\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 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)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) $$ \noindent defined by $Z_a\in {\cal Z}\to Z_a\subset \Delta^{n+k}\times X$. \par\smallskip\noindent\sl Case $n=1$. \rm Consider the union $\hat {\cal C}_0$ of thouse components of ${\cal C}_{f,2C}$ which intersect ${\cal C}_0$. At least one of thouse components, say ${\cal K}$, containes two points $s_1$ and $s_2$ s.t. $Z_{s_1}$ projects onto $\Delta^k_0$ and $Z_{s_2}$ projects onto $\Delta^k_s$ with $s\not= 0$. Consider the restriction ${\cal Z}\mid_{{\cal K}}$ of the universal space onto ${\cal K}$. This is an irreducible complex space of finite dimension. Take points $z_1\in Z_{s_1}$ and $z_2\in Z_{s_2}$ and join them by an analytic disk $\phi :\Delta \to {\cal Z}\mid_{{\cal K}}$, $\phi (0)=z_1, \phi (1/2)=z_2$. Then the composition $\psi = \pi\circ F\circ \phi :\Delta \to \Delta $ is not degenerate because $\psi (0)=0\not= s=\psi (1/2)$. Thus $\psi $ is proper and obviously so is the map $F:{\cal Z}\mid_{\phi (\Delta )}\to F({\cal Z}\mid_{\phi (\Delta )})\subset \Delta^{n+k}\times X$. Thus $F:{\cal Z}\mid_{\phi (\Delta )})$ is an analytic set in $U\times \Delta^k\times X$ for small enough $U$ extending $\Gamma_f$ by the reason of dimension. \par\smallskip\noindent\sl Case $n\ge 2$. \rm Consider an increasing family of complex spaces $\{ {\cal C}_{f,C}: C\ge C_0 \} $, where $C_0$ is from the Theorem 1. Note that for $C_1 = \int_{\Delta }\Delta \psi \cdot \int_{ Z_{\phi (s)}}\Omega^{k,k} = \int_{{\cal Z}\mid_{\phi (\Delta )}}dd^c(\pi^* \psi )\wedge \Omega^{k,k} = $$ $$ = \int_{\phi (\Delta )}\pi^*\psi \wedge dd^c\Omega^{k,k} \le 0. $$ \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 $=\overline{}$ 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 $\phi_1,...,\phi_k\in {\cal D}^{1,0}(\Omega )$ $$ \ge 0. $$ \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 $$ \Vert T\Vert (U)=sup\{ \vert \vert :\phi\in {\cal D}^k(U), \Vert \phi (x)\Vert \le 1, x\in U\}. \eqno(2.1.4) $$ 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 $$ <\tilde T,\phi >=\lim_{n\rightarrow \infty } \eqno(2.1.5) $$ \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 $$ \mu_1(B\cap K) = -\lim_{k\rightarrow \infty }\int_Bv_k\cdot \mu w_e^n = - \lim_{k\rightarrow \infty } = $$ $$ = -\lim_{k\rightarrow \infty } \le 0, \eqno(2.1.7) $$ \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 $\Delta_z(t)=\{ (z,\lambda ):\vert \lambda \vert 0$, 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 $1-r0$ 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 $a 0$. 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 $$ = \int_{\Delta^{n+1}} \tilde T\wedge (dd^c\Vert z'\Vert^2)^{n-1}\wedge dd^c\phi = $$ $$ = \int_{\Delta^{n-1}} (dd^c\Vert z'\Vert^2)^{n-1} \int_{\Delta^2}(\tilde T)_{z'}\wedge dd^c\phi = = \int_{\Delta^{n-1}} (dd^c\Vert z'\Vert^2)^{n-1} \int_{\Delta^2}\tilde T_{z'}\wedge dd^c\phi \le $$ $$ \le \int_{\Delta^{n-1}} (dd^c\Vert z'\Vert^2)^{n-1} \int_{\Delta^2}dd^c(\tilde T)_{z'}\wedge \phi \le 0. $$ 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 = \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) $$ \noindent see [Bk] or [D-M]. Let $\Omega $ be the maximal open subset of $\Delta^k$ such that the multivalued map $s$, which is given by its graph $S^0$ takes exactly $N$ different values (and $N$ is maximal). First we shall prove that $S^0\cap (\Omega \times \Delta^l)$ is analytic. Let further $\Omega_1$ be some simply connected open subset of $\Omega $. Then $s\mid_{\Omega_1}$ decomposes to $N$ well-defined single valued maps $s^1, ...,s^N$. So it is enough to consider the case when $s$ is single valued. Put $s(z')=(s_1(z'),...,s_l(z'))$. Note that in this case $\nu_{K,\bar J,z'}= \delta_{\{ z^{''}-s(z')\} }$. We can write for the coefficients of our current $R$ and for $\phi \in C^{\infty }(\Delta^{k+l})$ such that $\pi (\supp \phi )\subset \subset \Delta^k$ that $$ = \int_{\Delta^k }\bar \phi (z',s(z'))d\mu_{K,\bar J}(z'). \eqno(2.6.9) $$ If we choose $\phi $ not depending on $z^{''}:=(z_{k+1},...,z_{k+l})$ then (2.6.9) gives $$ = \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= = = $$ $$ = \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 = = = = $$ $$ = 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) + + \bar Q(z) + O(\Vert z\Vert^3), \eqno(2.5.1) $$ \noindent where $Q(z)$ is a holomorphic polynomial, $$ - 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 $0\le a0$. 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