w=(w_1,\ldots,w_{n+m})\in \cb^{n+m}$, we introduce the Hermitian
norm $\|w\|=(|w_1|^2+\cdots+|w_{n+m}|^2)^{1/2}$.  Let
$$\Omega_1=\big\{y\in N_{X_{0}}:\|\sigma_0(y)\|<1\,,\,\,\hbox{\rm rank}_y\psi
= n+m\big\}\,.$$
Note that for each $z\in \pb^{n+m}$, $\psi^{-1} (z) \cap \Omega_1$ is
finite since it is a 0-dimensional algebraic set; it is empty if
$z_0=0$.  Let $\Omega_0$ denote the set of $y\in \Omega_1$ such that
$$y'\in N_{X_{0}},~~\psi(y')=\psi(y),~~y'\neq y~~\Longrightarrow~~
\|\sigma_0(y')\|>\|\sigma_0(y)\|\,.$$
Clearly, $X_0\subset\Omega_0$.  By the Tarski-Seidenberg principle,
$\Omega_0$ is a semi-algebraic set.  We now verify that $\Omega_0$
is open in $N_{X_{0}}$: Let $y\in \Omega_0$.  Suppose on the contrary
that there is a sequence $\{y_\nu\}\subset\Omega_1-\Omega_0$ such that
$y_\nu\rightarrow y$.  Then we can find $y'_\nu \in N_{X_{0}}$ such
that $y'_\nu \neq y_\nu$, $\psi(y'_\nu) = \psi(y_\nu)$, and
$\|\sigma_0(y'_\nu)\|\leq \|\sigma_0(y_\nu)\|$.  Thus
$\{\sigma_0(y'_\nu)\}$ and $\{\psi(y'_\nu)\}$ are bounded in
$\cb^{n+m}$, and therefore $\{\pi_0(y'_\nu)\} = \{\psi(y'_\nu) -
\sigma_0(y'_\nu)\}$ is bounded.  Thus after passing to a subsequence,
we can assume that $y'_\nu\rightarrow y'\in N_{X_{0}}$.  We then have
$\psi(y')=\psi(y)$ and $\|\sigma_0(y')\|\leq \|\sigma_0(y)\|$.  Since
rank$_y\psi = n+m$, there is a neighborhood $W$ of $y$ in $N_{X_{0}}$
such that $\psi|_W$ is injective. Then for $\nu$ sufficiently large,
$y_\nu\in W$ and $y'_\nu\not\in W$. Therefore $y'\neq y$.  This
contradicts the assumption that $y\in \Omega_0$.  Therefore $\Omega_0$
is open.
 
Let $\psi_0=\psi_{|\Omega_0}$, and let $U=\psi_0(\Omega_0)$, an open
set in $\cb^{n+m}$.  By the definition of $\Omega_0$, $\psi_0$ is
injective and thus $\psi_0^{-1}:U\rightarrow \Omega_0$ is holomorphic.
Furthermore $\psi_0(X_0)=X_0\subset U$.  Again by the Tarski-Seidenberg
principle, $U$ is a semi-algebraic set.
 
Let $p_1,\ldots,p_N$ be homogeneous polynomials in $\cb[z_0,\ldots,z_{n+m}]$
generating the ideal of $X$.  Let $g,h$ be the real-algebraic functions on
$\pb^{n+m}$ given by
$$\eqalignno{
g(z) &= \sum^N_{i=1}{|p_i(z)|^2\over \|z\|^{2\,\deg p_{i}}}\,,&(3.3)\cr
h(z) &= {|z_0|^2\over \|z\|^2}\,.&(3.4)\cr}$$
Then $X_0=X\ssm H_0=g^{-1}(0)\ssm h^{-1}(0)$. By Lemma~3.5 (with
$M=\pb^{n+m}$), there exist $c>0$, $\lambda\in\nb$ such that
$$U\supset\big\{z \in \pb^{n+m}:g(z)<ch(z)^\lambda\big\}\,.\eqno(3.5)$$
 
Next, we describe the approximation process. Let $\wt f:\Delta
\rightarrow \cb^{n+m+1}\ssm\{0\}$ be such that $\eta\circ\wt f=f$,
where $\eta:\cb^{n+m+1}\ssm\{0\}\rightarrow \pb^{n+m}$. The existence
of $\wt f$ is easily shown by lifting first $f$ as a meromorphic map
with values in a given affine chart of~$\pb^{n+m}$, and then by
multiplying with a suitable Weierstrass product to cancel the poles.
Choose $0<\varepsilon <{1-R\over 2}$ such that, writing $D_j=\Delta
_\varepsilon (t_j)$, we have $D_i\cap D_j=\emptyset$ for $i\neq j$.
Write $\wt f=(\wt f^0,\ldots,\wt f^{n+m})$, and note that
$h\circ f = |\wt f^0|^2/\|\wt f\|^2$.  Let $\mu_j$ denote the
order of vanishing of $\wt f^0$ at $t_j$, for $1\leq j\leq \ell$.
We choose a sequence of interpolating polynomial maps $\wt
f_\nu:\cb\rightarrow \cb^{n+m+1}$ (see for example [Wa60]) such that
\smallskip
\item{i)} $\wt f_\nu \rightarrow \wt f$ uniformly on $\ov\Delta
_{1-\varepsilon}$,
\smallskip
\item{ii)} $J^k_0(\wt f_\nu) = J_0^k(\wt f)$ for all $\nu$,
\smallskip
\item{iii)} $J^{\mu_j\lambda-1}_{t_j}(\wt f_\nu) = J^{\mu_j\lambda-1}
_{t_j}(\wt f)$ for all $j,\nu$.
\smallskip\noindent
Let $f_\nu =\eta\circ \wt f_\nu:\cb\rightarrow \pb^{n+m}$.  We claim that
there is a positive integer $\nu_0$ such that
$$f_\nu (\ov\Delta_R\ssm\{t_1,\ldots,t_\ell\})\subset U~~~~
\hbox{\rm for}~~\nu>\nu_0\,.\eqno(3.6)$$
Since
$$f_\nu\big(\ov\Delta_R\ssm{\textstyle\bigcup}D_j\big)\subset U~~~~
\hbox{\rm for}~~\nu\gg0\,,$$
to verify (3.6) it suffices to show that
$$f_\nu (D_j\ssm\{t_j\})\subset U~~~~\hbox{\rm for}~~\nu\gg0\eqno(3.7)$$
for $1\leq j\leq \ell$.  By (i)
$$p_i \circ \wt f_\nu \rightarrow p_i\circ \wt f = 0$$
uniformly on $\ov\Delta _{1-\varepsilon}$, for $1\leq i\leq N$.  By (iii),
$$(p_i\circ\wt f_\nu)^{(\alpha)}(t_j) = 0~~~~\hbox{\rm for}~~0\le
\alpha<\mu_j\lambda \,.$$
for all $i,j,\nu$.  Therefore by Lemma~3.3
$${p_i\circ \wt f_\nu(t_j+\zeta)\over \zeta^{\mu_j\lambda}}
\rightarrow 0\eqno(3.8)$$
uniformly for $0<|\zeta|\leq \varepsilon$, and thus
$${g\circ f_\nu(t_j +\zeta)\over |\zeta|^{2\mu_{j}\lambda }}\rightarrow 0
\eqno(3.9)$$
uniformly for $0<|\zeta|\leq\varepsilon$.  By Lemma 3.3 applied to the
sequence $\{(\wt f_\nu^0-\wt f^0)(t_j+\zeta)\}$ with $s=\mu_j$,
we can find a constant $c>0$ such that
$$|\wt f^0_\nu(t_j+\zeta)|\geq c|\zeta|^{\mu_{j}}~~~~\hbox{\rm for}~~
|\zeta|\leq\varepsilon~~\hbox{\rm and}~~\nu\gg 0\eqno(3.10)$$
and in particular,
$$D_j\cap f_\nu^{-1}(A) = \{t\in D_j:h\circ f_\nu (t) = 0\} =
\{t_j\}~~~~\hbox{\rm for}~~\nu\gg 0\eqno(3.11)$$
for $1\leq j\leq\ell$. We conclude from (3.4), (3.9) and (3.10) that
$${g\circ f_\nu\over (h\circ f_\nu)^\lambda }\rightarrow 0\eqno(3.12)$$
uniformly on $D_j\ssm\{t_j\}$. Equation (3.7) then follows from (3.5) and
(3.12).
 
By (3.6) we can define the holomorphic maps
$$F_\nu = \pi_0\circ\psi^{-1}_0 \circ f_\nu :\ov\Delta_R \ssm
\{t_1,\ldots,t_\ell\}\rightarrow X\,,$$
for $\nu \geq \nu_0$.  Since $f_\nu$, $\psi$, and $\pi_0$ are algebraic,
$F_\nu$ is Nash algebraic and thus extends to a Nash algebraic map on
$\ov\Delta_R$.  It remains to show that $F_\nu\rightarrow f$
uniformly.  Clearly, $F_\nu\rightarrow f$ uniformly on every compact
set disjoint from $\{t_1,\ldots,t_\ell\}$.  Let $1\leq j\leq \ell$ be
fixed; it suffices to show that $F_\nu \rightarrow f$ uniformly on
$\Delta _\delta (t_j)$ for some $\delta >0$.  Assume without loss of
generality that $f(t_j)=(0:1:0:\ldots:0)$, let $H_1=\{z\in
\pb^{n+m}:z_1=0\}$, and consider the biholomorphic map
$\varphi:\pb^{n+m}\ssm H_1\rightarrow \cb^{n+m}$ given by
$$\varphi(z_0:\ldots:z_{n+m}) = (z_0/z_1,z_2/z_1,\ldots,z_{n+m}/z_1)\,.$$
Let $B_1=\{w\in \cb^{n+m}:\|w\|<1\}$, $B_0=\big\{w\in \cb^{n+m}:
\|w\|<{1\over 3}\big\}$, and let $B'_\alpha=\varphi^{-1}(B_\alpha)$,
for $\alpha=0, 1$.  We note that
$$\inf\big\{\|x-x'\|:x\in B'_0 \cap \cb^{n+m},~x'\in\cb^{n+m}\ssm B'_1\big\}
>1\,.\eqno(3.13)$$
We now claim that
$$\pi_0(\psi^{-1}_0(U\cap B'_0))\subset B'_1\,.\eqno(3.14)$$
Suppose on the contrary that there is $x\in U\cap B'_0$ such that
$\pi_0\circ \psi^{-1}_0(x) = x'\in \cb^{n+m} \ssm B'_1$.  Let $y =
\psi^{-1}_0(x) \in \Omega_0$. Then
$$x=\psi(y)=\pi_0(y)+\sigma_0(y)=x'+\sigma_0(y)\,,$$
and therefore
$$\|x-x'\|=\|\sigma_0(y)\|<1\,,$$
which contradicts (3.13). Thus (3.14) is verified. Since $\varphi\circ
f(t_j)=0$, we can choose $\delta \in{}]0,\varepsilon[$ such that writing
$K=\ov\Delta _\delta (t_j)$, we have $\varphi\circ f(K)\subset B_0$ and thus
$$\varphi\circ f_\nu(K)\subset B_0~~~~\hbox{\rm for}~~\nu\gg0\,.$$
Therefore by (3.6)
$$f_\nu(K\ssm\{t_j\})\subset U \cap B'_0~~~~\hbox{\rm for}~~\nu\gg0\,.$$
Hence by (3.14)
$$F_\nu(K\ssm\{t_j\})\subset B'_1\,,$$
and hence by the removable-singularities theorem in one-variable
$$(\varphi\circ F_\nu)(K)\subset B_1~~~~\hbox{\rm for}~~\nu\gg 0\,.$$
Since $\varphi\circ F_\nu\rightarrow \varphi\circ f$ uniformly on
$\partial K$, it follows that $\varphi\circ F_\nu\rightarrow\varphi\circ f$
uniformly on $K$.  Since $\varphi$ extends to an automorphism of
$\pb^{n+m}$, we conclude that $F_\nu\rightarrow f$ uniformly on all of
$\ov\Delta_R$.\finpr
\vskip12pt 
 
\titre 4. Complete pluripolar sets and $L^2$ approximation theorem|
 
One difficulty in proving the general case of the approximation
theorem is that we have to deal with Stein submanifolds of projective
manifolds, and such a submanifold $S$ might not be
contained in an affine open set. To overcome this difficulty, we first shrink
$S$ a little bit and apply a small perturbation to make $S$ smooth up to the
boundary with essential singularities at each boundary point. Then $S$
becomes in some sense very far from being algebraic, but with an additional
assumption on $S$ it is possible to show that $S$ is actually contained in an
affine Zariski open subset (Corollary 4.7). The technical tools needed are
H\"ormander's $L^2$ estimates for $\ov\partial$ and the notion of complete
pluripolar set.
 
\than Definition 4.1|Let $Y$ be a complex manifold.
\smallskip
\item{\rm(i)} A function $u:Y\to[-\infty,+\infty[$ is said to be
quasi-plurisubharmonic $($quasi-psh for short$)$ if $u$ is locally equal to
the sum of a smooth function and of a plurisubharmonic (psh) function,
or equivalently, if $\sqrt{-1}\partial\op u$ is bounded below by a continuous
real $(1,1)$-form. \smallskip \item{\rm(ii)} A closed set $P$ is said to be
complete pluripolar in~$Y$ if there exists a quasi-psh function $u$ on $Y$
such that $P=u^{-1}(-\infty)$.
\smallskip\fintha
 
\rem Remark 4.2|Note that the sum and the maximum of two quasi-psh
functions is quasi-psh.  (However, the decreasing limit of a sequence
of quasi-psh functions may not be quasi-psh, since any continuous
function is the decreasing limit of smooth functions and there are many
continuous non-quasi-psh functions, e.g., $-\log^+|z|$.)  The usual
definition of pluripolar sets deals with psh functions rather than
quasi-psh functions; our choice is motivated by the fact that we want
to work on compact manifolds, and of course, there are no nonconstant
global psh functions in that case.  It is elementary to verify that if
$u$ is a quasi-psh function on a Stein manifold $S$, then $u+\varphi$
is psh for some smooth, rapidly growing function $\varphi$ on $S$, and
hence our definition of complete pluripolar sets coincides with the
usual definition in the case of Stein manifolds. Finally, we remark
that the word ``complete'' refers to the fact that $P$ must be the full
polar set of~$u$, and not only part of it.
 
\than Lemma 4.3|Let $P$ be a closed subset in a complex manifold~$Y$.
\smallskip
\item{\rm(i)} If $P$ is complete pluripolar in~$Y$,
there is a quasi-psh function $u$ on $Y$ such that $P=u^{-1}(-\infty)$
and $u$ is smooth on $Y\ssm P$.
\smallskip
\item{\rm(ii)} $P$ is complete pluripolar in $Y$ if and
only if there is an open covering $(\Omega_j)$ of $Y$ such that
$P\cap\Omega_j$ is complete pluripolar in~$\Omega_j$.
\smallskip
\fintha
 
\proof We first prove result (i) locally, say on a ball $\Omega=B(0,r_0)
\subset\cb^n$, essentially by repeating the arguments given in Sibony
[Sib85]. Let $P$ be a closed set in~$\Omega$, and let
$v$ be a psh function on $\Omega$ such that $P=v^{-1}(-\infty)$.
By shrinking $\Omega$ and subtracting a constant from $v$, we may assume
$v\leq 0$. Select a convex increasing function $\chi:[0,1]\to\rb$ such that
$\chi(t)=0$ on $[0,1/2]$ and $\chi(1)=1$. We set
$$w_k=\chi\big(\exp(v/k)\big).$$
Then $0\leq w_k\le1$, $w_k$ is plurisubharmonic on $\Omega$ and
$w_k=0$ in a neighborhood of~$P$. Fix a family $(\rho_\varepsilon)$
of smoothing kernels and set $\Omega'=B(0,r_1)\compact\Omega$.
For each~$k$, there exists $\varepsilon_k>0$ small such that
$w_k\star\rho_{\varepsilon_k}$ is well defined on $\Omega'$ and vanishes on
a neighborhood of $P\cap\Omega'$. Then
$$h_k=\max\big\{w_1\star\rho_{\varepsilon_1},\ldots,w_k\star
\rho_{\varepsilon_k}\big\}$$
is an increasing sequence of continuous psh functions such that
$0\leq h_k\leq 1$ on $\Omega'$ and $u_k=0$ on a neighborhood of
$P\cap\Omega'$.  The inequalities
$$h_k\geq w_k\star\rho_{\varepsilon_k}\geq w_k=\chi\big(\exp(v/k)\big)$$
imply that $\lim h_k=1$ on $\Omega'\ssm P$. By Dini's lemma, $(h_k)$
converges uniformly to~$1$ on every compact subset of~$\Omega'\ssm P$.
Thus, for a suitable subsequence $(k_\nu)$, the series
$$h(z):=|z|^2+\sum_{\nu=0}^{+\infty}(h_{k_\nu}(z)-1)\eqno(4.1)$$
converges uniformly on every compact subset of $\Omega'\ssm P$ and defines
a strictly plurisubharmonic function on $\Omega'$ which is continuous on
$\Omega'\ssm P$ and such that $h=-\infty$ on $P\cap\Omega'$. Richberg's
regularization theorem [Ri68] implies that there is a psh function $u$
on $\Omega'$ such that $u$ is smooth on $\Omega'\ssm P$ and $h\leq u\leq h+1$.
Then $u^{-1}(-\infty)=P\cap\Omega'$ and Property (i) is proved
on~$\Omega'$.
 
(ii) The ``only if'' part is clear, so we just consider the ``if'' part.
By means of (i) and by taking a refinement of the covering,
we may assume that we have open sets
$\wt\Omega_j\supset\!\supset\Omega_j\supset\!\supset\Omega'_j$ where
$(\Omega'_j)$ is a locally finite open
covering of~$Y$ and for each $j$ there is a psh function $u_j$
on $\wt\Omega_j$ such that
$P\cap\wt\Omega_j=u_j^{-1}(-\infty)$ with $u_j$ smooth on~$\wt\Omega_j\ssm P$.
We define inductively a strictly decreasing sequence
$(t_\nu)$ of real numbers by setting $t_0=0$ and
$$t_{\nu+1}\leq\inf\big\{\bigcup_{j,k\leq\nu}
u_j\big(\,\ov\Omega_j\cap\ov\Omega_k\ssm
u_k^{-1}([-\infty,t_\nu[)\big)\big\}\;.\eqno(4.2)$$
The infimum is finite since $u_j$ and $u_k$ have the same poles. Now,
if we take in addition $t_{\nu+1}-t_\nu<t_\nu-t_{\nu-1}$, there
exists a smooth convex function $\chi:\rb\to\rb$ such that
$\chi(t_\nu)=-\nu$.  Choose functions $\varphi_j\in C^\infty(\Omega_j)$
such that $\varphi_j \equiv 0$ on $\Omega'_j$,\break
$\varphi_j \leq 0$ on $\Omega_j$, and $\varphi_j(z)\to -\infty$ as $z
\to \partial\Omega_j$.  We set $$v_j=\chi\circ u_j+\varphi_j\;.$$
Then $v_j$ is quasi-psh on $\Omega_j$.  Let
$$v(z)=\max_{\Omega_j\ni z}~~v_j(z),~~~~z\in V\;.\eqno(4.3)$$
Clearly $v^{-1}(-\infty)=P$.  To show that $v$ is quasi-psh it suffices to
verify that if $z_0\in \partial \Omega_k$, then for $z\in\Omega_k\ssm P$
sufficiently close to $z_0$ we have $v_k(z)<v(z)$.  This is obvious if
$z_0\not\in P$ since $v_k\to -\infty$ as $z\to z_0$.
So consider $z_0\in \partial \Omega_k\cap P\cap \Omega'_j$.  Suppose that
$z\in \Omega_k\cap\Omega'_j\ssm P$ is sufficiently close to $z_0$ so that
$$\max\{u_j(z),u_k(z)\}<\min\{t_j,t_k\}~~~\hbox{and}~~~\varphi_k(z)\leq
-3\;.$$
Then we have $u_j(z)\in[t_{\nu+1},t_\nu[$ and
$u_k(z)\in[t_{\mu+1},t_\mu[$ with indices $\mu,\nu\geq\max\{j,k\}$. 
Then by (4.2), $u_j(x)\geq t_{\mu+2}$ and $u_k(x)\geq t_{\nu+2}$; hence
$|\mu-\nu|\leq 1$ and $|\chi\circ u_j(z)-\chi\circ u_k(z)|\leq 2$.
Therefore
$$v(z)\geq v_j(z)=\chi\circ u_j(z)\geq\chi\circ u_k(z)-2\geq v_k(z)+1\;,$$
which completes the proof of (ii).
 
To obtain the global case of property (i), we replace the max function
in (4.3) by the regularized max functions

$${\textstyle\max_\varepsilon}=\max{}\star\rho^m_\varepsilon:
\rb^m\rightarrow \rb\;,$$
where $\rho^m_\varepsilon$ is of the form
$\rho_\varepsilon^m(x_1,\ldots,x_m)=\varepsilon^{-m}\rho(x_1/\varepsilon)
\cdots\rho(x_m/\varepsilon)$. This form of smoothing kernel ensures that
$$x_m\leq\max\{x_1,\ldots,x_{m-1}\}-2\varepsilon\Rightarrow
{\textstyle \max_\varepsilon}(x_1,\ldots,x_m)={\textstyle \max_\varepsilon}
(x_1,\ldots,x_{m-1})\,.$$
(Of course, $\max_\varepsilon $ is convex and is invariant under
permutations of variables.) The function $v_\varepsilon $ obtained by
using $\max_\varepsilon $ in (4.3) with any $\varepsilon \leq 1/2$ is
quasi-psh on $Y$, smooth on $Y\ssm P$, and has polar set $P$.
\finpr\vskip7pt
 
Now we show that for any Stein submanifold $S$ of a complex manifold~$Y$,
there are complete pluripolar graphs of sections $K\to N_S$ over arbitrary
compact subsets~$K\subset S$.
 
\than Proposition 4.4|Let $S$ be a $($not necessarily closed$)$ Stein
submanifold of a complex manifold~$Y$, with $\dim S<\dim Y$, and let
$K$ be a holomorphically convex compact subset of~$S$.
Let $\psi:U\to V$ be as in $(2.1)\,${\rm(i)}.
Then there is a continuous section $g:K\to N_S$ which is
holomorphic in the interior~$K^\circ$ of~$K$, such that
$g(K)\subset U$ and the sets $K_\varepsilon= \psi(\varepsilon g(K))$ are
complete pluripolar in~$Y$ for $0<\varepsilon\leq 1$.\fintha
 
Here $\varepsilon g(K)$ denotes the $\varepsilon$-homothety of the
compact section $g(K)\subset N_S$; thus $K_\varepsilon$ converges to $K$ as
$\varepsilon$ tends to~$0$.
 
\proof By 4.3(ii), the final assertion is equivalent to proving
that $\varepsilon g(K)$ is complete pluripolar in $N_S$,
and we may of course assume that $\varepsilon =1$.  The
holomorphic convexity of $K$ implies the existence of a
sequence $(f_j)_{j\geq 1}$ of holomorphic functions on $S$
such that $K=\cap \{|f_j|\leq 1\}$.  By Lemma 2.3, there is
a finite sequence $(g_k)_{1\leq k\leq N}$ of holomorphic
sections of $N_S$ without common zeroes.  We consider the
sequence $(j_\nu, k_\nu)_{\nu\geq 1}$ of pairs of positive
integers obtained as the concatenation $\Lambda_1,
\Lambda_2,\ldots,\Lambda_\ell,\ldots$ of the finite sequences
$$\Lambda_\ell = (j_q, k_q)_{\nu(\ell)\leq
q<\nu(\ell+1)}=\left((1,1),\ldots,(1,N),\ldots,(\ell,
1),\ldots,(\ell,N)\right)$$
(where $\nu(\ell) = N\ell(\ell-1)/2+1)$.  We define $g$ to
be the generalized ``gap sequence"
$$g(z) = \sum^\infty_{\nu=1} e^{-p_{\nu}}
f_{j_{\nu}}(z)^{p_{\nu}^{2}}g_{k_{\nu}}(z)~~~~\forall z\in K\;,\eqno(4.4)$$
where $(p_\nu)$ is a strictly increasing sequence of
positive integers to be defined later.  The series (4.4)
converges uniformly on $K$ (with respect to a Hermitian
metric on $N_S$) and thus $g$ is continuous on $K$ and
holomorphic on $K^\circ$.  To construct our psh function with
polar set $g(K)$ we first select a smooth Hermitian metric
$\|~~\|$ on $N_S$ such that $\zeta \mapsto \log\|\zeta\|$
is psh on $N_S$ (e.g., take $\|\zeta\|^2 = \sum
|g^\star_i(z)\cdot \zeta|^2$ for $\zeta\in N_{S,z}$, where
$(g^\star_i)$ is a collection of holomorphic sections of $N_S^\star$
without common zeroes).  We consider the global holomorphic sections
$$s_q = \sum^q_{\nu=1} e^{-p_{\nu}} f_{j_{\nu}}^{p_{\nu}^{2}}
g_{k_\nu}\eqno(4.5)$$ of $N_S$, which converge to $g$ uniformly on
$K$ exponentially fast.  We consider the psh functions $u_\ell$ on $N_S$
given by
$$u_\ell (\zeta) = \max_{\nu(\ell)\leq q<\nu (\ell+1)}
{1\over p^4_q} \log \|\zeta - s_q (z) \|\;,~~~~\forall
\zeta\in N_{S,z},\;\; \forall z\in S\;.\eqno(4.6)$$
We shall show that the infinite sum
$$u = \sum^\infty_{\ell=1} \max (u_\ell,-1)\eqno(4.7)$$
is psh on $N_S$ with $u^{-1} (-\infty) = g(K)$.  For
$z\in K$, $\zeta = g(z) \in g(K)$, we obtain from (4.4) the
estimate
$$\|\zeta-s_q(z)\|\leq Ce^{-p_{q+1}}$$
where $C$ is independent of the choice of $z\in K$.  Thus
if we choose the $p_q$ with $p_{q+1}\geq 2p^4_q$, we have
$u_\ell < -1$ on $g(K)$ for $\ell$ large, and thus
$u\equiv -\infty$ on $g(K)$.  Next we show that $u$ is
psh.  For this it suffices to show that
$\sum^\infty_{\ell=1} \sup_Au_\ell<+\infty$ for all compact
subsets $A$ of $N_S$.  Let $S_q$ be an exhausting sequence
of compact subsets of $S$.  When $\zeta\in N_{S,z}$ with
$z\in S_q$, we have
$$\log\|\zeta-s_q(z)\|\leq \log (1+\|\zeta\|) + (p^2_q + 1)
\log (1+M_q)$$
where $M_q$ is the maximum of $\|g_1\|,\ldots,\|g_N\|,
|f_1|,\ldots,|f_q|$ on $S_q$.  In addition to our previous
requirement, we assume that $p_q \geq \log (1+M_q)$ and hence
$$u_\ell (\zeta)\leq {2 + \log
(1+\|\zeta\|) \over p_{\nu(\ell)}},\;\;\forall \zeta\in
N_{S, z},\;\;\forall z\in S_{\nu(\ell)}\;.$$
Since $p_{\nu(\ell)} \geq p_\ell \geq 2^{\ell-1}$ (in fact, the growth is at
least doubly exponential), it follows that the series $\sum u_\ell$ converges
to a psh function $u$ on $N_S$.
 
We finally show that $u(\zeta)$ is finite outside $g(K)$. Fix a point
$\zeta \in N_{S, z} - g(K)$.  If $z\in K$, then the logarithms
appearing in (4.6) converge to $\log\|\zeta-g(z)\|>-\infty$, and hence
$u(\zeta)>-\infty$. Now suppose $z\not\in K$. Then one of the values
$f_j(z)$ has modulus greater than~$1$. Thus for $\ell$ sufficiently
large, there is a $q=q(\ell)$ in the range $\nu(\ell)\leq
q<\nu(\ell+1)$ such that $|f_{j_{q}}(z)|>1$ and $\|g_{k_{q}}(z)\| =
\max\|g_i(z)\|>0$. Assume further that $|f_{j_{q}}(z)|=\max_{1\leq
i\leq\ell}|f_i(z)|$. For this choice of $q$ we have
$$\|\zeta-s_q(z)\|\geq e^{-p_{q}}|f_{j_{q}}(z)|^{p_{q}^{2}}|g_{k_{q}}(z)|-
{|f_{j_{q}}(z)|^{p_{q-1}^2+1}|g_{k_{q}}(z)|\over|f_{j_{q}}(z)|-1}-
\|\zeta\|\;.$$
Since $p_q\geq 2p_{q-1}^4$, we easily conclude that $\|\zeta -
s_{q(\ell)} (z) \|>1$ for $\ell$ sufficiently large and hence $u_\ell
(z)$ is eventually positive. Therefore $u(z)>-\infty$; hence $u^{-1}
(-\infty)=g(K)$, and so $g(K)$ is complete pluripolar in $N_S$.\finpr
 
\rem Remark|If one chooses $p_\nu$ to be larger than the norm of the
first $\nu$ derivatives of $f_1,\ldots,f_\nu$ on $K$ in the above
proof, the section $g$ will be smooth on $K$.
 
\than Proposition 4.5|Let $K\i S\i Y$ be as in
Proposition 4.4, and let $I$ be a finite subset of the
interior $K^\circ$ of $K$.  Then for every positive integer
$m$ we can select the section $g:K\rightarrow N_S$ in
Proposition 4.4 with the additional property that the
$m$-jet of $g$ vanishes on $I$ $($and thus $K_\varepsilon $
is tangent to $K$ of order $m$ at all points of $I)$.\fintha
 
\proof The proof of Proposition 4.4 used only the fact that
the sections $g_1\ld g_N$ do not vanish simultaneously on
$S\ssm K$. Hence, we can select the $g_j$ to vanish at the
prescribed order $m$ at all points of $I$.\finpr\vskip7pt
 
Next, we prove the following simple approximation theorem based on
H\"ormander's $L^2$ estimates (see Andreotti-Vesentini [AV65] and
[H\"o66]).
 
\than Proposition 4.6|Let $X$ be a projective algebraic
manifold and let $L$ be an ample line bundle on~$X$. Let $P$ be a
complete pluripolar set in $X$ such that $L$ is trivial on a
neighborhood $\Omega$ of~$P$; i.e., there is a nonvanishing section
$s\in H^0(\Omega,L)$. Then there is a smaller neighborhood $V\subset
\Omega$ such that every holomorphic function $h$ on  $\Omega$ can be
approximated uniformly on $V$ by a sequence of global sections $h_\nu\in
H^0(X,L^{\otimes\nu})$; precisely, $h_\nu s^{-\nu}\to h$ uniformly
on~$V$. \fintha
 
\proof By Lemma 4.3(i), we can choose a quasi-psh
function $u$ on $X$ with $u^{-1}(-\infty)=P$ and $u$~smooth on $X\ssm P$.
Fix a K\"ahler metric $\omega$ on $X$ and a Hermitian metric on $L$
with positive definite curvature form $\Theta(L)=\sqrt{-1}
\partial\op\varphi$, where $e^{-\varphi}$ is the weight representing the
metric of~$L$. Precisely, let $(U_\alpha)_{0\leq\alpha\leq N}$ be an open
cover of $X$ with trivializing sections $s_\alpha\in H^0(U_\alpha,L)$.  We let
$\|s_\alpha\|^2 = e^{-\varphi_\alpha}$ so that for $v=v^\alpha s_\alpha(z)\in
L_z$ we have $\|v\|^2=|v^\alpha|^2 e^{-\varphi_\alpha(z)}$.  In the following,
we shall take $U_0=\Omega$ and $s_0=s$.  We shall also drop the index $\alpha$
and denote the square of the norm of $v$ simply by $\|v\|^2=|vs^{-1}|^2
e^{-\varphi(z)}$.
 
By multiplying $u$ with a small positive constant,
we may assume that $\Theta(L)+\sqrt{-1}\partial\op
u\geq\varepsilon\omega$ for some $\varepsilon>0$. Then, for $\nu\in\nb$ large,
H\"ormander's $L^2$ existence theorem (as given in [De92, Theorem 3.1])
implies that for every $\op$-closed $(0,1)$-form $g$ with values in
$L^{\otimes\nu}$ such that $$\int_X\|g\|^2e^{-\nu u}dV_\omega<+\infty$$
(with $dV_\omega={}$K\"ahler volume element), the equation $\op f=g$ admits
a solution $f$ satisfying the estimate
$$\int_X|fs^{-\nu}|^2e^{-\nu(\varphi+u)}dV_\omega
=\int_X\|f\|^2e^{-\nu u}dV_\omega\le{1\over\lambda_\nu}
\int_X\|g\|^2e^{-\nu u}dV_\omega,$$
where $\lambda_\nu$ is the minimum of the eigenvalues of
$$\nu\big(\Theta(L)+\sqrt{-1}\partial\op u\big)+{\rm Ricci}(\omega)$$
with respect to $\omega$ throughout~$X$. Clearly
$\lambda_\nu\geq\varepsilon\nu+\rho_0$
where $\rho_0\in\rb$ is the minimum of the
eigenvalues of Ricci$(\omega)$. We apply the existence theorem to the
$(0,1)$-form $g=\op(h\theta s^\nu)=h\op\theta s^\nu$, where $\theta$
is a smooth cut-off function such that
$$\eqalign{
&\theta=1~~~~\hbox{\rm on}~~\big\{z\in\Omega\,:\,\varphi(z)+u(z)\le
c\big\},\cr
&\Supp\,\theta\subset\big\{z\in\Omega\,:\,\varphi(z)+u(z)<c+1\big\}
\compact \Omega\cr}$$
for some sufficiently negative $c<0$. Then we get a
solution of the equation $\op f_\nu=h\op\theta s^\nu$ on $X$ with
$$\int_X|f_\nu s^{-\nu}|^2e^{-\nu(\varphi+u)}dV_\omega
\le{1\over\varepsilon\nu+\rho_0}
\int_X|h|^2|\op\theta|^2e^{-\nu(\varphi+u)}dV_\omega.$$
Since $\Supp(\op\theta)\subset \{c\leq\varphi(z)+u(z)\leq c+1\}$, we infer
$$e^{-\nu(c-1)}\int_{\{\varphi+u<c-1\}}|f_\nu s^{-\nu}|^2dV_\omega
\le{e^{-\nu c}\over\varepsilon\nu+\rho_0}\int_\Omega
|h|^2|\op\theta|^2dV_\omega.$$
Thus $f_\nu s^{-\nu}\to 0$ in $L^2(\{\varphi+u<c-1\})$.  Since the
$f_\nu s^{-\nu}$ are holomorphic functions on $\{\varphi+u<c-1\}$, it
follows that $f_\nu s^{-\nu}\to 0$ uniformly on the neighborhood
$V=\{\varphi+u<c-2\}\compact\Omega$ of~$P$.  Hence $h_\nu=h\theta
s^\nu-f_\nu$ is a holomorphic section of $L^{\otimes\nu}$ such that
$h_\nu s^{-\nu}$ converges uniformly to~$h$ on~$V$.\finpr
 
\than Corollary 4.7|Let $X$ be a projective algebraic manifold and let
$S$ be a Stein submanifold with $\dim S<\dim X$, such that there exists
an ample line bundle $L$ on $X$ with trivial restriction~$L_{|S}$.
Let $K$ be a holomorphically convex compact subset of~$S$, and let
$(K_\varepsilon)_{0<\varepsilon\leq 1}$ be the complete pluripolar
sets constructed in Proposition~$4.4$. Then
for each sufficiently small $\varepsilon$, there is a positive integer
$\nu(\varepsilon)$ and a section $h_{\nu(\varepsilon)}\in
H^0(X,L^{\otimes\nu(\varepsilon)})$ such that $K_\varepsilon$ does not
intersect the divisor of~$h_{\nu(\varepsilon)}$; i.e., $K_\varepsilon$ is
contained in the affine Zariski open set
$X\ssm h_{\nu(\varepsilon)}^{-1}(0)$.\fintha
 
\proof By Lemma~2.2, $L$ is trivial on a neighborhood $\Omega$ of~$S$.
For $\varepsilon>0$ small, we have $K_\varepsilon\subset\Omega$ and we can
apply Proposition~4.6 to get a uniform approximation $h_{\nu(\varepsilon)}
\in H^0(X,L^{\otimes\nu(\varepsilon)})$ of the function $h=1$ such that
$|h_{\nu(\varepsilon)}s^{-\nu(\varepsilon)}-h|<1/2$ near~$K_\varepsilon$.
Then $K_\varepsilon\cap h_{\nu(\varepsilon)}^{-1}(0)=\emptyset$.\finpr
 
\rem Remark 4.8|The existence of an ample line bundle on $X$ with
trivial restriction to $S$ is automatic if
$H^1(S,\oc_S^\star)=H^2(S,\zb)=0$, since in that case every holomorphic
line bundle on $S$ is trivial. In Corollary~4.7, the conclusion that
$K$ admits small perturbations $K_\varepsilon$ contained in affine
Zariski open sets does not hold any longer if the hypothesis on the
existence of an ample line bundle $L$ with $L_{|S}$ trivial is omitted.
In fact, let $S$ be a Stein manifold and let $K$ be a holomorphically
convex compact set such that the image of the restriction map
$H^2(S,\zb)\to H^2(K,\zb)$ contains a nontorsion element. Then there
exists a holomorphic line bundle $E$ on $S$ such that all positive
powers $E_{|K}^{\otimes m}$ are topologically nontrivial. Let
$g_0,\dots,g_N\in H^0(S,E)$ be a collection of sections which generate
the fibres of $E$ and separate the points of~$S$. By the proof of
Corollary~2.4, we find a holomorphic embedding $\varphi:S\to\pb^N$ into
some projective space, such that $E\simeq\varphi^\star\oc(1)$. Since
$\varphi^\star\oc(m)_{|K}\simeq E_{|K}^{\otimes m}$, we infer that
$\oc(m)_{|\varphi(K)}$ is topologically nontrivial for all~$m>0$. It
follows that $\varphi(K)$ cannot be contained in the complement
$\pb^N\ssm H$ of a hypersurface of degree~$m$, because $\oc(m)$ is
trivial on these affine open sets. The same conclusion holds for
arbitrary homotopic deformations $\varphi(K)_\varepsilon$
of~$\varphi(K)$.
 
To give a specific example, let $S = \cb^\star \times \cb^\star$ and
$K=\partial \Delta \times \partial \Delta$.  We consider the torus
$T=\cb /\{1,\sqrt{-1}\}\zb$, which has a holomorphic embedding
$g:T\rightarrow \pb^2$.  Let $\varphi:S\rightarrow \pb^8$ be the
holomorphic embedding given by $\varphi (z, w) = \psi (g(t), (1:z:w))$
where $\psi:\pb^2 \times \pb^2\rightarrow \pb^8$ is the Segre embedding
and $t = {\sqrt{-1} \over 2\pi} \log z + {1\over 2\pi} \log w$. Since
$\varphi (K)$ is a $C^\infty$ deformation of the elliptic curve $A
=\psi (g(T)\times \{(1:0:0)\})\i \pb^8$, then for any deformation
$\varphi(K)_\varepsilon $ of $\varphi(K)$ and for any algebraic
hypersurface $H\i\pb^8$ we have $\varphi(K)_\varepsilon \cdot H=A\cdot H>0$
(where $\cdot$ denotes the intersection product in the homology of
$\pb^8$), and therefore $\varphi (K)_\varepsilon \cap H\neq \emptyset$.

\rem Remark 4.9|Under the hypotheses of Corollary~4.7, it is not true
that $S$ is contained in a Zariski open subset of~$X$, even if we
assume that~$H^2(S,\zb)=0$. In fact, a simple counterexample is given
by a dense embedding $S=\cb\hookrightarrow X$ of the complex line into an
abelian surface~$X\,$; every germ of analytic curve which does not intersect
the line $S$ must have its tangents parallel to $S$, hence there cannot
exist any closed curve avoiding~$S$. We do not know in general whether
every compact subset of $S$ is contained in a Zariski open subset of~$X$.
\vskip12pt

\bigtitle 5. Algebraic approximation on Runge domains|
of affine algebraic manifolds|
 
In this section, we prove the main approximation theorem 1.1 for
holomorphic mappings on a Runge domain $\Omega$ in an affine algebraic
manifold.  We also prove a related result on the exhaustion of holomorphic
vector bundles on $\Omega$ by algebraic vector bundles (Proposition~5.3).
 
To prove Theorem 1.1, we first make a reduction to the following situation:
\smallskip(i) $f$ is an embedding;\vskip0pt
(ii) there is an ample line bundle $L$ on $X$ such that $f^\star L$ is
trivial on~$\Omega$.
\smallskip\noindent In fact, (i) is achieved by replacing $f$ with
$$f'=(i_{|\Omega},f):\Omega\to X'=\wh S\times X$$
where $i:S\to\wh S$ is a smooth projective compactification of~$S$
($\wh S$ exists by Hironaka's desingularization theorem [Hi64]).
Now, if (ii) does not hold for~$X'$, we fix an ample line bundle $L'$
on $X'$ and get by Corollary~2.4 applied to~$E=f^{\prime\star}L^{\prime-1}$
a holomorphic map $\varphi:\Omega\to\pb^m$ such that
$f^{\prime\star}L^{\prime-1}= \varphi^\star\oc(1)$. We set
$$f''=(\varphi,f'):\Omega\to X''=\pb^m\times X'$$
and let $L''\to X''$ be the total tensor product $L''=\oc(1)\stimes L'$.
Then $L''$ is ample on $X''$ and $f^{\prime\prime\star}L''=\oc_\Omega$.
Now, suppose that the conclusion holds for $f'':\Omega\to X''$, i.e.
$f''=\lim f''_\nu$ where $f''_\nu:\Omega_\nu\to X''$ is a Nash algebraic
map with the same $m_\nu$-jet as~$f''$ at all points~$t_{j,\nu}$.
Let $p:X''\to X$ be the projection. The maps $f_\nu=p\circ f''_\nu:
\Omega_\nu \to X$ converge to~$f=p\circ f''$ and satisfy the
conclusions of the theorem. Thus Theorem 1.1 is reduced to the
following:
 
\than Proposition 5.1|Let $\Omega$ be a Runge domain in an affine
algebraic manifold $S$, and let $(\Omega_\nu)$ be an increasing
sequence of relatively compact domains $\Omega_\nu\compact\Omega$ with
$\Omega=\bigcup\Omega_\nu$. Suppose $f:\Omega\to X$ is an embedding
into a projective algebraic manifold $X$ $($with $\dim S<\dim X)$ such
that there exists an ample line bundle $L$ on~$X$ with trivial restriction
$f^\star L$ to~$\Omega$. Then one can find sections $h_\nu\in
H^0(X,L^{\otimes\mu(\nu)})$ and Nash algebraic embeddings $f_\nu:\Omega_\nu
\to X\ssm h_\nu^{-1}(0)$ converging to~$f$ $($and satisfying the other
conclusions of Theorem 1.1 as well$)$.\fintha
 
\proof By enlarging the $\Omega_\nu$ if necessary, we can assume that
the $\Omega_\nu$ are Runge domains in $\Omega$ (and hence in $S$). By
Corollary~4.7 applied to the compact set
$$K_\nu=\hbox{holomorphic hull of $f(\ov\Omega_{\nu+1})$ in the Stein
variety $f(\Omega)\subset X$},$$
we get an embedding $j_\varepsilon: K_\nu\to X$ of the form
$j_\varepsilon(z)= \psi(\varepsilon g(z))$, converging to $\Id_{K_\nu}$
as $\varepsilon$ tends to~$0$, such that $j_\varepsilon(K_\nu)$ is
contained in an affine Zariski open subset~$Y_{\nu,\varepsilon}=X\ssm
h_{\nu,\varepsilon}^{-1}(0)$, $h_{\nu,\varepsilon}\in
H^0(X,L^{\otimes\mu(\nu,\varepsilon)})$. Moreover, by Proposition~4.5,
we can take $j_\varepsilon$ tangent to $\Id_{K_\nu}$ at order $m_\nu$ at
all points~$t_{j,\nu}$. We first approximate $f$ by $F_\nu=j_{\varepsilon
_\nu}\circ f$ on $\Omega_{\nu+1}$, choosing $\varepsilon_\nu$ so small
that $\sup_{\Omega_{\nu+1}}\delta(F_\nu(z),f(z))<1/\nu$ with respect to
some fixed distance $\delta$ on~$X$. We denote
$Y_\nu=Y_{\nu,\varepsilon_\nu}$ in the sequel. By construction,
$F_\nu(\Omega_{\nu+1})\subset j_{\varepsilon_\nu}(K_\nu)\subset Y_\nu$.
 
Let $Y_\nu\subset\cb^N$ be an algebraic embedding of $Y_\nu$ in some affine
space. Then $F_\nu$ can be viewed as a mapping
$$F_\nu=(F_{1,\nu}\ld F_{N,\nu}):\Omega_{\nu+1}\to\cb^N,~~~~
F_\nu(\Omega_{\nu+1})\subset Y_\nu.$$
By Lemma~2.1, there is a retraction $\rho_\nu:V_\nu\to Y_\nu$ defined on a
neighborhood $V_\nu$ of~$Y_\nu$, such that the graph $\Gamma_{\rho_\nu}$
is contained in an algebraic variety $A_\nu$ with $\dim A_\nu=N$.
Now, since $\Omega_{\nu+1}$ is a Runge domain in~$S$, there are polynomial
functions $P_{k,\nu}$ in the structure ring $\cb[S]$, such that
$F_{k,\nu}-P_{k,\nu}$ is uniformly small on $\ov\Omega_\nu$
for each $k=1\ld N$. Of course, we can make a further finite interpolation
to ensure that $P_{k,\nu}-F_k$ vanishes at the prescribed order $m_\nu$
at all points~$t_{j,\nu}$. We~set
$$P_\nu=(P_{1,\nu}\ld P_{N,\nu}):S\to\cb^N,~~~~f_\nu=\rho_\nu\circ P_\nu
:\Omega_\nu\to Y_\nu\;,$$ and take our approximations $P_{k,\nu}$
sufficiently close to $F_{k,\nu}$, in such a way that $f_\nu$ is well defined
on $\Omega_\nu$ (i.e. $P_\nu(\Omega_\nu)\subset V_\nu$), is an embedding of
$\Omega_\nu$ into $Y_\nu$, and satisfies $\delta(f_\nu,F_\nu)<1/\nu$
on~$\Omega_\nu$. Then $\delta(f_\nu,f)<2/\nu$ on~$\Omega_\nu$. Now,
the graph of~$f_\nu$ is just the submanifold
$$\Gamma_{f_\nu}=(\Omega_\nu\times Y_\nu)\cap(P_\nu\times\Id_{Y_\nu})^{-1}
(\Gamma_{\rho_\nu}).$$
Each connected component of $\Gamma_{f_\nu}$ is contained in an irreducible
component $G_{\nu,j}$ of dimension $n=\dim S$ of the algebraic variety
$(P_\nu\times\Id_{Y_\nu})^{-1}(A_\nu)$, hence $\Gamma_{f_\nu}$ is
contained in the $n$-dimensional algebraic variety $G_\nu=\bigcup
G_{\nu,j}$.\finpr
 
\than Corollary 5.2|Let $\Omega_\nu\compact\Omega\subset S$ be as in
Proposition 5.1. Suppose that $H^2(\Omega; \zb)=0$. Then for every
holomorphic embedding $f:\Omega\to X$ into a projective algebraic manifold
$X$ (with $\dim S<\dim X$), one can find affine Zariski open sets
$Y_\nu\subset X$ and Nash algebraic embeddings $f_\nu :\Omega_\nu\to Y_\nu$
converging to~$f$.\fintha
 
The ideas of the proof of Proposition 5.1 can be used to yield the
following related result on the exhaustion of holomorphic vector bundles by
their algebraic counterparts. A similar result has been announced by
Tancredi and Tognoli ([TT92], Theorem~4.1).
 
\than Proposition 5.3|Let $\Omega$ be a Runge domain in an $n$-dimensional
affine algebraic manifold $S$ and let $E\to\Omega$ be a holomorphic vector
bundle. Then for every relatively compact domain $\Omega_0\compact\Omega$
there exist an $n$-dimensional projective algebraic manifold $Z$, a
Nash algebraic embedding $i:\Omega_0\hookrightarrow Z$, and an algebraic
vector bundle $\wt E\to Z$ such that $i^\star\wt E$ is isomorphic
to $E_{|\Omega_0}$. Moreover, $Z$ can be chosen in such a way that there
exists an ample line bundle $L\to Z$ with trivial restriction $i^\star L$.
\fintha
 
\proof By Corollary~2.4, there is a holomorphic map $\varphi:\Omega\to G$
into a Grassmannian, such that $E\simeq\varphi^\star Q$ where $Q$ is the
universal quotient vector bundle. We construct a holomorphic map of the form
$$f=(\varphi,\varphi'):\Omega\to X= G\times\pb^N,$$ where
$\varphi':S\to\pb^N$ is chosen in such a way that $f$ is an embedding and $X$
has an ample line bundle $L$ with trivial pull-back $f^\star L$ (using the
same argument as in the reduction of Theorem~1.1 to Proposition~5.1).  Then
$E\simeq f^\star Q_X$, where $Q_X$ is the pull-back of $Q$ to~$X$.
 
Choose a Runge domain $\Omega_1$ with $\Omega_0\compact\Omega_1\compact\Omega$.
By Proposition 5.1, $f_{|\Omega_1}$ is a uniform limit of Nash algebraic maps
$f_\nu:\Omega_1\to X$. Hence the images $f_\nu(\Omega_1)$ are contained in
algebraic subvarieties $A_\nu\subset X$ of dimension $n=\dim S$.  Let
$f'=f_{\mu|\Omega_0}$, where $\mu$ is chosen to be sufficiently large so that
$f'$ is homotopic to~$f_{|\Omega_0}$.  Then the holomorphic vector bundle
$E':=f'^\star Q_X$ is topologically isomorphic to~$E_{|\Omega_0}$. By~a
theorem of Grauert [Gr58], $E'\simeq E_{|\Omega_0}$ as holomorphic vector
bundles.  Let $\sigma:A''_\mu\to A_\mu$ be the normalization of~$A_\mu$ and
let $f'':\Omega_0\to A''_\mu$ be the map such that $f'=\sigma\circ f''\,$;
since $f'$ is an embedding into~$X$, we see that $f''(\Omega_0)$ is
contained in the set of regular points of~$A''_\mu$. (The reason for the
introduction of the normalization is that $f'(\Omega_0)$ can contain multiple
points of~$A_\mu$.) By Hironaka's resolution of singularities [Hi64], there
exists a desingularization $\pi:Z\to A''_\mu$ with center contained in
the singular locus of~$A''_\mu$. Let $i:\Omega_0 \hookrightarrow Z$
denote the embedding given by $f''=\pi\circ i$. We note that the
exceptional divisor of $Z$ does not meet $i(\Omega_0)$. The algebraic
vector bundle $\wt E:=(\sigma\circ\pi)^\star Q_X$ then satisfies
$$i^\star\wt E=(\sigma\circ\pi\circ i)^\star Q_X=f^{\prime\star}Q_X
=E'\simeq E_{|\Omega_0}.$$
 
Finally, we note that the line bundle $L'':=\sigma^\star L_{|A_\mu}$ is ample
on $A''_\mu$ and thus there is an embedding $A''_\mu\subset\pb^m$ such that
$\oc(1)_{|A''_\mu}=L^b$ for some $b\in\nb$ (where $\oc(1)$ is the hyperplane
section bundle in $\pb^m$).  We can suppose that $\pi$ is an embedded
resolution of singularities with respect to this embedding; i.e., there is a
modification $\pi:\wt X\to\pb^m$ such that $\wt X$ is smooth and $Z\subset\wt
X$ is the strict transform of $A''_\mu$. Then there is an exceptional divisor
$D\geq 0$ of the modification $\pi:\wt X\to\pb^m$ such that
$\wt L:=\pi^\star\oc(a)\otimes\oc(-D)$ is ample on $\wt X$ for $a\gg 0$.
(We can take $D=\pi^\star[\pi(H)]-[H]$, for an ample hypersurface
$H\subset\wt X$ that does not contain any component of the exceptional divisor
of $\wt X$.) We can assume that $\mu$ has been chosen sufficiently large so
that $f'^\star L\simeq (f^\star L)_{|\Omega_0}$ is trivial.  By construction
$i^\star\oc_Z(D)$ is trivial, and hence so is $i^\star(\wt L_{|Z})$.\finpr
 
\rem Remark 5.4|Here, it is not necessary to
use Grauert's theorem in its full strength. The special case we need
can be dealt with by means of the results obtained in Section~2. In
fact, for $\mu$ large, there is a one parameter family of holomorphic
maps $F:\Omega_1\times U\to X$, where $U$ is a neighborhood of the
interval $[0,1]$ in~$\cb$, such that $F(z,0)=f(z)$ and $F(z,1)=f_\mu(z)$
on~$\Omega_1\,$; this can be seen by taking a Stein neighborhood $V$ of
$f(\Omega)$ in~$X$ and a biholomorphism $\varphi=\psi^{-1}$ from a
neighborhood of the diagonal in $V\times V$ onto a neighborhood of the
zero section in the normal bundle (using Lemma~2.1$\,$(i)); then
$F(z,w)=\psi\big(w\,\varphi(f(z),f_\mu(z))\big)$ is the required family.
Now, if $Y$ is a Stein manifold and $U$ is connected, Lemma~2.2 and
an easy connectedness argument imply that all slices over $Y$ of a
holomorphic vector bundle $B\to Y\times U$ are isomorphic, at least
after we take restriction to a relatively compact domain in~$Y$. Hence
$$E'=f_\mu^\star Q_{X|\Omega_0}=(F^\star Q_X)_{|\Omega_0\times\{1\}}
\simeq(F^\star Q_X)_{|\Omega_0\times\{0\}}=f^\star Q_{X|\Omega_0}
\simeq E_{|\Omega_0}.$$
 
\rem Remark 5.5|In general, the embedding $i:\Omega_0\hookrightarrow Z$
in Proposition~5.3 will not extend to a (univalent) rational map
$S\merto Z$, but will instead extend as a multi-valued branched map. 
For example, let $S=X\ssm H$, where $H$ is a smooth hyperplane section
of a projective algebraic manifold $X$ with some nonzero Hodge numbers
off the diagonal.  By Cornalba-Griffiths [CG75,~\S~21] (see also
Appendix~2 in [CG75]) there is a holomorphic vector bundle $E'\to S$
such that the Chern character $\ch(E')\in H^{\rm even}(S,\qb)$ is not
the restriction of an element of $H^{\rm even}(X,\qb)$ given by
(rational) algebraic cycles.  Choose $\Omega_0\compact\Omega\subset S$
such that the inclusion $\Omega_0\hookrightarrow S$ induces an
isomorphism $H^{\rm even}(S,\qb)\approx H^{\rm even}(\Omega_0,\qb)$,
and let $E=E'_{|\Omega}$. Now let $i:\Omega_0\hookrightarrow Z$ and
$\wt E\to Z$ be as in the conclusion of Proposition~5.3.  We assert
that $i$ cannot extend to a rational map on $S$. Suppose on the
contrary that $i$ has a (not necessarily regular) rational extension
$i':X\merto Z$.  By considering the graph $\Gamma_{i'}\subset X\times
Z$ and the projections $\pr_1:\Gamma_{i'}\to X$, $\pr_2:\Gamma_{i'}\to Z$,
we obtain a coherent algebraic sheaf $\fc=\pr_{1\star}
\oc(\hbox\pr_2^\star \wt E)$ on $X$ with restriction $\fc_{|\Omega_0}
=i^\star\wt E\simeq E_{|\Omega_0}$, and therefore $\ch({\cal
F})_{|\Omega_0}=\ch(E')_{|\Omega_0}$. Now $\ch(\fc)$ is given by
algebraic cycles in $X$, but by the above, $\ch(\fc)_{|S}=\ch(E')$,
which contradicts our choice of $E'$. In particular,
$E_{|\Omega_0}$ is not isomorphic to an algebraic vector bundle on~$S$.

\rem Problem 5.6|The method of proof used here leads to the following
natural question. Suppose that $f:\Omega\to X$ is given as in
Theorem~1.1, and assume in addition that $\dim X\ge 2\dim S+1$.
Is it true that $f$ can be approximated by Nash algebraic embeddings
$f_\nu:\Omega_\nu\to X$ such that $f_\nu(\Omega_\nu)$ is contained in
a {\sl nonsingular} algebraic subvariety $A_\nu$ of $X$ with
$\dim A_\nu=\dim f_\nu(\Omega_\nu)=\dim S\,$? If~$\dim X=2\dim S$, does
the same conclusion hold with $f_\nu$ being an immersion and $A_\nu$ an
immersed nonsingular variety with simple double points$\,$?

Our expectation is that the answer should be positive in both cases, because
there is a priori enough space in~$X$ to get the smoothness of
$f_\nu(\Omega_\nu)$ by a generic choice of~$f_\nu$, thanks to a Whitney
type argument. The difficulty is to control the singularities introduced
by the Nash algebraic retractions. If we knew how to do this, the technical
point of using a desingularization of $A_\mu$ in the proof of Corollary~5.3
could be avoided; one would take $Z=A_\mu$ and $i=f_{\mu|\Omega_0}:
\Omega_0\hookrightarrow Z$, and Proposition~5.1 would yield immediately the
stronger conclusion that $i(\Omega_0)$ is contained in an affine Zariski open
set $Z\ssm H$, with $H$ ample.\vskip6pt
 
\bigtitle 6. Exhaustion of Stein manifolds by Runge|
domains of affine algebraic manifolds|
 
The methods developed in this paper are used in this section to give a
new proof of Stout's Theorem 1.4 by methods substantially different from
those in Stout [St84] and in Tancredi-Tognoli [TT89].  To prove
Theorem~1.4, we first show the existence of proper embeddings
of arbitrarily large Stein domains into affine algebraic manifolds,
such that the normal bundle of the embedding is trivial. Secondly, when
such an embedding is given, the embedded manifold is a global complete
intersection, so it can be approximated through an approximation of its
defining equations by polynomials. We begin with an elementary lemma
about Runge neighborhoods.

\than Lemma 6.1|Let $S$ be a closed submanifold of a Stein manifold~$Y$.
Then there is a fundamental system of neighborhoods $\Omega$ of $S$ which
are Runge open sets in~$Y$.
\fintha
 
\proof Let $f_1=\ldots=f_N=0$ be a finite system of defining equations
of $S$ in~$Y$, and let $u$ be a strictly plurisubharmonic smooth
exhaustion function of~$Y$. For every convex increasing function
$\chi\in C^\infty(\rb)$, we set
$$\Omega_\chi=\Big\{\sum_{1\leq j\leq N}|f_j(z)|^2e^{\chi(u(z))}<1\Big\}.$$
Since $\Omega_\chi$ is a sublevel set of a global psh function on~$Y$,
we infer that $\Omega_\chi$ is a Runge domain in~$Y$ by
[H\"o66,~Th.~5.2.8]. Clearly $\{\Omega_\chi\}$ is a fundamental system of
neighborhoods of~$S$.\finpr
 
\than Lemma 6.2|Let $S$ be a Stein manifold. For any relatively compact
Stein domain $S_0\compact S$, there is a proper embedding $j:S_0\to A$
into an affine algebraic manifold, such that the image $j(S_0)$ has
trivial normal bundle $N_{j(S_0)}$ in~$A$.
\fintha
 
\proof By the Bishop-Narasimhan embedding theorem, we can suppose that
$S$ is a closed submanifold of an affine space~$\cb^p$. Since any exact
sequence of vector bundles on a Stein manifold splits, $T_S\oplus
N_{S/\cb^p}$ is isomorphic to the trivial bundle $T_{\cb^p|S}=S\times
\cb^p$. Hence, to make the normal bundle of $S$ become trivial, we need
only to embed a Stein neighborhood of $S_0$ (in $\cb^p$) into an
algebraic manifold so that its normal bundle when restricted to $S_0$
is isomorphic to~$T_S$.
 
For this, we choose by Lemma~2.1 a holomorphic retraction
$\rho:\Omega\to S$ from a neighborhood $\Omega$ of $S$ in $\cb^p$
onto~$S$. We can suppose $\Omega$ to be a Runge open subset of~$\cb^p$,
by shrinking $\Omega$ if necessary (apply Lemma~6.1). Let
$\Omega_0\compact\Omega$ be a Runge domain in $\cb^p$ containing~$\ov
S_0$. By Proposition 5.3 applied to the holomorphic vector bundle
$\rho^\star T_S$ on~$\Omega$, there is an open embedding $\varphi:\Omega_0\to
Z$ into a projective algebraic manifold~$Z$, an algebraic vector bundle
$E\to Z$ such that $\varphi^\star E\simeq(\rho^\star T_S)_{|\Omega_0}$,
and an ample line bundle $L\to Z$ such that $\varphi^\star L$ is
trivial. We consider the composition of embeddings
$$S\cap\Omega_0\lhra\Omega_0\buildo{\displaystyle\varphi}\over\lhra Z
\lhra E\lhra \wh E,$$
where $Z\hookrightarrow E$ is the zero section and $\wh E=P(E\oplus\cb)$
is the compactification of $E$ by the hyperplane at infinity in each fibre.
As $N_{Z/\wh E}\simeq E$, we find
$$\eqalign{
(N_{Z/\wh E})_{|S\cap\Omega_0}&\simeq(\varphi^\star E)_{|S\cap\Omega_0}
\simeq(\rho^\star T_S)_{|S\cap\Omega_0}=(T_S)_{|S\cap\Omega_0}\,,\cr
N_{S\cap\Omega_0/\wh E}&=(N_{S/\cb^p})_{|S\cap\Omega_0}\oplus
(N_{Z/\wh E})_{|S\cap\Omega_0}\cr
&\simeq(N_{S/\cb^p}\oplus T_S)_{|S\cap\Omega_0}
\simeq(S\cap\Omega_0)\times\cb^p\,.\cr}$$
Now, $\wh E$ is equipped with a canonical line bundle $\oc_{\wh E}(1)$,
which is relatively ample with respect to the fibres of the projection
$\pi:\wh E\to Z$. It follows that there is an integer $m\gg 0$ such
that $\wh L:=\pi^\star L^{\otimes m}\otimes\oc_{\wh E}(1)$ is ample
on~$\wh E$. Now, the zero section of $E$ embeds as
$P(0\oplus\cb)\subset P(E\oplus\cb) =\wh E$, hence $\oc_{\wh
E}(1)_{|Z}\simeq Z\times\cb$. Since $\varphi^\star L$ is trivial, we
infer that $\wh L_{|S\cap\Omega_0}$ is also trivial. Apply Corollary~4.7
to $X=\wh E$ with the ample line bundle $\wh L\to\wh E$, and to some
holomorphically convex compact set $K$ in the submanifold
$S\cap\Omega_0\subset\wh E$, such that~$K\supset\ov S_0$. We infer the
existence of a holomorphic embedding $i:S_0\to Y$ into an affine
Zariski open subset $Y$ of~$\wh E$. By construction, this embedding is
obtained from the graph of a section $K\to N_{S\cap\Omega_0/\wh E}\,$,
hence $$N_{S_0/Y}\simeq (N_{S\cap\Omega_0/\wh E})_{|S_0}\simeq
S_0\times\cb^p.$$ It remains to show that we can find a {\sl proper}
embedding $j$ satisfying this property. We simply set
$$j=(i,i'):S_0\la A=Y\times\cb^q,$$
where $i':S_0\to\cb^q$ is a proper embedding of $S_0$ into an affine space.
Clearly, $j$ is proper. Moreover
$$N_{j(S_0)/Y\times\cb^q}\simeq(T_Y\oplus\cb^q)/\Im(di\oplus di')$$
with $di$ being injective, hence we get an exact sequence
$$0\la\cb^q\la N_{j(S_0)/Y\times\cb^q}\la N_{i(S_0)/Y}\la 0\,.$$
As $N_{i(S_0)/Y}$ is trivial, we conclude that
$N_{j(S_0)/Y\times\cb^q}$ is also trivial.\finpr
 
\than Lemma~6.3|Let $S_0$ be a closed Stein submanifold of an affine
algebraic manifold~$A$, such that $N_{S_0/A}$ is trivial. For every
Runge domain $S_1\compact S_0$, there exists a closed algebraic
submanifold $Y\subset A$ such that $S_1$ is biholomorphic to a Runge
open set in~$Y$.
\fintha
 
\proof By Lemmas~2.1 and 6.1, we can find a Runge neighborhood $\Omega$ of
$S_0$ in~$A$ and a holomorphic retraction $\rho:\Omega\to S_0$. Let
$\ic$ be the ideal sheaf of $S_0$ in~$A$. Then $\ic/\ic^2$ is a coherent
sheaf supported on~$S_0$, and its restriction to $S_0$ is isomorphic
to the conormal bundle $N_{S_0/A}^\star$. Since this bundle is trivial by
assumption, we can find $m=\codim S_0$ global generators $g_1\ld g_m$
of $\ic/\ic^2$, as well as liftings $\wt g_1\ld\wt g_m\in H^0(A,\ic)$
($A$ is Stein, thus $H^1(A,\ic^2)=0$). The system of equations $\wt g_1
=\ldots=\wt g_m=0$ is a regular system of equations for $S_0$ in a
sufficiently small neighborhood. After shrinking~$\Omega$, we can
suppose that
$$S_0=\big\{z\in\ov\Omega\,:\,\wt g_1(z)=\ldots=\wt g_m(z)=0\big\},~~~~
d\wt g_1\wedge\ldots\wedge d\wt g_m\ne 0~~\hbox{\rm on}~\ov\Omega.$$
Let $P_{j,\nu}$ be a sequence of polynomial functions on~$A$ converging
to $\wt g_j$ uniformly on every compact subset of~$A$. By Sard's
theorem, we can suppose that the algebraic variety $Y_\nu=\bigcap_
{1\leq j\leq m}P_{j,\nu}^{-1}(0)$ is nonsingular (otherwise, this can
be obtained by adding small generic constants $\varepsilon_{j,\nu}$
to~$P_{j,\nu}$). It is then clear that $Y_\nu\cap\Omega$ converges to
$S_0$ uniformly on all compact sets. In particular, the restriction
$$\rho_\nu:Y_\nu\cap\rho^{-1}(S_1)\la S_1$$
of $\rho$ is a biholomorphism of $Y_\nu\cap\rho^{-1}(S_1)$ onto $S_1$
for $\nu$ large. This shows that $S_1$ can be embedded as an open subset
of $Y_\nu$ via~$\rho_\nu^{-1}$.
 
It remains to show that $Y_\nu\cap\rho^{-1}(S_1)$ is a Runge domain
in~$Y_\nu$ for $\nu$~large. Let $h$ be a holomorphic function
on~$Y_\nu\cap\rho^{-1}(S_1)$. Then $h\circ\rho_\nu^{-1}$ is a
holomorphic function on~$S_1$. Since $S_1$ is supposed to be a Runge
domain in~$S_0$, we have $h\circ\rho_\nu^{-1}=\lim h'_k$ where $h'_k$
are holomorphic functions on $S_0$. Then $h'_k\circ\rho$ is holomorphic
on~$\Omega$, and its restriction to $Y_\nu\cap\rho^{-1}(S_1)$ converges
to $h$ uniformly on compact subsets. However, $\Omega$ is also a Runge
domain in~$A$, so we can find a holomorphic (or even algebraic)
function $h_k$ on~$A$ such that $|h_k-h'_k\circ\rho|<1/k$ on
$\ov{Y_\nu\cap\rho^{-1}(S_1)}$, which is compact in $\Omega$ for $\nu$~large.
Then the restriction of $h_k$ to $Y_\nu$ converges to $h$ uniformly on all
compact subsets of~$Y_\nu\cap\rho^{-1}(S_1)$, as desired.\finpr\vskip7pt
 
\noindent{\sl Proof of Theorem~1.4}\pointir
Take a Stein domain $S_0\compact S$ such that $\Omega\compact S_0$.
Then $S_1=\Omega$ is also a Runge domain in~$S_0$, and Theorem~1.4 follows
from the combination of Lemmas~6.2 and 6.3.\finpr\vskip7pt
 
We can now use Theorem~1.4 and the techniques from this section to infer
Theorem~1.5 from Proposition~5.3.
 
\noindent{\sl Proof of Theorem~1.5}\pointir By Theorem~1.4, we can take
a Runge domain $\wt\Omega\compact S$ such that
$\wt\Omega\supset\!\supset \Omega$ and embed $\wt\Omega$ as a Runge
domain in an affine algebraic manifold~$M$. By applying Proposition~5.3 with
$M$ playing the role of $S$, $\wt\Omega$ the role of $\Omega$, and $\Omega$
the role of $\Omega_0$, we obtain a projective algebraic
manifold $Z$ containing $\Omega$ and an algebraic vector bundle $\wt E\to
Z$ such that $\wt E_{|\Omega}\simeq E_{|\Omega}$. Moreover, there is an ample
line bundle $L$ on $Z$ such that $L_{|\Omega}$ is trivial.
 
Only one thing remains to be proved, namely that $\Omega$ can be taken
to be a Runge subset in an affine open subset $Z\ssm D$. (As mentioned
earlier, this would be immediate if Problem~5.6 had a positive answer.)
To reach this situation, we modify the construction as follows.
Embed $\Omega$ as $\Omega\times\{0\}\subset Z\times\pb^1$. This embedding
has trivial normal bundle and moreover
$L\stimes\oc_{\pb^1}(1)_{|\Omega\times\{0\}}$ is trivial. By
Corollary~4.7, for every holomorphically convex compact
set~$K\subset\Omega$, there is a holomorphic map
$g:K^\circ\to\cb\subset\pb^1$ which is smooth up to the boundary, such that
the graph of $g$ is contained in an affine open set $Z_1\ssm D_1$, where
$D_1$ is an ample divisor of $Z_1:=Z\times\pb^1$ in the linear system
$m|L\stimes\oc_{\pb^1}(1)|$, $m\gg 0$. Let $\Omega'\subset K^\circ$ be a
Runge open subset of~$\Omega$ and let $j_N:\Omega'\to\cb^N$ be a proper
embedding. We consider the embedding
$$j'=(\Id_{\Omega'},g,j_N):\Omega'\la Z_2:=Z\times\pb^1\times\pb^N.$$
Its image $j'(\Omega')$ is a closed submanifold of the affine open set
$(Z_1\ssm D_1)\times\cb^N=Z_2\ssm D_2$, where
$D_2=(D_1\times\pb^N)+(Z_1\times \pb^{N-1}_\infty)$ is ample in~$Z_2$.
Moreover, the normal bundle of $j$ is trivial (same argument as at the
end of the proof of Lemma~6.2). Let $\Omega''\compact \Omega'$ be an
arbitrary Runge domain in~$\Omega'$. By the proof of Lemma~6.3, we can
find a nonsingular algebraic subvariety $Z''\subset Z_2$ such that
$\Omega''$ is biholomorphic to a Runge open set in $Z''\ssm (Z''\cap D_2)$,
via a retraction $\rho$ from a neighborhood of $j'(\Omega')$
onto~$j'(\Omega')$. Let $E_2=\pr_1^\star\wt E\to Z_2$ and let
$E''$ be the restriction of $E_2$ to~$Z''$. Since $\rho^\star E_2\simeq E_2$
on a neighborhood of $j'(\Omega')$ by Lemma~2.2, we
infer that $E''_{|\Omega''}\simeq E_{|\Omega''}$. Therefore the conclusions
of Theorem~1.5 hold with $\Omega''$, $Z''$, $D''=Z''\cap D_2$ and $E''$ in
place of $\Omega$, $Z$, $D$ and $\wt E$. Since $\Omega''$ can be taken to
be an arbitrary Runge open set in~$S$, Theorem~1.5 is proved.\finpr\vskip7pt
 
\noindent{\sl Proof of Theorem~1.6}\pointir Without loss of generality,
we can assume that the $\Omega_\nu$ are Runge open sets in~$S$
(otherwise, just enlarge $\Omega_\nu$ into a Runge open set
$\Omega'_\nu\compact S$, in such a way that $\Omega'_{\nu-1}\compact
\Omega'_\nu$). By Theorem~1.4, each $\Omega_\nu$ is holomorphic to
a Runge domain in an affine algebraic manifold~$Y_\nu$. Theorem~1.1
applied to $\Omega=\Omega_{\nu+1}\subset S=Y_{\nu+1}$
yields a sequence of approximations $f_{\nu,k}$ of $f_{|\Omega_{\nu+1}}$
on arbitrarily large Runge domains $\Omega_{\nu,k}\compact\Omega_{\nu+1}$.
The conclusion follows by taking a suitable diagonal
sequence~$f_{\nu,k_\nu}$.\finpr
\vskip30pt

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\tenpoint$~$\hfill (December 22, 1992)
\end
