Meromorphic quotients for some holomorphic G-actions.
Daniel Barlet\footnote{Institut Elie Cartan
Alg\`{e
1505.06932.tex
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\begin{document}

\date{12/05/15}

\author{Daniel Barlet\footnote{Institut Elie Cartan, Alg\`{e}bre et G\'eom\`{e}trie,\newline
Universit\'e de Lorraine, CNRS UMR 7502   and  Institut Universitaire de France.}.}


\title{Meromorphic quotients for some holomorphic G-actions.}

\maketitle

\parag{Abstract} Using mainly  tools from  [B.13]  and [B.15] we give a necessary and sufficient condition in order that a holomorphic  action of a connected complex Lie group $G$ on a reduced  complex space $X$  admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert  that, when $G = K.B$ \  with $B$   a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action  implies, assuming moreover that there exists a $G-$invariant  Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.

\parag{AMS classification} 32 M 05, 32 H 04, 32 H 99, 57 S 20.

\parag{Key words} Strongly quasi-proper map, strongly quasi-proper meromorphic quotient, Holomorphic G-action, finite type cycles,
%  escape at infinity, infinite breaking,

\tableofcontents

\section{Introduction}
In this article we explain how the tools developed in [M.00], [B.08], [B.13]  and [B.15] can be applied to produce in suitable cases a meromorphic quotient of a holomorphic action of a connected complex Lie group $G$ on a reduced  complex space $X$. This uses the notion of {\bf strongly quasi-proper map} introduced in {\it loc. cit.} and our first goal is to give three hypotheses, called ${\rm [H.1], [H.2], [H.3]}$ on the group action which are {\bf equivalent} to the existence of a {\bf strongly quasi-proper meromorphic quotient}, notion defined in the section 1.2. 


The proof of this equivalence is the content of proposition \ref{necessary} and theorem \ref{mero. quot.}. Then we discuss these hypotheses and give a simple sufficient condition ${\rm [H.1str]}$ which implies ${\rm [H.1]}$. The existence theorem for a strongly quasi-proper meromorphic quotient under our three assumptions is applied to prove the following results :

\begin{thm}\label{G = K.B}
Assume that we have a holomorphic action of a  connected complex Lie group $G$ on a reduced complex space $X$. Assume that  $G = K.B$ where $K$ is a compact (real) subgroup of $G$ and $B$ a connected complex closed subgroup of $G$. Assume that the action of $B$ on $X$  satisfies   the condition ${\rm [H.1str]}$ on a $G-$invariant Zariski open dense subset $\Omega$ in $X$, and  the conditions ${\rm [H.2]}$ and ${\rm [H.3]}$. Then the $G-$action satisfies ${\rm [H.1str],[H.2]}$ and ${\rm [H.3]}$ ; so it has a strongly quasi-proper meromorphic quotient.
\end{thm}


\parag{Acknowledgements}  An important part of this article comes from discussions with Peter Heinzner during a stay in Bochum. I want to thank him for his help  and for his hospitality.


\section{Strongly quasi-proper meromorphic quotients.}

\subsection{Preliminaries.}

For the definition of the topology on the space $\mathcal{C}_{n}^{f}(X)$ of finite type $n-$cycles in $X$ and its relationship with the (topological) space $\mathcal{C}_{n}^{loc}(X)$ we refer to [B-M],  [B.13] and [B.15].\\
For the convenience of the reader we recall shortly here the definition of a geometrically f-flat map (GF map) and of a strongly quasi-proper map (SQP map) between irreducible complex spaces and we give  a short summary on some properties of the SQP maps. For more details on these notions see [B.13] and [B.15].

\begin{defn}\label{GF} A holomorphic map $f : M \to N$ between two  irreducible irreducible complex spaces is called a {\bf geometrically f-flat map (a GF-map for short)} if the following conditions are fullfilled :
\begin{enumerate}[i)]
\item The map is quasi-proper equidimensionnal and surjective.\\
 Let $n : \dim M - \dim N$.
\item There exists a holomorphic map $\varphi : N \to \mathcal{C}^{f}_{n}(M)$\footnote{that is to say a f-analytic family of finite type $n-$cycles in $M$ parametrized by $N$.} such that for $y$ generic in $N$ the cycle $\varphi(y)$ is reduced and equal to the set-theoretic fiber $f^{-1}(y)$ of $f$ at $y$.
\end{enumerate}
A holomorphic map $f : M \to N$ between two  irreducible irreducible complex spaces will be {\bf strongly quasi-proper (SQP map for short)} if there exists a modification\footnote{a modification is, by definition,  always proper.} 
  $\tau : \tilde{N} \to N$ such that the strict transform\footnote{By definition $\tilde{M}$ is the irreducible component of $M\times_{N}\tilde{N}$ which surjects on $\tilde{N}$ and $\tilde{f}$ is induced by the projection.} $\tilde{f} : \tilde{M} \to \tilde{N}$ of $f$ by $\tau$ is a GF map.\\
  A meromorphic map $M \dasharrow N$ will be called strongly quasi-proper when the projection on $N$ of its  graph is a SQP map.
\end{defn}

Note that a GF map has, by definition, a {\it holomorphic fiber map} and that a SQP  holomorphic (or meromorphic) map has a {\it meromorphic fiber map} via the composition of the holomorphic fiber map of $\tilde{f}$ with the (holomorphic) direct image map $\tau_{*} :  \mathcal{C}^{f}_{n}(\tilde{M}) \to  \mathcal{C}^{f}_{n}(M)$. Of course, a SQP holomorphic map is quasi-proper, but the converse is not true. The notion of strongly quasi-proper map is stable by modification of the target space, property which is not true in general for a quasi-proper map having ``big fibers''  (see [B.15]).\\

Let $\pi : M \to N$ be a SQP map between irreducible complex spaces and define $n := \dim M - \dim N$. By definition of a SQP map, we can find a Zariski open dense subset $N_{0}$ in $N$ and a holomorphic map $\varphi_{0} : N_{0} \to \mathcal{C}_{n}^{f}(M)$ such that 
\begin{enumerate}[i)]
\item For each $y$ in $N_{0}$ we have the equality of subsets $\vert \varphi_{0}(y)\vert = \pi^{-1}(y)$.
\item For $y$ generic in $N_{0}$ the cycle $\varphi_{0}(y)$ is reduced.
\end{enumerate}
Let $\Gamma \subset N_{0}\times \mathcal{C}_{n}^{f}(M)$ be the graph of $\varphi_{0}$. Then also by definition of a SQP map, the closure $\bar \Gamma$ of $\Gamma$ in $N \times \mathcal{C}_{n}^{f}(M)$ is proper over $N$. Then, using the semi-proper direct image theorem 2.3.2 of [B.15], this implies that $\tilde{N} := \bar \Gamma$ is an irreducible complex space (locally of finite dimension) with the structure sheaf induced by the sheaf of holomorphic functions on $N \times \mathcal{C}_{n}^{f}(M)$. Moreover the natural projection $\tau : \tilde{N} \to N$ is a (proper) modification.\\
Let $\tilde{M} := M\times_{N,str} \tilde{N}$ the strict transform of $M$ by $\tau$, that is to say the irreducible component of $M\times_{N}\tilde{N}$ containing the graph of  $\pi_{0}$, the restriction of $\pi$ to the open set $\pi^{-1}(N_{0})$\footnote{this graph is a Zariski open set in $M\times_{N}\tilde{N}$ which is irreducible as $N$ is irreducible.}. Then let $\tilde{\pi} : \tilde{M} \to \tilde{N}$ the strict transform of $\pi$ by the modification  $\tau$, which is induced on $\tilde{M}$  by the natural projection of $M\times_{N}\tilde{N}$ onto $\tilde{N}$. The set-theoretical fiber at $\tilde{y} := (y,C) \in \tilde{N}$ of $\tilde{\pi}$ is the subset $\vert C\vert \times\tilde{y}$ in $\tilde{M}$. The map $\psi : \tilde{N} \to  \mathcal{C}_{n}^{f}(\tilde{M})$ \ given by $(y, C) \mapsto C \times\{\tilde{y}\}$ is holomorphic and satisfies $\vert \psi(\tilde{y})\vert =\tilde{\pi}^{-1}(\tilde{y})$ for all $\tilde{y}$ in $\tilde{N}$. Moreover $\psi(\tilde{y})$ is a reduced cycle for generic $\tilde{y}$ in $ \tilde{N}$. So the map $\tilde{\pi}$ is geometrically f-flat. It is the {\bf canonical GF-flattning} of $\pi$.\\
Then we have an isomorphism induced by $\psi$
$$ \psi : \tilde{N} \longrightarrow \mathcal{C}_{n}^{f}(\tilde{\pi})  $$
where $ \mathcal{C}_{n}^{f}(\tilde{\pi})      := \{ C \in  \mathcal{C}_{n}^{f}(\tilde{M}) \ / \  \exists \tilde{y} \in \tilde{N} \ s. t. \ \vert C\vert \subset \tilde{\pi}^{-1}(\tilde{y})\}$ is a closed analytic subset of $ \mathcal{C}_{n}^{f}(\tilde{M})$ (see [B.15] proposition 2.1.7.);  the inverse map is induced by the holomorphic map $\hat{\pi} : \mathcal{C}_{n}^{f}(\tilde{\pi}) \to \tilde{N}$ which associates to $\gamma \in \mathcal{C}_{n}^{f}(\tilde{\pi}) $ the point in $\tilde{N}$ whose $\tilde{\pi}-$fiber contains $\gamma$ (see the proposition 2.1.7 of [B.15]).\\
The direct image of $n-$cycles by $\tau$ gives a holomorphic map $\tau_{*} :  \mathcal{C}_{n}^{f}(\tilde{M}) \to \mathcal{C}_{n}^{f}(M)$ which sends $\tilde{N} \simeq \mathcal{C}_{n}^{f}(\tilde{\pi})$ in $\mathcal{C}_{n}^{f}(\pi)$. Let us show that it is an isomorphism of $\tilde{N}$ onto its image in $\mathcal{C}_{n}^{f}(\pi)$ :\\
We have an obvious holomorphic map $\tilde{N} \to \mathcal{C}_{n}^{f}(\pi)$ given by $(y,C) \mapsto C$. We have also a holomorphic map $\mathcal{C}_{n}^{f}(\pi) \to N \times \mathcal{C}_{n}^{f}(M)$ given by $C \mapsto (\hat{\pi}, C)$ where $\hat{\pi} : \mathcal{C}_{n}^{f}(\pi) \to N$ is the map associating to $C \in \mathcal{C}_{n}^{f}(\pi) $ the point $y \in N$ such that $\vert C\vert \subset \pi^{-1}(y)$. This proves our claim.\\
Remark that $\mathcal{C}_{n}^{f}(\pi) $ is not, in general, a complex space (locally of finite dimension).\\

\subsection{Action of $G$ on $\mathcal{C}_{n}^{f}(X)$.}

Let $G$ be a Lie group. We shall say that $G$ acts {\bf continuously holomorphically} on the reduced complex space $X$ when the action $f : G \times X \to X$ is a continuous map such that for each $g \in G$ fixed, the map $x \mapsto f(g,x)$ is a (biholomorphic)  automorphism of $X$. Then there is a natural action of $G$ induced on the set $\mathcal{C}_{n}^{f}(X)$ of finite type $n-$cycles given by $(g,C) \mapsto g_{*}(C)$ where we denote $g_{*}(C)$ the direct image of the cycle $C$ by the automorphism of $X$ associated to $g \in G$. When $G$ is a complex Lie group and the map $f$ is holomorphic  we shall say that the action is {\bf completely holomorphic}.
 
 \begin{prop}\label{holomorphy}
 The action of $G$ on $\mathcal{C}_{n}^{f}(X)$  is continuously holomorphic. This precisely means that the map $G \times\mathcal{C}_{n}^{f}(X) \to \mathcal{C}_{n}^{f}(X)$ given by  $(g,C) \mapsto g_{*}(C)$  is continuous and that, for any f-analytic family of $n-$cycles $(C_{s})_{s \in S}$ in $X$ parametrized by a reduced complex space $S$, the family
 $g_{*}(C_{s})_{(g, s) \in \{g\}\times S}$ is f-analytic for each fixed $g\in G$ . If $G$ is complex Lie group and the action is completely holomorphic, the action of  $G$ on $\mathcal{C}_{n}^{f}(X)$ is completely holomorphic, so for  any f-analytic family of $n-$cycles $(C_{s})_{s \in S}$ in $X$ parametrized by a reduced complex space $S$, the family
 $g_{*}(C_{s})_{(g, s) \in G\times S}$ is f-analytic.
 \end{prop}
 
 \parag{Proof} First we prove the continuity of the action of $G$ on $\mathcal{C}_{n}^{loc}(X)$. To apply the theorem IV 2.5.6 de [B-M] it is enough to see that the map $ F : G \times X \to G\times X$ given by $(g, x) \mapsto (g, g.x)$ is proper. But if $L \subset G$ and $K \subset X$ are compacts sets, we have $F^{-1}(L\times K) \subset L \times (L^{-1}.K)$ which is a compact set in $G \times X$.\\
  The only point left to prove the continuity statement for the topology of $\mathcal{C}_{n}^{f}(X)$, assuming that the continuity for the topology of $\mathcal{C}_{n}^{loc}(X)$ is obtained as follows :\\
 Let $W$ be a relatively compact open set in $X$ and $\mathcal{W}$ be the open set in $\mathcal{C}_{n}^{f}(X)$ of cycles $C$ such any irreducible component of $C$ meets $W$. Then we want to show that the set of $(g, s) \in G \times S$ such that $g_{*} (C_{s})$ lies in $\mathcal{W}$ is an open set in $G \times S$. As the topology of $\mathcal{C}_{n}^{f}(X)$ has a countable basis\footnote{This is a corollary of the fact that this is true for $\mathcal{C}_{n}^{loc}(X)$ (see [B-M] ch.IV) as the topology of $X$ has a countable basis of open sets; see  [B.15] for details.} it is enough to show that if a sequence $(g_{\nu}, s_{\nu})$ converges to $(g, s)$ with $g_{*}(C_{s}) \in \mathcal{W}$ then for $\nu \gg 1$ we have also $(g_{\nu})_{*}(C_{s_{\nu}})\in \mathcal{W}$. If this not the case, we can choose for infinitely many $\nu$ an irreducible component  $\Gamma_{\nu}$ of $(g_{\nu})_{*}(C_{s_{\nu}})$ which does not meet $W$.  Up to pass to a sub-sequence, we may assume that the sequence $\Gamma_{\nu}$ converges in $\mathcal{C}_{n}^{loc}(X)$ to a cycle $\Gamma$ which does not meet $W$ and is contained in $g_{*}(C_{s})$. This is a simple consequence of the continuity of the $G-$action on $\mathcal{C}^{loc}_{n}(X)$ and the characterization of compact subsets in $\mathcal{C}^{loc}_{n}(X)$ (see [B-M] ch.IV). As any irreducible component of $g_{*}(C_{s})$ meets $W$ this implies that $\Gamma$ is the empty $n-$cycle. This means that for any compact $K$ in $X$ there exists an integer $\nu(K)$ such that for $\nu \geq \nu(K)$ we have $\Gamma_{\nu} \cap K = \emptyset$. Choose now a compact neighbourhood $L$ of $g(K)$. For $\nu$ large enough we shall have $K\subset g_{\nu}^{-1}(L)$. This comes from the fact that the automorphisms $g_{\nu}^{-1}$ converge to $g^{-1}$ in the compact-open topology. Then this implies that for $\nu \geq \nu(L)$ the irreducible component $g_{\nu}^{-1}(\Gamma_{\nu})$ of $C_{s_{\nu}}$ does not meet $K$. Then, when $s_{\nu} \to s$ the cycles $C_{s_{\nu}}$ does not converge to $C_{s}$ for the topology of $\mathcal{C}_{n}^{f}(X)$ because we have some ``escape at infinity'' in  a well choosen sub-sequence. Contradiction.$\hfill \blacksquare$\\
 
\begin{lemma}\label{action}
Assume now that in the situation above we have a Lie group $H$ acting continuously holomorphically on $M$. Assume that $\pi$ satisfies $\pi(h.x) = \pi(x)$ for each $x \in M$. Then the strict transform $\tilde{M}$ of $M$ by the canonical modification $\tau : \tilde{N} \to N$ giving the canonical GF flattning of $\pi$ has a natural continuous holomorphic action of $H$ such that the projection $\tilde{M} \to M$ is $H-$equivariant. Moreover the GF map $\tilde{\pi} : \tilde{M} \to \tilde{N}$ satisfies $\tilde{\pi}(h.\tilde{x}) = \tilde{\pi}(\tilde{x})$ for each $(h,\tilde{x}) \in H \times \tilde{M}$.
\end{lemma}

\parag{proof} We have a natural action of $H$ on $\mathcal{C}_{n}^{f}(M)$ which is continuous and holomorphic for each fixed $h \in H$. As the action of $H$ is trivial on the set of fibers of $\pi$, the action of $H$ is trivial on $\tilde{N} $. Define the action of $H$ on $M\times_{N}\tilde{N}$ by the formula
$$ h.(x,(\pi(x), C)) = (h.x, \pi(x), h_{*}(C)) .$$
It is easy to see that this is a continuous holomorphic action, and that it leaves $\tilde{M}$ globally invariant. Now we have for each $h \in H$ and each $(x,\pi(x), C) \in \tilde{M}$ :
$$ \tilde{\pi}\big(h.(x, (\pi(x), C))\big) = \tilde{\pi}\big(h.x, (\pi(x), h_{*}(C))\big) = (\pi(x), h_{*}(C)) = (\pi(x), C) = \tilde{\pi}(x, (\pi(x), C)) $$
because a limit of $H-$invariant cycles is an $H-$invariant cycle.$\hfill \blacksquare$\\

We shall also use the following simple tool from the cycle's space.

\begin{prop}\label{non inf. br.}

Let \ $M$ \ be a reduced complex space and \ $(X_{s})_{s\in S}$ \ a f-continuous family of \ $d-$dimensional finite type cycles parametrized by a  compact subset \ $S$ in $\mathcal{C}_{d}^{f}(M)$. Let \ $(C_{t})_{t \in T}$ \  be a f-continuous family  of finite type non empty  \ $n-$dimensional cycles  in $M$ parametrized by a subset $T$  in $\mathcal{C}_{n}^{f}(M)$ which is compact in \ $\mathcal{C}_{n}^{loc}(M)$. We assume the following condition :
\begin{itemize}
\item  There exists an open dense set $T'$  in $T$ such that  each \ $C_{t}, t \in T'$ \ is equal to the union some \ $X_{s}$. \hfill $(@@)$
\end{itemize}
Then $T$ is a compact subset in $\mathcal{C}^{f}_{n}(M)$.
\end{prop}


\parag{Proof} First remark that, as \ $S$ \ is compact, there exists a compact set \ $L \subset M$ \ such that any irreducible component of any \ $X_{s}$ \ meets \ $L$.\\
Let $(t_{m})_{m \in \mathbb{N}}$ be a  sequence of points in $T'$ converging to a point $t \in T$ and denote by $C_{m}$ the cycle $C_{t_{m}}$ for short and $C_{t} = C_{\infty}$.
Now choose for each \ $m$ \ an irreducible component \ $\Gamma_{m}$ \ of some \ $X_{s_{m}}$ \ contained in \ $C_{m}$. Up to pass to a subsequence, we may assume that \ $\Gamma_{m}$ \ converges in \ $\mathcal{C}_{d}^{f}(M)$ \ to a cycle \ $\Gamma$ \ which is non empty (it contains a point in \ $L$) \ and included in \ $\vert C_{\infty}\vert$. So  \ $C_{\infty}$ \ is not the empty cycle. \\
Let \ $x$ \ be a generic point of an irreducible component \ $D$ \ of \ $C_{\infty}$. Then, up to pass to a subsequence, we may  choose a sequence \ $(x_{m})$ \ of points respectively in \ $C_{m}$ \ which converges to \ $x$. Choose for each \ $m$ \ an irreducible component \ $\Gamma_{m}$ \ of some \ $X_{s_{m}} \subset \vert C_{m}\vert$ \ which contains \ $x_{m}$. This is possible because of condition \ $(@@)$. Now, again up to pass to a subsequence, we may assume that the sequence \ $(\Gamma_{m})_{m \in \mathbb{N}}$ \ converges in \ $\mathcal{C}_{d}^{f}(M)$ \ to a cycle \ $\Gamma$ \ containing the point \ $x$ \ and contained in \ $\vert C_{\infty}\vert$. Note that $\vert \Gamma\vert$ is contained in some $\vert X_{s_{\infty}}\vert$ as we may assume, by compactness of $S$,  that  the sequence $(s_{m})$ converges to $s_{\infty}\in S$. Then we have $\vert X_{s_{\infty}}\vert \subset \vert C_{\infty}\vert$. As \ $D$ \ is the only irreducible component of \ $C_{\infty}$ \ containing \ $x$, it contains at least an irreducible component of  $\vert X_{s_{\infty}}\vert$ containing $x$,  and so \ $D$ \ meets \ $L$. So we have proved that \ $C_{\infty}$ \ is not the the empty \ $n-$cycle and that any irreducible component of \ $C_{\infty}$ \ meets the compact set \ $L$. This is enough to conclude thanks to the proposition 3.2.2 in [B.15]. $\hfill \blacksquare$\\








\subsection{Definition of SQP meromorphic quotient.}

We shall consider a complex connected Lie group $G$ and a completely holomorphic action of $G$ on an irreducible complex space $X$. It is given, by definition, by a holomorphic map $f : G\times X \to X$
such that $f(g.g',x) = f(g, f(g'.x))$ for all $g, g' \in G$ and $x \in X$, assuming that for each $g \in G$ the holomorphic map $x \mapsto f(g,x)$ is an automorphism of $X$, and that $f(1,x) = x$ for all $x \in X$.

\begin{def}\label{SQP-mero-quot.}
A {\bf strongly quasi-proper meromorphic quotient} (we shall say a {\bf SQP-meromorphic quotient} for short) for such an action $f : G\times X \to X$ will be the following data:
\begin{enumerate}
\item a $G-$modification\footnote{This means that we have a completely  holomorphic $G-$action on $\tilde{X}$ and that the modification $\tau$ is $G-$equivariant.} $\tau : \tilde{X} \to X$ with center $\Sigma$.
\item a holomorphic $G-$invariant GF map \ $q : \tilde{X} \to Q$ \ where $Q$ is an irreducible complex space.
\item an analytic $G-$invariant  subset $Y \subset X$ containing $\Sigma$, with no interior point in $X$. We shall denote $\tilde{Y} := \tau^{-1}(Y)$, $\Omega := X \setminus Y, \tilde{\Omega} :=\tau^{-1}(\Omega)$ and $Q' := q(\tilde{\Omega})$. Note that, as $q$ is an open surjective map, $Q'$ is open and dense in $Q$.
\end{enumerate}
Now we ask that these data satisfy the following properties :
\begin{enumerate}[i)]
\item The restriction to $ \Omega$ of the map $q\circ \tau^{-1}$ is a GF map onto the dense open set  $Q'$ in $Q$ and there is an open dense set $Q''$ in $Q'$ such that each fiber  of $q\circ \tau^{-1}$  at a  point in $Q''$ is equal to a $G-$orbit  in $\Omega$.
\item There exists an open dense subset $\Omega_{0} \subset \Omega$ such that for each $\tilde{x}$ in $ \tilde{\Omega}_{0} := \tau^{-1}(\Omega_{0})$ the {\bf closure} $\overline{G.\tilde{x}}$  of  $G.\tilde{x}$ in $\tilde{X}$ is exactly the set $q^{-1}(q(\tilde{x}))$. 
\end{enumerate}
\end{def}

\begin{prop}\label{universal}
Let $G$ be a complex connected Lie group which acts completely holomorphically on an irreducible complex space $X$. Assume that we have a SQP meromorphic quotient for this action, given by a modification $\tau : \tilde{X} \to X$ and a $G-$invariant  GF map $q : \tilde{X} \to Q$. Then let $\psi : Q \to \mathcal{C}_{n}^{f}(X)$ be the holomorphic map obtained by the composition  of the fiber map of the GF map $q$ and the direct image map for $n-$cycles by the modification $\tau$. Define $Q_{u}:= \psi(Q)$. Then we have the following properties:
\begin{enumerate}
\item $Q_{u}$ is a closed analytic subset in $  \mathcal{C}_{n}^{f}(X)$ which is an irreducible complex space (locally of finite dimension) with the structure sheaf induced by the sheaf of holomorphic functions on $ \mathcal{C}_{n}^{f}(X)$.
\item Let $\tilde{X}_{u}$ be the graph of the meromorphic map $ q_{u}: X \dasharrow Q_{u}$ given by the holomorphic  map  $\psi\circ q : \tilde{X} \to Q_{u}$ and let $\tau_{u}: \tilde{X}_{u} \to X$ and $q_{u} : \tilde{X}_{u} \to Q_{u}$  be the projections on $X$ and $Q_{u}$ respectively  of this graph. Then $(\tau_{u}, q_{u})$ is also a SQP meromorphic quotient for the given $G-$action.
\item For any  SQP meromorphic quotient $(\tau, q)$  there exists a unique holomorphic surjective map $\eta : Q \to Q_{u} $ such that the meromorphic maps $q : X \dasharrow Q$ and $q_{u} : X \dasharrow Q_{u}$ 
satisfies $\eta\circ q = q_{u}$.
\item For any G-invariant holomorphic map $h : X \to Y$ there exists a holomorphic map $H : Q_{u} \to Y$ such that  $h\circ\tau_{u} = H \circ q_{u}$.
\end{enumerate}
\end{prop}

\bigskip

\begin{defn}\label{univ.}
In the situation of the previous theorem the SQP meromorphic quotient for the given $G-$action defined by  $(\tau_{u}, q_{u})$  will be called {\bf the minimal SQP meromorphic quotient} of this $G-$action.
\end{defn}

So the proposition above says that the existence of a SQP meromorphic quotient for the given $G-$action implies the existence and uniqueness of a minimal meromorphic quotient for this $G-$action.

\parag{Proof} To prove the point 1) we shall prove that the map $\psi\circ q : \tilde{X} \to  \mathcal{C}_{n}^{f}(X)$ is semi-proper.\\
Let $C \not= \emptyset$ be in $\mathcal{C}_{n}^{f}(X)$ and fix a relatively compact open set $W$ in $X$ meeting all irreducible components of $C$. The subset $\mathcal{W}$ of $\mathcal{C}_{n}^{f}(X)$ of cycles $C'$  such that any irreducible component of $C'$ meets $W$ is an open set containing $C$. Now $ q(\tau^{-1}(\bar W))$ is a compact set in $Q$, as $\tau$ is proper. Take any $y \in Q$ such that $C' : =   \psi(y)$ is in $\mathcal{W}$. The point $y$ is the limit in $Q$ of points $y_{\nu} \in q(\tilde{\Omega}_{0})$ such that the fiber of $Q$ at $y$ is limit in $\mathcal{C}_{n}^{f}(\tilde{X})$ of the fibers $q^{-1}(y_{\nu}) = \overline{G.\tilde{x_{\nu}}}$ where, for $\nu \gg 1$, we can choose $\tilde{x}_{\nu}$ in $\tilde{\Omega}_{0} \cap \tau^{-1}(W)$. Up to pass to a sub-sequence, we may assume that $\tilde{x}_{\nu}$ converges to a point $\tilde{x}$ in  $\tau^{-1}(\bar W)$. Then the continuity of $q$ implies that $q(\tilde{x}) = y$ and $C'$ is the limit of $\overline{G.\tilde{x}_{\nu}}$. So $\vert C'\vert$ is in  the image by $\psi$ of the compact set $q(\tau^{-1}(\bar W))$ and this  gives the semi-properness of $\psi\circ q$.\\
Now the  direct image theorem 2.3.2 in [B.15] shows that $Q_{u}$ is an irreducible complex space (locally of finite dimension) and the point 1) is proved.\\
To prove the second point we have to show that the map $q_{u} : \tilde{X}_{u} \to Q_{u}$ is a GF map. By definition $ \tilde{X}_{u}$ is the closure in $X \times Q_{u}$ of the graph of the map $q_{\vert \Omega_{0}}$ where $\Omega_{0}$ is an open dense set in $X$ such that for any point $x \in \Omega_{0}$ we have $\psi(q(x)) = \overline{G.x}$ (as $\Omega_{0}$ is disjoint from the center of the modification $\tau$ we identify here $\Omega_{0}$ and $\tilde{\Omega}_{0} := \tau^{-1}(\Omega_{0})$). Then by irreducibility of $Q_{u}$ and $X$ the closed analytic subset $ \tilde{X}_{u} \subset X \times Q_{u}$ is equal to the graph of the tautological family of cycles in $X$ parametrized by $Q_{u} \subset \mathcal{C}_{n}^{f}(X)$. This proves the point 2).\\
Now consider a  SQP meromorphic quotient of the given action given by the maps $\tau : \tilde{X} \to X$ and $q : \tilde{X} \to Q$. Let $\psi : \tilde{X} \to \mathcal{C}_{n}^{f}(X)$ the composition of the holomorphic map classifying the fibers of the GF map $q$ with the direct image of $n-$cycles by the modification $\tau$. Then, by the construction in the proof of the point 1), we know that $\psi(Q) = Q_{u}$ and then the map
$\psi$ induces a surjective holomorphic map $\eta : Q \to Q_{u}$. We may  assume that $\tilde{X}$ is in fact the graph of the meromorphic map $q : X \dasharrow Q$. Then, as $\tilde{X}_{u}$ is the graph of the meromorphic map $q_{u} : X \dasharrow Q_{u}$ the holomorphic map $\id_{X}\times \eta : X \times Q \to  X \times Q_{u}$ sends $\tilde{X}$ to $\tilde{X}_{u}$ because this is true over a dense open set in $X$ where the maps $q$ and $q_{u}$ are holomorphic and satisfy $q^{-1}(q(x)) = q_{u}^{-1}(q_{u}(x)) = \overline{G.x}$. This complete the proof of  3).\\
Consider now a G-invariant holomorphic map $h : X \to Y$. Recall (see [B.15] proposition 2.1.7) that $\mathcal{C}_{n}^{f}(h)$, the subset of cycles $C \in \mathcal{C}_{n}^{f}(X)$ which are contained in a fiber of $h$ is a closed analytic subset in $\mathcal{C}_{n}^{f}(X)$. On an open dense subset of $Q_{u}$ we know that a point corresponds to a cycle with support $\overline{G.x}$ for some $x \in X$. Then this means that an open dense subset in $Q_{u}$ is contained in $\mathcal{C}_{n}^{f}(h)$. As this subset is closed we obtain $Q_{u}\subset \mathcal{C}_{n}^{f}(h)$. Now there is a holomorphic map $\hat{h} : \mathcal{C}_{n}^{f}(h) \to Y$ associating to a cycle $C$  the point in $Y$ such $\vert C\vert \subset h^{-1}(y)$ (see [B.15] proposition 2.1.7). This induces a holomorphic map $ H : Q_{u} \to Y$, and it is clear that the relation $h\circ\tau_{u} = H \circ q_{u}$ is true on a dense open set, so everywhere.$\hfill \blacksquare$





\subsection{Good points, good open set.}

Let \ $X$ \ be a reduced complex space and \ $G$ \ be a connected  complex Lie group.
Let \ $f : G \times X \to X$ \ be a holomorphic action of \ $G$ \ on \ $X$. 

 \begin{defn}\label{good}
 We shall say that a point \ $x \in X$ \ \ is a { \bf good point}  for the action \ $f$ \ if the following condition is satisfied
 \begin{itemize}
 \item For each compact set \ $K$ \ in \ $X$ \ there exists an open  neighbourhood \ $V$ \ of \ $x$ \ and a compact set \ $L$ \  in \ $G$ \ such that if \ $y \in V$ \  and \ $g \in G$ \ are such that \ $g.y \in K$,  there exists \ $\gamma \in L$ \ with \ $\gamma.y = g.y$
 \end{itemize}
 We shall say  that the action of $G$ on $X$ is {\bf good} when each point in $X$ is a good point. If $\Omega$ is  a $G-$invariant open set in  $ X$, we shall say that $\Omega$ is a {\bf a good open set }  for the action \ $f$ \  when all  points in $\Omega$ are good points for the $G-$action given by  \ $f$ {\bf restricted to $\Omega$}.\\
 \end{defn}
 
\parag{Remarks}\begin{enumerate}
\item If \ $x \in X$ \ is a good point, then for any \ $g_0 \in G$ \ $g_0.x$ \ is also a good point : for \ $K$ \ given, choose \ $g_0.V$ \ as neighbourhood of \ $g_0.x$ \ and  the compact set \ $L.g_0^{-1} \subset G$ \  to satisfy the needed conditions.
\item If \ $\Omega$ \ is a  good open set, then points in $\Omega$ are not in general good points for the action on $X$.
\item If \ $\Omega$ \ is a  good open set and $W \subset \Omega$ a G-invariant open set, then $W$ is a good open set.
\item If $M$ is a compact set of good points in $X$ for any compact set $K$ in $X$ we can find a neighbourhood $V$ of $M$ in $X$ and a compact set $L$ in $G$ such that for any point $y \in V$ and any $g \in G$ such that $g.y \in K$ there exists $\gamma \in L$ with $\gamma.y = g.y$. This is easily obtained by a standard compactness argument. We shall say that a  compact set of good points is {\bf uniformely good}.\\
\end{enumerate}

\begin{lemma}\label{Good 1}
Let $x$ be a point in $X$. Then $x$ is a good point for the $G-$action on $X$ if and only if the map $F_{X} : G \times X \to X \times X$ given by $(g,x) \mapsto (x, g.x)$ is semi-proper at each point of $\{x\}\times X$. As a consequence  a $G-$invariant open set $\Omega$ in $X$ is a good open set for the $G-$action if and only if the map $ F_{\Omega} : G \times \Omega \to \Omega \times \Omega $ given by $(g, x) \mapsto (x, g.x)$ is semi-proper.
 \end{lemma}
 
 \parag{Proof} Let $x \in X$ be a good point and fix any $z \in X$. To prove that the map $F_{X}$ is semi-proper at  $(x, z)$ choose compact neighbourhoods $V_{0}$ and $K$ of $x$ and $z$ in $X$ and apply the definition of a good point to the compact set $K$. So we can find a neighbourhood $V$ of $x$, that we may assume to be contained in $V_{0}$, and a compact set $L$ in $G$ such that for any $y \in V$ such that $g.y\in K$ we have a $\gamma \in L$ with $g.y = \gamma.y$. Then we have $F_{X}(G \times X) \cap (V\times K) = F_{X}(L \times V_{0}) \cap (V\times K) $ and $L \times V_{0}$ is a compact set in $G \times X$.\\
 Conversely, assume that the map $F_{X}$ is semi-proper at each point of $ \{x\}\times X$. Take a compact set $K$ in $X$ and apply the semi-properness to each point $(x, z)$ where $z$ is in $K$. For each $z \in K$ we obtain  open neighbourhoods $V_{z}$ and $W_{z}$  of $x$ and $z$ in $X$ and a compact set $L_{z}\times M_{z}$ in $G\times X$ such that 
 $$F_{X}(G \times X)\cap (V_{z}\times W_{z}) = F_{X}(L_{z}\times M_{z}) \cap (V_{z}\times W_{z}) .$$
 Extract a finite sub-cover $W_{1}, \dots, W_{N}$  of $K$ by the open sets $W_{z}$ and define the compact set  $L : = \cup_{i \in [1, N]} L_{z_{i}}$ and the neighbourhood  $V := \cap_{i \in [1,N]} V_{z_{i}}$ of $x$. Then if $y$ is in $V$  and $g.y$ is in $K$ there exists $i \in [1,N]$ such that $g.y \in W_{i}$. As $y$ is in $V_{i}$ we can find a $\gamma \in L_{z_{i}}\subset L$ with $F_{X}(g, y) = F_{X}(\gamma, y)$ and this implies that $x$ is a good point. \\
 The second assertion is an easy consequence of the first one.$\hfill \blacksquare$\\


\begin{prop}\label{Good 2}
Let \ $G$ \ be a connected  complex Lie group. Let \ $f : G \times X \to X$ \ be an holomorphic action of \ $G$ \ on a reduced complex space \ $X$. Then we have the following properties :
\begin{enumerate}[i)]
\item If \ $x$ \ is a good point for \ $f$ \ the orbit \ $G.x$ \ is a closed analytic subset of \ $X$.
\item If \ $x$ \ is a good point for \ $f$ \ and if \ $(G.x) \cap K = \emptyset $ \ where \ $K \subset X$ \ is a compact set, there exists a neighbourhood \ $V$ \ of \ $x$ \ in \ $X$ \ such that \ $(\overline{ G.x'}) \cap K = \emptyset$ \ for any \ $x'$ \ in \ $V$ ($x'$ is not assume here to be a good point).
\item If \ $x$ \ is a good point for \ $f$ \ there exists a neighbourhood \ $V$ \ of \ $x$ \ such that any good point \ $x' \in V$ \ has an orbit which is a closed analytic subset of the same dimension than \ $G.x$.
\item Let \ $\Omega$ \ be   a good connected open set  in $X$ which is normal  and let \ $n$ \ be the dimension of \ $G.x$ \ for \ $x \in \Omega$. Then there exists a holomorphic map\footnote{This means that we have an $f-$analytic family of $n-$cycles in \ $\Omega$ \ parametrized by \ $\Omega$.} $\varphi : \Omega \to \mathcal{C}_n^f(\Omega) $   given generically by \ $\varphi(x) : = G.x$ \ as a reduced $n-$cycle in \ $\Omega$.
\item When we have a good open set $\Omega$  in $X$ which is normal, there exists a quasi-proper equidimensional holomorphic quotient of $\Omega$ for the action restricted to $\Omega$. 
\end{enumerate}
\end{prop}

\parag{Proof} We already proved that $x$ is a good point if the map $G \to X$ given by $g \mapsto g.x$ s semi-proper in lemma 1.3.2. Now Kuhlmann's theorem  [K.64], [K.66]  gives that \ $f_x(G) = G.x$ \ is a closed analytic subset of \ $X$. This proved i)\\

Assume ii) is not true ;  then we have a compact set \ $K$ \ such that \ $(G.x)\cap K = \emptyset$ \ and a sequence \ $(x_{\nu})_{\nu \in \mathbb{N}}$ \ converging to \ $x$ \ and such that \ $(\overline{G.x_{\nu}}) \cap K$ \ is not empty for each \ $\nu$. Fix a compact neighbourhood $\tilde{K}$ of $K$ such that $(G.x) \cap \tilde{K} = \emptyset$. This is possible thanks to i). Pick a point \ $y_{\nu} = \lim_{\alpha \to \infty} \ g_{\nu,\alpha}.x_{\nu}$ \ in \ $(\overline{G.x_{\nu}}) \cap K$ \ for each \ $\nu$. Up to pass to a subsequence we may assume that sequence\ $(y_{\nu})$ \ converges to \ $y \in K$ \ when \ $\nu \to + \infty$. So, for $\alpha \geq \alpha(\nu)$, we can assume that $g_{\nu,\alpha}.x_{\nu}$ is in $\tilde{K}$. But, as \ $x$ \ is a good point, for the given \ $\tilde{K}$ \  there exists a neighbourhood \ $V$ \ of \ $x$ \ and a compact set \ $L \subset G$ \ as in the definition. We may assume that \ $x_{\nu}$ \ is in \ $V$ \ for \ $\nu \geq \nu_0$ \ and so we may find, for \ $\nu \geq \nu_0, \alpha \geq \alpha(\nu)$, elements  \ $\gamma_{\nu, \alpha} \in L$ \ such that \ $\lim_{\alpha \to \infty} \gamma_{\nu,\alpha}.x_{\nu} = y_{\nu} \quad \forall \nu \geq \nu_0$.\\
Up to pass to a subsequence for each given $\nu \geq \nu_{0}$, we may assume that the sequence\ $(\gamma_{\nu, \alpha})$ \ converges to some \ $\gamma_{\nu} \in L$.  And again, that the sequence $\gamma_{\nu}$ converges to some $\gamma \in L$. So the continuity of \ $f$ \ gives  \ $y_{\nu} = \gamma_{\nu}.x_{\nu} \to \gamma.x = y \in \tilde{K}$ \ giving a contradiction because we assume \ $(G.x) \cap \tilde{K} = \emptyset$. This proves ii).\\

Let \ $E : = (U,B, j)$ \ be a \ $n-$scale on \ $\Omega$ \ adapted to the $n-$cycle \ $G.x$. Then the compact set \ $K : = j^{-1}(\bar U\times \partial B)$ \ does not meet \ $G.x$, by definition of an adapted scale. Using ii), there exists a neighbourhood \ $V$ \ of \ $x$ \ such that for any \ $x' \in V$ \ we have \ $(\overline{G.x' })\cap K = \emptyset$. As for a good point \ $x' \in V$ \ we know that \ $G.x'$ \ is a closed analytic subset, the \ $n-$scale is then adapted to \ $G.x'$. This implies that the dimension of \ $G.x'$ \ is at most equal to \ $n$. But the semi-continuity of the dimension of the stabilizers implies that the dimension of \ $G.x' \simeq G\big/St(x')$ \ is at least equal to \ $n = \dim (G\big/St(x))$. This proves iii).\\

Remark that for any $x' \in V$ such that $\overline{G.x'}$ is a closed analytic subset in $\Omega$, the previous  proof  shows also that $\overline{G.x'}$ is of pure dimension $n$.\\

To prove iv)  fix a good connected open set \ $\Omega$ \ and define
$$ Z : = \{ (g,x,y) \in G\times \Omega \times \Omega  \ / \  y = g.x .\}  $$
This is a closed analytic subset in $G\times \Omega\times \Omega$. Let us show that the projection \ $p : Z \to \Omega \times \Omega$ \ is semi-proper. Pick a point \ $(x,y) \in \Omega\times \Omega$ \ and choose compact neighbourhoods  \ $V$ \ and \ $K$ \ respectively of \ $x$  and of \ $y$ \ in \ $ \Omega$. Using the fact that any open set  \ $\Omega' \subset\subset \Omega$ \ is  uniformely good, for the compact \ $K$ \ in \ $\Omega$  we find a compact set \ $L$ \ in \ $G$ \ such that for \ $x' \in \Omega'$ \ and \ $g \in G$ \ with \ $g.x' \in K$ \ there exists \ $\gamma \in L$ \ with \ $\gamma.x' = g.x'$. Choosing \ $\Omega'$ \ containing \ $V$ this implies  that we have
$$ p(Z) \cap (V \times K) = p\big((L \times V \times K) \cap Z\big). $$
So the projection \ $p$ \ is semi-proper. Its image \ $p(Z)$ \ is then a closed analytic subset of \ $\Omega \times \Omega$ \ by Kuhlmann's theorem [K.64], [K.66]. But now the projection
$$ \pi : p(Z) \to \Omega $$
is \ $n-$equidimensional, thanks to iii), and has irreducible generic fibers on a normal basis \ $\Omega$. So its fibers (with generic multiplicity equal to $1$) define an analytic family of \ $n-$cycles of \ $X$ \ parametrized by \ $\Omega$. It is clearly \ $f-$ analytic because each fiber is irreducible{\footnote{But some multiplicities may occur.} and we have an holomorphic section because each \ $x $ \ lies in \ $G.x$.\\
To prove v) let us prove that the holomorphic map $\varphi : \Omega \to \mathcal{C}_{n}^{f}(\Omega)$ classifying the fibers of $p(Z)$ is semi-proper.  Fix a point $C \in \mathcal{C}_{n}^{f}(\Omega)$ and a point $x_{1}, \dots, x_{k} $ in each irreducible component of $\vert C\vert$. Let $W$ a relatively compact open neighbourhood of $\{x_{1}, \dots , x_{k}\}$ in $\Omega$ and let $\mathcal{W}$ be the open set in $\mathcal{C}_{n}^{f}(\Omega)$ of cycles such that each irreducible component meets $W$. Let  $C'$ be in $\mathcal{W} \cap \varphi(\Omega)$ we know that if $C' = \varphi(z)$ that  $\vert C'\vert = G.z$. So $G.z$ has to meet $W$ and we can choose $y $ in the compact set $ \bar W$ such that $C' =  \varphi(y)$  and this gives the semi-properness of $\varphi$. Now the semi-proper direct image theorem 2.3.2 of [B.15] implies that the image $Q$  of $\varphi$ is a finite dimensional complex space. \\
Then the holomorphic map $q : \Omega \to Q$ which is a quasi-proper equidimensional holomorphic quotient for the action $f$ on $\Omega$ as {\bf each} fiber of $q$ is set-theoretically  a $G-$orbit.$\hfill\blacksquare$\\

\subsection{The conditions ${\rm [H.1], [H.2]} $ and ${\rm [H.3]}.$}

Now we shall consider the following conditions on the action \ $f$.

\begin{itemize}
\item{\bf  There exists a $G-$invariant  dense open set $\Omega_{1}$  in $ X$  which admits a GF- holomorphic quotient}. \hfill ${\bf [H.1]}$
\end{itemize}

Recall that this means that there exists a $G-$invariant  geometrically f-flat holomorphic map $q : \Omega_{1} \to Q_{1}$ onto a reduced complex space $Q_{1}$  such that each fiber of $q$ over a point in $Q_{1}$ is set-theoretically an orbit in $\Omega_{1}$.\\


The following  stronger condition will be useful in the sequel:

\begin{itemize}
\item{\bf  There exists a $G-$invariant  dense  open set $\Omega$  in $ X$  which is   good for the action    $f$  (on $\Omega$)} .\hfill ${\bf [H.1str]}$
\end{itemize}

An immediate consequence of the proposition \ref{Good 2} is that the condition ${ \rm[H.1str]}$  below is sufficient to satisfy ${\rm [H.1}]$ by taking $\Omega_{1}$ as the set of normal points in $\Omega$ (which is open dense and $G-$invariant in $\Omega$). \\

Now assume [H.1] and  define  $\mathcal{R} : = \{ (x,y) \in \Omega_{1} \times \Omega_{1} \ / \  y \in G.x\} $. It is a closed analytic set in \ $\Omega_{1}\times \Omega_{1}$. We shall denote \ $\overline{\mathcal{R}}$ \ the closure of \ $\mathcal{R}$ \ in \ $X \times X$. Our second assumption will be :
\begin{itemize}
\item  {\bf The  subset $\overline{\mathcal{R}}$   is analytic  in  $X \times X$ and there exists an open dense subset $\Omega_{0} \subset \Omega_{1}$ such that for each $x \in \Omega_{0}$ we have} 
\begin{equation*}
{\bf \overline{G.x} = \overline{\mathcal{R}} \cap(\{x\}\times X)}. \qquad  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad  {\bf [H.2]}
\end{equation*}
\end{itemize}

Remark that the first projection $p_{1} : \overline{\mathcal{R}}\cap (\Omega_{0}\times X) \to \Omega_{0}$ is quasi-proper because we have a holomorphic section of this map (with irreducible fibers, thanks to $[H.2]$) which is given by $x \mapsto (x, x)$.\\
Assuming that $\Omega_{0}$ contains only normal points\footnote{This not restrictive, as we may always assume that $X \setminus \Omega_{0}$ contains the non normal points in $X$. We shall always assume that $\Omega_{0}$ is normal in the sequel, without any more comment.} in $X$, the equidimensionality and quasi-properness  on $\Omega_{0}$ of the projection of $\overline{\mathcal{R}}$ implies that there exists a holomorphic map 
$$\bar \varphi_{0} : \Omega_{0} \to \mathcal{C}_{n}^{f}(X)  $$
where the supports are  given by $x \mapsto \overline{G.x}$ and where the multiplicity is generically equal to $1$.\\
We shall denote \ $\Gamma \subset \Omega_{0} \times \mathcal{C}_n^f(X) $ \ the graph of the map \ $\bar \varphi_{0}$ \ and \ $\bar \Gamma$ \ its closure in \ $X \times \mathcal{C}_n^f(X) $. We shall denote \ $\theta : \bar \Gamma \to X$ \ the map induced by the first projection. Our last hypothesis is :
\begin{itemize}
\item {\bf The  map  $\theta : \bar\Gamma \to X$  is  proper} . \hfill ${\bf [H.3]}$
\end{itemize}

This hypothesis ${\rm [H.3]}$, assuming ${\rm [H.1]}$ and ${\rm [H.2]}$ is in fact equivalent to ask that the projection $\overline{\mathcal{R}} \to X$ is strongly quasi-proper: this is an immediate consequence of the  proposition 3.2.2 of [B.15].\\

The following proposition shows that these conditions ${\rm [H.1], [H.2]}$ and ${\rm [H.3]}$ are necessary for the existence of a SQP-meromorphic quotient for a completely holomorphic action of $G$ on $X$.

\begin{prop}\label{necessary}
Assuming that the completely holomorphic action $f : G \times X \to X$ of the connected complex Lie group $G$ on the irreducible complex space $X$ has a SQP-meromorphic quotient, then the conditions ${\rm [H.1], [H.2]}$ and ${\rm [H.3]}$ are satisfied.
\end{prop}

\parag{Proof} The conditions to be a SQP-meromorphic quotient gives an open set $\Omega_{1}$ which is dense, $G-$stable and which admits a GF holomorphic quotient for the action on $\Omega_{1}$. So ${\rm [H.1]}$ is clear.\\
Let $S$ be the graph of the equivalence relation given by $q$ on $\tilde{X}$. Then the proper direct image $(\tau\times \tau)(S)$ is $\overline{\mathcal{R}}$ and so the condition ${\rm [H.2]}$ is satisfied as soon as we can find an open and dense subset $\Omega_{0}$ in $\Omega_{1}$ such that for each $x \in \Omega_{0}$ we have the equality 
$$\overline{G.x} = \overline{\mathcal{R}} \cap(\{x\}\times X). $$
But this property is given by the condition ii) in the definition of a SQP-meromorphic quotient.\\
The composition of $q$ with the classifying map $Q \to \mathcal{C}_{n}^{f}(\tilde{X})$ for the fibers of $q$ gives a holomorphic map $\tilde{\psi} : \tilde{X} \to \mathcal{C}_{n}^{f}(\tilde{X})$. Composed with the direct image map, which is holomorphic (see [B.M] ch.IV ; the ``quasi-proper'' part of this result is easy, as $\tau$ is proper) $\tau_{*} :\mathcal{C}_{n}^{f}(\tilde{X}) \to \mathcal{C}_{n}^{f}(X)$, we obtain the fact that $\overline{\mathcal{R}}$ is  strongly quasi-proper on $X$ which is the condition  ${\rm [H.3]}$.$\hfill \blacksquare$
 

 
 \subsection{Existence of SQP-meromorphic quotient}
 
Now we shall prove that  conditions \ ${\rm [H.1], [H.2]}$ \  and \ ${\rm [H.3]}$ \ on a completely holomorphic action of a connected complex Lie group $G$  on an irreducible complex space $X$  are sufficient for the existence of a SQP meromorphic quotient. 
 
 \begin{thm}\label{mero. quot.}
Under the hypothesis \ ${\rm [H.1], [H.2]}$ \  and \ ${\rm [H.3]}$ \ there exists a proper  $G-$equivariant  modification \ $\tau : \tilde{X} \to X$ \ with center contained in \ $X \setminus \Omega_{0}$\footnote{The dense open subset $\Omega_{0} \subset \Omega_{1}$ is defined in the condition ${\rm [H.2]}$} and a   geometrically f-flat holomorphic map
$$ q :  \tilde{X} \to  Q$$
on a  reduced complex space, which give a strongly quasi-proper meromorphic quotient for the given \ $G-$action.
\end{thm}

Of course the complex space \ $\tilde{X}$ \ is the topological space \ $\bar \Gamma$ \ with a structure of a reduced  complex space such that the projection on \ $X$ \ is a proper modification. Then the space \ $Q$ \ is the  image of \ $\tilde{X}$ \ in \ $ \mathcal{C}_{n}^{f}(X)$. So we need some semi-proper direct image theorem for such a map to prove this result. Such a result is the content of  the theorem 2.3.2 of [B.15]

\parag{Proof}  The first remark is that the hypothesis \ $[H. 3]$ \ says that the projection \ $p : \bar \Gamma \to X$ \ is a proper topological modification of \ $X$. But to apply directly the part ii)  of the theorem 2.3.6 of [B.13] to the projection \ $p_{1} : \overline{\mathcal{R}} \to X$ \ we need quasi-properness of this map. This is given by the proposition 3.2.2  of [B.15] as we have the condition $[H.3]$.\\
Then we obtain a proper (holomorphic) modification with center in \ $\Sigma \subset X \setminus \Omega_{0}$,  $\tau :  \tilde{X} \to X$,  and a \ $f-$analytic family of cycles in \ $X$ \   parametrized by \ $\tilde{X}$ \ extending the family \ $(\overline{G.x})_{x \in \Omega_{0}}$, corresponding to a ``holomorphic'' map extending \ $\bar \varphi_{0}$ :
$$\tilde{\varphi} : \tilde{X} \to \mathcal{C}_{n}^{f}(X) .$$
Now let us prove that this map \ $\tilde{\varphi}$ \ is quasi-proper\footnote{This makes sense as the fibers are closed analytic subsets of  \ $\tilde{X}$.}. This will allow us to apply the theorem 2.3.2  of {\it loc. cit.} \ and to define the reduced complex space \ $Q $ as the image $ \tilde{\varphi}(\tilde{X})$. Then it will be easy to check that the map \ $\tilde{\varphi} : \tilde{X} \to Q$ \ is a strongly quasi-proper  meromorphic quotient for the \ $G-$action we consider.\\
If \ $C_{0}$ \ is in \ $\mathcal{C}_{n}^{f}(X)$ \ and  is not the empty cycle, choose a relatively compact open set \ $W$ \ in \ $X$ \ such that any irreducible component of \ $\vert C_{0}\vert$ \ meets \ $W$. Then let \ $\mathcal{W}$ \ be the open set in \ $\mathcal{C}_{n}^{f}(X)$ \ defined by the condition that any irreducible component of \ $C$ \ meets \ $W$ \ for \ $C \in \mathcal{W}$. Then we shall prove that there exists a compact set \ $K$ \ in \ $\tilde{X}$ \ such that any irreducible component of the fiber of $\tilde{\varphi}$ at a point in $\mathcal{W} \cap \tilde{\varphi}(\tilde{X})$ meets $K$.  Let \ $K : = \tau^{-1}(\bar W)$. If \ $(y, C)$ \ is in \ $\tilde{X}$\footnote{Recall that, as a topological space, \ $\tilde{X} = \bar \Gamma$.} with \ $C \in  \mathcal{W}$, each irreducible component of \ $C$ \ meets \ $W$. But the fiber of \ $\tilde{\varphi}$ \ at \ $C$ \ is equal to \ $\vert C\vert$, and the quasi-properness is proved. $\hfill \blacksquare$\\
 


\section{Application.}


\subsection{The sub-analytic lemma.}

We shall use the following lemma (see  [G-M-O]) in our application.

\begin{lemma}\label{Bishop}
Let $M$ be a reduced complex space and $Y \subset M$ a closed analytic subset with no interior point in $M$. Let $R$   be a closed (complex)  analytic subset in  $ M \setminus Y$ such that $\bar R$ is a sub-analytic set in $M$. Then $\bar R$ is a (complex) analytic subset in $M$.
\end{lemma}

This important lemma is a consequence of Bishop's theorem (see [Bi.64]) and of a classical result on sub-analytic subsets (see [G-M-O]).

\subsection{Proof of the theorem \ref{G = K.B}.}

Now we shall assume that $G$ is a connected complex Lie group such that $G = K.B$ where $B$ is a closed  complex connected subgroup of $G$ and $K$ a compact real subgroup of $G$.


The first condition ${\rm [H.1str]}$ for the $G-$action is given by the following lemma :

\begin{lemma}\label{good B to good G}
In the situation of the  theorem \ref{G = K.B}, assume that we have a  \ $G-$invariant  open set $\Omega$ which is a good open set  for the $B-$action, then $\Omega$ is a good open set for the $G-$action.
\end{lemma}

\parag{Proof} Consider a point $x \in \Omega$ and a compact set $M$ in $\Omega$. Then there exists a neighbourhood $V$ of $x$ in $\Omega$ and a compact set $L$ in $B$ such that $b.y \in M$ for some $y \in V$ and some $b \in B$  implies that we can find $\beta \in L$ with $b.y = \beta.y$. Now assume that $M$ is $K-$invariant (here we use the $G-$invariance of $\Omega$) and that $g.y$ is in $M$ for some $g \in G$ \ and some $y \in V$. Write $g= k.b$ for some $k \in K$ and $b \in B$. Then $b.y$ is again in $M$ so we can find $\beta \in L$ with $\beta.y = b.y$ and then $ g.y = k.\beta.y$ with $k.\beta \in K.L$ which is a compact set in $G$.
So $x$ is a good point for the $G-$action on $\Omega$. $\hfill \blacksquare$\\

The corollary of the next lemma will give the first part of ${\rm [H.2]}$ for the $G-$action assuming that we have a  $G-$invariant  dense  good Zariski open set $\Omega$ for the $B-$action with the condition ${\rm [H.2]}$ for the $B-$action. In order to use the sub-analytic lemma \ref{Bishop} in this situation we shall need the following lemma.

\begin{lemma}\label{H.2 i)}
Let $\Omega$ be an open  $G-$invariant good set for the $B-$action, and then also good for the $G-$action thanks to the previous lemma.\\
 Let $\chi : K\times X\times X \to X\times X$ the map given by  $(k,x,y) \mapsto (k.x,y)$ and let \\
  $p : K\times X\times X \to X\times X$ be the natural  projection. Then we have
$$ p(\chi^{-1}(\overline{\mathcal{R}_{B}})) = \overline{\mathcal{R}_{G}} $$
where we define
 $$\mathcal{R}_{B} := \{(x,y) \in \Omega\times \Omega \ / \  B.x = B.y\} \quad {\rm and} \quad \mathcal{R}_{G} := \{(x,y) \in \Omega\times \Omega \ / \  G.x = G.y \} ,$$
 and where the closures are taken in $X\times X$.
 \end{lemma}
 
 \parag{Proof} Remark first that
  $$ p(\chi^{-1}(\mathcal{R}_{B})) = \{(x,y) \in \Omega\times \Omega \ / \  \exists k \in K \quad� B.k.x = B.y\}.$$
So $(x, y) \in p(\chi^{-1}(\mathcal{R}_{B}))$ implies   $y \in B.k.x \subset G.x$ and also $k.x \in B.y$;  we conclude that $x$ is in $K.B.y = G.y$.
 This gives the inclusion $ p(\chi^{-1}(\mathcal{R}_{B}))  \subset \mathcal{R}_{G}$. The opposite inclusion is easy because $G.x = G.y$ implies that $x \in K.B.y$ so there exists $k \in K$ such that $k.x \in B.y$.\\
 Now the map $\chi$ and  $p$ are proper, so we obtain the inclusion $ p(\chi^{-1}(\overline{\mathcal{R}_{B}})) \subset \overline{\mathcal{R}_{G}}$.\\
 Let $(x,y) := lim_{\nu\to \infty} (x_{\nu}, y_{\nu})$ where $(x_{\nu}, y_{\nu})$ is in $\mathcal{R}_{G}$ for each $\nu \in \mathbb{N}$. Then for each $\nu$  the exists $k_{\nu} \in K, b_{\nu} \in B$ such that $x_{\nu} = k_{\nu}.b_{\nu}.y_{\nu}$; then $(k_{\nu}^{-1}, x_{\nu}, y_{\nu}) \in \mathcal{R}_{B}$. Up to pass to a subsequence we may assume that  the sequence $(k_{\nu})$ converges to some $k \in K$. As $(k_{\nu}^{-1},x_{\nu},y_{\nu})$ is in $\chi^{-1}(\mathcal{R}_{B})$ for each $\nu$, the point $(k^{-1},x,y)$ is in 
  $$\overline{\chi^{-1}(\mathcal{R}_{B})} = \chi^{-1}(\overline{\mathcal{R}_{B}})$$
  and so $(x,y)$ is in $p(\chi^{-1}(\overline{\mathcal{R}_{B}})) $ proving the opposite inclusion.$\hfill \blacksquare$\\
 
 \begin{cor}\label{H.2 ii)}
 In the situation of the previous lemma, assume that $X \setminus \Omega$ is a (complex) analytic subset;  then  if the subset $\overline{\mathcal{R}_{B}}$ is (complex)  analytic in $X\times X$, the subset $\overline{\mathcal{R}_{G}}$ is also  a (complex) analytic subset of $X \times X$.
 \end{cor}
 
 \parag{Proof} Note first that the maps $\chi$ and $p$ are real analytic, so assuming that $\overline{\mathcal{R}_{B}}$ is analytic implies that $ p(\chi^{-1}(\overline{\mathcal{R}_{B}}))$ is sub-analytic. Then, as we know that $\mathcal{R}_{G}$ is an irreducible locally closed complex analytic subset, the conclusion follows from the lemma \ref{Bishop}, as our assumption that $\Omega$ is a Zariski (dense) open set in $X$ implies that $\Omega\times \Omega$ is Zariski open (and dense) in $X\times X$.$\hfill \blacksquare$\\
 
A first step to prove the quasi-properness of $\overline{\mathcal{R}_{G}}$ is our next result.
 
 \begin{lemma}\label{adherence orbites}
 Let assume that the $B-$action $f$ on $X$ satisfies ${\rm [H.1]}$ and ${\rm [H.2]}$. Let $\Omega_{0} \subset \Omega_{1}$ be an open set on which the fiber at any $x \in \Omega_{0}$  of $\overline{\mathcal{R}_{B}}$ is equal to $\overline{B.x}$ (with some multiplicity). Then the fiber at any $x \in \Omega_{0}$ of $\overline{\mathcal{R}_{G}}$ is equal to $\overline{G.x}$ (with some multiplicity).
 \end{lemma}
 
 \parag{Proof} As we know that the map  $x \mapsto \overline{B.x}$, with generic multiplicity $1$, extends to is a f-analytic family of cycles of $X$ parametrized by $\Omega_{0}$, for each sequence $(x_{\nu})_{\nu \in \mathbb{N}}$ of points in $\Omega_{0}$ converging to a point $x \in \Omega_{0}$ we have (with suitable multiplicity)  $\overline{B.x} = \lim_{\nu \to \infty} \overline{B.x_{\nu}}$ in the topology of $\mathcal{C}_{d}^{f}(X)$. We shall show that this implies, also with suitable multiplicity, the equality $\overline{G.x} = \lim_{\nu \to \infty} \overline{G.x_{\nu}}$ in the topology of $\mathcal{C}_{n}^{f}(X)$. As we have $G = K.B$ with $K$ compact, for any $y \in X$ we have $\overline{G.y} = K.\overline{B.y}$. So the inclusion of $\lim_{\nu \to \infty} \overline{G.x_{\nu}}$ in the fiber at $x$ of $\overline{\mathcal{R}_{G}}$ is clear. The point is to prove the opposite inclusion. Let $y$ be a point in the fiber at $x \in \Omega_{0}$ of $\overline{\mathcal{R}_{G}}$. It is a limit of a sequence $y_{\nu} \in G.x_{\nu}$ where $x_{\nu} \in \Omega_{0}$ converges to $x$. Write $y_{\nu} = k_{\nu}.b_{\nu}.x_{\nu}$ with $k_{\nu} \in K$ and $b_{\nu} \in B$. Up to pass to a subsequence, we may assume that the sequence  $(k_{\nu})$ converges to a point $k \in K$. So we have $k^{-1}.y$ which is the limit of the sequence $b_{\nu}.x_{\nu}$. We obtain that $k^{-1}.y$ is in the limit of $\overline{B.x_{\nu}}$ which has support equal to $\overline{B.x}$. Then $y$ is in $K.\overline{B.x} = \overline{G.x}$, concluding the proof.$\hfill \blacksquare$\\
 
 \parag{Proof of the theorem \ref{G = K.B}} The fact that the $G-$action satisfies ${\rm [H.1str]}$ is consequence of lemma \ref{good B to good G}. The analyticity of $\overline{\mathcal{R}_{G}}$ in $X \times X$ is proved at corollary \ref{H.2 ii)}. The lemma \ref{adherence orbites} gives a dense open set $\Omega_{0}$ where the fiber of the projection $p_{1}$ of  $\overline{\mathcal{R}_{G}}$ at each point $x \in \Omega_{0}$ is equal to $\overline{G.x}$ as a set.
 This implies the quasi-properness of $p_{1}$ over $\Omega_{0}$, because $x$ is in $G.x$ and $G$ is connected;  assuming (which is not restrictive) that $\Omega_{0}$ is normal, we obtain a holomorphic map 
 $$ \Phi : \Omega_{0} \longrightarrow \mathcal{C}_{n}^{f}(X)$$
 where the support of $\Phi(x)$ is equal to $\overline{G.x}$  for each $x \in \Omega_{0}$ and  with generic multiplicity equal to $1$. This complete the proof of ${\rm [H.2]}$ for the $G-$action.\\
 Thanks to proposition 3.2.2 of [B.15],  to prove ${\rm [H.3]}$  it is enough to show that the closure of the graph $\Gamma_{G}$ of $\Phi$ in $X\times \mathcal{C}_{n}^{f}(X)$ is proper on $X$.\\
  The projection $p_{B}: \overline{\mathcal{R}}_{B} \to X$ is strongly quasi-proper so, for $V$ compact in $X$, the limits of cycles (with convenient multiplicity) $\overline{B.x}$ for $x \in V \cap \Omega_{0}$ stay in a compact subset $S_{0}$ in $\mathcal{C}_{d}^{f}(X)$. Then let $S := K.S_{0}$; this is again a compact subset in $\mathcal{C}_{d}^{f}(X)$. But the subset $T$ of  limits of the generic fibers of the projection $p_{G}: \overline{\mathcal{R}}_{G} \to X$ for $x \in V$ is a compact set of $\mathcal{C}_{n}^{loc}(X)$ thanks to [B.13] theorem 2.3.6 i).
 For $x \in V \cap \Omega_{0}$ we have $p_{G}^{-1}(x) = \overline{G.x}$, and let $T'$ be the open dense set in $T$ described by the cycles $\bar \varphi_{0}(x), x \in V \cap \Omega_{0}$. As we know that for $x$ in $\Omega_{0}$ we have $\vert \bar \varphi_{0}(x) \vert = \overline{G.x}$ we know that each of them is an union of $d-$cycles in $S$, thanks to the lemma \ref{K-action} below. The proposition \ref{non inf. br.} implies that the set of  limits of $\overline{G.x_{\nu}}$ for $x_{\nu} \in V \cap \Omega_{0}$ is a compact set in $\mathcal{C}^{f}_{n}(X)$, proving [H.3].$\hfill \blacksquare$
 
 



\begin{lemma}\label{K-action}
Let $K$ be a compact Lie group with a holomorphic action on a reduced complex space $M$. Let $X$ be an irreducible complex space and let  $\varphi : X \to \mathcal{C}^{f}_{d}(M)$ and $\psi : X \to  \mathcal{C}^{f}_{n}(M)$ two holomorphic maps. Assume that for each $x$ in  a dense open subset $\Omega_{0}$ in $X$ we have
\begin{equation*}
 \vert \psi(x)\vert = \cup_{k\in K} \ k.\vert \varphi(x)\vert = K.\vert \varphi(x)\vert  \tag{@@@}
 \end{equation*}
where $K$ acts holomorphically on $\mathcal{C}^{f}_{d}(M)$ via the action of $K$ on $M$.\\
Then the relation $(@@@)$ holds for each $x \in X$.
\end{lemma}

\parag{proof} Let $x \in X$ and choose a sequence $(x_{\nu})_{\nu \geq 0}$ in $\Omega_{0}$ converging to $x$ and let $y$ a point in $\vert \psi(x)\vert$. Then we can choose a sequence $(y_{\nu})_{\nu \geq 0}$ of points in $\vert \psi(x_{\nu})\vert$ converging to $y$. As $x_{\nu}$ is in $\Omega_{0}$ we can write $ y_{\nu} = k_{\nu}.z_{\nu}$ with $z_{\nu}\in \vert \varphi(x_{\nu})\vert$ and $k_{\nu} \in K$. Up to pass to a subsequence we can assume that the sequence $(k_{\nu})$ converges to some $k \in K$. So the sequence $(z_{\nu})$ converges to $k^{-1}.y$ which is in $\vert \varphi(x)\vert$. This gives that $y$ is in $k.\vert \varphi(x)\vert$ and we have proved the inclusion $\vert \psi(x)\vert \subset K.\vert \varphi(x)\vert$.\\
Conversely, if $z$ is in $\vert \varphi(x)\vert$ and $k$ is in $K$, write again $z = \lim z_{\nu}$ with $z_{\nu} \in \vert \varphi(x_{\nu})\vert$ where the sequence $(x_{\nu})$ of points in $\Omega_{0}$ converges to $x$. We have $k.z_{\nu}\in \vert \psi(x_{\nu})\vert$, and then $k.z$ is in $\vert\psi(x)\vert$ conluding the proof.$\hfill \blacksquare$\\


\newpage

 \section{Bibliography}
 
 \begin{itemize}
 
  \item{[B.75]} Barlet, D. {\it Espace analytique r\'{e}duit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie}, Fonctions de plusieurs variables complexes, II (S\'{e}minaire Fran\c cois Norguet, 1974-1975), pp.1-158. Lecture Notes in Math., Vol. 482, Springer, Berlin, 1975

 
 \item{[B.78]} Barlet, D. \textit{Majoration du volume des fibres g\'en\'eriques et forme g\'eom\'{e}trique du th\'eor\`{e}me d'aplatissement}, Seminaire Lelong-Skoda 78-79, Lect. Notes in Math. 822, p.1-17, Springer-Verlag, Berlin.
 
  \item{[B.08]} Barlet, D. \textit{Reparam\'etrisation universelle de familles f-analytiques de cycles et f-aplatissement g\'eom\'etrique} Comment. Math. Helv. 83 (2008),\\ p. 869-888.

 
 \item{[B.13]} Barlet, D. \ {\it Quasi-proper meromorphic equivalence relations}, Math. Z. (2013), vol. 273, p. 461-484.
 
 \item�[B.15] Barlet, D. \ {\it Strongly quasi-proper maps and f-flattning theorem},\\ arXiv.1504.01579v1 [math. CV].
 
 \item{[Bi.64]} Bishop, E. {Condition for the analyticity of certain sets}, Michigan Math. J. 11 (1964), p.289-304.
 
 \item{[B-M]} Barlet, D. et Magn\'usson,J. {\it Cycles analytiques complexes I : Th\'eor\`{e}mes de pr\'eparation des cycles}, Cours Sp\'ecialis\'es 22, Soci\'et\'e math\'ematique de France, (2014).
 
  \item{[K.64]} Kuhlmann, N. \textit{{\"U}ber holomorphe Abbildungen komplexer R{\"a}ume} Archiv der Math. 15 (1964), p.81-90.
 
 \item{[K.66]} Kuhlmann, N. \textit{Bemerkungen {\"u}ber  holomorphe Abbildungen komplexer R{\"a}ume} Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Ged{\"a}achtnisfeier K. Weierstrass (1966), p.495-522.
 
 \item{[G-M-O]}  B. Gilligan, C. Miebach and K. Oeljeklaus {\it Homoheneous K{\"a}hler and Halmitonian manifolds}, Math. Ann. (2011), p.889-901.
 
  \item{[M.00]} Mathieu, D. \textit{Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations}, Ann. Inst. Fourier (Grenoble) t. 50, fasc.4 (2000) p.1155-1189.
  
 \item{[R.57]} Remmert, R. \textit{Holomorphe und meromorphe Abbildungen komplexer R{\"a}ume}, Math. Ann. 133 (1957), p.328-370.


 
 \end{itemize}

\end{document}

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\newpage

\begin{prop}\label{corresp.}
In the situation of the previous lemma assume moreover that
\begin{itemize}
\item The map $\varphi$ is semi-proper.
\item For $x, y$ in $X$ if we have $\vert \psi(x)\vert = \vert \psi(y)\vert$ then $\psi(x) = \psi(y)$.
\end{itemize}
Then the map $(\varphi, \psi)$ is semi-proper and the projection of the image $\Gamma := (\varphi,\psi)(X)$ on $\varphi(X)$  is proper.
\end{prop}

\parag{Proof} Fix $x \in X$ and consider $z := \varphi(x)$;  the semi-properness of $\varphi$ gives a open neighbourhood $V$ of $z$ in $\mathcal{C}^{f}_{d}(M)$ and a compact set $L$ in $X$ such that $\varphi(X) \cap V = \varphi(L) \cap V$. Now consider for $w := \psi(x)$. We shall show that $$(\varphi, \psi)(L) \cap (V\times \mathcal{C}^{f}_{n}(M)) = (\varphi, \psi)(X) \cap (V\times \mathcal{C}^{f}_{n}(M)).$$
 If $(\varphi, \psi)(y)$ is in $(V\times \mathcal{C}^{f}_{n}(M))$, then there exists $y' \in L$ such that $\varphi(y') = \varphi(y)$. But, thanks to the lemma \ref{K-action}, we have
 $$ \vert \psi(y)\vert = K.\vert \varphi(y)\vert = K.\vert \varphi(y')\vert = \vert \psi(y')\vert $$
 and so $\psi(y') = \psi(y)$. Then we have $y' \in L$ such that $(\varphi, \psi)(y') = (\varphi, \psi)(y)$, and the map $(\varphi, \psi)$ is semi-proper.\\
 As we know now that $\Gamma$ and $\varphi(X)$ are locally compact, to prove the properness of the projection it is enough to show that for any compact subset $A$ in $\varphi(X)$ the subset $\Gamma \cap (A \times (\mathcal{C}^{f}_{n}(M))$ is contained in $A \times L$ where $L$ is a compact set in $\mathcal{C}^{f}_{n}(M)$. The semi-properness of $\varphi$ gives us, a compact set $\Lambda$ in $X$ such that $\varphi(\Lambda)\cap A = A$. Then consider $(y, z) \in \Gamma$ with $y \in A$. There exists $x \in X$ with $y = \varphi(x)$ and $\psi(x) = z$ and there exists also $x' \in \Lambda$  such that $y = \varphi(x')$. As we have $ K.\vert \varphi(x)\vert = \vert\psi(x)\vert = \vert \varphi(x')\vert = \vert \psi(x')\vert$, we have $\psi(x) = \psi(x')$ from our assumption. So we have $z \in L := \psi(\Lambda)$ which is a compact subset of $\mathcal{C}^{f}_{n}(M)$. So the projection $\Gamma \to \varphi(X)$ is proper.$\hfill \blacksquare$\\
 
 Remark that, thanks to the semi-proper direct image theorem (see [B.15]), $\Gamma$ is a (locally finite dimensional) irreducible complex space and the projection is an bijective proper  map. So $\Gamma$ is the graph of a continuous meromorphic map $f :\varphi(X) \to \mathcal{C}^{f}_{n}(M)$. Assuming that $\psi$ is also semi-proper, then $\psi(X)$ is also an irreducible (locally finite dimensional) complex space and f is a meromorphic continuous map between $\varphi(X)$ and $\psi(X)$.
 
 \begin{cor}\label{two quotients}
 In the situation of the theorem \ref{G = K.B}, we have a natural meromorphic map $f: Q_{B} --\to Q_{G}$ which is proper (and $K-$invariant ?).
 \end{cor}
 
 \parag{Hors texte} Le point � eclaircir est le fait que $\psi(x)$ et $\psi(x')$ ont m�me support seulement si ces deux cycles sont �gaux. La pr�sence d'un ouvert dense sur lequel on a $\varphi(x)$ et $\psi(x)$ qui sont r�duits et v�rifient $K.\varphi(x) = \psi(x)$.


 \subsection{The condition $[H.3bis]$ and infinite breaking.}
 To comfort the reader that it is possible to use our main result, we shall give an other formulation of this hypothesis, which seems more easy to check on concrete examples and which is equivalent to the hypothesis \ $[H 3]$ \ in our situation above, assuming  \ $[H 1]$ \ and \ $[H 2]$ :
\begin{itemize}
\item For any sequence \ $(x_{m})_{m \geq 0}$ \ of points in \ $\Omega_{0}$ \ converging to a point \ $x \in X$ \ and such that \ $\overline{G.x_{m}}$ \ has a limit in \ $\mathcal{C}_{n}^{loc}(X)$ \ then this limit has only finitely many irreducible components.$\hfill{\bf (H 3bis)}$
\end{itemize}

We shall say that a sequence $(C_{m})_{m \in \mathbb{N}}$ of finite type cycles converging to a cycle $C$ in $\mathcal{C}_{n}^{loc}(X)$ presents an {\bf infinite breaking} when the cycle $C$ has infinitely many irreducible components. This means that $C$ is not in $\mathcal{C}_{n}^{f}(X)$.\\

To understand that the condition $[H.3bis]$ may be often manageable  recall that we have :
\begin{enumerate}[i)]
\item  The closure \ $\widehat{\Gamma}$ \ of \ $\Gamma$ \ in \ $X\times \mathcal{C}_{n}^{loc}(X)$ \ {\bf is always proper over \ $X$} \  thanks to the analyticity of $\overline{\mathcal{R}}$ which is given by the first part of \ $[H 2]$ \ (see [B.13]  th. 2.3.6 ).
\item  The empty cycle cannot be a limit of such a sequence \ $\overline{G.x_{m}}$ \ because if \ $K$ \ is a compact neighbourhood of \ $x \in X$ \ any \ $\overline{G.x_{m}}$ \ meets \ $K$ \ for \ $m \gg 1$. So there is no {\it escape at infinity}\footnote{See [B.13] for this notion.} in such a situation.
\item In case of an infinite breaking, either the limit cycle has a finite number of connected components and  the singular set of such a limit has dimension \ $\leq n-1$ \ and is \ $G-$stable with infinitely many irreducible components of dimension at most $n-1$, or the limit cycle contains an infinite number of closure of (disjoint)  orbits.
\end{enumerate}
So the phenomenon of ``infinite breaking'' may be detected at several levels.\\
We give an useful consequence of the fact  that infinite breaking does not appear.

\begin{prop}\label{f-conv.}
Let \ $M$ \ be a reduced complex space and consider a sequence of finite type \ $n-$cycles \ $(C_{m})_{m \geq 0}$ \ in \ $M$ \ with the following properties :
\begin{enumerate}[i)]
\item there exists a compact set \ $K$ \ in \ $M$ \ such that each irreducible component of each \ $C_{m}$ \ meets \ $K$.
\item The sequence \ $(C_{m})_{m \geq 0}$ \ converges for the topology of \ $\mathcal{C}_{n}^{loc}(M)$ \ to a \ $n-$cycle \ $C$.
\item The cycle \ $C$ \ is in \ $\mathcal{C}_{n}^{f}(M)$.
\end{enumerate}
Then the sequence \ $(C_{m})_{m \geq 0}$ \ converges to \ $C$ \ in the topology of \ $\mathcal{C}_{n}^{f}(M)$.
\end{prop}

Note that if, for each $m \geq 0$ the subset  $\vert C_{m}\vert$ is irreducible and contains a point $x_{m}$ such that the sequence $(x_{m})_{m\geq 0}$ converges in $M$, then the condition i) is satisfied. Moreover when we have a holomorphic action of $G$ satisfying [H.1] and [H.2] and  $C_{m} = \overline{G.x_{m}}$ where the sequence $(x_{m})_{m\geq 0}$ of points in $\Omega_{0}$ converges, the condition i) and  ii) will always be satisfied (see theorem 2.3.6 of [B.13]).

\parag{Proof} We begin by proving the case where each $C_{m}$ is irreducible. Note first that \ $C$ \ is not the empty \ $n-$cycle because, up to pass to a subsequence, we may assume that for each \ $m \geq 0$ \ we can choose a point in \ $x_{m} \in C_{m} \cap K$ \ in order that the sequence \ $(x_{m})_{m \geq 0}$ \ converges to a point \ $x \in K$. Then we have \ $x \in \vert C\vert$.\\
Let \ $W$ be an open set in \ $M$ \ which meets each irreducible component of \ $\vert C\vert$, and let \ $E :=(U, B, j)$ \ be a scale on \ $W$ \ adapted to \ $C$ \ and such that \ $\deg_{E}(C) \geq 1$. This is possible because \ $C$ \ has at least one irreducible component and such an irreducible component meets \ $\Omega$. So for \ $m \geq m(E)$ \ the scale \ $E$ \ will be adapted to \ $C_{m} $ \ and we shall have \ $\deg_{E}(C_{m}) = \deg_{E}(C) \geq 1$. This shows that for \ $m \geq m(E)$ \ the compact  set \ $j^{-1}(\bar U\times \bar B)$ \ meets the unique  irreducible component of \ $C_{m}$. So any subsequence of the sequence \ $(C_{m})_{m \geq 0}$ \ has a subsequence which converges to \ $C$ \ in the sense of \ $\mathcal{C}_{n}^{f}(M)$. This is enough to conclude the proof in this case.\\
Consider now the general case. For any choice of a sequence \ $(\Gamma_{m})_{m \geq 0}$ \ where \ $\Gamma_{m}$ \ is an irreducible component of \ $C_{m}$, it lies in a relatively compact subset of \ $\mathcal{C}_{n}^{loc}(M)\setminus \{\emptyset\}$. Up to pass to a subsequence we may assume that \ $(\Gamma_{m})$ \ converges to a non empty cycle \ $\Gamma \leq C$ \ in the topology of \ $\mathcal{C}_{n}^{loc}(M)$. Then \ $\Gamma$ \ is in \ $\mathcal{C}_{n}^{f}(M)$. Now take any open set \ $W \subset M$ \ such that any irreducible component of \ $C$ \ meets \ $W$. If there is infinitely many \ $\Gamma_{m}$ \ such that \ $\Gamma_{m}$ \ does not meets \ $W$, the reasonning above shows, up to pass to a subsequence, that we may suppose that the sequence \ $\Gamma_{m}$ \ converges  in the topology of \ $\mathcal{C}_{n}^{loc}(M)$ \ to a {\bf non empty} cycle \ $\Gamma \leq C$. But any irreducible component of \ $\Gamma$ \ has to meet \ $W$ \ giving a contradiction as \ $\Gamma_{m}\cap W = \emptyset$ \ by construction.
So, for \ $m \gg 1$ \ any irreducible component of \ $C_{m}$ \ meets \ $W$ \ and the sequence \ $C_{m}$ \ converges to \ $C$ \ in the topology of \ $\mathcal{C}_{n}^{f}(M)$.$\hfill \blacksquare$\\


\begin{cor}\label{compact}
Let \ $M$ \ be a reduced complexe space and \ $A \subset \mathcal{C}_{n}^{f}(M) \setminus \{\emptyset\}$. Assume that the following conditions are fullfilled :
\begin{enumerate}[i)]
\item The set \ $A$  is compact in \ $\mathcal{C}_{n}^{loc}(M)$. 
\item There exists a compact set \ $K \subset M$ \ such for any irreducible component \ $\Gamma$ \ of some  \ $C \in A$ \ the intersection \ $\Gamma \cap K$ \ is not empty.
\end{enumerate}
Then  \ $A$ \ is  compact in \ $\mathcal{C}_{n}^{f}(M)$.
\end{cor}

Note that conversely any compact subset in $\mathcal{C}_{n}^{f}(M) \setminus \{\emptyset\}$ is compact in $\mathcal{C}_{n}^{loc}(M)$ and satisfies $(@)$ : choose for each $C \in A$ an open set $\mathcal{U}_{C}\cap \Omega(W_{C})$ containig $C$. As $A$ is compact we can extract a finite subcovering given by $C_{1}, ..., C_{N}$ and take $K := \overline{\cup_{j=1}^{N} \ W_{j}}$.


\parag{Proof} Note that any limit of a sequence in \ $A$ \ for the topology of \ $\mathcal{C}_{n}^{loc}(M)$ \ is not the empty cycle, as it contains a point in \ $K$.  Now for any sequence in \ $A$ \ we may find a sub-sequence converging for the topology of \ $\mathcal{C}_{n}^{loc}(M)$ \  to a cycle in \ $\mathcal{C}_{n}^{f}(M)$. So, applying the proposition \ref{f-conv.}, we conclude that such a subsequence converges for the topology of \ $\mathcal{C}_{n}^{f}(M)$.$\hfill \blacksquare$

\parag{Remark} For the general comparison between compactness in the space \ $\mathcal{C}_{n}^{f}(M)$ \ with its natural topology and the topology induced by \ $\mathcal{C}_{n}^{loc}(M)$ \ the first main trouble is the phenomenon of {\bf``escape at infinity''} which corresponds to the fact that in the limit process an irreducible component goes to infinity, that is to say that an irreducible component goes out of any given compact set.\\
Here this first ``problem'' is avoided by the assumption that any  irreducible component of any cycle  meets a fixed compact set. So no component can ``escape at infinity''. \\
But another phenomenon appears here (more visible because the first problem disappears) : it is the {\bf  ``infinite breaking''}, corresponding to the fact that  the limit of irreducible cycles may be a cycle with infinitely many irreducible components. It is very easy to see  that this phenomenon may appear already in the case of smooth hypersurfaces in \ $\C^{n}$ \ (see below). It would be very interesting to have a simple  topological sufficient condition  to avoid such a phenomenon. For instance let me ask the following question :
\parag{Question} Let \ $f : M \to D $ \ a holomorphic map of an irreducible normal complex space (a connected complex manifold ?) such that for any \ $z \not= 0$ \ the fiber \ $f^{-1}(z)$ \ is a complex manifold with an uniform bound\footnote{For instance an uniform bound for the dimension of the vector spaces  $H^{1}(f^{-1}(z), \C)$, when $ 0 < \vert z\vert \ll 1$ ?} for the cohomology of the fibers of $f$  for $z$ near enough to  $0$. Then is  the number of irreducible components of \ $f^{-1}(0)$ \ finite ? $\hfill \square$

\parag{An example of infinite breaking}

Let \ $ f : \C^{2} \to \C $ \ the holomorphic function given by \ $f(x,y) = x.\sin(y)$. Then the critical set of \ $f$ \ is the discrete sequence \ $(0,k.\pi), k \in \mathbb{Z}$. The only critical value is \ $0$. For any \ $\varepsilon \not= 0$ \ the smooth curve \ $\{ f = \varepsilon \}$ \ is isomorphic to \ $\C \setminus \pi.\mathbb{Z}$. So its first cohomology group has  infinite rank.The \ $0-$fiber has infinitely many irreducible components : $\{y = k.\pi\}, k \in \mathbb{Z}$ \ and also $\{x = 0\}$. \\

To conclude this paragraph we shall prove the following lemma.

\begin{lemma}\label{H3bis}
Under the hypotheses \ $[H1]$ \ and \ $[H 2]$ \ the hypothesis \ $[H 3bis]$ \ is equivalent to\ $[H 3]$.
\end{lemma}
\parag{Proof} As [H.3] implies clearly [H.3bis], assume that [H.3bis] is satisfied (with also [H.1] and [H.2]). Let \ $x_{0} $ \ be a point in \ $ Y : = X \setminus \Omega_{0}$ \ and choose a compact neighbourhood \ $K$ \ of \ $x_{0}$ \ in \ $X$. Then define
$$ A : = \{ \overline{G.x}, x \in \Omega_{0} \cap K \} \subset \mathcal{C}_{n}^{f}(X) .$$
Then  the point i) of the theorem 2.3.6 of [B.13]  implies that \ $A$ \ is relatively compact in \ $\mathcal{C}_{n}^{loc}(X)$, thanks to [B.78] and Bishops'theorem. The hypothesis \ $[H 3bis]$ \ implies that \ $\bar A$ \ is in fact a subset of \ $ \mathcal{C}_{n}^{f}(X)$. So the condition iii) of the corollary \ref{compact} is satified. The condition i) of this corollary is obviously satisfied. But the condition ii) is also fulfilled as any element in \ $A$ \ is irreducible and meets \ $K$. So the corollary \ref{compact} allows to conclude that \ $\bar A$ \ is compact in \ $\mathcal{C}_{n}^{f}(X)$. Now the properness of the projection \ $\bar \Gamma \to X$ \ follows because \ $X$ \ is locally compact. $\hfill \blacksquare$\\




