\documentclass[12pt]{amsart}

\headheight=6.15pt
\textheight=574pt
\textwidth=432pt
\oddsidemargin=19pt
\evensidemargin=19pt
\topmargin=14pt

\begin{document}

\makeatletter
\let\over\@@over
\makeatother
\let\Bbb\mathbb
\let\cal\mathcal
\def\cc{{\Bbb C}}
\def\pp{{\Bbb P}}
\def\eps{\varepsilon}
\def\zz{{\Bbb Z}}
\def\rr{{\Bbb R}}
\def\ss{{\Bbb S}}
\def\bb{{\Bbb B}}
\def\qq{{\Bbb Q}}

\def\area{\operatorname{area}}
\def\Aut{\operatorname{Aut}}
\def\cl{\operatorname{cl}}
\def\Hol{\operatorname{Hol}}
\def\hatHol{\widehat{\Hol}}
\def\id{\operatorname{id}}
\def\supp{\operatorname{supp}}
\def\Sym{\operatorname{Sym}}
\def\sym{\operatorname{sym}}
\def\Vol{\operatorname{Vol}}
\def\vol{\operatorname{vol}}

\def\longtitle{} %This gets added to the title of 0.4 when the section
                 %starts, but not in the table of contents

\title[Continuity principle and extension of meromorphic mappings]
{Continuity principle and extension properties of meromorphic
mappings with values in non K\"ahler manifolds}

\curraddr{Universit\'e de Lille-I\\
U.F.R. de Math\'ematiques\\
Villeneuve d'Ascq Cedex\\
59655 France}

\email{ivachkov@gat.univ-lille1.fr}

\address{IAPMM Acad Sci. of Ukraine, 
Naukova 3/b, 290053 Lviv, 
Ukraine}

\author{S. Ivashkovich}
\thanks{Research during the author's stay
at MSRI was supported in part by NSF grant DMS-9022140.}
\date{April 1997}

\subjclass{32D15} 
\keywords{meromorphic map, Continuity principle, Hartogs extension
theorem, spherical shell, complex Plateau problem.}

\begin{abstract}
In this paper we are proving an analogue of E. Levi Continuity Principle
for meromorphic mappings with values in general complex spaces. We also
describe the singularities of meromorphic mappings into complex spaces
carrying pluriclosed Hermitian metric forms and geometrical obstructions
to removing them.
\end{abstract}

\maketitle

{\footnotesize
\vspace{-24pt}
\tableofcontents
}

\section*{0. Introduction}

\subsection*{0.1. Continuity principle}

Let us start with recalling the ''Continuity principle'' for meromorphic
functions
due to  E. Levi, see [Lv]. Consider a meromorphic function $f$, which is
defined on a ring domain $\Delta^n\times A(r,1)$. Here $A(r,1)=\{ z_{n+1}\in
\cc :r<\vert z_{n+1}\vert <1\}$ is an annulus in $\cc $ and $\Delta^n $ is a
unit polydisk. Denote by $S$ the set of points $s\in \Delta^n$, such that
the restriction $f_s:=f\mid_{\{ s\} \times A(r,1)}$ is well defined and
extends as a meromorphic function of one variable onto the whole disk
$\Delta $.

\par\smallskip\noindent\bf
Theorem (E. Levi). \it If $S$ is not contained in a countable union of
locally closed proper analytic subsets of $\Delta^n$ then $f$
meromorphically extends onto the whole polydisk $\Delta^{n+1}$.
\par\smallskip\rm
Recall that by a locally closed analytic subset of $\Delta^n$ one means
an analytic set in some open subset of $\Delta^n$.

In this paper  we are interested wether this theorem can be generalised to
the case of  meromorphic mappings with values in more or less arbitrar
complex spaces.

Recall that a meromorphic mapping $f:D\to X$ between normal complex spaces
$D$ and $X$ is given by a holomorphic map $f:D\setminus I(f)\to X$ of the
complement to an analytic set $I(f)$ of codimension at least two, such that
the closure $\bar\Gamma_f$ of the graph of $f$ is an analytic set in the
product $D\times X$. One requires also that $\bar\Gamma_f$ should be proper
over $D$, i.e. restriction $\pi\mid_{\bar\Gamma_f}:\bar\Gamma_f\to D$ of
natural projection $\pi :D\times X\to D$ onto $\bar\Gamma_f$ is proper. As
$I(f)$ one takes usually the \it minimal \rm analytic subset of $D$ such
that $f$ is \it holomorphic \rm on $D\setminus I(f)$ and calles $I(f)$ the
set of points of indeterminancy of $f$.

In the case $X=\cc\pp^1$ this gives exactly a
meromorphic function on $D$, see [Re].

One cannot expect for the direct generalisation of the E. Levi theorem to
the mappings into general complex spaces as the following couple of
examples show.

\noindent
1. Let $X=\cc^2\setminus \{ 0\} /(z\sim 2z)$ be a Hopf surface. And let
$\pi :\cc^2\setminus \{ 0\} \to X$ be a natural projection. Embedd the unit
bidisk $i:\Delta^2\to \cc^2$ in a standart way, and define $f=\pi \circ i
:\Delta^2\setminus \{ 0\} \to X$. Then for all $s\in S=\Delta\setminus \{ 0
\} $ $f_s$ is holomorphic on $\Delta $, but $f_0$ doesn't extends to zero.

\noindent
2. In [Hs-1] A. Hirschowitz constructed an example of compact complex  surface
$X$ (of class $VII_0$) and holomorphic mapping $\pi:\bb^2_*\to X$ from the
puctured ball into $X$, such that for any complex curve $C\ni 0$ the
restriction $\pi\mid_{C\setminus  \{ 0\}}$ extends holomorphically onto
$C$, but $\pi $ cannot be meromorphically extended to origin. If we put
$f=\pi \circ i:\Delta^2_*\to X$ as above, we obtain a mapping $f:\Delta
\times \Delta_*\to X$ such that for all $s\in \Delta $ $f_s$ is holomorphic
on $\Delta $, but $f$ is not a meromorphic mapping of the couple $(z_1,z_2)$.

\noindent
3. In the example of M. Kato, see [Ka-1] or consrtruction of our Example 1, $S$
can be made equal to $\Delta\cap \{ z:\vert z-1\vert <1\}$, and for all
$s\in \Delta\setminus S$ $f_s$ doesn't extends holomorphically onto $\Delta $.
I.e. the maximal set $S$ of points for which $f_s$ extends onto $\Delta $
can be ``essentially'' smaller then $\Delta $!

Hovewer, let us follow the main steps of the proof of E. Levi theorem.

\noindent\sl
Step 1 . Localisation. \rm For an integer $p$ denote by $S_p$ the set
os thouse $s\in S$ that the extension $f_s$ is well defined and has at most
$p$ poles counting with multiplicities. Then one immediately sees that
there is a point $z_0\in \Delta^n$ and a natural $p$ such that for any
neighborhood $U\ni z_0$  the set $S_p\cap U$ is not contained in a proper
analytic subset of $U$.

\noindent\sl
Step 2. ``Lewisches Einsatz''. \rm Take such $z_0$ as above and take
$\eps_1>\eps_2 >0$ small enough to find a neighborhood of $U\ni z_0$ such that
$f$ is \it holomorphic \rm in $\bar U\times A(1-\eps_1 , 1-\eps_2)$. Then $f$
meromorphically
extends onto $U\times \Delta $.

\noindent\sl
Step 3. Globalisation. \rm If $U\not=\Delta^n$ one can (obviously) allwayse
find $z_1
\in \partial U\cap \Delta^n$ and $p$ such that the set $U_p$ of $z\in U$
such that $f_z$ has at most $p$ poles is not analytic in any neighborhood
of $z_1$. This shows that the maximal $U$ such that $f$ is meromorphic on
$U\times \Delta $ is equal to $\Delta^n$.

\rm Our first observation is that the statement of \sl Step 2 (i.e. main
ingredient of E. Levi Theorem) \rm
remains true
for the mappings into an ``almost'' \it arbitrary \rm complex space instead of
$\cc\pp^1$.
One only need to express the ``boundedness of the number of poles'' in
other terms. Taking into account that the winding number of $f_s\mid_{
\partial\Delta }$ is uniformly bounded, say by $N$, one easily sees that
to say that
$f_s$ takes the value $\infty $ not more then $p$ times is equivalent
to say that the area of the image of $f_s$, counted with multiplicities,
is not more then $N\cdot (p+1)$. Here $\cc\pp^1$ is equipped with the usual
spherical metric of total area one.

\par\smallskip\noindent\bf
Definition 0.1. \it We say that a complex space $X$ is disk-convex in
dimension $k$ if for any compact $K\subset\subset X$ there is another
compact $\hat K$ such that for every meromorphic mapping $f:\bar\Delta^k
\to X$ with $f(\partial \Delta^k)\subset K$ one has
$f(\bar\Delta^k)\subset \hat K$.
\par\smallskip\rm
All compact spaces are of course disk-convex. More generally all $k$-
convex spaces are disk-convex in dimension $k$.

Put $A^k(r,1)=\{ z\in \cc^k: r<\Vert z\Vert <1\} $ and $A^k_s(r,1):=\{ s\}
\times A^k(r,1)$ for $s\in \Delta^n$. Let $f:\Delta^n\times A^k(r,1)\to X$
be a holomorphic mapping into a normal complex space $X$. Denote by $S$
the set of points $s\in \Delta^n$ such that the restriction
$f_s:=f\mid_{A^k_s(r,1)}$ extends meromorphically onto the polydisk
$\Delta^k_s:=\{ s\}\times \Delta^k$.

\par\smallskip\noindent\bf
Theorem 1 \sl (Continuity principle). \it
Let $f:\Delta^n\times A^k(r,1)\to X$
be a holomorphic mapping into a normal, disk-convex in dimension $k$
complex space $X$.
Suppose that there is a constant $C_0<\infty$ and a compact $K\subset X$
such that for $s$ in some
subset $S\subset \Delta^n$, which is thick at origin:

(a) the restriction  $f_s:=f\mid_{A^k_s(r,1)}$ is well defined and extends
meromorphically
onto the polydisk  $\Delta^k_s:=\{ s\}\times \Delta^k$,  and
$\Vol (\Gamma_{f_s})\le C_0$ for all $s\in S$;

(b) $f(\Delta^n\times A^k(r,1))\subset K$ and $f_s(\Delta^k)\subset K$
for all $s\in S$.

\noindent
Then:

1ø. If $n=1$ then there is a neighborhood $U\ni 0$ in $\Delta $ such that
$f$ extends meromorphically onto $U\times \Delta^k$.

2ø. If $n\ge 2$ and $X$ has bounded cycle geometry in dimension $k$, then
again there is a neighborhood $U\ni 0$ in $\Delta^n$ and a meromorphic
extension of $f$ onto $U\times \Delta^k$.

\par\smallskip\rm Here $\Gamma_{f_s}$ denotes the graph of $f_s$ and the volumes
are taken with respect to some Hermitian  metric $h$ on $X$
and a standart Euclidean metric on $\cc^k$. The condition of finitness clearly
doesn't depends on the particular choice of the metrics. As usually saying
that a set $S\subset \Delta^n$ is thick at the point $z_0$  we mean that
for any
neighborhood $U\ni z_0$ $S\cap U$ is not contained in a proper analytic
subset of $U$.

Denote by ${\cal B}_k(X)$ the Barlet space of compact analytic cycles
of dimension $k$ in $X$. This is an analytic space, which has not more
then countable number of components.

\par\smallskip\noindent\bf
Definition 0.2. \it We say that a complex space $X$ has bounded cycle
geometry in dimension $k$ if all connected components of the Barlet space
${\cal B}_k(X)$ are compact.

\rm In other words $X$ has bounde cycle geometry in dimension $k$ if all
\it irreducible \rm components of ${\cal B}_k(X)$ are compact and all
connected components of ${\cal B}_k(X)$ are just the finite unions of
irreducible ones.
Note again
that the property to have bounded cycle geometry doesn't depend on the
choice of Hermitian metric.

There are two points to discuus here. The first one is the boundedness
of cycle geometry condition in the case $n\ge 2$ on the contrary to the
case $n=1$. The second - local character of the extension obtained, on the
contrary to the classical case $X=\cc\pp^1$, i.e. what are the obstructions
to the globalisation (Step 3)?

Cocerning the first point 
we  shall construct a following example, showing that if $n\ge 2$ the
condition of boundedness of cycle geometry cannot be removed. Denote by
$z_1,z_2,z_3,z_0$ the coordinates in $\cc^4=\cc^3_z\times \cc^1_{z_0},
z=(z_1,z_2,z_3)$.
\par\smallskip\noindent\bf
Example 1. \it There is a compact complex $4$-dimensional manifold
$X$  and a holomorphic mapping $f:\Delta^3\times A^1(r,1)\to X$ such that:

(1) for any $s\in S=\{ (z_1,z_2,z_3)\in \Delta^3: \vert z_1\vert^2>
\vert z_2\vert^2 + \vert z_3\vert^2\} $ the restriction $f_s=f\mid_{A^1_s
(r,1)}$ holomorphically extends onto $\Delta_s$;

(2) for any $t>1$ there is a constant $C_t<\infty $ such that for all
$s\in S_t=\{ (z_1,z_2,z_3)\in \Delta^3: \vert z_1\vert^2>t\cdot (
\vert z_2\vert^2 + \vert z_3\vert^2)\} $ one has $\area (\Gamma_{f_s})
\le C_t$;

(3) but for all $z\in \Delta^3\setminus \bar S =\{ (z_1,z_2,z_3)\in \Delta^3
: \vert z_1\vert^2<
\vert z_2\vert^2 + \vert z_3\vert^2\} $ every point of the innner
circle of the annulai $A^1_z(r,1):=\{ z_0\in \Delta_z: 1>\vert z_0\vert^2 >
\vert z_2\vert^2 +
\vert z_3\vert^2 - \vert z_1\vert^2 \} $ consists of essentially
singular points of $f_z:A^1_z(r,1)\to X$, here $r^2=
\vert z_2\vert^2 + \vert z_3\vert^2 - \vert z_1\vert^2 $. 

\par\smallskip\rm
In particular this $f$ doesn't extend meromorphically to any open set
of the type $U\times \Delta $, where $U$ is a neighborhood of the origin
in $\Delta^3$. The cycle geometry of this $X$ is not bounded in dimension
one. If one wishes to have an example of such type with $n=2$, one can take
a restriction of $f$ onto $\{ z_3=0\} $.

Our idea for the proof of the Continuity principle is rufly the following.
First, adapting the Barlet construction to our (noncompact) case, one
shows that there is a finite dimensional normal complex space ${\cal C}_f$,
parametrising (analytically) the cycles $Z$ in $\Delta^{n+k}\times X$
which:
\par\smallskip

1) project onto $\{ z\} \times \Delta^k$ for some $z$ in the neighborhood
of zero in $\Delta^n$;

2) $Z\cap (\Delta^n\times A^k(r,1)\times X)\subset \Gamma_f$.
\par\smallskip
From the condition on the thickness of $S$ in the Theorem 1 one gets
that $\dim {\cal C}_f=n$. Take now the  universal space ${\cal Z}\to
{\cal C}_f$ and the evaluation map  $F:{\cal Z}\to \Delta^{n+k}\times X$.
For a neighborhood $U\ni 0$ denote ${\cal Z}_U= (\pi\circ F)^{-1}(U)$. 
Here $\pi :\Delta^{n+k}\times X\to \Delta^n$ is a natural projection. If
there exists a neighborhood $U\ni 0$  in $\Delta^n$
such that $F\mid_{{\cal Z}_U}:{\cal Z}_U\to U\times \Delta^k\times X$ is
proper then  $F({\cal Z}_U)$ will be an analytic subset in $U\times
\Delta^k\times X$ extending the graph of $f\mid_{U\times A^k(r,1)}$.


For $n=1$ $F$ is allwayse proper, because a nonconstant holomorphic
map from the unit disk to the complex space is alwayse proper. For
$n\ge 2$ the condition of boundedness of cycle geometry naturally
comes out. There is one more case, important in the applications,
when $F$ is proper.
\par\smallskip\noindent\bf
Corollary 1. \it Let $f:\Delta^n\times A^k(r,1)\to X$ be a holomorphic
map into a normal, disk-convex in dimension $k$ complex space $X$.
Suppose that:

(1) for every $s\in \Delta^n$ outside of thin set, the restriction
$f_s$ extends meromorphically onto $\Delta_s^k$;

(2) there is a compact $K\subset\subset X$ such that $f_s(\Delta_s^k)
\subset K$ for all $s$ and $f(\Delta^n\times A^k(r,1))\subset K$;

(3) the volumes of the graphs $\Gamma_{f_s}$ are uniformly bounded in
$\Delta^n$, i.e. there exists $C_0<\infty $ s.t.
$\vol(\Gamma_{f_s})\le C_0$ for all $s$.

\noindent
Then $f$ meromorphically extend onto $\Delta^{n+k}$.
\par\smallskip\rm
Note that here one doesn't needs the boundedness of cycle geometry
of $X$.
It is worth, probably to point out one case when the boundedness
of cycle geometry is satisfied automatically - when $k=\dim X - 1$.
Really the cycle space of divisors is allwayse compact (provided $X$
is compact).

\subsection*{0.2. Hartogs-type extension theorem and spherical shells}

The second point, which should be discussed in concern with the
Continuity principle, is the local character of the extension obtained.
Namely
a mapping into a general space extends only to $U\times \Delta^k$. While
in the case $X=\cc\pp^1$ $U=\Delta^n $. And in fact in general the maximal
$U$ such that the map can be extended onto $U\times \Delta^k$ can be
smaller then $\Delta^n$, as we had see above.


The reason is that the volume of $\Gamma_{f_z}$ can tend to  $+\infty $
when $z$ approach $\partial U\cap \Delta^n$. Take for exapmle the Hopf
surface
$X=(\cc^2\setminus \{ 0\} )/(z\sim 2z)$ and a canonical projection
$f :\cc^2\setminus \{ 0\}\to X$. Concider its restriction onto $\Delta^2
\setminus \{0\} $. $(1,1)$-form $w=i/2{dz_1\wedge d\bar z_1+
dz_2\wedge d\bar z_2\over \Vert z\Vert^2}$ defines a canonical metric on
$X$. One easily sees that $\area (\Gamma_{f_s})\sim \log \vert s
\vert $ when $s\to 0$.

Such type of behavoir  means a violation of the Hartogs
extension phenomenon for the mapppings into this $X$. Denote by

$$
H^k_n(r):=\{ (z',z^{''})\in \Delta^{n+k}:1-r<\Vert z{''}\Vert <1 or
\Vert z'\Vert <r\} =
$$
$$
= \Delta^n\times A^k(1-r,1)\cup \Delta^n(r)\times \Delta^k\eqno(0.1)
$$
the $k$-concave Hartogs figure in $\cc^{n+k}$.
\par\smallskip\noindent\bf
Definition 0.3. \it We say that the meromorphic mappings into the space $X$
have the
Hartogs-type extension property in bidimension $(n,k)$ if any meromorphic
map $f:H^k_n(r)\to X$ extends meromorphically onto $\Delta^{n+k}$.


\rm So let us look now for the reasons of falue of Hartogs-type extension of
meromorphic mappings
with values in (normal) complex spaces. When the image space is K\"ahler
there are no obstructions for the Hartogs-type extension, see [Iv-3]. Here we
shall try to
propose a general conjecture  and state a result in the
direction of proving it. Our point of depart will be the following observation:

\it every compact complex manifold of dimension $k+1$ carries a Hermitian

metric form $w$ with $dd^cw^k=0$.

\rm Really, the
condition to carry $dd^c$-closed strictly positive $(k,k)$-form for a
compact complex
manifold is alternative to that of carrying a   bidimension $(k+1,
k+1)$-current $T$ with $dd^cT\ge 0$ but $\not \equiv 0$. This in the case of
$\dim X=k+1$ is a nonconstant plurisubharmonic function, which on
compact $X$ doesn't exist.

Let us introduce the class ${\cal G}_k$ of normal complex spaces,
carrying
a nondegenerate positive $dd^c$-closed strictly positive $(k,k)$-forms.
Note that the sequence
$\{ {\cal G}_k\}$ is rather exaustive: ${\cal G}_k$ contains all compact
complex
manifolds of dimension $k+1$.

Note also that compact spaces from ${\cal G}_k$
have bounded cycle geometry in dimension $k$, see 1.4. We conjecture that
meromorphic mappings into the spaces of class ${\cal G}_k$ are ''almost
Hartogs-extendable'' in bidimension $(n,k)$ for all $n\ge 1$:

\par\smallskip\noindent\sl
Cojecture. \it Every meromorphic map $f:H^k_n(r)\to X$, where $X\in
{\cal G}_k$ and is disk-convex in dimension $k$, extends to a meromorphic
map from $\Delta^{n+k}\setminus A$
to $X$,
where $A$ is an analytic subvariety of $\Delta^{n+k}$ (may be empty) of pure
codimension
$k+1$. Moreover, if $A\not=\emptyset $, then for every sphere $\ss^{2k+1}$
embedded into $\Delta^{n+k}\setminus A$ in such a way that $[\ss^{2k+1}]
\not=0$ in $H_{2k+1}(\Delta^{n+k}\setminus A,\zz)$, $f(\ss^{2k+1})$
also is not homologous to zero in $X$.
\par\smallskip\rm
In this paper we shall prove this conjecture in the case $k=1$.
\par\smallskip\noindent\bf
Definition 0.4. \it Let us call a Hermitian form $w$ on $X$ plurinegative if
$dd^cw\le 0$.
\par\smallskip\rm
The class of normal complex spaces admiting plurinegative Hermitian metrik
form we shall denote by ${\cal P}_{-}$.
\par\smallskip\noindent\bf
Theorem 2. \it Let $f:H^1_n(r)\to X$ be a meromorphic map into a disk-convex
complex space $X$ which admits a plurinegative Hermitian metrik form. Then:

(1) $f$ extends to a meromorphic map  $\hat f:\Delta^{n+1}\setminus A
\to X$, where $A$ is closed $(n-1)$-polar subset of $\Delta^{n+1}$.

(2) If moreover, $w$ is pluriclosed then $A$ is a analytic subvariety of
$\Delta^{n+1}$ of pure codimension two (may be empty). If $A\not= \emptyset
$ then for every sphere $\ss^3$ embedded into $\Delta^{n+1}
\setminus A$ in such a way that $[\ss^3]\not=0$ in $H_3(\Delta^
{n+1}\setminus A,\zz)$, its image $f(\ss^{3})$ also is not homologous to zero
in $X$.


\par\smallskip\noindent\bf
Remarks. \rm 1. One can estimate the number of irreducible components of
the singularity set $A$ in this theorem meating a  compact subset $P\subset
\subset \Delta^{n+1}$. Namely, let a compact $K\subset X$, which contains
$\cl [f(P\setminus S)]$, is chosen to be a finite  subcomplex of $CW$-
complex $X$. Choose a point $z'\in \Delta^{n-1}$ such that $A$ intersects
$\Delta^2_z:=\{ z\} \times \Delta^2$ by discrete set $A_{z'}$. Let $A_{z'}
\cap \partial P=\emptyset $. Then

$$
\mid A_{z'}\cap P\mid \le \vert \int_{\partial (P\cap \Delta^2_{z'})}
d^cw\vert \cdot [\inf\{ \vert \int_{\gamma }d^cw\vert :\gamma \in H_3(K,\zz ),
\int_{\gamma }d^cw\not=0\} ]^{-1}.\eqno(0.2)
$$


In other words the number of branches of singular set (and moreover, their
existence)
is bounded by the differential geometry of $X$. Remark that the subset $\{
\vert \int_{\gamma }d^cw
\vert : \gamma \in H_3(K,\zz ), \int_{\gamma }d^cw\not=0\}\subset \rr $ is
separated from zero, see (2.2.14).

\noindent
2. Let us call a spherical shell of dimension $k+1$ in complex
space
$X$  an image $\Sigma $ of the standard sphere $\ss^{2k+1}\subset
\cc^{k+1}$
under the meromorphic map of some neighborhood of $\ss^{2k+1}$ into $X$,
such
that $\Sigma $ is not homologous to zero in $X$. This notion is close to 
the notion of the global spherical shell, introduced by Kato, see [Ka-3]. 
Thus we obtain the following
\par\smallskip\noindent\bf
Corollary 2. \it Let $X$ be a disk-convex complex space which possess a
pluriclosed Hermitian metric form. Then the following is equivalent:

(a) $X$ possesses a meromorphic extension property in bidimension $(n,1)$
for all $n\ge 1$,

and thus in all bidimensions $(n,k)$.

(b) $X$ contains no two-dimensional spherical shells.

\par\smallskip\noindent\rm
3. A wide class of complex manifolds without two-dimensional
spherical shells is  for example a class of such manifolds $X$ for which the
Hurewicz homomorphism  $\pi_3(X)\to H_3(X,Z)$ vanishes.

\noindent 4. {\sl Corollary 2} for the case when $X$ is
compact complex surface was proved in [Iv-2] using the classification of 
surfaces.

\noindent 5. A number of applications will show that characterisation of the 
obstructions for the extendibility of meromorphic mappings in terms of 
spherical shells is useful.
\par\smallskip
Theorem 2 was proved in [Iv-2] under an additional (very restrictive)
assumption: \it manifold $X$ doesn't containe rational curves. \rm In
this case meromorphic maps into $X$ are just \it holomorhpic \rm . Let
us explaine the difference. One can prove, see [Iv-4], that if \it
holomorphic \rm mappings into a complex space $X$ are Hartogs extendable
in bidimension $(n,k)$, then they are Hartogs extendable in bidimension
$(n+1,k)$. Thus, for example, to prove the Hartogs-type extension
for holomorphic mappings under the assumption of absence of rational curves,
it is sufficient to prove the extendibility
of holomorphic mappings with values in $X$ from $H^1_1(r)$ onto the unit
bidisk.

Hovewer such reduction fail for the meromorphic maps, i.e. in the presnce
of rational curves.
In [Iv-4] we had
constructed the following example, shoving that
increasing of dimension in general for meromorphic mappings doesn't
works. 

\par\smallskip
\noindent
\bf{Example 2}. \it There exists a compact complex three-fold $X$ such that:

(a) For every domain $D$ in $\cc^2$ every meromorphic mapping $f : 
D\longrightarrow X$ extends to a meromorphic mapping $\hat f : \hat D 
\longrightarrow X$. Here $\hat D$ stands for the envelope of holomorphy of 
$D$.

(b) But there exists a meromorphic mapping $F : B^3 \setminus \{ 0\} 
\longrightarrow X$ from punctured threeball into $X$ which does not extend 
to the origin.

\par\smallskip\rm
This example shows that for the meromorphic mappings the Hartogs-type
extendibility in bidimension $(1,1)$ doesn't imply the extendibility
in bidimension $(2,1)$ and, what is more suprusing, it doesn't imply
the extendibility in bidimension $(1,2)$! Of course this $X$ doesn't
carries a pluruiclosed metric form.




Continuity principle will be applied in the proof of Theorem 2 in the
following context. First we shall prove in 2.2 this theorem for n=1. Then
in the case $n\ge 2$ first extend the mapping onto the set $\Delta^n\setminus
S$, where $S$ is closed (n-1)-polar subset of $\Delta^{n+1}$. Taking an
appropriate projections we shall find ourselves in the assumtion of the
Corollary 1 with $k=2$. (In fact we cold also use the C.P.with $k=1$ together with an 
observation  that spaces carrying a plurinegative
metrik-forms have bounded cycle geometry in dimension one.) This will give
the result.

In fact extension of the map from the complement of the thin set makes
the major diffuculty here. In K\"ahler case it was done by Y.-T. Siu using
his theorem on analyticity of upper level sets of Lelong numbers for the closed
positive currents. In our case currents (i.e. preimages of the metric form
under the mapping $f$ ) are not longer closed, but only pluriclosed, even
only plurinegative. For such currents the Theorem of Siu is not longer
valid. And this was a motivation for us to develope the approach based on
the Continuity principle.


\subsection*{0.3. K\"ahler case: Griffiths approach revisited}

As the title of this article says and as we had just explained above,
our object here are meromorphic mappings into \it non K\"ahler \rm
complex manifolds. Hovewer let us make one remark, which is probably
interesting especially in the K\"ahler case. First result on
extension of meromorphic mappings is due to P. Griffiths, who proved
in [Gr] that
every holomorphic mapping from a punctured ball $\bb^n_*\subset \cc^n$
into a compact K\"ahler manifold extends meromorphically to origin.
He conjectured then that if $A$ is a codimension two analytic subvariety
in complex manifold $D$ and $G$ a sufficiently small neighborhood of
$A$ then any meromorphic map  $f:D\setminus \bar G\to X$ into a compact
K\"ahler manifold $X$ extends meromorphically onto the whole $D$. He also
proposed an approach to the proof of this statement in the case when
the ampping $f$ is defined on $D\setminus A$. Namely, he proposed
to estimate the integrals
$$
\int_{K\setminus A}(f^*w)^q, q=1,...,n
$$
\noindent
where $K$ is a relatively compact subdomain in $D$, and $f^*w$ is a pull
back (as a current) of the K\"ahler form $w$ from $X$. This whould meen
the estimate of the volume of the graph of $f$ in the neighborhood of
$A\times X$. So the application of the Bishop extension theorem for
analytic sets would give the statement. Indeed, his proof in the case
$A=\{ point\} $ was exactly this. In fact, as one can see in the proof
of the Theorem 2  (and it was known in the K\"ahler case), 
for $q=1,2$ one can estimate thouse integrals in the same way as Griffiths
did. But then B. Shiffman and Taylor constructed an example of a bedegree
(1,1) current $T$ in $\Delta^3\setminus \{ line \} $ such that
$\int_{\Delta^3\setminus \{ line \} }T^3=\infty $, see [Si-3]. As we
already had mentioned the codimension two singularities where succefully
removed by Y.-T. Siu in [Si-3] using a different approach.

Hovewer our Continuity Principle says exactly that it is enough to
estimate just $\int_{K\setminus A}f^*w$! Thus we obtain another
proof of the theorem of Siu, which works also in non K\"ahler case.

There was an attempt in [Sb] to realise the Griffiths approach by means
of Fubini theorem. Unfortunatly it containes an unrecouverable gap.

\def\longtitle{: meromorphic correspondences, complex Plateau problem,
coverings of compact complex manifolds}

\subsection*
{0.4. Applications, generalisations, open questions\protect\longtitle}

It is natural to consider also the extension of meromorphic mappings from 
\it singular \rm spaces. This is equivalent to considering a multivalued 
meromorphic correspondences from smooth domains, and this to single valued 
maps into symmetric powers of the image space, see 3.1 for details. However 
one pays price for such a reductions. In this direction we construct in 3.2  
the following
\par\smallskip\noindent\bf
Example 3. \it There is a compact complex surface $X$ such that:

(a) every meromorphic map $f:H^2(r)\to X$ extends meromorphically onto 
$\Delta^2$, but

(b) there exists a two-valued meromorphic correspondence $Z$ between 

punctured two-ball $\bb^2_*\in \cc^2$ and $X$ which cannot be extended 
to origin.
\par\smallskip\noindent\rm
The reason is in fact, that $\sym ^2(X)$ containes a two-dimensional
spherical shell, while $X$ not.
In 3.3 we restate our results for multivalued meromorphic correspondences 
between (singular) complex spaces.


Results and techniques of this paper can be applied to several questions in 
complex analysis and geometry. We shall give two such applications.  

Recall that a complex Plateau problem for a compact real submanifold
$M$ of a complex manifold $X$ consists in finding an analytic chain $A
\subset X\setminus M$ with ``boundary'' $M$, see 2.5 for details.
\par\smallskip\noindent\bf
Corollary 3. \it Let $M$ be a strongly pseudoconvex, maximally complex 
compact $CR$-manifold in a disk-convex complex manifold $X$ from class 
${\cal G}_1$. Suppose that $M$ bounds an abstract smooth Stein domain. 

(a) If $\dim M\ge 5$ then the complex Plateau problem for $M\subset X$ has a 
solution.

(b) If $\dim M=3$ then the complex Plateau problem for $M\subset X$ has a 
solution iff $M$ is 

homologous to zero in $X$.  
\par\smallskip\noindent\bf
Remarks. \rm 1. Let $H^2:=\cc^2\setminus \{ 0\} /z\sim 2z$ be a Hopf surface. 
Take $M$ to be an image of a standard unit sphere from $\cc^2$ under the 
natural projection $\pi :\cc^2\setminus \{ 0\} \to H^2$. $M$ is not homologous 
to zero in $H^2$ (i.e. it is a spherical shell in $H^2$!), so a complex Plato 
problem has no solution for this $M$.

In the case (a), i.e. when $\dim M\ge 5$ the spherical shells in $X$ are 
not an obstructions for finding a film with boundary $M$ because we have 
``enough concavity''.

\noindent 2. Consider a Hopf three-fold  $H^3:=\cc^3\setminus \{ 0\} /(z\sim 
2z)$.
In this case take a sphere $\ss^3$ in a hyperplane $\{ z_1=0\} $. Its image 
$M$ under the natural projection will be homologous to zero but will not 
bound any analytic set in $H^3$. The reason here is that $H^3$ doesn't 
belongs to ${\cal G}_1$ but only to ${\cal P}_-$.

\noindent
3. If one doesn't requires strict pseudoconvexity of a ``contour'' $M$ then 
counterexamples are known already in $\cc\pp^3$, see [Db].

\noindent 4. The condition on $M$ to bound an abstract Stein smooth Stein 
domain is really restrictive in dimension 3, while for bigger dimension 
one has the Rossi theorem guaranteeing the existence (but in general not 
smooth) abstract Stein domain with boundary $M$, see [Rs]. 

The proof consists in extension of a $CR$-embedding of $M$ into $X$ onto the 
Stein domain bounded by this $M$. We do it along the levels of appropriate 
plurisubharmonic Morse exhaustion function, see 2.5.
\par\smallskip
Consider now a domain $D$ in complex manifold $\Omega $, which covers 
(without ramifications) a compact complex manifold $X$.
\par\smallskip\noindent\bf
Corollary 4. \it Suppose that $X\in {\cal G}_1$. Then: 

(a) if $\dim X\ge 3$ then $D$ is equal to a Levi-pseudoconvex domain minus 
a (possibly empty) variety of pure codimension two; 

(b) if $X$ is K\"ahler then $D$ is Levi-pseudoconvex.
\par\smallskip\rm
Part (b) was conjectured in [C-H] and was stated without proof in [Iv-3] as
a simple corollary of the Hartogs-type extension theorem for mappings into
K\"ahler manifolds.
The proof is obtained by applying the extension results to the covering map 
$D\to X$, see 3.5.

It is interesting to consider the case when $\Omega $ is compact and $D=
\Omega \setminus E$, where $E$ is removable, i.e. holomorphic functions from
$V\setminus E$ extend onto $V$ for the Stein open subsets $V\subset \Omega $.
For example if the Haussdorff $(\dim _{\rr }\Omega -2)$-measure of $E$
is zero. Then, provided $X$ carries a pluriclosed (negative) metric, one gets a
further information about $E$, ex. analyticity or pluripolarity. Such
examples really occur, see [Ka-2], [La].

In the end of this paper in \S 4 we give some open questions.

\section*{1. Continuity principle.}

\subsection*{1.1. Cycle space associated to a meromorphic map}

We shall need some notions and results from the theory of cycle spaces 
developed by D.Barlet, see [Ba]. Partially recalling thouse facts we adapt
them to our situation. For the english spelling of
the Barlet terminology we refer to [F].

Recall that an analytic cycle of dimension $k$ in complex space $Y$ is a 
formal sum $Z= 
\sum_jn_jZ_j$, where $\{ Z_j\} $ is a locally finite sequence of  analytic 
subsets (allwayse of pure dimension 
$k$) and $n_j$ are positive integers called multiplicities of $Z_j$.
$\vert Z\vert :=\bigcup_jZ_j$-support of $Z$. All complex spaces in this
paper are reduced, normal and countable at infinity.

With a given  meromorphic mapping $f:\Delta^n\times A^k(r,1)\to X$, satifying 
conditions of the Theorem 1 we shall associate the following space 
of cycles. Fix some $0<c<1$. Consider a set ${\cal C}^{'}_{f,C}$ of all
analytic cycles $Z$ in $Y:=\Delta
^{n+k}\times X$ of pure dimension $k$, such that:

(a) $Z\cap [\Delta^n\times A^k(r,1)] = \Gamma_{f_{z'}}\cap \{ z\} \times
A^k(r,1)\times X$ for
some $z'\in \Delta^n(c)$. This means, in particular, that for this $z^{'}$
$f_{z'}$ extends meromorphically from $A^k_{z'}(r,1)$ onto $\Delta^k_{z'}$.

(b) $\vol(Z)< C$, where $C$ is a some constant, $C>C_0$, $C_0$
beeing from Theorem 1.


Define $\bar {\cal C}_{f,C}$ to be a closure of ${\cal C}^{'}_{f,C}$ 
in the usual topology of currents, see below. We shall show that ${\cal C}
_{f,C}:=\{ Z\in \bar {\cal C}_{f,C}: \vol(Z)<C\} $
is an analytic space of finite dimension in the neighborhood of each
of its points.



\par\smallskip
Let $Z$ be an analytic cycle of dimension $k$ in (reduced, normal) complex 
space $Y$. In our applications $Y$ will be $\Delta^{n+k}\times X$. By a 
coordinate chart  adapted to $Z$ we shall understand an  open neighborhood $V$
in $Y$ such that $V\cap \vert Z\vert \not=\emptyset $ together with an 
isomorphism $j $ of $V$ onto a closed subvariety $\tilde V$ in the
neighborhood
of $\bar\Delta^k\times \bar\Delta^q$, such that $j^{-1}(\bar\Delta^k
\times
\partial \Delta^q)\cap \vert Z\vert =\emptyset $. We shall denote such a chart 
by $(V,j )$. By an image $j (Z)$ of cycle $Z$ under isomorphism $j $
(or by any
other isomorphism) we shall understand the image of underlying analytic set 
together with multiplicities.

Sometimes we shall, following Barlet, denote: $\Delta^k=U, \Delta^q=B$ and
give to the quadriple $(V,j,U,B)$ the name \it scale \rm adapted to
$Z$.

If $\pi :\cc^k\times \cc^q\to \cc^k$ is a natural
projection, then $\pi \mid_{j(Z)}:j(Z)\to \Delta^k$ is a branching
covering of degree 
say $d$. Number $q$ depends on the embedding dimension of $Y$ (or $X$ in our 
case). We shall skeep sometimes $j$ in our notations. The branched
covering
$\pi \mid_Z:=\pi :Z\cap (\Delta^k\times \Delta^q)\to
\Delta^k$ defines in a natural way a mapping $\phi :\Delta^k\to \Sym ^d
(\Delta^q)$ -                                                              
symmetric power of $\Delta^q$ of degree $d$ - as $\phi (z)=(\pi\mid_Z)^{-1}(z)$. This 
allouds represent a cycle $Z\cap \Delta^{k+q}$ with $\vert Z\vert \cap 
(\bar\Delta^k\times \partial\Delta^q)=\emptyset $, as a graph of $d$-valued 
holomorphic map.


Let $S$ be a normal complex space.
\par\smallskip\noindent\bf
Definition 1.1.1. \rm A holomorphic map $\Phi :S\times \bar\Delta^k\to \Sym 
^d(\Delta^q)$ we shall call an analytic family of $k$-dimensional subvarities 
in $\Delta^k\times \Delta^q$ parametrised by $S$. 

We shall need the following Changing of the projection Theorem, due to Barlet. Let 
$(V,j,U,B)$ be a scale on $\bar\Delta^k\times\Delta^q$ adapted to the
cycle
$Z$. Let $Z$ be included in an analytic family  $\Phi $ in $\Delta^k\times
\Delta^q$ as above, i.e.
there is a holomorphic map $\Phi :S\times \bar\Delta^k\to \Sym ^{d}(\Delta 
^q)$ such 
that $\Phi (s_0,\cdot )$ represents $Z$ for some $s_0\in S$.  Then for some
neighborhood $S_0\ni s_0$
$V$ is adapted to all $Z_s,s\in S_0$. In particular this defines 
a map $\Psi :S_0\times U\to \Sym ^{d_1}(B)$.
\par\smallskip\noindent\bf
Theorem (D. Barlet). \it Mapping $\Psi $ is holomorphic.
\par\smallskip\rm 
\noindent For the proof see [Ba], Chapitres I and II.  Here finite
dimensionality and normality of $S$ are essential.

Withough loss of generality we suppose that our mapping  $f$ is defined
on $\Delta^n\times A^k(r,b)$ with $b>1$. Now each
$Z\in {\cal C}_{f,C}$ can be covered by a \it finite \rm number of adapted
neighborhoods $(V_{\alpha },j_{\alpha })$. Their union $\bigcup_{\alpha }V_
{\alpha }$
we shall denote by $W_Z$. Taking this covering $(V_{\alpha },j_{\alpha })$ 
to be small enough, we can  suppose that:

\par\smallskip\noindent
(c)  if $V_{\alpha_1}\cap V_{\alpha_2}\not=\emptyset $ then on every
irreducible component of the intersection $Z\cap V_{\alpha_1}\cap V_{\alpha
_2}$ a point $x_1$ is fixed such that  

$(c_1)$ either there is a polycilinder
neighborhood $\Delta_1^k\subset \Delta^k$ of $\pi_1(j_1(x_1))$ such that
the
chart $V_{12}=j_{\alpha_1}^{-1}(\Delta_1^k\times \Delta^q)$ is
adapted to $Z$ and is
contained in $V_2$, here $V_{12}$ is given the same embedding
$j_{\alpha_1}$;

$(c_2)$ or this is fulfilled for $V_2$ instead of $V_1$. 

\noindent
(d) if $V_{\alpha }\ni y$ with $p(y)\in \bar\Delta^n(c)\times A^k({r+1
\over 2},1)$ then $p(\bar V_{\alpha })\subset \bar\Delta^n({c+1\over 2})
\times A^k(r,1)$. 
\par\smallskip
Here we denote by $p :\Delta^{n+k}\times X\to \Delta^{n+k}$ and by
$\pi :\Delta^n\times \Delta^k\times X\to \Delta^n$ the natural
projections. Case $(c_1)$ can be  fulfiled when embedding dimension of
$V_{\alpha_1}$ is smaller or equal of that of $V_{\alpha_2}$, and $(c_2)$
in the opposite case, see [B].


\par\smallskip
Note that we have allwayse that $p(Z)=\Delta^k_{z_0}$ for some $z_0\in
\Delta^k(c)$ if $Z\in {\cal C}_{f,C}$, because $Z$ is a limit of
$Z_n\in {\cal C}^{'}_{f,C}$.


The subsets $W_Z$ together with the topology 
of convergence on compact subsets on $\Hol (\Delta^k,\Sym ^d(\Delta^q))
$ define a (metrizable) topology on our cycle space ${\cal C}_{f,C}$. 

Each cycle $Z\in {\cal C}_{f,C}$ can be viewed as a current of integration and
thus
${\cal C}_{f,C}$ can be also equiped with the corresponding locally flat topology.
Cycles $Z_n$ are converging in this topology to $Z$ if for any smooth $(k,k)$
-form $w$ with compact support in $Y$ 

$$
\int_{Z_n} w \to \int_{Z} w .
$$
The both topologies are equivalent, see [F]. For us will be important that 
$\bar {\cal C}_{f,C}$ is compact. This generalisation of Bishop's theorem is
due  to
Harvey and Shiffman, see [H-S].

Family of cycles $Z_s$ parametrised by normal complex space $S$ we call \it 
analytic \rm if for any $s_0\in S$, and any coordinate chart $(V,j)$ adapted
to $Z_{s_0}$ the family $Z_s\cap V$ is analytic in $V$ for $s$ in the
neighborhood of $s_0$ in the sence of Definition 1.1.1.

We shall need a criterium for the analyticity of changing the projections map
in the case when the parameter space $S$ is of infinite dimension.

Denote by $S_m(\cc^q)$  (resp. by $\Lambda^i(\cc^q)$)  the $m$-th symmetric
(resp. exterior) pover of $\cc^q$. Put $F_i:=Hom(\Lambda^i(\cc^k), \Lambda^i
(\cc^q))$.

Let $(S,s_0)$ be a germ of Banach analytic space and let $$\Phi :(S,s_0)
\to H(\bar\Delta^k, \Sym ^d(\Delta^q))$$ be a germ of analytic map.

Fix some $s\in S$ and denote by $Z_s$ the analytic set in $\Delta^k\times
\Delta^q$ defined by $\Phi_s:=\Phi (s,\cdot ):\Delta^k\to \Sym ^d
(\Delta^q)$. Let $R(Z_s)$ the locus of ramification of $\pi\mid_{Z_s}:Z_s
\to \Delta^k$. For $z\in \Delta^k\setminus R(Z_s)$ define 
$$
T_m^i(\Phi_s):\Delta^k\to F_i\otimes S_m(\cc^p)\eqno(1.1.1)
$$
as
$$
T_m^i(\Phi_s)(z)=\Sigma_{j=1}^d\Lambda^i(D\Phi_{s,j}(z))\otimes
\Phi_{s,j}(z)^m.\eqno(1.1.2)
$$
\noindent
Here $\Phi_j, j=1,...,d$ are the local branches of $\Phi $. $T_m^i(\Phi_s)$
extend holomorphically onto the whole polydisk $\Delta^k$, see [B]
Proposition 1, Chap.2. This defines
a maps 
$$
T_m^i(\Phi ):S\times\Delta^k\to F_i\otimes S_m(\cc^p).\eqno(1.1.3)
$$
Following Barlet we call the family \it isotropic \rm if thouse maps are
analytic for all $i,m$. The statement of Canging the Projection Theorem
remains true for
the case of Banach analytic sets $S$, provided the family $\Phi $ is
isotropic, see Theorem 4 in [B]. In fact one needs to check analyticity
only for $1\le m\le d-1$. Note also that $F_i=0$ if $i>\max \{ k,q\} $.

Now let $E=(V,j,U,B)$ be a scale on the complex space $Y$.
Let us denote by $\Hol _Y(\bar U,\sym ^d(B))$ the Banach analytic set
of all $d$-sheeted analytic subsets on $\bar U\times B$, which are contained
in $j(Y)$. We need to make the tautological family $\Hol _Y(\bar U,
\sym ^d(B))\times U\to \sym ^d(B)$ isotropic. Put ${\cal F}:=
\Sigma_iF_i$ and ${\cal S}:=\Sigma_mS_m(\cc^q)$. Fix some polydisk $U'
\subset\subset U$ and call the data $E=(V,j,U,U',B)$ the \it double scale
\rm on $Y$. Consider a continuous map
$$
T:\Hol (\bar U,\sym ^d(B))\to \Hol (\bar U',{\cal F}\otimes
{\cal S})
$$
\noindent
given by
$$
h\in \Hol (\bar U,\sym ^d(B))\to \Sigma_{i,m}T^i_m(h)\eqno(1.1.4)
$$
Denote by  $\hatHol (\bar U,\sym ^d(B))$ the graph  $\Gamma_T$
of (1.1.4) in
$\Hol (\bar U, \sym ^d(B))\times \Hol (\bar U', {\cal F}\otimes
{\cal S})$.


Let
$\hatHol _Y(\bar U,
\sym ^d(B))$ be the restriction of $\Gamma_T$
onto $\Hol _Y(\bar U,\sym ^d(B))$. Then the set 
$\hatHol _Y(\bar U,
\sym ^d(B))$ is a Banach analytic subset of
$\Hol (\bar U,\sym ^d(B))\times \Hol (U',{\cal F}\otimes
{\cal S})$, and the tautological family
$$
\hatHol (\bar U,\sym ^d(B))\times U\to \sym ^d(B)
$$
\noindent
is (obviously) isotropic! See [B].
\par\smallskip\noindent\bf
Definition 1.1.2. \it The family ${\cal Z}$ of analytic cycles in an open set
$W\subset Y$, para\-metrised by a Banach analytic set $S$, is called analytic
in the neighborhood of $s_0\in S$ if for any scale $V$ adapted to $Z_{s_0}$
there is a neighborhood $U\ni s_0$ s.t. $\{ {\cal Z}_s:s\in U\}$ is isotropic
in $V$.\rm

\subsection*{1.2. Analyticity of ${\cal C}_{f,C}$}

Let $f:\Delta^n\times A^k(r,1)\to X$ be our map. Denote by ${\cal C}_0$ the 
subset of $\bar {\cal C}_{f,C}$ consisting of cycles which are limits of
$\{ \Gamma_
{f_{s_n}}\} $ for $s_n\rightarrow 0,s_n\in S$. This is a compact subset (by 
Bishop's theorem) of the topological space ${\cal C}_{f,2C}$. For every cycle
$Z\in
{\cal C}_0$ define its neighborhood $W_Z$ as above. Let $W_{Z_1},...,W_{Z_N}$ be 
a finite covering of ${\cal C}_0$. Remark that there is an $\eps_0>0$ such that
for any $s\in S\cap \Delta^n(\eps_0)$ we have $\Gamma_{f_s}\subset \bigcup_{
j=1}^NW_{Z_j}$.



We whant to show now that ${\cal C}_{f,2C}$ is an analytic space of finite
dimension in the neighborhood of ${\cal C}_0$. It is enough to prove this 
for the $W_{Z_1}$ for example.

Let $W_{Z_1}=\bigcup_{\alpha =1}^{N_1} V_{\alpha } $. We divide $V_{\alpha }$
-s into two types.
\par\smallskip
\noindent
Type 1. For such $V_{\alpha }$ as in (d)  put 
$$
H_{\alpha }:=\bigcup_z
\{ [\Gamma_{f_z}\cap A^k(r,1)\times X]\cap V_{\alpha }\} 
\subset \Hol _Y(\Delta^k, \Sym ^{d_{\alpha }}(\Delta^p)).\eqno(1.2.1)
$$
\noindent
Union is
taken over all $z\in \Delta^n$ for which $V_{\alpha }$ is adapted to
$\Gamma_{f_z}$.

\noindent
Type 2. For all others $V_{\alpha }$ we put $H_{\alpha }:=\hatHol _Y
(\bar\Delta^k,
\Sym ^{d_{\alpha }}(\Delta^p))$.
\par\smallskip
All $H_{\alpha }$ are open sets in complex Banach spaces and, for $V_{\alpha }
$ of first
type they are of dimension $n$ and smooth. 

For every irreducible component of $V_{\alpha }\cap V_{\beta }\cap Z_1$ we
fix some point
$x_{\alpha \beta l}$ on this component (indice $l$ indicates the component),
and some
chart $V_{\alpha }\cap V_{\beta }\supset (V_{\alpha \beta l},\phi_{\alpha
\beta l})\ni x_{\alpha \beta l}$ adapted to this
component as in (c). Put $H_{\alpha \beta l}:=\hatHol (\Delta^k,
\Sym ^{d_{\alpha \beta l}}(\Delta^p))$.

Consider a finite products $\Pi_{(\alpha )}H_{\alpha }$ and $\Pi_{(\alpha
\beta l)}H_{\alpha \beta l}$. In the second
product we take only triples with $\alpha <\beta $. They are Banach analytic
spaces and by
Changing the projection Theorem of Barlet, for each pair $\alpha <\beta $ we
have two
holomorphic mappings $\Phi_{\alpha \beta }:H_{\alpha }\to \Pi_{(l)}H_
{(\alpha \beta l)}$
and $\Psi_{\alpha \beta }:H_{\beta }\to
\Pi_{(l)}H_{\alpha \beta l}$. This defines two holomorphic maps $\Phi ,\Psi
:\Pi_{(\alpha )}
H_{\alpha }\to \Pi_{\alpha <\beta ,l}H_{\alpha \beta l}$. Kernel ${\cal A}_1$
of this pair, i.e. thouse
$h=\{ h_\alpha \} $
that $\Phi (h)=\Psi (h)$, consists exactly from analytic cycles in a
neighborhood $W_{Z_1}$ of $Z_1$. This kernel is a Banach analytic set and
moreover, the family ${\cal A}_1$ is an analytic family in $W_{Z_1}$.
\par\smallskip\noindent\bf
Lemma 1.2.1. \it ${\cal A}_1$ is of finite dimension.
\par\smallskip\noindent\sl
Proof of the Lemma. \rm Take a smaller covering $\{ V_{\alpha }^{'},
j_\alpha\} $ of
$Z_1$. Namely $V_{\alpha }^{'}=V_{\alpha } $ for the $V_{\alpha }$ of the
first type and $V_{\alpha }^{'}=
j_{\alpha }^{-1}(\Delta^k_{1-\eps }\times \Delta^p)$ for the second. In
the same
manner we obtain a Banach analytic set ${\cal A}_1^{'}$. We have a holomorphic
mapping $K:{\cal A}_1\to {\cal A}_1^{'}$ defined by the restrictions. Diffe
rential $dK\equiv K$ of this map is a compact operator. We also have an  in
verse map $F$ because of the isotropy of the family ${\cal A}_1$, see above.
Thus $id-dK\circ dF$ is Fredholm. Because ${\cal A}_1^{'}\subset \{ h\in
\Pi_{(i)}H_i^{'}: (id-K\circ F)(h)=0\} $ we obtain that ${\cal A}_1^{'}$ is
an analytic subset in the complex manifold of finite dimension.
\par\smallskip
\hfill{Lemma is proved}
\par\smallskip
The same holds for any point in $\Delta^n$ instead of zero. Thus
${\cal C}_{f,C}$ is finite dimensional analytic space.

\subsection*{1.3. Proof of the Continuity principle}

Now we are prepared to prove our Continuity principle.
Consider a universal family ${\cal Z}:=\{ Z_a:a\in {\cal C}_{f,2C_0}\} $,
this time constant $C_0$ is taken from Theorem 1. This is
complex space of finite dimension . We have an evaluation map
$$
F:{\cal Z}\to \Delta^{n+k}\times X\eqno(1.3.1)
$$
\noindent
defined by $Z_a\in {\cal Z}\to Z_a\subset \Delta^{n+k}\times X$.
\par\smallskip\noindent\sl
Case $n=1$. \rm Consider the union $\hat {\cal C}_0$ of thouse components
of ${\cal C}_{f,2C}$ which intersect ${\cal C}_0$. At least one of thouse
components, say ${\cal K}$, containes two points $s_1$ and $s_2$ s.t.
$Z_{s_1}$ projects onto $\Delta^k_0$ and $Z_{s_2}$ projects onto
$\Delta^k_s$ with $s\not= 0$. Consider the restriction ${\cal Z}\mid_{{\cal
K}}$ of the universal space onto ${\cal K}$. This is an irreducible
complex space of  finite dimension. Take points $z_1\in Z_{s_1}$
and $z_2\in Z_{s_2}$ and join them by an analytic disk $\phi :\Delta \to
{\cal Z}\mid_{{\cal K}}$, $\phi (0)=z_1, \phi (1/2)=z_2$. Then the
composition $\psi = \pi\circ F\circ \phi :\Delta \to \Delta $ is not
degenerate because $\psi (0)=0\not= s=\psi (1/2)$. Thus $\psi $ is proper
and obviously so is the map $F:{\cal Z}\mid_{\phi
(\Delta )}\to F({\cal Z}\mid_{\phi (\Delta )})\subset \Delta^{n+k}\times
X$. Thus  $F:{\cal Z}\mid_{\phi (\Delta )})$ is an analytic set in $U\times
\Delta^k\times X$ for small enough $U$ extending $\Gamma_f$ by the reason
of dimension.
\par\smallskip\noindent\sl
Case $n\ge 2$. \rm
Consider an increasing family of complex spaces $\{ {\cal C}_{f,C}: C\ge C_0
\} $, where $C_0$ is from the Theorem 1. Note that for $C_1<C_2$
${\cal C}_{f,C_1}$ is an open subset of ${\cal C}_{f,C_2}$. This
allouds correctly define an irreducible components of the analytic space
${\cal C}_f:=\bigcup_{C\ge C_0}{\cal C}_{f,C}$.

As above by ${\cal C}_0$ denote the compact subset of ${\cal C}_f$, which
consists from the limits of $\Gamma_{f_s}$, $s\to 0,s\in S$, $\vol
(\Gamma_{f_s})\le C_0$. Clearly only finite number of components of
${\cal C}_f$ intersect ${\cal C}_0$. Denote them by ${\cal K}_1,...,
{\cal K}_N$.

Note that if all this components 
would have zero dimension it would contradict the thickness of $S$ at
origin. Take any component, say  ${\cal K}_1$ of
positive dimension. If evaluation map $F$ restricted onto ${\cal Z}\mid_
{{\cal K}_1}$ is proper, then again $F({\cal Z}\mid_{{\cal K}_1})$ is an
analytic set in the neighborhood $U\times \Delta^k\times X$. If for all
${\cal K}_j$ $F({\cal Z}\mid_{{\cal K}_j})$ is proper then, because we only
have only finitely many this components , we shall have
a neighborhood $U\ni 0$ in $\Delta^n$ and analytic set
$F({\cal Z}\mid_{\bigcup K_j})$ in $U\times \Delta^k\times X$. Would this
set be of dimension less then $n+k$, it would contradict the thickness
of $S$ at zero. Thus in this case we have again an extension
$F({\cal Z}\mid_{\bigcup K_j}$ of $\Gamma_f$ onto $U\times \Delta^k
\times X$.

It remaine to prove that for any ${\cal K}_j$ the
restriction of $F$ onto ${\cal Z}\mid_{{\cal K}_j}$ is proper. Suppose
now that there is a component, say ${\cal K}_1$, such that $F$ restricted to
${\cal Z}\mid_{{\cal K}_1}$ is not proper. Consider a map $\phi :{\cal K}_1
\to \Delta^n$ given by $\phi (s) = \pi (F(Z_s))$. If $\phi^{-1}(0)$ is
compact, then there are a neighborhoods $W_1\supset \phi^{-1}(0)$ in
${\cal K}_1$ and $V_1\ni 0$ in $\Delta^n$ 
such that $\phi\mid_W:W_1\to V_1\subset \Delta^n$ is proper.
Thus $F:{\cal Z}\mid_{W_1}\to V_1\times \Delta^k\times X$ is
proper. If this is
the case for all  ${\cal K}_1,...,{\cal K}_N$ then again
$F:{\cal Z}\mid_{\bigcup W_j}\to \bigcap V_j\times
\Delta^k\times X$ is proper and we are done.

So, suppose that $\phi^{-1}(0)$ is not compact in ${\cal K}_1$. Each
$Z_s, s\in \phi^{-1}(0)$ has a form $\{ 0\} \times B_s\cup \Gamma_{f_0}$,
where $B_s$ is a compact cycle in $X$. Thus an appropriate connected
component of $\phi^{-1}(0)$ parametrizes a noncompact connected and
closed subvariety in ${\cal B}_k(X)$. This contradicts the bounded cycle
geometry condition on $X$.

\par\smallskip
\hfill{Theorem 1 is proved.}
\par\smallskip\noindent\sl
Proof of the Corollary 1. \rm In the proof of Corollary 1 we shall need
some resuts on the meromorphic families of analytic subsets. They will
be used also in the proof of Theorem 2. That's why we had collected them
in 2.3. For the proofs of thouse results we refer to [Iv-4].

Denote by $\nu_j$ the minimal volume of compact $j$-dimensional analytic
subset in $K\subset\subset X$ - compact from Corollary 1. Put
$$
\nu =\min \{ vol(A_{k-j}\cdot\nu_j: j=1,...,k\} , \eqno(1.3.2)
$$
\noindent
where $A_{k-j}$ runs over all $(k-j)$-dimensional analytic subsets of
$\Delta^k$, which intersect $\Delta^k_r$. Denote by $W$ the maximal open
subset of $\Delta^n$ such that $f$ meromorphically extends onto
$\Delta^n\times A^k(r,1)\cup W\times \Delta^k$. Put $S=\Delta^n\setminus
W$. Let
$$
S_l=\{ z\in S: vol(\Gamma_{f_z}\le l\cdot {\nu \over 2}\}.\eqno(1.3.3)
$$
Lemma 2.3.2 tells us that $S_{l+1}\setminus S_l$ are pluripolar and by the
Josefson theorem so is $S$.


Consider a function  $v(z) = \vol(
\Gamma_{f_z})$. We know that $v\ge 0,v\le C_0$ and $v$ is lower semicontinuous
. The latter follows from Lemma 2.3.1.(4). Let $v^*$ be its  upper
regularisation, i.e. $v^*=\lim\sup_{z\to z_0}v(z)$. Then $v^*$ is upper
semi-continuous
and bounded by $C_0$. Consider an analytic space

$$
{\cal C}_{f,2C_0,c}:=\{ Z\in {\cal C}_{f,2C_0}: \Vert \pi (Z)\Vert <c\}
,\eqno(1.3.4)
$$
\noindent
where $0<c<1$ is fixed. Consider the following relatively compact in
${\cal C}_{f,2C_0}$ and closed in ${\cal C}_{f,2C_0,c}$ set:
$$
{\cal C}_{f,v^*+{\nu \over 4},c} = \{ Z\in {\cal C}_{f,2C_0,c}: \vol(Z)\le
v^*[\pi (Z)]+{\nu \over 4}\},\eqno(1.3.5)
$$
\noindent
where, as above $\pi : \Delta^n\times\Delta^k\times X\to \Delta^n$ is a
natural projection. Function $\vol(Z)$ is contunious on ${\cal C}_
{f,2C_0}$. Thus for any $Z_0\in \bar {\cal C}_{f,v^*+{\nu \over 4},c}$ the
set $W_{Z_0,{\nu \over 4}}:=\{ Z: \vert \vol(Z)-\vol(Z_0)\vert <
{\nu \over 4}\} $ is an open neighborhood of $Z_0$ in ${\cal C}_{f,2C_0}$.
Closures and neighborhoods are taken in ${\cal C}_{f,2C_0,c}$. Put
$$
W_{f,v^*,{\nu \over 2}}:=\bigcup_{Z_0\in \bar {\cal C}_{f,v^*+{\nu \over 4},c}}
W_{Z_0,{\nu \over 4}}\eqno(1.3.6)
$$
\noindent
and
$$
W_{f,v^*,{3\nu \over 4}}:=\bigcup_{Z_0\in \bar {\cal C}_{f,v^*+{\nu \over 4},c}}
W_{Z_0,{\nu \over 2}}.\eqno(1.3.7)
$$
We are going to show now that $\bar W_{f,v^*,{\nu \over 2}} =
W_{f,v^*,{3\nu \over 4}}$. 

Find a point $z_0\in W$ which has a neighborhood, say $V$, such that
for all $z\in V$ $\vert vol(\Gamma_{f_z})-vol(\Gamma_{f_{z_0}})\vert
<{\nu \over 2}$. Such $z_0$ exists because $W$ is not pluripolar.
Lemma 2.3.2
shows that the analytic spaces $W_{f,v^*,{\nu \over 2}}$ and
$W_{f,v^*,{3\nu \over 4}}$ coincide with $\bigcup_{z\in V}\Gamma_{f_z}$
in the neighborhood of $z_0$.
The same argument shows that the following is true. Take the irreudcible
components of analytic spaces  $W_{f,v^*,{\nu \over 2}}$ and $W_{f,v^*,
{3\nu \over 4}}$ which contain $\bigcup_{z\in V}\Gamma_{f_z}$. We denote
them in the same manner
$W_{f,v^*,{\nu \over 2}}$ and $W_{f,v^*,
{3\nu \over 4}}$ respectively. The holomorphis mapping $\pi \circ F:
W_{f,v^*,{\nu \over 2}}\to \Delta^n$ is surjective. The same for  $\pi \circ
F: W_{f,v^*, {3\nu \over 4}}\to \Delta^n$. There is a pluripolar set
$\hat S\supset S$ such that   $\pi \circ F: \bar W_{f,v^*,{\nu \over 2}}
\setminus (\pi \circ F)^{-1}(\hat S)\to \Delta^n\setminus \hat S $ is
biholomorphic. And the same for $\pi \circ F: W_{f,v^*, {3\nu \over 4}}
\setminus (\pi \circ F)^{-1}(\hat S) \to \Delta^n \setminus \hat S$.
But this means that $W_{f,v^*,{3\nu \over 4}}\setminus W_{f,v^*,{\nu \over 2}}
\subset (\pi \circ F)^{-1}(\hat S)$. The latter is a pluripolar
subset of an irreducible analytic space, and thus has dense complement.
So the statement is proved.

Note now that if $0<c_1<c$ then
$\bar W_{f,v^*,{\nu \over 2}}\cap (\pi \circ F)^{-1}(\bar\Delta^n_{c_1})$
is compact in $W_{f,v^*, {3\nu \over 4}}$. This gives the properness
of $\pi\circ F$ restricted to
$W_{f,v^*, {3\nu \over 4}}$, and thus the properness of $F$ there.
\par\smallskip
\hfill{q.e.d.}

\subsection*{1.4. Spaces from ${\cal G}_k$ have bounded cycle geometry}

Let us proove here that spaces from ${\cal G}_k$ have bounded cycle
geometry in dimension $k$. In fact somewhat stronger statement is true.
Denote by
${\cal P}^k_{-}$ the class of complex spaces which carry a strictly positive $(k,k)$-
forms $\Omega^{k,k}$ which are $dd^c$-negative, i.e. $dd^c\Omega^{k,k}\le 0$.
\par\smallskip\noindent\bf
Proposition 1.4.1. \it Let $X\in {\cal P}^k_{-}$ be disk-convex in dimension $k$
and let ${\cal K}$ be an irreducible component of ${\cal B}_k(X)$. Then 
\par\smallskip
1) ${\cal K}$ is compact.
\par\smallskip
2) If $\Omega^{k,k}$ os some $dd^c$-negative $(k,k)$-form on $X$ then
$\int_{Z_s}\Omega^{k,k}\equiv const$ for $s\in {\cal K}$.
\par\smallskip
3) $X$ has bounded cycle geometry.
\par\smallskip\noindent\sl
Proof. \rm Let $F:{\cal Z}\mid_{{\cal K}}\to X$ be the evaluation map, and
let $\Omega^{k,k}$ be a sitrictly positive $dd^c$-negative $(k,k)$-form on
$X$. Then $\int_{Z_s}\Omega^{k,k}$ measures the volume of $Z_s$. Let us
prove that the function $v(s)=\int_{Z_s}\Omega^{k,k}$ is plurisuperharmonic
on ${\cal K}$. Take an analytic disk $\phi :\Delta \to {\cal K}$. Then by
Stokes theorem and reasons of bedegree for any nonnegative test function
$\psi \in
{\cal D}(\Delta )$ :
$$
<\psi ,\Delta \phi^*(v)> = \int_{\Delta }\Delta \psi \cdot \int_{
Z_{\phi (s)}}\Omega^{k,k} = \int_{{\cal Z}\mid_{\phi (\Delta )}}dd^c(\pi^*
\psi )\wedge \Omega^{k,k} =
$$
$$
= \int_{\phi (\Delta )}\pi^*\psi \wedge dd^c\Omega^{k,k}
\le 0.
$$
\noindent
Here $\pi :{\cal Z}\mid_{{\cal K}}\to {\cal K}$ is a natural projection.
So $\Delta \phi^*(v)\le 0$ for any analytic disk in ${\cal K}$ in the sence
of distributions. Thus $v$ is plurisuperharmonic.

Note that by Harvey-Shiffman generalisation of Bishops theorem $v(s)\to
\infty $ while $s\to \partial {\cal K}$. So by the minimum principle
$v\equiv const$ and ${\cal K}$ is compact again by Bishop theorem.
\par\smallskip\noindent
2) The same computation shows that $\int_{Z_s}\Omega^{k,k}$ is
plurisuperharmonic for any $dd^c$-negative $(k,k)$-form. While ${\cal K}$
is proved to be compact, we get the statement.
\par\smallskip\noindent
3) Let ${\cal R}$ be any connected component of ${\cal B}_k(X)$. Write
${\cal R}=\bigcup_j {\cal K}_j$. From (1) we have that $v$ is constant on
${\cal R}$. So if $\{ {\cal K}_j\} $ is not finite then ${\cal R}$ has an
accumulation point $s=\lim s_j$, where all $s_j$ belong to different components
${\cal K}_j$ of ${\cal R}$. This contradicts to the fact that ${\cal B}_k(X)$
is a complex space.
\par\smallskip
\hfill{q.e.d.}


\subsection*{1.5. Construction of Example 1} 

We start with recalling one example due to M. Kato, see [Ka-1].

In $\cc\pp^3$ with homogeneous coordinates $[z_0:z_1:z_2:z_3]$ consider a
domain $D= \{ z\in \cc\pp^3: \vert z_0\vert^2+\vert z_1\vert^2>\vert z_2\vert
^2+\vert z_3\vert^2\} $. Group $Sp(1,1)$ naturally acts on $\cc\pp^3$ and
preserves $D$, i.e. $g(D)=D$ for all $g\in Sp(1,1)$. Action of $Sp(1,1)$ is
transitive on $D$ and Kato proved, using the result of Vinberg, that there
exist a discrete subgroup $\Gamma\subset Sp(1,1)$, acting properly and
discontinuously on $D$, and such that $D/\Gamma =X^3$ is a compact complex
manifold, see [Ka-1] for details.

Projective plane $\cc\pp^2=\{ z_3=0\} $ intersects $D$ by the complement to
the unit ball  $[z_0:z_1:z_2]\in \cc\pp^2:\vert z_2\vert^2<\vert z_0\vert^2+
\vert z_1\vert^2\} $. If $\pi :D\to X^3$ is a natural projection, then its
restriction $\pi\mid_{\cc\pp^2\cap D}:\cc\pp^2\setminus \bar\bb^4\to X^3$
defines a holomorphic map from the complement to the closed unit ball to
$X^3$, which has singularity at each point of $\partial\bb^4$!
\par\smallskip

Blow up $\cc\pp^4$ at origin of its affine part. Denote by $\cc\pp^4_0$
the resulting manifold. There is a natural holomorphic projection.
$p:\cc\pp^4_0\to \cc\pp^3$-concidered as an exeptional divisor. $\Gamma $
beeing a group of $4\times 4$ matrices acts naturally on affine part
$\cc^4$ of $\cc\pp^4$. This action obviously extends onto $\cc\pp^4$ and
lifts onto $\cc\pp^4_0$. Moreover the actions of $\Gamma $ on $\cc\pp^3$
and $\cc\pp^4_0$ are equivariant with respect to the projection $p$.
Put $\hat D:=p^{-1}(D)$ and $\hat X:=\hat D/\Gamma $. If take now
$f:=\{ z\in \cc^4: \vert z_0\vert^2+\vert z_1\vert^2>\vert z_2\vert^2+
\vert z_3\vert^2\} \to \hat X$ to be a restriction of quotient map, we get
our example.






\section*{2. Hartogs-type extension and spherical shells}
\subsection*{2.1. Generalities on pluripotential theory}

We start with some well known facts from pluripotential theory . Let $D$ be
an open subset of $\cc^n$ and $S$ subset of $D$.
Consider the following class of functions:
\par\smallskip
$${\cal U}(S,D) = \{ u\in {\cal P}_{+}(D) : u|_S\ge 1\}\eqno
(2.1.1)
$$
\par\smallskip
\noindent
where by ${\cal P}_{+}(D)$ we denote the class of nonnegative
plurisuperharmonic functions in $D$.
\par\smallskip
\noindent
\bf Definition 2.1.1. \rm The  lower regularization $w_*$ of the function
\par\smallskip
$$
w(\zeta ,S,D) = \inf \{u(\zeta ) :
u\in {\cal U}(S,D)\}\eqno(2.1.2)
$$
is called a ${\cal P}$- measure of $S$ in $D$, i.e.
$$
w_*(z,S,D)=\lim_{\zeta \rightarrow z}\inf w(\zeta ,S,D) \eqno(2.1.3)
$$
\par\smallskip\noindent 
Note that $w_{\ast }$ is plurisuperharmonic in $D$ .
\par\smallskip
\noindent
\bf Definition 2.1.2. \rm A point $s_0\in S$ is called a locally regular point
of $S$ if $w^*(s_0,S\cap \Delta^n (s_0,\varepsilon ),\Delta^n (s_0,\varepsilon ))
= 1$ for all $\varepsilon >0$.

We shell also say that the set $S$ is locally regular at $s_0$.


\rm
\par\smallskip
Recall that subset $S\subset D $ is called pluripolar 
if there exists a plurisubharmonic function $u$ in $D ,u\not\equiv -\infty $, 
such
that $u\mid_S \equiv -\infty $. $S$ is complete pluripolar if $S=\{ z: u(z)=
-\infty \} $. We shall repeatedly use the following statement:

\it if subset $S\subset D $ ($D$ is now pseudoconvex) is not locally regular 
at all its points then $S$ 

is pluripolar, 

\noindent 
\rm see [B-T],[Sd]. We shall repeteadly use  the following immediate corollary from the 
famous Josefson theorem, see [Kl]: 
\par\smallskip\noindent\bf
\it Let $\Omega $ be a pseudoconvex set in $\cc^n $, and let $S_n$ be a 
sequence of subsets of $\Omega $ such that:

1) $S_1$ is closed and pluripolar in $\Omega $;

2) $S_{n+1}\subset \Omega\setminus S_n$ is closed in $\Omega \setminus S_n$
and pluripolar;



\noindent
Then $S:=\bigcup_{n=1}^{\infty }S_n$ is pluripolar in $\Omega $.
\par\smallskip\noindent
\rm 

\rm Denote by ${\cal D}^{k,k}(\Omega )$ the space of $C^{\infty }$-forms of
bidegree $(k,k)$ with compact support on complex manifold $\Omega $. $\phi \in
{\cal D}^{k,k}(\Omega )$ is real if $\bar \phi =\phi $.
The dual space ${\cal D}_{k,k}(\Omega )$ is the space of currents of
bidimension $(k,k)$ (bidegree $(n-k,n-k)$, $n=\dim \Omega $).
$T\in {\cal D}_{k,k}(\Omega )$ is real if $<T,\bar \phi >=\overline{<T,\phi >}$
for all $\phi \in {\cal D}^{k,k}(\Omega )$.
\par\smallskip\noindent\bf
Definition 2.1.3. \rm Current $T\in {\cal D}_{k,k}(\Omega )$ is called
positive if for all $\phi_1,...,\phi_k\in {\cal D}^{1,0}(\Omega )$
$$
<T,{i\over 2}\phi_1\wedge \bar\phi_1\wedge ...\wedge {i\over 2}\phi_k\wedge 
\bar\phi_
k>\ge 0.
$$
\par\smallskip
$T$ is negative if $-T$ is positive. 
\par\smallskip\noindent\bf
Definition 2.1.4. \rm We shall say that the current $T\in {\cal D}_{k,k}
(\Omega )$ is pluripositive (-negative) if $T$ is positive and $dd^cT$ is
positive (-negative).

$T$ is pluridefinite if it is either pluripositive or plurinegative.
\par\smallskip\noindent\bf
Definition 2.1.5. \rm The current $T$ (not necessarily positive) is pluriclosed
if $dd^cT=0$.
\par\smallskip
Let $\phi =\Sigma_J\phi_J(x)dx_J\in {\cal D}^k(\Omega )$, where $\Omega $ open
subset of $\rr^n$. Euclidean norm of $\phi $ at  $x$ is
$$
\Vert \phi (x)\Vert = (\Sigma_{\vert J\vert =k}\vert \phi_J(x)\vert^2)^{1/2}.
$$
Further if $T\in {\cal D}_k(\Omega )$ and $U$ is open in $\Omega $ then the
mass of $T$ in $U$ is a number
$$
\Vert T\Vert (U)=sup\{ \vert <T,\phi>\vert :\phi\in {\cal D}^k(U), \Vert \phi
(x)\Vert \le 1, x\in U\}. \eqno(2.1.4)
$$
Let $K\in \Omega $ be closed and $T\in {\cal D}_k(\Omega \setminus K)$. We say
that $T$ has locally finite mass in the neighborhood of $K$ if for any open
relatively compact $U\in \Omega $ $\Vert T\Vert (U\setminus K)<\infty $. In
this case a trivial extension $\tilde T$ of $T$ onto $\Omega $ is defined in
the following way. Take a sequence $u_n\in C^{\infty }(\Omega )$, $0\le u_n\le
1, u_n\equiv 0 $ in the neighborhood of $K$, and $u_n\nearrow \chi_{\Omega
\setminus K}$ uniformly on compacts in $\Omega \setminus K$. Here $\chi_A$ is
a characteristic function of the set $A$. Then for $\phi \in {\cal D}^k(\Omega
)$ put
$$
<\tilde T,\phi >=\lim_{n\rightarrow \infty }<u_nT,\phi > \eqno(2.1.5)
$$
\noindent
This correctly defines a current on $\Omega $, see [Lg].

If $K$ is complete pluripolar compact in strictly pseudoconvex domain $\Omega
\subset \cc^n$ and $T$ closed, positive current on $\Omega \setminus K$, then
$T$ has locally finite mass in the neighborhood of $K$, see [Iv-2], {\sl Lemma
2.1 }.
\par\smallskip\noindent\bf
Lemma 2.1.1. \it (a) Let $K$ be a complete, pluripolar compact in strictly
pseudoconvex domain $\Omega \in \cc^n$ and $T$ be a pluridefinite current
of bidegree (1,1)  on $\Omega \setminus K$ of locally finite mass in the
neighborhood of $K$, and such that $dT$ has coefficients measures in $\Omega 
\setminus K$. Then $dd^c\tilde T$ has coefficients measures on $\Omega $.

(b) If $K$ is of Haussdorff dimension zero and $n=2$ then
$\chi_K\cdot dd^c\tilde T$
is negative.
\par\smallskip\noindent\sl
Proof. \rm Part (a) of this lemma is proved in [Iv-2], Proposition 2.3 for
the
currents of bidimension (1,1) (the
condition on $dT$ was forgotten there). If $T$ is of bedegree (1,1) then
take $T\wedge (dd^c\Vert z\Vert^2)^{n-2}$ to get the same conclusion.

(b) Let $\{ u_n\} $ be a sequence of smooth p.s.h.functions in $\Omega $,
equal to zero in the neighborhood of $K$, $0\le u_n\le 1$ and such that
$u_n\nearrow \chi_{\Omega \setminus K}$ uniformly on compacts in $\Omega
\setminus K$, see {\sl Lemma 1.2} from [Sb]. Put $v_n=u_n-1$.

Let $w_e$ denote the Euclidean volume-form in $\cc^2$. Put $\widetilde{dd^cT}=
\mu_0\cdot w_e^n$. Then $\mu_0$ is a measure on $\Omega $. According to
part (a) the distribution  $\mu $ defined from $dd^c\tilde T=\mu \cdot w_e^n$
is a measure. Write
$$
\mu =\chi_K\cdot \mu + \chi_{\Omega \setminus K}\cdot \mu \eqno(2.1.6)
$$
\noindent Where obviously $\chi_{\Omega \setminus K}=\mu_0$. The measure $\chi_
K\cdot \mu $ we denote by $\mu_1$. We shall prove that the measure $\mu_1$ is
nonpositive.

Take a ball in $\cc^n$ centered at $s_0\in K$ such that $\partial B\cap K=
\emptyset $. One has
$$
\mu_1(B\cap K) = -\lim_{k\rightarrow \infty }\int_Bv_k\cdot \mu w_e^n =
- \lim_{k\rightarrow \infty }<v_k,dd^c\tilde T> =
$$
$$
= -\lim_{k\rightarrow \infty }<dd^cv_k,\tilde T> \le 0, \eqno(2.1.7)
$$
\noindent
because $\tilde T$ is positive and $dd^cv_k\ge 0$. So for any such ball we
have
$$
\mu_1(B\cap K)\le 0 \eqno(2.1.8)
$$
All that left, is to use the following Vitali-type theorem for a general 
measures,
see [Fd], p.154 . Let $D$ be an open set in $\cc^n$ and $\sigma $ a finite
positive Borel measure on $D$. Let further ${\cal B}$ be a family of closed
balls of positive radii such, that for any point $x\in D$ the family ${\cal B}$
contains the balls of arbitrarily small radii centered at $x$. Then one can
find a countable subfamily $\{ B_i\} $ of balls in ${\cal B}$ such that
$$
\sigma (D\setminus \bigcup_{(i)}B_i) = 0. \eqno(2.1.9)
$$
Represent our measure $\mu_1$ as a difference $\mu_1=\mu_1^+ - \mu_1^-$ of two
nonnegative measures. Take some relatively compact open subset $D\subset \Omega
$. As ${\cal B}$ take the family of all balls that $\partial B\cap K=\emptyset 
$.
Because of zero dimensionality of $K$ it is a Vitali-type covering. Let $\{ B_i
\} $ are such that $\mu_1^+ (D\setminus \bigcup_{(i)}B_i) = 0$. Then
$\mu_1^+(D) = \mu_1^+(D\setminus \bigcup_{(i)}B_i) + \sum_{(i)}\mu_1^+(B_i) =
\sum_{(i)}\mu_1^+(B_i)$. Consequently
$$
\mu_1(D) = \mu_1^+(D) - \mu_1^-(D) \le \mu_1^+(\bigcup_{(i)}B_i) -
\mu_1^-(\bigcup_{(i)}B_i) =
$$
$$
= \sum_i\mu_1^+(B_i) - \sum_i\mu_1^-(B_i) = \sum_i\mu_1(B_i)\le 0 \eqno(2.1.
10)
$$
\noindent by (2.1.8). Thus $\mu_1(D)\le 0 $ for any relatively compact open 
$D$ in $\Omega $. So
the measure $\mu_1 $ is negative.
\par\smallskip

\hfill{q.e.d.}
\par\smallskip\noindent\bf
Remark. \rm As we shall see in the proof of the Theorem 2 (Lemma 2.6.1) the
conclusion of the part (b) of Lemma 2.1.1 remains true for arbitrary $n$ and
complete, pluripolar closed set of Haussdorff dimension $2n-4$. 
\par\bigskip
\par\bigskip
Recall, see [Sb], that a subset $K\subset \Omega $ is called (complete)
$p$-polar if for any $a\in \Omega $ there exists a neighborhood $V$ of $a$
and such coordinates $z_1,...,z_n$ in $V$ that the sets
$$
K_{z^0_I} = K\cap \{ z_{i_1}=z_{i_1^0},...,z_{i_p}=z_{i_p^0}\} \eqno(2.1.11)
$$
\noindent
are (complete) pluripolar in subspaces $V_{z_i^0}:=\{ z\in V:
z_{i_1}=z_{i_1^0},...,z_{i_p}=z_{i_p^0}\} $ for almost all $z_I^0=(z_{i_1}^0,
...,z_{i_p}^0)\in \pi^I(V)$, where $I$ runs over a finite set of multiindices
with $\vert I\vert =p$, such that $\{ (\pi^{I})^*w_e\} _I $ generates the
space of bedegree $(p,p)$-forms. Here $\pi^I(z_1,...,z_n)=(z_{i_1},...,
z_{i_p})$ stands for the projection onto
the space of variables $(z_{i_1},...,z_{i_p})$.

\subsection*{2.2. Metrics on complex spaces and proof of Theorem 2 in dimension two}

A Hermitian metric form on the complex space $X$ we define in the following 
way. Let an open covering $U_{\alpha }$ of $X$ is given together with proper 
holomorphic injections $\phi_{\alpha }:U_{\alpha }\to V_{\alpha }$ into a 
domains $V_{\alpha }\subset \cc^{n(\alpha )}$. Let $U_{\alpha }^{'}$ be the 
images of $U_{\alpha }$. Let $\{ w_{\alpha }\} $ are positive (1,1)-forms on 
$V_{\alpha }$. $\{ w_{\alpha }\} $ defines a Hermitian metric form on $X$ if 
$(\phi_{\alpha }\circ \phi_{\beta }^{-1})^*w_{\beta }=w_{\alpha }$ for all 
$\alpha ,\beta $. Note that  $\phi_{\alpha }\circ \phi_{\beta }^{-1}$ is 
defined in some neighborhood of $\phi_{\beta }(U_{\alpha }\cap U_{\beta })$ 
in $\cc^{n(\beta )}$. We say that the metric $w$ is K\"ahler if $dw_{\alpha
}=0$ on $V_{\alpha }$ for all $\alpha $.
\par\smallskip\noindent\bf
Definition 2.2.1. \rm $w$ is pluriclosed (negative) if $dd^cw_{\alpha }=0$ 
($dd^cw_{\alpha }\le 0$) in $V_{\alpha }$  for all $\alpha $. 

Let a meromorphic mapping $f:H^2(r)\to X$ from two-dimensional Hartogs
figure into a disk-convex complex space is given. Let $w$ be a plurinegative
metric form on $X$. By $W$ denote the maximal open subset of the unit disk
$\Delta $ such that $f$ meromorphically extends onto $H_W(r):=W\times \Delta
\cup \Delta \times A_{1-r,1}$. Here by $A_{1-r,1}$ we denote the annulus
$\{ z\in \cc :1-r<\vert z\vert <1\} $. Let $I(f)$ be the fundamental set of 
$f$ and by 
$\hat f$
denote the mapping $\hat f(z)=(z,f(z))$ into the graph. For $z\in W$
define
$$
\mu_t(z) = \area  \hat f(\Delta_z(t)) = \int_{\Delta_z(t)}(dd^c\vert
\lambda \vert ^2+f\mid_{\Delta_z(t)}^*w).\eqno(2.2.1)
$$
\noindent
Here $\Delta_z(t)=\{ (z,\lambda ):\vert \lambda \vert <t\} $.
We start with the following simple observation. Denote by $\nu_1=\nu_1(K)$ 
the infimum of areas of compact complex curves which are contained in 
compact $K\subset X$. $\nu_1>0$, see {\sl Lemma 2.3.1} below.
\par\smallskip\noindent\bf
Lemma 2.2.1. \it Let $f:\Delta \times A_{1-r,1}\to X$ be a holomorphic mapping 
into a disk-convex complex space $X$. Suppose that for a sequence of points 
$\{ s_n\} \subset \Delta $, $s_n\rightarrow 0$, the following holds:

(a) $f_s:=f\mid_{\{ s\} \times A_{1-r,1}}$ extends holomorphically onto $\Delta
_s:=\{ s\} \times \Delta $;

(b) For  a compact $K$  in $X$, which contains $f[(\Delta (1/2)\times A_{1-
2/3\cdot r,1-1/3\cdot r})\cup \bigcup_{(n)}\{ s_n\} \times \Delta_{1-1/3\cdot 
r}]$, one has 
$$
\vert \area  \hat f(\Delta_{s_n}(1-1/3\cdot r)) - \area  \hat 
f(\Delta_0(1-1/3\cdot r)\vert \le {1\over 2}\cdot \nu_1(K). \eqno(2.2.2)
$$
\noindent
Then $f$ holomorphically extend onto $V\times \Delta $ for some open $V\ni 0$.
\par\smallskip\noindent\sl
Proof. \rm First of all let us show that ${\cal H}-\lim_{n\rightarrow \infty }
\hat f(\bar\Delta_{s_n}(1-1/3\cdot r))=\hat f(\bar\Delta_0(1-1/3\cdot r))$, 
i.e. the sequence of graphs $\{ \hat f(\bar\Delta_{s_n}(1-1/3\cdot r))\} $ 
converges in Hausdorff metric to the graph of the limit. If not, there would 
be a subsequence (still denoted as $\{ \hat f(\bar\Delta_{s_n}(1-1/3\cdot r))
\} $), such that  ${\cal H}-\lim_{n\rightarrow \infty }
\hat f(\bar\Delta_{s_n}(1-1/3\cdot r))=\hat f(\bar\Delta_0(1-1/3\cdot r))\cup 
\bigcup_{j=1}^NC_j$, where $\bigcup_{j=1}^NC_j$ is a union of compact curves, 
see {\sl Lemma 2.3.1} below. Thus by (2.1.2)  $\area  \hat f(\bar\Delta_
{s_n}(1-1/3\cdot r))\ge \area \hat f(\bar\Delta_0(1-1/3\cdot r))+N\cdot 
\nu_1(K)$. This contradicts (2.2.2).

Take a Stein neighborhood $V$ of  $\hat f(\bar\Delta_0(1-1/3\cdot r))$, see 
[Si-1]. Then for $\delta >0$ small enough we have that $f(\Delta_{\delta }
\times A_{1-1/3r-\delta ,1-1/3r+\delta })\subset V$ and $f(\Delta_{s_n}(1-1/3r)
)\subset V$ if $s_n\in \Delta_{\delta }$. From Hartogs theorem for holomorphic 
functions we get that $f$ extends to a holomorphic map from $\Delta_{\delta }
\times \Delta_{1-1/3r-\delta }$ to $V$. 
\par\smallskip
\hfill{q.e.d.}

\par\smallskip
\noindent
\bf Lemma 2.2.2. \it If the metric form $w$ is plurinegative and $W$ is
maximal then $\partial W\cap \Delta $ is complete polar in $\Delta $.
\par\smallskip\noindent\sl
Proof. \rm Take a point $z_0\in \partial W\cap \Delta $. Choose  relatively
compact in $\Delta $ neighborhood $U$ of $z_0$ and $1-r<t<1$ such that
$I(f)\cap \bar U\times \partial \Delta(t)=\emptyset $. Denote by $\phi=i\phi^
{\alpha \beta }dz_{\alpha }\wedge d\bar z_{\beta }$ the current $f^*+dd^c
\Vert z\Vert^2$.
The area function from (1.2.1) can be written now as
$$
\mu_t(z_1) = i\cdot \int_{\vert z_2\vert \le t}\phi_{22}(z_1,z_2)dz_2\wedge
d\bar z_2 .\eqno(2.2.3)
$$
The condition that $dd^c\phi $ is negative means that
$$
{\partial ^2\phi_{11} \over \partial z_2\partial\bar z_2}+
{\partial ^2\phi_{22} \over \partial z_1\partial\bar z_1}-
{\partial ^2\phi_{12} \over \partial z_2\partial\bar z_1}-
{\partial ^2\phi_{21} \over \partial z_1\partial\bar z_2}\le 0 \eqno(2.2.4)
$$
\noindent
on $H_W(r)$. Now we can estimate the Laplacian of $\mu_t $:
$$
\Delta \mu (z_1) = i\int_{\vert z_2\vert \le t}
{\partial ^2\phi_{22} \over \partial z_1\partial\bar z_1}dz_2\wedge d\bar z_2 
\le i\int_{\vert z_2\vert \le t}(-{\partial ^2\phi_{11} \over \partial z_2
\partial\bar z_2}+{\partial ^2\phi_{12} \over \partial z_2\partial\bar z_1} + 
{\partial ^2\phi_{21} \over \partial z_1\partial\bar z_2})dz_2\wedge d\bar z_2
= 
$$
$$
= i\int_{\vert z_2\vert = t}{\partial \phi_{11} \over \partial z_2
}dz_2 + i\int_{\vert z_2\vert =t}{\partial \phi_{12} \over \partial\bar z_1}
d\bar z_2  - i\int_{\vert z_2\vert =t}{\partial \phi_{21} \over \partial z_
1} dz_2 = \psi (z_1).\eqno(2.2.5)
$$
Inequality (2.2.5) holds for $z_1\in U\cap W$. But the right hand side $\psi $ 
is smooth in the whole $U$. Let $\Psi $ be the smooth solution of $\Delta \Psi 
=\psi $ in $U$. Put $\hat \mu (z)=\mu_t(z)-\Psi (z)$. Then $\hat \mu $ is 
superharmonic and bounded from below in $U\cap W$, after shrinking of $U$. 

Denote further by $E$ the set of points $z_1\in \partial W\cap U$ such that 
$\mu_t(z)\rightarrow +\infty $ as $z\in W,z\rightarrow z_1$. Note that $\hat 
\mu (z)$ also tends to $+\infty $ in this case. For any point $z_{\infty }\in 
[\partial W\cap U]\setminus E$ we can find a sequence $\{ z_n\} \subset W,z_n 
\rightarrow z_{\infty }$ such that $\mu_t(z_n)\le C$. So by {\sl Lemma 2.2.1} 
$f\mid_{\Delta_{z_{\infty }}\setminus \Delta_{z_{\infty }}(1-r)}$ extends onto 
$\Delta_{z_{\infty }}$. Thus we can define 
$$
\mu_t(z) = \area  \hat f(\Delta_z(t)) = \int_{\Delta_z(t)}(dd^c\vert
\lambda \vert ^2+f\mid_{\Delta_{z_{\infty }}(t)}^*w).\eqno(2.2.6)
$$
Let $\nu_1$ be from {\sl Lemma 2.2.1}. Set $E_j=\{ z\in \partial W\cap U:
\mu_t(z)\le {j\over 2}\nu_1\} $ for $j=1,2,...$. From {\sl Lemma 2.3.1} we 
see that $E_j$ are closed subsets of $\partial W\cap U$, $E_j\subset E_{j+1}$ 
and we have that $\partial W\cap U = E\cup \bigcup_{j=1}^{\infty }E_j$. 

Further from {\sl Lemma 2.2.1}  we see that 
$E_{j+1}\setminus E_j$ is a discrete subset of $U\setminus E_j$, say $U
\setminus E_j = \{ a_{ij}\} $. Now put 
$$
u_1(z) =-\sum_{i,j}c_{ij}\log \vert z-a_{ji}\vert. \eqno(2.2.7)
$$
Here the positive constants $c_{ij}$ are chosen in such a manner that  
$\sum_{i,j}c_{ij}< +\infty $. Then $u_1(z)$ is superharmonic in $U$,  
$u_1(z)\rightarrow +\infty $ as $z\rightarrow \bigcup_{j=1}^{\infty }E_j$ 
and $u_1(z)\not= +\infty $ for all $z\in U\cap W$. Now put $u_2(z)= \hat 
\mu (z) + u_1(z)$. Note that $u_2$ is superharmonic in $W\cap U$ and $u_2(z) 
\rightarrow +\infty $ as $z\rightarrow \partial W\cap U$. Define 
$$
u_n(z) = \min \{ n, u_2(z)\} \eqno(2.2.8)
$$
\noindent 
for $n\ge 3$. Note that $u_n$ are superharmonic in $U$, because $u_n\equiv n$ 
in 
the neighborhood of $\partial W\cap U$. Put now $u(z) =\lim_{n\rightarrow 
\infty }u_n(z)$. Then $u$ is superharmonic in $U$ as a nondecreasing limit of 
superharmonic functions. Using the fact that $\hat \mu $ is 
finite on $W$, we obtain that $u(z)=u_2(z)\not= +\infty $ for any $z\in U\cap 
W$ and $u\mid_{U\setminus W}\equiv +\infty $, i.e.  $\partial W\cap \Delta $ is
{\it complete} polar in $\Delta $. 
So the {\sl Lemma } is proved.
\par\smallskip
\hfill{q.e.d.}

\par\smallskip
In what follows we shall use the fact that the closed set of zero harmonic 
measure on the plain has zero Hausdorff dimension, see [Gl].

\noindent 
(a) Let us finish the proof of the part (1) of {\sl Theorem 2} in
dimension two. Put $S_1=\Delta
\setminus W$ where $W$ is a maximal domain in $\Delta $ such that our map $f$ 
extends meromorphically onto $H_W(r)$. We had proved that $S_1$ is of harmonic 
measure zero. In particular $S_1$ is zero dimensional. For any $\delta >0$ we 
can find $0<\delta_1<\delta $ such that $\partial \Delta_{1-\delta_1}\cap S_1
=\emptyset $. Now we can change coordinates $z_1,z_2$ and consider the 
Hartogs figure $H = \{ (z_1,z_2)\in \Delta^2: 1-r<\vert z_2\vert <1, \vert z_1
\vert <1 $ or $\vert z_2\vert <1, 1-\delta_1-\eps <\vert z_1\vert <1-\delta_1+
\eps \} $, where $\eps $ is small enough. Applying once more  {\sl Lemma 
2.2.1} we extend $f$ onto $\Delta \times (\Delta \setminus S_2)$ where $S_2$ 
is of harmonic measure zero and obtain the statement of part (1) of 
{\sl Theorem 2} in the case of dimension two.

\noindent 
(b) Suppose now that our metric form $w$ on $X$ is pluriclosed. Adding to 
$S:=S_1\times S_2$ the discrete in $\Delta^2\setminus S$ set of points of 
indeterminacy of $f$ we can suppose that $f$ is holomorphic on $\Delta^2
\setminus S$. Denote by $T$ the positive (1,1) current (in fact smooth form) 
$f^*w$ on $\Delta^2\setminus S$. By {\sl Lemma 3.3} from [Iv-2] we have that 
$T$ has locally summable coefficients on the whole $\Delta^2$ and  
$dd^c\tilde T$ is a negative measure supported on $S$.
\par\smallskip\noindent\bf
Lemma 2.2.3. \it Suppose that the metric form $w$ is pluriclosed and take a 
ball $B\subset\subset \Delta^2$ such that $\partial B\cap S=\emptyset $. 

(i) If $f(\partial B)$ is homologous to zero in $X$ then $dd^c\tilde T=0$ on 
$B$. 

(ii) If $dd^c\tilde T=0$ then $f$ meromorphically extends onto $B$.
\par\smallskip 
\rm In [Iv-2] {\sl Lemma 4.4 }] this statement is proved for the case 
when $S\cap B=\{ 0\} $. One can easily check that the same proof holds for the 
case when $S\cap B$ is closed zero dimensional.

So statement $(2)$ of {\sl Theorem 2} is proved in this case.
\par\smallskip\noindent 
Let us prove the Remark 1, stated after the Theorem 2 in Introduction.
Take a relatively compact open subset $P\subset \Delta^2$ with smooth
boundary  and choose a finite subcomplex $K$ of CW-complex $X$ to contain the 
$\cl [f(\bar P\setminus S)]$. Let $\theta_1,...,\theta_N$ be the generators
of $H^3(K,\zz )$ and $\psi_1,...,\psi_L$ be the generators of $H_3(K,\zz )$. 
Take a real numbers $r_1,...,r_N$ such that 
$$
d^cw = r_1\theta_1 +...+r_N\theta_N \eqno(2.2.9)
$$
Take a ball $B\subset\subset \Delta^2$ with $\partial B\cap S=\emptyset $. 
Then there are integers $z_1,...,z_L$ such that 
$$
f(\partial B) = z_1\psi_1+...+z_L\psi_L \eqno(2.2.10)
$$
\noindent 
in $H_3(K,\zz )$. For the measure $\mu $ defined from $dd^c\tilde T=\mu \cdot 
({i\over 2})^2 
dz\wedge d\bar z$ we have that 
$$
\mu (B\cap S)= \int_Bdd^c\tilde T = \int_{\partial B}d^cT.\eqno(2.2.11)
$$ 
Using that, we can write
$$
\mu (B\cap S) = \int_{f(\partial B)}d^cw = \sum_{k=1}^N\sum_{i=1}^Lz_ir_k 
\int_{\psi_i}\theta_k .\eqno(2.2.12)
$$
\noindent
Put $c^{ik}=\int_{\psi_i}\theta_k\in \zz $. Now if we put $\tilde z^k = 
\sum_{i=1}^Lz_ic^{ik} \in \zz $ then 
$$
\mu (B\cap S) = \sum_{k=1}^N\tilde z^kr_k \eqno(2.2.13)
$$
So for any such a ball $\mu (B\cap S)$ is a linear combination with integer 
coefficients of a given reals $r_1,...,r_N$. By induction one easily prove 
that there is an $\eps =\eps (r_1,..,r_N)>0$ such that each linear combination 
of $r_1,...,r_N$ with integer coefficients is either zero or greater with 
modulus then $\eps $. We denote this $\eps $ as $\eps (w,K)$.

Now we obtain that 
$$
\vert S\cap P\vert \le \eps (w,K)^{-1}\cdot \vert \int_{
\partial P}d^cw \vert . \eqno(2.2.14)
$$              
\par\smallskip
\hfill{q.e.d.}


\subsection*{2.3. Meromorphic families of analytic sets}

When one studies  {\it meromorphic} mappings from domains of dimension more 
then two, one needs some analog of {\sl Lemma 2.2.1} for ``meromorphic 
polydisks''. That will be given in this paragraph, see {\sl Lemmas 2.3.1}
and 2.3.2. For the proof of this two Lemmas we refer to [Iv-4].

Fix a complex space $X$, equipped with some Hermitian metric $h$. By $w_h$, or 
simply by $w$ denote the $(1,1)$-form canonically associated with $h$.
Let $\Delta ^q$ be a polydisk in $\cc^q$ with standard 
Euclidean metric $e$. The associated form will be denoted by $w_e = dd^c 
\Vert z\Vert ^2 = i/2 \sum_{j=1}^qdz_j\wedge d\bar z_j $. By $p_1:\Delta ^q
\times X \longrightarrow \Delta ^q$ and $p_2:\Delta ^q\times X\longrightarrow 
X$ we denote the projections onto the first and second factors. On the product 
$\Delta \times X$ we consider the metric form $w = p^{\ast }_1w_e + p^{\ast }_2
w_h$.
\par\smallskip
\noindent
\bf Definition 2.3.1. \rm  By a meromorphic $q$-disk in the complex space $X$ 
we shall understand a meromorphic mapping $\phi :\Delta ^q \longrightarrow X $ 
, which is defined in some neighbourhood of the closure $\bar \Delta ^q$.
\par\smallskip
It will be convenient for us to consider instead of mappings $\phi : \Delta ^q 
\longrightarrow X$ their graphs $\Gamma _{\phi }$. By $\hat \phi = (z,\phi 
(z))$ we shall denote the mapping into the graph $\Gamma _{\phi } \subset 
\Delta ^q\times X$. The volume of the graph $\Gamma _{\phi }$ of the mapping 
$\phi $ is given by 
\par\smallskip
$$
\vol(\Gamma _{\phi }) = \int_{\Gamma _{\phi }}w^q = 
\int_{\Delta ^q}(\phi ^{\ast }w_h + dd^c\Vert z\Vert ^2)^q \eqno(2.3.1)
$$
\par\smallskip
\noindent
Here by $\phi ^{\ast }w_h$ we denote the preimage of $w_h$ under $\phi $, i.e.
$\phi ^{\ast }w_h = (p_1)_{\ast }p^{\ast }_2w_h$.

Recall that the Hausdorff distance between two subsets $A$ and $B$ of the 
metric space $(Y,\rho )$ is a number $\rho (A,B) = \inf \{ \varepsilon : A^
{\varepsilon } \supset B, B^{\varepsilon }\supset A\} $. Here by $A^
{\varepsilon }$ we denote the $\varepsilon $-neighborhood of the set $A$, i.e. 
$A^{\varepsilon } = \{y\in Y: \rho (y,A) < \varepsilon \}$.

Further, let $\{ \phi _r\} $ be the sequence of meromorphic mappings of the 
complex space $D$ into the complex space $X$.
\par\smallskip
\noindent
\bf Definition 2.3.2. \rm   We shall say that $\{\phi _r\}$ converge on the 
compacts
in $D$ to the meromorphic mapping $\phi :D\longrightarrow X $, if for every 
relatively compact open $D_1 \subset \subset D$ the graphs $\Gamma _{\phi _r} 
\cap (D_1\times X)$ converge in the Hausdorff metric on $D_1\times X$ to the 
graph $\Gamma _{\phi }\cap (D_1\times X)$.

First we shall  prove the following
\par\smallskip
\noindent
\bf Lemma 2.3.1. \it Let $\{\phi _r\}$ be a sequence of meromorphic $q$-disks 
in 
complex space $X$. Suppose that there exists a compact $K\subset X$ and a 
constant $C<\infty $ such that:

a) $\phi _r(\Delta ^q)\subset K$ for all $r$;

b) $\vol(\Gamma _{\phi _r})\le C$ for all $r$.
\par\smallskip
\noindent
Then there exists a subsequence $\{\phi _{r_j}\}$ and a proper analytic set $A
\subset \Delta ^q$ such that:

1) the sequence $\{\Gamma _{\phi _{r_j}}\}$ converges in the Hausdorff metric 
to the analytic subset $\Gamma $ of $\Delta ^q\times X$ of pure dimension $q$;

2) $\Gamma = \Gamma _{\phi }\cup \hat \Gamma $,where $\Gamma _{\phi }$ is the 
graph of some meromorphic mapping $\phi :\Delta ^q \longrightarrow X$,and 
$\hat \Gamma $ is a pure $q$-dimensional analytic subset of $\Delta ^q\times X$
,mapped by the projection $p_1$ onto $A$;

3) $\phi _{r_j}\longrightarrow \phi $ on compacts in $\Delta ^q\setminus A$;

4) one has 
$$
\lim_{j \longrightarrow \infty }\vol(\Gamma _{\phi _{r_j}})\ge 
\vol(\Gamma _{\phi }) + \vol(\hat \Gamma ).\eqno(2.3.2)
$$

5) For every $1\le p\le dimX - 1$ there exists a positive constant $\nu _p 
= \nu _p(K,h)$ such that the volume of every pure $p$-dimensional compact 
analytic subset of $X$ which is contained in $K$ is not less then $\nu _p$.

6) Put $\hat \Gamma = \bigcup_{p=0}^{q-1}\Gamma _p $, where $\Gamma _p$ is a 
union of all irreducible components of $\hat \Gamma $ such that $\dim  
[p_1(\Gamma _p)] = p$. Then 
\par\smallskip
$$
\vol_{2q}(\hat \Gamma ) \ge \sum_{p=0}^{q-1}\vol_{2p}(A_p)\cdot 
\nu _{q-p}\eqno(2.3.3)
$$
\par\smallskip
\noindent
where $A_p = p_1(\Gamma _p)$.
\par\smallskip
\noindent
\par\medskip \rm
Let  $S$ be a set. By $\Delta ^q(b)$ we always denote a polydisk of radii $b$ 
in $\cc^n$ centered at origin. Polydisk is equipped with the usual Euklidean 
metric from $\cc^n$.
\par\smallskip
\noindent 
\bf Definition 2.3.3. \rm By a family of $q$-dimensional polydisks in complex 
space $X$ we shell understand an subset ${\cal F}\subset S\times \Delta ^q(b )
\times X $ such that, for every $s\in S$ the set  $ {\cal F}_s = {\cal F}\cap 
\{s\}\times 
\Delta ^q(b)\times X$ is a graph of a meromorphic mapping of $\Delta ^q(b)$ 
into $X$.

Suppose further that the set $S$ is equipped with topology and let our space 
$X$ be equipped with some Hermitian metric $h$.
\par\smallskip
\noindent
\bf Definition 2.3.4. \rm We shall say that the family ${\cal F}$ is 
continuous 
at  point $s_0\in S$ if ${\cal H}-\lim_{s\rightarrow s_0}
{\cal F}_s = {\cal F}_{s_0}$.

Here by ${\cal H}-\lim_{s\rightarrow s_0} {\cal F}_s $ 
we denote the limit of closed subsets of ${\cal F}_s$ in the Haussdorff
metric on $\Delta ^q(b)\times X$. ${\cal F}$ is continuous if it is 
continuous 
at each point of $S$. If $\Omega $ is open in $\Delta ^q(b)$ then the 
restriction ${\cal F}_{\Omega }$ is naturally defined as ${\cal F}\cap (S 
\times \Omega \times X)$.

When $S$ is a complex space itself, we give the following 
\par\smallskip
\noindent
\bf Definition 2.3.5. \rm Call the family ${\cal F}$ meromorphic if the 
closure $\hat {\cal F}$ of the set 
${\cal F}$ is an analytic subset of $S\times \Delta ^q(b) \times X$.


Consider a 
meromorphic mapping $f: \Delta^p \times \Delta^q(a) \longrightarrow X$ into a 
complex space $X$. Let $S$ be some closed subset of $\Delta^p $ and $s_0\in S$ 
some accumulation point of $S$. Suppose that for each $s\in S$ the 
restriction $f_s = f|_{\{ s\} }\times \Delta^q(a) $ meromorphically extends 
onto 
a $q$-disk $\Delta ^q(b)$. Let, as in Lemma 2.3.1  $\nu _j$ denotes the minima 
of 
volumes of $j$-dimensional compact analytic subsets contained in some compact 
$K\subset X$. Fix some $a<c<b$ and put 
\par\smallskip
$$\nu = \min \{ \vol(A_{q-j})\cdot \nu _j : j = 1,...,q\},\eqno(2.3.6)$$
\par\smallskip
\noindent
where $A_{q-j}$ are running over all $(q-j)$-dimensional analytic subsets of 
$\Delta ^q(b)$, intersecting $\bar \Delta ^q(c)$. Here $a<c<b$. Clearly 
$\nu > 0$. In the 
following {\sl Lemma } the volumes of graphs over polydisks $\Delta (b)$ 
are taken.
\par\smallskip
\noindent
\bf Lemma 2.3.2. \it Suppose  that there exists 
a neighbourhood $U\ni s_0$ in $
\Delta^p $ such that, for all $s_1,s_2\in S\cap U$ 
\par\smallskip
$$
| \vol(\Gamma _{f_{s_1}}) - \vol(\Gamma _{f_{s_2}}) 
| < \nu /2 \eqno(2.3.7)
$$ 
\par\smallskip
\noindent
Then the family $\{ \Gamma_{f_s}: s\in U\} $ is continuous at $s_0$. If,
moreover $s_0$ is a locally regular point of $S$ then  there exists
a neighbourhood $V_c\ni s_0$ in $\Delta^p $ such, that $f$ 
meromorphically extends onto $V_c\times \Delta ^q(c)$.

\subsection*{2.4. Proof of Theorem 2 in higher dimensions}

Proof. \rm (1) Let $f:H^1_n(r)\to X$ be our map. For an open subset $W
\subset \Delta^n$ denote by 
$$
H_W(r) = (W\times \Delta )\cup (\Delta^n\times A_{1-r,1}) 
$$ 
\noindent 
the Hartogs figure {\it over} $W$. 
\par\smallskip 
\noindent\sl 
Step 1. \it $f$ extends to a holomorphic map of $\bigcup_{z^{'}\in\Delta^{n-1}
_r\setminus R_1}(\Delta_{z^{'}}^2\setminus S_{z^{'}})$ into $X$, where $R_1$ is 
contained in locally finite union of locally closed proper 
subvarieties of $\Delta^{n-1}_r$ and $S_{z^{'}}$ is zerodimensional and 
pluripolar in $\Delta^2_{z^{'}}$.
\par\smallskip\noindent\sl 
Proof of Step 1. \rm 
For $z'=(z_1,...,z_{n-1})\in \Delta^{n-1}_r$ by $H^2_{z'}(r)$ denote the 
two-dimensional Hartogs domain $\{ z'\} \times H^2(r)$ in bidisk $\Delta^2_{z'}
=
\{ z'\} \times \Delta^2\in \cc^{n+1}$. Shrinking $H_n^1(r)$ if necessary, 
we can suppose that $I(f)$ consists of finitely many irreducible components. 
Denote by $R_1$ the set of $z'\in \Delta^{n-1}_r$ such that $\dim [H^2_
{z'}\cap I(f)]>0$. $R_1$ is clearly contained in  finite union of 
locally closed proper analytic subsets of $\Delta_r^{n-1}$. For $z'\in \Delta^
{n-1}_r\setminus R_1$ by 2.1 $f\mid_{H^2_{z'}(r)}$ extends to a 
holomorphic map $f_{z'}:\Delta^2_{z'}\setminus S_{z'}\to X$, where $S_{z'}$ is 
zerodimensional and complete pluripolar in $\Delta^2_{z'}$. Also $S_{z'}
\supset \Delta^2_{z'}\cap I(f)$.

Take a point $z'\in \Delta^{n-1}_r\setminus R_1$ and a point $z_n\in \Delta 
\setminus \pi_n(S_{z'})$. Here $\pi_n:\{ z'\} \times \Delta \times\Delta \to 
\{ z'\} 
\times \Delta $ is the projection onto the variable $z_n$. Take a domain $U  
\subset\subset \{ z'\} \times \Delta \times \{ 0\} $ which is biholomorphic 
to the unit disk, doesn't contains points from $\pi_n(S_{z'})$ and contains 
the points $u:=(z',0,0) $ and $v:=(z',z_n,0)$. We take also $U$ to intersect
$A(r,1)$.
If $\{ z'\} \times \{ 0\}$ is
in $\pi_n(S_{z'}$ the take as $u$ some point close to $(z',0,0)$ in $\{ z'\} 
\times \Delta $. Find a Stein neighbourhood $V$ of the graph $\Gamma_{f\mid_
{\{ z'\} \times \bar U\times \Delta }}$. Let $w\in \partial U\cap A(r,1)$ be
some point. We have that $f(\{ z',w\} \times \Delta )\subset V$ and 
$f(\{ z'\} \times \partial U\times \Delta )\subset V$. So the usual
continuity principle for holomorphic functions 
gives us the {\it holomorphic } extension of $f$ to the neighborhood of $\{
z'\} \times \bar
U\times \Delta $ in $\Delta^{n+1}$. Changing little bit a bend of $z_{n+1}$ 
and repeating the arguments as above  we obtain a holomorphic   extension of  
$f$ onto the neighborhood of $\{ z'\} \times (\Delta \setminus S_{z'})$ for 
each $z'\in \Delta^{n-1}_r\setminus R_1$.      
\par\smallskip\noindent\sl 
Step 2. \it $f$ extends holomorphically onto $(\Delta^{n-1}_r\times \Delta^2)
\setminus R$, where $R$ is a closed subset of $\Delta^{n-1}_r\times \Delta^2$ 
of Hausdorff codimension $4$.

\par\smallskip\noindent\sl 
Proof of Step 2. \rm Consider a subset $R_2\subset R_1$ consigning of such 
$z^{'}\in \Delta^{n-1}_r$ that $\dim [H^2_{z^{'}}\cap I(f)]=2$, i.e. 
$H^2_{z^{'}}\subset I(f)$. This is a finite union of locally closed 
subvarieties  of $\Delta^{n-1}_r$ of complex codimension at least two. Thus 
$\bigcup_{z^{'}\in R_2}\Delta^2_{z^{'}}$ has Hausdorff codimension at least 
four.

For $z^{'}\in R_1\setminus R_2=\{ z^{'}\in \Delta^{n-1}_r:\dim [H^2_{z^{'}}
(r)\cap I(f)]=1\} $ using {\sl Theorem 1} we can extend $f_{z^{'}}$
holomorphically onto
$\Delta^2_{z^{'}}$ minus zerodimensional polar set. Repeating the arguments 
from {\sl Step 1} we can extend $f$ holomorphically  to the neighborhood of 
$\Delta^2_{z^{'}}\setminus C_{z^{'}}$ in $\Delta^{n-1}_r\times \Delta^2$. 
Here $C_{z^{'}}$ is a complex curve containing one dimensional components 
of $H^2_{z^{'}}(r)\cap I(f)$. 

$\bigcup_{z^{'}\in R_1\setminus R_2}C_{z^{'}}$ has Hausdorff codimension 
at least four. Thus the proof of {\sl Step 2} will be completed when we 
put $R=\bigcup_{z^{'}\in R_1\setminus R_2}C_{z^{'}}\cup \bigcup_{z^{'}\in 
R_2}\Delta^2_{z^{'}}$.

\par\smallskip\noindent\sl 
Step 3. We shall state this step in the form of the {\sl Lemma}. 
\par\smallskip\noindent\sl
Lemma 2.6.1. \it There is a closed (n-1)-polar subset $R_0\subset R$ and a 
holomorphic extension of $f$ onto $(\Delta^{n-1}_r\times \Delta^2)\setminus 
R_0$ such that the current $T:=f^*w$ has locally summable coefficients in the 
neighborhood of $R_0$. Moreover $dd^c\tilde T$ is negative.
\par\smallskip\rm 
Take a point $z_0\in R$ and using the fact that $R$ is of Hausdorff 
codimension four in $\cc^{n+1}$, find a neighbourhood $V\ni z_0$ with a 
coordinate system $(z_1,...,z_{n+1})$ such that $V=\Delta^{n-1}\times \Delta^2
$ is those coordinates and for all $z'\in \Delta^{n-1}$ one has  $R\cap 
\partial \Delta^2_{z'}=0$. By 2.1  
the restrictions $f_{z'}$ extend holomorphically onto $\Delta^2_{z'}\setminus 
R_0(z')$, where $R_0(z')$ are closed complete pluripolar in $\Delta^2_{z'}$ of 
Hausdorff dimension zero. By the arguments similar to that ones from {\sl 
Step1 } $f$ extends holomorphically to the neighborhood of $V\setminus R_0$, 
$R_0:=\bigcup_{z'\in \Delta^{n-1}}R_0(z')$.

Consider now a current $T=f^*w$ defined on $(\Delta^{n-1}\times \Delta^2)
\setminus R$. Note that $T$ is smooth, positive and $dd^cT\le 0$ there. By the 
part ($b_2$) of {\sl Lemma 1.1.3} every restriction $T_{z'}:=T\mid_{\Delta^2_
{z'}}\in L^1_{loc}(\Delta^2_{z'})$, $z'\in \Delta^{n-1}$. We shall use the 
following Oka-type inequality for plurinegative currents, proved in [F-Sb]: 

\it there is a constant $C_{\rho }$, such that for any plurinegative current 
$T$ in $\Delta^2$ one has 
$$
\Vert T\Vert (\Delta^2) + \Vert dd^cT\Vert (\Delta^2)\le C_{\rho }\Vert T
\Vert (\Delta^2\setminus \bar\Delta^2_{\rho }).\eqno(2.4.1)
$$
\noindent
Here $0<\rho <1$. \rm 

Use (2.4.1) for the trivial extensions $\tilde T_{z'}$ of $T_{z'}$, which are 
plurinegative by ($b$) of {\sl Lemma 2.1.1}, to obtain that masses $\Vert 
\tilde T_{z'}\Vert (\Delta^2)$ are uniformly bounded on $z'$ on compacts in 
$\Delta^{n-1}$. On $L^1$ the mass norm coincides with the $L^1$-norm. So 
taking the second factor in $\Delta^{n-1}\times \Delta^2$ with different bends
and using Fubini theorem we obtain that $T\in L^1_{loc}(\Delta_r^{n-1}\times 
\Delta^2)$.

All that left to prove is that $dd^c\tilde T$ is negative. It is enough to 
prove that for any collection $L$ of $(n-1)$ linear functions $\{ l_1,...,l_
{n-1}
\} $ the measure $dd^c\tilde T\wedge {i\over 2}\partial l_1\wedge\bar
\partial l_1\wedge ...\wedge {i\over 2}\partial l_{n-1}\wedge \bar\partial l_
{n-1} $ is nonpositive, see [Hm]. Complete those functions to the coordinate 
system 
$\{ z_1=l_1,...,z_{n-1}=l_{n-1},z_n,z_{n+1}\} $ and note that for almost all 
$z'\in \Delta^{n-1}$ the set $\Delta^2_{z'}\cap R_0$ is of Hausdorff dimension 
zero. Thus $\tilde T\mid_{z'}$ is plurinegative for all such $z'$. Take a 
nonnegative function $\phi \in {\cal D}(\Delta^{n+1})$. We have that 
$$
<dd^c(\tilde T\wedge (dd^c\Vert L\Vert^2)^{n-1}),\phi > = \int_{\Delta^{n+1}}
\tilde T\wedge (dd^c\Vert z'\Vert^2)^{n-1}\wedge dd^c\phi =   
$$
$$
=  \int_{\Delta^{n-1}} (dd^c\Vert z'\Vert^2)^{n-1}
\int_{\Delta^2}(\tilde T)_{z'}\wedge  dd^c\phi =    
= \int_{\Delta^{n-1}} (dd^c\Vert z'\Vert^2)^{n-1}
\int_{\Delta^2}\tilde T_{z'}\wedge  dd^c\phi \le 
$$
$$
\le \int_{\Delta^{n-1}} (dd^c\Vert z'\Vert^2)^{n-1}
\int_{\Delta^2}dd^c(\tilde T)_{z'}\wedge  \phi \le 0.
$$
We had used here Fubini theorem for $L^1$-functions, the fact that $(\tilde T)
_{z'}=\tilde T_{z'}$ for currents from $L^1_{loc}$ which are smooth outside 
of suitably situated set $R_0$, and finally the plurinegativity of $\tilde T
_{z'}$. 

Thus $\tilde T$ is plurinegative. Putting $S=R_0$ we get the statement of (a) 
for $\Delta^{n-1}_r\times \Delta^2$ instead of $\Delta^{n+1}$. But this 
obviously  implies it for $\Delta^{n+1}$.
\par\smallskip\noindent
(2) Now suppose that the metric form $w$ is pluriclosed. Write $V=B^{n-1}
\times B^2$ for some neighborhood of point $a\in S$ such that $\pi \mid_S : 
S\to B^2$ is proper and $S_{z'}=S\cap B^2_{z'}$ is zerodimensional pluripolar 
compact in $B^2_{z'}$ for all $z^{'}\in B^{n-1}$.

For every $z'\in B^{n-1}$ let $S^0_{z'}$ be a finite, by {\sl Theorem 1}, set 
of points $s\in B^2_{z'}$ such that $dd^c\tilde T_{z'}$ has nonzero mass at $s$.
$\vert S^0_{z'}\vert $ is {uniformly bounded on $z'$ on $\bar B^{n-1}$. 
The points 
$s\in S^0_{z'}$ could be also characterized by the condition that for any 
3-sphere $S_r(s)\subset B^2_{z'}$ centered at $s$ $\int_{S_r(s)}d^cT_{z'}= 
C_s<0$. Number $C_s$ doesn't depend on $r$ sufficiently small. From this one 
immediately gets the closeness of the set $S^0:=\bigcup_{z'\in B^{n-1}}S^0_
{z'}$.
\par\smallskip\noindent\sl
Step 4. \it $f$ extends meromorphically onto 
$V\setminus S^0$. 
\par\smallskip\noindent\sl
Proof of Step 4. \rm Let $b\in B^2_{z'}\setminus S^0_{z'}$. Find a neighborhood
$W\cong B^{n-1}\times B^2$ of $b$ such that $W\cap S_0=\emptyset $ and 
$\pi_2\mid_S:S\cap W\to B^2$ is proper. Here $S$ is a minimal closed subset 
of $V$ such that $f:V\setminus S \to X$ is holomorphic, see part (1).

First we prove the following:
\par\smallskip\noindent\bf
Lemma 2.6.2. \it Suppose that the metric form $w$ on $X$ is pluriclosed and for
all $z'\in B^{n-1}$ $f(\partial B^2_{z'})\sim 0$ in $X$. Then: 

(i) $dd^c\tilde T=0$ in the sense of distributions. 

(ii) There is a $(1,0)$-current $\gamma $ in $W$, smooth in $W\setminus R_0$, 
such that $\tilde T=i(\partial\bar\gamma -\bar\partial\gamma )$. 
\par\smallskip\noindent\sl
Proof. \rm (i) Let $\tilde T_{\eps }$ be a smoothing of $\tilde T$ by 
convolution. Then $\tilde T_{\eps }$ are plurinegative and $\tilde T_{\eps }
\to T$ in ${\cal D}_{n,n}(W)$. We have that 
$$
\int_Wdd^c\tilde T_{\eps }\wedge (dd^c\Vert z'\Vert^2)^{n-1}= 
\int_{\partial W}d^c\tilde T_{\eps }\wedge (dd^c\Vert z'\Vert^2)^{n-1} = 
$$
$$
= \int_{\partial B^{n-1}\times B^2}d^cT_{\eps }\wedge (dd^c\Vert z'\Vert^2)
^{n-1} + \int_{B^{n-1}\times \partial B^2}d^cT_{\eps }\wedge (dd^c\Vert z'\Vert
^2)^{n-1}. \eqno(2.6.2)
$$
The first integral vanishes by the degree reason. So 
$$
\Vert dd^c\tilde T_{\eps }\wedge (dd^c\Vert z'\Vert^2)^{n-1}\Vert (W) = 
-\int_{B^{n-1}}(dd^c\Vert z'\Vert^2)^{n-1}\int_{\partial B^2_{z'}}d^c\tilde T_{
\eps }. \eqno(2.6.3)
$$
Remark now that $\int_{\partial B^2_{z'}}d^c\tilde T_{\eps }\to 
\int_{\partial B^2_{z'}}d^cT_{\eps }=\int_{f(\partial B^2_{z'})}d^cw=0$ 
because $f(\partial B^2_{z'})\sim 0$ in $X$. So the right hand side of (2.6.3) 
tends to zero as $\eps \searrow 0$. We get that 
$$
\Vert dd^c\tilde T\wedge (dd^c\Vert z'\Vert^2)^{n-1}\Vert (W) = 
\lim_{\eps \searrow 0}\Vert dd^c\tilde T_{\eps }\wedge (dd^c\Vert z'\Vert^2)
^{n-1}\Vert (W)=0.\eqno(2.6.4)
$$
\noindent
Taking sufficiently many such coordinate systems we see that $\Vert dd^c\tilde 
T\Vert (W)=0$. 
\par\smallskip\noindent
(ii) $\partial \tilde T$ is a $\bar\partial $-closed and $\partial $-closed 
$(2,1)$-current. So, if $\phi \in {\cal D}_{n-2,2}(W)$ is $\partial $-closed 
and such that $\bar\partial \phi = \tilde T$ then $\phi $ is smooth on 
$W\setminus S$ by elliptic regularity of $\bar\partial $. We have now 
$d\tilde T=\partial \tilde T+\bar\partial \tilde T=\bar\partial \phi+\partial 
\bar\phi $. Thus $d(\tilde T-\phi \bar\phi )=0$. So $\tilde T-\phi -\bar\phi $
is $d$-closed current of degree two on $W$. Consider an elliptic system in 
$W$:
$$
d\gamma =\tilde T-\phi -\bar\phi 
$$
$$
d^*\gamma =0 \eqno(2.6.5)
$$
\noindent
Then $\gamma $ has a solution in $W$. Indeed, let $\gamma_1$ be any solution 
of the first equation. Find a distribution $\gamma $ on $W$ with $*d*d\gamma 
=\Delta \gamma = *d*\gamma_1$ and put $\gamma_2 =\gamma_1-d\gamma $. 
$\gamma_2 $
is smooth on $W\setminus S$ because $\Delta \gamma_2 = d^*d\gamma_2 + 
dd^*\gamma_2 = d^*(\tilde T-\phi + \bar\phi )$ and $\gamma_2$ satisfies 
(3.4.6). Write $\gamma_2 =\gamma^{1,0}+\bar\gamma^{1,0}$. Then we have 
$\partial \gamma^{1,0}=-\phi $ and $\bar\partial 
\bar\gamma ^{0,1}=-\bar\phi $, so 
$$
\tilde T = d\gamma_2 +\phi + \bar\phi = \partial \bar \gamma ^{1,0} + 
\bar\partial \gamma^{0,1} \eqno(2.6.6)
$$
\noindent
with $\gamma^{1,0}$ having needed regularity.
\par\smallskip
\hfill{q.e.d.}
\par\bigskip\noindent\bf
Lemma 2.6.3. \it If $\tilde T$ is pluriclosed then the volumes $\Gamma_{f_z}
\cap \bb^2_z\times X$ are uniformly bounded for $z\in \bb^{n-1}_r$.
\par\smallskip\noindent\sl
Proof. \rm 
Now let $\gamma^{1,0}$ be as in (2.6.6). Smoothing by convolutions we still 
have 
$\tilde T_{\eps }=\partial \bar \gamma^{1,0}_{\eps }+\bar\partial \gamma^{1,0}
_{\eps }$. Take $0<r_1<r_2$ in such a way that $\partial (B^{n-1}_r\times B^2
_{r_2})\cap S\subset \partial B^{n-1}_r\times B^2_{r_2}$ for all $r<r_1$ , 
then:

$$
\int_{B^{n-1}_r\times B^2_r\setminus S} T^2\wedge (dd^c\Vert z'\Vert
^2)^{n-1}\le \int_{B^{n-1}_r\times B^2_{r_2}\setminus S} T^2\wedge (dd^c\Vert 
z'\Vert^2)^{n-1} = 
$$

$$
= \lim_{\eps \searrow 0}\int_{B^{n-1}_r\times B^2_{r_2}\setminus S}
\tilde T^2_{\eps }\wedge (dd^c\Vert z'\Vert^2)^{n-1}\le 
\lim_{\eps \searrow 0}\int_{B^{n-1}_r\times B^2_{r_2}}
\tilde T^2_{\eps }\wedge (dd^c\Vert z'\Vert^2)^{n-1} = 
$$
$$
= \lim_{\eps \searrow 0}\int_{B^{n-1}_r\times B^2_{r_2}}
(\partial \bar\gamma^{1,0}_{\eps }+\bar\partial \gamma^{1,0}_{\eps })^2\wedge 
(dd^c\Vert z'\Vert^2)^{n-1}\le 
$$
$$
\le \lim_{\eps \searrow 0}\int_{B^{n-1}_r}(dd^c\Vert z'\Vert^2)^{n-1}
\int_{B^2_{r_2}}d(\bar\gamma^{1,0}_{\eps }+\gamma^{1,0}_{\eps })\wedge 
d(\bar\gamma^{1,0}_{\eps }+\gamma^{1,0}_{\eps }) = 
$$
$$
= \lim_{\eps \searrow 0}\int_{B^{n-1}_r}(dd^c\Vert z'\Vert^2)^{n-1}
\int_{B^2_{r_2}}(\bar\gamma^{1,0}_{\eps }+\gamma^{1,0}_{\eps })\wedge 
d(\bar\gamma^{1,0}_{\eps }+\gamma^{1,0}_{\eps })= 
$$
$$
= \int_{B^{n-1}_r}(dd^c\Vert z'\Vert^2)^{n-1}
\int_{B^2_{r_2}}(\bar\gamma^{1,0}+\gamma^{1,0})\wedge 
d(\bar\gamma^{1,0}+\gamma^{1,0})\le c\cdot r^{2(n-1)}. \eqno(2.6.7)
$$
In the second inequality we had used the positivity of $T$. In the third the 
fact that $\bar\partial \bar\gamma_{\eps }^{1,0}\wedge \partial \gamma_{\eps }
^{1,0}$ is positive and $\bar\partial \bar\gamma_{\eps }^{1,0}\wedge \bar
\partial \bar\gamma_{\eps }^{1,0}=0$. Finally $\gamma_{\eps }^{1,0}\to \gamma 
^{1,0}$ on $\bar B^{n-1}_r\times \partial B^2_{r_2}$ while $\gamma ^{1,0}$ is 
smooth there. This gives a needed bound for the $\int_{B^{n-1}_r\times B^2_r
\setminus S}T^2\wedge (dd^c\Vert z'\Vert^2)^{n-1}$.

\par\smallskip
\hfill{q.e.d.}
\par\smallskip

{\sl Theorem 1} gives us now the proof of {\sl Step 4}.
\par\smallskip\noindent\sl
Step 5. \it The set $S^0$ is analytic of pure codimension two.
\par\smallskip\noindent\sl
Proof of Step 5. \rm We had proved that $f$ meromorphically extends onto 
$V\setminus S^0$, where $V\cong B^{n-1}\times B^2$ and $S^0$ is a graph of $N$- 
valued continuous mapping of $B^{n-1}$ to $B^2$. $N$- valued means that 
$\vert S^0_{z'}\vert \le N$ for every $z'\in \bar B^{n-1}$ and there exist 
$z_0^{'}$ such $\vert S^0_{z'}\vert =N$. Continuous - simply that $S^0$ is 
closed. Remark that $\tilde T$ is $L^1$-current on $V$ with $dd^c\tilde T\le 
0$ and supported on $S^0$.
\par\smallskip\noindent\bf
Lemma 2.6.4. \it Let $S^0$ be the graph of $N$- valued continuous mapping 
of $\bar\Delta^k$ to $\Delta^l$ and let $R$ be a closed positive current in 
$\Delta^{k+l}$ of bidimension $(k,k)$ supported on $S^0$. Then $S^0$ is a pure
$k$-dimensional analytic variety in $\Delta^{k+l}$.
\par\smallskip\noindent\sl
Proof. \rm Write $R=R_{K,\bar J}({i\over 2})^k{\partial \over \partial z^K}
\wedge {\partial \over \partial \bar z^J}$, where $K$ and $J$ are multiindices 
of length $k$. Consider a measures $R_{K,\bar J}$. Disintegrate this measures 
with respect to the natural projection $\pi :\Delta^k\times \Delta^l\to 
\Delta^k$, see [Bk], p.58. Denote by $\mu_{K,\bar J}=\pi_*(R_{K,\bar J})$ their 
direct images. Then disintegration means that one has probability measures 
$\nu_{K,\bar J,z'}$ on $\Delta^l_{z'}:=\{ z'\} \times \Delta^l$ with the 
property that for every continuous function $h$ in $\Delta^{k+l}$ 
$$
<R_{K,\bar J},h> = \int_{\Delta^k}(\int_{\Delta^l_{z'}}\bar h\mid_{\Delta^
l_{z'}}d\nu_{K,\bar J,z'})d\mu_{K,\bar J}, \eqno(2.6.8)
$$
\noindent
see [Bk] or [D-M]. 


Let $\Omega $ be the maximal open subset of $\Delta^k$ such that the 
multivalued map $s$, which is given by its graph $S^0$ takes exactly $N$ 
different values (and $N$ is maximal). First we shall prove that $S^0\cap 
(\Omega \times \Delta^l)$ is analytic.

Let further $\Omega_1$ be some simply connected open subset of $\Omega $.
Then $s\mid_{\Omega_1}$ decomposes to $N$ well-defined single valued maps $s^1,
...,s^N$. So it is enough to consider the case when $s$ is single valued. Put 
$s(z')=(s_1(z'),...,s_l(z'))$. Note that in this case $\nu_{K,\bar J,z'}=
\delta_{\{ z^{''}-s(z')\} }$. We can write for the coefficients of our current 
$R$ 
and for $\phi \in C^{\infty }(\Delta^{k+l})$ such that $\pi (\supp  \phi 
)\subset \subset \Delta^k$ that 
$$
<R_{K,\bar J},\phi > = \int_{\Delta^k }\bar \phi (z',s(z'))d\mu_{K,\bar J}(z').
\eqno(2.6.9)
$$
If we choose $\phi $ not depending on $z^{''}:=(z_{k+1},...,z_{k+l})$ then 
(2.6.9) gives 
$$
<R_{K,\bar J},\phi > = \int_{\Delta^k}\bar\phi (z')d\mu_{K,\bar J}(z'). 
\eqno(2.6.10)
$$
From the closeness of $R$ we obtain that 
$$
0=<R,d[({i\over 2})^k\phi (z')
dz_1\wedge ...dz_{p-1}\wedge dz_{p+1}\wedge ...\wedge dz_k\wedge d\bar z_J]>
=<R_{1...(p-1)(p+1)...k,\bar J},{\partial \phi \over \partial z_p}> =      
$$
$$ 
= \int_{\Delta^k}{\partial \bar\phi \over \partial \bar z_p}d\mu_{K,\bar J}.
$$

So $\mu_{K,\bar J}(z_1) = c_{K,\bar J}(z_1)\cdot ({i\over 2})^kdz'\wedge d\bar 
z'$, where $c_{K,\bar J}$ are holomorphic for $K=(1,...,k)$ and all $J$. In 
particular $c_{1...k,\bar 1...\bar k}$ is constant. Now take the $(k-1,k)$- 
forms $\psi_{q\bar p} = \phi (z')\cdot \bar z_p\cdot ({1\over 2})^kdz_1\wedge 
...\wedge dz_{q-1}\wedge dz_{q+1}\wedge ...\wedge dz_k\wedge d\bar K$. We 
have 
$$
0 = <R,d\psi_{q,\bar p}> = <R,{\partial \phi \over \partial z_q}\cdot \bar z_p
({i\over 2})^kdz^K\wedge d\bar z^K> = <R_{K,\bar K},{\partial \phi \over 
\partial z_q}\bar z_{\bar p}> = 
$$
$$
= c_{1\bar 1} \int_{\Delta }{\partial \bar\phi \over \partial \bar z_q}(z')
\cdot s_p(z')({i\over 2})^kdz'\wedge d\bar z',
$$
\noindent 
i.e. $s_p$ are holomorphic. 

Thus we had proved that $s$ is $N$-valued analytic map of $\Omega $ into 
$\Delta^l$. Considering appropriate discriminants and using Rado's Theorem, 
we obtain analyticity of $s$ on the whole $\Delta^k$.

If for some sphere $S^3$ imbedded into $\Delta^{n+1}\setminus S^0$ $f(S^3)$
is
homologous to zero in $X$ then one can extend $f$ through one of the branches 
of $S^0$ using the same arguments as in the proof of {\sl Step 4}.
\par\smallskip
\hfill{q.e.d.}

\subsection*{2.5. Complex Plateau problem for three-dimensional contours}

We shall apply the results of this paragraph to the complex Plateau problem.
Namely, we shall prove part (b) of the {\sl Corollary 3} from 
{\sl Introduction}.

Recall that complex Plateau problem for a real compact submanifold $M$ of 
complex manifold $X$ consists in finding a complex analytic subset $A\subset 
X\setminus M$ such that $\partial [A]=[M]$ in the sense of currents. The 
necessary condition on $M$ for complex Plateau problem to have a solution 
is the maximal complexity of $M$, i.e. $M$ should be a $CR$-submanifold 
of $X$ of $\dim _{CR} M=p$, where $\dim_{\rr }M=2p+1$. In the case when 
$X$ is Stein this is also sufficient, see [H-L]. Already in the case $X=
\cc\pp^3$ the maximal complexity of the ``contour'' $M$ is not sufficient 
any more, [Dl].

We suppose that $M$ bounds an abstract Stein domain, i.e. there is a 
complex manifold $D $ with boundary $M$ such that $D \setminus 
M$ is Stein, and the $CR$-embedding $f:M\to X$ is given. All that we  need 
to prove is that $f$ extends meromorphically onto $D $. Clearly we 
can suppose that $f$ is already holomorphically extended to some neighborhood 
of $M$ in $D $.
\par\smallskip\noindent\bf
Proposition 2.5.1. \it Let $(D, M)$ be as above and suppose additionally 
that $\dim M=3$.

(a) Then any $CR$-map $f:M\to X$, where $X$ is a disk-convex complex space 
admitting 

a pluriclosed Hermitian metric form, extends meromorphically onto 
$D \setminus S$. Here $S$ 

is a finite subset of $D$.

(b) If $f(M)$ is homologous to zero in $X$, or if $X$ doesn't contain 
spherical shells, then 

$S$ is empty.
\par\smallskip\noindent\sl 
Proof. \rm Let $\rho :D \to [0,1]$ be a strictly plurisubharmonic (and thus 
Morse) exhausting function. Denote by $D^{+}_{\eps }=\{ z\in D: \rho (z)> 
\eps \} $. 
Let ${\cal E}$ be the set of such $\eps $ that $f$ can be meromorphically 
extended onto $D^{*}_{\eps }\setminus S_{\eps }$, where $S_{\eps }$ is a 
discrete set. ${\cal E}$ is obviously closed and nonempty. 
All we need to prove is that ${\cal E}$ is open. 

Let $\eps_0 =\inf \{ \eps \in {\cal E}\} $. If $\eps_0$ is a regular value 
of $\rho $ then the needed result immediately follows from part (2) of 
{\sl Theorem 2}.

Consider the case of not regular value $\eps_0$ of $\rho $. Denote by 
$M_{\eps_0}=\{ z:\rho (z)=\eps_0\} $-the critical level set. Fix a critical 
point 
$z_0\in M_{\eps_0}$. All we need to prove is that for any neighborhood 
$W$ of $z_0$ the envelope of holomorphy of $W\cap D^{+}_{\eps }$ contains 
some neighborhood of $z_0$. For convenience we can suppose that $z_0=0$ and 
$\eps_0=0$. Write 
$$
\rho (z) = Q(z) + <z,z> + \bar Q(z) + O(\Vert z\Vert^3), \eqno(2.5.1)
$$
\noindent where $Q(z)$ is a holomorphic polynomial, $<z,z>$ - Hermitian form -
Levi form of $\rho $. By linear coordinate change we transform $< , >$ to the 
sum of squares of absolute values. Then by unitary coordinate change we transform 
$Q$ to the some of squares with real nonnegative coefficients. Now (2.5.1) 
has a form
$$
\rho (z)= \sum_{j=1}^pa_jz_j^2 + \sum_{j=1}^pa_j\bar z_j^2 + \sum_{j=1}^n 
\vert z_j\vert^2 + O(\Vert z\Vert^3). \eqno(2.5.2)
$$
\noindent In coordinates $z_j=x_j+iy_j$ we rewrite (2.5.2) as follows
$$
\rho (z) = 2\sum_{j=1}^pa_j(x_j^2-y_j^2) + \sum_{j=1}^n(x_j^2+y_j^2) + 
O(\Vert z\Vert^3) = 
$$
$$
= \sum_{j=1}^p[(1+2a_j)x_j^2 + (1-2a_j)y_j^2] + \sum_{j=p+1}^n(x_j^2+y_j^2) 
+ O(\Vert z\Vert^3). \eqno(2.5.3)
$$
\noindent Renumerate the coordinates in such a way that $a_j\ge {1\over 2}$ 
for $j=1,...,q$ and $a_j<1/2$ for $j=q+1,...,p$. Then 
$$
\rho (z) \ge \sum_{j=1}^q[(2a_j+1)x_j^2-(2a_j-1)y_j^2] + \delta\cdot \sum_
{j=q+1}^p\vert z_j\vert^2 + O(\Vert z\Vert^3) \ge 
$$
$$
\ge \sum_{j=}^q[(2a_j-\delta_1+1)x_j^2-(2a_j+\delta_1-1)y_j^2] + \delta\cdot 
\sum_{j=q+1}^p\vert z_j\vert^2  := \rho_1(z), \eqno(2.5.4)
$$
\noindent for some $\delta >0$ and $\delta_1$ can be chosen arbitrarily small 
for small $\Vert z\Vert $. While obviously $D^+:=\{ z\in \bb^n: \rho_1(z)>0\} 
\subset D^+_{\eps_0}$, all we need is to prove the following
\par\smallskip\noindent\sl
Lemma 2.5.2. \it The envelope of holomorphy of $D^+$ contains the origin.
\par\smallskip\noindent\sl
Proof. \rm Consider two cases.

\noindent\sl
Case 1: $q\le n-1$. \rm In this case $D^+$ contains the following Hartogs 
figure:
$$
H := \{ z\in \bb^n: \sum_{j=1}^q[(2a_j-\delta_1+1)x_j^2 - (2a_j+\delta_1-1)
y_j^2]>0, \delta \cdot \sum_{j=q+1}^n\vert z_j\vert^2<1 
$$
or
$$
\sum_{j=1}^q(2a_j-\delta_1+1)x_j^2 - (2a_j+\delta_1-1)y_j^2]>-\eps , \delta
\cdot \sum_{j=q+1}^n\vert z_j\vert^2>\eps \} .
$$
\noindent
The envelope of holomorphy of $H$ obviously contains the origin.
\par\smallskip\noindent\sl
Case 2: $q=n$. \rm In this case 
$$
D^+ = \{ z\in \bb^n: \sum_{j=1}^n[(2a_j-\delta_1+1)x_j^2 - (2a_j+\delta_1-1)
y_j^2]>0\} .\eqno(2.5.5)
$$
\noindent Put $b_j=2a_j-\delta_1+1$, $c_j=2a_j+\delta_1-1$, $j=1,...,n$. For 
small $\delta-1$, $b_j>c_j$. Write (2.5.5) in the form 
$$
D^+ = \{ z\in \bb^n: \sum_{j=1}^nb_jx_j^2 >  \sum_{j=1}^nc_jy_j^2\} .
\eqno(2.5.6)
$$
\noindent In the new coordinates $z_j\to \sqrt{b_j}z_j$ (2.5.6) take a form 
$$
D^+ = \{ z\in \bb^n:  \sum_{j=1}^nx_j^2 >  \sum_{j=1}^n\delta_jy_j^2\} , 
\eqno(2.5.7)
$$
\noindent where $\delta_j = {c_j\over b_j}<1$, $j=1,...,n$. Put $\delta_0:= 
\max \{ \delta_1,...,\delta_n\} <1$. Then 
$$
D^+\supset D^+_1=\{ z\in \bb^n: \Vert x\Vert^2>\delta_0\cdot \Vert y\Vert^2\} 
.\eqno(2.5.8)
$$
\noindent The set $D_1^+$ contains clearly the following complete ``tube torus''
$$
T = \{ x+iy\in  \cc^n: \Vert x\Vert =1, \Vert y\Vert \le 1/\delta_0\} , 
\eqno(2.5.9)
$$
\noindent
where $1/\delta_0 := \eta >1$. We shall prove that already the envelope of 
holomorphy of $T$ contains the origin. For this consider the following 
continuous family of complex hypersurfaces
$$
C_t = \{ z\in \cc^n: z_1^2+...+z_n^2=t\} \eqno(2.5.10)
$$
\noindent or 
$$
C_t = \{ x+iy\in \cc^n:  \Vert x\Vert^2-\Vert y\Vert^2=t, (x,y)=0 \} , 
\eqno(2.5.11)
$$
\noindent where $(x,y)=x_1y_1+...+x_ny_n$. Consider the intersections of $C_t$ 
with a ball of radii $1+\eta^2$:
$$
\tilde C_t = \{ x+iy \in \bb^n_{1+\eta^2 }: \Vert x\Vert -\Vert y\Vert =t, 
(x,y)=0\} .\eqno(2.5.12)
$$
\noindent This is a continuous family of irreducible analytic hypersurfaces in 
$\bb^n_{1+\eta }$ such that 
$$
\tilde C_{1+\eta^2 } = \{ x+iy\in \bb^n_{1+\eta^2 }: \vert x\Vert^2-\Vert y
\Vert^2=1+\eta^2, (x,y)=0\} = 
$$
$$
= \{ x+iy\in \bb^n_{1+\eta^2}: \Vert x\Vert^2+\Vert y\Vert^2=1+\eta^2=\Vert x
\Vert^2-\Vert \Vert^2, (x,y)=0\} = 
$$
$$
\{ x+iy\in \bb^n_{1+\eta^2}: \Vert x\Vert^2=1+\eta^2, y=0\} \subset T,
$$ 
\noindent but $\tilde M_0\ni 0$. By continuity principle the envelope of 
holomorphy of $T$ contains the origin.
\par\smallskip
\hfill{q.e.d}
\par\smallskip\noindent\sl
End of the proof of Corollary 3b. \rm

So, as in the case of regular value, we can extend our map $f$ meromorphically 
to the neighborhood of the critical level $M_{\eps_0}$ minus discrete set.
As a result we obtain the extension $\hat f$ of our map onto $\bar D\setminus 
S$ where $S$ is a finite subset of $\bar D$ not intersecting $M=\partial D$.
If we put $T:= f^*w$ then $dd^c\tilde T$ is nonpositive measure supported on 
$S$. We have 
$$
\int_Sdd^c\tilde T = \int_Ddd^c\tilde T = \int_{\partial D}d^cT = \int_{f(
\partial D)}d^cw = \int_Md^cw=0,
$$ \noindent if $M$ is homologous to zero in $X$, or if $X$ doesn't contain 
spherical shells.
\par\smallskip
\hfill{Corollary 3b is proved.}


\section*{3. Generalisations, applications, open questions}

In this last section of the paper we shall give some more applications 
of the results and (in fact more) techniques used here. 

\subsection*{3.1. Multivalued mappings and singular domains of definition}

Let $D$ be a domain in complex space $\Omega$ and $x_0\in \partial D$ be a 
boundary point. $D$ is said to be $q$-concave at $x_0$ if there is a 
neighborhood $U\supset x_0$ and smooth function $\rho :U\to \rr $ such that 

1) $D\cap U  = \{ x\in U:\rho (x)<0\} $;

2) Levi form of $\rho $ at $x_0$ has at least $n-q+1$ negative eigenvalues.
\par\smallskip\noindent 
Here $n=\dim \Omega $. By the Projection Lemma of Siu, see [Si-T], if 
$x_0$ is a $q$-concave boundary point of $D$, $q\le n-1$, one can find a 
neighborhoods $U\ni x_0$ and $V\ni 0\in \cc^n$ and a proper holomorphic map 
$\pi :(U,x_0)\to (V,0)$ such that $\pi (D\cap V)$ will contain a Hartogs 
figure $H$, whose associated polydisk $P$ contains the origin. Let $d$ be 
the branching number of $\pi $.

Now suppose that a meromorphic map $f:D\to X$ is given, where $X$ is another 
complex space. $\pi^{-1}\circ f$ defines a $d$-valued meromorphic 
correspondence between $V$ and $X$.
\par\smallskip\noindent\bf
Definition 3.1.1. \rm A $d$-valued meromorphic correspondence between complex 
spaces $V$ and $X$ is an irreducible analytic subset $Z\subset V\times X$ such 
that restriction $p_1\mid_Z$ of the natural projection onto the first factor 
on $Z$ is proper, surjective and generically $d$ to one.
\par\smallskip
Thus the extension of $f$ onto the neighborhood of $x_0$ is equivalent to the 
extension of $Z$ from $H$ to $P$. Clear that if $f$ was also a correspondence 
it will produce no additional complications. Thus we should discuss how far the 
problem of extending of correspondences go from the extension of mappings. 

Let $Z$ be a $d$-valued meromorphic correspondence between the Hartogs figure 
$H$ and $X$. $Z$ defines in a natural way a mapping $f_Z:H\to \Sym ^d(X)$ 
- symmetric power of $X$ of order $d$. Clearly the extension of $Z$ onto $P$ 
is equivalent to the extension of $f_Z$ onto $P$. If $X$ was for example a 
K\"ahler manifold. Then by [V] $\Sym ^d(X)$ is a K\"ahler space. So 
meromorphic correspondences with values in K\"ahler manifolds are extendable 
through pseudo-concave  boundary points.

For the manifolds from class ${\cal G}_1$ this is no longer the case, if even 
they doesn't contain spherical shells.

\subsection*{3.2. Construction of Example 2: appearance of shells in symmetric products}

In this section we shall construct the following 
\par\smallskip\noindent\bf
Example 2. \it There is a compact complex (elliptic) surface $X$ such that:

(a) every meromorphic map $f:H^2(r)\to X$ extends meromorphically onto 
$\Delta^2$, but

(b) there exists a two-valued meromorphic correspondence $Z$ between 
$\cc^2_*$ and $X$ 

which cannot be extended  to origin.
\par\smallskip\rm 
Consider a standard Hopf surface $H=\cc^2\setminus \{ 0\} /(z\sim 2\cdot z)$.
By $\pi :H\to \cc\pp^1$ denote the standard projection.  
Let $\phi : C\to \cc\pp^1$ be a nonconstant meromorphic function on the 
Riemann surface $C$ of positive genus. $\phi $ will be a $d$-sheeted 
ramified covering of $\cc\pp^1$ by $C$. If we take $C$ to be a torus we can 
have such $\phi $ with $d=2$. Following Kodaira we shall construct 
an elliptic surface over $C$ in the following way. Put 
$$
X_1 = \{ (z,y)\in C\times H : \phi (z)=\pi (y)\}.\eqno(3.2.1)
$$
\noindent
Elliptic structure on $X_1$ is given by the restriction onto $X_1$ of the 
natural projection $p_1:C\times H\to C$. Note that restriction onto $X_1$ 
of the natural projection $p_2:C\times H\to H$ gives us an $d$-sheeted 
covering $p_2\mid_{X_1}$ of $H$ by $X_1$ which preserves the elliptic 
structure. Let $n:X\to X_1$ be a normalization of $X_1$. Then $X$ is a 
smooth elliptic surface over $C$ with elliptic fibration $p:=p_1\mid_{X_1}
\circ n:X\to C$ and $F:=p_2\mid_{X_1}\circ n :X\to H$ will be a  $d$- 
sheeted covering.

$Z:=F^{-1}\circ \pi : \cc^2_*\to X$ is $d$-valued meromorphic correspondence 
between  $\cc^2_*$ and $X$, which cannot be extended to origin, because 
the projection $\pi :\cc^2_*\to H$ cannot be extended meromorphically to 
zero.

On the other hand in [Iv-1] it was proved that meromorphic mappings from 
$H^2(r)$ to an elliptic surface over a Riemann surface of positive genus 
are extendable onto $\Delta^2$.     
\par\smallskip
One can interpret this example in the way that $\Sym ^2(X)$ could have 
a spherical shell if even $X$ has not.

\subsection*{3.3. Extension theorems for  meromorphic correspondences}

In view of discussion in 3.1 and 3.2 we can restate our results for
meromorphic correspondences.
\par\smallskip\noindent\bf
Definition 3.3.1. \rm By a branched spherical shell of degree $d$  in a 
complex space $X$ we shall understand an image $\Sigma $ of $\ss^3\subset \cc^2
$ under the $d$-valued meromorphic correspondence between some neighborhood 
of $\ss^3$ and $X$ such that $\Sigma \not\sim 0$ in $X$.
\par\smallskip\noindent\bf
Corollary 3.3.1. \it Let $Z$ be a meromorphic correspondence from the domain 
$D$ in complex space $\Omega $ into the  disk-convex 
complex space $X\in {\cal G}_1$ and let $x_0$ be a concave boundary point of 
$D$. Then $Z$  extends onto some neighborhood of $x_0$ in $\Omega $ minus 
(possibly empty) complex variety $C$ of pure codimension two. If $X$ doesn't
contain  branched
spherical shells then $C=\emptyset $.
\par\smallskip\noindent\bf
Remark. \rm We would like to point  out here that branched shell could be a 
much violate object that not branched one. For this purpose we
consider a smooth 
complex curve $C$ in the neighborhood of $\bar \bb^2(2)\setminus \bb^2(1/2)$ 
which doesn't extend to $\bb^2(1)$. Let $\pi :W\to \bb^2(2)\setminus \bar\bb^2
(1/2)$ be a covering branched along $C$. The images of $\hat S:=\pi^{-1}(\ss^3)
$ in the complex spaces could be a branched shells which doesn't bound 
an abstract Stein domains. 


\subsection*{3.4. Complex Plateau problem for higher-dimensional contours}

First let us prove part (a) of {\sl Corollary 3} from {\sl Introduction}.
Actually only in the end of the proof in 2.5 we used the fact that the 
dimension of $D$ is two. The following proposition clearly enables us to 
finish the proof also of  part (a) of {\sl Corollary 3} i.e for $\dim D\ge 3$.
\par\smallskip\noindent\bf
Proposition 3.4.1. \it Every holomorphic map $f$ from $H_1^n(r)$ to disk-convex 
complex space $X\in {\cal G}_1$ extends meromorphically onto $\Delta^n$ 
provided $n\ge 2$. 
\par\smallskip\noindent\bf
Remark. \rm In the {\sl Example 2} (see Introduction) the map $F:\bb^3\to 
X$ was \it not holomorphic \rm ! We shall essentially use the condition of 
holomorphicity of $f$ in the proof of this {\sl Proposition }.
\par\smallskip\noindent\sl
Proof. \rm It will be convenient for us simultaneously with the proof of the 
main statement of the {\sl Proposition} to prove also the following weaker  
statement. Denote by $A^n(a,b):=\Delta^n(b)\setminus \Delta^n(a)$, for $0\le 
a<b$. 

\par\smallskip
\it Every holomorphic map $f:A^n({1\over 2},1)\to X$, where $X$ from {\sl 
Proposition 5.1.1}, 

extends meromorphically onto $\Delta^n$, provided $n\ge 2$ and $f(\partial 
\Delta^n_{3/4})$ is  

homologous to zero in $X$.  
\par\smallskip\rm
We shall prove both statements by  induction on $n$. For $n=2$ the  
second statement follows directly from {\sl Theorem 1}. So it is sufficient 
to prove that for 
any $n\ge 2$ from the second statement follows the statement of {\sl 
Proposition } for this $n$.

So let a holomorphic mapping $f:H_1^n(r)\to X$ is given. For every $z\in 
\Delta $ restriction $f_z$ of $f$ onto $\Delta^n_z:=\{ z\} \times \Delta^n$ is 
holomorphic on $A^n(r,1)$. So, by the assumption $f_z$ meromorphically 
extends onto $\Delta^n$, because $f(\partial\Delta^n_z)\sim f(\partial\Delta^
n_0)\sim 0$ in $X$! Lemma 2.3.2  
immediately gives us (after shrinking $\Delta^{n+1}$ and taking different bends 
of $z_2,...,z_{n+1}$) the meromorphic extension of $f$ onto $\Delta^{n+1}
\setminus S$. Where $S$ is zerodimensional pluripolar compact in $\Delta^{n+1}
$.

Because $I(f)$ is an analytic set of positive 
dimension outside of zerodimensional set, $\overline{I(f)}$ is analytic 
in $\Delta^{n+1}\setminus H_1^n$, and thus empty. So the fundamental set 
of $f$ is discrete in $\Delta^{n+1}\setminus S$.

Put $T:=f^*w$, where $w$ is pluriclosed metric form on $X$. $T$ has locally 
summable coefficients in $\Delta^{n+1}$ and its trivial extension $\tilde T$ 
is plurinegative with $dd^c\tilde T$ supported on $S$. Observe that
$\tilde T=T$ is pluriclosed outside of $S$.

{\sl Lemma 2.6.2} tells us that $dd^c\tilde T=0$ and moreover there is a 
(1,0)-current $\gamma $ in any given ball $W\subset \Delta^{n+1}$, $\partial 
W\cap S=\emptyset $, smooth on $W\setminus S$, such that $\tilde T=i(\partial 
\bar\gamma -\bar\partial \gamma )$ . Remark that the conditions of {\sl Lemma 
2.6.2} are satisfied, because $S$ is zerodimensional and $n+1\ge 3$. All that 
remained is to repeat the arguments from the proof of {\sl Lemma 2.6.3} to 
estimate the volume of the graph of $f$ in the neighborhood of $S$. Namely 

$$
\Vol (\Gamma_{f\mid_{W\setminus S}}) = \int_{W\setminus S}(T + dd^c\Vert 
z\Vert^2)^{n+1} = \sum_{j=0}^{n+1}C_{n+1}^j\int_{W\setminus S} T^j\wedge 
(dd^c\Vert z'\Vert^2)^{n+1-j}\le 
$$

$$
\le C\cdot \int_{W\setminus S} T^j\wedge (dd^c\Vert 
z'\Vert^2)^{n+1-j} = C\cdot \lim_{\eps \searrow 0}\int_{W\setminus S}\tilde 
T^j_{\eps }\wedge (dd^c\Vert z'\Vert^2)^{n+1-j}\le 
$$

$$ 
\le C\cdot  \lim_{\eps \searrow 0}\int_W\tilde T^j_{\eps }\wedge (dd^c\Vert z'
\Vert^2)^{n+1-j} = C\cdot \lim_{\eps \searrow 0}\int_W(\partial \bar\gamma^
{1,0}_{\eps }+\bar\partial \gamma^{1,0}_{\eps })^j\wedge  
(dd^c\Vert z'\Vert^2)^{n+1-j} = 
$$

$$
= C\cdot \lim_{\eps \searrow 0}\int_W(d(\bar\gamma^{1,0}_{\eps }+\gamma^{1,0}
_{\eps }))^j\wedge (dd^c\Vert z'\Vert^2)^{n+1-j} = 
$$

$$
= C\cdot \lim_{\eps 
\searrow 0}
\int_{\partial W}(\bar\gamma_{\eps }^{1,0}+\gamma_{\eps }^{1,0})\wedge 
d(\bar\gamma^{1,0}_{\eps }+\gamma^{1,0}_{\eps })^{j-1}\wedge (dd^c\Vert z'
\Vert^2)^{n+1-j} = 
$$

$$
= C\cdot \int_{\partial W}(\bar\gamma^{1,0}+\gamma^{1,0})\wedge d(\bar\gamma^
{1,0}+\gamma^{1,0})^{j-1}\wedge (dd^c\Vert z'\Vert^2)^{n+1-j} <  \infty 
$$
\par\smallskip
From Bishop theorem we get an extension of the graph of $f$ onto $\Delta
^{n+1}$.
\par\smallskip
\hfill{q.e.d.}


\subsection*{3.5. Coverings of compact complex manifolds}

Let $X$ be a compact complex manifold and let $D$ be a domain in complex 
manifold $\Omega $ which covers $X$. This means that there is some subgroup 
$G$ of the group $\Aut(D)$ of biholomorphic automorphisms of $D$, 
acting properly and discontinuously on $D$ without fixed points, such that 
$D/G=X$. 

Before proving the {\sl Corollary 4} we want to recall that locally 
pseudoconvex domain in 
Stein manifold is Stein (Oka), and moreover locally pseudoconvex domain 
in $\cc\pp^n$ is also Stein, provided it is different from $\cc\pp^n$ itself, 
see [Ks] and [Hs-2].
\par\smallskip\noindent\sl 
Proof of Corollary 4. \rm Let $f:D\to X$ be a covering map. Suppose that one 
can find a point 
$p\in \partial D$ such that for any neighborhood $V$, biholomorphic to a ball, 
$V\cap D$ is not pseudoconvex.  
By Docquer-Grauert theorem,[D-G], this means that there exists an embedding
$\phi :H_{n-1}^1\to V$ such that $\phi (\Delta^n)\cap \partial D$ is
nonempty, $n=\dim D$. But $f\circ\phi $ extends meromrophically onto
$\Delta^n\setminus S$, where $S$ is: 
\par\smallskip
(a) locally finite union of subvarieties of pure codimension two by
{\sl Theorem 2} ;

(b) empty, by [Iv-3].  
\par\smallskip
This means that locally $\partial D\subset I(f)\cup S$.

Suppose we can find a point $p\in (I(f)\setminus S)\cap D$. Take 
an analytic disk $\psi :\Delta \to U$ through $p$, i.e.$\psi (0)=p$, 
which is not contained in $I(f)$. Composition $f\circ \psi :\Delta \to X$ 
is holomorphic. Put $x:=(f\circ \psi )(o)$. Choose a neighborhood $G\ni x$
such that \it every \rm  branch of $f^{-1}$ is single valued on $G$. While 
$f\circ  f^{-1}=\id $ on $G$, the map $f$ must be \it holomorphic 
\rm in the neighborhood of $p$. Really, take $p_1\in \psi (\Delta )
\setminus I(f)$ and close enough to $p$ to have $x_1=f(p_1)\in G$. Then the
branch of $f^{-1}$ which sends $x_1$ to $p_1$ must send $x$ to $p$.
Contradiction.

Thus $I(f)=\emptyset $ and $D\supset (U\setminus S)$. 


\par\smallskip
\hfill{q.e.d.}

\section*{4. Open questions}
\par\medskip\rm
In this paragraph we list some open problems which seem to us be
of significant interest.
\par\smallskip\noindent\sl
Problem 1. \rm Prove the conjecture stated in Itroduction.
\par\smallskip\noindent\sl
Problem 2. \rm Let the complex manifold $D$ is defined as two-sheeted
cover of $\Delta^2\setminus \rr^2$, i.e. $D$ is ``nonschlicht'' domain
over $\cc^2$. Does there exist a compact compex manifold $X$ and
a holomorphic (meromorphic) mapping $f:D\to X$ which separates the
points?

Note that the results of this paper imply that such $X$ if exists cannot
possed a plurinegative metrik form. Thus examples could occur starting
from $\dim X\ge 3$.
\par\smallskip
In the following problems the space $X$ is equipped with some Hermitian
metrik. On the subsets of $\cc^n$ the metrik is allwayse $dd^c\Vert z
\Vert^2$.
\par\smallskip\noindent\sl
Problem 3. \rm Consider a class ${\cal J}_R$ of meromorphic mappings
$f:\Delta^k\to X$, $X$ beeing compact, such that 

(a) $\Vert Df\Vert\ge R>0$. Here $\Vert Df\Vert $ denotes the norme of the
differential of $f$;

(b) $\vol(f_s(\Delta^k)\le C_1$ for all $s\in \Delta^k$.

Prove that there is a constant $C_2=C_2(X,R,C_1)$, not depending on $f$, such
that $\vol(\Gamma_{f_s})\le C_2$.
\par\smallskip\noindent\sl
Problem 4. \rm Let $f:\Delta^k_*\to X$ be a meromorphic mapping from a
punctured polydisk into a compact complex space $X$. Suppose that
$\vol f(\Delta^k_*)<\infty $. Prove that $f$ meromorphically extends
to zero.
\par\smallskip\noindent\sl
Problem 5. \rm  Let $f:\Delta^{k+1}_*\to X\in {\cal G}_k$ be a meromorphic
map from punctured $(k+1)$-disk into a compact complex space from class
${\cal G}_k$, see Introduction. Prove that $\vol(f(A^k(r,1))=
O(log^{{k+1\over k}}({1\over r}))$. In particular for equidimensional maps
$f:\Delta^n_*\to X^n$ one allwayse should have $\vol(f(A^n(r,1))=
O(log^{{n+1\over n}}({1\over r}))$.
\par\smallskip\noindent\sl
Problem 6. \rm Fix some $0<r<1$ and some constant $R$. Fix also a compact
complex space $X$. Consider the following class ${\cal F}_R$ of meromorphic
mappings from $f:\Delta^n\to X$:

\it (a) $\vol_{2n}(\Gamma_f\cap (A^n(r,1)\times X))\le R$;

(b) for every $k$-disk $\Delta^k_z$ which has sides parallel to the coordinate
planes (i.e. $\Delta^k_z=\{ z\} \times \Delta^k$, where $z\in \Delta^{n-k}$)
$\vol_{2k}(\Gamma_{f\mid_{\Delta^k\cap A^n(r,1)}})\le R$.
\par\smallskip\noindent\rm
6a. Prove that for any constant $l$ there is a constant $A$ such that for
any $f\in {\cal F}_R$ satisfying $\vol_{2k}(\Gamma_{f_z})\le l$ for
all restrictions $f_z$ of $f$ onto the $k$-disks $\Delta^k_z$ one 
has $\vol_{2n}(\Gamma_f)\le A$.
\par\smallskip\noindent
6b. Vice versa: for any constant $a$ there is a constant $L$ such that
forany $f\in {\cal F}_R$ such that $\vol_{2n}(\Gamma_f)\le a$ one
has $\vol_{2k}(\Gamma_{f_s})\le L$ for all $\Delta^k_z$.
\par\smallskip\noindent\sl
Problem 7. \rm Let $X$ is a compact complex manifold carrying a plurinegative
metrik form, and let $f:\Delta^3\setminus S\to X$ is a meromrohpic mapping.
Suppose that $S$ is a minimal closed subset of $\Delta^3$ such that $f$
extends onto $\Delta^3\setminus S$. Prove that each connected component
of $S$ is a complex curve.
\par\smallskip\noindent\sl
Problem 8. \rm Let $K$ a complet pluripolar compact in pseudoconvex domain
in $\cc^n, n\ge 2$. Let $T$ a positive current on $D\setminus K$ such that
$dd^cT\le O$ on $D\setminus K$.
\par\smallskip
(a) Prove that $T$ has locally finite mass in the neighborhood of $K$.

(b) Suppose that $T$ has locally finite mass in the neighborhood of $K$.
Let $\tilde T$ its trivial extension. Prove that $dd^c\tilde T\le O$.



\spaceskip=4pt plus3.5pt minus 1.5pt
\spaceskip=5pt plus4pt minus 2pt
\font\csc=cmcsc10
\font\tenmsb=msbm10
\newdimen\length
\newdimen\lleftskip
\lleftskip=2.5\parindent%  to be adjusted !!!
\length=\hsize \advance\length-\lleftskip
%
\def\entry#1#2#3#4\par{\parshape=2  0pt  \hsize%
\lleftskip \length%
\noindent\hbox to \lleftskip%
{\bf[#1]\hfill}{\csc{#2 }}{\sl{#3}}#4%
\par\medskip% to be adjusted
}
\ifx \twelvebf\undefined \font\twelvebf=cmbx12\fi


\section*{References}
\footnotesize

\entry{A-S}{Andreotti A., Stoll W.:}{Extension of Holomorphic Maps.} Ann.\ Math.
{\bf72}, 312-348, (1960).

\entry{Ba}{Barlet D.:}{Espace analytique reduit des cycles analytiques 
complexes compacts d'un espace analytique complexe de dimension finie .} 
Semiar Norguet IX, Lect. Notes Math., {\bf 482}, 1-157, (1975).

\entry{B-T}{Bedford E., Taylor B.:}{ A new capacity for plurisubharmonic 
functions.}  Acta\ Math. (1982) {\bf149}, 1-40, (1982).

\entry{Bk}{Bourbaki N.:}{Elements de Mathematiques. Integration vectorielle.}
Hermann (1959).

\entry{C-H}{Carlson , Harvey R.:}{Coverings of compact complex manifolds.}


\entry{D-M}{Dellacherie C., Meyer P.-A.:}{Probabilit‚s et potentiel.} Publ.
\ Inst.\ Math.\ Univ.\ Strabourg, {\bf XV}, Hermann (1975).

\entry{Dl}{Dloussky G.:}{Envelopes d'holomorphie et prolongements 
d'hypersurfaces.} Sem.\ P.Lelong, LNM, {\bf578}, 217-235 (1977).

\entry{D-G}{Docquier F., Grauert H.:}{ Levisches Problem und Rungescher
Satz f\"ur Teilgebiete Steinscher Mannigfaltikeiten.} Math. Ann. {\bf 140},
94-123, (1960).

\entry{Db}{Dolbeault P.:}{Geometric measure theory and the calculus of 
variations.} Proc.\ Symp.\ Pure\ Math. {\bf44}, 171-205, (1986).

\entry{Fd}{Federer :}{Geometric measure theory.} Berlin, \ Springer (1969).

\entry{F-Sb}{Fornaess J.-E., Sibony N.:}{Oka's unequality for currents and 
applicatins.} to \ appear \ in \ Math.\ Ann .


\entry{Ga}{Gauduchon P.:}{Les metriques standard d'une surface a premier 
nombre de Betti  pair.} Asterisque.\ Soc.\ Math.\ France.  {\bf126}, 129-135, 
(1985).

\entry{Gl}{Golysin G.:}{Geometric function theory.} Nauka,\ Moscow (1966).

\entry{Gr}{Griffiths P.:}{Two theorems on extensions of holomorphic mappings
.} Invent.\ math. {\bf14}, 27-62, (1971).

\entry{H}{Harvey R.:}{Holomorphic chains and their boundaries.} Proc.\ Symp.\ 
Pure\ Math. {\bf30}, Part I, 307-382 (1977).

\entry{H-S}{Harvey R., Shiffman B.:}{A characterization of holomorphic chains
.} Ann.\ Math. {\bf99}, 553-587, (1974).

\entry{H-L}{Harvey R., Lawson H.:}{An intrinsic characterisation of K\"ahler 
manifolds.} Invent.\ math. {\bf74}, 169-198, (1983).


\entry{Hs-1}{Hirschowitz A.:}{Les deux types de m\'eromorphie diff\'erent.} 
J.\ reine \& \ ang. \ Math., {\bf313} , 157 - 160, (1980).

\entry{Hs-2}{Hirschowitz A.:}{Pseudoconvexite au-dessus d'espaces plus ou 
moins homogen\`es.} Invent.\ Math. {\bf26}, 303-322, (1974).

\entry{Hm}{H\"ormander L.:}{Notions of convexity.} Birkh\"auser (1994).

\entry{Iv-1}{Ivashkovich S.:}{Rational curves and Extensions of Holomorphic 
mappings.} Proc.\ Symp.\ Pure \ Math. {\bf52} Part 1, 93-104, (1991).

\entry{Iv-2}{Ivashkovich S.:}{Spherical shells as obstructions for the 
extension of  holomorphic mappings.} The\ Journal\ of\ Geometric\ Analysis. 
{\bf2}, N 4, 351-371, (1992).

\entry{Iv-3}{Ivashkovich S.:}{The Hartogs-type extension theorem for the 
meromorphic maps into compact K\"ahler manifolds.} Invent.\ math. {\bf109}
, 47-54, (1992).

\entry{Iv-4}{Ivashkovich S.:}{One example in concern with extension and
separate analyticity properties of meromorphic mappings.} Submited for
the publication (1997).


\entry{Ka-1}{Kato M.:}{Examples on an Extension Problem of Holomorphic Maps 
and Holomorphic 1-Dimensional Foliations.} Tokyo\ Journal\ Math. {\bf13}, n 1,
139-146, (1990).

\entry{Ka-2}{Kato M.:}{Compact quotient manifolds of domains in a complex
3-dimensional projective space and the Lebesgue measure of limit sets.}
Tokyo\ Journal\ Math. {\bf19}, 99-119 (1996).

\entry{Ka-3}{Kato M.:}{Compact complex manifolds containing "global"
spherical shells I.} Proc. Intl. Symp. Algebraic Geometry, Kyoto,
45-84 (1977).

\entry{Kl}{Klimek M.:}{Pluripotential theory.} London.\ Math.\ Soc.\ Monographs
,\ New\ Series 6, (1991).

\entry{Ks}{Kiselman C.:}{On entire functions of exponential type and
indicators of analytic functionals.} Acta Math. {\bf117}, 1-35 (1967).

\entry{La}{L\`arusson F.:}{Compact quotients of large domains in complex
projective space.} Preprint (1997).

\entry{Lg}{Lelong P.:}{Plurisubharmonic Functions and Positive Differential 
Forms. Gordon and Breach.} New-York. (1969) 78 p.

\entry{Lv}{Levi E.:}{Studii sui punti singolari essenziali delle funzioni
analitiche di due o pi\'u variabili complesse.} Annali di Mat. pura ed appl.
{\bf17}, n 3, 61-87 (1910).


\entry{Re}{Remmert R.:}{Holomorphe und meromorphe Abbildungen komplexer 
R\"aume.} Math.\ Ann. {\bf133}, 328-370, (1957).

\entry{Rs}{Rossi H.:}{Attaching analytic spaces to an analytic space along 
a pseudoconvex boundary.} Proc.\ Conf.\ Complex\ Analysis, Monneapolis, 
Springer-Verlag, 242-256 (1965).

\entry{Sa}{Sadullaev A.:}{Plurisubharmonic measures and capacities on 
complex manifolds.} Russian\ Math.\ Surv. {\bf36}, No 4 , 61-119, (1981).

\entry{Sh-1}{Shiffman B.:}{Extension of Holomorphic Maps into Hermitian 
Manifolds.} Math.\ Ann. {\bf194}, 249-258, (1971).

\entry{Sh-2}{Shiffman B.:}{Separately meromorphic mappings into compact 
K\"ahler manifolds.} Contributions to Complex Analysis and Analytic Geometry
(H. Skoda and J.-M.Tr‚preau, eds) Vieweg, Braunschweig, Germany, 243-250 
(1994).

\entry{Sb}{Sibony N.:}{Quelques problemes de prolongement de courants en 
analyse complexe.} Duke\ Math.\ J. {\bf52}, 157-197, (1985).

\entry{Si-1}{Siu Y.-T.:}{Every stein subvariety admits a stein neighborhood.} 
Invent.\ Math. {\bf38}, N 1, 89-100, (1976).

\entry{Si-2}{Siu Y.-T.}{Techniques of Extension of Analytic Objcts.} New \ York 
, Dekker (1974).

\entry{Si-3}{Siu Y.-T.}{Extension of meromorphic maps into K\"ahler
manifolds.} Ann.\ Math. {\bf102}, 421-462 (1975).

\entry{Si-T}{Siu Y.-T., Trautmann G.:}{Gap-Sheaves and Extension of Coherent 
Analytic Subsheaves.} Lect.\ Notes\ Math. {\bf172}, Springer-Verlag, (1971).

\entry{Sk-1}{Skoda H.:}{Sous-ensembles analytiques d'odre fini ou infini dans 
$\cc^n$.} Bull.\ Soc.\ MAth.\ France {\bf100}, 353-408 (1972).


\entry{Sk-2}{Skoda H.:}{Prolongement des courans, positifs, ferm\'es de masse 
finie.} Invent.\ math. {\bf66}, 361-376, (1982).

\entry{St}{Stolzenberg G.:}{Volumes, Limits and Extension of Analytic 
Varieties.} Springer \ Verlag (1966).

\entry{V}{Varouchas J.:}{Stabilit\'e de la classe des vari\'et\'es
K\"ahl\'eriennes
par certain morphismes propres.} Invent.\ math. {\bf77}, 117-127 (1984).

\end{document}
