% Regularization of closed positive currents
% of type (1,1) by the flow of a Chern connection

\magnification=1200
\input broc.mac

\def\ic{{\cal I}}
\def\jc{{\cal J}}
\def\mc{{\cal M}}
\def\oc{{\cal O}}
\def\ld{,\ldots,}
\def\bu{{\scriptstyle\bullet}}
\def\gge{\raise1.5pt\hbox{$\scriptscriptstyle\ge$}}
\let\wt=\widetilde
\let\ov=\overline
\def\op{\ov\partial}
\def\loc{{\rm loc}}
\def\un{{\bf 1}}
\def\Id{{\rm Id}}
\def\ac{{\rm ac}}
\def\re{\mathop{\rm Re}\nolimits}
\def\exph{\mathop{\rm exph}\nolimits}
\let\la=\longrightarrow
\let\Llra=\Longleftrightarrow
\def\lraww{\mathrel{\rlap{$\longrightarrow$}\kern-1pt\longrightarrow}}

%------------------------------------------------------------------


\centerline{\bftwelve Regularization of closed positive currents}
\centerline{\bftwelve of type (1,1) by the flow of a Chern connection}
\vskip8pt
{\noindent\hangindent4cm\hangafter-1
{\bf Jean-Pierre Demailly

}}

{\noindent\hangindent2cm\hangafter-2{\sl
Universit\'e de Grenoble I, Institut Fourier, BP 74\hfill\break
U.R.A. 188 du C.N.R.S., F-38402 Saint-Martin d'H\`eres

}}
\vskip27pt
\hfill{\sl Dedicated to Professor Pierre Dolbeault on the occasion of
his retirement}
\vskip7pt

\titre 1. Introduction|

Let $X$ be a compact $n$-dimensional complex manifold and let $T$ be a
closed positive current of bidegree $(1,1)$ on $X$.  In general, $T$
cannot be approximated by closed positive currents of class $C^\infty$:
a necessary condition for this is that the cohomology class  $\{T\}$ is
numerically effective in the sense that \hbox{$\int_Y\{T\}^p\ge 0$} for
every $p$-dimensional subvariety $Y\subset X$. For example, if
$E\simeq\pb^{n-1}$ is the exceptional divisor of a one-point blow-up
$X\to X'$, then $T=[E]$ cannot be positively approximated: for every
curve $C\subset E$, we have \hbox{$\int_C\{E\}=\int_C c_1(\oc(-1))<0$}.
However, we will see that it is always possible to approximate a closed
positive current $T$ of type $(1,1)$ by closed real currents admitting
a small negative part, and that this negative part can be estimated in
terms of the Lelong numbers of $T$ and the geometry of~$X$.

Let $\alpha$ be a smooth closed $(1,1)$-form representing the same
$\partial\op$-cohomology class as $T$ and let $\psi$ be a quasi-psh
function on $X$ (that is, a function which is locally the sum of a
plurisubharmonic function and a smooth function) such that
$T=\alpha+{i\over\pi}\partial\op\psi$.  Such a decomposition exists 
even when $X$ is non-K\"ahler, since we can always find an open
covering $(\Omega_k)$ of $X$ such that $T=\smash{{i\over\pi}}
\partial\op\psi_k$ over $\Omega_k$, and construct a global
$\psi=\sum\theta_k\psi_k$ by means of a partition of unity~$(\theta_k)$
(note that $\psi-\psi_k$ is smooth on~$\Omega_k$). If $\psi_\varepsilon$
is an approximation of $\psi$, then $T_\varepsilon=\alpha+{i\over\pi}
\partial\op\psi_\varepsilon$ is an approximation of $T$.  We are thus
led to study a regularization process for quasi-psh functions. In this
context, we prove the following result.

\tha Theorem 1.1|Let $T$ be a closed almost positive $(1,1)$-current
and let $\alpha$ be a smooth real $(1,1)$-form in the same
$\partial\op$-cohomology class as $T$, i.e. $T=\alpha+{i\over\pi}\partial\op
\psi$ where $\psi$ is an almost psh function.
Let $\gamma$ be a continuous real $(1,1)$-form such that $T\ge\gamma$.
Suppose that $T_X$ is equipped with a smooth hermitian metric $\omega$
such that the Chern curvature form $\Theta(T_X)$ satisfies
$$\big(\Theta(T_X)+u\otimes\Id_{T_X}\big)(\theta\otimes\xi,\theta\otimes\xi)
\ge 0~~~~\forall\theta,\xi\in T_X~~~\hbox{\rm with}~~
\langle\theta,\xi\rangle=0,$$
for some continuous nonnegative $(1,1)$-form $u$ on~$X$. Then 
there is a family of closed almost positive $(1,1)$-currents
$T_\varepsilon=\alpha+{i\over\pi}\partial\op\psi_\varepsilon$,
$\varepsilon\in{}]0,\varepsilon_0[$, such that $\psi_\varepsilon$
is smooth over~$X$, increases with~$\varepsilon$,
and converges to $\psi$ as $\varepsilon$ tends to~$0$ $($in particular,
$T_\varepsilon$ is smooth and converges weakly to $T$ on~$X)$,
and such that
\item{\rm(i)}$T_\varepsilon\ge\gamma-\lambda_\varepsilon u
-\delta_\varepsilon\omega$~~ where:
\item{\rm(ii)}$\lambda_\varepsilon(x)$ is an increasing family of
continuous functions on $X$ such that $\lim_{\varepsilon\to 0}
\lambda_\varepsilon(x)=\nu(T,x)$~ $($Lelong number of $T$ at~$x)$
at every point,
\item{\rm(iii)}$\delta_\varepsilon$ is an increasing family of positive
constants such that $\lim_{\varepsilon\to 0}\delta_\varepsilon=0$.
\fintha

More precise results are given in Theorems~4.1 and 6.1. Such
approximations can in turn be used to obtain various estimates of
intersection theory [De91b], or asymptotic inequalities for Dolbeault
cohomology~[De90]. They can be also applied to study compact complex
manifolds with partially semipositive curvature in the sense of
Griffiths (see Section~5); in that case, we prove for instance that
every effective divisor is nef (i.e. numerically effective), and that
the variety is projective if and only if it is Moishezon.

Our proof uses some ideas already developed in~[De82], although more
general and more precise results will be obtained here. The main idea
is to use a convolution kernel constructed by means of the
exponential map associated to a Chern connection on~$T_X$.
To get precise estimates of the Hessian forms involved, we determine
the Taylor expansion of the exponential map at order~$3$.
The third order coefficients can be calculated explicitly in terms of
the curvature tensor of the metric. What is perhaps most remarkable is
that we have been ultimately able to find the complete Taylor expansion
of the exponential convolution kernel (Proposition~3.8); this is indeed
possible because we use a modified exponential map which is made fiberwise
quasi-holomorphic (see Section~2). Finally, we apply Kiselman's singularity
attenuation technique [Ki78], [Ki79] in combination with our main estimates
to define a partial regularization process for closed
\hbox{$(1,1)$-currents}: in that way, the Lelong numbers can be killed
up to any given level (Theorem~6.1).

Further techniques based on H\"ormander's $L^2$ existence theorems
[H\"o66] are explained in our recent papers [De91a], [De92]; they lead
to similar estimates, but with a numerical hypothesis instead of a
curvature hypothesis: namely, $u$ should then be a closed real
$(1,1)$-form such that the cohomology class $c_1(\oc_{T_X}(1))+\pi^\star
u$ is nef on the total space of the projective bundle
\hbox{$P(T^\star_X)\buildo\pi\over\to X$} of hyperplanes in~$T_X$. This
condition, which is more natural than a curvature hypothesis from the
point of view of algebraic geometry, is also more general than the
Griffiths semipositivity of $\Theta(T_X)+u\otimes\Id_{T_X}$. However,
it is not clear how the above numerical condition can be related to the
partial semipositivity hypothesis made in Theorem~1.1 (see the comments
after Definition~5.1); for instance, the partial semipositivity
hypothesis is void for curves. Therefore, both types of hypotheses seem
to have their own domain of applicability. Moreover, the techniques
developed here are considerably simpler and in some sense more precise
and more explicit, so we felt interesting to explain this simpler
method, which is probably easier to extend to currents of higher
bidegrees. The main ideas of this work have been worked out during a
stay of the author at Bayreuth University in November~1989. The author
wishes to thank this Institution for its hospitality.

\titre 2. Exponential map associated to the Chern connection|

Suppose that the manifold $X$ is equipped with a smooth hermitian
metric \hbox{$\omega=i\sum\omega_{lm}dz_l\wedge d\ov z_m$.} Denote by
$D$ the Chern connection of $T_X$ and by $\Theta(T_X)={i\over 2\pi}D^2$
the curvature tensor. We define an exponential map  $\exp:T_X\la X$ as
follows: if $\zeta\in T_{X,z}$, then $\exp_z(\zeta)$  is the position
at time $t=1$ of the curve $t\mapsto u(t)$ starting at  $u(0)=z$ with
initial tangent vector $u'(0)=\zeta$ and satisfying the  second order
differential equation $D(du/dt)=0$ (parallel translation with respect
to $D$). If~$\omega$ is K\"ahler, the Chern connection coincides with
the Levi-Civita connection, so exp is given in that case by the
riemannian geodesics; otherwise, exp differs from the usual riemannian
exponential map. For any $x\in X$, fix analytic coordinates
$(z_1\ld z_n)$ centered at $x$ such that $(\partial/\partial z_l)$ is
an orthonormal basis of $T_X$ at $x$. Consider the Taylor expansion of
second order
$$\eqalign{
\omega_{lm}(z)=	
\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle=
\delta_{lm}+\sum_j(a_{jlm}z_j+\ov a_{jml}\ov z_j)
&+\sum_{j,k}(b'_{jklm}z_jz_k+\ov b'_{jkml}\ov z_j\ov z_k)\cr
&+\sum_{j,k}c'_{jklm}z_j\ov z_k+O(|z|^3).\cr}$$
We may always arrange that the antisymmetry relation $a_{jlm}=-a_{ljm}$ holds;
\hbox{otherwise} the change of variables $z_m=z_m'-{1\over 4}
\sum(a_{jlm}+a_{ljm})z'_jz'_l$ yields coordinates $(z'_l)$ with this property.
If~$\omega$ is K\"ahler, the symmetry of $a_{jlm}=\partial\omega_{lm}/\partial
z_j$ in $j,l$ implies $a_{jlm}=0\,$; in that case $b'_{jklm}$ is also
symmetric in $j,k,l$ and a new change of variables
$z_m=z_m'-{1\over 3}\sum b'_{jklm}z'_jz'_kz'_l$ gives $b'_{jklm}=0$ likewise.
The holomorphic frame of $T_X$ defined by
$$e_l={\partial\over\partial z_l}-\sum_m\Big(\sum_j a_{jlm}z_j+
\sum_{j,k}b'_{jklm}z_jz_k\Big){\partial\over\partial z_m}$$
satisfies
$$\leqalignno{
\langle e_l,e_m\rangle
&=\delta_{lm}-\sum_{j,k}c_{jklm}z_j\ov z_k+O(|z|^3),&(2.1)\cr
{\partial\over\partial z_l}
&=e_l+\sum_m\Big(\sum_j a_{jlm}z_j+\sum_{j,k}b_{jklm}z_jz_k+O(z^3)\Big)e_m
&(2.2)\cr}$$
with $c_{jklm}=-c'_{jklm}-\sum_p a_{jlp}\ov a_{kmp}$ and
$b_{jklm}=b'_{jklm}+\sum_p a_{jlp}a_{kpm}$. We may of course suppose
that $b_{jklm}=b_{kjlm}$. Also, by a modification of the third
order terms in $(e_l)$, we can suppose that no term $O(z^3)$ appears
in (2.2). The formula $\partial\langle e_l,e_m\rangle=
\langle De_l,e_m\rangle$ easily gives the expression of $De_l$, $D^2e_l$
and $\Theta(T_X)_{x}$:
$$\leqalignno{
De_l&=-\sum_{j,k,m}c_{jklm}\ov z_k\,dz_j\otimes e_m+O(|z|^2),\cr
\Theta(T_X)_{x}&={i\over 2\pi}\sum_{j,k,l,m}c_{jklm}\,dz_j\wedge d\ov z_k
\otimes e^\star_l\otimes e_m.&(2.3)\cr}$$
Given a vector field $\zeta=\sum\zeta_l\,\partial/\partial z_l$ in $T_X$,
we denote by $(\xi_l)$ the components of $\zeta$ with respect to the basis
$(e_l)$, thus $\zeta=\sum\xi_le_l$. By (2.2) we have
$$\xi_m=\zeta_m+\sum_{j,l}a_{jlm}z_j\zeta_l+\sum_{j,k,l}b_{jklm}z_jz_k\zeta_l.
\leqno(2.4)$$
In the K\"ahler case everything is much simpler, we take
$e_l=\partial/\partial z_l$ and $\xi_m=\zeta_m$. In general,
the Chern connection $D$ is given by $D\zeta=D\big(\sum\zeta_l\,
\partial/\partial z_l\big)$ with
$$\leqalignno{
D\Big({\partial\over\partial z_l}\Big)&=
-\sum_{j,k,m}c_{jklm}\ov z_k\,dz_j\otimes e_m
+\sum_{j,m}a_{jlm}dz_j\otimes e_m\cr
&\phantom{{}=-\sum_{j,k,m}c_{jklm}\ov z_k\,dz_j\otimes e_m}
+2\sum_{j,k,m}b_{jklm}z_k\,dz_j\otimes e_m+O(|z|^2)\,dz\cr
&=-\sum_{j,k,m}(c_{jklm}\ov z_k-2b_{jklm}z_k)
dz_j\otimes{\partial\over\partial z_m}\cr
&\phantom{{}={}}+\sum_{j,m}\Big(a_{jlm}-\sum_{k,p}a_{jlp}a_{kpm}z_k\Big)
dz_j\otimes{\partial\over\partial z_m}+O(|z|^2)\,dz,\cr
D\zeta&=\sum_m d\zeta_m\otimes{\partial\over\partial z_m}
-\sum_{j,k,l,m}(c_{jklm}\ov z_k-2b_{jklm}z_k)
\zeta_l dz_j\otimes{\partial\over\partial z_m}&(2.5)\cr
&\phantom{{}={}}+\sum_{j,l,m}\Big(a_{jlm}-\sum_{k,p}a_{jlp}a_{kpm}z_k\Big)
\zeta_ldz_j\otimes{\partial\over\partial z_m}+O(|z|^2){\cdot}\zeta\,dz.\cr}$$
Consider a curve $t\mapsto u(t)$. By a substitution of variables 
$z_j=u_j(t)$, $\zeta_l=du_l/dt$ in formula (2.5), the equation 
$D(du/dt)=0$ becomes
$${d^2u_m\over dt^2}=\sum_{j,k,l}\big(c_{jklm}\ov u_k(t)-
2b_{jklm}u_k(t)\big){du_j\over dt}{du_l\over dt}
+O\big(|u(t)|^2\big){\cdot}\Big({du\over dt}\Big)^2\,;\leqno(2.6)$$
the contribution of the terms $\sum a_{jl\bu}\zeta_ldz_j$ is zero
by the antisymmetry relation; moreover the remainder term only 
contains $\cb$-quadratic terms in
$du/dt$. The initial condition $u(0)=z$, $u'(0)=\zeta$ gives
$u_m(t)=z_m+t\zeta_m+O(t^2|\zeta|^2)$, hence
$${d^2u_m\over dt^2}=\sum_{j,k,l}\big(c_{jklm}(\ov z_k+t\ov\zeta_k)-2b_{jklm}
(z_k+t\zeta_k)\big)\zeta_j\zeta_l+O\big(|\zeta|^2(|z|+|\zeta|)^2\big).$$
Two successive integrations yield
$$\eqalign{
u_m(t)=z_m+t\zeta_m+\sum_{j,k,l}c_{jklm}&\Big({t^2\over 2}\ov z_k
+{t^3\over 6}\ov\zeta_k\Big)\zeta_j\zeta_l\cr
-2\sum_{j,k,l}b_{jklm}&\Big({t^2\over 2}z_k+{t^3\over 6}
\zeta_k\Big)\zeta_j\zeta_l+O\big(t^2|\zeta|^2(|z|+|\zeta|)^2\big).\cr}$$
An iteration of this procedure (substitution in (2.6) followed by an
integration) easily shows that all terms but the first two in the
Taylor expansion of $u_m(t)$ contain $\cb$-quadratic factors of the
form~$\zeta_j\zeta_l$. Let us substitute $\zeta_m$ by its expression
in terms of $z,\xi$ deduced from (2.4). We find that
$\exp_z(\zeta)=u(1)$ has a third order expansion
$$~~\exp_z(\zeta)_m=g_m(z,\xi)+\sum_{j,k,l}c_{jklm}
\Big({1\over2}\ov z_k+{1\over 6}\ov\xi_k\Big)
\xi_j\xi_l+O\big(|\xi|^2(|z|+|\xi|)^2\big)\leqno(2.7)$$
where
$$\eqalign{
g_m(z,\xi)=z_m+\xi_m&{}-\sum_{j,l}a_{jlm}z_j\xi_l+
\sum_{j,k,l,p}a_{jlp}a_{kpm}z_jz_k\xi_l\cr
&{}-\sum_{j,k,l}b_{jklm}\Big(z_jz_k\xi_l+z_k\xi_j\xi_l
+{1\over 3}\xi_j\xi_k\xi_l\Big)\cr}$$
is a holomorphic polynomial of degree $3$ in $z,\xi$
and where the remainder involves $\cb$-quadratic factors $\xi_j\xi_l$ in
all terms. In the K\"ahler case we simply have $\xi_m=\zeta_m$ and
$g_m(z,\xi)=z_m+\xi_m$.

The exponential map is unfortunately non holomorphic. However, we can make
it quasi-holomorphic with respect to $\zeta$ as follows:
for~$z$ fixed, we consider the formal power series obtained by eliminating
all monomials in the Taylor expansion of $\zeta\mapsto\exp_z(\zeta)$ at the
origin which are not holomorphic with respect to~$\zeta$. This defines
in a unique way a jet of infinite order along the zero section of~$T_X$.
E.~Borel's theorem shows that there is a smooth map
$$T_X\la X,~~~~(z,\zeta)\longmapsto\exph_z(\zeta)$$
such that its jet at $\zeta=0$ coincides with the ``holomorphic''
part of $\zeta\mapsto\exp_z(\zeta)$ (of course, this map is defined
only up to an addition of flat $C^\infty$ functions along the zero section
of~$T_X$). Moreover, (2.7) implies that
$$~~\exph_z(\zeta)_m=g_m(z,\xi)+{1\over2}\sum_{j,k,l}c_{jklm}\ov z_k
\xi_j\xi_l+O\big(|\xi|^2(|z|+|\xi|)^2\big)\leqno(2.8)$$
By including in $g_m$ all holomorphic monomials of partial degree at most $2$
in $z$ and $N$ in $\xi$ ($N\ge 2$ being a given integer), we get
holomorphic polynomials $h_m(z,\xi)$ of linear part $z_m+\xi_m$ and
total degree $N+2$, such that
$$\exph_z(\zeta)_m=h_m(z,\xi)+
O\big(\ov z,z\ov z,\ov z\ov z,|z|^3,\xi^{N-1}\big)\xi^2.$$
Here a notation as $O\big(\ov z,z\ov z,\ov z\ov z,|z|^3,\xi^{N-1}\big)\xi^2$
indicates an arbitrary function in the ideal of $C^\infty$ functions
generated by monomials of the form $\ov z_k\xi_l\xi_m$, $z_i\ov z_j\xi_l
\xi_m$, $\ov z_i\ov z_j\xi_l\xi_m$, $z^\alpha\ov z^\beta\xi_l\xi_m$ and
$\xi^\gamma$, for all multi-indices $|\alpha|+|\beta|=3$ and $|\gamma|=N+1$
(the notation $|z|^3$ thus stands for an arbitrary monomial of degree $3$
in $z$, $\ov z$, so that $O(|z|^3)$ is compatible with the usual Landau
notation). By the implicit function theorem applied to the mapping
$h=(h_m)_{1\le m\le n}$ we thus get:

\tha Proposition 2.9|Let $\omega$ be a smooth hermitian metric on~$X$.
There exists a $C^\infty$ map
$$T_X\la X,~~~~(x,\zeta)\longmapsto\exph_x(\zeta),~~~~\zeta\in T_{X,x}$$
with the following properties:
\smallskip
\item{\rm(i)} For every $x\in X$, $\exph_x(0)=x$ and $d_\zeta\exph_x(0)=
\Id_{T_{X,x}}$.
\smallskip
\item{\rm(ii)} For every $x\in X$, the map $\zeta\mapsto\exph_x(\zeta)$ has a
holomorphic Taylor expansion at $\zeta=0$. Moreover, with respect to~$\omega$,
there are local normal coordinates $(z_1,\ldots,z_n)$ on $X$ centered
at~$x$ and holomorphic normal coordinates $(\xi_j)$ on the fibers of $T_X$
near~$x$ such that
$$\exph_z(\zeta)=h_x\big(z,\rho_x(z,\xi)\big),$$
where $h_x(z,\xi)$ is a holomorphic polynomial map of degree 2 in $z$ and
of degree $N$~in~$\xi$, and where $\rho_x:\cb^n\times\cb^n\to\cb^n$ is a
smooth map such that
$$\eqalign{
h_{x,m}(z,\xi)&=z_m+\xi_m-\sum_{j,l}a_{jlm}z_j\xi_l
+\sum_{j,k,l,p}a_{jlp}a_{kpm}z_jz_k\xi_l\cr
&\phantom{{}=z_m+\xi_m}-\sum_{j,k,l}b_{jklm}\Big(z_jz_k\xi_l+z_k\xi_j\xi_l
+{1\over 3}\xi_j\xi_k\xi_l\Big)\cr
&\phantom{{}=z_m+\xi_m}+O\big((|z|+|\xi|)^4\big),\cr
\rho_{x,m}(z,\xi)&=\xi_m+\sum_{2\le|\alpha|\le N}\Big(
\sum_k d_{\alpha km}\xi^\alpha\ov z_k
+\sum_{j,k}e_{\alpha jkm}\xi^\alpha z_j\ov z_k\Big)\cr
&\phantom{{}=z_m+\xi_m}
+O\big(\ov z^2,|z|^3,\xi^{N-1}\big)\xi^2.\cr}$$
\smallskip
\item{\rm(iii)} For $\alpha=(0,\ldots,1_j,\ldots,1_l,\ldots,0)$ of degree~$2$,
we have $d_{\alpha km}={1\over 2}c_{jklm}$ where $(c_{jklm})$ is the curvature
tensor of $\omega$ at~$x$.\smallskip
\fintha

Of course, if the hermitian metric $\omega$ is real analytic, all the above
expansions are convergent, hence $\exph_z(\zeta)$ is real analytic and
holomorphic in $\zeta$ in a neighborhood of the zero section of~$T_X$.
By taking $N=\infty$, we obtain a real analytic map $\rho(x,\xi)$ which
is holomorphic in $\xi$, so the above remainder term becomes
$O\big(\ov z^2,|z|^3\big)\xi^2$.

\titre 3. Regularization of quasi-psh functions|

We now come to the main idea. Select a 
cut-off function $\chi:\rb\to\rb$ of class $C^\infty$ such that
$$\chi(t)>0~~\hbox{\rm for}~~t<1,~~~~\chi(t)=0~~\hbox{\rm for}~~t\ge 1,
~~~~\int_{v\in\cb^n}\chi(|v|^2)\,d\lambda(v)=1.$$
If $\psi$ is a quasi-psh function on $X$, we set
$$\psi_\varepsilon(z)={1\over\varepsilon^{2n}}
\int_{\zeta\in T_{X,z}}\psi\big(\exph_z(\zeta)\big)\,
\chi\Big({|\zeta|^2\over\varepsilon^2}\Big)\,d\lambda(\zeta),
~~~~\varepsilon>0.\leqno(3.1)$$
Here $d\lambda$ denotes the Lebesgue measure on $\cb^n$, resp. on the 
hermitian space $\big(T_{X,z},\omega(z)\big)$. For $w\in\cb$ with 
$|w|=\varepsilon$, we have $\psi_\varepsilon(z)=\Psi(z,w)$ with
$$\Psi(z,w)=\int_{\zeta\in T_{X,z}}\psi\big(\exph_z(w\zeta)\big)\,
\chi(|\zeta|^2)\,d\lambda(\zeta).\leqno(3.2)$$
The change of variable $y=\exph_z(w\zeta)$ expresses $w\zeta$ as a smooth
function of $y,z$ in a neighborhood of the diagonal in $X\times X$.
Hence $\Psi$ is smooth over $X\times\{0<|w|<\varepsilon_0\}$ for some
$\varepsilon_0>0$. We are going to compute $\partial\op\Psi$ over this
set and estimate its negative part when $|w|$ is small. For this, we
fix a point $x\in X$ and use the coordinates $(z,\xi)$ on $T_X$ introduced
in~\S$\,$2; for simplicity, we omit the index $x$ in the notation of
$h_x$ and~$\rho_x$. By (2.1), we have
$$\leqalignno{
~~~~|\zeta|^2&=\sum_m|\xi_m|^2-\sum_{j,k,l,m}c_{jklm}\,z_j\ov z_k\xi_l\ov\xi_m
+O(|z|^3)|\xi|^2,&(3.3)\cr
~~~~d\lambda(\zeta)&={1\over 2^nn!}(i\partial\op|\zeta|^2)^n&(3.4)\cr
&=\Big(1-\sum_{j,k,l}c_{jkll}\,z_j\ov z_k+O(|z|^3)\Big){i\over 2}d\xi_1
\wedge d\ov\xi_1\wedge\ldots\wedge{i\over 2}d\xi_n\wedge d\ov\xi_n.\cr}$$
In (3.2), we make the change of variables $s=w^{-1}\rho(z,w\xi)$, hence
we can write
$\exph_z(w\zeta)=h(z,ws)$. By (2.9) we get
$$\leqalignno{
\qquad s_m=\xi_m+\sum_{2\le|\alpha|\le N}\Big(
\sum_k d_{\alpha km}w^{|\alpha|-1}\xi^\alpha\ov z_k&{}+
\sum_{j,k}e_{\alpha jkm}w^{|\alpha|-1}\xi^\alpha z_j\ov z_k\Big)&(3.5)\cr
&{}+O\big(\ov z^2,|z|^3,w^{N-1}\xi^{N-1}\big)w\xi^2.\cr}$$
Therefore
$$\leqalignno{
\qquad\xi_m=s_m-\sum_{2\le|\alpha|\le N}\Big(
\sum_k d_{\alpha km}w^{|\alpha|-1}s^\alpha\ov z_k&{}+
\sum_{j,k}e_{\alpha jkm}w^{|\alpha|-1}s^\alpha z_j\ov z_k\Big)&(3.6)\cr
&{}+O\big(\ov z^2,|z|^3,w^{N-1}s^{N-1}\big)ws^2\cr}$$
and $\xi=s+O(w^Ns^{N+1})$ for $z=0$. After a substitution in (3.2), (3.3),
(3.4) we get
$$\Psi(z,w)=\int_{\cb^n}\psi\big(h(z,ws)\big)\,
\chi\big(A(z,w,s)\big)\,B(z,w,s)\,d\lambda(s)\leqno(3.7)$$
where
$$\eqalign{
A(z,w,s)&=\sum_m|s_m|^2-\sum_{j,k,l,m}c_{jklm}z_j\ov z_ks_l\ov s_m\cr
&\qquad{}-2\re\sum_{\alpha,k,m}d_{\alpha km}w^{|\alpha|-1}s^\alpha
\ov s_m\ov z_k\cr
&\qquad{}-2\re\sum_{\alpha,j,k,m}e_{\alpha jkm}w^{|\alpha|-1}s^\alpha\ov s_m
z_j\ov z_k\cr
&\qquad{}+\sum_{\alpha,\beta,j,k,m}d_{\alpha km}\ov{d_{\beta jm}}\,
w^{|\alpha|-1}\ov w^{|\beta|-1}s^\alpha\ov s^\beta z_j\ov z_k\cr
&\qquad{}+O\big(z^2,\ov z^2,|z|^3,|w|^{N-1}|s|^{N-1}\big)|w||s|^3,\cr}$$
$$\eqalign{
B(z,w,s)&=1-\sum_{j,k,l}c_{jkll}z_j\ov z_k\cr
&\qquad{}-2\re\sum_{\alpha,k,m}d_{\alpha km}w^{|\alpha|-1}\alpha_m
s^{\alpha-\un_m}\ov z_k\cr
&\qquad{}-2\re\sum_{\alpha,j,k,m}e_{\alpha jkm}w^{|\alpha|-1}\alpha_m
s^{\alpha-\un_m}z_j\ov z_k\cr
&\qquad{}+\sum_{\alpha,\beta,j,k,l,m}d_{\alpha km}\ov{d_{\beta jl}}\,
w^{|\alpha|-1}\ov w^{|\beta|-1}\alpha_m\beta_ls^{\alpha-\un_m}\ov s^{\beta-\un_l}
z_j\ov z_k\cr
&\qquad{}+O\big(z^2,\ov z^2,|z|^3,|w|^{N-1}|s|^{N-1}\big)|w||s|\,;\cr}$$
here $(\un_m)_{1\le m\le n}$ denotes the standard basis of $\zb^n$,
hence $s^{\un_m}=s_m$.

\tha Proposition 3.8|For any integer $N\ge 2$ and any
$(\theta,\eta)\in T_{X,x}\times\cb$, the Hessian form
of $\Psi$ at $(x,w)\in X\times\cb$ satisfies an estimate
$$\eqalign{
\partial\op\Psi_{(x,w)}[\theta,\eta]^2=\int_{\zeta\in T_{X,x}}
\partial\op\psi\cdot\big(\tau\wedge\ov\tau&{}+|w|^2V\big)_{\exph_x(w\zeta)}\,
\chi(|\zeta|^2)\,d\lambda(\zeta)\cr
&{}+O\big(|w|^{N-1}\big)[\theta,\eta]^2\cr}$$
where $\tau$ $($resp. $V)$ is a vector field $($resp. a $(1,1)$-vector
field$)$ depending smoothly on the parameters $x,w$ and linearly $($resp.
quadratically$)$ on $\theta,\eta$, and where the notation $[\tau]^2$
stands for the $(1,1)$-vector $\tau\wedge\ov\tau\in\Lambda^{1,1}T_X$.
The vector fields $\tau$, $V$ are given at $y=\exph_x(w\zeta)$ by
$$\eqalign{
\tau_y&=
\partial\exph_{(x,w\zeta)}\big(\theta^h+\eta\zeta^v+|w|^2\Xi^v_y\big),\cr
V_y&=
\partial\exph_{(x,w\zeta)}(U^v-|w|^2\Xi^v\wedge\ov{\hbox{$\Xi^v$}}\big)_y,\cr}$$
where $\theta^h,\zeta^v\in T(T_X)_{(x,w\zeta)}$ are respectively the
horizontal lifting of $\theta$ with respect to the connection $D$ and
the vertical tangent vector associated to $\zeta$, and where $\Xi$,~$U$
can be expressed in the coordinates {\rm 2.9$\,$(ii)} by
$$\eqalign{
\Xi_y&=\sum_{\alpha,j,l,m}{1\over\chi(|\zeta|^2)}{\partial\over\partial
\ov\zeta_l}\big(\chi_1(|\zeta|^2)\ov \zeta^{\alpha-\un_m}\big)
\ov{d_{\alpha jl}}\,{\alpha_m\over|\alpha|}\,\ov w^{|\alpha|-2}
\theta_j\,{\partial\over\partial z_m},\cr
U_y&=\sum_{l,m}{1\over 2}(U_{m,l}+\ov{U_{l,m}})
{\partial\over\partial z_m}\wedge{\partial\over\partial\ov z_l},\cr
U_{m,l}&=-{\chi_1(|\zeta|^2)\over\chi(|\zeta|^2)}\Big\{
\sum_{j,k}c_{jklm}\theta_j\ov\theta_k\cr
&\phantom{{}=-{\chi_1(|s|^2)\over\chi(|s|^2)}\Big\{}
+2\sum_{\alpha,j,k}e_{\alpha jkm}w^{|\alpha|-1}{\alpha_l\over|\alpha|}
\zeta^{\alpha-\un_l}\theta_j\ov\theta_k\cr
&\phantom{{}=-{\chi_1(|s|^2)\over\chi(|s|^2)}\Big\{}
+2\sum_{\alpha,k}d_{\alpha km}(|\alpha|-1)w^{|\alpha|-2}
{\alpha_l\over|\alpha|}\zeta^{\alpha-\un_l}\eta\ov\theta_k\Big\}\cr
&\phantom{{}={}}+\sum_{\alpha,\beta,j,k}d_{\alpha km}\ov{d_{\beta jl}}\,
w^{|\alpha|-1}\ov w^{|\beta|-1}\zeta^\alpha
\rlap{\hbox{$\ov{\phantom{\zeta}}$}}\zeta^\beta\theta_j\ov\theta_k.
\cr}$$
Here $(c_{jklm})$ is the curvature tensor of the given hermitian metric
$\omega$ on~$X$, and $\chi_1$ denotes the primitive
$\chi_1(t)=\int_{+\infty}^t\chi(u)\,du$ of $\chi$ such that
$\chi_1(t)=0$ for $t\ge 1$. Moreover, $\alpha,\beta\in\nb^n$ run over
all multi-indices such that $2\le|\alpha|,~|\beta|\le N$. The~formula is
exact when $\omega$ is real analytic and $N=\infty$.\fintha

\remarque Proof|A brute force differentiation of (3.7) at $z=0$ gives
$$\leqalignno{
\partial\op\Psi_{(x,w)}[\theta,\eta]^2
&=\int_{\cb^n}\partial\op(\psi\circ h)_{(0,ws)}[\theta,\eta s]^2\,
\chi\big(A(0,w,s)\big)\,B(0,w,s)\,d\lambda(s)\cr
&\qquad{}-\int_{\cb^n}\op(\psi\circ h)_{(0,ws)}[\theta,\eta s]\,
E_{(w,s)}[\theta,\eta]\,d\lambda(s)&(3.9)\cr
&\qquad{}-\int_{\cb^n}\partial(\psi\circ h)_{(0,ws)}[\theta,\eta s]\,
\ov{E_{(w,s)}[\theta,\eta]}\,d\lambda(s)\cr
&\qquad{}-\int_{\cb^n}\psi\circ h(0,ws)\,F_{(w,s)}[\theta,\eta]^2\,d\lambda(s)
&(3.9')\cr
&&\hbox{\rm where}\cr
E_{(w,s)}&=-\partial_{(z,w)}\Big(\chi\big(A(z,w,s)\big)B(z,w,s)\Big),\cr
F_{(w,s)}&=-\partial\op_{(z,w)}\Big(\chi\big(A(z,w,s)\big)B(z,w,s)
\Big)\cr}$$
at $z=0$, $|w|<\varepsilon_0$. Clearly, terms $w^pz\ov z$ in $A$ or $B$ play
no role in $E_{(w,s)}\,$, while terms $w^p\ov z$ contribute either linearly
as $dw\wedge d\ov z$ differentials of~$F_{(w,s)}$ or quadratically as
$dz\wedge d\ov z$ differentials. This gives
$$\leqalignno{
E_{(w,s)}[\theta,\eta]
&=\chi'(|s|^2)\sum_{\alpha,j,l}\ov{d_{\alpha jl}}\,\ov w^{|\alpha|-1}
\ov s^\alpha s_l\theta_j\cr
&\qquad{}+\chi(|s|^2)\sum_{\alpha,j,l}\ov{d_{\alpha jl}}\,\ov w^{|\alpha|-1}
\alpha_l\ov s^{\alpha-\un_l}\theta_j
+O(|w|^{N-1}|s|^N)[\theta,\eta],\phantom{\sum_j}\cr
F_{(w,s)}[\theta,\eta]^2
&=\chi'(|s|^2)\sum_{j,k,l,m}c_{jklm}s_l\ov s_m\theta_j\ov\theta_k
+\chi(|s|^2)\sum_{j,k,l}c_{jkll}\theta_j\ov\theta_k\cr
&\qquad{}+2\re\bigg\{\chi'(|s|^2)\sum_{\alpha,j,k,m}
e_{\alpha jkm}w^{|\alpha|-1}s^\alpha\ov s_m\theta_j\ov\theta_k\cr
&\qquad{}+\chi(|s|^2)\sum_{\alpha,j,k,m}e_{\alpha jkm}
w^{|\alpha|-1}\alpha_ms^{\alpha-\un_m}\theta_j\ov\theta_k\cr
&\qquad{}+\chi'(|s|^2)\sum_{\alpha,k,m}
d_{\alpha km}(|\alpha|-1)w^{|\alpha|-2}s^\alpha\ov s_m\eta\ov\theta_k\cr
&\qquad{}+\chi(|s|^2)\sum_{\alpha,k,m}
d_{\alpha jkm}(|\alpha|-1)w^{|\alpha|-2}\alpha_ms^{\alpha-\un_m}
\eta\ov\theta_k\bigg\}\cr
&\qquad{}-\chi''(|s|^2)\sum_{\alpha,\beta,j,k,l,m}d_{\alpha km}
\ov{d_{\beta jl}}\,w^{|\alpha|-1}\ov w^{|\beta|-1}s^\alpha\ov s^\beta
s_l\ov s_m\theta_j\ov\theta_k\cr
&\qquad{}-\chi'(|s|^2)\sum_{\alpha,\beta,j,k,m}d_{\alpha km}
\ov{d_{\beta jm}}\,w^{|\alpha|-1}\ov w^{|\beta|-1}s^\alpha\ov s^\beta
\theta_j\ov\theta_k\cr
&\qquad{}-\chi'(|s|^2)\sum_{\alpha,\beta,j,k,l,m}d_{\alpha km}
\ov{d_{\beta jl}}\,w^{|\alpha|-1}\ov w^{|\beta|-1}
s^\alpha\ov s_m\beta_l\ov s^{\beta-\un_l}\theta_j\ov\theta_k\cr
&\qquad{}-\chi'(|s|^2)\sum_{\alpha,\beta,j,k,l,m}d_{\alpha km}
\ov{d_{\beta jl}}\,w^{|\alpha|-1}\ov w^{|\beta|-1}\alpha_ms^{\alpha-\un_m}
\ov s^\beta s_l\theta_j\ov\theta_k\cr
&\qquad{}-\chi(|s|^2)\sum_{\alpha,\beta,j,k,l,m}d_{\alpha km}
\ov{d_{\beta jl}}\,w^{|\alpha|-1}\ov w^{|\beta|-1}\alpha_m\beta_l
s^{\alpha-\un_m}\ov s^{\beta-\un_l}\theta_j\ov\theta_k\cr
&\qquad{}+O(|w|^{N-2}|s|^N)[\theta,\eta]^2.\cr}$$
We find
$$\leqalignno{
E_{(w,s)}[\theta,\eta]
&=\sum_{l,m}{\partial^2\over\partial\ov s_l\partial s_m}
\Big(\chi_1(|s|^2)\sum_{\alpha,j}\ov{d_{\alpha jl}}\,\ov w^{|\alpha|-1}
{\alpha_m\over|\alpha|}\ov s^{\alpha-\un_m}\theta_j\Big)\cr
&\kern125pt{}+O(|w|^{N-1}|s|^N)[\theta,\eta],\phantom{\sum_j}\cr}$$
$$\leqalignno{
F_{(w,s)}[\theta,\eta]^2
&=\sum_{l,m}{\partial^2\over\partial\ov s_l\partial s_m}
\Big(\chi_1(|s|^2)\sum_{j,k}c_{jklm}\theta_j\ov\theta_k\Big)\cr
&\qquad{}+2\re\Big\{\sum_{l,m}{\partial^2\over\partial\ov s_l\partial s_m}
\Big(\chi_1(|s|^2)\sum_{\alpha,j,k}e_{\alpha jkm}w^{|\alpha|-1}
{\alpha_l\over|\alpha|}s^{\alpha-\un_l}\theta_j\ov\theta_k\Big)\cr
&\qquad{}+\sum_{l,m}{\partial^2\over\partial s_l\partial\ov s_m}
\Big(\chi_1(|s|^2)\sum_{\alpha,k}d_{\alpha km}(|\alpha|-1)w^{|\alpha|-2}
{\alpha_l\over|\alpha|}\ov s^{\alpha-\un_l}\eta\ov\theta_k\Big)\Big\}\cr
&\qquad{}-\sum_{l,m}{\partial^2\over\partial\ov s_l\partial s_m}
\Big(\chi(|s|^2)\sum_{\alpha,\beta,j,k}d_{\alpha km}\ov{d_{\beta jl}}\,
w^{|\alpha|-1}\ov w^{|\beta|-1}s^\alpha\ov s^\beta\theta_j\ov\theta_k\Big)\cr
&\qquad{}+O(|w|^{N-2}|s|^N)[\theta,\eta]^2.\cr}$$
In all these expansions, the remainder terms $O(\bu)$
involve uniform constants when the origin $x$ of coordinates belongs to a
compact subset of a coordinate patch.  By the mean value properties of
plurisubharmonic functions, we have
$$\int_{|s|<1}|\psi(x+ws)|\,d\lambda(s)\le C\big(1+\big|\log|w|\big|\big)$$
locally uniformly in $x$. An integration by parts with compact supports
yields
$$\int_{|s|<1}\partial(\psi\circ h)_{(0,ws)}\,O(|w|)\,d\lambda(s)=
\int_{|s|<1}\psi\circ h(0,ws)\,d\lambda(s)=O\big(\log|w|\big),$$
hence the remainder terms $O(|w|^{N-1})$ in $E_{(w,s)}$ and $O(|w|^{N-2})$
in $F_{(w,s)}$ give contributions of order at most $O(|w|^{N-2}\log|w|)$ in
$\partial\op\Psi$ as $|w|$ tends to $0$. An integration by parts in $(3.9)$
and $(3.9')$ gives
$$\leqalignno{
\partial\op\Psi_{(x,w)}&[\theta,\eta]^2=\int_{\cb^n}
\partial\op(\psi\circ h)_{(0,ws)}\cdot\Big\{(\theta,\eta s)\wedge
\ov{(\theta,\eta s)}+|w|^2(0,\Xi)\wedge\ov{(\theta,\eta s)}\cr
&{}+|w|^2(\theta,\eta s)\wedge\ov{(0,\Xi)}+
|w|^2(0,U)\Big\}\,\chi\big(A(0,w,s)\big)\,B(0,w,s)\,d\lambda(s)\cr
&{}+O\big(|w|^{N-2}\log|w|\big)[\theta,\eta]^2&(3.10)\cr}$$
with
$$\eqalign{
\Xi&=\sum_{\alpha,j,l,m}{1\over\chi(|s|^2)}{\partial\over\partial\ov s_l}
\big(\chi_1(|s|^2)\ov s^{\alpha-\un_m}\big)
\ov{d_{\alpha jl}}\,{\alpha_m\over|\alpha|}\,\ov w^{|\alpha|-2}
\theta_j\,{\partial\over\partial z_m},\cr
U&=\sum_{l,m}{1\over 2}(U_{m,l}+\ov{U_{l,m}})
{\partial\over\partial z_m}\wedge{\partial\over\partial\ov z_l},\cr}$$
$$\leqalignno{
U_{m,l}&=-{\chi_1(|s|^2)\over\chi(|s|^2)}\Big\{
\sum_{j,k}c_{jklm}\theta_j\ov\theta_k\cr
&\phantom{{}=-{\chi_1(|s|^2)\over\chi(|s|^2)}\Big\{}
+2\sum_{\alpha,j,k}e_{\alpha jkm}w^{|\alpha|-1}{\alpha_l\over|\alpha|}
s^{\alpha-\un_l}\theta_j\ov\theta_k\cr
&\phantom{{}=-{\chi_1(|s|^2)\over\chi(|s|^2)}\Big\{}
+2\sum_{\alpha,k}d_{\alpha km}(|\alpha|-1)w^{|\alpha|-2}
{\alpha_l\over|\alpha|}s^{\alpha-\un_l}\eta\ov\theta_k\Big\}\cr
&\phantom{{}={}}+\sum_{\alpha,\beta,j,k}d_{\alpha km}\ov{d_{\beta jl}}\,
w^{|\alpha|-1}\ov w^{|\beta|-1}s^\alpha\ov s^\beta\theta_j\ov\theta_k.\cr}$$
The choice $\chi(t)={C\over(1-t)^2}\exp\big({1\over t-1}\big)$
for $t<1$ gives $\chi_1(t)=-C\exp\big({1\over t-1}\big)$, so
$\chi_1(t)/\chi(t)=(1-t)^2$ is smooth and bounded, and our vector
fields $\Xi$, $U$ are smooth. We can write
$$\eqalign{
(\theta,\eta s)\wedge\ov{(\theta,\eta s)}&{}+|w|^2(0,\Xi)\wedge
\ov{(\theta,\eta s)}+|w|^2(\theta,\eta s)\wedge\ov{(0,\Xi)}+|w|^2(0,U)\cr
&=(\theta,\eta s+|w|^2\Xi)\wedge\ov{(\theta,\eta s+|w|^2\Xi)}
+(0,U-|w|^2\Xi\wedge\ov\Xi),\cr}$$
therefore (3.10) implies the formula in proposition~3.8 with
$$\eqalign{
\tau&=dh_{(0,ws)}(\theta,\eta s+|w|^2\Xi),\cr
V&=dh_{(0,ws)}(0,U-|w|^2\Xi\wedge\ov\Xi).\cr}$$
Since $\exph_z(\zeta)=h(z,\rho(z,\xi)$, $\rho(0,\xi)=\xi+O(\xi^{N+1})$ and
$\partial\rho_{(0,\xi)}=d\xi+O(\xi^N)d\xi$ by (2.9), we infer that the
$(1,0)$-differential of $\exph$ at $(x,\zeta)\in T_X$ is
$$\partial\exph_{(x,\zeta)}=dh_{(0,\xi)}+O(|\xi|^N)d\xi$$
modulo the identification of the tangent spaces $T(T_X)_{(x,\zeta)}$ and
$T(T_{\cb^n})_{(0,\xi)}$ given by the coordinates $(z,\xi)$ on~$T_X$.
However, these coordinates are precisely those which
realize the splitting $T(T_X)_{(x,\zeta)}=(T_{X,x})^h\oplus(T_{X,x})^v$ with 
respect to the connection $D$. Since $s=\xi+O(w^N\xi^{N+1})$ and $\xi=\zeta$
at $z=0$, we get
$$\eqalign{
\tau&=\partial\exph_{(x,w\zeta)}
(\theta^h+\eta\zeta^v+|w|^2\Xi^v)+O(|w|^N|\zeta|^N),\cr
V&=\partial\exph_{(x,w\zeta)}(U^v-|w|^2\Xi^v\wedge\ov{\hbox{$\Xi^v$}})
+O(|w|^N|\zeta|^N).\cr}$$
We can drop the terms $O(|w|^N)$ in $\tau$ and $V$ because
$$\leqalignno{
\int_{|\zeta|<1}\partial\op\psi\big(\exph_x(w\zeta)\big)\,d\lambda(\zeta)
&={1\over|w|^{2n}}\int_{|\zeta|<|w|}\partial\op\psi\big(\exph_x(\zeta)\big)\,
d\lambda(\zeta)\cr
&=O(|w|^{-2})&(3.11)\cr}$$
by the usual estimates on Lelong numbers (see e.g. [Le69]), thus
$$\int_{|\zeta|<1}\partial\op\psi\big(\exph_x(w\zeta)\big)\,O\big(|w|^N\big)\,
d\lambda(\zeta)=O\big(|w|^{N-2}\big).$$
After substituting $\zeta$ to $s$ in the formal expression of $\Xi$ and
$U$, we get precisely the formulas given in Proposition~3.8. It remains
to explain why the remainder term $O(|w|^{N-2}\log|w|)$ is in fact of the
type $O(|w|^{N-1})$. To see this, we increase $N$ by two units and estimate
the additional terms in the expansions, due to the contribution of all
multi-indices $\alpha$ with $|\alpha|=N+1$ or $N+2$. It is easily seen that
the additional terms in $\Xi$ and $U$ are $O(|w|^{N-1})$, so they are
$O(|w|^{N+1})$ in $\tau$ and~$|w|^2V$. The contribution of these terms to
$\partial\op\Psi_{(x,w)}$ is thus of the form
$$\int_{|\zeta|<1}\partial\op\psi\big(\exph_x(w\zeta)\big)\,
O\big(|w|^{N+1}\big)\,d\lambda(\zeta)=O\big(|w|^{N-1}\big).~~~\finpr$$

\titre 4. Approximation theorem and estimates|

A straightforward consequence of our computations is the following
appro\-ximation theorem for closed positive currents.

\tha Theorem 4.1|Let $X$ be a compact $n$-dimensional manifold equipped
with a smooth hermitian metric~$\omega$. Fix a smooth semipositive
$(1,1)$-form $u$ on $X$ such that the Chern curvature tensor $\Theta(T_X)
\in{\rm Herm}(T_X\otimes T_X)$ of $(T_X,\omega)$ satisfies
$$\big(\Theta(T_X)+u\otimes\Id_{T_X}\big)(\theta\otimes\xi,\theta\otimes\xi)
\ge 0$$
for all $\theta,\xi\in T_X$ such that $\langle\theta,\xi\rangle=0$.
Let $T=\alpha+{i\over\pi}\partial\op\psi$ be a closed real current 
where $\alpha$ is a smooth closed real $(1,1)$-form and $\psi$ is 
quasi-psh. Suppose that $T\ge\gamma$ for some real $(1,1)$-form
$\gamma$ with continuous coefficients. As~$w$ tends to $0$ and
$x$~runs over~$X$, there is a uniform lower bound
$$\alpha_x[\theta]^2+{i\over\pi}\partial\op\Psi_{(x,w)}[\theta,\eta]^2\ge
\gamma_x[\theta]^2-\lambda(x,|w|)\,u_x[\theta]^2-\delta(|w|)\,|\theta|^2-
{1\over\pi}K(|\theta||\eta|+|\eta|^2)$$
where $K>0$ is a sufficiently large constant, $\delta(t)$ a continuous
increasing function with $\lim_{t\to 0}\delta(t)=0$, and
$$\lambda(x,t)=t\,{\partial\over\partial t}\big(\Psi(x,t)+Kt^2\big).$$
The above derivative $\lambda(x,t)$ is a nonnegative continuous function on
$X\times{}]0,\varepsilon_0[$ which is increasing in $t$ and such that
$$\lim_{t\to 0}\lambda(x,t)=\nu(T,x)~~~~\hbox{\rm(Lelong number of
$T$ at $x$).}$$
In particular, the currents $T_\varepsilon=\alpha+{i\over\pi}\partial\op
\Psi(\bu,\varepsilon)$ are smooth closed real currents converging weakly
to $T$ as $\varepsilon$ tends to~$0$, such that
$$T_\varepsilon\ge\gamma-\lambda(\bu,\varepsilon)\,u-\delta(\varepsilon)
\,\omega.$$
\fintha\vskip-6pt

\remarque Proof|It suffices to prove the estimate for $|w|<
\varepsilon(\delta)$, with $\delta>0$ fixed in place of~$\delta(|w|)$.
Also, the estimate is local on $X$. If we change $\psi$ into
$\psi+\psi_0$ with a smooth function $\psi_0$, then $\alpha$ is changed
into $\alpha-{i\over\pi}\partial\op\psi_0$ and $\Psi$ into
$\Psi+\Psi_0$, where $\Psi_0$ is a smooth function on $X\times\cb$ such
that $\Psi_0(z,w)=\psi_0(z)+O(|w|^2)$. It follows that the estimate
remains unchanged up to a term $O(1)|\eta|^2$. We can thus work on  a
small coordinate open set $\Omega\subset X$ and choose $\psi_0$ such 
that $\gamma-(\alpha-{i\over\pi}\partial\op\psi_0)$ is positive
definite and small at~$x$, say equal to ${\delta\over 4}\omega_x$.
After shrinking~$\Omega$ and making the change $\psi\mapsto\psi+\psi_0$,
we may in fact suppose that $T=\alpha+{i\over\pi}\partial\op\psi$
on~$\Omega_{x,\delta}\subset\Omega$, where $\alpha$ satisfies
$\gamma_x-\alpha_x={\delta\over 4}\omega_x$ and $\gamma-{\delta\over 2}
\omega\le\alpha\le\gamma$ on~$\Omega_{x,\delta}$. In particular,
${i\over\pi}\partial\op\psi\ge\gamma-\alpha$ and $\psi$ is
plurisubharmonic on~$\Omega_{x,\delta}$. All we have to show is that
$${i\over\pi}\partial\op\Psi_{(x,w)}[\theta,\eta]^2\ge
-\lambda(x,|w|)\,u_x[\theta]^2-{\delta\over 2}
|\theta|^2-{1\over\pi}K(|\theta||\eta|+|\eta|^2)\leqno(4.2)$$
for $|w|<w_0(\delta)$ small. For this, we apply
proposition 3.8 at order $N=2\,$; order $2$ is enough for our purposes
since we will neglect all terms in $\partial\op\Psi$ which converge
to $0$ with~$w$, especially all $O(|w|)$ terms. By (3.11), we can
neglect all terms of the form $\partial\op\psi(\exph_x(w\zeta))
O(|w|^3)$ under the integral sign. Up~to such terms,
$\partial\op\psi\cdot\big(\tau\wedge\ov\tau+|w|^2V\big)_{\exph_x(w\zeta)}
\chi(|\zeta|^2)$ is equal to
$$\eqalign{
-|w|^2\chi_1(|\zeta|^2)\re\sum_{l,m}{\partial^2\psi\over\partial\ov z_l
\partial z_m}&\Big\{{\chi(|\zeta|^2)\over-|w|^2\chi_1(|\zeta|^2)}
\ov\tau_l\tau_m+\sum_{j,k}c_{jklm}\theta_j\ov\theta_k\cr
&\quad{}+2\sum_{|\alpha|=2,k}d_{\alpha km}(|\alpha|-1)w^{|\alpha|-2}
{\alpha_l\over|\alpha|}\zeta^{\alpha-\un_l}\eta\ov\theta_k\Big\}\cr
\ge-|w|^2\chi_1(|\zeta|^2)\sum_{l,m}{\partial^2\psi\over
\partial\ov z_l\partial z_m}&\Big\{{1\over|w|^2}\ov\tau_l\tau_m
+\sum_{j,k}c_{jklm}\Big(\theta_j\ov\theta_k+{1\over 2}\zeta_j\eta\ov\theta_k
+{1\over 2}\ov\zeta_k\theta_j\ov\eta\Big)\Big\}\cr
=-|w|^2\chi_1(|\zeta|^2)\sum_{l,m}{\partial^2\psi\over
\partial\ov z_l\partial z_m}&\Big\{{1\over|w|^2}\ov\tau_l\tau_m
+\sum_{j,k}c_{jklm}\tau_j\ov\tau_k\cr
&\quad{}-\sum_{j,k}c_{jklm}\Big({1\over 2}\zeta_j\eta\ov\theta_k
+{1\over 2}\ov\zeta_k\theta_j\ov\eta+\zeta_j\ov\zeta_k\eta\ov\eta\Big)
\Big\}\cr}$$
in view of 2.9$\,$(iii) and of the inequality $0\le-\chi_1\le\chi\,$; in the
last equality, we have also used the fact that $\tau=\theta+\eta\zeta+O(|w|)$.

By (3.11), the mixed terms $\theta_j\ov\eta$, $\eta\ov\theta_k$ and the
terms $\eta\ov\eta$ give rise to contributions bounded below by
$-K'(|\theta||\eta|+|\eta|^2)$. Hence we get the estimate
$$\leqalignno{
&{i\over\pi}\partial\op\Psi_{(x,w)}[\theta,\eta]^2\ge\cr
&{1\over\pi}|w|^2\int_{\cb^n}-\chi_1(|\zeta|^2)\sum_{j,k,l,m}
{\partial^2\psi\over\partial\ov z_l\partial z_m}\big(\exph_x(w\zeta)\big)\,
\Big(c_{jklm}+{1\over|w|^2}\delta_{jm}\delta_{kl}\Big)\tau_j\ov\tau_k\,
d\lambda(\zeta)\cr
&\kern120pt -K'\big(|\theta||\eta|+|\eta|^2\big).&(4.3)\cr}$$
We need a lemma.

\tha Lemma 4.4|Suppose that the curvature assumption of Theorem~4.1 is
satisfied. Then for every~$\varepsilon>0$, there is a constant
$M_\varepsilon>0$ such that
$$\sum_{j,k,l,m}{1\over 2\pi}(c_{jklm}+M_\varepsilon\delta_{jm}\delta_{kl})
\tau_j\ov\tau_k\xi_l\ov\xi_m+
\sum_{j,k,l}u_{jk}\tau_j\ov\tau_k\xi_l\ov\xi_l+\varepsilon|\tau|^2|\xi|^2\ge 0$$
for all tangent vectors $\tau,\xi$.\fintha

Let us consider the hermitian form $H$ on $T_X\otimes T_X$ defined by
$$H(\tau\otimes\xi,\tau\otimes\xi)=
\sum_{j,k,l,m}{1\over 2\pi}c_{jklm}\tau_j\ov\tau_k\xi_l\ov\xi_m+
\sum_{j,k,l}u_{jk}\tau_j\ov\tau_k\xi_l\ov\xi_l+\varepsilon|\tau|^2|\xi|^2.$$
Let $\mu$ be the infimum of $H(\tau\otimes\xi,\tau\otimes\xi)$ on the
compact set \hbox{$\{|\tau|=1\}\times\{|\xi|=1\}$}. By our curvature
assumption we have $H(\tau\otimes\xi,\tau\otimes\xi)\ge\varepsilon$ when
$\tau\perp\xi=0$ and $|\tau|=|\xi|=1$, therefore
$H(\tau\otimes\xi,\tau\otimes\xi)\ge 0$ on some neighborhood
$|\langle\tau,\xi\rangle|<r_\varepsilon$ of that set. It follows that
$$H(\tau\otimes\xi,\tau\otimes\xi)+{|\mu|\over r_\varepsilon^2}
|\langle\tau,\xi\rangle|^2\ge0$$
for all $|\tau|=|\xi|=1$. Lemma~4.4 follows with
$M_\varepsilon=|\mu|/r_\varepsilon^2$.\finpr

Let us apply inequality 4.4 to each
vector $\xi$ in a basis of eigenvectors of $\partial\op\psi$, multiply
by the corresponding (nonnegative) eigenvalue and take the sum. We get
$${1\over 2\pi}\sum_{j,k,l,m}{\partial^2\psi\over\partial\ov z_l\partial z_m}
\big(c_{jklm}+M_\varepsilon\delta_{jm}\delta_{kl}\big)\tau_j\ov\tau_k+
\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}\,
\big(u[\tau]^2+\varepsilon|\tau|^2\big)\ge 0.$$
Combining this with (4.3) for $|w|^2<1/M_\varepsilon$, we infer
$$\eqalign{
{i\over\pi}\partial\op&\Psi_{(x,w)}[\theta,\eta]^2\cr
&~~{}\ge -2|w|^2\int_{\cb^n}-\chi_1(|\zeta|^2)\sum_l
{\partial^2\psi\over\partial z_l\partial\ov z_l}\big(\exph_x(w\zeta)\big)\,
\big(u_x[\tau]^2+\varepsilon|\tau|^2\big)d\lambda(\zeta)\cr
&\kern120pt -K'\big(|\theta||\eta|+|\eta|^2\big)\cr
&~~{}\ge -\Big\{2|w|^2\int_{\cb^n}-\chi_1(|\zeta|^2)\sum_l
{\partial^2\psi\over\partial z_l\partial\ov z_l}\big(\exph_x(w\zeta)\big)\,
d\lambda(\zeta)\Big\}\big(u_x[\theta]^2+\varepsilon|\theta|^2\big)\cr
&\kern120pt -K''\big(|\theta||\eta|+|\eta|^2\big)\cr}$$
by (3.11) again, in combination with the equality
$\tau=\theta+\eta\zeta+O(|w|)$). The change of variables $\zeta\mapsto s$
defined by $\exph_x(w\zeta)=x+ws$ yields $\zeta=s+O(w^2s^3)$ by 2.9$\,$(ii),
hence choosing $\varepsilon\ll\delta$ small enough we get
$${i\over\pi}\partial\op\Psi_{(x,w)}[\theta,\eta]^2\ge
-\lambda_\Omega(x,|w|)\,u_x[\theta]^2-{\delta\over 3}|\theta|^2
-K\big(|\theta||\eta|+|\eta|^2\big)$$
where
$$\lambda_\Omega(x,|w|)=2|w|^2\int_{\cb^n}-\chi_1(|s|^2)\sum_l
{\partial^2\psi\over\partial z_l\partial\ov z_l}(x+ws)\,d\lambda(s).$$
By definition, the Lelong number $\nu(\psi,x)$ is $\lim_{r\to 0}\nu(\psi,x,r)$
where
$$\nu(\psi,x,r)={1\over\pi^{n-1}r^{2n-2}/(n-1)!}\int_{B(x,r)}{2\over\pi}
\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}\,d\lambda(z).$$
Therefore we have
$$\nu(\psi,x,|w|r)={1\over\pi^{n-1}r^{2n-2}/(n-1)!}|w|^2\int_{|s|<r}
{2\over\pi}\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}(x+ws)\,
d\lambda(s).$$
As $-\chi_1(|s|^2)=2\int_{|s|}^{+\infty}\chi(r^2)\,r\,dr$, the Fubini
formula gives
$$\eqalign{
\lambda_\Omega(x,|w|)&=4|w|^2\int_0^{+\infty}\Big\{\int_{|s|<r}
\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}(x+ws)\,d\lambda(s)
\Big\}~\chi(r^2)\,r\,dr\cr
&={2\pi^n\over(n-1)!}\int_0^1\nu(\psi,x,|w|r)\,\chi(r^2)\,r^{2n-1}\,dr,\cr
\lambda_\Omega(x,t)&=\int_{\cb^n}\nu(\psi,x,t|s|)\,\chi(|s|^2)\,d\lambda(s).
\cr}$$
Hence $\lambda_\Omega(x,t)$ is smooth in $(x,t)$, increasing in $t$ and 
$\lim_{t\to 0}\lambda_\Omega(x,t)=\nu(\psi,x)=\nu(T,x)$, as desired.

The expected estimate (4.2) thus holds on $\Omega$ with $\lambda_\Omega(x,t)$
in place of $\lambda(x,t)$; we only have to show that $\lambda_\Omega(x,t)-
\lambda(x,t)$ is small. By (4.2), $\Psi(x,w)+K|w|^2$ is plurisubharmonic
in~$w$, and so is a convex function of $\log|w|\,$; since $\Psi(x,t)$ is
bounded above as $t$ tends to $0$, the sum $\Psi(x,t)+Kt^2$ which is convex
in $\log t$ must be increasing in $t$. Therefore
$$\lambda(x,t)={\partial\over\partial\log t}\big(\Psi(x,t)+Kt^2\big)$$
is a nonnegative increasing function of $t$. When we put $\theta=0$,
Proposition~3.8 gives $\Xi=V=0$ and $\tau=\partial\exph_{(x,w\zeta)}(\eta
\zeta^v)$, thus
$$\eqalign{
{\partial^2\Psi\over\partial w\partial\ov w}(x,w)&=
\int_{\zeta\in T_{X,x}}\partial\op\psi_{\exph_x(w\zeta)}[\zeta]^2\,
\chi(|\zeta|^2)\,d\lambda(\zeta)+O(|w|^{N-1})\cr
&=\int_{\cb^n}\partial\op\psi_{x+ws}[s]^2\,\chi(|s|^2)\,d\lambda(s)+O(1),
\cr}$$
again by the change of variable $\exph_x(w\zeta)=x+ws$, for which
$\zeta=s+O(w^2s^3)$.
Since $\partial^2/\partial w\partial\ov w=(t^{-1}\partial/\partial t)
\circ(t\,\partial/\partial t)$ for a function of $w$ depending only on 
$t=|w|$, a multiplication by $t$ followed by an integration implies
$$\leqalignno{
\qquad t{\partial\Psi(x,t)\over\partial t}&=t{\partial\over\partial t}
\int_{\cb^n}\psi(x+ts)\,\chi(|s|^2)\,d\lambda(s)+O(t^2)&(4.5)\cr
&=\int_{\cb^n}\nu(\psi,x,t|s|)\,\chi(|s|^2)\,d\lambda(s)+O(t^2)
=\lambda_\Omega(x,t)+O(t^2)\,;\cr}$$
here, we used the well known fact that $\nu(\psi,x,t)$ is equal to the 
derivative $\partial/\partial\log t$ of the mean value of $\psi$ on 
the sphere $S(x,t)$. Hence $\lambda_\Omega(x,t)-\lambda(x,t)=O(t^2)$
and the first estimate in 4.1 follows. Now,
it is clear that $\psi_\varepsilon$ converges to $\psi$ in
$L^1_{\rm loc}$, so $T_\varepsilon$ converges weakly to~$T\,$;
note also that $\psi_\varepsilon+K\varepsilon^2$ is increasing
in $\varepsilon$ by the above arguments. The proof is complete.\finpr

\remarque {\petcap Remark 4.6}|An integration of the first equality in
(4.5) also shows that
$$\Psi(x,t)=\int_{|s|<1}\psi(x+ts)\,\chi(|s|^2)\,d\lambda(s)+O(t^2)$$
relatively to the system of normal coordinates at~$x$ defined in \S$\,$2.

\remarque {\petcap Remark 4.7}|The estimates obtained in Theorem~4.1 can
be slightly improved by setting
$$\wt\Psi(x,w)=\Psi(x,w)+|w|,~~~~
\wt\lambda(x,t)=t\,{\partial\over\partial t}\big(\wt\Psi(x,t)\big).$$
Indeed $\wt\lambda(x,t)=\lambda(x,t)+t-2Kt^2$
is increasing in $t$ and larger than $\lambda(x,t)$ for $t$~small,
hence $\wt\Psi(x,w)$ is convex and increasing in $\log|w|$, while
$$\partial\op\wt\Psi_{(x,w)}[\theta,\eta]^2=
\partial\op\Psi_{(x,w)}[\theta,\eta]^2+{|\eta|^2\over 4|w|}.$$
Since $K|\theta||\eta|\le 2K^2|w||\theta|^2+|\eta|^2/(8|w|)$,
we get for $|w|<\varepsilon_0$ small enough a lower bound of the form
$$\alpha_x[\theta]^2+{i\over\pi}\partial\op\wt\Psi_{(x,w)}[\theta,\eta]^2\ge
\gamma_x[\theta]^2-\wt\lambda(x,|w|)\,u_x[\theta]^2-\wt\delta(|w|)\,
|\theta|^2,\leqno(4.8)$$
where $\lim_{t\to 0}\wt\lambda(x,t)=\nu(T,x)$ and
$\lim_{t\to 0}\wt\delta(t)=0$, $\wt\delta$ being continuous and
increasing.

\titre 5. Compact manifolds with partially semipositive curvature|

In case the curvature of the ambient manifold~$X$ satisfies suitable
semipositivity assumptions, our smoothing theorem~4.1 yields interesting
geometric consequences.

\tha Definition 5.1|We say that $X$ has partially semipositive
curvature $($in the sense of Griffiths$)$ if there exists a hermitian
metric $\omega$ on~$X$ such that the associated Chern curvature tensor
$\Theta(T_X)$ satisfies
$$\Theta(T_X)(\theta\otimes\xi,\theta\otimes\xi)\ge 0$$
for all $\theta,\xi\in T_X$ with $\langle\theta,\xi\rangle=0.$\fintha

The standard Griffiths semipositivity assumption for $T_X$ would
require that the above inequality is satisfied for {\sl all pairs}
$\theta,\xi\in T_X$. This is the case if $X$ is a compact complex
homogeneous manifold, since the tangent bundle is then generated by
global sections; a fortiori, such manifolds satisfy the property in
Def.~5.1. However, the partial semipositivity condition is
substantially weaker. In fact, every complex curve has partially
semipositive curvature, but a curve has a metric of semipositive
curvature if and only if it is of genus $0$ or~$1$. This observation
gives rise immediately to higher dimensional examples; in fact let
$X=\pb^1\times C$ where $C$ is a curve of genus $g\ge 2$, equipped with
its unique metric of constant curvature~$-1$. The Fubini-Study metric
on $\pb^1$ has curvature~$+2$, hence we have
$$\langle\Theta(T_X)(\theta\otimes\xi),\theta\otimes\xi\rangle=
2|\theta_1|^2|\xi_1|^2-|\theta_2|^2|\xi_2|^2,$$ where the subscripts
$1,2$ denote respectively the first and second projection of a tangent
vector; clearly the difference is nonnegative when \hbox{$\langle
\theta_1,\xi_1\rangle+\langle\theta_2,\xi_2\rangle=0$}, thus
$\pb^1\times C$ has partially semipositive curvature. It would
be interesting to have a more algebraic understanding of the notion of
partial semipositivity. This is certainly a very difficult problem;
recall in this context the deep unsolved conjecture of Griffiths
that ampleness is equivalent to Griffiths positivity.

The following results were already obtained in [De92] under the
assumption that $T_X$ is nef (i.e. $\oc_{T_X}(1)$ is nef
over~$P(T^\star_X))$. It is well known that Griffiths semipositivity
implies nefness, and that the converse is not true; however, if
Griffiths' conjecture holds, nefness would be equivalent to the fact
that the curvature tensor $\Theta(T_X)$ can be made larger than
$-\varepsilon$ for every $\varepsilon>0$ (see [DPS92]). The above
examples show that partial semipositivity covers different situations.

\tha Proposition 5.2|Let $(X,\omega)$ be a compact hermitian manifold with
partially semipositive curvature.
\smallskip
\item{\rm(i)} For any closed real $(1,1)$-current $T$ on $X$ such that
$T\ge\gamma$ for some conti\-nuous real $(1,1)$-form $\gamma$ on $X$,
there is a family $T_\varepsilon$ of smooth approximations in the same
$\partial\op$-cohomology class as $T$ and converging weakly to~$T$
as $\varepsilon$ tends to $0$, such that $T_\varepsilon\ge\gamma-
\delta_\varepsilon\omega$ with $\lim_{\varepsilon\to 0}\delta_\varepsilon=0$.
\smallskip
\item{\rm(ii)} Every closed positive $(1,1)$-current $T$ is numerically
effective in the sense of {\rm[De92]}$\,$; in particular, if $X$ is K\"ahler,
the De Rham cohomology class $\{T\}$ satisfies $\int_Y\{T\}^p\ge 0$ for
every $p$-dimensional analytic subset $Y\subset X$.
\smallskip\fintha

\remarque Proof|(i) follows immediately from Theorem~4.1 by taking $u=0$.

To get (ii), we apply (i) with $\gamma=0$. We then obtain $T_\varepsilon\ge
-\delta_\varepsilon\omega$. Let $\omega_0$ be a K\"ahler metric on~$X$
($\omega_0$ need not be related to $\omega$). Multiplying $\omega_0$ by a
suitable constant we get $\omega_0\ge\omega$ and so $T_\varepsilon+
\delta_\varepsilon\omega_0\ge 0$. As $\{T\}=\{T_\varepsilon\}$, this implies
$$\int_Y\big(\{T\}+\delta_\varepsilon\{\omega_0\}\big)^p=
\int_Y\big(T_\varepsilon+\delta_\varepsilon\omega_0\big)^p\ge 0,$$
thus $\int_Y\{T\}^p\ge 0$ in the limit.\finpr\vskip7pt

Proposition 5.2 implies that a compact complex manifold with partially
semipositive curvature must be minimal in the sense that it does not
contain any divisor $D$ which can be blown down to some lower dimensional
variety. Indeed, in the latter case, there is always a curve $C\subset D$
such that $D\cdot C=\int_C\{D\}<0$.

\tha Proposition 5.3|Let $X$ be a compact complex manifold with
partially semipositive curvature. Then $X$ is K\"ahler if and only if
$X$ is in the Fujiki class ${\cal C}$ $(={}$ class of complex varieties
which are bimeromorphic to K\"ahler manifolds$)$, and $X$ is projective
if and only if $X$ is Moishezon.\fintha

\remarque Proof|Let $X$ be a manifold in the Fujiki class ${\cal C}$
such that $T_X$ has partially semipositive curvature. We know by Fujiki
[Fu78] that $X$ has a smooth K\"ahler modification $\mu:\wt X\to X$.
Let $\beta$ be a K\"ahler metric on $\wt X$ and let $T=\mu_\star\beta$
be the direct image current. If $\omega$ is a hermitian metric on $X$,
we have $\beta\ge c\,\mu^\star\omega$ on $\wt X$ for some small
constant $c>0$, thus by taking the direct image we get
$T=\mu_\star\beta\ge c\omega$ on~$X$. Now Proposition 5.2$\,$(i)
produces a smooth approximation $T_\varepsilon$ of $T$ such that
$T_\varepsilon\ge c\omega-\delta_\varepsilon\omega \ge {c\over
2}\omega$ for $\varepsilon$ small enough. Thus $T_\varepsilon$ is a
K\"ahler metric and $X$ is K\"ahler. In particular, if $X$ is
Moishezon, we can take $\beta$ to be the curvature form of an ample
line bundle $\oc(D)$ over~$\wt X$. Then $T$ and $T_\varepsilon$ are
cohomologous to $\mu_\star([D])=[D']$ where $D'=\mu(D)$ is the image of
$D$ in~$X$. This implies that $\oc(D')$ is ample and therefore $X$ is
projective algebraic (we could also have applied the well-known result
of Moishezon that every K\"ahler Moishezon manifold is projective
algebraic). The converse implications are trivial.\finpr

\titre 6. Singularity attenuation process for closed (1,1)-currents|

If $T$ is a closed positive or almost positive current on a complex
manifold~$X$, we denote by
$E_c(T)$ the $c$-upperlevel set of Lelong numbers:
$$E_c(T)=\big\{x\in X\,;\,\nu(T,x)\ge c\big\},~~~~c>0.$$
A~well-known theorem of [Siu74] asserts that $E_c(T)$ is a closed
analytic subset of~$X$. A combination of Theorem~4.1 with Kiselman's
singularity attenuation technique yields the following partial
regularization process. The sets $E_c(T)$ appear precisely to be
the obstructions to smoothing when no loss of positivity is admitted.

\tha Theorem 6.1|Let $T$ be a closed almost positive $(1,1)$-current
and let $\alpha$ be a smooth real $(1,1)$-form in the same
$\partial\op$-cohomology class as $T$, i.e. $T=\alpha+{i\over\pi}\partial\op
\psi$ where $\psi$ is an almost psh function.
Let $\gamma$ be a continuous real $(1,1)$-form such that $T\ge\gamma$.
Suppose that $T_X$ is equipped with a smooth hermitian metric $\omega$
such that the Chern curvature form satisfies
$$\big(\Theta(T_X)+u\otimes\Id_{T_X}\big)(\theta\otimes\xi,\theta\otimes\xi)
\ge 0~~~~\forall\theta,\xi\in T_X~~~\hbox{\rm with}~~
\langle\theta,\xi\rangle=0,$$
for some continuous nonnegative $(1,1)$-form $u$ on~$X$. Then for every
$c>0$, there is a family of closed almost positive $(1,1)$-currents
$T_{c,\varepsilon}=\alpha+{i\over\pi}\partial\op\psi_{c,\varepsilon}$,
$\varepsilon\in{}]0,\varepsilon_0[$, such that $\psi_{c,\varepsilon}$
is smooth on $X\ssm E_c(T)$, increasing with respect to~$\varepsilon$,
and converges to $\psi$ as $\varepsilon$ tends to~$0$ $($in particular,
the current $T_{c,\varepsilon}$ is smooth on $X\ssm E_c(T)$ and converges
weakly to $T$ on~$X)$, and such that
\item{\rm(i)}$T_{c,\varepsilon}\ge\gamma-\min\{\lambda_\varepsilon,c\}u
-\delta_\varepsilon\omega$~~
where:
\item{\rm(ii)}$\lambda_\varepsilon(x)$ is an increasing family of
continuous functions on $X$ such that $\lim_{\varepsilon\to 0}
\lambda_\varepsilon(x)=\nu(T,x)$ at every point,
\item{\rm(iii)}$\delta_\varepsilon$ is an increasing family of positive
constants such that
$\lim_{\varepsilon\to 0}\delta_\varepsilon=0$,
\item{\rm(iv)}$\nu(T_{c,\varepsilon},x)=\big(\nu(T,x)-c\big)_+$
at every point $x\in X$.\fintha

\remarque Proof|Following an idea of Kiselman~[Ki79], we let
$\psi_{c,\varepsilon}$ be the Legendre transform
$$\psi_{c,\varepsilon}(x)=\inf_{|w|<1}\Big(\wt\Psi(x,\varepsilon w)+
{\varepsilon\over 1-|w|^2}-c\log|w|\Big),$$
where $\wt\Psi(x,w)=\Psi(x,w)+|w|$ is the function defined in Remark~4.7.
It is clear that $\psi_{c,\varepsilon}$ is increasing in~$\varepsilon$ and that
$$\lim_{\varepsilon\to 0}\psi_{c,\varepsilon}(x)=\wt\Psi(x,0_+)=\Psi(x,0_+)=
\psi(x).$$
Moreover, as $\wt\Psi(x,w)$ is convex and increasing~in $t=\log|w|$, the
function
$$t\longmapsto\Phi_{c,\varepsilon}(x,t):=
\wt\Psi(x,\varepsilon t)+{\varepsilon\over 1-t^2}-c\log t$$
is strictly convex in $\log t$ and tends to $+\infty$ as $t$ tends to~$1$.
It follows that the infimum is attained for $t=t_0(x)\in{}[0,1[$
given either by the zero of the $\partial/\partial\log t$ derivative:
$$\wt\lambda(x,\varepsilon t)+{2\varepsilon t^2\over (1-t^2)^2}-c=0$$
when $\nu(T,x)=\lim_{t\to 0}\wt\lambda(x,t)<c$,
or by $t_0(x)=0$ when $\nu(T,x)\ge c$ (in which case the above strictly
convex function is increasing on the whole interval $[0,1[$). Since the
$\partial/\partial\log t$ derivative is itself strictly increasing in~$t$,
the implicit function theorem shows that $t_0(x)$ depends smoothly on $x$
on~$X\ssm E_c(T)=\{\nu(T,x)<c\}$, hence
$\psi_{c,\varepsilon}(x)=\Phi_{c,\varepsilon}(x,t_0(x))$ is actually smooth
on~$X\ssm E_c(T)$.

Now, fix a point $x\in X\ssm E_c(T)$ and $t_1>t_0(x)$. For all $z$
in a neighborhood $V$ of $x$ we still have $t_0(z)<t_1$, thus
$$\psi_{c,\varepsilon}(z)=\inf_{|w|<t_1}\Big(\wt\Psi(z,\varepsilon w)+
{\varepsilon\over 1-|w|^2}-c\log|w|\Big)~~~~\hbox{\rm on}~~V.$$
By (4.8), all functions involved in that infimum have a complex Hessian
in $(z,w)$ bounded below by
$$\gamma_z-\alpha_z-\wt\lambda(z,\varepsilon t_1)u_z
-\wt\delta(\varepsilon t_1)\omega_z.$$
Hence $\partial\op\psi_{c,\varepsilon}$ has the same
lower bound (this is an easy consequence of Kiselman's principle [Ki78]
that an infimum $\inf_w u(z,w)$ of plurisubharmonic functions
depending only on $\re w$ is plurisubharmonic in~$z$).
By taking $t_1$ arbitrarily close to $t_0(x)$ and by
shrinking $V$, the lower bound comes arbitrary close to
$$\gamma_x-\alpha_x-\wt\lambda\big(x,\varepsilon t_0(x)\big)\,u_x
-\wt\delta\big(\varepsilon t_0(x)\big)\omega_x\ge
\gamma_x-\alpha_x-\min\{\wt\lambda(x,\varepsilon),c\}\,u_x
-\wt\delta(\varepsilon)\omega_x,$$
because $\wt\lambda(x,\varepsilon t_0(x))=c-2\varepsilon
t_0(x)^2/(1-t_0(x)^2)^2\le c$ and $\wt\lambda(x,t)$, $\wt\delta(t)$ are
increasing in~$t$. Therefore we get
$$\alpha+{i\over\pi}\partial\op\psi_{c,\varepsilon}\ge\gamma
-\min\{\wt\lambda(\bu,\varepsilon),c\}\,u-\wt\delta(\varepsilon)\omega$$
on $X\ssm E_c(T)$. However, as the lower bound is a continuous
$(1,1)$-form and as $\psi_{c,\varepsilon}$ is quasi-psh, it is
immediate to check that the lower bound extends to~$X$ by continuity.
Hence properties 6.1$\,$(i), (ii), (iii) are proved.

Remark~4.6 finally shows that $\psi_{c,\varepsilon}$ differs locally
from Kiselman's usual Legendre transform only by bounded terms, thus
$$\nu(\psi_{c,\varepsilon},x)=\big(\nu(\psi,x)-c\big)_+$$
at every point~$x\in X$ by Kiselman's results [Ki78], [Ki79]. Therefore
property 6.1$\,$(iv) also holds.\finpr

\vskip28pt
\centerline{\bf References}
\vskip8pt
\eightpoint
\lettre KMM87| 

\article De82|Demailly, J.-P|Estimations $L^2$ pour l'op\'erateur $\op$
d'un fibr\'e vectoriel holomorphe semi-positif au dessus d'une vari\'et\'e
k\"ahl\'erienne compl\`ete|Ann. Sci. Ec. Norm. Sup.|15|1982|457-511|

\divers De90|Demailly, J.-P|Singular hermitian metrics on positive line
bundles|{\rm Proceedings of the Bayreuth conference ``Complex algebraic
varieties'', April~2-6, 1990, edited by K.~Hulek, T.~Peternell,
M.~Schneider, F.~Schreyer, Lecture Notes in Math. ${\rm
n}^\circ\,$1507, Springer-Verlag, {\oldstyle 1992}}|

\divers De91a|Demailly, J.-P|A numerical criterion for very ample line
bundles|Pr\'epubli\-cation ${\rm n}^\circ\,$153, Institut Fourier, Univ. 
Grenoble~I, September {\oldstyle 1991}, to appear in J. Differential Geom|

\divers De91b|Demailly, J.P|Monge-Amp\`ere operators, Lelong numbers
and intersection theory|{\rm Pr\'epub. Inst. Fourier ${\rm n}^\circ\,$173,
Mai {\oldstyle 1991}, 70p, to appear in the Proceedings ``Complex
Analysis and Geometry'', CIRM, Trento (Italy), edited by V.~Ancona
and A.~Silva}|

\article De92|Demailly, J.-P|Regularization of closed positive currents
and Intersection Theory|J. Alg. Geom.|1|1992|361-409|

\divers DPS91|Demailly, J.P., Peternell, Th., Schneider, M|Compact
complex manifolds with numerically effective tangent bundles|{\rm
Pr\'epub. Inst. Fourier ${\rm n}^\circ\,$184, Octobre
{\oldstyle 1991}, 48p}|

\article Fu78|A. Fujiki|Closedness of the Douady spaces of compact K\"ahler
spaces|Publ. R.I.M.S., Kyoto Univ.|14|1978|1--52|

\livre H\"o66|H\"ormander, L|An introduction to Complex Analysis in several 
variables|North-Holland Math. libr., vol.7, 
Amsterdam, London|1966{\rm, third edition,} 1991|

\article Ki78|Kiselman, C.O|The partial Legendre transformation for 
plurisubharmonic functions|Invent. Math.|49|1978|137-148|

\article Ki79|Kiselman, C.O|Densit\'e des fonctions plurisousharmoniques|Bull.
Soc. Math. France|107|1979|295-304|

\article Le57|Lelong, P|Int\'egration sur un ensemble analytique complexe|Bull.
Soc. Math. France|85|1957|239-262|

\livre Le69|Lelong, P|Plurisubharmonic functions and positive differential
forms|Gordon and Breach, New-York, and Dunod, Paris|1969|

\article Siu74|Siu, Y.T|Analyticity of sets associated to Lelong numbers and
the extension of closed positive currents|Invent. Math.|27|1974|53-156|
\tenpoint\vskip3pt
$~$\hfill(January 22, 1993)

\end


%-----------------------------------------------------------------------
% Ancienne version


\centerline{\bftwelve Regularization of closed positive currents}
\centerline{\bftwelve of type (1,1) by the flow of a Chern connection}
\vskip8pt
{\noindent\hangindent4cm\hangafter-1
{\bf Jean-Pierre Demailly

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e de Grenoble I\hfill\break
Institut Fourier, BP 74\hfill\break
U.R.A. 188 du C.N.R.S.\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip27pt

\titre 1. Introduction|

Let $X$ be a compact $n$-dimensional complex manifold and let $T$ be a
closed positive current of bidegree $(1,1)$ on $X$.  In general, $T$
cannot be approximated by closed positive currents of class $C^\infty$:
a necessary condition for this is that the cohomology class  $\{T\}$ be
numerically effective in the sense that \hbox{$\int_Y\{T\}^p\ge 0$} for
every $p$-dimensional subvariety $Y\subset X$. For example, if
$E\simeq\pb^{n-1}$ is the exceptional divisor of a one-point blow-up
$X\to X'$, then $T=[E]$ cannot be positively approximated: for every
curve $C\subset E$, we have \hbox{$\int_C\{E\}=\int_C c_1(\oc(-1))<0$}.
However, we will see that it is always possible to approximate a closed
positive current $T$ of type $(1,1)$ by closed real currents admitting
a small negative part, and that this negative part can be estimated in
terms of the Lelong numbers of $T$ and the geometry of~$X$ (see
Theorem~4.1).

Let $\alpha$ be a smooth closed $(1,1)$-form representing the same
$\partial\op$-cohomology class as $T$ and let $\psi$ be a quasi-psh
function on $X$ (that is, a function which is locally the sum of a
plurisubharmonic function and a smooth function) such that
$T=\alpha+{i\over\pi}\partial\op\psi$.  Such a decomposition exists 
even when $X$ is non-K\"ahler, since we can always find an open
covering $(\Omega_k)$ of $X$ such that $T=\smash{{i\over\pi}}
\partial\op\psi_k$ over $\Omega_k$, and construct a global
$\psi=\sum\theta_k\psi_k$ by means of a partition of unity~$(\theta_k)$
(note that $\psi-\psi_k$ is smooth on~$\Omega_k$). If $\psi_\varepsilon$
is an approximation of $\psi$, then $T_\varepsilon=\alpha+{i\over\pi}
\partial\op\psi_\varepsilon$ is an approximation of $T$.  We are thus
led to study a regularization process for quasi-psh functions.  For
this we use some ideas already developed in~[De82], although more
general and more precise results will be obtained here.

 Further techniques based on H\"ormander's $L^2$ existence theorems
[H\"o66] or Kiselman's Legendre transform [Ki78] are explained in our
recent papers [De91], [De92]; they lead to similar estimates but
only require a numerical hypothesis instead of a curvature hypothesis.
However, the techniques involved are considerably more difficult, so we
felt interesting to present here our earlier method, which is simpler
and probably easier to extend to higher bidegrees. The main ideas of
this work have been worked out during a stay of the author at Bayreuth
University in November~1989. The author wishes to thank this
Institution for its hospitality.

\titre 2. Exponential map associated to the Chern connection|

Suppose that the manifold $X$ is equipped with a smooth hermitian metric
\hbox{$\omega=i\sum\omega_{lm}dz_l\wedge d\ov z_m$.} Denote by $D$ the
Chern connection of $T_X$ and by $\Theta(T_X)={i\over 2\pi}D^2$  the curvature
tensor. We define an exponential map  $\exp:T_X\la X$ as follows: if
$\zeta\in T_{X,z}$, then $\exp_z(\zeta)$  is the position at time $t=1$ of
the curve $t\mapsto u(t)$ starting at  $u(0)=z$ with initial tangent
vector $u'(0)=\zeta$ and satisfying the  second order differential
equation $D(du/dt)=0$ (parallel translation with respect to $D$).
If~$\omega$ is K\"ahler the Chern connection coincides with the
Levi-Civita connection, so the exponential map is given in that case
by the riemannian geodesics. For any $x\in X$, fix analytic
coordinates $(z_1\ld z_n)$ centered at $x$ such that
$(\partial/\partial z_l)$ is an orthonormal basis of $T_X$ at $x$.
Consider the Taylor expansion of second order
$$\eqalign{
\omega_{lm}(z)=	
\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle=
\delta_{lm}+\sum_j(a_{jlm}z_j+\ov a_{jml}\ov z_j)
&+\sum_{j,k}(b'_{jklm}z_jz_k+\ov b'_{jkml}\ov z_j\ov z_k)\cr
&+\sum_{j,k}c'_{jklm}z_j\ov z_k+O(|z|^3).\cr}$$
We may always arrange that the antisymmetry relation $a_{jlm}=-a_{ljm}$ holds;
\hbox{otherwise} the change of variables $z_m=z_m'-{1\over 4}
\sum(a_{jlm}+a_{ljm})z'_jz'_l$ yields coordinates $(z'_l)$ with this property.
If~$\omega$ is K\"ahler, the symmetry of $a_{jlm}=\partial\omega_{lm}/\partial
z_j$ in $j,l$ implies $a_{jlm}=0\,$; in that case $b'_{jklm}$ is also
symmetric in $j,k,l$ and a new change of variables
$z_m=z_m'-{1\over 3}\sum b'_{jklm}z'_jz'_kz'_l$ gives $b'_{jklm}=0$ likewise.
The holomorphic frame of $T_X$ defined by
$$e_l={\partial\over\partial z_l}-\sum_m\Big(\sum_j a_{jlm}z_j+
\sum_{j,k}b'_{jklm}z_jz_k\Big){\partial\over\partial z_m}$$
satisfies
$$\leqalignno{
\langle e_l,e_m\rangle
&=\delta_{lm}-\sum_{j,k}c_{jklm}z_j\ov z_k+O(|z|^3),&(2.1)\cr
{\partial\over\partial z_l}
&=e_l+\sum_m\Big(\sum_j a_{jlm}z_j+\sum_{j,k}b_{jklm}z_jz_k+O(z^3)\Big)e_m
&(2.2)\cr}$$
with $c_{jklm}=-c'_{jklm}-\sum_p a_{jlp}\ov a_{kmp}$ and
$b_{jklm}=b'_{jklm}+\sum_p a_{jlp}a_{kpm}$. We may of course suppose
that $b_{jklm}=b_{kjlm}$. Also, by a modification of the third
order terms in $(e_l)$, we can suppose that no term $O(z^3)$ appears
in (2.2). The formula $\partial\langle e_l,e_m\rangle=
\langle De_l,e_m\rangle$ easily gives the expression of $De_l$, $D^2e_l$
and $\Theta(T_X)_{x}$:
$$\leqalignno{
De_l&=-\sum_{j,k,m}c_{jklm}\ov z_k\,dz_j\otimes e_m+O(|z|^2),\cr
\Theta(T_X)_{x}&={i\over 2\pi}\sum_{j,k,l,m}c_{jklm}\,dz_j\wedge d\ov z_k
\otimes e^\star_l\otimes e_m.&(2.3)\cr}$$
Given a vector field $\zeta=\sum\zeta_l\,\partial/\partial z_l$ in $T_X$,
we denote by $(\xi_l)$ the components of $\zeta$ with respect to the basis
$(e_l)$, thus $\zeta=\sum\xi_le_l$. By (2.2) we have
$$\xi_m=\zeta_m+\sum_{j,l}a_{jlm}z_j\zeta_l+\sum_{j,k,l}b_{jklm}z_jz_k\zeta_l.
\leqno(2.4)$$
In the K\"ahler case everything is much simpler, we take
$e_l=\partial/\partial z_l$ and $\xi_m=\zeta_m$. In general,
the Chern connection $D$ is given by $D\zeta=D\big(\sum\zeta_l\,
\partial/\partial z_l\big)$ with
$$\leqalignno{
D\Big({\partial\over\partial z_l}\Big)&=
-\sum_{j,k,m}c_{jklm}\ov z_k\,dz_j\otimes e_m
+\sum_{j,m}a_{jlm}dz_j\otimes e_m\cr
&\phantom{{}=-\sum_{j,k,m}c_{jklm}\ov z_k\,dz_j\otimes e_m}
+2\sum_{j,k,m}b_{jklm}z_k\,dz_j\otimes e_m+O(|z|^2)\,dz\cr
&=-\sum_{j,k,m}(c_{jklm}\ov z_k-2b_{jklm}z_k)
dz_j\otimes{\partial\over\partial z_m}\cr
&\phantom{{}={}}+\sum_{j,m}\Big(a_{jlm}+\sum_{k,p}a_{jlp}a_{kpm}z_k\Big)
dz_j\otimes{\partial\over\partial z_m}+O(|z|^2)\,dz,\cr
D\zeta&=\sum_m d\zeta_m\otimes{\partial\over\partial z_m}
-\sum_{j,k,l,m}(c_{jklm}\ov z_k-2b_{jklm}z_k)
\zeta_l dz_j\otimes{\partial\over\partial z_m}&(2.5)\cr
&\phantom{{}={}}+\sum_{j,l,m}\Big(a_{jlm}+\sum_{k,p}a_{jlp}a_{kpm}z_k\Big)
\zeta_ldz_j\otimes{\partial\over\partial z_m}+O(|z|^2).\zeta dz.\cr}$$
Consider a curve $t\mapsto u(t)$. By a substitution of variables 
$z_j=u_j(t)$, $\zeta_l=du_l/dt$ in formula (2.5), the equation 
$D(du/dt)=0$ becomes
$${d^2u_m\over dt^2}=\sum_{j,k,l}\big(c_{jklm}\ov u_k(t)-
2b_{jklm}u_k(t)\big){du_j\over dt}{du_l\over dt}
+O\big(|u(t)|^2\big).\Big({du\over dt}\Big)^2\,;\leqno(2.6)$$
the contribution of the terms $\sum a_{jl\bu}\zeta_ldz_j$ is zero
by the antisymmetry relation; moreover the remainder term only 
contains $\cb$-quadratic terms in
$du/dt$. The initial condition $u(0)=z$, $u'(0)=\zeta$ gives
$u_m(t)=z_m+t\zeta_m+O(t^2|\zeta|^2)$, hence
$${d^2u_m\over dt^2}=\sum_{j,k,l}\big(c_{jklm}(\ov z_k+t\ov\zeta_k)-2b_{jklm}
(z_k+t\zeta_k)\big)\zeta_j\zeta_l+O\big(|\zeta|^2(|z|+|\zeta|)^2\big).$$
Two successive integrations yield
$$\eqalign{
u_m(t)=z_m+t\zeta_m+\sum_{j,k,l}c_{jklm}&\Big({t^2\over 2}\ov z_k
+{t^3\over 6}\ov\zeta_k\Big)\zeta_j\zeta_l\cr
-2\sum_{j,k,l}b_{jklm}&\Big({t^2\over 2}z_k+{t^3\over 6}
\zeta_k\Big)\zeta_j\zeta_l+O\big(t^2|\zeta|^2(|z|+|\zeta|)^2\big).\cr}$$
An iteration of this procedure (substitution in (2.6) followed by an
integration) easily shows that all terms but the first two in the
Taylor expansion of $u_m(t)$ contain $\cb$-quadratic factors of the
form~$\zeta_j\zeta_l$. Let us substitute $\zeta_m$ by its expression
in terms of $z,\xi$ deduced from (2.4). We find that
$\exp_z(\zeta)=u(1)$ has a third  order expansion
$$~~\exp_z(\zeta)_m=g_m(z,\xi)+{1\over2}
\sum_{j,k,l}c_{jklm}\Big(\ov z_k+{1\over 3}\ov\xi_k\Big)
\xi_j\xi_l+O\big(|\xi|^2(|z|+|\xi|)^2\big)\leqno(2.7)$$
where $g_m(z,\xi)$ is a holomorphic polynomial of degree $3$ in $z,\xi$
with linear part $g_{m,1}(z,\xi)=z_m+\xi_m$, and where the remainder
involves $\cb$-quadratic factors $\xi_j\xi_l$ in all terms. In the K\"ahler
case we simply have
$\xi_m=\zeta_m$ and $g_m(z,\xi)=z_m+\xi_m$.

\titre 3. Regularization of quasi-psh functions|

We now come to the main idea. Select a 
cut-off function $\chi:\rb\to\rb$ of class $C^\infty$ such that
$$\chi(t)>0~~\hbox{\rm for}~~t<1,~~~~\chi(t)=0~~\hbox{\rm for}~~t\ge 1,
~~~~\int_{v\in\cb^n}\chi(|v|^2)\,d\lambda(v)=1.$$
If $\psi$ is a quasi-psh function on $X$, we set
$$\psi_\varepsilon(z)={1\over\varepsilon^{2n}}
\int_{\zeta\in T_{X,z}}\psi\big(\exp_z(\zeta)\big)\,
\chi\Big({|\zeta|^2\over\varepsilon^2}\Big)\,d\lambda(\zeta),
~~~~\varepsilon>0.\leqno(3.1)$$
Here $d\lambda$ denotes the Lebesgue measure on $\cb^n$, resp. on the 
hermitian space $\big(T_{X,z},\omega(z)\big)$. For $w\in\cb$ with 
$|w|=\varepsilon$, we have $\psi_\varepsilon(z)=\Psi(z,w)$ with
$$\Psi(z,w)=\int_{\zeta\in T_{X,z}}\psi\big(\exp_z(w\zeta)\big)\,
\chi(|\zeta|^2)\,d\lambda(\zeta).\leqno(3.2)$$
The change of variable $y=\exp_z(w\zeta)$ expresses $w\zeta$ as a smooth
function of $y,z$ in a neighborhood of the diagonal in $X\times X$.
Hence $\Psi$ is smooth over $X\times\{0<|w|<\varepsilon_0\}$ for some
$\varepsilon_0>0$. We are going to compute $\partial\op\Psi$ over this
set and estimate its negative part when $|w|$ is small. By (2.1), we have
$$\leqalignno{
~~~~|\zeta|^2&=\sum_m|\xi_m|^2-\sum_{j,k,l,m}c_{jklm}\,z_j\ov z_k\xi_l\ov\xi_m
+O(|z|^3)|\xi|^2,&(3.3)\cr
~~~~d\lambda(\zeta)&={1\over 2^nn!}(i\partial\op|\zeta|^2)^n&(3.4)\cr
&=\Big(1-\sum_{j,k,l}c_{jkll}\,z_j\ov z_k+O(|z|^3)\Big){i\over 2}d\xi_1
\wedge d\ov\xi_1\wedge\ldots\wedge{i\over 2}d\xi_n\wedge d\ov\xi_n.\cr}$$
In (3.2), we make the change of variables $\zeta\mapsto\xi\mapsto s\in\cb^n$
defined by
$$\exp_z(w\zeta)=g(z,ws),~~~~g(z,\xi)=\big(g_m(z,\xi)\big)_{1\le m\le n}.$$
Since all terms in the asymptotic expansion of $\exp_z(\zeta)-g(z,\xi)$
given by (2.7) contain a factor $\xi_l$, we see that
$w^{-1}\big(\exp_z(w\zeta)-g(z,w\xi)\big)$ is smooth with respect to all
variables. From this and the fact that $g(z,ws)-g(z,w\xi)\sim w(s-\xi)$, 
we infer that the variable $s$ depends smoothly on $\xi,z,w$ and has 
a Taylor expansion
$$s_m=\xi_m+{1\over 2}\sum_{j,k,l}c_{jklm}\Big(\ov z_k+{1\over 3}\ov w\,
\ov\xi_k\Big)w\xi_j\xi_l+O\big(|w||\xi|^2(|z|+|w||\xi|)^2\big).\leqno(3.5)$$
Therefore
$$\xi_m=s_m-{1\over 2}\sum_{j,k,l}c_{jklm}\Big(\ov z_k+{1\over 3}\ov w\,
\ov s_k\Big)ws_js_l+O\big(|w||s|^2(|z|+|w||s|)^2\big).\leqno(3.6)$$
After a substitution in (3.2), (3.3), (3.4) we get
$$\Psi(z,w)=\int_{\cb^n}\psi\big(g(z,ws)\big)\,
\chi\big(A(z,w,s)\big)\,B(z,w,s)\,d\lambda(s)\leqno(3.7)$$
where
$$\eqalign{
A(z,w,s)&=\sum_m|s_m|^2-\sum_{j,k,l,m}c_{jklm}
\Big(z_j\ov z_k+{1\over 2}w s_j\ov z_k+{1\over 2}\ov wz_j\ov s_k+
{1\over3}w\ov ws_j\ov s_k\Big)s_l\ov s_m\cr
&\phantom{{}=\sum_m|s_m|^2-\sum_{j,k,l,m}c_{jklm}
\Big(z_j\ov z_k+{1\over 2}w s_j\ov z_k}+O\big((|z|+|w||s|)^3|s|^2\big)\cr
B(z,w,s)&=1-\sum_{j,k,l}c_{jkll}\Big(z_j\ov z_k+{1\over 2}w s_j\ov z_k+
{1\over 2}\ov wz_j\ov s_k+{1\over3}w\ov ws_j\ov s_k\Big)\cr
&\phantom{{}=1}-\sum_{j,k,l}c_{jklj}\Big({1\over 2}w\ov z_k s_l+
{1\over 6}w\ov w\,\ov s_k s_l\Big)\cr
&\phantom{{}=1}-\sum_{j,k,m}c_{jkkm}\Big({1\over 2}\ov w z_j\ov s_m+
{1\over 6}w\ov ws_j\ov s_m\Big)+O\big((|z|+|w||s|)^3\big).\cr}$$
\vskip-4pt

\tha Proposition 3.8|For $(\theta,\eta)\in T_{X,x}\times\cb$, the Hessian form
of $\Psi$ at $(x,w)\in X\times\cb$ satisfies
$$\eqalign{
\partial\op\Psi_{(x,w)}[\theta,\eta]^2=\int_{\zeta\in T_{X,x}}
\big(\partial\op\psi[\tau]^2&+|w|^2\partial\op\psi.V\big)_{\exp_x(w\zeta)}
\,\chi(|\zeta|^2)\,d\lambda(\zeta)\cr
&+O\big(|w|\,\big|\log|w|\big|\big)[\theta,\eta]^2\cr}$$
where $\tau$ $($resp. $V)$ is a vector field $($resp. a $(1,1)$-vector
field$)$ depending smoothly on the parameters $x,w$ and linearly $($resp.
quadratically$)$ on $\theta,\eta$, and where the notation $[\tau]^2$
stands for the $(1,1)$-vector $\tau\wedge\ov\tau\in\Lambda^{1,1}T_X$.
The vector fields $\tau$, $V$ are given at $y=\exp_x(w\zeta)$ by
$$\tau(y)=\partial\exp_{(x,w\zeta)}.(\theta^h+\eta\zeta^v)+|w|^2\beta(y)$$
where $\theta^h,\zeta^v\in T(T_X)_{(x,w\zeta)}$ are respectively the
horizontal lifting of $\theta$ with respect to the connection $D$ and
the vertical tangent vector associated to $\zeta$, and
$$\eqalign{
\beta(y)={}&\sum_{j,k,l,m}c_{jklm}\Big({1\over 2}\theta_j+{1\over 6}\zeta_j
\eta\Big)\ov\zeta_k\zeta_l\,{\partial\over\partial z_m}\cr
&+{\chi_1(|\zeta|^2)\over\chi(|\zeta|^2)}\sum_{j,k,m}\Big\{
c_{jkkm}\Big({1\over 2}\theta_j+{1\over 3}\zeta_j\eta\Big)
-c_{kmjk}\,{1\over 6}\zeta_j\eta\Big\}{\partial\over\partial z_m},\cr
V(y)=&-{\chi_1(|\zeta|^2)\over\chi(|\zeta|^2)}\sum_{j,k,l,m}c_{jklm}
\Big(\theta_j\ov\theta_k+{1\over 2}\zeta_j\ov\theta_k\eta+{1\over 2}\ov \zeta_k
\theta_j\ov\eta+{1\over 3}\zeta_j\ov\zeta_k\eta\ov\eta\Big)
{\partial\over\partial z_m}\wedge{\partial\over\partial\ov z_l}\cr
&+{\chi_2(|\zeta|^2)\over\chi(|\zeta|^2)}\sum_{j,l,m}(c_{jmlj}+c_{ljjm})
{1\over 6}\eta\ov\eta\,{\partial\over\partial z_m}\wedge
{\partial\over\partial\ov z_l},\cr}$$
where $\chi_1(t)=\int_{+\infty}^t\chi(u)\,du$ and 
$\chi_2(t)=\int_{+\infty}^t\chi_1(u)\,du$ are primitives of $\chi$ with
$\chi_j(t)=0$ for $t\ge 1$.\fintha

\remarque Proof|A brute force differentiation of (3.6) at $z=0$ gives
$$\leqalignno{
\partial\op\Psi_{(x,w)}[\theta,\eta]^2
={}&\int_{\cb^n}\partial\op(\psi\circ g)_{(0,ws)}[\theta,\eta s]^2\,
\chi\big(A(0,w,s)\big)\,B(0,w,s)\,d\lambda(s)\cr
&-\int_{\cb^n}\op(\psi\circ g)_{(0,ws)}[\theta,\eta s]\,
E_{(w,s)}[\theta,\eta]\,d\lambda(s)&(3.9)\cr
&-\int_{\cb^n}\partial(\psi\circ g)_{(0,ws)}[\theta,\eta s]\,
\ov{E_{(w,s)}[\theta,\eta]}\,d\lambda(s)\cr
&-\int_{\cb^n}\psi\circ g(0,ws)\,F_{(w,s)}[\theta,\eta]^2\,d\lambda(s)
&(3.9')\cr
&&\hbox{\rm where}\cr
E_{(w,s)}[\theta,\eta]
=\chi'&(|s|^2)\sum_{j,k,l,m}c_{jklm}\Big({1\over 2}\ov w\,\ov s_k\theta_j
+{1\over 3}\ov ws_j\ov s_k\eta\Big)s_l\ov s_m\cr
{}+\chi&(|s|^2)\sum_{j,k,l}c_{jkll}\Big({1\over 2}\ov w\,\ov s_k\theta_j
+{1\over 3}\ov ws_j\ov s_k\eta\Big)\cr
{}+\chi&(|s|^2)\sum_{j,k,l}c_{jklj}{1\over 6}\ov w\,\ov s_ks_l\eta\cr
{}+\chi&(|s|^2)\sum_{j,k,m}c_{jkkm}\Big({1\over 2}\ov w\,\ov s_m\theta_j
+{1\over 6}\ov ws_j\ov s_m\eta\Big)+O(|w|^2|s|^2)[|\theta|+|s||\eta|],\cr}$$
$$\eqalign{
F_{(w,s)}[\theta,\eta]^2
=\chi'&(|s|^2)\sum_{j,k,l,m}c_{jklm}\Big(\theta_j\ov\theta_k+
{1\over 2}s_j\ov\theta_k\eta+{1\over 2}\ov s_k\theta_j\ov\eta
+{1\over 3}s_j\ov s_k\eta\ov\eta\Big)s_l\ov s_m\cr
{}+\chi&(|s|^2)\sum_{j,k,l}c_{jkll}\Big(\theta_j\ov\theta_k+
{1\over 2}s_j\ov\theta_k\eta+{1\over 2}\ov s_k\ov\theta_j\ov\eta
+{1\over 3}s_j\ov s_k\eta\ov\eta\Big)\cr
{}+\chi&(|s|^2)\sum_{j,k,l}c_{jklj}\Big({1\over 2}s_l\ov\theta_k\eta
+{1\over 6}\ov s_k s_l\eta\ov\eta\Big)\cr
{}+\chi&(|s|^2)\sum_{j,k,m}c_{jkkm}\Big({1\over 2}\ov s_m\theta_j\ov\eta
+{1\over 6}s_j\ov s_m\eta\ov\eta\Big)+O(|w||s|)[|\theta|+|s||\eta|]^2.\cr}$$
In all these expansions, the remainder terms $O(\bu)$
involve uniform constants when the origin $x$ of coordinates belongs to a
compact subset of a coordinate patch.  By the mean value properties of
plurisubharmonic functions, we have
$$\int_{|s|<1}|\psi(x+ws)|\,d\lambda(s)\le C\big(1+\big|\log|w|\big|\big)$$
locally uniformly in $x$. An integration by parts with compact supports
yields
$$\int_{\cb^n}\partial(\psi\circ g)_{(0,ws)}\,O(|w|^2)\,d\lambda(s)=
\int_{\cb^n}\psi\circ g(0,ws)\,O(|w|)\,d\lambda(s)=O\big(|w|\log|w|\big),$$
hence the remainder terms $O(|w|^\bu)$ in $E_{(w,s)}$, $F_{(w,s)}$ give
a negligible contribution in $\partial\op\Psi$ as $|w|$ tends to $0$. 
We find
$$\leqalignno{
E_{(w,s)}[\theta,\eta]=\sum_m{\partial\over\partial s_m}\Big\{
&\chi(|s|^2)\sum_{j,k,l}c_{jklm}\Big({1\over 2}\ov w\,\ov s_k\theta_j
+{1\over 3}\ov w s_j\ov s_k\eta\Big)s_l\cr
{}-\chi_1(|s|^2)\sum_{j,l}c_{jmlj}\,{1\over 6}\,\ov w s_l&\eta
+\chi_1(|s|^2)\sum_{j,k}c_{jkkm}\Big(
{1\over 2}\ov w\theta_j+{1\over 6}\ov w s_j\eta\Big)s_l\Big\}+O(|w|^2)\cr
F_{(w,s)}[\theta,\eta]^2=\sum_{l,m}{\partial^2\over\partial\ov s_l\partial s_m}
\Big\{&\chi_1(|s|^2)\sum_{j,k}c_{jklm}\Big(\theta_j\ov\theta_k{+}
{1\over 2}s_j\ov\theta_k\eta{+}{1\over 2}\ov s_k\theta_j\ov\eta 
{+}{1\over 3}s_j\ov s_k\eta\ov\eta\Big)\cr
{}-{}&\chi_2(|s|^2)\Big(\sum_j c_{jmlj}+\sum_k c_{lkkm}\Big)
{1\over 6}\eta\ov\eta\Big\}+O(|w|).\cr}$$
An integration by parts in $(3.9)$ and $(3.9')$ gives
$$\eqalign{
\partial\op\Psi_{(x,w)}&[\theta,\eta]^2=\int_{\cb^n}
\partial\op(\psi\circ g)_{(0,ws)}.\Big([\theta,\eta s]^2
+|w|^2(0,\alpha)\wedge\ov{(\theta,\eta s)}\cr
&{}+|w|^2(\theta,\eta s)\wedge\ov{(0,\alpha)}+
|w|^2(0,U)\Big)\,\chi\big(A(0,w,s)\big)\,B(0,w,s)\,d\lambda(s)\cr
&{}+O\big(|w|\log|w|\big)[\theta,\eta]^2\cr}$$
where
$$\eqalign{
\alpha={}&\sum_{j,k,l,m}c_{jklm}\Big({1\over 2}\theta_j+{1\over 3}s_j
\eta\Big)\ov s_k s_l\,{\partial\over\partial z_m}\cr
&+{\chi_1(|s|^2)\over\chi(|s|^2)}\Big\{\sum_{j,k,m}c_{jkkm}\Big({1\over 2}
\theta_j+{1\over 3}s_j\eta\Big){\partial\over\partial z_m}\,{-}\sum_{j,l,m}
c_{jmlj}\,{1\over 6}s_l\eta\,{\partial\over\partial z_m}\Big\}+O(|w|),\cr
U=&-{\chi_1(|s|^2)\over\chi(|s|^2)}\sum_{j,k,l,m}c_{jklm}
\Big(\theta_j\ov\theta_k+{1\over 2}s_j\ov\theta_k\eta+{1\over 2}\ov s_k
\theta_j\ov\eta+{1\over 3}s_j\ov s_k\eta\ov\eta\Big)
{\partial\over\partial z_m}\wedge{\partial\over\partial\ov z_l}\cr
&+{\chi_2(|s|^2)\over\chi(|s|^2)}\sum_{j,l,m}(c_{jmlj}+c_{ljjm})
{1\over 6}\eta\ov\eta\,{\partial\over\partial z_m}\wedge
{\partial\over\partial\ov z_l}+O(|w|).\cr}$$
Observe that the choice $\chi_2(t)=C\exp\big(1/(t-1)\big)$ for $t<1$ gives
smooth and bounded quotients 
$${\chi_1(t)\over\chi(t)}=-{(1-t)^2\over 3-2t},~~~~
{\chi_2(t)\over\chi(t)}={(1-t)^4\over 3-2t}~~~\hbox{\rm for}~~t<1,$$
thus our vector fields are smooth. We get the formula in proposition~3.8
with
$$\eqalign{
\tau&=dg_{(0,ws)}(\theta,\eta s+|w|^2\alpha)
       =dg_{(0,ws)}(\theta,\eta s)+|w|^2\alpha+O(|w|^3)\cr
V&=dg_{(0,ws)}(0,U-|w|^2\alpha\wedge\ov\alpha)=U+O(|w|).\cr}$$
Since $g(z,ws)=\exp_z(w\zeta)$, a differentiation yields
$$dg_{(0,ws)}(\theta,\eta s)=d\exp_{(x,w\zeta)}.(\theta,w\delta)$$
where $(\theta,\delta)\in T(T_X)$ is the image of the tangent vector
$(\theta,w^{-1}\eta s)$ in the coordinate change $(z,s)\mapsto(z,\zeta)$.
However, the coordinates $(z,\xi)$ on $T_X$ are precisely those which
realize the splitting $T(T_X)_{(x,\zeta)}=(T_{X,x})^h\oplus(T_{X,x})^v$ with 
respect to the connection $D$. In these coordinates, we find, thanks to~(3.6):
$$\eqalign{
\delta_m&=d\xi_m.(\theta,w^{-1}\eta s)\cr
&=w^{-1}\eta s_m-\sum_{j,k,l}c_{jklm}\Big\{{1\over 2}\Big(\ov\theta_k+
{1\over 3}\ov\eta\,\ov s_k\Big)ws_js_l+{1\over 3}\ov w\,\ov s_k\eta s_js_l\Big\}
+O\big(|w|^2\big).\cr}$$
If we combine this with (3.5) and observe that $\xi_m=\zeta_m$ for
$z=0$, we get
$$w\delta_m=\eta\Big(\zeta_m-{1\over 6}|w|^2\sum_{j,k,l}c_{jklm}\ov\zeta_k
\zeta_j\zeta_l\Big)-w^2\sum_{j,k,l}c_{jklm}\Big({1\over 2}\ov\theta_k+
{1\over 6}\ov\eta\ov\zeta_k\Big)\zeta_j\zeta_l+O\big(|w|^3\big).$$
Since $\exp_z(\zeta)_m=z_m+\xi_m+O\big(|\xi|(|z|+|\xi|)\big)$, we find
$\op\exp_{(x,w\zeta)}=O(|w|)$ and $\partial\exp_{(x,w\zeta)}=dz+d\xi+
O(|w|)$. Therefore
$$\eqalign{
dg_{(0,ws)}(\theta,\eta s)&=\partial\exp_{(x,w\zeta)}.(\theta^h+\eta\zeta^v)
-{1\over 6}\eta|w|^2\sum_{j,k,l}c_{jklm}\ov\zeta_k\zeta_j\zeta_l\,
{\partial\over\partial z_m}+O\big(|w|^3\big),\cr
\tau&=\partial\exp_{(x,w\zeta)}.(\theta^h+\eta\zeta^v)+|w|^2\beta~~~~~
\hbox{\rm with}\cr
\beta&=\alpha-{1\over 6}\eta\sum_{j,k,l}c_{jklm}\ov\zeta_k\zeta_j\zeta_l\,
{\partial\over\partial z_m}+O(|w|).\cr}$$
Since $\zeta_m=\xi_m=s_m+O(|w|^2|s|)$ for $z=0$ by~(3.6), we infer 
from the expressions of $\alpha$ and $U$ that the vector fields $\tau$ and
$V$ are precisely those given in proposition~3.8. We can drop the terms
$O(|w|)$ in $\beta$ and $V$ because
$$\eqalign{
\int\partial\op\psi\big(\exp_x(w\zeta)\big)\,O\big(|w|^3\big)\,
d\lambda(\zeta)&=\int\psi\big(\exp_x(w\zeta)\big)\,O(|w|)\,d\lambda(\zeta)\cr
&=O\big(|w|\log|w|\big).\finpr\cr}$$

\titre 4. Main approximation theorem|

A straightforward consequence of our computations is the following
approximation theorem for closed positive currents.

\tha Theorem 4.1|Let $X$ be a compact $n$-dimensional manifold equipped
with a smooth hermitian metric~$\omega$. Fix a smooth semipositive
$(1,1)$-form $u$ on $X$ such that the Chern curvature of $(T_X,\omega)$
satisfies $\Theta(T_X)+u\otimes{\rm Id}_{T_X}\ge 0$ in the sense of Griffiths. 
Let $T=\alpha+{i\over\pi}\partial\op\psi$ be a closed real current 
where $\alpha$ is a smooth closed real $(1,1)$-form and $\psi$ is 
quasi-psh. Suppose that $T\ge\gamma$ for some real $(1,1)$-form
$\gamma$ with continuous coefficients. As~$w$ tends to $0$ and
$x$ runs over~$X$, there is a uniform lower estimate
$$\alpha_x[\theta]^2+{i\over\pi}\partial\op\Psi_{(x,w)}[\theta,\eta]^2\ge
\gamma_x[\theta]^2-\lambda(x,|w|)\,u_x[\theta]^2-\delta(|w|)\,|\theta|^2-
{1\over\pi}K|\eta|^2$$
where $K$ is a constant, $\lim_{t\to 0}\delta(t)=0$ and where 
$\lambda(x,t)=t\,\partial/\partial t\big(\Psi(x,t)+Kt^2\big)$ is a 
nonnegative continuous function on $X\times]0,\varepsilon_0[$ which is 
increasing in $t$ and such that
$$\lim_{t\to 0}\lambda(x,t)=\nu(T,x)~~~~\hbox{\rm(Lelong number of
$T$ at $x$).}$$
In particular, the currents $T_\varepsilon=\alpha+{i\over\pi}\partial\op
\psi_\varepsilon$ are smooth closed real currents converging weakly
to $T$ as $\varepsilon$ tends to~$0$, such that
$$T_\varepsilon\ge\gamma-\lambda(\bu,\varepsilon)\,u
-\delta(\varepsilon)\,\omega.$$
\fintha\vskip-6pt

\remarque Proof|The estimate is local on $X$. If we change $\psi$ into
$\psi+\psi_0$ with a smooth function $\psi_0$, then $\alpha$ is changed
into $\alpha-{i\over\pi}\partial\op\psi_0$ and $\Psi$ into
$\Psi+\Psi_0$, where $\Psi_0$ is a smooth function on $X\times\cb$ such
that $\Psi_0(z,w)=\psi_0(z)+O(|w|^2)$. It follows that the estimate
remains unchanged up to a term $O(1)|\eta|^2$. We can thus work on  a
small coordinate open set $\Omega\subset X$ and choose $\psi_0$ such 
that $\psi+\psi_0$ is plurisubharmonic on~$\Omega$. Hence we may in
fact suppose that $T=\alpha+{i\over\pi}\partial\op\psi$ on
$\Omega$, where $\psi$ is plurisubharmonic and ${i\over\pi}\partial\op\psi
\ge\gamma-\alpha$.

We then use proposition 3.8 to compute ${i\over\pi}\partial\op\Psi_{(x,w)}
[\theta,\eta]^2$. The contribution given by the first term
$i\partial\op\psi[\tau]^2$ is nonnegative and bounded below by
$$\int_{\zeta\in\cb^n}(\gamma-\alpha)[\tau]^2\chi(|\zeta|^2)\,d\lambda(\zeta)
\ge(\gamma_x-\alpha_x)[\theta]^2-o(1)\,|\theta|^2-O(1)\,|\eta|^2$$
when $|w|$ tends to zero, thanks to the continuity of~$\gamma-\alpha$ and
the fact that $\tau=\theta+\eta\zeta+O(|w|)$ in coordinates.
By taking into account the second term $i\partial\op\psi.V$ and applying
the change of variable $\exp_x(w\zeta)=x+ws$ with respect to
the normal coordinates $(z_j)$ of section~2, we infer
$$\eqalign{
{i\over\pi}\partial\op\Psi_{(x,w)}[\theta,\eta]^2\ge
(\gamma_x-\alpha_x)[\theta]^2&{}+|w|^2\int_{\cb^n}\Big({i\over\pi}
\partial\op\psi.V\Big)(x+ws)\,\chi(|s|^2)\,d\lambda(s)\cr
&{}-o(1)|\theta|^2-O(1)|\eta|^2,\cr}$$
because all terms $\partial\op\psi(x+ws)\,O(|w|^3)$ can be neglected. 
Up to such terms, $(i\partial\op\psi.V\big)(x+ws)\,\chi(|s|^2)$ is
equal to
$$\eqalign{
&{}-\chi_1(|s|^2)\sum_{j,k,l,m}
{\partial^2\psi\over\partial\ov z_l\partial z_m}(x+ws)\,c_{jklm}
\Big(\theta_j\ov\theta_k+{1\over 2}s_j\ov\theta_k\eta+{1\over 2}\ov s_k
\theta_j\ov\eta+{1\over 3}s_j\ov s_k\eta\ov\eta\Big)\cr
&{}+\chi_2(|s|^2)\sum_{j,l,m}{\partial^2\psi\over\partial\ov z_l
\partial z_m}(x+ws)\,(c_{jmlj}+c_{ljjm}){1\over 6}\eta\ov\eta.\cr}$$
Let us first consider the contribution of the pure terms $\theta_j\ov\theta_k$:
$${1\over\pi}|w|^2\int_{\cb^n}-\chi_1(|s|^2)\sum_{j,k,l,m}
{\partial^2\psi\over\partial\ov z_l\partial z_m}(x+ws)\,c_{jklm}\theta_j
\ov\theta_k\,d\lambda(s).$$
By our curvature assumption
$$\sum_{j,k,l,m}{1\over 2\pi}c_{jklm}\theta_j\ov\theta_k\xi_l\ov\xi_m+
\sum_{j,k,l}u_{jk}\theta_j\ov\theta_k\xi_l\ov\xi_l\ge 0$$
for all tangent vectors $\theta,\xi$. Apply this inequality to each
vector $\xi$ in a basis of eigenvectors of $\partial\op\psi$, multiply
by the corresponding (nonnegative) eigenvalue and take the sum. We get
$${1\over 2\pi}\sum_{j,k,l,m}{\partial^2\psi\over\partial\ov z_l\partial z_m}
\,c_{jklm}\theta_j\ov\theta_k+
\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}\,u[\theta]^2\ge 0.$$
As $-\chi_1(|s|^2)\ge 0$, we conclude that the terms in $\theta_j\ov\theta_k$
are bounded below by $-\lambda_\Omega(x,|w|)\,u_x[\theta]^2$ where
$$\lambda_\Omega(x,|w|)=2|w|^2\int_{\cb^n}-\chi_1(|s|^2)\sum_l
{\partial^2\psi\over\partial z_l\partial\ov z_l}(x+ws)\,d\lambda(s).$$
By definition, the Lelong number $\nu(T,x)$ is $\lim_{r\to 0}\nu(T,x,r)$
where
$$\eqalign{
\nu(T,x,r)&={(n-1)!\,2^n\over(2\pi r^2)^{n-1}}\int_{B(x,r)}{1\over\pi}
\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}\,d\lambda(z),\cr
\nu(T,x,|w|r)&={2\,(n-1)!\over\pi^n r^{2n-2}}|w|^2\int_{|s|<r}
\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}(x+ws)\,d\lambda(s).
\cr}$$
As $-\chi_1(|s|^2)=2\int_{|s|}^{+\infty}\chi(r^2)\,r\,dr$, the Fubini
formula gives
$$\eqalign{
\lambda_\Omega(x,|w|)&=4|w|^2\int_0^{+\infty}\Big\{\int_{|s|<r}
\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}(x+ws)\,d\lambda(s)
\Big\}~\chi(r^2)\,r\,dr\cr
&={2\pi^n\over(n-1)!}\int_0^1\nu(T,x,|w|r)\,\chi(r^2)\,r^{2n-1}\,dr,\cr
\lambda_\Omega(x,t)&=\int_{\cb^n}\nu(T,x,t|s|)\,\chi(|s|^2)\,d\lambda(s).\cr}$$
Hence $\lambda_\Omega(x,t)$ is smooth in $(x,t)$, increasing in $t$ and 
$\lim_{t\to 0}\lambda_\Omega(x,t)=\nu(T,x)$,\break as desired. Similarly, 
the coefficient of $\eta\ov\eta$ is bounded below by
$$-C\int_{|s|<1}\sum_l{\partial^2\psi\over\partial z_l\partial\ov z_l}
(x+ws)\,d\lambda(s)=-C'\nu(T,x,|w|)$$
and $\nu(T,x,|w|)$ is uniformly bounded on a neighborhood of 
$\Omega\times\{0\}$ in $X\times\cb$.

We are left with the coefficients of the mixed terms $\theta_j\ov\eta$
or $\ov\theta_k\eta$. These coefficients are also uniformly bounded by
$\nu(T,x,|w|)$. Moreover, any weak limit of dilations
$s\mapsto\partial\op\psi(x+ws)$ as $w$ tends to $0$ is invariant by
complex homotheties (cf. e.g. [BDM89]). Since the integrals
contain terms $s_j$ or $\ov s_k$ which have zero integral over circles,
the limit of these coefficients as $w$ tends to $0$ must vanish.
Adding up everything, we finally get
$${i\over\pi}\partial\op\Psi_{(x,w)}[\theta,\eta]^2\ge
(\gamma_x-\alpha_x)[\theta]^2-\lambda_\Omega(x,|w|)u_x[\theta]^2
-\delta(|w|)\,|\theta|^2-K|\eta|^2.$$
The estimate of Theorem~4.1 thus holds on $\Omega$ with
$\lambda_\Omega(x,t)$ in place of $\lambda(x,t)$; we only have to show
that this change is unimportant. By the estimate just obtained
$\Psi(x,w)+K|w|^2$ is plurisubharmonic in $w$, and so is a convex
function of $\log|w|\,$; since $\Psi(x,t)$ is bounded above as $t$ 
tends to $0$, the sum $\Psi(x,t)+Kt^2$ which is convex in $\log t$ must 
be increasing in $t$. Therefore
$$\lambda(x,t)={\partial\over\partial\log t}\big(\Psi(x,t)+Kt^2\big)$$
is a nonnegative increasing function of $t$. Proposition~3.8 gives
$${\partial^2\Psi\over\partial w\partial\ov w}(x,w)=
\int_{\zeta\in T_{X,x}}\big(\partial\op\psi[\tau^v]^2\big)_{\exp_x(w\zeta)}
\chi(|\zeta|^2)\,d\lambda(\zeta)+O(1)$$
with $\tau^v\big(\exp_x(w\zeta)\big)=\partial\exp_{(x,w\zeta)}(\zeta^v)$,
because all curvature terms which contain a factor $O(|w|^2)$ ultimately
give a bounded contribution thanks to our previous arguments.
The change of variable $\exp_x(w\zeta)=x+ws$ yields $s=\zeta+O(w^2)$ and
$\tau^v=s+O(w^2)$, hence
$${\partial^2\Psi\over\partial w\partial\ov w}(x,w)=
\int_{\cb^n}\partial\op\psi_{x+ws}[s]^2\,\chi(|s|^2)\,d\lambda(s)+O(1).$$
Since $\partial^2/\partial w\partial\ov w=(t^{-1}\partial/\partial t)
\circ(t\,\partial/\partial t)$ for a function of $w$ depending only on 
$t=|w|$, a multiplication by $t$ followed by an integration implies
$$\eqalign{
t{\partial\Psi(x,t)\over\partial t}&=t{\partial\over\partial t}
\int_{\cb^n}\psi(x+ts)\,\chi(|s|^2)\,d\lambda(s)+O(t^2)\cr
&=\int_{\cb^n}\nu(T,x,t|s|)\,\chi(|s|^2)\,d\lambda(s)+O(t^2)
=\lambda_\Omega(x,t)+O(t^2)\,;\cr}$$
here, we used the well known fact that $\nu(T,x,t)$ is equal to the 
derivative $\partial/\partial\log t$ of the mean value of $\psi$ on 
the sphere $S(x,t)$. The first estimate in 4.1 follows. Now,
it is clear that $\psi_\varepsilon$ converges to $\psi$ in
$L^1_{\rm loc}$, so $T_\varepsilon$ converges weakly to~$T\,$;
note also that $\psi_\varepsilon+K\varepsilon^2$ is increasing
in $\varepsilon$ by the above arguments. The proof is complete.\finpr

\remarque Remark 4.2|An integration of the first equality in the last
formula also shows that
$$\Psi(x,t)=\int_{\cb^n}\psi(x+ts)\,\chi(|s|^2)\,d\lambda(s)+O(t^2).$$
\vfill\eject

\centerline{\bf References}
\vskip15pt
\eightpoint
\lettre KMM87| 

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\vskip30pt
\tenpoint\noindent(November 1989, revised September 9, 1991)

\end
