\magnification=1200
\input broccod.tex
\input ams.tex
\vsize=19.5cm
\parindent=0.5cm

\def\nb{{\Bbb N}}
\def\zb{{\Bbb Z}}
\def\qb{{\Bbb Q}}
\def\rb{{\Bbb R}}
\def\cb{{\Bbb C}}
\def\pb{{\Bbb P}}

\def\fr{{\cal F}}
\def\hr{{\cal H}}
\def\ir{{\cal I}}
\def\oc{{\cal O}}

\def\mmr{{\goth m}}
\def\ov{\overline}
\def\wh{\widehat}
\def\re{{\rm Re}\,}
\def\im{{\rm Im}\,}
\def\dbar{\smash{\ov\partial}}
\def\ddbar{\smash{\partial\ov\partial}}
\def\ssubset{\subset\!\subset}
\def\ci{{C^\infty}}
\def\la{\longrightarrow}
\def\ld{,\,\ldots\,,}
\def\wt{\widetilde}

\def\thskip{\phantom{\llap{\raise-5pt\hbox{$)$}}}\hfill\break}
\def\pav#1|{\vskip15pt{\bf #1.}\vskip8pt}
\def\pointir{.--- }
\def\th#1|#2\endth{\vskip3pt{\bf#1}\pointir {\sl#2}}
\def\proof{{\sl Proof}\pointir}
\def\endpr{~\hbox{$\square$}}

\def\article#1|#2|#3|#4|#5|#6|#7|
{{\leftskip=7mm\noindent
\hangindent=2mm\hangafter=1
\llap{[#1]\hskip1.35em}{\petcap #2}\pointir {\sl #3}, {\rm #4}, 
\nobreak{\bf #5} ({\oldstyle #6}), \nobreak #7.\par}}

\def\book#1|#2|#3|#4|#5|
{{\leftskip=7mm\noindent
\hangindent=2mm\hangafter=1
\llap{[#1]\hskip1.35em}{\petcap #2}\pointir{\sl #3}, #4, 
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\def\miscel#1|#2|#3|#4|
{{\leftskip=7mm\noindent
\hangindent=2mm\hangafter=1
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\centerline{\bf COHOMOLOGY OF q--CONVEX SPACES IN TOP DEGREES}

\vskip 0.5cm
{\noindent\hangindent4cm\hangafter-1
{\sl Jean-Pierre DEMAILLY

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
Laboratoire associ\'e au C.N.R.S.
n\kern-2pt{\rm\char'27}188,\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip30pt 
\eightpoint 
{\bf Abstract}\pointir It is shown that every strongly $q$--complete
subvariety of a complex analytic space has a fundamental system of
strongly $q$--complete neighborhoods.  As a consequence, we find a
simple proof of Ohsawa's result that every non compact irreducible
$n$--dimensional analytic space is strongly $n$--complete. Finally,
it is shown that $L^2$--cohomology theory readily implies Ohsawa's 
Hodge decomposition and Lefschetz isomorphism theorems for
absolutely $q$--convex manifolds.
\tenpoint
\vskip30pt

\pav 1. Introduction|

Let $(X,\oc_X)$ be a complex analytic space, possibly non reduced. 
Recall that a function $\varphi$ on $X$ is said to be strongly
$q$--convex in the sense of Andreotti-Grauert [A-G] if there exists a
covering of $X$ by open patches $A_\lambda$ isomorphic to closed
analytic sets in open sets $\Omega_\lambda\subset\cb^{N_\lambda}$~,
$\lambda\in I$~, such that each restriction $\varphi_{\restriction
A_\lambda}$ admits an extension $\wt\varphi_\lambda$ on $\Omega_\lambda$
which is strongly $q$--convex, \ie such that $i\ddbar\wt\varphi_\lambda$
has at most $q-1$ negative or zero eigenvalues at each point
of $\Omega_\lambda$~.  The strong $q$--convexity property is easily shown
not to depend on the covering nor on the embeddings
$A_\lambda\subset\Omega_\lambda$~. 

The space $X$ is said to be strongly $q$--complete, resp.  strongly
$q$--convex, if $X$ has a smooth exhaustion function $\varphi$ such that
$\varphi$ is strongly $q$--convex on $X$~, resp.  on the complement
$X\setminus K$ of a compact set $K\subset X$~.  The main new result of
this paper is~:

\th Theorem 1|Let $Y$ be an analytic subvariety in a complex space
$X$~.  If $Y$ is strongly $q$--complete, then $Y$ has a fundamental
family of strongly $q$--complete neighborhoods $V$ in $X$~.\endth

The special case of Stein neighborhoods ($q=1$) has been proved long ago
by Y.T.~Siu [S3]. The special case when $q=\dim Y+1$ is due to
D.~Barlet, who used it in the study of the convexity of spaces of
cycles (cf. [Ba]). This case is also a conse\-quence of
the results of T.~Ohsawa [Oh2], who obtained simultaneously a proof
for $q=\dim Y$~. Somewhat surprisingly, our proof of 
the general case is much simpler that the original proof of Siu for the 
Stein case, and also probably simpler than the partial proofs of
Barlet and Ohsawa.  The
main idea is to extend an exhaustion of $Y$ to a neighborhood by means
of a patching procedure.  However, up to our knowledge, the extension
can be done only after the exhaustion of $Y$ has been slightly modified
in a neighborhood of the singular set (cf.  theorem 4).  Theorem 1 
follows now rather easily from the fact that any subvariety $Y$ is the
set of $-\infty$ poles of an ``almost plurisubharmonic" function 
(a function whose complex Hessian has locally bounded negative part).  
Theorem 1 can be used to obtain a short proof of Ohsawa's results on
$n$--convexity of $n$--dimensional complex spaces~:

\th Theorem 2 {\rm (Ohsawa [Oh2])}|Let $X$ be a complex space such that
all irreducible components have dimension $\le n$~.\thskip
{\rm (a)}~~$X$ is always strongly $(n+1)$--complete.\thskip
{\rm (b)}~~If $X$ has no compact irreducible component of 
dimension $n$~, then $X$ is strongly $n$--complete.\thskip
{\rm (c)}~~$\,$If $X$ has only finitely many irreducible 
components of dimension $n$~, then $X$ is strongly $n$--convex.
\endth

The main step consists in proving that a $n$--dimensional connected non
compact manifold always has a strongly subharmonic exhaustion function
with respect to any hermitian metric (a result due to Greene and Wu [G-W]).
The proof is then completed by induction on $n$~, using theorem 1.  

These results will usually be applied in connection with the
Andreotti-Grauert theorem [A-G].  Let $\fr$ be a coherent sheaf over an
analytic space $X$~.  The Andreotti-Grauert theorem asserts that
$H^q(X,\fr)$ is finite dimensional if $X$ is strongly\break $q$--convex
and equal to zero if $X$ is strongly $q$--complete.  When $\dim X\le
n$~, a combination with theorem 2 yields~:

\noindent$\bullet$ $H^n(X,\fr)=0$ if $X$ has no compact $n$--dimensional
component~;\hfill\break
$\bullet$ $\dim H^n(X,\fr)<\infty$ if $X$ has only finitely 
many ones.

\noindent The special case of these statements when $\fr$ is a vector 
bundle over a manifold goes back to Malgrange [Ma]. The general case 
was first completed by Siu [S1,S2], with a direct but much more complicated
method.

Finally, we show that Ohsawa's Hodge decomposition theorem for
an absolutely $q$--convex K\"ahler manifold $M$ is a direct consequence 
of Hodge decomposition for $L^2$ harmonic forms~; the key fact is the
observation that any smooth form of degree $k\ge n+q$ becomes 
$L^2$ for some suitably modified K\"ahler metric~; thus $H^k(M,\cb)$
can be considered as a direct limit of $L^2$--cohomology groups. The
Lefschetz isomorphism on $L^2$--cohomology groups then produces in the
limit an isomorphism from the cohomology with compact supports onto the
cohomology without supports.\eject

\th Theorem 3 {\rm (Ohsawa [Oh1], [O-T])}|Let $(M,\omega)$ be a K\"ahler 
$n$--dimensional manifold. Suppose that $M$ is absolutely $q$--convex, 
i.e. admits a smooth plurisubharmonic exhaustion function that is 
strongly $q$--convex on $M\setminus K$ for some compact set $K$~. Set
$\Omega^r=\oc(\Lambda^rT^\star M)$. Then the De Rham cohomology groups 
with arbitrary $($resp. compact$)$ supports have decompositions
$$\eqalign{
&H^k(M,\cb)\simeq\bigoplus_{r+s=k}H^s(M,\Omega^r)~,~~~~
H^r(M,\Omega^s)\simeq\ov{H^s(M,\Omega^r)}~,~~~~k\ge n+q~,\cr
&H^k_c(M,\cb)\simeq\bigoplus_{r+s=k}H^s_c(M,\Omega^r)~,~~~~
H^r_c(M,\Omega^s)\simeq\ov{H^s_c(M,\Omega^r)}~,~~~~k\le n-q~,\cr}$$
and these groups are finite dimensional. Moreover, there is a
Lefschetz isomorphism
$$\omega^{n-r-s}\wedge{\scriptstyle\bullet}~~:~~H^s_c(M,\Omega^r)
\la H^{n-r}(M,\Omega^{n-s})~,~~~~r+s\le n-q~.$$
\endth

Observe that the finiteness statement holds as soon as $X$ is
strongly $q$--convex (this is a consequence of Morse theory
for the De Rham groups and a consequence of the Andreotti-Grauert
theorem for the Dolbeault groups). By an example of Grauert and 
Riemenschneider [G-R] (cf. also [Oh1]), neither Hodge
decomposition nor Hodge symmetry necessarily hold on a strongly 
$q$--convex manifold in degrees $\ge n+q$ or $\le n-q$~: if $V$ is a
positive rank $q$ vector bundle over a projective $m$--fold $Y$~, 
then the space $X$ equal to $P(V\oplus\oc)=V\cup V_\infty$ minus 
the unit ball bundle $\ov B(V)$ is $q$--convex, however with $n=q+m$
it can be checked that
$$H^2_c(X,\cb)=\cb~,~~~~H^0_c(X,\Omega^2)=0~,~~~~
H^2_c(X,\oc)\supset H^1(Y,V^\star)~,$$
and there are examples where $q=m\ge 2$ and $H^1(Y,V^\star)$ is arbitrarily
large. 

The author wishes to express warm thanks to Prof. K. Diederich for
valuable remarks which contributed to improve the original version 
of this article.

\pav 2.  Existence of $q$--convex neighborhoods|

The first step in the proof of theorem 1 is the following 
approximate extension theorem for strongly $q$--convex functions.

\th Theorem 4|Let $Y$ be an analytic set in a complex space $X$
and $\psi$ a strongly $q$--convex $\ci$ function on $Y$~. For every
continuous function $\delta>0$ on $Y$~, there exists a strongly $q$--convex
$\ci$ function $\varphi$ on a neighborhood $V$ of $Y$ such that
$\psi\le\varphi_{\restriction Y}<\psi+\delta$~.\endth

\proof Without loss of generality, we may assume $Y$ closed in $X$~. 
Let $Z_k$ be the union of all irreducible components of dimension $\le k$
of one of the sets $Y$~, $Y_{\rm sing}$~, $(Y_{\rm sing})_{\rm sing}$~, 
$\ldots$~.  It is clear that $Z_k\setminus Z_{k-1}$ is a smooth
$k$--dimensional submanifold of $Y$ (possibly empty) and that 
$\bigcup Z_k=Y$~.  We shall prove by induction on $k$ the following 
statement~:

{\sl There exists a $\ci$ function $\varphi_k$ on $X$ which is strongly
$q$--convex along $Y$ and on a closed neighborhood $\ov V_k$ of $Z_k$
in $X$~, such that $\psi\le\varphi_{k\restriction Y}<\psi+\delta$~.}

We first observe that any smooth extension $\varphi_{-1}$ of $\psi$ to
$X$ satisfies the requirements with $Z_{-1}=V_{-1}=\emptyset$~.  Assume that
$V_{k-1}$ and $\varphi_{k-1}$ have been constructed.  Then $Z_k\setminus
V_{k-1}\subset Z_k\setminus Z_{k-1}$ is contained in $Z_{k,{\rm reg}}$~. 
The closed set $Z_k\setminus V_{k-1}$ has a locally finite covering 
$(A_\lambda)$ in $X$ by open coordinate patches
$A_\lambda\subset\Omega_\lambda\subset\cb^{N_\lambda}$ in which
$Z_k$ is given by equations $z'_\lambda=(z_{\lambda,k+1}\ld
z_{\lambda,N_\lambda})=0$~. Let $\theta_\lambda$ be $\ci$ functions
with compact support in $A_\lambda$ such that $0\le\theta_\lambda
\le 1$ and $\sum\theta_\lambda=1$ on $Z_k\setminus V_{k-1}$~.  We set
$$\varphi_k(x)=\varphi_{k-1}(x)+\sum~\theta_\lambda(x)\,\varepsilon_\lambda^3
\,\log(1+\varepsilon_\lambda^{-4}|z'_\lambda|^2)~~~~{\rm on}~~X~.$$ 
For $\varepsilon_\lambda>0$ small enough, we will have
$\psi\le\varphi_{k-1\restriction Y}\le\varphi_{k\restriction Y}<
\psi+\delta$~. Now, we check that $\varphi_k$ is still strongly
$q$--convex along $Y$ and near every point $x_0\in\ov V_{k-1}$~,
and that $\varphi_k$ becomes strongly $q$--convex near every point 
$x_0\in Z_k\setminus V_{k-1}$~. We may assume that $x_0\in{\rm Supp}\,
\theta_\mu$ for some $\mu$~, otherwise $\varphi_k$ coincides with
$\varphi_{k-1}$ in a neighborhood of $x_0$~. Select $\mu$ and a 
small neighborhood $W\ssubset\Omega_\mu$ of $x_0$ such that

\noindent(a) if $x_0\in Z_k\setminus V_{k-1}$ then
$\theta_\mu(x_0)>0$ and $A_\mu\cap W\ssubset\{\theta_\mu>0\}$~;

\noindent(b) if $x_0\in A_\lambda$ for some $\lambda$ (there is a finite
set $I$ of such $\lambda$'s),
then $A_\mu\cap W\ssubset A_\lambda$ and $z_{\lambda\restriction A_\mu\cap W}$
has a holomorphic extension $\wt z_\lambda$ to $\ov W$~;

\noindent(c) if $x_0\in\ov V_{k-1}$ then $\varphi_{k-1
\restriction A_\mu\cap W}$ has a strongly $q$--convex extension 
$\wt\varphi_{k-1}$ to $\ov W\,;$

\noindent(d) if $x_0\in Y\setminus\ov V_{k-1}$~, $\varphi_{k-1
\restriction Y\cap W}$ has a strongly $q$--convex extension 
$\wt\varphi_{k-1}$ to $\ov W\,.$

\noindent Otherwise take an arbitrary smooth extension 
$\wt\varphi_{k-1}$ of $\varphi_{k-1\restriction A_\mu\cap W}$ to $\ov W$
and let $\smash{\wt\theta_\lambda}$ be an extension of $\theta_{\lambda
\restriction A_\mu\cap W}$ to $\ov W$~. Then
$$\wt\varphi_k=\wt\varphi_{k-1}+\sum\wt\theta_\lambda\,\varepsilon_\lambda^3\,
\log(1+\varepsilon_\lambda^{-4}|{\wt z}^{\,\prime}_\lambda|^2)$$
is an extension of $\varphi_{k\restriction A_\mu\cap W}$ to $\ov W$~,
resp. of $\varphi_{k\restriction Y\cap W}$ to $\ov W$ in case (d).
As the function $\log(1+\varepsilon_\lambda^{-4}
|{\wt z}^{\,\prime}_\lambda|^2)$ is plurisubharmonic and as its first
derivative $\langle{\wt z}^{\,\prime}_\lambda,
d{\wt z}^{\,\prime}_\lambda\rangle\,(\varepsilon_\lambda^4+
|{\wt z}^{\,\prime}_\lambda|^2)^{-1}$ is bounded by ${\rm O}
(\varepsilon_\lambda^{-2})$~, we see that
$$i\ddbar\wt\varphi_k\ge i\ddbar\wt\varphi_{k-1}-{\rm O}
({\scriptstyle\sum}\varepsilon_\lambda)~.$$ 
Therefore, for $\varepsilon_\lambda$ small enough, $\wt\varphi_k$
remains $q$--convex on $\ov W$ in cases (c) and (d). Since
all functions ${\wt z}^{\,\prime}_\lambda$ vanish along $Z_k\cap W$~,
we have
$$i\ddbar\wt\varphi_k\ge i\ddbar\wt\varphi_{k-1}
+\sum_{\lambda\in I}\theta_\lambda\,\varepsilon_\lambda^{-1}\,i\ddbar
|{\wt z}^{\,\prime}_\lambda|^2\ge i\ddbar\wt\varphi_{k-1}
+\theta_\mu\,\varepsilon_\mu^{-1}\,i\ddbar|z'_\mu|^2$$
at every point of $Z_k\cap W$~.
Moreover $i\ddbar\wt\varphi_{k-1}$ has at most $(q-1)$--negative
or zero eigenvalues on $TZ_k$ since $Z_k\subset Y$~, whereas
$i\ddbar|z'_\mu|^2$ is positive definite in the normal directions to
$Z_k$ in $\Omega_\mu$~.  In case (a), we thus find that $\wt\varphi_k$ 
is strongly $q$--convex on $\smash{\ov W}$ for $\varepsilon_\mu$ small 
enough~; we also observe that only finitely many conditions are
required on each $\varepsilon_\lambda$ if we choose a locally finite
covering of $\bigcup{\rm Supp}\,\theta_\lambda$ by neighborhoods
$W$ as above. Therefore, for $\varepsilon_\lambda$ small enough,
$\varphi_k$ is strongly $q$--convex on a
neighborhood $\smash{\ov V'_k}$ of $Z_k\setminus V_{k-1}$~.  The
function $\varphi_k$ and the set $V_k=V_{k-1}\cup V'_k$ satisfy the
requirements at order $k$~.  It is clear that we can choose the sequence
$\varphi_k$ stationary on every compact subset of $X$~; the limit
$\varphi$ and the open set $V=\bigcup V_k$ fulfill theorem 4.\endpr

The second step is the existence of almost psh (plurisubharmonic)
functions having poles along a prescribed analytic set.  By an almost
psh function on a manifold, we mean a function that is locally equal to
the sum of a psh function and of a smooth function, or equivalently, a
function whose complex Hessian has bounded negative part. On a complex
space, we require that our function can be locally extended as an
almost psh function in the ambient space of an embedding.

\th Lemma 5|Let $Y$ be an analytic subvariety in a complex 
space $X$~. There exists an almost plurisubharmonic function $v$ on $X$
such that $v\in C^\infty(X\setminus Y)$ and $v=-\infty$ on $Y$
$($with logarithmic poles along $Y)$.\endth

\proof Since $\ir_Y\subset\oc_X$ is a coherent subsheaf, there is a
locally finite covering of $X$ by patches $A_\lambda$ isomorphic to
analytic sets in balls $B(0,r_\lambda)\subset\cb^{N_\lambda}$~, such
that $\ir_Y$ admits a system of generators $g_\lambda=(g_{\lambda,j})$
on a neighborhood of each set $\ov A_\lambda$~.  We set
$$\eqalign{
&v_\lambda(z)=\log|g_\lambda(z)|^2-{1\over r^2_\lambda
-|z-z_\lambda|^2}~~~~{\rm on}~~A_\lambda~,\cr
&v(z)=m\big(\ldots\,,v_\lambda(z),\,\ldots\big)~~~~ {\rm
for}~~\lambda~~{\rm such~that~~}A_\lambda\ni z~,\cr}$$ 
where $m$ is a regularized max function defined as follows~: select a
smooth function $\rho$ on $\rb$ with support in $[-1/2,1/2]$~, such that
$\rho\ge 0$~, $\int_\rb\rho(u)du=1$~, $\int_\rb u\rho(u)\,du=0$~, and set
$$m(t_1\ld t_p)=\int_{\rb^p}\max\{t_1+u_1\ld t_p+u_p\}
\prod_{1\le j\le p}\rho(u_j)\,du_j~.$$
It is clear that $m$ is increasing in all variables and convex,
thus $m$ preserves plurisubharmonicity. Moreover, we have
$$m(t_1\ld t_j\ld t_p)=m(t_1\ld \wh t_j\ld t_p)$$
as soon as $t_j<\max\{t_1\ld t_{j-1},t_{j+1}\ld t_p\}-1$~.
As the generators $(g_{\lambda,j})$ can be expressed in terms of
one another on a neighborhood of $\ov A_\lambda\cap \ov A_\mu$~, we see
that the quotient $|g_\lambda|/|g_\mu|$ remains bounded on this set. 
Therefore none of the values $v_\lambda(z)$ for $A_\lambda\ni z$ and
$z$ near $\partial A_\lambda$ contributes to the value of $v(z)$~,
since $1/(r_\lambda^2-|z-z_\lambda|^2)$ tends to $+\infty$ on
$\partial A_\lambda$~. It follows that $v$ is
smooth on $X\setminus Y$~; as each $v_\lambda$ is almost psh on
$A_\lambda$~, we also see that $v$ is almost psh on $X$~.\endpr
\vskip5pt

{\sl Proof of theorem 1}\pointir By theorem 4 applied to a strongly
$q$--convex exhaustion of $Y$ and $\delta=1$~, there exists a strongly
$q$--convex function $\varphi$ on a neighborhood $W_0$ of $Y$ such that
$\varphi_{\restriction Y}$ is an exhaustion.  Let $W_1$ be a
neighborhood of $Y$ such that $\ov W_1\subset W_0$ and such that
$\varphi_{\restriction\ov W_1}$ is an exhaustion.  We are going to show
that every neighborhood $W\subset W_1$ of $Y$ contains a strongly
$q$--complete neighborhood $V$~.  If $v$ is the function given by
lemma 5, we set
$$\wt v=v+\chi\circ\varphi~~~~{\rm on~~}\ov W$$
where $\chi:\rb\to\rb$ is a smooth convex increasing function.  If
$\chi$ grows fast enough, we get $\wt v>0$ on $\partial W$ and the
$(q-1)$--codimensional subspace on which $i\ddbar\varphi$ is positive 
definite (in some ambient space) is also positive definite for
$i\ddbar\wt v$ provided that $\chi'$ be large enough to compensate the
bounded negative part of $i\ddbar v$~. Then $\wt v$ is
strongly $q$--convex.  Let $\theta$ be a smooth convex increasing
function on $]-\infty,0[$ such that $\theta(t)=0$ for $t<-3$ and
$\theta(t)=-1/t$ on $]-1,0[$~.  The open set $V=\{z\in W\,;\,\wt v(z)<0\}$
is a neighborhood of $Y$ and $\smash{\wt\psi}=\varphi+\theta\circ\wt v$ 
is a strongly $q$--convex exhaustion of $V$~.\endpr

\pav 3. q--convexity properties in top degrees|

It is obvious by definition that a $n$--dimensional complex manifold $M$ is
strongly $q$--complete for $q\ge n+1$~. If $M$ is connected
and non compact, this property also holds for $q=n$~, i.e. there is
a smooth exhaustion $\psi$ on $M$ such that $i\ddbar\psi$ has at least
one positive eigenvalue everywhere. In fact, one
can even show that $M$ has strongly subharmonic exhaustion functions.
Let $\omega$ be an arbitrary hermitian metric on $M$~.
We consider the Laplace operator $\Delta_\omega$ defined by
$$\Delta_\omega v={\rm Trace}_\omega(i\ddbar v)=\sum_{1\le j,k\le n}
\omega^{jk}(z){\partial^2v\over\partial z_j\partial\ov z_k}$$
where $(\omega^{jk})$ is the conjugate of the inverse matrix of 
$(\omega_{jk})$~. Observe that $\Delta_\omega$ coincides with the usual
Laplace-Beltrami operator only if $\omega$ is K\"ahler. We will say that $v$
is {\sl strongly $\omega$--subharmonic} if $\Delta_\omega v>0$~.  Clearly,  
this property implies that $i\ddbar v$ has at least one positive 
eigenvalue at each point, i.e. that $v$ is strongly $n$--convex. 
Moreover, since
$$\Delta_\omega \chi(v_1\ld v_s)=\sum_j{\partial\chi\over\partial t_j}(v_1\ld
v_s)\,\Delta_\omega v_j+\sum_{j,k}{\partial^2\chi\over\partial t_j\partial
t_k}(v_1\ld v_s)\,\langle \partial v_j,\partial v_k\rangle_\omega,$$
subharmonicity has the advantage of being preserved by all convex increasing 
combinations, whereas a sum of strongly $n$--convex functions is not
necessarily $n$--convex. We shall need the following partial converse.

\th Lemma 6|If $\psi$ is strongly $n$--convex on $M$~, there is a
hermitian metric $\omega$ such that $\psi$ is strongly
subharmonic with respect to $\omega$~.\endth

\proof Let $U_\lambda\ssubset U'_\lambda$~, $\lambda\in\nb$~, be
locally finite coverings of $M$ by open balls equipped with coordinates
such that $\partial^2\psi/\partial z_1\partial\ov z_1>0$ on 
$\smash{\ov U}'_\lambda$~. By induction on $\lambda$~, we construct a
hermitian metric $\omega_\lambda$ on $M$ such that $\psi$ is strongly
$\omega_\lambda$--subharmonic on $U_0\cup\ldots\cup U_{\lambda-1}$~.
Starting from an arbitrary $\omega_0$~, we obtain $\omega_\lambda$
from $\omega_{\lambda-1}$ by increasing the coefficient
$\omega^{1\,1}_{\lambda-1}$ in $(\omega^{jk}_{\lambda-1})=
(\omega_{\lambda-1,kj})^{-1}$ on a neighborhood of 
$\smash{\ov U}_\lambda$~. Then $\omega=\lim\omega_\lambda$ is the
required metric.\endpr

\th Lemma 7|Let $U,W\subset M$ be open sets such that for every
connected component $U_s$ of $U$ there is a connected component
$W_{t(s)}$ of $W$ such that $W_{t(s)}\cap U_s\ne\emptyset$ and
$W_{t(s)}\setminus\ov U_s\ne\emptyset$~.  Then there exists a function
$v\in\ci(M,\rb)$~, $v\ge 0$~, with support contained in $\ov U\cup\ov
W$~, such that $v$ is strongly $\omega$--subharmonic and $>0$ on
$U$~.\endth

\proof We first prove that the result is true when $U,W$ are small cylinders
with the same radius and axis. Let $a_0\in M$ be a given
point and $z_1\ld z_n$ holomorphic coordinates centered at $a_0$~. 
We set $\re z_j=x_{2j-1}$~, $\im z_j=x_{2j}$~,
$x'=(x_2\ld x_{2n})$ and $\omega=\sum\wt\omega_{jk}(x)dx_j\otimes dx_k$.
Let $U$ be the cylinder $|x_1|<r$~,
$|x'|<r$~, and $W$ the cylinder $r-\varepsilon<x_1<r+\varepsilon$~,
$|x'|<r$~. There are constants $c,C>0$ such that
$$\sum\wt\omega^{jk}(x)\xi_j\xi_k\ge c|\xi|^2~~~{\rm and}~~~
\sum|\wt\omega^{jk}(x)|\le C~~~{\rm on}~~\ov U~.$$
Let $\chi\in\ci(\rb,\rb)$ be a nonnegative
function equal to $0$ on $]-\infty,-r]\cup[r+\varepsilon,+\infty[$ and 
strictly convex on $]-r,r]$~. We take explicitly $\chi(x_1)=(x_1+r)
\exp(-1/(x_1+r)^2\big)$ on $]-r,r]$ and
$$v(x)=\chi(x_1)\exp\big(1/(|x'|^2-r^2)\big)~~~{\rm on}~~U\cup W~,~~~
v=0~~~{\rm on}~~M\setminus(U\cup W)~.$$
We have $v\in\ci(M,\rb)$~, $v>0$ on $U$~, and a simple computation gives
$$\eqalign{
{\Delta_\omega v(x)\over v(x)}&=
\wt\omega^{11}(x)\big(4(x_1+r)^{-5}-2(x_1+r)^{-3}\big)\cr
&+\sum_{j>1}\wt
\omega^{1j}(x)\big(1+2(x_1+r)^{-2}\big)(-2x_j)(r^2-|x'|^2)^{-2}\cr 
&+\kern-2pt\sum_{j,k>1}\wt\omega^{jk}(x)\Big( x_jx_k\big(4-8(r^2-|x'|^2)
\big)-2(r^2-|x'|^2)^2\delta_{jk}\Big)(r^2-|x'|^2)^{-4}.
\cr}$$
For $r$ small, we get
$${\Delta_\omega v(x)\over v(x)}\ge 2c(x_1+r)^{-5}-C_1(x_1+r)^{-2}|x'|
(r^2-|x'|^2)^{-2}+(2c|x'|^2-C_2r^4)
(r^2-|x'|^2)^{-4}$$
with constants $C_1,C_2$ independent of $r$~.  
The negative term is bounded by\break $C_3(x_1+r)^{-4}+c|x'|^2(
r^2-|x'|^2)^{-4}$~, hence
$$\Delta_\omega v/v(x)\ge c(x_1+r)^{-5}+(c|x'|^2-C_2r^4)(r^2-|x'|^2)^{-4}~.$$
The last term is negative only when $|x'|<C_4r^2$~, in which case
it is bounded by $C_5r^{-4}<c(x_1+r)^{-5}$~. Hence
$v$ is strongly $\omega$--subharmonic on $U$~.

Next, assume that $U$ and $W$ are connected. Then $U\cup W$ is connected. 
Fix a point $a\in W\setminus\ov U$~. If $z_0\in U$ is given, we choose a 
path $\Gamma\subset U\cup W$ from $z_0$ to $a$\break
which is piecewise linear with respect to holomorphic coordinate
patches. Then we can find a finite sequence of
cylinders $(U_j,W_j)$ of the type described above, $1\le j\le N$~,
whose axes are segments contained in $\Gamma$~, such that 
$$U_j\cup W_j\subset U\cup W~,~~~\ov W_j\subset U_{j+1}~~~{\rm and}~~~
z_0\in U_0~,~~~a\in W_N\subset W\setminus\ov U~.$$
For each such pair, we have a function $v_j\in\ci(M)$ with support in
$\ov U_j\cup\ov W_j$~, $v_j\ge 0$~, strongly $\omega$--subharmonic and $>0$ on
$U_j$~. By induction, we can find constants $C_j>0$ such that
$v_0+C_1v_1+\cdots+C_j v_j$ is strongly $\omega$--subharmonic on
$U_0\cup\ldots\cup U_j$ and $\omega$--subharmonic on $M\setminus\ov W_j$~. 
Then
$$w_{z_0}=v_0+C_1v_1+\ldots+C_Nv_N\ge 0$$
is $\omega$--subharmonic on $U$ and strongly $\omega$--subharmonic
$>0$ on a neighborhood $\Omega_0$ of the given point $z_0$~. Select
a denumerable covering of $U$ by such neighborhoods $\Omega_p$ and set
$v(z)=\sum\varepsilon_pw_{z_p}(z)$ where $\varepsilon_p$ is a sequence
converging sufficiently fast to $0$ so that $v\in\ci(M,\rb)$~. Then $v$
has the required properties.

In the general case, we find for each pair $(U_s,W_{t(s)})$ a function
$v_s$ with support in $\smash{\ov U_s\cup\ov W_{t(s)}}$~, strongly
$\omega$--subharmonic and $>0$ on $U_s$~.  Any convergent series
$v=\sum\varepsilon_sv_s$ yields a function with the desired
properties.\endpr

\th Lemma 8|Let $X$ be a connected, locally connected and
locally compact topological space. If $U$ is a relatively compact
open subset of $X$~, we let $\wt U$ be the union of $U$ with all compact
connected components of $X\setminus U$~. Then $\wt U$ is open and
relatively compact in $X$~, and $X\setminus\smash{\wt U}$ has only finitely
many connected components, all non compact.\endth

\proof A rather easy exercise of general topology. Intuitively, $\wt U$
is obtained by ``filling the holes" of $U$ in $X$~.\endpr

\th Theorem 9 {\rm (Greene-Wu [G-W])}|Every $n$--dimensional connected 
non compact complex manifold $M$ has a strongly subharmonic exhaustion 
function with respect to any hermitian metric $\omega$~. In particular,
$M$ is strongly\break $n$--complete.
\endth

\proof Let $\varphi\in\ci(M,\rb)$ be an arbitrary exhaustion function. 
There exists a sequence of connected smoothly bounded open sets
$\Omega'_\nu\ssubset M$ with $\smash{\ov\Omega'_\nu}\subset\Omega'_{\nu+1}$
and $M=\bigcup\Omega'_\nu$~.  Let $\Omega_\nu=\smash{\wt\Omega'_\nu}$ be
the relatively compact open set given by lemma 8.\break  Then
$\smash{\ov\Omega_\nu}\subset\Omega_{\nu+1}$~, $M=\bigcup\Omega_\nu$ 
and $M\setminus\Omega_\nu$ has no compact connected component.  We set
$$U_1=\Omega_2~,~~~~U_\nu=\Omega_{\nu+1}\setminus\ov{\Omega_{\nu-2}}
~~~{\rm for}~~\nu\ge 2~.$$ 
Then $\partial U_\nu=\partial\Omega_{\nu+1}\cup\partial\Omega_{\nu-2}$~;
any connected component $U_{\nu,s}$ of $U_\nu$ has its boundary
$\partial U_{\nu,s}\not\subset\partial\Omega_{\nu-2}$~, otherwise $\ov
U_{\nu,s}$ would be open and closed in $M\setminus\Omega_{\nu-2}$~,
hence $\ov U_{\nu,s}$ would be a compact connected component of
$M\setminus\Omega_{\nu-2}$~.  Therefore $\partial U_{\nu,s}$ intersects
$\partial\Omega_{\nu+1}\subset U_{\nu+1}$~. If $U_{\nu+1,t(s)}$
is a connected component of $U_{\nu+1}$ containing a point of
$\partial U_{\nu,s}$~, then $U_{\nu+1,t(s)}\cap U_{\nu,s}\ne
\emptyset$ and $U_{\nu+1,t(s)}\setminus\ov U_{\nu,s}\ne\emptyset$~.  
Lemma 7 implies that there is a nonnegative function
$v_\nu\in\ci(M,\rb)$ with support in $U_\nu\cup U_{\nu+1}$~, which is
strongly $\omega$--subharmonic on $U_\nu$~.  An induction yields
constants $C_\nu$ such that
$$\psi_\nu=\varphi+C_1v_1+\cdots+C_\nu v_\nu$$ 
is strongly $\omega$--subharmonic on $\ov{\Omega_\nu}\subset
U_0\cup \ldots\cup U_\nu$~, thus $\psi=\varphi+\sum C_\nu v_\nu$ is a
strongly $\omega$--subharmonic exhaustion function on
$M$~.\endpr\vskip7pt

By an induction on the dimension, the above result can be generalized 
to an arbitrary complex space, as was first shown by T.~Ohsawa [Oh2].

{\sl Proof of theorem 2} (a,b)\pointir By induction on 
$n=\dim X$~. For $n=0$~, 
property (b) is void and (a) is obvious (any function can then be
considered as strongly $1$--convex). Assume that (a) has been proved
in dimension $\le n-1$~. Let $X'$ be the union
of $X_{\rm sing}$ and of the irreducible components of $X$ of dimension
at most $n-1$~, and $M=X\setminus X'$ the $n$--dimensional part of
$X_{\rm reg}$~. As $\dim X'\le n-1$~, the induction hypothesis shows
that $X'$ is strongly $n$--complete. By theorem 1,
there exists a strongly $n$--convex exhaustion function $\varphi'$
on a neighborhood $V'$ of $X'$~. Take a closed neighborhood
$\ov V\subset V'$ and an arbitrary exhaustion $\varphi$ on $X$ that extends 
$\varphi'_{\smash{\restriction\ov V}}$~. Since every function on a
$n$--dimensional manifold is strongly $(n+1)$--convex~, we conclude
that $X$ is at worst $(n+1)$--complete, as stated in (a).

In case (b), the hypothesis means that the connected components $M_j$ 
of $M=X\setminus X'$ have non compact closure $\ov M_j$ in $X$~.
On the other hand, lemma 6 shows that there exists a hermitian metric
$\omega$ on $M$ such that $\varphi_{\restriction M\cap V}$ is
strongly\break $\omega$--subharmonic. Consider the open sets 
$U_{j,\nu}\subset M_j$ provided by lemma 10 below. By the arguments 
already used in theorem 9, one can find a strongly $\omega$--subharmonic
exhaustion $\psi=\varphi+\sum_{j,\nu}C_{j,\nu}v_{j,\nu}$ on $X$~,
with $v_{j,\nu}$ strongly\break $\omega$--subharmonic on
$U_{j,\nu}$~, Supp$\,v_{j,\nu}\subset U_{j,\nu}\cup U_{j,\nu+1}$ and
$C_{j,\nu}$ large. Then $\psi$ is strongly $n$--convex on $X$~.

\th Lemma 10|For each $j$~, there exists a sequence of open sets 
$U_{j,\nu}\ssubset M_j$~, $\nu\in\nb$~, such that\thskip
{\rm(a)}~~$M_j\setminus V'\subset\bigcup_\nu U_{j,\nu}$~ and
$(U_{j,\nu})$ is locally finite in $\ov M_j~;$\thskip
{\rm(b)}~~for every connected component $U_{j,\nu,s}$ of $U_{j,\nu}$
there is a connected component $U_{j,\nu+1,t(s)}$ of $U_{j,\nu+1}$
such that $U_{j,\nu+1,t(s)}\cap U_{j,\nu,s}\ne\emptyset$ and
$U_{j,\nu+1,t(s)}\setminus\smash{\ov U_{j,\nu,s}}\ne\emptyset$~.\endth
\vskip4pt

By lemma 8 applied to the space $\ov M_j$~, there exists a sequence of 
relatively compact connected open sets $\Omega_{j,\nu}$ in $\ov M_j$ such 
that $\ov M_j\setminus\Omega_{j,\nu}$ has no compact connected component, 
$\ov\Omega_{j,\nu}\subset\Omega_{j,\nu+1}$ and
$\ov M_j=\bigcup\Omega_{j,\nu}$~. We define a compact set 
$K_{j,\nu}\subset M_j$ and an open set $W_{j,\nu}\subset\ov M_j$
containing $K_{j,\nu}$ by
$$K_{j,\nu}=(\ov\Omega_{j,\nu}\setminus\Omega_{j,\nu-1})\setminus V'~,
~~~~W_{j,\nu}=\Omega_{j,\nu+1}\setminus\ov\Omega_{j,\nu-2}~.$$
By induction on $\nu$~, we construct an open set
$U_{j,\nu}\ssubset W_{j,\nu}\setminus X'\subset M_j$ and a finite set 
$F_{j,\nu}\subset \partial U_{j,\nu}\setminus\ov\Omega_{j,\nu}$~.
We let $F_{j,-1}=\emptyset$~. 
If these sets are already constructed for $\nu-1$~, the
compact set $K_{j,\nu}\cup F_{j,\nu-1}$ is contained in the open
set $W_{j,\nu}$~, thus contained in a finite union of connected components 
$W_{j,\nu,s}$~. We can write $K_{j,\nu}\cup F_{j,\nu-1}=
\bigcup L_{j,\nu,s}$ where $L_{j,\nu,s}$ is contained in 
$W_{j,\nu,s}\setminus X'\subset M_j$~. The open set $W_{j,\nu,s}\setminus X'$
is connected and non contained in $\ov\Omega_{j,\nu}\cup
L_{j,\nu,s}$~, otherwise its closure $\ov W_{j,\nu,s}$ would have no
boundary point $\in\partial\Omega_{j,\nu+1}$~, thus would be open and 
compact in $\ov M_j\setminus\Omega_{j,\nu-2}$~, contradiction. 
We select a point $a_s\in (W_{j,\nu,s}\setminus X')\setminus(
\ov\Omega_{j,\nu}\cup L_{j,\nu,s})$ and a smoothly bounded connected 
open set $U_{j,\nu,s}\ssubset W_{j,\nu,s}\setminus X'$ containing
$L_{j,\nu,s}$ with $a_s\in\partial U_{j,\nu,s}$~. 
Finally, we set $U_{j,\nu}=\bigcup_s U_{j,\nu,s}$
and let $F_{j,\nu}$ be the set of all points $a_s$~. By construction, we
have $U_{j,\nu}\supset K_{j,\nu}\cup F_{j,\nu-1}$~, thus 
$\bigcup U_{j,\nu}\supset\bigcup K_{j,\nu}=M_j\setminus V'$~, and $\partial
U_{j,\nu,s}\ni a_s$ with $a_s\in F_{j,\nu}\subset U_{j,\nu+1}$~.
Property (b) follows.\endpr
\vskip8pt

{\sl Proof of theorem 2} (c)\pointir Let $Y\subset X$ be the
union of $X_{\rm sing}$ with all irreducible components of $X$ that are
non compact or of dimension $<n$~.  Then $\dim Y\le n-1$~, so $Y$ is
$n$--convex and theorem 1 implies that there is an exhaustion
function $\psi\in\ci(X,\rb)$ such that $\psi$ is strongly $n$--convex on
a neighborhood $V$ of $Y$~.  Then the complement $K=X\setminus V$ is
compact and $\psi$ is strongly $n$--convex on $X\setminus K$~.\endpr

\pav 4. A simple proof of Ohsawa's Hodge decomposition theorem|

Let $M$ be a complex $n$--dimensional manifold admitting a K\"ahler
metric $\omega$ and a strongly $q$--convex
plurisubharmonic exhaustion function $\psi$~. For any
convex increasing function $\chi\in\ci(\rb,\rb)$~, we consider the
new K\"ahler metric
$$\omega_\chi=\omega+i\ddbar(\chi\circ\psi)=\omega+\chi'(\psi)\,
i\ddbar\psi+\chi''(\psi)\,i\partial\psi\wedge\dbar\psi$$
and the associated geodesic distance $\delta_\chi$~.
Then the norm of $\chi''(\psi)^{1/2}d\psi$ with respect to 
$\omega_\chi$ is less than $1$~, thus if $\rho$ is a primitive of
$(\chi'')^{1/2}$ we have
$$\big|\rho(\psi(x))-\rho(\psi(y))\big|\le\delta_\chi(x,y)~.$$
Hence $\omega_\chi$ is complete as soon as
$\lim_{+\infty}\rho(t)=+\infty$~, that is $\int_0^{+\infty}
\chi''(t)^{1/2}dt=+\infty$.\break In the sequel, we always assume that
$\chi$ grows sufficiently fast at infinity so that this condition
is fulfilled. We denote by $\smash{L^{2,(k)}_\chi(M)=\bigoplus_{r+s=k}
L^{2,(r,s)}_\chi(M)}$ the space of $L^2$ forms of degree $k$ with respect
to the metric $\omega_\chi$~, by $\hr^k_\chi(M)$ the subspace of 
$L^2$ harmonic forms of degree $k$ with respect to the associated
Laplace-Beltrami operator $\Delta_\chi=dd^\star_\chi+d^\star_\chi d$
and by $\smash{\hr^{r,s}_\chi(M)}$ the space of $L^2$--harmonic forms of 
bidegree $(r,s)$ with respect to 
$\smash{\ov\square_\chi=\dbar\,\dbar^\star_\chi+\dbar^\star_\chi\dbar}$~. 
As $\omega_\chi$ is K\"ahler, we have $\smash{\square_\chi}=
\ov\square_\chi={1\over 2}\Delta_\chi$~, hence
$$\hr^k_\chi(M)=\bigoplus_{r+s=k}\hr^{r,s}_\chi(M)~,
~~~~\hr^{s,r}_\chi(M)=\ov{\hr^{r,s}_\chi(M)}\leqno(1)$$
for each $k=0,1\ld 2n$~. Since $\omega_\chi$ is complete,
we have orthogonal decompositions
$$\leqalignno{
L^{2,(r,s)}_\chi(M)&=\hr^{r,s}_\chi(M)\oplus
\ov{{\rm Im}^{r,s}\,\dbar_\chi\strut}\oplus
\ov{{\rm Im}^{r,s}\,\dbar^\star_\chi\strut}\cr
{\rm Ker}^{r,s}\,\dbar_\chi&=\hr^{r,s}_\chi(M)\oplus
\ov{{\rm Im}^{r,s}\,\dbar_\chi\strut}~,&(2)\cr}$$
where $\dbar_\chi$ is the unbounded $\dbar$ operator acting on $L^2$
forms with respect to $\omega_\chi$ and where $\ov{{\rm Im}^{r,s}}$ 
means closure of the range (in the specified bidegree). 
In particular $\hr^{r,s}_\chi(M)$ is isomorphic to the quotient
${\rm Ker}^{r,s}\,\dbar_\chi/\smash{\ov{{\rm Im}^{r,s}\,\dbar_\chi\strut}}$~.
Of course, similar results also hold for $\Delta_\chi$--harmonic forms.

\th Lemma 11|Let $u$ be a form of type $(r,s)$ with $L^2_{\rm loc}$
coefficients on $M$~. If $r+s\ge n+q$~, then 
$u\in\smash{L^{2,(r,s)}_\chi(M)}$
as soon as $\chi$ grows sufficiently fast at infinity.\endth

\proof At each point $x\in M$~, there is an orthogonal basis
$(\partial/\partial z_1\ld\partial/\partial z_n)$ of $T_x X$ in which
$$\omega=i\sum_{1\le j\le n}dz_j\wedge d\ov z_j~,~~~~
\omega_\chi=i\sum_{1\le j\le n}\lambda_jdz_j\wedge d\ov z_j~,$$
where $\lambda_1\le\ldots\le\lambda_n$ are the eigenvalues
of $\omega_\chi$ with respect to $\omega$~. Then the volume elements
$dV=\omega^n/2^nn!$ and $dV_\chi=\omega_\chi^n/2^n n!$ are related by
$$dV_\chi=\lambda_1\ldots\lambda_n\,dV$$
and for a $(r,s)$--form $u=\sum_{I,J}u_{I,J}dz_I\wedge d\ov z_J$ we find
$$|u|^2_\chi=\sum_{|I|=r,|J|=s}\Big(\prod_{k\in I}\lambda_k
\prod_{k\in J}\lambda_k\Big)^{-1}|u_{I,J}|^2~,$$
in particular
$$|u|^2_\chi dV_\chi\le{\lambda_1\ldots\lambda_n\over
\lambda_1\ldots\lambda_r\,\lambda_1\ldots\lambda_s}\,|u|^2dV
={\lambda_{r+1}\ldots\lambda_n\over\lambda_1\ldots\lambda_s}\,|u|^2dV~.$$
On the other hand, we have upper bounds
$$\lambda_j\le 1+C_1\chi'(\psi)~,~~1\le j\le n-1~,~~~~
\lambda_n\le 1+C_1\chi'(\psi)+C_2\chi''(\psi)$$
where $C_1(x)$ is the largest eigenvalue of $i\ddbar\psi(x)$ and 
$C_2(x)=|\partial\psi(x)|^2$~;
to find the $n-1$ first inequalities, we need only apply the
minimax principle on the kernel of $\partial\psi$~. As $i\ddbar\psi$
has at most $q-1$ zero eigenvalues on $X\setminus K$~, 
the minimax principle also gives lower bounds
$$\lambda_j\ge 1~,~~1\le j\le q-1~,~~~~
\lambda_j\ge 1+c\,\chi'(\psi)~,~~q\le j\le n~,$$
where $c(x)\ge 0$ is the $q$--th eigenvalue of $i\ddbar\psi(x)$ and
$c(x)>0$ on $X\setminus K$~. Assuming $\chi'\ge 1$~, we infer easily
$$\eqalign{
{|u|^2_\chi dV_\chi\over|u|^2dV}&\le{\big(1+C_1\chi'(\psi)\big)^{n-r-1}
\big(1+C_1\chi'(\psi)+C_2\chi''(\psi)\big)\over
\big(1+c\chi'(\psi)\big)^{s-q+1}}\cr
&\le C_3\big(\chi'(\psi)^{n+q-r-s-1}+
\chi''(\psi)\chi'(\psi)^{n+q-r-s-2}\big)~~~~{\rm on}~~X\setminus K~.\cr}$$
For $r+s\ge n+q$~, this is less than
$$C_3\big(\chi'(\psi)^{-1}+\chi''(\psi)\chi'(\psi)^{-2}\big)~,$$
and it is easy to show that this quantity can be made arbitrarily
small when $\chi$ grows sufficiently fast at infinity on $M$~.\endpr

It is a well-known result of Andreotti-Grauert [A-G] that the 
natural topology on the cohomology
groups $H^k(M,\fr)$ of a coherent sheaf $\fr$ over a strongly\break
$q$--convex manifold is Hausdorff for $k\ge q$~. If $\fr=\oc(E)$
is the sheaf of sections of a holomorphic vector bundle, this topology
is given by the Fr\'echet topology on the Dolbeault complex of
$L^2_{\rm loc}$ forms with $L^2_{\rm loc}$ $\dbar$--differential.
In particular, the morphism
$${\rm Ker}^{r,s}\dbar_\chi\la H^s(M,\Omega^r)$$
is continuous and has a closed kernel, and therefore this kernel
contains $\ov{{\rm Im}^{r,s}\,\dbar_\chi\strut}$~. We thus obtain
a factorization
$$\hr^{r,s}_\chi(M)\simeq{\rm Ker}^{r,s}\,\dbar_\chi/
\smash{\ov{{\rm Im}^{r,s}\,\dbar_\chi\strut}}\la H^s(M,\Omega^r)~.$$
Consider the direct limit
$$\lim_{\displaystyle\la\atop\scriptstyle\chi}
\hr^{r,s}_\chi(M)\la H^s(M,\Omega^r)\leqno(3)$$
over the set of smooth convex increasing functions $\chi$ with the ordering
$$\chi_1\preccurlyeq\chi_2\iff \chi_1\le\chi_2~~{\rm and}~~
L^{2,(k)}_{\chi_1}(M)\subset L^{2,(k)}_{\chi_2}(M)~~{\rm for}~~k=r+s~;$$
this ordering is filtering by the proof of lemma 13. It is well known 
that the De Rham cohomology groups are always Hausdorff,
hence there is a similar morphism
$$\lim_{\displaystyle\la\atop\scriptstyle\chi}
\hr^k_\chi(M)\la H^k(M,\cb)~.\leqno(4)$$
The first decomposition in theorem 3 follows now from (1) and the 
following simple lemma.

\th Lemma 12|The morphisms $(3)$, $(4)$ are one-to-one for $k=r+s\ge n+q$~.
\endth

\proof Let us treat for example the case of (3). Let $u$ be
a smooth $\dbar$--closed form of bidegree $(r,s)$~, $r+s\ge n+q$~.
Then there is a choice of $\chi$ for which 
$u\in\smash{L^{2,(r,s)}_\chi(M)}$~, so 
$u\in\smash{{\rm Ker}^{r,s}\,\dbar_\chi}$ and (3) is surjective.
If a class $\{u\}\in\hr^{r,s}_{\chi_0}(M)$ is mapped to zero in 
$H^s(M,\Omega^r)$~, we can write $u=\dbar v$ for some smooth 
form $v$ of bidegree $(r,s-1)$~. In the case $r+s>n+q$~, we have 
$v\in\smash{L^{2,(r,s-1)}_\chi(M)}$ for
some $\chi\succcurlyeq\chi_0$~. Hence the class of $u=\dbar_\chi v$ in
$\hr^{r,s}_\chi(M)$ is zero and (3) is injective. When $r+s=n+q$~, the 
form $v$ need not lie in any space $\smash{L^{2,(r,s-1)}_\chi(M)}$~,
but it suffices to show that $u=\dbar v$ is in the closure of 
${\rm Im}^{r,s}\,\dbar_\chi$ for some $\chi$~. Let $\theta\in\ci(\rb,\rb)$
be a cut-off function such that $\theta(t)=1$ for $t\le 1/2$~, 
$\theta(t)=0$ for $t\ge 1$ and $|\theta'|\le 3$~. Then 
$$\dbar\big(\theta(\varepsilon\psi)v\big)=\theta(\varepsilon\psi)\dbar v
+\varepsilon\theta'(\varepsilon\psi)\dbar\psi\wedge v~.$$
By the proof of lemma 11, there is a continuous function $C(x)>0$
such that $|v|^2_\chi dV_\chi\le C\big(1+\chi''(\psi)/\chi'(\psi)\big)
|v|^2dV$~, whereas $|\dbar\psi|^2_\chi\le 1/\chi''(\psi)$ by the
definition of $\omega_\chi$~. Hence we see that
$$\int_M|\theta'(\varepsilon\psi)\dbar\psi\wedge v|^2_\chi dV_\chi
\le 9\int_M C\big(1/\chi''(\psi)+1/\chi'(\psi)\big)|v|^2dV$$
is finite for $\chi$ large enough, and
$\dbar\big(\theta(\varepsilon\psi)v\big)$ converges to $\dbar v=u$
in $L^{2,(r,s)}_\chi(M)$~.\endpr

By Poincar\'e-Serre duality, the groups $H^k_c(M,\cb)$ and
$H^s_c(M,\Omega^r)$ with compact supports are dual to $H^{2n-k}(M,\cb)$
and $H^{n-s}(M,\Omega^{n-r})$ as soon as the latter groups are Hausdorff
and finite dimensional. This is certainly true for $k=r+s\le n-q$~,
thus we also obtain a Hodge decomposition
$$H^k_c(M,\cb)\simeq\bigoplus_{r+s=k}H^s_c(M,\Omega^r)~,~~~
H^r_c(M,\Omega^s)\simeq\ov{H^s_c(M,\Omega^r)}~,~~~k\le n-q~.\leqno(5)$$
As in Ohsawa [Oh1], it is easy to prove that the Lefschetz isomorphism
$$\omega_\chi^{n-r-s}\wedge{\scriptstyle\bullet}~~:~~\hr^{r,s}_\chi(M)
\la\hr^{n-s,n-r}_\chi(M)\leqno(6)$$
yields in the limit an isomorphism from the cohomology with compact
support onto the cohomology without supports. Indeed, the natural
morphism
$$H^s_c(M,\Omega^r)\la{\rm Ker}^{r,s}\,\dbar_\chi/
\smash{\ov{{\rm Im}^{r,s}\,\dbar_\chi\strut}}\simeq\hr^{r,s}_\chi(M)~,~~~~
r+s\le n-q\leqno(7)$$
is dual to $\hr_\chi^{n-r,n-s}(M)\la H^{n-s}(M,\Omega^{n-r})$~, which
is surjective for $\chi$ large by lemma 11 and the finite dimensionality
of the target space. Hence (7) is injective for $\chi$ large and after
composition with (6) we get an injection
$$H^s_c(M,\Omega^r)\la\hr^{n-s,n-r}_\chi(M)~.$$
If we take the direct limit over all $\chi$~, combine with the
isomorphism (3) and observe that $\omega_\chi$ has the same cohomology class
as $\omega$~, we obtain an injective map
$$\omega^{n-r-s}\wedge{\scriptstyle\bullet}~~:~~H^s_c(M,\Omega^r)
\la H^{n-r}(M,\Omega^{n-s})~,~~~~r+s\le n-q~.\leqno(8)$$
As both sides have the same dimension by Serre duality and Hodge
symmetry, this map must be an isomorphism. Since (8) can be factorized
through $H^s(M,\Omega^r)$ or through $H^{n-r}_c(M,\Omega^{n-s})$~,
we infer that the natural morphism
$$H^s_c(M,\Omega^r)\la H^s(M,\Omega^r)\leqno(9)$$
is injective for $r+s\le n-q$ and surjective for $r+s\ge n+q$~.
Of course, similar properties hold for the De Rham cohomology groups.
\vskip30pt

\centerline{\bf References}
\vskip20pt
\eightpoint
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