% Complex Analytic and Differential Geometry, Chapter III % J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France \input analgeom.mac \def\gge{{\scriptscriptstyle\ge}} \titlea{Chapter III}{\newline Positive Currents and Lelong Numbers} \begpet In 1957, P. Lelong introduced natural positivity concepts for currents of pure bidimension $(p,p)$ on complex manifolds. With every analytic subset is associated a current of integration over its set of regular points and all such currents are positive and closed. The important closedness property is proved here via the Skoda-El Mir extension theorem. Positive currents have become an important tool for the study of global geometric problems as well as for questions related to local algebra and intersection theory. We develope here a differential geometric approach to intersection theory through a detailed study of wedge products of closed positive currents (Monge-Amp\`ere operators). The Lelong-Poincar\'e equation and the Jensen-Lelong formula are basic in this context, providing a useful tool for studying the location and multiplicities of zeroes of entire functions on $\bbbc^n$ or on a manifold, in relation with the growth at infinity. Lelong numbers of closed positive currents are then introduced; these numbers can be seen as a generalization to currents of the notion of multiplicity of a germ of analytic set at a singular point. We prove various properties of Lelong numbers (e.g. comparison theorems, semi-continuity theorem of Siu, transformation under holomorphic maps). As an application to Number Theory, we prove a general Schwarz lemma in $\bbbc^n$ and derive from it Bombieri's theorem on algebraic values of meromorphic maps and the famous theorems of Gelfond-Schneider and Baker on the transcendence of exponentials and logarithms of algebraic numbers. \endpet \titleb{1.}{Basic Concepts of Positivity} \titlec{1.A.}{Positive and Strongly Positive Forms} Let $V$ be a complex vector space of dimension $n$ and $(z_1\ld z_n)$ coordinates on $V$. We denote by $(\partial/\partial z_1\ld \partial/\partial z_n)$ the corresponding basis of $V$, by $(dz_1\ld dz_n)$ its dual basis in $V^\star$ and consider the exterior algebra $$\Lambda V^\star_\bbbc=\bigoplus\Lambda^{p,q}V^\star,~~~~ \Lambda^{p,q}V^\star=\Lambda^pV^\star\otimes\Lambda^q\ovl{V^\star}.$$ We are of course especially interested in the case where $V=T_xX$ is the tangent space to a complex manifold $X$, but we want to emphasize here that our considerations only involve linear algebra. Let us first observe that $V$ has a canonical orientation, given by the $(n,n)$-form $$\tau(z)=\ii dz_1\wedge d\ovl z_1\wedge\ldots\wedge\ii dz_n\wedge d\ovl z_n= 2^n\,dx_1\wedge dy_1\wedge\ldots\wedge dx_n\wedge dy_n$$ where $z_j=x_j+iy_j$. In fact, if $(w_1\ld w_n)$ are other coordinates, we find $$\eqalign{ &dw_1\wedge\ldots\wedge dw_n=\det(\partial w_j/\partial z_k)\,dz_1\wedge\ldots \wedge dz_n,\cr &\tau(w)=\big|\det(\partial w_j/\partial z_k)\big|^2\,\tau(z).\cr}$$ In particular, a complex manifold always has a canonical orientation. More generally, natural positivity concepts for $(p,p)$-forms can be defined. \begstat{(1.1) Definition} A $(p,p)$-form $u\in\Lambda^{p,p}V^\star$ is said to be positive if for all $\alpha_j\in V^\star$, $1\le j\le q=n-p$, then $$u\wedge \ii\alpha_1\wedge\ovl\alpha_1\wedge\ldots\wedge \ii\alpha_q\wedge \ovl\alpha_q$$ is a positive $(n,n)$-form. A $(q,q)$-form $v\in\Lambda^{q,q}V^\star$ is said to be strongly positive if $v$ is a convex combination $$v=\sum\gamma_s~\ii\alpha_{s,1}\wedge\ovl\alpha_{s,1} \wedge\ldots\wedge \ii\alpha_{s,q}\wedge\ovl\alpha_{s,q}$$ where $\alpha_{s,j}\in V^\star$ and $\gamma_s\ge 0$. \endstat \begstat{(1.2) Example} \rm Since $\ii^p(-1)^{p(p-1)/2}=\ii^{p^2}$, we have the commutation rules $$\eqalign{ &\ii\alpha_1\wedge\ovl\alpha_1\wedge\ldots\wedge \ii\alpha_p\wedge\ovl\alpha_p= i^{p^2}\alpha\wedge\ovl\alpha,~~~~\forall\alpha=\alpha_1\wedge\ldots\wedge\alpha_p \in\Lambda^{p,0}V^\star,\cr &i^{p^2}\beta\wedge\ovl \beta\wedge i^{m^2}\gamma\wedge\ovl \gamma= i^{(p+m)^2}\beta\wedge \gamma\wedge\ovl{\beta\wedge \gamma},~~~~ \forall \beta\in\Lambda^{p,0}V^\star,~\forall \gamma\in\Lambda^{m,0}V^\star.\cr}$$ Take $m=q$ to be the complementary degree of $p$. Then $\beta\wedge \gamma=\lambda dz_1\wedge\ldots\wedge dz_n$ for some $\lambda\in\bbbc$ and $i^{n^2}\beta\wedge\gamma\wedge\ovl{\beta\wedge \gamma}= |\lambda|^2\tau(z)$. If we set $\gamma=\alpha_1\wedge\ldots\wedge\alpha_q$, we find that $i^{p^2}\beta\wedge\ovl \beta$ is a positive $(p,p)$-form for every $\beta\in\Lambda^{p,0}V^\star\,$; in particular, strongly positive forms are positive.\qed \endstat The sets of positive and strongly positive forms are closed convex cones, i.e. closed and stable under convex combinations. By definition, the positive cone is dual to the strongly positive cone via the pairing $$\cmalign{ &\Lambda^{p,p}V^\star\times&\Lambda^{q,q}V^\star&\lra\bbbc\cr &\hfill(u,&v)&\longmapsto u\wedge v/\tau,\cr}\leqno(1.3)$$ that is, $u\in\Lambda^{p,p}V^\star$ is positive if and only if $u\wedge v\ge 0$ for all strongly positive forms $v\in\Lambda^{q,q}V^\star$. Since the bidual of an arbitrary convex cone $\Gamma$ is equal to its closure $\ovl\Gamma$, we also obtain that $v$ is strongly positive if and only if $v\wedge u=u\wedge v$ is $\ge 0$ for all positive forms $u$. Later on, we will need the following elementary lemma. \begstat{(1.4) Lemma} Let $(z_1\ld z_n)$ be arbitrary coordinates on $V$. Then $\Lambda^{p,p}V^\star$ admits a basis consisting of strongly positive forms $$\beta_s=\ii\beta_{s,1}\wedge\ovl\beta_{s,1}\wedge\ldots \wedge\ii\beta_{s,p}\wedge\ovl\beta_{s,p},~~~~1\le s\le{n\choose p}^2$$ where each $\beta_{s,l}$ is of the type $dz_j\pm dz_k$ or $dz_j\pm \ii dz_k$, $1\le j,k\le n$. \endstat \begproof{} Since one can always extract a basis from a set of generators, it is sufficient to see that the family of forms of the above type generates $\Lambda^{p,p}V^\star$. This follows from the identities $$\cmalign{ &4dz_j\wedge d\ovl z_k=~~~&(dz_j+dz_k)&\wedge&(\ovl{dz_j+dz_k})&- &(dz_j-dz_k)&\wedge&(\ovl{dz_j-dz_k})\cr &\hfill+\ii&(dz_j+\ii dz_k)&\wedge&(\ovl{dz_j+\ii dz_k})&-\ii&(dz_j-\ii dz_k)&\wedge&(\ovl{dz_j-\ii dz_k}),\cr}$$ $$dz_{j_1}\wedge\ldots\wedge dz_{j_p}\wedge d\ovl z_{k_1}\wedge \ldots\wedge d\ovl z_{k_p}=\pm\bigwedge_{1\le s\le p}dz_{j_s}\wedge d\ovl z_{k_s}.\eqno{\square}$$ \endproof \begstat{(1.5) Corollary} All positive forms $u$ are real, i.e. satisfy $\ovl u=u$. In terms of coordinates, if $u=i^{p^2}\sum_{|I|=|J|=p}u_{I,J}\, dz_I\wedge d\ovl z_J$, then the coefficients satisfy the hermitian symmetry relation $\ovl{u_{I,J}}=u_{J,\,I}$. \endstat \begproof{} Clearly, every strongly positive $(q,q)$-form is real. By Lemma 1.4, these forms generate over $\bbbr$ the real elements of $\Lambda^{q,q}V^\star$, so we conclude by duality that positive $(p,p)$-forms are also real.\qed \endproof \begstat{(1.6) Criterion} A form $u\in\Lambda^{p,p}V^\star$ is positive if and only if its restriction $u_{\restriction S}$ to every $p$-dimensional subspace $S\subset V$ is a positive volume form on $S$. \endstat \begproof{} If $S$ is an arbitrary $p$-dimensional subspace of $V$ we can find coordinates $(z_1\ld z_n)$ on $V$ such that $S=\{z_{p+1}=\ldots=z_n=0\}$. Then $$u_{\restriction S}=\lambda_S\,\ii dz_1\wedge d\ovl z_1\wedge\ldots\wedge \ii dz_p\wedge d\ovl z_p$$ where $\lambda_S$ is given by $$u\wedge\ii dz_{p+1}\wedge d\ovl z_{p+1}\wedge\ldots\wedge \ii dz_n\wedge d\ovl z_n=\lambda_S\,\tau(z).$$ If $u$ is positive then $\lambda_S\ge 0$ so $u_{\restriction S}$ is positive for every $S$. The converse is true because the $(n-p,n-p)$-forms $\bigwedge_{j>p}\ii dz_j\wedge d\ovl z_j$ generate all strongly positive forms when $S$ runs over all $p$-dimensional subspaces.\qed \endproof \begstat{(1.7) Corollary} A form $u=\ii\sum_{j,k}u_{jk}\,dz_j\wedge d\ovl z_k$ of bidegree $(1,1)$ is positive if and only if $\xi\mapsto\sum u_{jk}\xi_j\ovl\xi_k$ is a semi-positive hermitian form on $\bbbc^n$. \endstat \begproof{} If $S$ is the complex line generated by $\xi$ and $t\mapsto t\xi$ the parametrization of $S$, then $u_{\restriction S}= \big(\sum u_{jk}\xi_j\ovl\xi_k\big)\,\ii dt\wedge d\ovl t$.\qed \endproof Observe that there is a canonical one-to-one correspondence between hermitian forms and real $(1,1)$-forms on $V$. The correspondence is given by $$h=\sum_{1\le j,k\le n}h_{jk}(z)\,dz_j\otimes d\ovl z_k\longmapsto u=\ii\sum_{1\le j,k\le n}h_{jk}(z)\,dz_j\wedge d\ovl z_k\leqno(1.8)$$ and does not depend on the choice of coordinates: indeed, as $\ovl h_{jk}= h_{kj}$, one finds immediately $$u(\xi,\eta)=\ii\sum h_{jk}(z)(\xi_j\ovl\eta_k-\eta_j\ovl\xi_k)= -2\Im h(\xi,\eta),~~~~\forall\xi,\eta\in TX.$$ Moreover, $h$ is $\ge 0$ as a hermitian form if and only if $u\ge 0$ as a $(1,1)$-form. A diagonalization of $h$ shows that every positive $(1,1)$-form $u\in\Lambda^{1,1}V^\star$ can be written $$u=\sum_{1\le j\le r}\ii\gamma_j\wedge\ovl\gamma_j~,~~~~\gamma\in V^\star, ~~r={\rm rank~of~~}u,$$ in particular, every positive $(1,1)$-form is strongly positive. By duality, this is also true for $(n-1,n-1)$-forms. \begstat{(1.9) Corollary} The notions of positive and strongly positive $(p,p)$-forms coincide for $p=0,1,n-1,n$.\qed \endstat \begstat{(1.10) Remark} \rm It is not difficult to see, however, that positivity and strong positivity differ in all bidegrees $(p,p)$ such that $2\le p\le n-2$. Indeed, a positive form $i^{p^2}\beta\wedge\ovl\beta$ with $\beta\in\Lambda^{p,0}V^\star$ is strongly positive if and only if $\beta$ is decomposable as a product $\beta_1\wedge\ldots\wedge\beta_p$. To see this, suppose that $$i^{p^2}\beta\wedge\ovl\beta=\sum_{1\le j\le N}i^{p^2}\gamma_j\wedge\ovl\gamma_j$$ where all $\gamma_j\in\Lambda^{p,0}V^\star$ are decomposable. Take $N$ minimal. The equality can be also written as an equality of hermitian forms $|\beta|^2=\sum|\gamma_j|^2$ if $\beta,\gamma_j$ are seen as linear forms on $\Lambda^pV$. The hermitian form $|\beta|^2$ has rank one, so we must have $N=1$ and $\beta= \lambda\gamma_j$, as desired. Note that there are many non decomposable $p$-forms in all degrees $p$ such that $2\le p\le n-2$, e.g. $(dz_1\wedge dz_2+dz_3\wedge dz_4)\wedge\ldots\wedge dz_{p+2}\,$: if a $p$-form is decomposable, the vector space of its contractions by elements of $\bigwedge^{p-1}V$ is a $p$-dimensional subspace of $V^\star$; in the above example the dimension is $p+2$. \endstat \begstat{(1.11) Proposition} If $u_1\ld u_s$ are positive forms, all of them strongly positive $($resp. all except perhaps one$)$, then $u_1\wedge\ldots\wedge u_s$ is strongly positive $($resp. positive$)$. \endstat \begproof{} Immediate consequence of Def.~1.1. Observe however that the wedge product of two positive forms is not positive in general (otherwise we would infer that positivity coincides with strong positivity).\qed \endproof \begstat{(1.12) Proposition} If $\Phi:W\lra V$ is a complex linear map and $u\in\Lambda^{p,p}V^\star$ is $($strongly$)$ positive, then $\Phi^\star u\in\Lambda^{p,p}W^\star$ is $($strongly$)$ positive. \endstat \begproof{} This is clear for strong positivity, since $$\Phi^\star(\ii\alpha_1\wedge\ovl\alpha_1\wedge\ldots\wedge \ii\alpha_p\wedge\ovl\alpha_p)=\ii\beta_1\wedge\ovl\beta_1\wedge\ldots\wedge \ii\beta_p\wedge \ovl\beta_p$$ with $\beta_j=\Phi^\star\alpha_j\in W^\star$, for all $\alpha_j\in V^\star$. For $u$ positive, we may apply Criterion~1.6: if $S$ is a $p$-dimensional subspace of $W$, then $u_{\restriction\Phi(S)}$ and $(\Phi^\star u)_{\restriction S}=(\Phi_\restriction S)^\star u_{\restriction\Phi(S)}$ are positive when $\Phi_{\restriction S}: S\lra\Phi(S)$ is an isomorphism; otherwise we get $(\Phi^\star u)_{\restriction S}=0$.\qed \endproof \titlec{1.B.}{Positive Currents} The duality between the positive and strongly positive cones of forms can be used to define corresponding positivity notions for currents. \begstat{(1.13) Definition} A current $T\in\cD'_{p,p}(X)$ is said to be positive $($resp. strongly positive$)$ if $\langle T,u\rangle\ge 0$ for all test forms $u\in\cD_{p,p}(X)$ that are strongly positive $($resp. positive$)$ at each point. The set of positive $($resp. strongly positive$)$ currents of bidimension $(p,p)$ will be denoted $$\cD^{\prime+}_{p,p}(X),~~~~\hbox{\rm resp.}~~\cD^{\prime\oplus}_{p,p}(X).$$ \endstat It is clear that (strong) positivity is a local property and that the sets $\cD^{\prime\oplus}_{p,p}(X)\subset\cD^{\prime+}_{p,p}(X)$ are closed convex cones with respect to the weak topology. Another way of stating Def.~1.13 is: \medskip \noindent{\it $T$ is positive $($strongly positive$)$ if and only if $T\wedge u\in\cD'_{0,0}(X)$ is a positive measure for all strongly positive $($positive$)$ forms $u\in\ci_{p,p}(X)$.} \medskip \noindent This is so because a distribution $S\in\cD'(X)$ such that $S(f)\ge 0$ for every non-negative function $f\in\cD(X)$ is a positive measure. \begstat{(1.14) Proposition} Every positive current $T=i^{(n-p)^2}\sum T_{I,J}\,dz_I \wedge d\ovl z_J$ in $\cD^{\prime +}_{p,p}(X)$ is real and of order $0$, i.e. its coefficients $T_{I,J}$ are complex measures and satisfy $\ovl{T_{I,J}}=T_{J,\,I}$ for all multi-indices $|I|=|J|=n-p$. Moreover $T_{I,I}\ge 0$, and the absolute values $|T_{I,J}|$ of the measures $T_{I,J}$ satisfy the inequality $$\lambda_I\lambda_J\,|T_{I,J}|\le 2^p\sum_M~\lambda_M^2\,T_{M,M},~~~~ I\cap J\subset M\subset I\cup J$$ where $\lambda_k\ge 0$ are arbitrary coefficients and $\lambda_I=\prod_{k\in I}\lambda_k$. \endstat \begproof{} Since positive forms are real, positive currents have to be real by duality. Let us denote by $K=\complement I$ and $L=\complement J$ the ordered complementary multi-indices of $I,J$ in $\{1,2\ld n\}$. The distribution $T_{I,I}$ is a positive measure because $$T_{I,I}~\tau=T\wedge i^{p^2}dz_K\wedge d\ovl z_K\ge 0.$$ On the other hand, the proof of Lemma~1.4 yields $$\eqalign{ T_{I,J}~\tau&=\pm\,T\wedge i^{p^2}dz_K\wedge d\ovl z_L= \sum_{a\in(\bbbz/4\bbbz)^p}\varepsilon_a\,T\wedge\gamma_a ~~~~{\rm where}\cr \gamma_a&=\bigwedge_{1\le s\le p}{\ii\over 4}(dz_{k_s}+ i^{a_s}dz_{l_s})\wedge(\ovl{dz_{k_s}+i^{a_s}dz_{l_s}}),~~~~ \varepsilon_a=\pm 1,\pm i.\cr}$$ Now, each $T\wedge\gamma_a$ is a positive measure, hence $T_{I,J}$ is a complex measure and $$\eqalign{ |T_{I,J}|\,\tau&\le\sum_a T\wedge\gamma_a= T\wedge\sum_a\gamma_a\cr &=T\wedge\bigwedge_{1\le s\le p}\Big(\sum_{a_s\in\bbbz/4\bbbz}{\ii\over 4} (dz_{k_s}+i^{a_s}dz_{l_s})\wedge(\ovl{dz_{k_s}+i^{a_s}dz_{l_s}}) \Big)\cr &=T\wedge\bigwedge_{1\le s\le p}\big(\ii dz_{k_s}\wedge d\ovl z_{k_s}+ \ii dz_{l_s}\wedge d\ovl z_{l_s}\big).\cr}$$ The last wedge product is a sum of at most $2^p$ terms, each of which is of the type $i^{p^2}dz_M\wedge d\ovl z_M$ with $|M|=p$ and $M\subset K\cup L$. Since $T\wedge i^{p^2}dz_M\wedge d\ovl z_M=T_{\complement M,\complement M}\,\tau$ and $\complement M\supset\complement K\cap\complement L=I\cap J$, we find $$|T_{I,J}|\le 2^p\sum_{M\supset I\cap J}T_{M,M}.$$ Now, consider a change of coordinates $(z_1\ld z_n)=\Lambda w= (\lambda_1 w_1\ld\lambda_nw_n)$ with $\lambda_1\ld \lambda_n>0$. In the new coordinates, the current $T$ becomes $\Lambda^\star T$ and its coefficients become $\lambda_I\lambda_J\,T_{I,J}(\Lambda w)$. Hence, the above inequality implies $$\lambda_I\lambda_J\,|T_{I,J}|\le 2^p\sum_{M\supset I\cap J} \lambda_M^2\,T_{M,M}.$$ This inequality is still true for $\lambda_k\ge 0$ by passing to the limit. The inequality of Prop.~1.14 follows when all coefficients $\lambda_k$, $k\notin I\cup J$, are replaced by $0$, so that $\lambda_M=0$ for $M\not\subset I\cup J$.\qed \endproof \begstat{(1.15) Remark} \rm If $T$ is of order $0$, we define the {\it mass measure} of $T$ by $\|T\|=\sum|T_{I,J}|$ (of course $\|T\|$ depends on the choice of coordinates). By the Radon-Nikodym theorem, we can write $T_{I,J}=f_{I,J}\|T\|$ with a Borel function $f_{I,J}$ such that $\sum|f_{I,J}|=1$. Hence $$T=\|T\|\,f,~~~~{\rm where}~~f=i^{(n-p)^2}\sum f_{I,J}\,dz_I\wedge d\ovl z_J.$$ Then $T$ is (strongly) positive if and only if the form $f(x)\in \Lambda^{n-p,n-p}T^\star_xX$ is (strongly) positive at $\|T\|$-almost all points $x\in X$. Indeed, this condition is clearly sufficient. On the other hand, if $T$ is (strongly) positive and $u_j$ is a sequence of forms with constant coefficients in $\Lambda^{p,p}T^\star X$ which is dense in the set of strongly positive (positive) forms, then $T\wedge u_j=||T||\,f\wedge u_j$, so $f(x)\wedge u_j$ has to be a positive $(n,n)$-form except perhaps for $x$ in a set $N(u_j)$ of $\|T\|$-measure~$0$. By a simple density argument, we see that $f(x)$ is (strongly) positive outside the $\|T\|$-negligible set $N=\bigcup N(u_j)$. As a consequence of this proof, $T$ is positive (strongly positive) if and only if $T\wedge u$ is a positive measure for all strongly positive (positive) forms $u$ of bidegree $(p,p)$ with {\it constant coefficients} in the given coordinates $(z_1\ld z_n)$. It follows that if $T$ is (strongly) positive in a coordinate patch $\Omega$, then the convolution $T\star\rho_\varepsilon$ is (strongly) positive in $\Omega_\varepsilon= \{x\in\Omega\,;\,d(x,\partial\Omega)>\varepsilon\}$.\qed \endstat \begstat{(1.16) Corollary} If $T\in\cD'_{p,p}(X)$ and $v\in C^0_{s,s}(X)$ are positive, one of them $($resp. both of them$)$ strongly positive, then the wedge product $T\wedge v$ is a positive $($resp. strongly positive$)$ current. \endstat This follows immediately from Remark~1.15 and Prop.~1.11 for forms. Similarly, Prop.~1.12 on pull-backs of positive forms easily shows that positivity properties of currents are preserved under direct or inverse images by holomorphic maps. \begstat{(1.17) Proposition} Let $\Phi:X\lra Y$ be a holomorphic map between complex analytic manifolds. \medskip \item{\rm a)} If $T\in\cD^{\prime+}_{p,p}(X)$ and $\Phi_{\restriction {\rm Supp}\,T}$ is proper, then $\Phi_\star T\in\cD^{\prime+}_{p,p}(Y)$. \medskip \item{\rm b)} If $T\in\cD^{\prime+}_{p,p}(Y)$ and if $\Phi$ is a submersion with $m$-dimensional fibers, then $\,\Phi^\star T\in\cD^{\prime+}_{p+m,p+m}(X)$. \medskip \noindent Similar properties hold for strongly positive currents.\qed \endstat \titlec{1.C.}{Basic Examples of Positive Currents} We present here two fundamental examples which will be of interest in many circumstances. \titled{(1.18) Current Associated to a Plurisubharmonic Function} Let $X$ be a complex manifold and $u\in\Psh(X)\cap L^1_\loc(X)$ a plurisubharmonic function. Then $$T=\ii d'd''u=\ii\sum_{1\le j,k\le n}{\partial^2u\over\partial z_j\partial\ovl z_k}\,dz_j\wedge d\ovl z_k$$ is a positive current of bidegree $(1,1)$. Moreover $T$ is closed (we always mean here $d$-closed, that is, $dT=0$). Assume conversely that $\Theta$ is a closed real $(1,1)$-current on $X$. Poincar\'e's lemma implies that every point $x_0\in X$ has a neighborhood $\Omega_0$ such that $\Theta=dS$ with $S\in\cD'_1(\Omega_0,\bbbr)$. Write $S=S^{1,0}+S^{0,1}$, where $S^{0,1}=\ovl{S^{1,0}}$. Then $d''S=\Theta^{0,2}=0$, and the Dolbeault-Grothendieck lemma shows that $S^{0,1}=d''v$ on some neighborhood $\Omega\subset\Omega_0$, with $v\in\cD'(\Omega,\bbbc)$. Thus $$\eqalign{ S&=\ovl{d''v}+d''v=d'\ovl v+d''v,\cr \Theta&=dS=d'd''(v-\ovl v)=\ii d'd''u,\cr}$$ where $u=2\Re v\in\cD'(\Omega,\bbbr)$. If $\Theta\in\ci_{1,1}(X)$, the hypoellipticity of $d''$ in bidegree $(p,0)$ shows that $d'u$ is of class $\ci$, so $u\in\ci(\Omega)$. When $\Theta$ is positive, the distribution $u$ is a plurisubharmonic function (Th.~I.3.31). We have thus proved: \begstat{(1.19) Proposition} If $\Theta\in\cD^{\prime+}_{n-1,n-1}(X)$ is a closed positive current of bidegree $(1,1)$, then for every point $x_0\in X$ there exists a neighborhood $\Omega$ of $x_0$ and $u\in\Psh(\Omega)$ such that $\Theta=\ii d'd''u$.\qed \endstat \titled{(1.20) Current of Integration on a Complex Submanifold} Let $Z\subset X$ be a closed $p$-dimensional complex submanifold with its canonical orientation and $T=[Z]$. Then $T\in\cD^{\prime\oplus}_{p,p}(X)$. Indeed, every $(r,s)$-form of total degree $r+s=2p$ has zero restriction to $Z$ unless $(r,s)=(p,p)$, therefore we have $[Z]\in\cD'_{p,p}(X)$. Now, if $u\in\cD_{p,p}(X)$ is a positive test form, then $u_{\restriction Z}$ is a positive volume form on $Z$ by Criterion~1.6, therefore $$\langle[Z],u\rangle=\int_Z u_{\restriction Z}\ge 0.$$ In this example the current $[Z]$ is also closed, because $d[Z]=\pm [\partial Z]=0$ by Stokes' theorem.\qed \titlec{1.D.}{Trace Measure and Wirtinger's Inequality} We discuss now some questions related to the concept of area on complex submanifolds. Assume that $X$ is equipped with a hermitian metric $h$, i.e. a positive definite hermitian form $h=\sum h_{jk}dz_j\otimes d\ovl z_k$ of class $\ci$~; we denote by $\omega=\ii\sum h_{jk}dz_j\wedge d\ovl z_k \in\ci_{1,1}(X)$ the associated positive $(1,1)$-form. \begstat{(1.21) Definition} For every $T\in\cD^{\prime+}_{p,p}(X)$, the trace measure of $T$ with respect to $\omega$ is the positive measure $$\sigma_T={1\over 2^pp!}\,T\wedge\omega^p.$$ \endstat If $(\zeta_1\ld\zeta_n)$ is an orthonormal frame of $T^\star X$ with respect to $h$ on an open subset $U\subset X$, we may write $$\eqalign{ &\omega=\ii\sum_{1\le j\le n}\zeta_j\wedge\ovl\zeta_j,~~~~ \omega^p=i^{p^2}p!\,\sum_{|K|=p}\zeta_K\wedge\ovl\zeta_K,\cr &T=i^{(n-p)^2}\sum_{|I|=|J|=n-p}T_{I,J}\,\zeta_I\wedge\ovl\zeta_J,~~~~ T_{I,J}\in\cD'(U),\cr}$$ where $\zeta_I=\zeta_{i_1}\wedge\ldots\wedge\zeta_{i_{n-p}}$. An easy computation yields $$\sigma_T=2^{-p}\Big(\sum_{|I|=n-p}T_{I,I}\Big)\,\ii\zeta_1\wedge\ovl\zeta_1 \wedge\ldots\wedge \ii\zeta_n\wedge\ovl\zeta_n.\leqno(1.22)$$ For $X=\bbbc^n$ with the standard hermitian metric $h=\sum dz_j\otimes d\ovl z_j$, we get in particular $$\sigma_T=2^{-p}\Big(\sum_{|I|=n-p}T_{I,I}\Big)\, \ii dz_1\wedge d\ovl z_1\wedge\ldots\wedge \ii dz_n\wedge d\ovl z_n. \leqno(1.22')$$ Proposition 1.14 shows that the mass measure $||T||=\sum|T_{I,J}|$ of a positive current $T$ is always dominated by $C\sigma_T$ where $C>0$ is a constant. It follows easily that the weak topology of $\cD_p'(X)$ and of $\cD_p^{0\,\prime}(X)$ coincide on $\cD_p^{\prime+}(X)$, which is moreover a metrizable subspace: its weak topology is in fact defined by the collection of semi-norms $T\longmapsto |\langle T,f_\nu\rangle|$ where $(f_\nu)$ is an arbitrary dense sequence in $\cD_p(X)$. By the Banach-Alaoglu theorem, the unit ball in the dual of a Banach space is weakly compact, thus: \begstat{(1.23) Proposition} Let $\delta$ be a positive continuous function on $X$. Then the set of currents $T\in\cD_p^{\prime+}(X)$ such that $\int_X \delta\,T\wedge\omega^p\le 1$ is weakly compact. \endstat \begproof{} Note that our set is weakly closed, since a weak limit of positive currents is positive and $\int_X \delta\,T\wedge\omega^p= \sup\langle T,\theta\delta\omega^p\rangle$ when $\theta$ runs over all elements of $\cD(X)$ such that $0\le\theta\le 1$.\qed \endproof Now, let $Z$ be a $p$-dimensional complex analytic submanifold of $X$. We claim that $$\sigma_{[Z]}={1\over 2^pp!}[Z]\wedge\omega^p={\rm Riemannian~volume~measure ~on~~}Z.\leqno(1.24)$$ This result is in fact a special case of the following important inequality. \begstat{(1.25) Wirtinger's inequality} Let $Y$ be an oriented real submanifold of class $C^1$ and dimension $2p$ in $X$, and let $dV_Y$ be the Riemannian volume form on $Y$ associated with the metric $h_{\restriction Y}$. Set $${1\over 2^pp!}\omega^p_{\restriction Y}=\alpha\,dV_Y,~~~~\alpha\in C^0(Y).$$ Then $|\alpha|\le 1$ and the equality holds if and only if $Y$ is a complex analytic submanifold of $X$. In that case $\alpha=1$ if the orientation of $Y$ is the canonical one, $\alpha=-1$ otherwise. \endstat \begproof{} The restriction $\omega_{\restriction Y}$ is a real antisymmetric $2$-form on $TY$. At any point $z\in Y$, we can thus find an oriented orthonormal $\bbbr$-basis $(e_1,e_2\ld e_{2p})$ of $T_zY$ such that $$\eqalign{ {1\over 2}\omega&=\sum_{1\le k\le p}\alpha_k\,e^\star_{2k-1}\wedge e^\star_{2k}~~~{\rm on}~~T_zY,~~~~{\rm where}\cr \alpha_k&={1\over 2}\omega(e_{2k-1},e_{2k})=-\Im h(e_{2k-1},e_{2k}).\cr}$$ We have $dV_Y=e^\star_1\wedge\ldots\wedge e^\star_{2p}$ by definition of the Riemannian volume form. By taking the $p$-th power of $\omega$, we get $${1\over 2^pp!}\omega^p_{\restriction T_zY}=\alpha_1\ldots\alpha_p\, e^\star_1\wedge\ldots\wedge e^\star_{2p}=\alpha_1\ldots\alpha_p\,dV_Y.$$ Since $(e_k)$ is an orthonormal $\bbbr$-basis, we have $\Re h(e_{2k-1},e_{2k})=0$, thus $h(e_{2k-1},e_{2k})=-\ii\alpha_k$. As $|e_{2k-1}|=|e_{2k}|=1$, we get $$0\le|e_{2k}\pm Je_{2k-1}|^2=2\big(1\pm\Re h(Je_{2k-1},e_{2k})\big) =2(1\pm\alpha_k).$$ Therefore $$|\alpha_k|\le 1,~~~~|\alpha|=|\alpha_1\ldots\alpha_p|\le 1,$$ with equality if and only if $\alpha_k=\pm1$ for all $k$, i.e. $e_{2k}=\pm Je_{2k-1}$. In this case $T_zY\subset T_zX$ is a complex vector subspace at every point $z\in Y$, thus $Y$ is complex analytic by Lemma~I.4.23. Conversely, assume that $Y$ is a $\bbbc$-analytic submanifold and that $(e_1,e_3\ld e_{2p-1})$ is an orthonormal complex basis of $T_zY$. If $e_{2k}:=Je_{2k-1}$, then $(e_1\ld e_{2p})$ is an orthonormal $\bbbr$-basis corresponding to the canonical orientation and $${1\over 2}\omega_{\restriction Y}= \sum_{1\le k\le p}e^\star_{2k-1}\wedge e^\star_{2k},~~~~ {1\over 2^pp!}\omega^p_{\restriction Y}=e^\star_1\wedge\ldots\wedge e^\star_{2p} =dV_Y.\eqno\square$$ \endproof Note that in the case of the standard hermitian metric $\omega$ on $X=\bbbc^n$, the form $\omega=\ii\sum dz_j\wedge d\ovl z_j= d\big(\ii\sum z_j\,d\ovl z_j\big)$ is globally exact. Under this hypothesis, we are going to see that $\bbbc$-analytic submanifolds are always \hbox{\it minimal surfaces} for the Plateau problem, which consists in finding a compact subvariety $Y$ of minimal area with prescribed boundary~$\partial Y$. \begstat{(1.26) Theorem} Assume that the $(1,1)$-form $\omega$ is exact, say $\omega=d\gamma$ with $\gamma\in\ci_1(X,\bbbr)$, and let $Y,Z\subset X$ be $(2p)$-dimensional oriented compact real submanifolds of class $C^1$ with boundary. If $\partial Y=\partial Z$ and $Z$ is complex analytic, then $${\rm Vol}(Y)\ge{\rm Vol(Z)}.$$ \endstat \begproof{} Write $\omega=d\gamma$. Wirtinger's inequality and Stokes' theorem imply $$\eqalignno{ {\rm Vol}(Y)&\ge{1\over 2^pp!}\Big|\int_Y\omega^p\Big| ={1\over 2^pp!}\Big|\int_Y d(\omega^{p-1}\wedge\gamma)\Big| ={1\over 2^pp!}\Big|\int_{\partial Y}\omega^{p-1}\wedge\gamma\Big|,\cr {\rm Vol}(Z)&={1\over 2^pp!}\int_Z\omega^p={1\over 2^pp!}\int_{\partial Z} \omega^{p-1}\wedge\gamma=\pm{1\over 2^pp!}\int_{\partial Y} \omega^{p-1}\wedge\gamma.&\square\cr}$$ \endproof \titleb{2.}{Closed Positive Currents} \titlec{2.A.}{The Skoda-El Mir Extension Theorem} We first prove the Skoda-El Mir extension theorem (Skoda 1982, El Mir 1984), which shows in particular that a closed positive current defined in the complement of an analytic set $E$ can be extended through~$E$ if (and only~if) the mass of the current is locally finite near~$E$. El Mir simplified Skoda's argument and showed that it is enough to assume $E$ complete pluripolar. We follow here the exposition of Sibony's survey article (Sibony 1985). \begstat{(2.1) Definition} A subset $E\subset X$ is said to be complete pluripolar in $X$ if for every point $x_0\in X$ there exist a neighborhood $\Omega$ of $x_0$ and a function $u\in\Psh(\Omega)\cap L^1_\loc(\Omega)$ such that $E\cap\Omega=\{z\in\Omega~;~u(z)=-\infty\}$. \endstat Note that any closed analytic subset $A\subset X$ is complete pluripolar: if $g_1=\ldots=g_N=0$ are holomorphic equations of $A$ on an open set $\Omega\subset X$, we can take $u=\log(|g_1|^2+\ldots+|g_N|^2)$. \begstat{(2.2) Lemma} Let $E\subset X$ be a closed complete pluripolar set. If $x_0\in X$ and $\Omega$ is a sufficiently small neighborhood of $x_0$, there exists: \medskip \item{\rm a)} a function $v\in\Psh(\Omega)\cap\ci(\Omega\ssm E)$ such that $v=-\infty$ on $E\cap\Omega~;$ \medskip \item{\rm b)} an increasing sequence $v_k\in\Psh(\Omega)\cap\ci(\Omega)$, $0\le v_k\le 1$, converging uniformly to $1$ on every compact subset of $\Omega\ssm E$, such that $v_k=0$ on a neighborhood of $E\cap\Omega$. \smallskip \endstat \begproof{} Assume that $\Omega_0\compact X$ is a coordinate patch of $X$ containing $x_0$ and that $E\cap\Omega_0=\{z\in\Omega_0~;~u(z)=-\infty\}$, $u\in\Psh(\Omega_0)$. In addition, we can achieve $u\le 0$ by shrinking $\Omega_0$ and subtracting a constant to $u$. Select a convex increasing function $\chi\in\ci([0,1],\bbbr)$ such that $\chi(t)=0$ on $[0,1/2]$ and $\chi(1)=1$. We set $$u_k=\chi\big(\exp(u/k)\big).$$ Then $0\le u_k\le1$, $u_k$ is plurisubharmonic on $\Omega_0$, $u_k=0$ in a neighborhood $\omega_k$ of $E\cap\Omega_0$ and $\lim u_k=1$ on $\Omega_0\ssm E$. Let $\Omega\compact\Omega_0$ be a neighborhood of $x_0$, let $\delta_0=d(\Omega,\complement\Omega_0)$ and $\varepsilon_k\in{}]0,\delta_0[$ be such that $\varepsilon_kn-p$ at every point of $M\cap\Omega$.\break After a change of the ordering, we may assume for example that $\zeta_1=d'g_1$, $\zeta_2=d'g_2$, $\ldots$, $\zeta_{n-p+1}=d'g_{n-p+1}$ are linearly independent on $\Omega$ (shrink $\Omega$ if necessary). Complete $(\zeta_1\ld\zeta_{n-p+1})$ into a continuous frame $(\zeta_1\ld\zeta_n)$ of $T^\star X_{\restriction\Omega}$ and set $$\Theta=\sum_{|I|=|J|=n-p}\Theta_{I,J}\,\zeta_I\wedge\ovl\zeta_J~~~{\rm on}~~ \Omega.$$ As $\Theta$ and $d'\Theta$ have measure coefficients supported on $M$ and $g_k=0$ on $M$, we get $g_k\Theta=g_kd'\Theta=0$, thus $$d'g_k\wedge\Theta=d'(g_k\Theta)-g_kd'\Theta=0,~~~~1\le k\le m,$$ in particular $\zeta_k\wedge\Theta=0$ for all $1\le k\le n-p+1$. When $|I|=n-p$, the multi-index $\complement I$ contains at least one of the elements $1\ld n-p+1$, hence $\Theta\wedge\zeta_{\complement I}\wedge\ovl \zeta_{\complement J}=0$ and $\Theta_{I,J}=0$.\qed \endproof \begstat{(2.11) Corollary} Let $\Theta\in\cD'_{p,p}(X)$ be a normal current. If ${\rm Supp}\,\Theta$ is contained in an analytic subset $A$ of dimension $0$ such that $$C_1||\Theta||_K\le\int_K\Theta\wedge\beta^p\le C_2||\Theta||_K.$$ We denote by $L^1(K)$, resp. by $L^\infty(K)$, the space of integrable (resp. bounded measurable) functions on $K$ with respect to any smooth positive density on~$X$. \begstat{(3.3) Chern-Levine-Nirenberg inequalities {\rm(1969)}} For all compact subsets $K,L$ of $X$ with $L\subset K^\circ$, there exists a constant $C_{K,L}\ge0$ such that $$||dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T||_L\le C_{K,L}~ ||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}\,||T||_K.$$ \endstat \begproof{} By induction, it is sufficient to prove the result for $q=1$ and $u_1=u$. There is a covering of $L$ by a family of balls $B'_j\compact B_j\subset K$ contained in coordinate patches of $X$. Let $\chi\in\cD(B_j)$ be equal to $1$ on $\smash{\ovl B'_j}$. Then $$||dd^cu\wedge T||_{L\cap\ovl B'_j}\le C\int_{\ovl B'_j}dd^c u\wedge T \wedge\beta^{p-1}\le C\int_{B_j}\chi\,dd^c u\wedge T\wedge\beta^{p-1}.$$ As $T$ and $\beta$ are closed, an integration by parts yields $$||dd^cu\wedge T||_{L\cap\ovl B'_j}\le C\int_{B_j}u\,T\wedge dd^c\chi \wedge\beta^{p-1}\le C'||u||_{L^\infty(K)}||T||_K$$ where $C'$ is equal to $C$ multiplied by a bound for the coefficients of the smooth form $dd^c\chi\wedge\beta^{p-1}$.\qed \endproof \begstat{(3.4) Remark} \rm With the same notations as above, any plurisubharmonic function $V$ on $X$ satisfies inequalities of the type \medskip \item{\rm a)} $\quad||dd^cV||_L\le C_{K,L}\,||V||_{L^1(K)}$. \medskip \item{\rm b)} $\quad{\displaystyle\sup_L}\,V_+\le C_{K,L}\,||V||_{L^1(K)}$. \medskip \noindent In fact the inequality $$\int_{L\cap\ovl B'_j}dd^cV\wedge\beta^{n-1}\le\int_{B_j}\chi dd^cV\wedge \beta^{n-1}=\int_{B_j}Vdd^c\chi\wedge\beta^{n-1}$$ implies a), and b) follows from the mean value inequality. \endstat \begstat{(3.5) Remark} \rm Products of the form $\Theta=\gamma_1\wedge\ldots\wedge \gamma_q\wedge T$ with mixed $(1,1)$-forms $\gamma_j=dd^cu_j$ or $\gamma_j=dv_j\wedge d^cw_j+dw_j\wedge d^cv_j$ are also well defined whenever $u_j$, $v_j$, $w_j$ are locally bounded plurisubharmonic functions. Moreover, for $L\subset K^\circ$, we have $$||\Theta||_L\le C_{K,L}||T||_K\prod||u_j||_{L^\infty(K)} \prod||v_j||_{L^\infty(K)}\prod||w_j||_{L^\infty(K)}.$$ To check this, we may suppose $v_j,w_j\ge 0$ and $||v_j||=||w_j||=1$ in~$L^\infty(K)$. Then the inequality follows from (3.3) by the polarization identity $$2(dv_j\wedge d^cw_j+dw_j\wedge d^cv_j)=dd^c(v_j+w_j)^2 -dd^cv_j^2-dd^cw_j^2-v_jdd^cw_j-w_jdd^cv_j$$ in which all $dd^c$ operators act on plurisubharmonic functions. \endstat \begstat{(3.6) Corollary} Let $u_1\ld u_q$ be continuous $($finite$)$ plurisubharmonic functions and let $u_1^k\ld u_q^k$ be sequences of plurisubharmonic functions converging locally uniformly to $u_1\ld u_q$. If $T_k$ is a sequence of closed positive currents converging weakly to $T$, then \smallskip \item{\rm a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k \wedge T_k\lra u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \smallskip \item{\rm b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T_k\lra dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \endstat \begproof{} We observe that b) is an immediate consequence of a) by the weak continuity of $dd^c$. By using induction on $q$, it is enough to prove result a) when~$q=1$. If $(u^k)$ converges locally uniformly to a finite continuous plurisubharmonic function $u$, we introduce local regularizations $u_\varepsilon=u\star\rho_\varepsilon$ and write $$u^kT_k-uT=(u^k-u)T_k+(u-u_\varepsilon)T_k+u_\varepsilon(T_k-T).$$ As the sequence $T_k$ is weakly convergent, it is locally uniformly bounded in mass, thus $||(u^k-u)T_k||_K\le ||u^k-u||_{L^\infty(K)}||T_k||_K$ converges to $0$ on every compact set~$K$. The same argument shows that $||(u-u_\varepsilon)T_k||_K$ can be made arbitrarily small by choosing $\varepsilon$ small enough. Finally $u_\varepsilon$ is smooth, so $u_\varepsilon(T_k-T)$ converges weakly to~$0$.\qed \endproof Now, we prove a deeper monotone continuity theorem due to (Bedford-Taylor 1982) according to which the continuity and uniform convergence assumptions can be dropped if each sequence $(u_j^k)$ is decreasing and $T_k$ is a constant sequence. \begstat{(3.7) Theorem} Let $u_1\ld u_q$ be locally bounded plurisubharmonic functions and let $u_1^k\ld u_q^k$ be decreasing sequences of plurisubharmonic functions converging pointwise to $u_1\ld u_q$. Then \medskip \item{\rm a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k \wedge T\lra u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \medskip \item{\rm b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T\lra dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \endstat \begproof{} Again by induction, observing that a)~$\Longrightarrow$~b) and that a) is obvious for $q=1$ thanks to Lebesgue's bounded convergence theorem. To proceed with the induction step, we first have to make some slight modifications of our functions $u_j$ and $u_j^k$. As the sequence $\smash{(u_j^k)}$ is decreasing and as $u_j$ is locally bounded, the family $\smash{(u_j^k)_{k\in\bbbn}}$ is locally uniformly bounded. The results are local, so we can work on a Stein open set $\Omega\compact X$ with strongly pseudoconvex boundary. We use the following notations: \medskip \itemitem{\kern-\parindent\rlap{\hbox{(3.8)}}} let $\psi$ be a strongly plurisubharmonic function of class $\ci$ near $\smash{\ovl\Omega}$ with $\psi<0$ on $\Omega$ and $\psi=0$, $d\psi\ne 0$ on~$\partial\Omega\,;$ \smallskip \itemitem{\kern-\parindent\rlap{\hbox{$(3.8')$}}} we set $\Omega_\delta=\{z\in\Omega\,;\,\psi(z)<-\delta\}$ for all $\delta>0$. \Input epsfiles/fig_3_1.tex \vskip6mm \centerline{{\bf III-1} Construction of $v_j^k$} \vskip6mm \noindent After addition of a constant we can assume that $-M\le\smash{u_j^k}\le-1$ near~$\smash{\ovl\Omega}$. Let us denote by $\smash{(u_j^{k,\varepsilon})}$, $\varepsilon\in{}]0,\varepsilon_0]$, an increasing family of regularizations converging to $\smash{u_j^k}$ as $\varepsilon\to 0$ and such that $-M\le\smash{u^{k,\varepsilon}_j}\le-1$ on $\smash{\ovl\Omega}$. Set $A=M/\delta$ with $\delta>0$ small and replace $\smash{u_j^k}$ by $\smash{v_j^k=\max\{A\psi,u_j^k\}}$ and $\smash{u_j^{k,\varepsilon}}$ by $\smash{v_j^{k,\varepsilon}= \max_\varepsilon\{A\psi,u_j^{k,\varepsilon}\}}$ where $\max_\varepsilon=\max~\star~\rho_\varepsilon$ is a regularized max function. Then $\smash{v_j^k}$ coincides with $\smash{u_j^k}$ on $\Omega_\delta$ since $A\psi<-A\delta=-M$ on $\Omega_\delta$, and $\smash{v_j^k}$ is equal to $A\psi$ on the corona $\Omega\setminus\Omega_{\delta/M}$. Without loss of generality, we can therefore assume that all $\smash{u_j^k}$ (and similarly all $\smash{u_j^{k,\varepsilon}})$ coincide with $A\psi$ on a fixed neighborhood of $\partial\Omega$. We need a lemma. \endproof \begstat{(3.9) Lemma} Let $f_k$ be a decreasing sequence of upper semi-continuous functions conver\-ging to $f$ on some separable locally compact space $X$ and $\mu_k$ a sequence of positive measures converging weakly to $\mu$ on $X$. Then every weak limit $\nu$ of $f_k\mu_k$ satisfies $\nu\le f\mu$. \endstat Indeed if $(g_p)$ is a decreasing sequence of continuous functions converging to $f_{k_0}$ for some $k_0$, then $f_k\mu_k\le f_{k_0}\mu_k \le g_p\mu_k$ for $k\ge k_0$, thus $\nu\le g_p\mu$ as $k\to+\infty$. The monotone convergence theorem then gives $\nu\le f_{k_0}\mu$ as $p\to+\infty$ and $\nu\le f\mu$ as $k_0\to+\infty$.\qed \begproof{of Theorem 3.7 (end).} Assume that a) has been proved for $q-1$. Then $$S^k=dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge T\lra S=dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T.$$ By 3.3 the sequence $(u_1^k S^k)$ has locally bounded mass, hence is relatively compact for the weak topology. In order to prove a), we only have to show that every weak limit $\Theta$ of $u_1^k S^k$ is equal to $u_1 S$. Let $(m,m)$ be the bidimension of $S$ and let $\gamma$ be an arbitrary smooth and strongly positive~form of bidegree $(m,m)$. Then the positive measures $S^k\wedge\gamma$ converge weakly to $S\wedge\gamma$ and Lemma~3.9 shows that $\Theta\wedge\gamma\le u_1S\wedge\gamma$, hence $\Theta\le u_1S$. To get~the equality, we set $\beta=dd^c\psi>0$ and show that $\int_\Omega u_1 S\wedge\beta^m\le\int_\Omega\Theta\wedge\beta^m$, i.e. $$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\le \liminf_{k\to+\infty}\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k \wedge T\wedge\beta^m.$$ As $u_1\le u_1^k\le u_1^{k,\varepsilon_1}$ for every $\varepsilon_1>0$, we get $$\eqalign{\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr &\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2 \wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr &=\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge u_2dd^cu_3 \wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr}$$ after an integration by parts (there is no boundary term because $\smash{u_1^{k,\varepsilon_1}}$ and $u_2$ both vanish on $\partial\Omega$). Repeating this argument with $u_2\ld u_q$, we obtain $$\eqalign{ \int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr &\le\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge\ldots\wedge dd^c u_{q-1}^{k,\varepsilon_{q-1}}\wedge u_q T\wedge\beta^m\cr &\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2^{k,\varepsilon_2} \wedge\ldots\wedge dd^c u_q^{k,\varepsilon_q}\wedge T\wedge\beta^m.\cr}$$ Now let $\varepsilon_q\to 0\ld\varepsilon_1\to 0$ in this order. We have weak convergence at each step and $\smash{u_1^{k,\varepsilon_1}}=0$ on the boundary; therefore the integral in the last line converges and we get the desired inequality $$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\le\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge T\wedge\beta^m.\eqno{\square}$$ \endproof \begstat{(3.10) Corollary} The product $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ is symmetric with respect to $u_1\ld u_q$. \endstat \begproof{} Observe that the definition was unsymmetric. The result is true when $u_1\ld u_q$ are smooth and follows in general from Th.~3.7 applied to the sequences $u_j^k=u_j\star\rho_{1/k}$, $1\le j\le q$.\qed \endproof \begstat{(3.11) Proposition} Let $K,L$ be compact subsets of $X$ such that $L\subset K^\circ$. For any plurisubharmonic functions $V,u_1\ld u_q$ on $X$ such that $u_1\ld u_q$ are locally bounded, there is an inequality $$||Vdd^cu_1\wedge\ldots\wedge dd^cu_q||_L\le C_{K,L}\,||V||_{L^1(K)}||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}.$$ \endstat \begproof{} We may assume that $L$ is contained in a strongly pseudoconvex open set $\Omega=\{\psi<0\}\subset K$ (otherwise we cover $L$ by small balls contained in $K$). A suitable normalization gives $-2\le u_j\le -1$ on $K\,$; then we can modify $u_j$ on $\Omega\setminus L$ so that $u_j=A\psi$ on $\Omega\setminus\Omega_\delta$ with a fixed constant $A$ and $\delta>0$ such that $L\subset\Omega_\delta$. Let $\chi\ge 0$ be a smooth function equal to $-\psi$ on $\Omega_\delta$ with compact support in $\Omega$. If we take $||V||_{L^1(K)}=1$, we see that $V_+$ is uniformly bounded on $\Omega_\delta$ by 3.4~b); after subtraction of a fixed constant we can assume $V\le 0$ on $\Omega_\delta$. First suppose $q\le n-1$. As $u_j=A\psi$ on $\Omega\setminus\Omega_\delta$, we find $$\eqalign{ &\int_{\Omega_\delta}-V\,dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge\beta^{n-q}\cr &~\,{}=\int_\Omega V\,dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge\beta^{n-q-1}\wedge dd^c\chi-A^q\int_{\Omega\setminus \Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi\cr &~\,{}=\int_\Omega\chi\,dd^cV\wedge dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge\beta^{n-q-1}-A^q\int_{\Omega\setminus \Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi.\cr}$$ The first integral of the last line is uniformly bounded thanks to 3.3 and 3.4~a), and the second one is bounded by $||V||_{L^1(\Omega)}\le$ constant. Inequality 3.11 follows for $q\le n-1$. If $q=n$, we can work instead on $X\times\bbbc$ and consider $V,u_1\ld u_q$ as functions on $X\times\bbbc$ independent of the extra variable in~$\bbbc$.\qed \endproof{} \titleb{4.}{Case of Unbounded Plurisubharmonic Functions} We would like to define $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ also in some cases when $u_1\ld u_q$ are not bounded below everywhere, especially when the $u_j$ have logarithmic poles. Consider first the case $q=1$ and let $u$ be a plurisubharmonic function on $X$. The {\it pole set} of $u$ is by definition $P(u)=u^{-1}(-\infty)$. We define the {\it unbounded locus} $L(u)$ to be the set of points $x\in X$ such that $u$ is unbounded in every neighborhood of~$x$. Clearly $L(u)$ is closed and we have $L(u)\supset\smash{\ovl{P(u)}}$ but in general these sets are different: in fact, $u(z)=\sum k^{-2}\log(|z-1/k|+e^{-k^3})$ is everywhere finite in $\bbbc$ but $L(u)=\{0\}$. \begstat{(4.1) Proposition} We make two additional assumptions: \smallskip \item{\rm a)} $T$ has non zero bidimension $(p,p)$ (i.e. degree of $T<2n$). \smallskip \item{\rm b)} $X$ is covered by a family of Stein open sets $\Omega\compact X$ whose boundaries $\partial\Omega$ do not intersect $L(u)\cap\Supp\,T$. \smallskip \noindent Then the current $uT$ has locally finite mass in X. \endstat For any current $T$, hypothesis 4.1~b) is clearly satisfied when $u$ has a discrete unbounded locus $L(u)$; an interesting example is $u=\log|F|$ where $F=(F_1\ld F_N)$ are holomorphic functions having a discrete set of common zeros. Observe that the current $uT$ need not have locally finite mass when $T$ has degree $2n$ (i.e. $T$ is a measure); example: $T=\delta_0$ and $u(z)=\log|z|$\break in $\bbbc^n$. The result also fails when the sets $\Omega$ are not assumed to be Stein; example: $X={}$ blow-up of $\bbbc^n$ at $0$, $T=[E]={}$ current of integration on the exceptional divisor and $u(z)=\log|z|$ (see \S~7.12 for the definition of blow-ups). \begproof{} By shrinking $\Omega$ slightly, we may assume that $\Omega$ has a smooth strongly pseudoconvex boundary. Let $\psi$ be a defining function of $\Omega$ as in (3.8). By subtracting a constant to $u$, we may assume $u\le-\varepsilon$ on $\ovl\Omega$. We fix $\delta$ so small that $\ovl\Omega\ssm\Omega_\delta$ does not intersect $L(u)\cap\Supp\,T$ and we select a neighborhood $\omega$ of $(\ovl\Omega\ssm\Omega_\delta)\cap \Supp\,T$ such that $\ovl\omega\cap L(u)=\emptyset$. Then we define $$u_s(z)=\cases{ \max\{u(z),A\psi(z)\}&on $\omega$,\cr \max\{u(z),s\}&on $\Omega_\delta=\{\psi<-\delta\}$.\cr}$$ By construction $u\ge -M$ on $\omega$ for some constant $M>0$. We fix $A\ge M/\delta$ and take $s\le -M$, so $$\max\{u(z),A\psi(z)\}=\max\{u(z),s\}=u(z)~~~~\hbox{\rm on}~~ \omega\cap\Omega_\delta$$ and our definition of $u_s$ is coherent. Observe that $u_s$ is defined on $\omega\cup\Omega_\delta$, which is a neighborhood of $\ovl\Omega\cap\Supp\,T$. Now, $u_s=A\psi$ on $\omega\cap(\Omega\ssm\Omega_{\varepsilon/A})$, hence Stokes' theorem implies $$\eqalign{ \int_\Omega dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}&{}- \int_\Omega Add^c\psi\wedge T\wedge(dd^c\psi)^{p-1}\cr &=\int_\Omega dd^c\big[(u_s-A\psi)T\wedge(dd^c\psi)^{p-1}\big]=0\cr}$$ because the current $[\ldots]$ has a compact support contained in $\ovl\Omega_{\varepsilon/A}$. Since $u_s$ and $\psi$ both vanish on $\partial\Omega$, an integration by parts gives $$\eqalign{ \int_\Omega u_sT\wedge(dd^c\psi)^p &=\int_\Omega\psi dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}\cr &\ge-||\psi||_{L^\infty(\Omega)}\int_\Omega T\wedge dd^cu_s\wedge (dd^c\psi)^{p-1}\cr &=-||\psi||_{L^\infty(\Omega)}A\int_\Omega T\wedge(dd^c\psi)^p.\cr}$$ Finally, take $A=M/\delta$, let $s$ tend to $-\infty$ and use the inequality $u\ge-M$ on~$\omega$. We obtain $$\eqalign{ \int_\Omega u\,T\wedge(dd^c\psi)^p &\ge -M\int_\omega T\wedge(dd^c\psi)^p+\lim_{s\to-\infty} \int_{\Omega_\delta}u_sT\wedge(dd^c\psi)^p\cr &\ge -\big(M+||\psi||_{L^\infty(\Omega)}M/\delta\big) \int_\Omega T\wedge(dd^c\psi)^p.\cr}$$ The last integral is finite. This concludes the proof.\qed \endproof \begstat{(4.2) Remark} \rm If $\Omega$ is smooth and strongly pseudoconvex, the above proof shows in fact that $$||uT||_{\ovl\Omega}\le{C\over\delta}||u||_{L^\infty( (\ovl\Omega\ssm\Omega_\delta)\cap\Supp\,T)}||T||_{\ovl\Omega}$$ when $L(u)\cap\Supp\,T\subset\Omega_\delta$. In fact, if $u$ is continuous and if $\omega$ is chosen sufficiently small, the constant $M$ can be taken arbitrarily close to $||u||_{L^\infty( (\ovl\Omega\ssm\Omega_\delta)\cap\Supp\,T)}$. Moreover, the maximum principle implies $$||u_+||_{L^\infty(\ovl\Omega\cap\Supp\,T)}= ||u_+||_{L^\infty(\partial\Omega\cap\Supp\,T)},$$ so we can achieve $u<-\varepsilon$ on a neighborhood of $\ovl\Omega \cap\Supp\,T$ by subtracting $||u||_{L^\infty((\ovl\Omega\ssm\Omega_\delta) \cap\Supp\,T)}+2\varepsilon$ $[$Proof of maximum principle: if $u(z_0)>0$ at $z_0\in\Omega\cap\Supp\,T$ and $u\le 0$ near $\partial\Omega\cap\Supp\,T$, then $$\int_\Omega u_+T\wedge(dd^c\psi)^p=\int_\Omega\psi dd^cu_+\wedge T \wedge(dd^c\psi)^{p-1}\le 0,$$ a contradiction$]$.\qed \endstat \begstat{(4.3) Corollary} Let $u_1\ld u_q$ be plurisubharmonic functions on $X$ such that $X$ is covered by Stein open sets $\Omega$ with $\partial\Omega\cap L(u_j)\cap\Supp\,T=\emptyset$. We use again induction to define $$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T= dd^c(u_1dd^c u_2\ldots\wedge dd^c u_q\wedge T).$$ Then, if $u_1^k\ld u_q^k$ are decreasing sequences of plurisubharmonic functions converging pointwise to $u_1\ld u_q$, $q\le p$, properties~$(3.7\,{\rm a,b})$ hold. \endstat \Input epsfiles/fig_3_2.tex \vskip6mm \centerline{{\bf III-2} Modified construction of $v_j^k$} \vskip6mm \begproof{} Same proof as for Th.~3.7, with the following minor modification: the max procedure $v_j^k:=\max\{u_j^k,A\psi\}$ is applied only on a neighborhood $\omega$ of $\Supp\,T\cap(\ovl\Omega\ssm\Omega_\delta)$ with $\delta>0$ small, and $u_j^k$ is left unchanged near \hbox{$\Supp\,T\cap\ovl\Omega_\delta$.} Observe that the integration by part process requires the functions $u_j^k$ and $\smash{u_j^{k,\varepsilon}}$ to be defined only near $\ovl\Omega\cap\Supp\,T$.\qed \endproof \begstat{(4.4) Proposition} Let $\Omega\compact X$ be a Stein open subset. If $V$ is a plurisubharmonic function on $X$ and $u_1\ld u_q$, $1\le q\le n-1$, are plurisubharmonic functions such that $\partial\Omega\cap L(u_j)=\emptyset$, then $Vdd^cu_1\wedge\ldots \wedge dd^cu_q$ has locally finite mass in $\Omega$. \endstat \begproof{} Same proof as for 3.11, when $\delta>0$ is taken so small that $\Omega_\delta\supset L(u_j)$ for all $1\le j\le q$.\qed \endproof Finally, we show that Monge-Amp\`ere operators can also be defined in the case of plurisubharmonic functions with non compact pole sets, provided that the mutual intersections of the pole sets are of sufficiently small Hausdorff dimension with respect to the dimension $p$ of~$T$. \begstat{(4.5) Theorem} Let $u_1\ld u_q$ be plurisubharmonic functions on~$X$. The currents $u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T$ and $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ are well defined and have locally finite mass in~$X$ as soon as $q\le p$ and $$\cH_{2p-2m+1}\big(L(u_{j_1})\cap\ldots\cap L(u_{j_m})\cap\Supp\,T\big)=0$$ for all choices of indices $j_1<\ldots0$ small. If $0\notin A_j$ for some~$j$, we choose instead $W_j$ to be a small neighborhood of~$0$ such that $\ovl W_j\subset (\ovl B'\times(1-\delta)B'')\ssm A_j\,$; property (4.8) is then automatically satisfied. Let $\chi_j\ge 0$ be a function with compact support in $W_j$, equal to $1$ near $K\cap A_j$ if $A_j\ni 0$ (resp. equal to $1$ near $0$ if $A_j\not\ni 0$) and let $\chi(z')\ge 0$ be a function equal to $1$ on $1/2\,B'$ with compact support in~$B'$. Then $$\int_{B'\times B''}dd^c(\chi_1u_1)\wedge\ldots\wedge dd^c(\chi_qu_q) \wedge T\wedge\chi(z')\,(dd^c|z'|^2)^s=0$$ because the integrand is $dd^c$ exact and has compact support in $B'\times B''$ thanks to~(4.8). If we expand all factors $dd^c(\chi_ju_j)$, we find a term $$\chi_1\ldots\chi_q\chi(z')dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T\ge 0$$ which coincides with $dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T$ on a small neighborhood of $0$ where $\chi_j=\chi=1$. The other terms involve $$d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j+u_jdd^c\chi_j$$ for at least one index $j$. However $d\chi_j$ and $dd^c\chi_j$ vanish on some neighborhood $V'_j$ of~$K\cap A_j$ and therefore $u_j$ is bounded on $\smash{\ovl B'\times\ovl B''}\ssm V'_j$. We then apply the induction hypothesis $({\rm b}_{q-1})$ to the current $$\Theta=dd^cu_1\wedge\ldots\wedge\wh{dd^cu_j}\wedge \ldots\wedge dd^cu_q\wedge T$$ and the usual Chern-Levine-Nirenberg inequality to the product of $\Theta$ with the mixed term $d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j$. Remark~3.5 can be applied because $\chi_j$ is smooth and is therefore a difference $\chi^{(1)}_j-\chi^{(2)}_j$ of locally bounded plurisubharmonic functions in $\bbbc^n$. Let $K'$ be a compact neighborhood of $\smash{ \ovl B'\times\ovl B''}$ with $K'\subset K^\circ$, and let $V_j$ be a neighborhood of $K\cap A_j$ with $\ovl V_j\subset V'_j$. Then with $L':=(\smash{\ovl B'\times\ovl B''})\ssm V'_j\subset(K'\ssm V_j)^\circ$ we obtain $$\eqalign{ ||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c&\chi_j)\wedge \Theta||_{\ovl B'\times \ovl B''} =||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c\chi_j)\wedge\Theta||_{L'}\cr &\le C_1||u_j||_{L^\infty(K'\ssm V_j)}||\Theta||_{K'\ssm V_j},\cr ||\Theta||_{K'\ssm V_j}\le||\Theta||_{K'} &\le C_2||u_1||_{L^\infty(K\ssm V_1)}\ldots\wh{||u_j||}\ldots ||u_q||_{L^\infty(K\ssm V_q)}||T||_K.\cr}$$ Now, we may slightly move the unitary basis in $\bbbc^n$ and get coordinate systems $z^m=(z^m_1\ld z^m_n)$ with the same properties as above, such that the forms $$(dd^c|z^{m\prime}|^2)^s={s!\over\pi^s} \ii\,dz^m_1\wedge d\ovl z^m_1\wedge\ldots\wedge\ii\,dz^m_s \wedge d\ovl z^m_s,~~~~1\le m\le N$$ define a basis of $\bigwedge^{s,s}(\bbbc^n)^\star$. It follows that all measures $$dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T\wedge \ii\,dz^m_1\wedge d\ovl z^m_1\wedge\ldots\wedge\ii\,dz^m_s\wedge d\ovl z^m_s$$ satisfy estimate $({\rm b}_q)$ on a small neighborhood $L$ of~$0$. \endproof \begproof{of Step 2.} We argue in a similar way with the integrals $$\eqalign{ \int_{B'\times B''}&\chi_1u_1dd^c(\chi_2u_2)\wedge\ldots dd^c(\chi_qu_q) \wedge T\wedge\chi(z')(dd^c|z'|^2)^s\wedge dd^c|z_{s+1}|^2\cr &=\int_{B'\times B''}|z_{s+1}|^2dd^c(\chi_1u_1)\wedge \ldots dd^c(\chi_qu_q)\wedge T\wedge\chi(z')(dd^c|z'|^2)^s.\cr}$$ We already know by $({\rm b}_q)$ and Remark~3.5 that all terms in the right hand integral admit the desired bound. For $q=1$, this shows that $({\rm b}_1)\Longrightarrow({\rm a}_1)$. Except for $\chi_1\ldots\chi_q\chi(z')\,u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T$, all terms in the left hand integral involve derivatives of~$\chi_j$. By construction, the support of these derivatives is disjoint from~$A_j$, thus we only have to obtain a bound for $$\int_Lu_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\alpha$$ when $L=\ovl B(x_0,r)$ is disjoint from $A_j$ for some $j\ge 2$, say $L\cap A_2=\emptyset$, and $\alpha$ is a constant positive form of type~$(p-q,p-q)$. Then $\ovl B(x_0,r+\varepsilon)\subset K^\circ\ssm\ovl V_2$ for some $\varepsilon>0$ and some neighborhood $V_2$ of~$K\cap A_2$. By the max construction used e.g. in Prop.~4.1, we can replace $u_2$ by a plurisubharmonic function $\wt u_2$ equal to $u_2$ in $L$ and to $A(|z-x_0|^2-r^2)-M$ in $\ovl B(x_0,r+\varepsilon)\ssm B(x_0,r+ \varepsilon/2)$, with $M=||u_2||_{L^\infty(K\ssm V_2)}$ and $A=M/\varepsilon r$. Let $\chi\ge 0$ be a smooth function equal to $1$ on $B(x_0,r+\varepsilon/2)$ with support in $B(x_0,r)$. Then $$\eqalign{ \int_{B(x_0,r+\varepsilon)}u_1&dd^c(\chi\wt u_2)\wedge dd^cu_3\wedge \ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr &=\int_{B(x_0,r+\varepsilon)}\chi\wt u_2dd^cu_1\wedge dd^cu_3\wedge\ldots \wedge dd^cu_q\wedge T\wedge\alpha\cr &\le O(1)~||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)} ||T||_K\cr}$$ where the last estimate is obtained by the induction hypothesis $({\rm b}_{q-1})$ applied to $dd^cu_1\wedge dd^cu_3\wedge\ldots\wedge dd^cu_q\wedge T$. By construction $$dd^c(\chi\wt u_2)=\chi\,dd^c\wt u_2+(\hbox{\rm smooth terms involving $d\chi$})$$ coincides with $dd^cu_2$ in $L$, and $({\rm a}_{q-1})$ implies the required estimate for the other terms in the left hand integral.\qed \endproof \begstat{(4.9) Proposition} With the assumptions of Th.~$4.5$, the analogue of the monotone convergence Theorem {\rm 3.7~(a,b)} holds. \endstat \begproof{} By the arguments already used in the proof of Th.~3.7 (e.g. by Lemma~3.9), it is enough to show that $$\eqalign{ \int_{B'\times B''}&\chi_1\ldots\chi_q\,u_1\wedge dd^cu_2\wedge \ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr &\le\liminf_{k\to+\infty}\int_{B'\times B''}\chi_1\ldots\chi_q\, u^k_1dd^cu^k_2\wedge\ldots\wedge dd^cu^k_q\wedge T\wedge\alpha\cr}$$ where $\alpha=\chi(z')(dd^c|z'|^2)^s$ is closed. Here the functions $\chi_j$, $\chi$ are chosen as in the proof of Step~1 in~4.7, especially their product has compact support in $B'\times B''$ and $\chi_j=\chi=1$ in a neighborhood of the given point~$x_0$. We argue by induction on $q$ and also on the number $m$ of functions $(u_j)_{j\ge 1}$ which are unbounded near~$x_0$. If $u_j$ is bounded near $x_0$, we take $W_j'' \compact W'_j\compact W_j$ to be small balls of center $x_0$ on which $u_j$ is bounded and we modify the sequence $u_j^k$ on the corona $W_j\ssm W''_j$ so as to make it constant and equal to a smooth function $A|z-x_0|^2+B$ on the smaller corona~$W_j\ssm W'_j$. In that case, we take $\chi_j$ equal to $1$ near~$\smash{\ovl W'_j}$ and $\Supp\,\chi_j\subset W_j$. For every $\ell=1\ld q$, we are going to check that $$\eqalign{ \liminf_{k\to+\infty}\int_{B'\times B''} \chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u_\ell)\wedge dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha\cr {}\le\liminf_{k\to+\infty}\int_{B'\times B''} \chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u^k_\ell)\wedge dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha.\cr}$$ In order to do this, we integrate by parts $\chi_1u^k_1dd^c(\chi_\ell u_\ell)$ into $\chi_\ell u_\ell dd^c(\chi_1u^k_1)$ for $\ell\ge 2$, and we use the inequality $u_\ell\le u^k_\ell$. Of course, the derivatives $d\chi_j$, $d^c\chi_j$, $dd^c\chi_j$ produce terms which are no longer positive and we have to take care of these. However, $\Supp\,d\chi_j$ is disjoint from the unbounded locus of $u_j$ when $u_j$ is unbounded, and contained in $W_j\ssm\smash{\ovl W'_j}$ when $u_j$ is bounded. The number $m$ of unbounded functions is therefore replaced by $m-1$ in the first case, whereas in the second case $u^k_j=u_j$ is constant and smooth on $\Supp\,d\chi_j$, so $q$ can be replaced by $q-1$. By induction on $q+m$ (and thanks to the polarization technique~3.5), the limit of the terms involving derivatives of $\chi_j$ is equal on both sides to the corresponding terms obtained by suppressing all indices~$k$. Hence these terms do not give any contribution in the inequalities.\qed \endproof We finally quote the following simple consequences of Th.~4.5 when $T$ is arbitrary and $q=1$, resp. when $T=1$ has bidegree $(0,0)$ and $q$ is arbitrary. \begstat{(4.10) Corollary} Let $T$ be a closed positive current of bidimension $(p,p)$ and let $u$ be a plurisubharmonic function on $X$ such that $L(u)\cap\Supp\,T$ is contained in an analytic set of dimension at most $p-1$. Then $uT$ and $dd^cu\wedge T$ are well defined and have locally finite mass in~$X$.\qed \endstat \begstat{(4.11) Corollary} Let $u_1\ld u_q$ be plurisubharmonic functions on $X$ such that $L(u_j)$ is contained in an analytic set $A_j\subset X$ for every~$j$. Then $dd^cu_1\wedge\ldots\wedge dd^cu_q$ is well defined as soon as $A_{j_1}\cap\ldots\cap A_{j_m}$ has codimension at least $m$ for all choices of indices $j_1<\ldots0$ such that $$[Z_1]\wedge\ldots\wedge[Z_q]=\sum m_k[C_k].$$ The integer $m_k$ is called the multiplicity of intersection of $Z_1\ld Z_q$ along the component~$C_k$. \endstat \begproof{} The wedge product has bidegree $(q,q)$ and support in $C=\bigcup C_k$ where $\codim C=q$, so it must be a sum as above with $m_k\in\bbbr_+$. We check by induction on $q$ that $m_k$ is a positive integer. If we denote by $A$ some irreducible component of $|Z_1|\cap\ldots\cap |Z_{q-1}|$, we need only check that $[A]\wedge[Z_q]$ is an integral analytic cycle of codimension $q$ with positive coefficients on each component $C_k$ of the intersection. However $[A]\wedge[Z_q]=dd^c(\log|f_q|~[A])$. First suppose that no component of $A\cap f_q^{-1}(0)$ is contained in the singular part $A_\sing$. Then the Lelong-Poincar\'e equation applied on $A_\reg$ shows that $dd^c(\log|f_q|~[A])=\sum m_k[C_k]$ on $X\ssm A_\sing$, where $m_k$ is the vanishing order of $f_q$ along $C_k$ in $A_\reg$. Since $C\cap A_\sing$ has codimension $q+1$ at least, the equality must hold on $X$. In general, we replace $f_q$ by $f_q-\varepsilon$ so that the divisor of $f_q-\varepsilon$ has no component contained in $A_\sing$. Then $dd^c(\log|f_q-\varepsilon|~[A])$ is an integral codimension $q$ cycle with positive multiplicities on each component of $A\cap f_q^{-1}(\varepsilon)$ and we conclude by letting $\varepsilon$ tend to zero.\qed \endproof \titleb{5.}{Generalized Lelong Numbers} The concepts we are going to study mostly concern the behaviour of currents or plurisubharmonic functions in a neighborhood of a point at which we have for instance a logarithmic pole. Since the interesting applications are local, we assume from now on (unless otherwise stated) that $X$ is a Stein manifold, i.e. that $X$ has a strictly plurisubharmonic exhaustion function. Let $\varphi:X\longrightarrow[-\infty,+\infty[$ be a continuous plurisubharmonic function (in general $\varphi$ may have $-\infty$ poles, our continuity assumption means that $e^\varphi$ is continuous). The sets $$\leqalignno{ S(r)&=\{x\in X\,;\,\varphi(x)=r\},&(5.1)\cr B(r)&=\{x\in X\,;\,\varphi(x)0$ the set $E_c=\{x\in X\,;\,\nu(T,x)\ge c\}$ is a closed set of locally finite $\cH_{2p}$ Hausdorff measure in $X$. \endstat \begproof{} By (5.7), we infer $\nu(T,a)=\lim_{r\to 0}\sigma_T\big(\ovl B(a,r)\big)p!/\pi^pr^{2p}$. The function $a\mapsto\sigma_T\big(\ovl B(a,r)\big)$ is clearly upper semicontinuous. Hence the decreasing limit $\nu(T,a)$ as $r$ decreases to $0$ is also upper semicontinuous in~$a$. This implies that $E_c$ is closed. Now, let $K$ be a compact subset in $X$ and let $\{a_j\}_{1\le j\le N}$, $N=N(\varepsilon)$, be a maximal collection of points in $E_c\cap K$ such that $|a_j-a_k|\ge 2\varepsilon$ for $j\ne k$. The balls $B(a_j,2\varepsilon)$ cover $E_c\cap K$, whereas the balls $B(a_j,\varepsilon)$ are disjoint. If $K_{c,\varepsilon}$ is the set of points which are at distance $\le\varepsilon$ of $E_c\cap K$, we get $$\sigma_T(K_{c,\varepsilon})\ge\sum\sigma_T\big(B(a_j,\varepsilon)\big) \ge N(\varepsilon)\,c\pi^p\varepsilon^{2p}/p!,$$ since $\nu(T,a_j)\ge c$. By the definition of Hausdorff measure, we infer $$\eqalignno{ \cH_{2p}(E_c\cap K)&\le\liminf_{\varepsilon\to 0} \sum\big(\hbox{\rm diam}\,B(a_j,2\varepsilon)\big)^{2p}\cr &\le\liminf_{\varepsilon\to 0}N(\varepsilon)(4\varepsilon)^{2p} \le{p!4^{2p}\over c\pi^p}\sigma_T(E_c\cap K).&\square\cr}$$ \endproof Finally, we conclude this section by proving two simple semi-continuity results for Lelong numbers. \begstat{(5.12) Proposition} Let $T_k$ be a sequence of closed positive currents of bidimension $(p,p)$ converging weakly to a limit~$T$. Suppose that there is a closed set $A$ such that $\Supp\,T_k\subset A$ for all~$k$ and such that $\varphi$ is semi-exhaustive on $A$ with $A\cap B(R)\compact X$. Then for all $rr$, hence the total mass $\mu_r(S(r))=\mu_r(B(s))$ is equal to the difference between the masses of $(dd^c\varphi)^n$ and $\bbbone_{X\setminus B(r)}(dd^c\varphi)^n$ over $B(s)$, i.e. $$\mu_r\big(S(r)\big)=\int_{B(r)}(dd^c\varphi)^n.\leqno(6.2)$$ \begstat{(6.3) Example} \rm When $(dd^c\varphi)^n=0$ on $X\setminus\varphi^{-1}(-\infty)$, formula (6.1) can be simplified into $\mu_r=(dd^c\varphi_{\gge r})^n$. This is so for $\varphi(z)=\log|z|$. In this case, the invariance of $\varphi$ under unitary transformations implies that $\mu_r$ is also invariant. As the total mass of $\mu_r$ is equal to $1$ by 5.10 and (6.2), we see that $\mu_r$ is the invariant measure of mass $1$ on the euclidean sphere of radius~$e^r$. \endstat \begstat{(6.4) Proposition} Assume that $\varphi$ is smooth near $S(r)$ and that $d\varphi\ne 0$ on $S(r)$, i.e. $r$ is a non critical value. Then $S(r)=\partial B(r)$ is a smooth oriented real hypersurface and the measure $\mu_r$ is given by the $(2n-1)$-volume form $(dd^c\varphi)^{n-1}\wedge d^c\varphi_{\restriction S(r)}$. \endstat \begproof{} Write $\max\{t,r\}=\lim_{k\to+\infty}\chi_k(t)$ where $\chi$ is a decreasing sequence of smooth convex functions with $\chi_k(t)=r$ for $t\le r-1/k$, $\chi_k(t)=t$ for $t\ge r+1/k$. Theorem 3.6 shows that $(dd^c\chi_k\circ\varphi)^n$ converges weakly to $(dd^c\varphi_{\gge r})^n$. Let $h$ be a smooth function $h$ with compact support near $S(r)$. Let us apply Stokes' theorem with $S(r)$ considered as the boundary of $X\setminus B(r)\,$: $$\leqalignno{ \int_X h(dd^c\varphi_{\gge r})^n&=\lim_{k\to+\infty} \int_X h(dd^c\chi_k\circ\varphi)^n\cr &=\lim_{k\to+\infty}\int_X -dh\wedge(dd^c\chi_k\circ\varphi)^{n-1} \wedge d^c(\chi_k\circ\varphi)\cr &=\lim_{k\to+\infty}\int_X -\chi'_k(\varphi)^n\,dh\wedge(dd^c\varphi)^{n-1} \wedge d^c\varphi\cr &=\int_{X\setminus B(r)}-dh\wedge(dd^c\varphi)^{n-1}\wedge d^c\varphi\cr &=\int_{S(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi+ \int_{X\setminus B(r)}h\,(dd^c\varphi)^{n-1}\wedge dd^c\varphi.\cr}$$ Near $S(r)$ we thus have an equality of measures $$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^{n-1}\wedge d^c \varphi_{\restriction S(r)}+\bbbone_{X\setminus B(r)}(dd^c\varphi)^n. \eqno{\square}$$ \endproof \begstat{(6.5) Jensen-Lelong formula} Let $V$ be any plurisubharmonic function on~$X$. Then $V$ is $\mu_r$-integrable for every $r\in{}]-\infty,R[$ and $$\mu_r(V)-\int_{B(r)}V(dd^c\varphi)^n= \int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt.$$ \endstat \begproof{} Proposition 3.11 shows that $V$ is integrable with respect to the measure $(dd^c\varphi_{\gge r})^n$, hence $V$ is $\mu_r$-integrable. By definition $$\nu(dd^cV,\varphi,t)=\int_{\varphi(z)0$. Then $B(r)$ is the polydisk of radii $(e^{r/\lambda_1}\ld e^{r/\lambda_n})$. If some coordinate $z_j$ is non zero, say $z_1$, we can write $\varphi(z)$ as $\lambda_1\log|z_1|$ plus some function depending only on the $(n-1)$ variables $z_j/z_1^{\lambda_1/\lambda_j}$. Hence $(dd^c\varphi)^n=0$ on $\bbbc^n\setminus\{0\}$. It will be shown later that $$(dd^c\varphi)^n=\lambda_1\ldots\lambda_n\,\delta_0.\leqno(6.11\,{\rm a})$$ We now determine the measures $\mu_r$. At any point $z$ where not all terms $|z_j|^{\lambda_j}$ are equal, the smallest one can be omitted without changing $\varphi$ in a neighborhood of $z$. Thus $\varphi$ depends only on $(n-1)$-variables and $(dd^c\varphi_{\ge r})^n=0$, $\mu_r=0$ near $z$. It follows that $\mu_r$ is supported by the distinguished boundary $|z_j|=e^{r/\lambda_j}$ of the polydisk $B(r)$. As $\varphi$ is invariant by all rotations $z_j\longmapsto e^{\ii\theta_j}z_j$, the measure $\mu_r$ is also invariant and we see that $\mu_r$ is a constant multiple of $d\theta_1\ldots d\theta_n$. By formula (6.2) and (6.11$\,$a) we get $$\mu_r=\lambda_1\ldots\lambda_n\,(2\pi)^{-n}d\theta_1\ldots d\theta_n. \leqno(6.11\,{\rm b})$$ In particular, the Lelong number $\nu(dd^cV,\varphi)$ is given by $$\nu(dd^cV,\varphi)=\lim_{r\to-\infty}{\lambda_1\ldots\lambda_n\over r} \int_{\theta_j\in[0,2\pi]}\kern-6pt V(e^{r/\lambda_1+\ii\theta_1} \ld e^{r/\lambda_n+\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}.$$ These numbers have been introduced and studied by (Kiselman 1986). We call them {\it directional Lelong numbers} with coefficients $(\lambda_1\ld\lambda_n)$. For an arbitrary current $T$, we define $$\nu(T,x,\lambda)=\nu\big(T,\log\max|z_j-x_j|^{\lambda_j}\big). \leqno(6.11\,{\rm c})$$ The above formula for $\nu(dd^cV,\varphi)$ combined with the analogue of Harnack's inequality (6.9$\,$d) for polydisks gives $$\leqalignno{ \nu(dd^cV,x,\lambda)&=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r} \int V(r^{1/\lambda_1}e^{\ii\theta_1}\ld r^{1/\lambda_n} e^{\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}\cr &=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r} \sup_{\theta_1\ld\theta_n}V(r^{1/\lambda_1}e^{\ii\theta_1}\ld r^{1/\lambda_n} e^{\ii\theta_n}).&(6.11\,{\rm d})\cr}$$ \endstat \titleb{7.}{Comparison Theorems for Lelong Numbers} Let $T$ be a closed positive current of bidimension $(p,p)$ on a Stein manifold $X$ equipped with a semi-exhaustive plurisubharmonic weight~$\varphi$. We first show that the Lelong numbers $\nu(T,\varphi)$ only depend on the asymptotic behaviour of $\varphi$ near the polar set $S(-\infty)$. In a precise way: \begstat{(7.1) First comparison theorem} Let $\varphi,\psi:X\longrightarrow[-\infty,+\infty[$ be continuous plurisubharmonic functions. We assume that $\varphi,\psi$ are semi-exhaustive on $\Supp\,T$ and that $$\ell:=\limsup{{\psi(x)}\over{\varphi(x)}}<+\infty~~~~{\it as}~~ x\in\Supp\,T~~{\it and}~~\varphi(x)\to -\infty.$$ Then $\nu(T,\psi)\le \ell^p\nu(T,\varphi)$, and the equality holds if $\ell=\lim\psi/\varphi$. \endstat \begproof{} Definition 6.4 shows immediately that $\nu(T,\lambda\varphi)=\lambda^p\nu(T,\varphi)$ for every scalar $\lambda>0$. It is thus sufficient to verify the inequality $\nu(T,\psi)\le\nu(T,\varphi)$ under the hypothesis $\limsup\psi/\varphi<1$. For all $c>0$, consider the plurisubharmonic function $$u_c=\max(\psi-c,\varphi).$$ Let $R_\varphi$ and $R_\psi$ be such that $B_\varphi(R_\varphi)\cap\Supp\,T$ and $B_\psi(R_\psi)\cap\Supp\,T$ be relatively compact in $X$. Let $r0$ large enough, we have $u_c=\varphi$ on $\varphi^{-1}([a,r])$ and Stokes' formula gives $$\nu(T,\varphi,r)=\nu(T,u_c,r)\ge\nu(T,u_c).$$ The hypothesis $\limsup\psi/\varphi<1$ implies on the other hand that there exists $t_0<0$ such that $u_c=\psi-c$ on $\{u_c0$ we have $$\big(dd^c\log\max_{1\le j\le n}|z_j|^{\lambda_j}\big)^n= \Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n= \lambda_1\ldots\lambda_n\,\delta_0.$$ \endstat \begproof{} In fact, our measures vanish on $\bbbc^n\ssm\{0\}$ by the arguments explained in example~6.11. Hence they are equal to $c\,\delta_0$ for some constant $c\ge 0$ which is simply the Lelong number of the bidimension $(n,n)$-current $T=[X]=1$ with the corresponding weight. The comparison theorem shows that the first equality holds and that $$\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n= \ell^{-n}\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\ell\lambda_j}\Big)^n$$ for all $\ell>0$. By taking $\ell$ large and approximating $\ell\lambda_j$ with $2[\ell\lambda_j/2]$, we may assume that $\lambda_j=2s_j$ is an even integer. Then formula (5.6) gives $$\eqalign{ &\int_{\sum|z_j|^{2s_j}0$, we thus get $$\eqalign{ \int_{Y_r^\star}\pi_0^\star T\wedge\omega_Y^{N+p} &=\sum_{0\le k\le p}\lambda_k\int_{B(0,r)\ssm\{0\}}T\wedge\beta^k\wedge (dd^c\log|z|)^{p-k}\cr &=\sum_{0\le k\le p}\lambda_k2^{-(p-k)}r^{-2(p-k)} \int_{B(0,r)\ssm\{0\}}T\wedge\beta^p<+\infty.\cr}$$ The Skoda-El Mir theorem 2.3 shows that the trivial extension $\wt\pi_0^\star T$ of $\pi_0^\star T$ is a closed positive current on~$Y_R$. Of course, the zero section $\pi^{-1}(0)$ might also carry some extra mass of the desired current~$\pi^\star T$. Since $\pi^{-1}(0)$ has codimension~$q$, this extra mass cannot exist when $q>n-p=\codim\pi^\star T$ and we simply set $\pi^\star T=\wt\pi_0^\star T$. On the other hand, if $q=n-p$, we set $$\pi^\star T:=\wt\pi_0^\star T+\nu(T,0)\,[\pi^{-1}(0)].\leqno(7.12)$$ We can now apply parallel slicing with respect to $\sigma:Y_R\to G(q,n)$, which is a submersion: for almost all $S\in G(q,n)$, there is a well defined slice $T_{\restriction S}= (\pi^\star T)_{\restriction\sigma^{-1}(S)}$. These slices coincide with the usual restrictions of $T$ to $S$ if $T$ is smooth. \begstat{(7.13) Theorem {\rm (Siu~1974)}} For almost all $S\in G(q,n)$ with $q\ge n-p$, the slice $T_{\restriction S}$ satisfies~ $\nu(T_{\restriction S},0)=\nu(T,0)$. \endstat \begproof{} If $q=n-p$, the slice $T_{\restriction S}$ consists of some positive measure with support in $S\ssm\{0\}$ plus a Dirac measure $\nu(T,0)\,\delta_0$ coming from the slice of $\nu(T,0)\,[\pi^{-1}(0)]$. The equality $\nu(T_{\restriction S},0)= \nu(T,0)$ thus follows directly from~(7.12). In the general case $q>n-p$, it is clearly sufficient to prove the following two properties: \smallskip \noindent a)~~ $\nu(T,0,r)=\displaystyle\int_{S\in G(q,n)} \nu(T_{\restriction S},0,r)\,dv(S)$~~ for all $r\in{}]0,R[\,$; \smallskip \noindent b)~~ $\nu(T_{\restriction S},0)\ge\nu(T,0)$~~ for almost all $S$. \smallskip \noindent In fact, a) implies that $\nu(T,0)$ is the average of all Lelong numbers $\nu(T_{\restriction S},0)$ and the conjunction with b) implies that these numbers must be equal to $\nu(T,0)$ for almost all $S$. In order to prove a) and b), we can suppose without loss of generality that $T$ is smooth on $B(0,R)\ssm\{0\}$. Otherwise, we perform a small convolution with respect to the action of ${\rm Gl}_n(\bbbc)$ on $\bbbc^n$: $$T_\varepsilon=\int_{g\in{\rm Gl}_n(\bbbc)}\rho_\varepsilon(g)\, g^\star T\,dv(g)$$ where $(\rho_\varepsilon)$ is a regularizing family with support in an $\varepsilon$-neighborhood of the unit element of ${\rm Gl}_n(\bbbc)$. Then $T_\varepsilon$ is smooth in $B(0,(1-\varepsilon)R)\ssm\{0\}$ and converges weakly to $T$. Moreover, we have $\nu(T_\varepsilon,0)=\nu(T,0)$ by (7.2) and $\nu(T_{\restriction S},0)\ge\limsup_{\varepsilon\to 0}\nu(T_{\varepsilon, \restriction S},0)$ by (5.12), thus a), b) are preserved in the limit. If $T$ is smooth on $B(0,R)\ssm\{0\}$, the slice $T_{\restriction S}$ is defined for all $S$ and is simply the restriction of $T$ to $S\ssm\{0\}$ (carrying no mass at the origin). \smallskip \noindent a) Here we may even assume that $T$ is smooth at $0$ by performing an ordinary convolution. As $T_{\restriction S}$ has bidegree $(n-p,n-p)$, we have $$\nu(T_{\restriction S},0,r)=\int_{S\cap B(0,r)}T\wedge\alpha_S^{q-(n-p)} =\int_{B(0,r)}T\wedge [S]\wedge\alpha_S^{p+q-n}$$ where $\alpha_S=dd^c\log|w|$ and $w=(w_1\ld w_q)$ are orthonormal coordinates on~$S$. We simply have to check that $$\int_{S\in G(q,n)}[S]\wedge\alpha_S^{p+q-n}\,dv(S)=(dd^c\log|z|)^p.$$ However, both sides are unitary and homothety invariant $(p,p)$-forms with Lelong number $1$ at the origin, so they must coincide by Lemma~7.11. \smallskip \noindent b) We prove the inequality when $S=\bbbc^q\times\{0\}$. By the comparison theorem~7.1, for every $r>0$ and $\varepsilon>0$ we have $$\leqalignno{ &\int_{B(0,r)}T\wedge\gamma_\varepsilon^p\ge\nu(T,0)~~~~\hbox{\rm where} &(7.14)\cr &\gamma_\varepsilon={1\over 2}dd^c\log(\varepsilon|z_1|^2+\ldots+ \varepsilon|z_q|^2+|z_{q+1}|^2+\ldots+|z_n|^2).\cr}$$ We claim that the current $\gamma_\varepsilon^p$ converges weakly to $$[S]\wedge\alpha_S^{p+q-n}= [S]\wedge\Big({1\over2}dd^c\log(|z_1|^2+\ldots+|z_q|^2)\Big)^{p+q-n}$$ as $\varepsilon$ tends to $0$. In fact, the Lelong number of $\gamma_\varepsilon^p$ at $0$ is $1$, hence by homogeneity $$\int_{B(0,r)}\gamma_\varepsilon^p\wedge(dd^c|z|^2)^{n-p}=(2r^2)^{n-p}$$ for all $\varepsilon,r>0$. Therefore the family $(\gamma_\varepsilon^p)$ is relatively compact in the weak topology. Since $\gamma_0=\lim\gamma_\varepsilon$ is smooth on $\bbbc^n\ssm S$ and depends only on $n-q$ variables ($n-q\le p$), we have $\lim\gamma_\varepsilon^p=\gamma_0^p=0$ on $\bbbc^n\ssm S$. This shows that every weak limit of $(\gamma_\varepsilon^p)$ has support in $S$. Each of these is the direct image by inclusion of a unitary and homothety invariant $(p+q-n,p+q-n)$-form on $S$ with Lelong number equal to $1$ at $0$. Therefore we must have $$\lim_{\varepsilon\to 0}\gamma_\varepsilon^p=(i_S)_\star (\alpha_S^{p+q-n})=[S]\wedge\alpha_S^{p+q-n},$$ and our claim is proved (of course, this can also be checked by direct elementary calculations). By taking the limsup in (7.14) we obtain $$\nu(T_{\restriction S},0,r+0)=\int_{\ovl B(0,r)}T\wedge[S]\wedge \alpha_S^{p+q-n}\ge\nu(T,0)$$ (the singularity of $T$ at $0$ does not create any difficulty because we can modify $T$ by a $dd^c$-exact form near $0$ to make it smooth everywhere). Property b) follows when $r$ tends to $0$.\qed \endproof \titleb{8.}{Siu's Semicontinuity Theorem} Let $X$, $Y$ be complex manifolds of dimension $n$, $m$ such that $X$ is Stein. Let $\varphi:X\times Y\longrightarrow[-\infty,+\infty[$ be a continuous plurisubharmonic function. We assume that $\varphi$ is {\it semi-exhaustive} with respect to $\Supp\,T$ , i.e. that for every compact subset $L\subset Y$ there exists $R=R(L)<0$ such that $$\{(x,y)\in\Supp\,T\times L\,; \,\varphi(x,y)\le R\}\compact X\times Y.\leqno(8.1)$$ Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$. For every point $y\in Y$, the function $\varphi_y(x):=\varphi(x,y)$ is semi-exhaustive on $\Supp\,T\,$; one can therefore associate with $y$ a generalized Lelong number $\nu(T,\varphi_y)$. Proposition~5.13 implies that the map $y\mapsto\nu(T,\varphi_y)$ is upper semi-continuous, hence the upperlevel sets $$E_c=E_c(T,\varphi)=\{y\in Y ;\nu(T,\varphi_y)\ge c\}\ ,\ c>0\leqno(8.2)$$ are closed. Under mild additional hypotheses, we are going to show that the sets $E_c$ are in fact analytic subsets of $Y$ (Demailly 1987a). \begstat{(8.3) Definition} We say that a function $f(x,y)$ is locally H\"older continuous with respect to $y$ on $X\times Y$ if every point of $X\times Y$ has a neighborhood $\Omega$ on which $$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$ for all $(x,y_1)\in\Omega$, $(x,y_2)\in\Omega$, with some constants $M>0$, $\gamma\in{}]0,1]$, and suitable coordinates on $Y$. \endstat \begstat{(8.4) Theorem {\rm(Demailly 1987a)}} Let $T$ be a closed positive current on $X$ and let $$\varphi:X\times Y\longrightarrow[-\infty,+\infty[$$ be a continuous plurisubharmonic function. Assume that $\varphi$ is semi-exhaustive on $\Supp\,T$ and that $e^{\varphi(x,y)}$ is locally H\"older continuous with respect to $y$ on~$X\times Y$. Then the upperlevel sets $$E_c(T,\varphi)=\{y\in Y;\nu(T,\varphi_y)\ge c\}$$ are analytic subsets of $Y$. \endstat This theorem can be rephrased by saying that $y\longmapsto\nu(T,\varphi_y)$ is upper semi-continuous with respect to the analytic Zariski topology. As a special case, we get the following important result of (Siu~1974): \begstat{(8.5) Corollary} If $T$ is a closed positive current of bidimension $(p,p)$ on a complex manifold~$X$, the upperlevel sets $E_c(T)=\{x\in X\,;\,\nu(T,x)\ge c\}$ of the usual Lelong numbers are analytic subsets of dimension $\le p$. \endstat \begproof{} The result is local, so we may assume that $X\subset\bbbc^n$ is an open subset. Theorem~8.4 with $Y=X$ and $\varphi(x,y)=\log|x-y|$ shows that $E_c(T)$ is analytic. Moreover, Prop.~5.11 implies $\dim E_c(T)\le p$.\qed \endproof \noindent{\bf(8.6) Generalization.} Theorem 8.4 can be applied more generally to weight functions of the type $$\varphi(x,y)=\max_j~\log\Big(\sum_k|F_{j,k}(x,y)|^{\lambda_{j,k}}\Big)$$ where $F_{j,k}$ are holomorphic functions on $X\times Y$ and where $\gamma_{j,k}$ are positive real constants; in this case $e^\varphi$ is H\"older continuous of exponent $\gamma=\min\{\lambda_{j,k},1\}$ and $\varphi$ is semi-exhaustive with respect to the whole space $X$ as soon as the projection ${\rm pr}_2:\bigcap F_{j,k}^{-1}(0)\lra Y$ is proper and finite. For example, when $\varphi(x,y)=\log\max|x_j-y_j|^\lambda_j$ on an open subset $X$ of $\bbbc^n$ , we see that the upperlevel sets for Kiselman's numbers $\nu(T,x,\lambda)$ are analytic in $X$ (a result first proved in (Kiselman~1986). More generally, set $\psi_\lambda(z)=\log\max|z_j|^{\lambda_j}$ and $\varphi(x,y,g)=\psi_\lambda\big(g(x-y)\big)$ where $x,y\in\bbbc^n$ and $g\in{\rm Gl}(\bbbc^n)$. Then $\nu(T,\varphi_{y,g})$ is the Kiselman number of $T$ at $y$ when the coordinates have been rotated by $g$. It is clear that $\varphi$ is plurisubharmonic in $(x,y,g)$ and semi-exhaustive with respect to $x$, and that $e^\varphi$ is locally H\"older continuous with respect to $(y,g)$. Thus the upperlevel sets $$E_c=\{(y,g)\in X\times{\rm Gl}(\bbbc^n)\,;\,\nu(T,\varphi_{y,g})\ge c\}$$ are analytic in $X\times{\rm Gl}(\bbbc^n)$. However this result is not meaningful on a manifold, because it is not invariant under coordinate changes. One can obtain an invariant version as follows. Let $X$ be a manifold and let $J^k\cO_X$ be the bundle of $k$-jets of holomorphic functions on~$X$. We consider the bundle $S_k$ over $X$ whose fiber $S_{k,y}$ is the set of $n$-tuples of $k$-jets $u=(u_1\ld u_n)\in(J^k\cO_{X,y})^n$ such that $u_j(y)=0$ and $du_1\wedge\ldots\wedge du_n(y)\ne 0$. Let $(z_j)$ be local coordinates on an open set $\Omega\subset X$. Modulo $O(|z-y|^{k+1})$, we can write $$u_j(z)=\sum_{1\le|\alpha|\le k}a_{j,\alpha}(z-y)^\alpha$$ with ${\rm det}(a_{j,(0\ld 1_k\ld 0)})\ne 0$. The numbers $((y_j),(a_{j,\alpha}))$ define a coordinate system on the total space of $S_{k\,\restriction\Omega}$. For $(x,(y,u))\in X\times S_k$, we introduce the function $$\varphi(x,y,u)=\log\max|u_j(x)|^{\lambda_j}= \log\max\Big|\sum_{1\le|\alpha|\le k}a_{j,\alpha}(x-y)^\alpha \Big|^{\lambda_j}$$ which has all properties required by Th.~8.4 on a neighborhood of the diagonal $x=y$, i.e. a neighborhood of $X\times_X S_k$ in $X\times S_k$. For $k$ large, we claim that Kiselman's directional Lelong numbers $$\nu(T,y,u,\lambda):=\nu(T,\varphi_{y,u})$$ with respect to the coordinate system $(u_j)$ at $y$ do not depend on the selection of the jet representives and are therefore canonically defined on~$S_k$. In fact, a change of $u_j$ by $O(|z-y|^{k+1})$ adds $O(|z-y|^{(k+1)\lambda_j})$ to $e^\varphi$, and we have $e^\varphi\ge O(|z-y|^{\max\lambda_j})$. Hence by (7.1) it is enough to take $k+1\ge\max\lambda_j/\min\lambda_j$. Theorem~8.4 then shows that the upperlevel sets $E_c(T,\varphi)$ are analytic in~$S_k$.\qed \medskip \titled{Proof of the Semicontinuity Theorem~8.4} As the result is local on $Y$, we may assume without loss of generality that $Y$ is a ball in $\bbbc^m$. After addition of a constant to $\varphi$, we may also assume that there exists a compact subset $K\subset X$ such that $$\{(x,y)\in X\times Y ;\varphi(x,y)\le 0\}\subset K\times Y.$$ By Th.~7.1, the Lelong numbers depend only on the asymptotic behaviour of $\varphi$ near the (compact) polar set $\varphi^{-1}(-\infty)\cap({\rm Supp T}\times Y)$. We can add a smooth strictly plurisubharmonic function on $X\times Y$ to make $\varphi$ strictly plurisuharmonic. Then Richberg's approximation theorem for continuous plurisubharmonic functions shows that there exists a smooth plurisubharmonic function $\wt\varphi$ such that $\varphi\le\wt\varphi\le\varphi+1$. We may therefore assume that $\varphi$ is smooth on $(X\times Y)\setminus\varphi^{-1}(-\infty)$. \medskip \noindent $\bullet$ {\bf First step}: {\it construction of a local plurisubharmonic potential.} Our goal is to generalize the usual construction of plurisubharmonic potentials associated with a closed positive current (Lelong 1967, Skoda 1972a). We replace here the usual kernel $|z-\zeta|^{-2p}$ arising from the hermitian metric of $\bbbc^n$ by a kernel depending on the weight $\varphi$. Let $\chi\in C^\infty(\bbbr,\bbbr)$ be an increasing function such that $\chi(t)=t$ for $t\le-1$ and $\chi(t)=0$ for $t\ge 0$. We consider the half-plane $H=\{z\in\bbbc\,;\,{\rm Re}z<-1\}$ and associate with $T$ the potential function $V$ on $Y\times H$ defined by $$V(y,z)=-\int^0_{{\rm Re}z}\nu(T,\varphi_y,t)\chi'(t)\,dt.\leqno(8.7)$$ For every $t>\Re z$, Stokes' formula gives $$\nu(T,\varphi_y,t)= \int_{\varphi(x,y)0$. If $Y$ is connected and $U_0\not\equiv-\infty$, we already conclude that the density set $\bigcup_{c>0}E_c$ is pluripolar in~$Y$. \medskip \noindent $\bullet$ {\bf Second step}: {\it application of Kiselman's minimum principle.} Let $a\ge 0$ be arbitrary. The function $$Y\times H\ni(y,z)\longmapsto\rho(y)+V(y,z)-a{\rm Re}z$$ is plurisubharmonic and independent of ${\rm Im}\,z$. By Kiselman's theorem~1.7.8, the Legendre transform $$U_a(y)=\inf_{r<-1}\big\{\rho(y)+V(y,r)-ar\big\}$$ is locally plurisubharmonic or $\equiv-\infty$ on $Y$. \begstat{(8.9) Lemma} Let $y_0\in Y$ be a given point. \medskip \item{\rm a)} If $a>\nu(T,\varphi_{y_0})$, then $U_a$ is bounded below on a neighborhood of $y_0$. \medskip \item{\rm b)} If $a<\nu(T,\varphi_{y_0})$, then $U_a(y_0)=-\infty$. \endstat \begproof{} By definition of $V$ (cf. $(8.7)$) we have $$V(y,r)\le-\nu(T,\varphi_y,r)\int_r^0\chi'(t)dt=r\nu(T,\varphi_y,r) \le r\nu(T,\varphi_y).\leqno(8.10)$$ Then clearly $U_a(y_0)=-\infty$ if $a<\nu(T,\varphi_{y_0})$. On the other hand, if $\nu(T,\varphi_{y_0})-\infty$ if $y\in Y\setminus E_c$ (apply Lemma~8.9). For any integer $k\ge 1$, let $w_k\in\ci(Y)$ be a plurisubharmonic regularization of $U_{c-1/k}$ such that $w_k\ge U_{c-1/k}$ on $Y$ and $w_k\le -2^k$ on\break $E_c\cap Y_k$ where $Y_k=\{y\in Y\,;\,d(y,\partial Y)\ge 1/k\}$. Then Lemma~8.9~a) shows that the family $(w_k)$ is uniformly bounded below on every compact subset of $Y\setminus E_c$. We can also choose $w_k$ uniformly bounded above on every compact subset of $Y$ because $U_{c-1/k}\le U_c$. The function $$w=\sum_{k=1}^{+\infty}~2^{-k}w_k$$ satifies our requirements.\qed \endproof \noindent $\bullet$ {\bf Third step}: {\it estimation of the singularities of the potentials} $U_a$. \begstat{(8.12) Lemma} Let $y_0\in Y$ be a given point, $L$ a compact neighborhood of $y_0$, $K\subset X$ a compact subset and $r_0$ a real number $<-1$ such that $$\{(x,y)\in X\times L;\varphi(x,y)\le r_0\}\subset K\times L.$$ Assume that $e^{\varphi(x,y)}$ is locally H\"older continuous in $y$ and that $$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$ for all $(x,y_1,y_2)\in K\times L\times L$. Then, for all $\varepsilon\in{}]0,1[$, there exists a real number $\eta(\varepsilon)>0$ such that all $y\in Y$ with $|y-y_0|<\eta(\varepsilon)$ satisfy $$U_a(y)\le\rho(y)+\big((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\big) \Big(\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\Big).$$ \endstat \begproof{} First, we try to estimate $\nu(T,\varphi_y,r)$ when $y\in L$ is near $y_0$. Set $$\left\{ \eqalign{\psi(x)&=(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2\cr \psi(x)&=\max\bigl(\varphi_y(x),(1-\varepsilon)\varphi_{y_0}(x)+ \varepsilon r-\varepsilon/2\bigr)\cr \psi(x)&=\varphi_y(x)\cr} \eqalign{&{~\rm if~}\cr &{~\rm if~}\cr &{~\rm if~}\cr} \eqalign{&\varphi_{y_0}(x)\le r-1\cr r-1\le&\varphi_{y_0}(x)\le r\cr r\le&\varphi_{y_0}(x)\le r_0\cr}\right.$$ and verify that this definition is coherent when $|y-y_0|$ is small enough. By hypothesis $$|e^{\varphi_y(x)}-e^{\varphi_{y_0}(x)}|\le M|y-y_0|^\gamma.$$ This inequality implies $$\eqalign{\varphi_y(x)&\le\varphi_{y_0}(x)+\log\bigl(1+M|y-y_0|^\gamma e^{-\varphi_{y_0}(x)}\bigr)\cr \varphi_y(x)&\ge\varphi_{y_0}(x)+\log\bigl(1-M|y-y_0|^\gamma e^{-\varphi_{y_0}(x)}\bigr).\cr}$$ In particular, for $\varphi_{y_0}(x)=r$, we have $(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=r-\varepsilon/2$, thus $$\varphi_y(x)\ge r+\log(1-M|y-y_0|^\gamma e^{-r}).$$ Similarly, for $\varphi_{y_0}(x)=r-1$, we have $(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2= r-1+\varepsilon/2$, thus $$\varphi_y(x)\le r-1+\log(1+M|y-y_0|^\gamma e^{1-r}).$$ The definition of $\psi$ is thus coherent as soon as $M|y-y_0|^\gamma e^{1-r}\le\varepsilon/2$ , i.e. $$\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\le r.$$ In this case $\psi$ coincides with $\varphi_y$ on a neighborhood of $\{\psi=r\}$ , and with $$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2$$ on a neighborhood of the polar set $\psi^{-1}(-\infty)$. By Stokes' formula applied to $\nu(T,\psi,r)$, we infer $$\nu(T,\varphi_y,r)=\nu(T,\psi,r)\ge\nu(T,\psi)= (1-\varepsilon)^p\nu(T,\varphi_{y_0}).$$ From (8.10) we get $V(y,r)\le r\nu(T,\varphi_y,r)$, hence $$\leqalignno{ U_a(y)&\le\rho(y)+V(y,r)-ar\le \rho(y)+r\big(\nu(T,\varphi_y,r)-a\big),\cr U_a(y)&\le\rho(y)+r\bigl((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\bigr). &(8.13)\cr}$$ Suppose $\gamma\log|y-y_0|+\log(2eM/\varepsilon)\le r_0$ , i.e. $|y-y_0|\le(\varepsilon/2eM)^{1/\gamma}e^{r_0/\gamma}\,$; one can then choose $r=\gamma\log|y-y_0|+\log(2eM/\varepsilon)$, and by $(8.13)$ this yields the inequality asserted in Th.~8.12.\qed \endproof \medskip \noindent $\bullet$ {\bf Fourth step}: {\it application of the H\"ormander-Bombieri-Skoda theorem.} The end of the proof relies on the following crucial result, which is a consequence of the H\"ormander-Bombieri-Skoda theorem (Bombieri~1970, Skoda~1972a, Skoda~1976); it will be proved in Chapter~8, see Cor.~8.?.?. \begstat{(8.14) Proposition} Let $u$ be a plurisubharmonic function on a complex mani\-fold~$Y$. The set of points in a neighborhood of which $e^{-u}$ is not integrable is an analytic subset of $Y$.\qed \endstat \begproof{of Theorem~8.4 (end).} The main idea in what follows is due to (Kiselman 1979). For $a,b>0$, we let $Z_{a,b}$ be the set of points in a neighborhood of which $\exp(-U_a/b)$ is not integrable. Then $Z_{a,b}$ is analytic, and as the family $(U_a)$ is increasing in $a$, we have $Z_{a',b'}\supset Z_{a'',b''}$ if $a'\le a''$, $b'\le b''$. Let $y_0\in Y$ be a given point. If $y_0\notin E_c$, then $\nu (T,\varphi_{y_0})0$. On the other hand, if $y_0\in E_c$ and if $a0$ that $$U_a(y)\le(1-\varepsilon)(c-a)\gamma\log|y-y_0|+C(\varepsilon)$$ on a neighborhood of $y_0$. Hence $\exp(-U_a/b)$ is non integrable at $y_0$ as soon as $b<(c-a)\gamma/2m$, where $m=\dim Y$. We obtain therefore $$E_c=\bigcap_{{\scriptstyle am_A}A\cap E_c(T).$$ However, for $c>m_A$, the intersection $A\cap E_c(T)$ is a proper analytic subset of~$A$.\qed \endproof \begstat{(8.16) Proposition} Let $T$ be a closed positive current of bidimension $(p,p)$ and let $A$ be an irreducible $p$-dimensional analytic subset of $X$. Then \hbox{$\bbbone_AT=m_A[A]$,} in particular $T-m_A[A]$ is positive. \endstat \begproof{} As the question is local and as a closed positive current of bidimension $(p,p)$ cannot carry any mass on a $(p-1)$-dimensional analytic subset, it is enough to work in a neighborhood of a regular point $x_0\in A$. Hence, by choosing suitable coordinates, we can suppose that $X$ is an open set in $\bbbc^n$ and that $A$ is the intersection of $X$ with a $p$-dimensional linear subspace. Then, for every point $a\in A$, the inequality $\nu(T,a)\ge m_A$ implies $$\sigma_T\big(B(a,r)\big)\ge m_A\,\pi^pr^{2p}/p!= m_A\sigma_{[A]}\big(B(a,r)\big)$$ for all $r$ such that $B(a,r)\subset X$. Now, set $\Theta=T-m_A[A]$ and $\beta=dd^c|z|^2$. Our inequality says that $\int\bbbone_{B(a,r)}\Theta\wedge\beta^p\ge 0$. If we integrate this with respect to some positive continuous function $f$ with compact support in $A$, we get $\int_X g_r\Theta\wedge\beta^p\ge 0$ where $$g_r(z)=\int_A\bbbone_{B(a,r)}(z)\,f(a)\,d\lambda(a) =\int_{a\in A\cap B(z,r)}f(a)\,d\lambda(a).$$ Here $g_r$ is continuous on $\bbbc^n$, and as $r$ tends to $0$ the function $g_r(z)/(\pi^pr^{2p}/p!)$ converges to $f$ on $A$ and to $0$ on $X\ssm A$, with a global uniform bound. Hence we obtain $\int\bbbone_Af\,\Theta\wedge\beta^p\ge 0$. Since this inequality is true for all continuous functions $f\ge 0$ with compact support in $A$, we conclude that the measure $\bbbone_A\Theta\wedge\beta^p$ is positive. By a linear change of coordinates, we see that $$\bbbone_A\Theta\wedge\Big(dd^c\sum_{1\le j\le n} \lambda_j|u_j|^2\Big)^n\ge 0$$ for every basis $(u_1\ld u_n)$ of linear forms and for all coefficients~\hbox{$\lambda_j>0$.} Take $\lambda_1=\ldots=\lambda_p=1$ and let the other $\lambda_j$ tend to~$0$. Then we get \hbox{$\bbbone_A\Theta\wedge\ii du_1\wedge d\ovl u_1\wedge\ldots\wedge du_p\wedge d\ovl u_p\ge 0$.} This implies $\bbbone_A\Theta\ge 0$, or equivalently $\bbbone_AT\ge m_A[A]$. By Cor.~2.4 we know that $\bbbone_AT$ is a closed positive current, thus $\bbbone_AT=\lambda[A]$ with $\lambda\ge 0$. We have just seen that $\lambda\ge m_A$. On the other hand, $T\ge\bbbone_AT=\lambda[A]$ clearly implies $m_A\ge\lambda$.\qed \endproof \begstat{(8.16) Siu's decomposition formula} If $T$ is a closed positive current of bidimension~$(p,p)$, there is a unique decomposition of $T$ as a (possibly finite) weakly convergent series $$T=\sum_{j\ge 1}\lambda_j[A_j]+R,~~~~\lambda_j>0,$$ where $[A_j]$ is the current of integration over an irreducible $p$-dimensional ana\-lytic set $A_j\subset X$ and where $R$ is a closed positive current with the property that $\dim E_c(R)0$. \endstat \begproof{of uniqueness.} If $T$ has such a decomposition, the $p$-dimensional components of $E_c(T)$ are $(A_j)_{\lambda_j\ge c}$, for $\nu(T,x)=\sum\lambda_j\nu([A_j],x)+\nu(R,x)$ is non zero only on $\bigcup A_j\cup\bigcup E_c(R)$, and is equal to $\lambda_j$ generically on $A_j$ $\big($more precisely, $\nu(T,x)=\lambda_j$ at every regular point of $A_j$ which does not belong to any intersection $A_j\cup A_k$, $k\ne j$ or to $\bigcup E_c(R)\big)$. In particular $A_j$ and $\lambda_j$ are unique. \endproof \begproof{of existence.} Let $(A_j)_{j\ge 1}$ be the countable collection of $p$-dimensional components occurring in one of the sets $E_c(T)$, $c\in\bbbq_+^\star$, and let $\lambda_j>0$ be the generic Lelong number of $T$ along $A_j$. Then Prop.~8.16 shows by induction on $N$ that $R_N=T-\sum_{1\le j\le N}\lambda_j[A_j]$ is positive. As $R_N$ is a decreasing sequence, there must be a limit $R=\lim_{N\to+\infty}R_N$ in the weak topology. Thus we have the asserted decomposition. By construction, $R$ has zero generic Lelong number along $A_j$, so $\dim E_c(R)0$.\qed \endproof It is very important to note that some components of lower dimension can actually occur in $E_c(R)$, but they cannot be subtracted because $R$ has bidimension $(p,p)$. A typical case is the case of a bidimension \hbox{$(n-1,n-1)$} current $T=dd^cu$ with $u=\log(|F_j|^{\gamma_1}+\ldots|F_N|^{\gamma_N})$ and $F_j\in\cO(X)$. In general $\bigcup E_c(T)=\bigcap F_j^{-1}(0)$ has dimension~$0$. Let $(Z_k)_{k\ge1}$ be the irreducible components of $Z$ of codimension $p$ exactly. Then there exist multiplicities $\lambda_k>0$ such that $$(dd^cu)^p=\sum_{k\ge 1}\lambda_k[Z_k]+R,$$ where $R$ is a closed positive current such that $\bbbone_ZR=0$ and $\codim E_c(R)>p$ for every $c>0$. Moreover the multiplicities $\lambda_k$ are integers if $\gamma_1\ld \gamma_N$ are integers, and $\lambda_k=\gamma_1\ldots\gamma_p$ if $\gamma_1\le\ldots\le\gamma_N$ and some partial Jacobian determinant of $(F_1\ld F_p)$ of order $p$ does not vanish identically along~$Z_k$. \endstat \begproof{} Observe that $(dd^cu)^p$ is well defined thanks to Cor.~4.11. The comparison theorem~7.8 applied with $\varphi(z)=\log|z-x|$, $v_1=\ldots=v_p=u$, $u_1=\ldots=u_p=\varphi$ and $T=1$ shows that the Lelong number of $(dd^cu)^p$ is equal to $0$ at every point of~$X\ssm Z$. Hence $E_c((dd^cu)^p)$ is contained in $Z$ and its $(n-p)$-dimensional components are members of the family $(Z_k)$. The asserted decomposition follows from Siu's formula~8.16. We must have $\bbbone_{Z_k}R=0$ for all irreducible components of~$Z$: when $\codim Z_k>p$ this is automatically true, and when $\codim Z_k=p$ this follows from (8.16) and the fact that $\codim E_c(R)>p$. If $\det(\partial F_j/\partial z_k)_{1\le j,k\le p}\ne 0$ at some point $x_0\in Z_k$, then $(Z,x_0)=(Z_k,x_0)$ is a smooth germ defined by the equations $F_1=\ldots=F_p=0$. If we denote $v=\log\sum_{j\le p}|F_j|^{\gamma_j}$ with $\gamma_1\le\ldots\le\gamma_N$, then $u\sim v$ near $Z_k$ and Th.~7.8 implies $\nu((dd^cu)^p,x)=\nu((dd^cv)^p,x)$ for all $x\in Z_k$ near $x_0$. On the other hand, if $G:=(F_1\ld F_p):X\to\bbbc^p$, Cor.~7.4 gives $$(dd^cv)^p=G^\star\Big(dd^c\log\sum_{1\le j\le p}|z_j|^{\gamma_j}\Big)^p= \gamma_1\ldots\gamma_p\,G^\star\delta_0=\gamma_1\ldots\gamma_p\,[Z_k]$$ near~$x_0$. This implies that the generic Lelong number of $(dd^cu)^p$ along $Z_k$ is $\lambda_k=\gamma_1\ldots\gamma_p$. The integrality of $\lambda_k$ when $\gamma_1\ld\gamma_N$ are integers will be proved in the next section.\qed \endproof \titleb{9.}{Transformation of Lelong Numbers by Direct Images} Let $F:X\to Y$ be a holomorphic map between complex manifolds of respective dimensions $\dim X=n$, $\dim Y=m$, and let $T$ be a closed positive current of bidimension $(p,p)$ on~$X$. If $F_{\restriction\Supp\,T}$ is proper, the direct image $F_\star T$ is defined by $$\langle F_\star T,\alpha\rangle=\langle T,F^\star\alpha\rangle \leqno(9.1)$$ for every test form $\alpha$ of bidegree $(p,p)$ on $Y$. This makes sense because $\Supp\,T\cap F^{-1}(\Supp\,\alpha)$ is compact. It is easily seen that $F_\star T$ is a closed positive current of bidimension $(p,p)$ on $Y$. \begstat{(9.2) Example} \rm Let $T=[A]$ where $A$ is a $p$-dimensional irreducible analytic set in $X$ such that $F_{\restriction A}$ is proper. We know by Remmert's theorem~2.7.8 that $F(A)$ is an analytic set in~$Y$. Two cases may occur. Either $F_{\restriction A}$ is generically finite and $F$ induces an \'etale covering $A\ssm F^{-1}(Z)\lra F(A)\ssm Z$ for some nowhere dense analytic subset $Z\subset F(A)$, or $F_{\restriction A}$ has generic fibers of positive dimension and $\dim F(A)<\dim A$. In the first case, let $s<+\infty$ be the covering degree. Then for every test form $\alpha$ of bidegree $(p,p)$ on $Y$ we get $$\langle F_\star[A],\alpha\rangle=\int_A F^\star\alpha=\int_{A\ssm F^{-1}(Z)} F^\star\alpha=s\int_{F(A)\ssm Z}\alpha=s\,\langle[F(A)],\alpha\rangle$$ because $Z$ and $F^{-1}(Z)$ are negligible sets. Hence $F_\star[A]= s[F(A)]$. On the other hand, if $\dim F(A)<\dim A=p$, the restriction of $\alpha$ to $F(A)_\reg$ is zero, and therefore so is this the restriction of $F^\star\alpha$ to $A_\reg$. Hence $F_\star[A]=0$.\qed \endstat Now, let $\psi$ be a continuous plurisubharmonic function on $Y$ which is semi-exhaustive on $F(\Supp\,T)$ (this set certainly contains $\Supp F_\star T$). Since $F_{\restriction\Supp\,T}$ is proper, it follows that $\psi\circ F$ is semi-exhaustive on $\Supp\,T$, for $$\Supp\,T\cap\{\psi\circ F0\,;\,\exists C>0,\,|H(z)|\ge C|z-x|^\alpha~ \hbox{\rm near}~x\big\},&({\rm b})\cr \ovl\mu_p(F,x)&=\inf_G{\sigma(G\circ F,x)^p\over\mu_p(G,0)},&({\rm c})\cr}$$ where $G$ runs over all germs of maps $(Y,y)\lra(\bbbc^n,0)$ such that $G\circ F$ is finite. \endstat \begproof{} If $F^{-1}(y)=\{x_1\ld x_N\}$, there is a neighborhood $W$ of $y$ and disjoint neighborhoods $V_j$ of $x_j$ such that $F^{-1}(W)=\bigcup V_j$. Then $F_\star T=\sum(F_{\restriction V_j})_\star T$ on $W$, so it is enough to consider the case when $F^{-1}(y)$ consists of a single point~$x$. Therefore, we assume that $F:V\to W$ is proper and finite, where $V$, $W$ are neighborhoods of $0$ in $\bbbc^n$, $\bbbc^m$ and $F^{-1}(0)=\{0\}$. Let \hbox{$G:(\bbbc^m,0)\lra(\bbbc^n,0)$} be a germ of map such that $G\circ F$ is finite. Hilbert's Nullstellensatz shows that there exists $\alpha>0$ and $C>0$ such that $|G\circ F(z)|\ge C|z|^\alpha$ near~$0$. Then the comparison theorem~7.1 implies $$\nu(G_\star F_\star T,0)=\nu(T,\log|G\circ F|)\le\alpha^p\nu(T,\log|z|)= \alpha^p\nu(T,0).$$ On the other hand, Th.~9.9 applied to $\Theta=F_\star T$ on $W$ gives $$\nu(G_\star F_\star T,0)\ge\mu_p(G,0)\,\nu(F_\star T,0).$$ Therefore $$\nu(F_\star T,0)\le{\alpha^p\over\mu_p(G,0)}\nu(T,0).$$ The infimum of all possible values of $\alpha$ is by definition $\sigma(G\circ F,0)$, thus by taking the infimum over $G$ we obtain $$\nu(F_\star T,0)\le\ovl\mu_p(F,0)\,\nu(T,0).\eqno{\square}$$ \endproof \begstat{(9.13) Example} \rm Let $F(z_1\ld z_n)=(z_1^{s_1}\ld z_n^{s_n})$, $s_1\le\ldots\le s_n$ as in~9.8. Then we have $$\mu_p(F,0)=s_1\ldots s_p,~~~~~~\ovl\mu_p(F,0)=s_{n-p+1}\ldots s_n.$$ To see this, let $s$ be the lowest common multiple of $s_1\ld s_n$ and let $G(z_1\ld z_n)=(z_1^{s/s_1}\ld z_n^{s/s_n})$. Clearly $\mu_p(G,0)=(s/s_{n-p+1})\ldots(s/s_n)$ and $\sigma(G\circ F,0)=s$, so we get by definition $\ovl\mu_p(F,0)\le s_{n-p+1}\ldots s_n$. Finally, if $T=[A]$ is the current of integration over the $p$-dimensional subspace $A=\{z_1=\ldots=z_{n-p}=0\}$, then $F_\star[A]= s_{n-p+1}\ldots s_n\,[A]$ because $F_{\restriction A}$ has covering degree $s_{n-p+1}\ldots s_n$. Theorem~9.12 shows that we must have $s_{n-p+1}\ldots s_n\le\ovl\mu_p(F,0)$, QED. If $\lambda_1\le\ldots\le\lambda_n$ are positive real numbers and $s_j$ is taken to be the integer part of $k\lambda_j$ as $k$ tends to $+\infty$, Theorems~9.9 and 9.12 imply in the limit the following: \endstat \begstat{(9.14) Corollary} For $0<\lambda_1\le\ldots\le\lambda_n$, Kiselman's directional Lelong numbers satisfy the inequalities $$\lambda_1\ldots\lambda_p\,\nu(T,x)\le\nu(T,x,\lambda)\le \lambda_{n-p+1}\ldots\lambda_n\,\nu(T,x).\eqno{\square}$$ \endstat \begstat{(9.15) Remark} \rm It would be interesting to have a direct geometric interpretation of~$\ovl\mu_p(F,x)$. In fact, we do not even know whether $\ovl\mu_p(F,x)$ is always an integer. \endstat \titleb{10.}{A Schwarz Lemma. Application to Number Theory} In this section, we show how Jensen's formula and Lelong numbers can be used to prove a fairly general Schwarz lemma relating growth and zeros of entire functions in~$\bbbc^n$. In order to simplify notations, we denote by $|F|_r$ the supremum of the modulus of a function $F$ on the ball of center $0$ and radius~$r$. Then, following (Demailly~1982a), we present some applications with a more arithmetical flavour. \begstat{(10.1) Schwarz lemma} Let $P_1\ld P_N\in\bbbc[z_1\ld z_n]$ be polynomials of degree~$\delta$, such that their homogeneous parts of degree $\delta$ do not vanish simultaneously except at~$0$. Then there is a constant \hbox{$C\ge 2$} such that for all entire functions $F\in\cO(\bbbc^n)$ and all $R\ge r\ge 1$ we have $$\log|F|_r\le\log|F|_R-\delta^{1-n}\nu([Z_F],\log|P|)\,\log{R\over Cr}$$ where $Z_F$ is the zero divisor of $F$ and $P=(P_1\ld P_N):\bbbc^n\lra \bbbc^N$. Moreover $$\nu([Z_F],\log|P|)\ge\sum_{w\in P^{-1}(0)}\ord(F,w)\,\mu_{n-1}(P,w)$$ where $\ord(F,w)$ denotes the vanishing order of $F$ at $w$ and $\mu_{n-1}(P,w)$ is the $(n-1)$-multiplicity of $P$ at $w$, as defined in $(9.5)$ and $(9.7)$. \endstat \begproof{} Our assumptions imply that $P$ is a proper and finite map. The last inequality is then just a formal consequence of formula (9.4) and Th.~9.9 applied to~$T=[Z_F]$. Let $Q_j$ be the homogeneous part of degree $\delta$ in~$P_j$. For~$z_0\in B(0,r)$, we introduce the weight functions $$\varphi(z)=\log|P(z)|,~~~~~~\psi(z)=\log|Q(z-z_0)|.$$ Since $Q^{-1}(0)=\{0\}$ by hypothesis, the homogeneity of $Q$ shows that there are constants $C_1,C_2>0$ such that $$C_1|z|^\delta\le|Q(z)|\le C_2|z|^\delta~~~~\hbox{\rm on}~~\bbbc^n. \leqno(10.2)$$ The homogeneity also implies $(dd^c\psi)^n=\delta^n\,\delta_{z_0}$. We apply the Lelong Jensen formula~6.5 to the measures $\mu_{\psi,s}$ associated with $\psi$ and to~$V=\log|F|$. This gives $$\mu_{\psi,s}(\log|F|)-\delta^n\log|F(z_0)|=\int_{-\infty}^sdt\int_{\{\psi\psi$ near $\{\psi=t-2\}$ and $(1-\varepsilon)\varphi+\varepsilon t-\varepsilon<\psi$ near $\{\psi=t\}$. Then Stokes' theorem implies $$\leqalignno{ \int_{\{\psi-C_5re^{(2-t)/\delta}+\varepsilon$ on $\{\psi=t-2\}$. Hence it is sufficient to take $\varepsilon=C_5re^{(2-t)/\delta}$. This number has to be $<1$, so we take $t\ge s_0\ge 2+\delta\log(C_5r)$. Moreover, (10.5) actually holds only if $P^{-1}(0)\subset\{\psi0$ a number $\omega_t(S)$ equal to the minimal degree of polynomials $P\in\bbbc[z_1\ld z_n]$ which vanish at order $\ge t$ at every point of~$S$. The obvious subadditivity property $$\omega_{t_1+t_2}(S)\le\omega_{t_1}(S)+\omega_{t_2}(S)$$ easily shows that $$\Omega(S):=\inf_{t>0}{\omega_t(S)\over t}= \lim_{t\to+\infty}{\omega_t(S)\over t}.$$ We call $\omega_1(S)$ the {\it degree} of $S$ (minimal degree of algebraic hypersurfaces containing~$S$) and $\Omega(S)$ the {\it singular degree} of~$S$. If we apply Cor.~10.6 to a polynomial $F$ vanishing at order $t$ on $S$ and fix $r=1$, we get $$\log|F|_R\ge t{\delta+n(n-1)/2\over n!}\log{R\over C}+\log|F|_1$$ with $\delta=\omega_1(S)$, in particular $${\rm deg}\,F\ge t{\omega_1(S)+n(n-1)/2\over n!}.$$ The minimum of ${\rm deg}\,F$ over all such $F$ is by definition~$\omega_t(S)$. If we divide by $t$ and take the infimum over~$t$, we get the interesting inequality $${\omega_t(S)\over t}\ge\Omega(S)\ge{\omega_1(S)+n(n-1)/2\over n!}. \leqno(10.7)$$ \begstat{(10.8) Remark} \rm The constant ${\omega_1(S)+n(n-1)/2\over n!}$ in (10.6) and (10.7) is optimal for $n=1,2$ but not for~$n\ge3$. It can be shown by means of H\"ormander's $L^2$ estimates (Waldschmidt 1978) that for every $\varepsilon>0$ the Schwarz lemma (10.6) holds with coefficient $\Omega(S)-\varepsilon\,$: $$\log|F|_r\le\log|F|_R-\ord(F,S)(\Omega(S)-\varepsilon)\log{R\over C_\varepsilon r},$$ and that $\Omega(S)\ge(\omega_u(S)+1)/(u+n-1)$ for every $u\ge 1\,$; this last inequality is due to (Esnault-Viehweg 1983), who used deep tools of algebraic geometry; (Azhari 1990) reproved it recently by means of H\"ormander's $L^2$ estimates. Rather simple examples (Demailly 1982a) lead to the conjecture $$\Omega(S)\ge{\omega_u(S)+n-1\over u+n-1}~~~~\hbox{for every}~~u\ge 1.$$ The special case $u=1$ of the conjecture was first stated by (Chudnovsky 1979). \endstat Finally, let us mention that Cor.~10.6 contains Bombieri's theorem on algebraic values of meromorphic maps satisfying algebraic differential equations (Bombieri 1970). Recall that an entire function $F\in\cO(\bbbc^n)$ is said to be of order $\le\rho$ if for every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that $|F(z)|\le C_\varepsilon\exp( |z|^{\rho+\varepsilon})$. A meromorphic function is said to be of order $\le\rho$ if it can be written $G/H$ where $G$, $H$ are entire functions of order $\le\rho$. \begstat{(10.9) Theorem {\rm(Bombieri 1970)}} Let $F_1\ld F_N$ be meromorphic functions on $\bbbc^n$, such that $F_1\ld F_d$, $n0$, there exist constants $C_1,C_2>0$, $r\ge 1$ and an infinite sequence $G_t$ of entire functions, $t\in T\subset \bbbn$ $($depending on $m$ and on the choice of the points $w_j)$, such that \smallskip \item{\rm a)} $G_t$ vanishes at order $\ge t$ at all points $w_1\ld w_m\,;$ \smallskip \item{\rm b)} $|G_t|_r\ge (C_1t)^{-t\,[K:\bbbq]}\,;$ \smallskip \item{\rm c)} $|G_t|_{R(t)}\le C_2^t$~~ where $R(t)=(t^{d-n}/\log t)^{1/(\rho_1+\ldots+\rho_d+\varepsilon)}$. \smallskip \endstat An application of Cor.~10.6 to $F=G_t$ and $R=R(t)$ gives the desired bound for the degree $\delta$ as $t$ tends to~$+\infty$ and $\varepsilon$ tends to~$0$. If $\delta_0$ is the largest integer which satisfies the inequality of Th.~10.9, we get a contradiction if we take $\delta=\delta_0+1$. This shows that $S$ must be contained in an algebraic hypersurface of degree $\delta\le\delta_0$.\qed \end