\magnification=1200 \vsize=20cm\voffset=3mm\hsize=13.5cm\hoffset=-0.1mm \parindent=0mm \parskip=6pt plus 2pt minus 2pt \input macros_ocr.tex %% Further macros \let\leq=\leqslant \let\geq=\geqslant \def\alg{{\mathop{\rm alg}}} \def\anal{{\mathop{\rm anal}}} \def\loc{{\mathop{\rm loc}}} \def\reg{{\mathop{\rm reg}}} \def\sing{{\mathop{\rm sing}}} \def\rang{\mathop{\rm rang}} \def\ord{\mathop{\rm ord}} \def\deg{\mathop{\rm deg}} \def\diam{\mathop{\rm diam}} \def\tr{\mathop{\rm tr}} \def\Ker{\mathop{\rm Ker}} \def\Im{\mathop{\rm Im}} \def\Hom{\mathop{\rm Hom}} \def\Herm{\mathop{\rm Herm}} \def\Ker{\mathop{\rm Ker}} \def\Im{\mathop{\rm Im}} \def\VM{\mathop{\rm VM}} \def\Supp{\mathop{\rm Supp}} \def\Ricci{\mathop{\rm Ricci}} \def\Langle{\langle\!\langle} \def\Rangle{\rangle\!\rangle} \def\?{\kern0.5pt} \def\longarrow#1{{\raise2.3pt\hbox to#1mm{\hrulefill}\kern-5pt\rightarrow}} \def\frac#1#2{{#1\over #2}} \font\goth=eufm10 \def\gm{\hbox{\goth m}} \footline{\hfill} \headline{\ifnum \pageno=1 \hfil\else \ifodd \pageno {\sevenbf \hfil Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines\hfil\folio} \else {\sevenbf \folio\hfil J.-P.\ Demailly, Mémoire de la Société Mathématique de France, n$\scriptstyle{}^\circ$19, 1985\hfil} \fi\fi} \def\statement{\bf} \def\bibitem#1{\item{\rlap{[#1]}\kern1cm}} {\sevenrm\baselineskip=8pt Mémoire de la Société Mathématique de France, n$\scriptstyle{}^\circ$19\\ Supplément au Bulletin de la S.M.F.\\ Tome 113, 1985, fascicule 2 \vskip1.5cm} \centerline{\hugebf Mesures de Monge-Ampère et} \medskip \centerline{\hugebf caractérisation géométrique des} \medskip \centerline{\hugebf variétés algébriques affines} \bigskip \centerline{par Jean-Pierre Demailly} \medskip \centerline{{\it Universit\'e de Grenoble I, Institut Fourier}} \centerline{{\it Laboratoire de Math\'ematiques associ\'e au C.N.R.S.\ n$^\circ$188}} \centerline{{\it BP 74, F-38402 Saint-Martin d'H\`eres, France}} \vskip1cm {\leftskip=9mm\rightskip=9mm {\bf Résumé.} À toute fonction d'exhaustion plurisousharmonique continue $\varphi$ sur un espace de Stein, nous associons une collection de mesures positives portées par les surfaces de niveau de $\varphi$, et définies à l'aide des opérateurs de Monge-Ampère au sens de Bedford et Taylor. Nous montrons que ces mesures jouent un rôle fondamental dans l'étude des propriétés de croissance et de convexité des fonctions plurisousharmoniques ou holomorphes. Lorsque le volume de Monge-Ampère de la variété est fini, un théorème d'algébricité de type Siegel s'applique aux fonctions holomorphes à croissance $\varphi$-polynomiale. Nous en déduisons que la finitude du volume de Monge-Ampère, associée à une minoration convenable de la courbure de Ricci, est une condition géométrique nécessaire et suffisante caractérisant les variétés algébriques affines. \medskip {\bf Abstract.} To every continuous plurisubharmonic exhaustion function $\varphi$ on a Stein space, we associate a collection of positive measures with support in the level sets of $\varphi$, defined by means of the Monge-Ampère operators in the sense of Bedford and Taylor. We show that these measures play a prominent part in the study of growth and convexity properties of plurisubharmonic or holomorphic functions. When the variety has finite Monge-Ampère volume, an algebraicity theorem of Siegel type holds for holomorphic functions with $\varphi$-polynomial growth. From this result, we deduce that the finiteness of Monge-Ampère volume, together with a suitable lower bound of the Ricci curvature, is a necessary and sufficient geometric condition characterizing affine algebraic varieties.\par} \footnote{}{\sevenrm 0037-9484/85 02 1 \$ 14.50 {\tt\copyright} Gauthier-Villars} \bigskip\bigskip \vfill\eject \strut\vskip1cm {\hugebf Table of contents} \bigskip {\bf 0. Introduction\dotfill~3} \medskip {\parindent=7.5mm\bigbf\baselineskip=15pt \item{A.} Measures of Monge-ampère and growth of the functions\\ plurisousharmoniques\dotfill~8\par} \smallskip 1. Common and functions plurisousharmoniques on the complex spaces\dotfill~8 2. Operators $(dd^c)^k$ and inequalities of Chern-Levine-Nirenberg\dotfill~14 3. Measures of Monge-Ampère and formula of Jensen\dotfill~21 4. Residual measure of $(dd^c\varphi)^n$ on $S(-\infty)$\dotfill~26 5. Principle of the maximum\dotfill~29 6. Ownerships of convexité of the functions psh\dotfill~31 7. Growth at the infinite of the functions psh\dotfill~37 8. Functions holomorphes $\varphi$ -polynomiales\dotfill~40 \bigskip {\parindent=7.5mm\bigbf\baselineskip=15pt \item{B.} Geometrical characterisation of the varieties\\ algébriques affines\dotfill~45\par} \smallskip 9.~Énoncé Of the criterion of algébricité\dotfill~45 10. Need of the conditions on the volume and the courbure\dotfill~48 11. Existence of a plongement on an open of a variety algébrique\dotfill~51 12. Quasi-surjectivité Of the plongement\dotfill~57 13. Démonstration of the criterion of algébricité (case lisse)\dotfill~65 14. Algébricité Of the complex spaces singular\dotfill~71\par {\parindent=6.5mm \item{15.} Appendix~: common and functions plurisousharmoniques at\\ minimum growth on a variety algébrique affine\dotfill~72\par} \bigskip {\bigbf Bibliographie\dotfill~82} \vfill\eject \section{0}{Introduction.} The present studio place se on the general frame of the spaces complex analytique. The first section is so consacrer a definition of the forms différentielles, common positive and functions plurisousharmoniques on a complex space $X$ éventuellement singular. Having given a plongement local of $X$ on an open $\Omega\subset\bC^N$, define prpers the forms différentielles on $X$ comme the restrictions at $X$ of the forms\lguil\?ambiantes\?\rguil\ On~$\Omega$~; the spaces of common deduce by duality comme on the case lisse.\medskip {\statement Definition 0.1.\pointir}{\it Be a function $V:X\to [-\infty,+\infty[$. {\parindent=6.5mm \vskip2pt \item{\rm(a)} $V$ have said plurisousharmonique $($ psh at abbreviated\/$)$ on $X$ if $V$ is locally restriction at $X$ of functions psh on the space ambiant~$\bC^N$. \vskip2pt \item{\rm(b)} $V$ have said faiblement psh if $V$ is locally intégrable and majorée on $X$, and if $dd^cV\geq 0$.\vskip2pt} Notation~$:$ states on here \vskip5pt \centerline{$\displaystyle d^c=i(\overline\partial-\partial),\quad\hbox{de sorte que}\quad dd^c=2i\,\partial\overline\partial.$}\medskip} All function psh is then faiblement psh, but at general a function faiblement psh no identifies se necessarily presque all around at a function psh. Display Prpers however that both coïncident notion when the space $X$ is locally irréductible. The démonstration of that outcome takes usage of deux ingredients : on the one hand the characterisation of the functions psh had to at Fornaess and Narasimhan [FN], on the other hand a théorème of elongation of the functions psh limited across the singular place of $X$ (who uses the resolution of the singularities). Study Prpers equally the transformation of the common positive closed and of the functions psh by own direct image. On the\S2, restart prpers essentially the developed method by Bedford and Taylor [BT2] for give an au courant positive sense $dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$ when the $\varphi_j$ are of the functions psh locally limited, and prpers generalise prpers au cas où the an of the functions (be $\varphi_i$ by example) no is locally limited. The classical inequalities of Chern-Levine-Nirenberg pouvoir then se énoncer comme tracks~: \medskip {\statement Théorème 0.2.\pointir}{\it For all open $\omega\compact X$ and all compact $K\subset\omega$ il there is of the constantes $C_1$, $C_2$ no dépendant that of $\omega$ and $K$ tel what have on the majorations of following mass~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle\int_K\Vert dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k\Vert\le C_1\Vert\varphi_1\Vert_{L^1(\omega)}\Vert\varphi_2\Vert_{L^\infty(\omega)}\ldots \Vert\varphi_k\Vert_{L^\infty(\omega)},$ \vskip2pt \item{\rm(b)} $\displaystyle\int_K\Vert \varphi_1\, dd^c\varphi_2\wedge\ldots\wedge dd^c\varphi_k\Vert\le C_2\Vert\varphi_1\Vert_{L^1(\omega)}\Vert\varphi_2\Vert_{L^\infty(\omega)}\ldots \Vert\varphi_k\Vert_{L^\infty(\omega)}.$\medskip}} Display Prpers finally on that situation the continuity séquentielle of the operators of Monge-Ampère $$ (\varphi_1,\ldots,\varphi_k)\longmapsto dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k\quad\hbox{et}\quad \varphi_1\,dd^c\varphi_2\wedge\ldots\wedge dd^c\varphi_k $$ for of the continuations décroissantes $\varphi_1^\nu,\ldots,\varphi_k^\nu$ of functions psh . Assumes On maintenant que $X$ is a space of Stein and that $X$ has catered of a function psh continuous exhaustive $\varphi:X\to[-\infty,R[$. Note Prpers then $$ B(r)=\big\{z\in X\,;\;\varphi(z) < r\big\},\qquad S(r) =\big\{z \in X\,;\; \varphi(z) = r\big\},\qquad r\in [-\infty,R[ $$ the\lguil\?pseudoboules\?\rguil\ And\lguil\?pseudosphères\?\rguil\ Associated at $\varphi$. HAS that data, display prpers what pouvoir on associate of natural way a collection of positive measures $\mu_r$, scopes by the sphère~$S(r)$, that urge prpers measures of Monge-Ampère associated at~$\varphi$. These have defined simply by $$ \mu_r(h) = \int_{S(r)} h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi,\qquad n = \dim X, $$ when $\varphi$ is of class $\cC^2$ and when $r$ is regular courage of~$\varphi$. Dans le cas où $\varphi$ is alone continuous, on has brought at use the definition of Bedford-Taylor for $(dd^c)^n$ and at state $$ \mu_r = \big(dd^c\max(\varphi,r)\big)^n-\bOne_{X\ssm B(r)}(dd^c\varphi)^n. $$ has On then a general formula of type Jensen-Lelong, whose démonstration is immediate consequences of the théorèmes of Stokes and of Fubini (cf.~\S3). \medskip {\statement Théorème 0.3.\pointir}{\it All function psh $V$ on $X$ is $\mu_r$ -intégrable quel that be $r < R$, and has on the formula $$ \int_{-\infty}^r dt\int_{B(t)}dd^cV\wedge(dd^c\varphi)^{n-1} = \mu_r(V) - \int_{B(r)}V(dd^c\varphi)^n. $$} Displays On besides that the measures $\mu_r$ dépendent continûment of $\varphi$ relatively at the continuations décroissantes. This allows see comme on the case $\cC^\infty$ that the family $(\mu_r)$ is the family of measures faiblement continues at left who désintègre the positive current $(dd^c\varphi)^{n-1}\wedge d\varphi\wedge d^c\varphi$ on the spheres~$S(r)$. The measures $\mu_r$ thus built jouir of a some number of of note natural ownership for the studio of the growth and of the convexité of the functions psh . The paragraph 4 studies the measure\lguil\?Residual\?\rguil\ $\mu_{-\infty}=\bOne_{S(-\infty)}(dd^c\varphi)^n$, scope by the at\-sem\-ble polar $S(-\infty)$. As (0.3), the measure $\mu_{-\infty}$ pouvoir also define comme the feeble limit of~$\mu_r$, when $r$ extends to~$-\infty$. Inspire of our former works [Of4,~De5] , display prpers that the measure $\mu_{-\infty}$ no depends essentially that of the behaviour asymptotique of $\varphi$ at the voisinage of $S(-\infty)$. Of that outcome découle the classical inequality $$ (dd^\varphi)^n\geq 2^n\sum_{x\in X}\nu(\varphi,x)^n\,\delta_x, \leqno(0.4) $$ where $\nu(\varphi,x)$ designates the number of Lelong of $\varphi$ entirely ~$x\in X$, and $\delta_x$ the measure of Dirac at $x$ (at a singular dot~$x$, that measure owes stand counted with equal multiplicity at the multiplicity of $X$ at~$x$ ). At the\S5 , display prpers that the measures $\mu_r$ check the principle of the maximum live at live of the functions psh, at know that for all function psh $V$ has on the equality : $$ \sup_{B(r)}V = \hbox{\rm sup essentiel de $V$ relativement à $\mu_r$}. \leqno(0.5) $$ The remarquable fact is that the equality has place although the support of $\mu_r$ pouvoir stand very lacunaire on $S(r)$, comme par exemple dans le cas où the pseudoboules $B(r)$ are of the polyèdres analytique. The paragraph 6 generalises at the present situation the ownerships of convexité classical owed at P. Lelong, relative at the half of functions psh on the bowls, sphère, polydisques~$\ldots$~. Display Prpers that the geometrical hypothesis natural who under-extends the validité of the ownerships of convexité is the fact that the function $\varphi$ be solution of the equation of Monge-Ampère homogène $(dd^c\varphi)^n\equiv 0$. Of precise manner~:\medskip {\statement Théorème 0.6.\pointir}{\it Assumes On that $(dd^c\varphi)^n\equiv 0$ on the open $\{\varphi > A\}$. Be $V$ a function psh on $X$. Then the sup of $V$ on $B(r)$, the half $\mu_r(V)$, and best generally the half at norm $L^p$, $r\mapsto[\mu_r(V_+^p)]^{1/p}$, are functions convexes increasing of $r\in{}]A,R]$.} \medskip The vérification of that outcome obtains se by of the elementary calculations of derived seconds, taking take part the formula of Jensen 0.3 and the théorèmes of Stokes and of Fubini. Best generally, show prpers a version with\lguil\?Parameter\?\rguil\ Of the théorème 0.6, relative at the measures $\mu_{y,r}$ on the fibres $\pi^{-1}(y)$ of a fibration holomorphe $\pi:X\to Y$. The function psh $\varphi$ data on $X$ has assumed exhaustive on the fibres and tel that $(dd^c\varphi)^n\equiv 0$ on the open $\{\varphi > A\}$, where $n$ is the dimension of the fibres. Then the half $\mu_{y,r}(V)$ and the half at norm $L^p$ are functions faiblement psh of the couple $(y, z)$ on $Y\times\bC$, if states on $r = \Re z$. Tirer easily the following extension of the théorème 0.6 at the produced spaces. \medskip {\statement Théorème 0.7.\pointir}{\it Are $X_1,\ldots,X_k$ of the spaces of Stein, catered of functions psh continuous exhaustive $\varphi_j:X_j\to[-\infty,R_j[$ tel that $(dd^c\varphi_j)^{n_j}\equiv 0$ on the open $\{\varphi_j>A_j\}$, $n_j = \dim X_j$. Then, if $V$ is psh on $X_1\times\cdots\times X_k$, the half at norm $L^p$ $$ M^p_V(r_1,\ldots,r_k) = \smash{ \Big[\mu_{r_1}\otimes\cdots\otimes\mu_{r_k}(V^p_+)\Big]^{1/p}} $$ is convexe simultaneously at the variables $(r_1,\ldots,r_k)\in \prod_{1\leq j\leq k}{}]A_j,R_j[$.} \medskip On the paragraphs 7 and 8 , take prpers the additional hypothesis that the volume of $X$ is at moderated growth at the infinite (the \lguil\?Ray\?\rguil\ $R$ Has assumed here equal at~$+\infty$ ). Of precise manner, assume prpers that $$ \lim_{r\to+\infty}{1\over r}\Vert\mu_r\Vert = 0 . \leqno(0.8) $$ Under that hypotheses, the formula of Jensen 0.3 involves the fundamental inequality following~: $$ \int_Xdd^cV\wedge(dd^c\varphi)^{n-1}\leq \liminf_{r\to+\infty}~{1\over r}\;\mu_r(V_+), \leqno(0.9) $$ of which découle a some number of outcomes in regard to the growth of the functions psh or the distribution of the courages of the functions holomorphes (comme suggest prpers the article\eject of N.~Sibony And P.M. Wong [SW]). At particular, all function psh or holomorphe limited on $X$ is constante. Having given a function holomorphe $f$ on $X$, define prpers on the other hand the\lguil\?Grade\?\rguil\ Of $f$ relatively at $\varphi$ by $$ \delta_\varphi(f) = \limsup_{r\to+\infty}~{1\over r}\;\mu_r(\log_+|f|),\qquad \leqno(0.10) $$ % and say prpers that $f$ is $\varphi$ -polynomiale if $\delta_\varphi(f)$ has ended. The inequality (0.9) trains whereas the order of cancellation of $f$ at a regular dot $a\in X$ checks the assessment : $$ \ord\nolimits_a(f) \leq C(a)\,\delta_\varphi(f). $$ By an elementary reasoning of algèbre linear owed at Siegel, prpers results the théorème of algébricité following (assumes on $X$ irréductible). \medskip {\statement Théorème 0.11.\pointir}{\it Be $K_\varphi(X)$ the body of the functions méromorphes of the form $f/g$ where $f$ and $g$ are $\varphi$ -polynomiales. Then~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $0\leq\deg\tr_\bC K_\varphi(X)\leq \dim_\bC X~;$ \vskip2pt \item{\rm(b)} If $\deg\tr_\bC K_\varphi(X) = \dim_\bC X$, then the body $K_\varphi(X)$ is of ended type.\medskip}} Comme particular case of that théorème, find prpers the outcome of W.~Stoll [St1] characterising the varieties algébriques on $\bC^N$ by the ownership that the growth of the area is minimum. The deuxième part B of that work has consacrer a characterisation of the varieties algébriques affines by a geometrical criterion \lguil\?intrinsèque\rguil, taking take part the finitude of the volume of Monge-Ampère and a minoration of the courbure of Ricci. Of precise manner, show prpers the following outcome : \medskip {\statement Théorème 0.12.\pointir}{\it Be $X$ a variety complex analytique, lisse, connexe, of dimension~$n$. Then $X$ is analytiquement isomorphe at a variety algébrique affine $X_\alg$ if and alone if $X$ checks the condition{\rm(c)} here-below and if $X$ owns a function of exhaustion $\varphi$ strictly psh of class $\cC^\infty$ tel who : {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle\Vol(X) = \int_X (dd^c\varphi)^n <+\infty~;$ \vskip2pt \item{\rm(b)} The courbure of Ricci of the métrique $\beta= dd^c(e^\varphi)$ admits a minoration of the form $$ \Ricci(\beta)\geq-{1\over 2}dd^c\psi, $$ with $\psi\in\cC^\infty(X,\bR)$, where $\psi\leq A\varphi + B$ and $A$, $B$ are of the constantes${}\geq 0~;$ \vskip2pt \item{\rm(c)} The spaces of cohomologie of grade pair $H^{2q}(X,\bR)$ are of ended dimension.\vskip2pt} The ring of the regular functions of the structure algébrique $X_\alg$ is then given by the intersection $K_\varphi(X)\cap\cO(X)$.\medskip} As a result of the works of W.~Stoll On the varieties strictly paraboliques (cf.\ [St2] and [Drunk]), D.~Burns Has stated the problem of the characterisation of the varieties algébriques affines at terms of functions of exhaustion having of the particular ownerships, checking by example the condition of homogeneity $(dd^c\varphi)^n\equiv 0$ out of a compact. The théorème 0.12 contributes a partial answer at that problem. Inscrire se on the other hand on the surname of the sufficient conditions obtained by Mok, Siu and Yau [SY], [MSY], [Mok1,2,3], although our hypotheses are noticeably different of those of the works précédemment cities. Our argumentation is of elsewhere analogous on her big lines at the tracked démarche by [Mok1,2,3]. The paragraph 10 shows the need of the conditions 0.12~(has,b,c) for all ensemble algébrique $X\subset\bC^N$. The function $\varphi$ is then given by $\varphi(z) = \log(1+|z|^2)$, so that the métrique $dd^c\varphi$ coincide with the métrique of Fubini-Study of the projectif space~$\bP^N$. Thanks to an explicit calculation of the courbure of Ricci, check prpers that the inequality of courbure (b) has place with $\psi = \log\sum_{K,L}|J_{K,L}|^2$, where the $J_{K,L}$ designate the déterminant jacobiens associé at a system of equations polynomiales of~$X$. Display Prpers besides by a contre-example that the condition of courbure (b) is very indispensable. The test of the suffisance of the conditions (has,b,c) takes the object of the\S11,12,13 . The general schéma of the démonstration is the following, Thanks to the assessments $L^2$ of Hörmander-Nakano-Skoda for the operator $\overline\partial$ and thanks to the hypothesis~(b), builds on a system $F =(f_1,\ldots,f_N)$ of functions holomorphes $\varphi$ -polynomiales who, out of an ensemble analytique $S\subset X$, defines a plongement of $X\ssm S$ on~$\bC^N$. Under the hypothesis (a) of finitude of the volume, the théorème of algébricité 0.11 involves that the grade of transcendence of the functions $f_1,\ldots,f_N$ is equal at $n = \dim X$. The morphisme $F$ forward therefore $X$ on a variety algébrique $M\subset\bC^N$ of dimension~$n$. The principal difficulty who subsister is then of test that the plongement is quasi-surjectif, c'est-à-dire that the open $\Omega= F(X\ssm S)$ is an open of Zariski of~$M$. Obtain Prpers that outcome displaying firstly that the $(1,1)$ -form $F_*(dd^c\varphi)$ prolonger se at a positive current closed $T$ of ended mass on~$M$, as $T = 0$ on $M\ssm\Omega$~; compte tenu de the assessments of mass who result of the building, this takes se essentially by the method of integration by developed parts by H.~Skoda [Sk5] and H.~El Mir [EM]. Afin de give a perception of the continuation of the reasoning, watch the already significant case where $N = n$, i.e.\ The case $M =\bC^n$. Il there is then a function psh $V$ on $\bC^n$ at minimum growth, i.e.\ $V(z)\leq C_1\log_+|z|+C_0$, tel that $dd^cV = T$. By building, the function $\tau = V - \varphi \circ F^{-1}$ is pluriharmonique on~$\Omega$, besides $\tau$ extends to $-\infty$ entirely of~$\partial\Omega$. Prpers results that the ensemble closed $M\ssm\Omega$ is pluripolaire. For display that $M\ssm\Omega$ is in fact an ensemble algébrique, our method consister check, using again the théorème of algébricité 0.11, that the $1$ -form holomorphe $h = \partial\tau$ prolonger se at a form méromorphe rational on $M = \bC^n$. Dans le cas où $M$ is a variety algébrique affine quelconque, the tie who exists on $\bC^n$ amid the $(1,1)$ -common positive closed $T$ of end projectif mass and the functions psh at minimum growth no applies se best. Pouvoir On however display (see the appendix\S15) that the inéquation différentielle $dd^c V\geq T$ resolves se always on $M$ with a solution psh $V$ tel that run $dd^cV - T$ be of class $\cC^\infty$ and at growth polynomiale. The end of the démonstration is then presque identical. The hypothesis (c), regarding prpers, serves at show that the \lguil\?topologie Of Zariski\?\rguil\ On $X$ is quasi-compact, and so that $X$ peut être recouvert by an ended number of opened of the form $X\ssm S$ (cf.\\S13). Prpers no know in fact if the hypotheses (c) is really indispensable. Signal lastly that the théorème 0.12 pouvoir stretch at the complex spaces at singu\-larités isolated (cf.\\S9), but the extension at the case quelconque raise of the difficulties who have studied at the\S14. \vskip1.5cm {\hugebf HAS. Measures of Monge-Ampère and\vskip0pt \strut\phantom{A.}croissance Of the functions plurisousharmoniques.\par} \vskip7mm \section{1}{Courants and functions plurisousharmoniques\\ on the complex spaces.} The aim of that paragraph is of give a definition of the common and of the functions psh on a space complex analytique éventuellement singular. The reader who wishes no consider on the continuation that the case lisse pouvoir spring straight at the\S2. Be $X$ a complex space reduced of pure dimension~$n$, $X_\reg$ (resp.\ $X_\sing$ ) The ensemble of her regular dots (resp.\ Singular). Comme the definitions that go prpers consider are local, pouvoir on without restriction assume that $X$ identifies se at an under-ensemble closed analytique of an open $\Omega\subset\bC^N$ at the half of a plongement $j:X\to\Omega$. Defines On then the space $\cC^k_{p,q}(X)$ of the $(p,q)$ -forms of class $\cC^k$ on $X$, $k\in\bN\cup\{\infty\}$, comme the image of the morphisme of restriction $$ j* :\cC^k_{p,q}(\Omega)\to\cC^k_{p,q}(X_\reg), $$ catered of the topologie quotient. If $j_1:X\to\Omega_1\subset\bC^{N_1}$ is other plongement, il there is (locally) of the applications holomorphes $f:\Omega\to\bC^{N_1}$ and $g:\Omega_1\to\bC^N$ tel that $j_1= f\circ j$ and $j = g\circ j_1$. The diagramme commutatif $$ \matrix{ X&\mathop{\longarrow{10}}\limits^{j} &f^{-1}(\Omega_1)\subset\Omega\cr \noalign{\vskip5pt} j_1\bigg\downarrow&\phantom{{}\longrightarrow{}}& \bigg\downarrow\Id\times f\cr \Omega_1\supset g^{-1}(\Omega)& \mathop{\longarrow{10}}\limits_{\textstyle g\times\Id}& \Omega\times\Omega_1,\cr}\kern40pt $$ display whereas the morphismes $j$ and $j_1$ induce very the same space-image $\cC^k_{p,q}(X)$, because $\Id\times f$ and $g\times\Id$ are of the plongements lisses closed. \medskip {\statement Definition 1.1.\pointir}{\it Designates On by $\cD_{p,q}(X)$ $($ resp.\ $\cD^k_{p,q}(X))$ The space of the $(p,q)$ -forms on $X$ of class $\cC^\infty$ $($ resp.\ $\cC^k)$ And at compact support, catered of the topologie confine induetive. The space dual $\cD'_{p,q}(X)$ is by definition the space of the common of bidimension $(p,q)$ and of bidegré $(n-p,n-q)$ on $X$. The common pertaining at the under-space $[\cD^k_{p,q}(X)]'$ have said common of order~$k$.} \medskip If $T\in [\cD^k_{p,q}(X)]'$, run $j_*T\in[\cD^k_{p,q}(\Omega)]'$ defined by $$ \langle j_*T,v\rangle = \langle T,j*v\rangle $$ for all form $v\in\cD^k_{p,q}(\Omega)$, is at support on $j(\Omega)$. Néanmoins, for $k\geq 1$, a current $\theta\in\smash{[\cD^k_{p,q}(\Omega)]'}$ at support on $j(\Omega)$ no provenir necessarily of a current $T$ defined on $X$, although $X$ is lisse. The operators différentiels $d$, $\partial$, $\overline\partial$ usuels and the operator of external multiplication by a form $\cC^\infty$ have stretched on the other hand by duality at the common, exactly comme on the cases lisse. Prpers would be particularly interesting of know reckon at general the groups of cohomologie local of the operators $d$ and $\overline\partial$~; prpers no know not even in fact if that groups are always nuls on the cases of singularities quelconques. \medskip {\statement Definition 1.2.\pointir}{\it A current $T\in\cD'_{p,p}(X)$ have said $($ faiblement$)$ positive if the current of bidegré $(n,n)$ $$ T\wedge i\alpha_1\wedge\overline\alpha_1\wedge\ldots \wedge i\alpha_p\wedge\overline\alpha_p $$ is a measure${}\geq 0$ for all system of $(1,0)$ -form $(\alpha_1,\ldots,\alpha_p)$ of class $\cC^\infty$ on~$X$.} \medskip Goes back at the same of say that run $j_*T$ is${}\geq 0$ on~$\Omega$~; at particular, a current $T\geq 0$ on $X$ is necessarily of order~$0$. Be now $F : X\to Y$ a morphisme of spaces analytique $X$, $Y$ of respective dimensions $n$, $m$. For assure that the morphisme reciprocal image $$ F^*:\cC^k_{p,q}(Y)\to \cC^k_{p,q}(X) $$ is very defined, prpers suffice of check the lemme tracking~: \medskip {\statement Lemme 1.3.\pointir}{\it Be $j : Y \to\Omega\subset\bC^N$ a plongement and $\alpha\in\cC^k_{p,q}(\Omega)$ a tel form that $\alpha_{|Y_\reg} = 0$. Then $F^*\alpha_{|x_\reg}= 0$.} \medskip {\it Démonstration}. Pouvoir On assume $X$ lisse and connexe. If $F(X)\not \subset Y_\sing$, then $F^{-1}(Y_\reg)$ is dense on $X$ and the outcome se ensuit by continuity. The alone difficulty is so the case where $F(X)\subset Y_\sing$. Reasoning by récurrence on the dimension of $Y$ and at décomposant $F$ under the form $$ X\mathop{\longrightarrow}\limits^{\scriptstyle F} Y_\sing\hookrightarrow Y $$ sees on who prpers suffice of consider instead of $F$ the case of the morphisme of inclusion $Y_\sing\hookrightarrow Y$. The lemme 1.3 results then of the continuity of $\alpha$ and of the lemme tracking~: \medskip {\statement Lemme 1.4.\pointir}{\it Be $a$ a regular dot on $Y_\sing$. Then il there is a continuation of dots $\{a_\nu\}\subset Y_\reg$, convergeant to~$a$, tel that on the grasmannienne of the $m$ -plans of $\bC^N$ the space tangent $T_{a_\nu}Y_\reg$ converge to a plan containing $T_aY_\sing$.} \medskip {\it Démonstration}. Is a consequence of the existence of stratifications of Whitney of~$Y$, see [Wh1] and [Wh2].\hfil\square\medskip Assume that the morphisme $F:X\to Y$ be own. Defines On then the application direct image $$ F_*:[\cD^k_{p,q}(X)]'\lra [\cD^k_{p,q}(Y)]' $$ by duality, stating for all running $T\in[\cD^k_{p,q}(X)]'$ and all form $\alpha\in\cD^k_{p,q}(Y)$ $$ \langle F_*T,\alpha\rangle = \langle T,F^*\alpha\rangle. $$ If $T$ is${}\geq 0$, prpers is clearly likewise for $F_*T$. Besides, the morphisme direct image $F_*$ commute with the operators $d$, $d^c$, $\partial$, $\overline\partial$. If $T$ is${}\geq 0$ closed, $F_*T$ is so also${}\geq 0$ closed. Come at now at the definition of the functions psh . \medskip {\statement Definition 1.5.\pointir}{\it Be $V : X\to [-\infty,+\infty[$ a function who no is identiquement $-\infty$ on any open of $X$. Say On that $V$ is plurisousharmonique on $X$ (psh at abbreviated) if, for all plongement local $j : X\hookrightarrow \Omega \subset \bC^N$, $V$ is locally restriction of a function psh on~$\Omega$.} \medskip J.E.~Fornaess And R.~Narasimhan Have given the fundamental characterisation following of the functions psh on a complex space. \medskip {\statement Théorème 1.6{\rm ([FN], th.~5.3.1)}\pointir}{\it A function $V:X \to [-\infty,+\infty[$ is psh on $X$ if and alone if~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $V$ is semi-continuous supérieurement~$;$ \vskip2pt \item{\rm(b)} For all application holomorphe $f : \Delta \to X$ of the disc unity on $X$, $V\circ f$ is under-harmonique or identiquement equalises at~$-\infty$ on~$\Delta$.\medskip}} Thanks to that outcome, pouvoir on easily generalise the théorème of elongation of Brelot at the case of the complex spaces. \medskip {\statement Théorème 1.7.\pointir}{\it Are $X$ a complex space locally irréductible and $Y \subset X$ an under-ensemble analytique of inner gap on $X$. Be $V$ a function psh on $X\ssm Y$, locally majorée at the voisinage of~$Y$. Then il there is a function psh $V^*$ on $X$ who prolonger~$V$, unique, data by $$ V*(y) = \limsup_{x\in X\ssm Y,\;x\to y}V(x),\qquad y\in Y. $$}\vskip-\parskip {\it Démonstration}. (a){\it Unicité of $V^*$}. Comme $V^*$ is semi-continuous supérieurement, has on for all $y \in Y$ $$ V^*(y) = \limsup_{x\in X\ssm Y,\;x\to y}V^*(x)\geq \limsup_{x\in X\ssm Y,\;x\to y}V(x). $$ Inversement, choose an application holomorphe $f:\Delta\to X$ tel that $f(0)=y$ and $f(\Delta)\not\subset Y$. Then $0$ has isolated on $f^{-1}(Y)$, and comme $V^*\circ f$ is psh on~$\Delta$, prpers comes $$ V^(y) = V^*(f(0)) = \limsup_{t\ne 0,\;t\to 0} V(f(t)) \leq \limsup_{x\in X\ssm Y,\;x\to y}V(x). $$ (b){\it Plurisousharmonicité of $V^*$}. The outcome is local on $X$. As the théorème of désingularisation of Hironaka [Hi], il there is a space lisse $X'$ and an own modification $\sigma : X'\to X$~; by definition, $\sigma$ is own and induces out of an ensemble analytique $Z \subset X$ an isomorphisme $$ \sigma : X'\ssm\sigma^{-1}(Z) \mathop{\longrightarrow}\limits^{\textstyle\sim} X\ssm Z. $$ For all $x\in X$, the fibre $\sigma^{-1}(x)$ is compact and connexe. At effect, if $\sigma^{-1}(x)$ be no connexe, the dot $x$ would have an opened voisinage $U$ irréductible ($X$ has assumed locally irréductible), as $\sigma^{-1}(U)$ be no connexe~; but then $U\ssm Z$ would be connexe and $\sigma^{-1}(U)\ssm \sigma^{-1}(Z)$ no connexe, those that is absurd. The function $V\circ\sigma$ is psh on\hbox{$X'\ssm \sigma^{-1}(Y)$} and locally majorée at the voisinage of~$\sigma^{-1}(Y)$. As the théorème of Brelot relative at the case lisse, $V\circ\sigma$ prolonger se at a function psh $V'$ on~$X'$. The function $V'$ is necessarily constante on the fibres $\sigma^{-1}(x)$, so $V'$ induced by passage at the quotient a function $V^*$ semi-continuous supérieurement on~$X$. Display maintenant que $V^*$ is psh on $X$ using the théorème~1.7. Having given a germ of application holomorphe $f:(\Delta,0) \to (X,x)$, il there is on $X'$ a germ of bend $(\Gamma',x')$ at the-above of the image $\Gamma = f(\Delta)$, so il there is an entire $k\in\bN^*$ and a germ $f' : (\Delta, 0)\to (X',x')$ tel that $f(t^k) = \sigma(f'(t))$. By continuation $V^*(f(t^k)) = V'(f'(t))$ is psh on~$(\Delta,0)$, those that trains that $V^*\circ f$ is also psh.\hfil\square\medskip {\statement Proposal 1.8.\pointir}{\it All function psh $V$ on $X$ is locally intégrable for the measure area of $X$ $($ relative at a plongement quelconque $j : X\to \Omega \subset \bC^N)$.} \medskip {\it Démonstration}. $V$ Being locally majorée by definition, pouvoir on assume $V \leq 0$. Quitte à take a rotation of the coordinate, prpers~existe entirely of $X$ a plongement local $j:X\hookrightarrow P$ on a polydisque of $\bC^N$ tel what have on of the projections $\pi^I:X\to P^I$ own at ended fibres on the $n$ -plans of coordinated $z_j$, $j\in I$, $I\subset\{1,\ldots,N\}$, $|I| = n$. Thus, for all $I$, il there is an ensemble analytique $S_I\subset P^I$ as the restriction $\pi^I: X\ssm (\pi^I)^{-1}(S_I)\to P^I\ssm S_I$ be a revêtement ended. The function $\pi^I_*V$ defined by $$ \pi^I_*V(y)(y) = \sum_{x\in(\pi^I)^{-1}(y)}V(x) $$ is psh${}\leq 0$ on $P^I\ssm S_I$, so prolonger se at a function psh $V_I$ on $P^I$ all entire. Comme the measure area of $X$ has given by $$ d\sigma_X = \sum_{|I|=n} (\pi^I)^* d\lambda_{\bC^n}, $$ where $d\lambda$ is the measure of Lebesgue, the conclusion results then since the $V_I$ are locally intégrables on $P^I$.\hfil\square \medskip The proposal 1.8 watch what pouvoir on consider all function psh on $X$ comme a current of bidegré~$(0,0)$. By regularisation of a local elongation of $V$ at $\bC^N$ and passage at the limit décroissante, checks on easily that the $(1,1)$ -running $dd^cV = 2i\partial\overline\partial V$ is positive on $X$. \medskip {\statement Definition 1.9.\pointir}{\it A function locally intégrable $V$ on $X$ have said faiblement psh if $V$ is locally majorée and if $dd^cV\geq 0$ at the sense of the common.} \medskip Against the case lisse, the hypothesis that $V$ be locally majorée is fundamental. Consider by example the. Bend paramétrée $(z_1,z_2) = (t^2,t^3)$ on $\bC^2$~; the function $V(t) = \Re(l/t)$ no is locally majorée at~$0$, cependant pouvoir on check that $dd^cV=0$ at the sense of the common (cf.\ Definition 1.1). Observe on the other hand who a function faiblement psh no identifies se necessarily presque all around at a function psh, comme displays prpers the example of the defined function on the bend $z_1z_2 = 0$ of $\bC^2$ by $V(z_1, 0) = 1$, $V(0, z_2) = 0$ if $z_2\ne 0$. Has On however the following outcome~: \medskip {\statement Théorème 1.10.\pointir}{\it Having given a function $V:X\to[-\infty+\infty[$, il there is équivalence amid the ownerships {\rm (a), (b), (c)} here- below : {\parindent=6.5mm \vskip2pt \item{\rm(a)} $V$ is faiblement psh on $X$. \vskip2pt \item{\rm(b)} $V$ coincides presque all around with a function $V_\reg$ psh on $X_\reg$, locally majorée at the voisinage of~$X_\sing$. \vskip2pt \item{\rm(c)} Il there is a function psh $\tilde V$ on the normalisation $\tilde X$ of $X$, tel that $\tilde V=V\circ\pi$ presque all around, where $\pi:\tilde X\to X$ is the morphisme natural. \vskip2pt} So that $V$ be equalises presque all around at a function psh on $X$, prpers falloir and prpers suffice that the following condition have realised~: for all $a\in X$, designates on by $V^*(a)$ the upper limit essential of $V(x)$ when $x\in X$ extends to $a$ and by $(X_j,a)$ the composantes irréductibles of the germ $(X,a)~;$ then $$ \mathop{\limsup\,{\rm ess}}_{x\in X_j\,;\;x\to a}~V(x) = V^*(a),\quad \forall j. $$ Under that hypothesis $V^*$ is psh on $X$ and $V = V^*$ presque all around.} \medskip {\it Démonstration}. (a)${}\Rightarrow{}$ (b). That involvement results forthwith of the definition 1.9 and of the very known case where $X$ is lisse. (b)${}\Rightarrow{}$ (a). Be $h_1=\ldots=h_m= 0$ of the local equations of $X_\sing$ on~$X$. Then for all $\varepsilon > 0$ the function $$ V_\varepsilon=\cases{ V_\reg + \varepsilon\log\big(|h_1|^2+\cdots+|h_m|^2\big)&sur $X_\reg$\cr \noalign{\vskip5pt} -\infty&sur $X_\sing$\cr} $$ is psh on $X$ as the théorème 1.6. Has On so $dd^cV_\varepsilon\geq 0$. Comme $dd^cV_\varepsilon$ converge faiblement to $dd^cV$ when $\varepsilon$ extends to~$0$, prpers se ensuit that $dd^cV\geq 0$. \medskip (b)${}\Rightarrow{}$ (c). The function $V_\reg\circ\pi$ is psh on $\tilde X\ssm\pi^{-1}(X_\sing)= \pi^{-1}(X_\reg)$, locally majorée at the voisinage of $\pi^{-1}(X_\sing)$, and $\tilde X$ is locally irréductible. The théorème 1.7 watch that $V_\reg\circ\pi$ prolonger se at a function psh $V$ on $X$. (c)${}\Rightarrow{}$ (b) Results since $\pi:\tilde X\ssm \pi^{-1}(X_\sing)\to X_\reg$ is an isomorphisme. As for the latter affirmation, the condition that prpers have given for the plurisousharmonicité of $V^*$ is evidently necessary. For see who is sufficient, observes on that the ensemble of the composantes irréductibles $(X_j,a)$ is at correspondence bijective with the ensemble of the dots $a_j$ of $\tilde X$ situated at the-above of $a$ (this result by example of Narasimhan [Nar], prop.~VI.2) and that $$ \tilde V(a_j) = \mathop{\limsup\,{\rm ess}}_{x\in X_j\,,\;x\to a}~V(x). $$ has On so by hypothesis $V^*\circ\pi = \tilde V$ entirely of $X$~; comme all application holomorphe $f : \Delta\to X$ raises se at an application $\tilde f :\Delta\to\tilde X$, the plurisousharmonicité of $\tilde V$ train that of $V^*$.\hfil\square \medskip {\statement Corollaire 1.11.\pointir}{\it If $X$ is locally irréductible and if $V$ is faiblement psh on $X$, then the defined function by $$ V^*(a) = \mathop{\limsup\,{\rm ess}}_{x\to a}~V(x),\qquad a \in X, $$ is psh on $X$ and $V = V^*$ presque all around.} \medskip {\statement Corollaire 1.12.\pointir}{\it If $V:X\to[-\infty,+\infty[$ is continuous and faiblement psh, then $V$ is psh.} \medskip For finish that section, examine prpers the transformation of the functions psh by direct image. \medskip {\statement Proposal 1.13.\pointir}{\it Be $F:X\to Y$ a morphisme own surjectif at ended fibres. {\parindent=6.5mm \vskip2pt \item{\rm(a)} If $V$ is a function faiblement psh on $X$, the function $F_*V$ defined by $$ F_*V(y) = \sum_{x\in F^{-1}(y)}V(x) $$ is faiblement psh on $Y$ and besides $$ dd^c(F_*V) = F_*(dd^cV). $$ \item{\rm(b)} assumes On besides that $Y$ is locally irréductible. If $V$ is psh and if the sum $\sum_{x\in F^{-1}(y)}V(x)$ has counted with multiplicities, then $F_*V$ is psh on~$Y$.\medskip}} {\it Démonstration}. (a) knows On that $F$ is a revêtement ramifié, i.e.\ Il there is an ensemble analytique $Z\subset$ as $$ F : X\ssm F^{-1}(Z)\to Y\ssm Z $$ be a revêtement of varieties lisses. Sees On whereas the definition of$F_*V$ coincide with that data for the direct images of common. Is clear that $F_*V$ is locally majorée on~$Y$, and the ownership $dd^c(F_*V) = F_*dd^cV\geq 0$ result since $F_*$ commute with the operators $d$ and~$d^c$. \medskip (b) Under the hypothesis $X$ locally irréductible, the cardinal of the fibre $F^{-1}(y)$, $y\in Y\ssm Z$, is locally constant at the voisinage of a dot of~$Z$. If besides $V$ is continuous, $F_*V$ prolonger se by continuity on~$Y$ across~$Z$, and the corollaire 1.12 watch that $F_*V$ is psh. On the general case, il there is for all fibre $F^{-1}(y) = \{x_1,\ldots x_m\}$ of the voisinage arbitrairement small $O_j$ of~$x_j$, $1\leq j\leq m$, and a voisinage $U$ of $y$ tel that\hbox{$F^{-1}(U) = O_1\cup\ldots\cup O_m$}. Type $V$ comme limit décroissante of functions psh continuous $V_k$ on a tel voisinage $F^{-1}(V)$. Prpers comes $$ F_*V=\lim_{k\to+\infty}F_*V_k\quad\hbox{sur}~U, $$ by continuation $F_*V$ is psh.\hfil\square \bigskip \section{2}{Opérateurs $\hbox{\bigbf(}\hbox{\bigbfit dd}^{\,\hbox{\bfit c}}\, \hbox{\bigbf)}^{\hbox{\bfit k}}$ and inequalities of Chern-Levine-Nirenberg.} On that section, recall prpers the definition of the operators of Monge-Ampère $(dd^c)^k$ entered by Bedford and Taylor [BT1], [BT2]. That definition allows give a sense au courant $dd^cV_1\wedge\ldots\wedge dd^cV_k$, when the $V_j$ are of the functions psh limited. Have Prpers need here of consider the case a little best general where the an of the functions $V_j$ pouvoir no stand limited, and redonner prpers on that frame a démonstration of the inequalities of Chern-Levine-Nirenberg [CLN]. Lastly, study prpers comme on [BT2] the continuity of the operator $(dd^c)^k$ relatively at the limits décroissantes of functions psh. Are $\varphi_1$, $\varphi_2$, $\ldots\,$, $\varphi_k$ of the functions psh locally limited on $X$ and $V$ a function psh quelconque. As Bedford-Taylor [BT2] pouvoir on define run${}\geq 0$ closed $dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$ by récurrence on $k$ stating $$ dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k= dd^c\big(\varphi_k\,dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_{k-1}\big). \leqno(2.1) $$ The positivité of the second member is evident by hypothesis of récurrence if $\varphi_k\in\cC^\infty(X)$~; the general case deduce by regularisation of $\varphi_k$ and passage at the feeble limit on the space of the common. Be $\Omega= \{\rho<0\}$ an open relatively compact on $X$, defined by a function $\rho$ strictly psh $\cC^\infty$ on a voisinage $\Omega'$ of $\Omega$ and tel that $d\rho\neq 0$ on~$\partial\Omega$. For all real $a > 0$ and all entire $0\leq k\leq n$ states on $$ \beta_k=|\rho|^{k+a}(dd^c\rho)^{n-k}+(k+a) |\rho|^{k-1+a} d\rho\wedge d^c\rho\wedge(dd^c\rho)^{n-k-1} $$ and designates on by $\Vert v\Vert_p$ the norm $L^p$ of a function $v$ on $\Omega$ relatively at the measure~$\beta_0$, $p\in[1,+\infty]$. The mass of the current (2.1) admits then the following assessments (cf.\ [CLN]). \medskip {\statement Théorème 2.2}.{\it Are $V$, $V_1,\ldots,V_k$ of the functions psh on $X$ tel that $V\leq 0$ and\hbox{$V_1\geq 0$, $\ldots$~, $V_k\geq 0$} on~$\Omega$. Then il there is of the constantes $C_j=C_j(k,a)\geq 0$, $j=1,2,3$, tel that {\parindent=6.5mm \vskip0pt \item{\rm (a)} $\displaystyle \int_\Omega \beta_{k+1}\wedge dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k\leq C_1\,\Vert V\Vert_1\,\Vert\varphi_1\Vert_\infty\cdots \Vert\varphi_k\Vert_\infty.$ \vskip-1pt \item{\rm (b)} $\displaystyle \int_\Omega \beta_k\wedge |V|\, dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k\leq C_2\,\Vert V\Vert_1\,\Vert\varphi_1\Vert_\infty\cdots \Vert\varphi_k\Vert_\infty.$ \vskip-1pt \item{\rm (c)} $\displaystyle \int_\Omega \beta_k\wedge dd^cV_1\wedge\ldots\wedge dd^cV_k\leq C_3\,\Vert V_1\Vert_k\,\Vert V_2\Vert_k\cdots\Vert V_k\Vert_k.$ \medskip}} {\it Démonstration}. Thanks to the lemme of approximation 2.4 here-below, pouvoir on assume that the $V_j$ and $\varphi_j$ are of class~$\cC^\infty$. An immediate calculation gives $$ \eqalign{ d^c\beta_k&=-2(k+a)\,|\rho|^{k-1+a}\,d^c\rho\wedge(dd^c\rho)^{n-k},\cr dd^c\beta_k&=2(k+a)\,\Big[-|\rho|^{k-1+a}\,(dd^c\rho)^{n-k+1}+ (k-1+a)\,d\rho\wedge d^c\rho\wedge(dd^c\rho)^{n-k}\Big],\cr} $$ of where the inequality amid forms $$ |dd^c\beta_k|\leq 2(k+a)\beta_{k-1}. $$ Note $I_k$, $J_k$ the integral (a), (b) respectively and $\psi_k = dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$. As the formula of integration by parts of the lemme 2.5 and kept account that $$ \beta_{k+1|\partial\Omega}=d^c\beta_{k+1|\partial\Omega}=0, $$ prpers comes $$ \eqalign{ I_k &=\int_\Omega dd^c\beta_{k+1}\wedge\varphi_k\,dd^cV\wedge\psi_{k-1}\cr &\leq 2(k+1+a)\,\Vert\varphi_k\Vert_\infty\int_\Omega\beta_k\wedge dd^cV\wedge\psi_{k-1}= 2(k+1+a)\,\Vert\varphi_k\Vert_\infty I_{k-1},\cr I_0 &=\int_\Omega V\,dd^c\beta_1\cr &\leq 2(1+a)\int_\Omega|V|\,|\rho|^a(dd^c\rho)^n \leq 2(1+a)\,\Vert V\Vert_1.\cr} $$ This show (a) by récurrence with $C_1(k.a)=2^{k+1}(1+a)\ldots(k+1+a)$, the inequality being of elsewhere checked although~$a=0$. On the other hand if $k\geq 1$ obtains on $$ \eqalign{ J_k &=\int_\Omega -\varphi_kdd^c(V\beta_k)\wedge\psi_{k-1}\cr &=\int_\Omega -\varphi_k\big(V\,dd^c\beta_k+\beta_k\wedge dd^cV+ 2\,dV\wedge d^c\beta_k\big)\wedge\psi_{k-1}\cr &=\int_\Omega\varphi_k\big(V\,dd^c\beta_k-\beta_k\wedge dd^cV\big) \wedge\psi_{k-1}+2V\,d\varphi_k\wedge d^c\beta_k\wedge\psi_{k-1}\cr} $$ integrating by parts the term $-2\varphi_k\,dV\wedge d^c\beta_k\wedge \psi_{k-1}$ at the deuxième line. Assumes On now $\varphi_k> 0$ and applies on the inequality of Cauchy-Schwarz at the composante of bidegré $(n-k+1, n-k+1)$ of the running $$ 2d\varphi_k\wedge d^c\beta_k=-4(k+a)\,|\rho|^{k-1+a}d\varphi_k\wedge d^c\rho \wedge(dd^c\rho)^{n-k}, $$ those that gives the majorant $$ \Big[4(k+a)^2\varphi_k|\rho|^{k-2+a}d\rho\wedge d^c\rho+ |\rho|^{k+a}{d\varphi_k\wedge d^c\varphi_k\over\varphi_k}\Big]\wedge (dd^c\rho)^{n-k}. $$ Prpers results $$ \eqalign{ J_k\leq\int_\Omega\varphi_k|V|\Big[&-dd^c\beta_k+ 4(k+a)^2|\rho|^{k-2+a}d\rho\wedge d^c\rho\wedge(dd^c\rho)^{n-k}\Big] \wedge\psi_{k-1}\cr &+\int_\Omega|V|\, |\rho|^{k+a}(dd^c\rho)^{n-k}\wedge{d\varphi_k\wedge d^c\varphi_k\over\varphi_k} \wedge\psi_{k-1}.\cr} $$ The form amid hooks on the first integral is equal at $$ 2(k+a)\Big[|\rho|^{k-1+a}(dd^c\rho)^{n-k+1}+ (k+1+a)|\rho|^{k-2+a}d\rho\wedge d^c\rho\wedge(dd^c\rho)^{n-k}\Big] \leq C_4\beta_{k-1} $$ with $C_4=C_4(k,a)={2(k+a)(k+1+a)\over k-1+a}$. Obtains On so finally $$ J_k\leq C_4\Vert\varphi_k\Vert_\infty J_{k-1} +\int_\Omega|V|\,\beta_k\wedge\psi_{k-1}\wedge {d\varphi_k\wedge d^c\varphi_k\over\varphi_k}. $$ Note $J'_k$ the integral obtained replacing $\varphi_k$ by $\varphi'_k= \exp(B\varphi_k)$ on $J_k$ and state $M = \Vert\varphi_k\Vert_\infty$. Prpers comes $$ \eqalign{ dd^c\varphi'&=e^{B\varphi_k}\big(B\,dd^c\varphi_k + B^2\,d\varphi_k\wedge d^c\varphi_k\big)\geq B\,e^{-BM}dd^c\varphi_k+{d\varphi'_k\wedge d^c\varphi'_k\over \varphi'_k},\cr B\,e^{-BM}J_k&\leq J'_k-\int_\Omega|V|\,\beta_k\wedge\psi_{k-1}\wedge {d\varphi'_k\wedge d^c\varphi'_k\over \varphi'_k} \leq C_4(k,a)\,e^{BM}J_{k-1}.\cr} $$ Comme $\inf_{B>0}{1\over B}e^{2BM}=2eM=2e\,\Vert\varphi_k\Vert_\infty$, this finishes the démonstration of (b) by récurrence on $k$, with the constante $$ C_2(k,a) = (4e)^k\;{k+a\over a}\,(2+a)\cdots (k+1+a). $$ For show (c), assume firstly that $V_1=V_2=\ldots = V_k = v \geq 0$. Comme $$ \Big(dd^cv^{{k\over k-1}}\Big)^{k-1}\geq v\Big({k\over k-1}\,dd^cv\Big)^{k-1}, $$ an integration by parts gives prpers $$ \int_\Omega\beta_k\wedge(dd^cv)^k=\int_\Omega dd^c\beta_k\wedge v\,(dd^cv)^{k-1}\le 2(k+a)\Big({k-1\over k}\Big)^{k-1} \int_\Omega\beta_{k-1}\wedge\Big(dd^cv^{{k\over k-1}}\Big)^{k-1}. $$ By récurrence on $k$ that inequality trains à son tour $$ \int_\Omega\beta_k\wedge(dd^cv)^k\leq 2^k(1+a)...(k+a) {(k-1)!\over k^{k-1}}\int_\Omega\beta_0v^k. $$ Replace now $v$ by $$ v = {V_1\over\Vert V_1\Vert_k}+\cdots+{V_k\over\Vert V_k\Vert_k}. $$ Prpers comes $$ {k!\over\Vert V_1\Vert_k\cdots\Vert V_k\Vert_k}\int_\Omega \beta_k\wedge dd^cV_1\wedge\ldots\wedge dd^cV_k\leq 2^k(1+a)...(k+a){(k-1)!\over k^{k-1}}\int_\Omega\beta_0v^k $$ while $\Vert v\Vert_k\leq k$. The inequality (c) se ensuit with $$ C_3(k,a) = 2^k(1+a)\ldots(k+a). \eqno\square $$ An immediate consequence of the théorème 2.2~(b) is that run $V\,dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$ is of mass locally ended on $X$~; at particular finds on the following outcome owed at Bedford-Taylor [BT2]. \medskip {\statement Corollaire 2.3.\pointir}{\it The measures coefficients of$dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$ no burden the groups pluripolaires $\{V = -\infty\}$.\hfil\square} \medskip The space $X$ being of Stein, il there is as J.E.~Fornaess And R.~Narasimhan [FN] Of the continuations décroissantes $V_m$, $\varphi_{j,m}$ of functions psh $\cC^\infty$ on $X$ tel that $$ V_m\to V,\qquad\varphi_{j,m}\to\varphi_j\qquad\hbox{pour $1\leq j\leq k$}. $$ {\statement Lemme 2.4.\pointir}{\it Il there is of the strictly increasing continuations of entire $m(\nu)$, $m_1(\nu)$, $\ldots$~, $m_k(\nu)$, $\nu\in\bN$, tel who at the senses of the convergence faible of the measures have on at the election the an or the autre of the ownerships of convergence here-below~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle dd^cV_{m(\nu)}\wedge dd^c\varphi_{1,m_1(\nu)}\wedge\ldots\wedge dd^c\varphi_{k,m_k(\nu)} \longrightarrow dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$ \vskip2pt \item{\rm(b)} $\displaystyle V_{m(\nu)}\wedge dd^c\varphi_{1,m_1(\nu)}\wedge\ldots\wedge dd^c\varphi_{k,m_k(\nu)} \longrightarrow V\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$. \medskip}} {\it Démonstration}. As the théorème 2.2 already showed on the case of the functions psh $\cC^\infty$ the continuations (a), (b) have limited locally at mass, and the limited parts of the space of the common of order~$0$ are métrisables for the topologie faible. On the case (a) observes on that $$ \varphi_{k,m}dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_{k-1} \longrightarrow \varphi_kdd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_{k-1} $$ by convergence monotone of $\varphi_{k,m}$ when $m\to+\infty$~; spending at the $dd^c$ has on so $$ dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_{k-1}\wedge dd^c\varphi_{k,m} \longrightarrow dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_{k-1}\wedge dd^c\varphi_k. $$ The topologie being métrisable, pouvoir on choose successively $m_k(\nu)=\nu$ afterwards $m_{k-1}(\nu)$, $\ldots$~, $m_1(\nu)$, $m(\nu)$ by récurrence on $\nu$ (resp.\ $m(\nu) =\nu$ Afterwards $m_1(\nu)$, $\ldots$~, $m_1(\nu)$ on the case (b)) for obtain the convergence wished.\hfil\square \medskip Énonçons Here en vue de references ultérieures the lemme of integration by parts that prpers have used. \medskip {\statement Lemme 2.5.\pointir}{\it If $u$ and $v$ are of the forms of class $\cC^2$ of bidegrés respective $(p,q)$ and $(n-p-1, n-q-1)$ with $p+q$ pair, then $$ \int_\Omega u\wedge dd^cv = \int_{\partial\Omega}u\wedge d^cv - d^cu\wedge v. $$} Suffice at effect of apply the théorème of Stokes at the form $$ d\big(u\wedge d^cv - d^cu\wedge v\big)= u\wedge dd^cv- dd^cu\wedge v+du\wedge d^cv+d^cu\wedge dv $$ and of observe that $$ du\wedge d^cv = i(\partial u\wedge\overline\partial v - \overline\partial u\wedge\partial v) = -d^cu\wedge dv.\eqno\square $$ Adapting the technical of [BT2] at the present situation, display prpers now the continuity of the operator $(dd^c)^k$ by report at the limits décroissantes of functions psh. \medskip {\statement Théorème 2.6.\pointir}{\it Are $\{\varphi_j^\nu\}_{\nu\in\bN} \subset L^\infty_\loc(X)$ and $\{V^\nu\}_{\nu\in\bN}$ of the continuations décroissantes of functions psh tel that $$ \varphi_j=\lim_{\nu\to+\infty}\varphi_j^\nu\in L^\infty_\loc(X),\qquad V=\lim_{\nu\to+\infty}V^\nu\not\equiv -\infty. $$ At the senses of the convergence faible of the measures, has on then {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle dd^cV^\nu\wedge dd^c\varphi_1^\nu\wedge \ldots\wedge dd^c\varphi_k^\nu\longrightarrow dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_k,$ \vskip2pt \item{\rm(b)} $\displaystyle V^\nu\wedge dd^c\varphi_1^\nu\wedge \ldots\wedge dd^c\varphi_k^\nu\longrightarrow V\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_k,$ \vskip2pt \item{\rm(c)} $\displaystyle \varphi_k^\nu dd^cV^\nu\wedge dd^c\varphi_1^\nu\wedge \ldots\wedge dd^c\varphi_{k-1}^\nu\longrightarrow \varphi_k\wedge dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_k.$ \medskip}} Test Prpers the théorème~2.6 simultaneously with the following ownership whom be a corollaire. \medskip {\statement Corollaire 2.7.\pointir}{\it {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle dd^c\big(V\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_k\big)= dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_k.$ \vskip2pt \item{\rm(b)} The common $V\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_k$ and $dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_k$ are symmetrical at~$\varphi_1,\ldots,\varphi_k$.\medskip}} {\it Démonstration}. Using the lemme 2.4 and a proceed evident of continuation diagonale, on se brings au cas où $V^\nu$, $\varphi_1^\nu$, $\ldots$~, $\varphi_k^\nu$ are of class $\cC^\infty$. Comme the ownerships 2.6~(has,b,c) are local, pouvoir on without loss of generality place on an open $\Omega = \{\rho < 0\}\compact X$. Goes On now bring at the situation where $\varphi_j$, $\varphi_j^\nu$ are of class $\cC^\infty$ at the voisinage of $\partial\Omega$ and nulles on~$\partial\Omega$, of way at pouvoir apply the formula of Stokes without terms of edge. Be $\bR^2\ni(u,v)\mapsto\lambda(u,v)$ a function $\cC^\infty$ convexe increasing at~$u$ and~$v$, who coincides with $\max(u,v)$ for $|u-v| > 1$. Then $\tilde\varphi_j^\nu= \lambda\big(\varphi_j^\nu-{2\over\varepsilon},\varepsilon^{-2}\rho\big)$ is psh~$\cC^\infty$, besides for $\varepsilon > 0$ enough small has on $$ \cases{ \tilde\varphi_j^\nu= \varphi_j^\nu - {2\over\varepsilon}& sur~ $\Omega_{3\varepsilon}=\{\rho<-3\varepsilon\}$,\cr \noalign{\vskip5pt} \tilde\varphi_j^\nu=\varepsilon^{-2}\rho& sur~ $\overline\Omega\ssm \Omega_\varepsilon=\{-\varepsilon\leq \rho\leq 0\}$.\cr} $$ pouvoir On so finally assume that $\varphi_j^\nu=\varphi_j=\varepsilon^{-2} \rho$ on the\lguil\?Tops\?\rguil\ $\overline\Omega\ssm \Omega_\varepsilon$ (And this quel that are $j$,~$\nu$ ). {\it Test of} 2.6~(a). Reasons On by récurrence on~$k$. As (2.1) prpers suffice of test that $$ \leqalignno{ \lim_{\nu\to+\infty}\int_\Omega \varphi_k^\nu\,dd^cV^\nu\wedge dd^c\varphi_1^\nu&\wedge \ldots\wedge dd^c\varphi_{k-1}^\nu\wedge dd^c\psi\cr &=\int_\Omega\varphi_k\,dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_{k-1} &(2.8)\cr} $$ for all form $\psi\in\cC^\infty_{n-k-1,n-k-1}(\overline\Omega)$ tel that $\psi_{|\partial\Omega}= 0$ (note that by hypothesis $\varphi^\nu_{k|\partial\Omega}= 0$ ). Quitte à replace $\psi$ successively by $\rho(dd^c\rho)^{n-k-1}$ and $\psi+A\rho(dd^c\rho)^{n-k-1}\kern-0.5pt,$ $A\gg 0$, pouvoir on assume $dd^c\psi\geq 0$ on~$\overline\Omega$. The inequality $\leq$ on (2.8) results then simply of the hypothesis of récurrence $$ dd^cV^\nu\wedge dd^c\varphi_1^\nu\wedge \ldots\wedge dd^c\varphi_{k-1}^\nu \longrightarrow dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_{k-1} $$ and of the théorème of convergence monotone. For test the inequality $\geq$ reverse, effects on of the integrations by successive parts at the half of the lemme 2.5~: $$ \eqalign{ \int_\Omega \varphi_k\,dd^cV&\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_{k-1}\wedge dd^c\psi\cr &\leq\int_\Omega \varphi_k^\nu\,dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_{k-1}\wedge dd^c\psi\cr &=\int_\Omega \varphi_{k-1}\,dd^cV\wedge dd^c\varphi_1\wedge \ldots\wedge dd^c\varphi_{k-2}\wedge dd^c\varphi_k^\nu\wedge dd^c\psi\cr &\leq\int_\Omega \varphi_{k-1}^\nu\,dd^cV\wedge dd^c\varphi_1\wedge \ldots \wedge dd^c\varphi_{k-2}\wedge dd^c\varphi_k^\nu\wedge dd^c\psi\cr &=\cdots\leq \int_\Omega \varphi_1^\nu\,dd^cV\wedge dd^c\varphi_2^\nu\wedge \ldots \wedge dd^c\varphi_k^\nu\wedge dd^c\psi\cr &=\int_\Omega V\,dd^c\varphi_1^\nu\wedge dd^c\varphi_2^\nu\wedge\ldots \wedge dd^c\varphi_k^\nu\wedge dd^c\psi\cr &\kern40pt{}- \int_{\partial\Omega} V\,d^c\varphi_1^\nu\wedge dd^c\varphi_2^\nu\wedge \ldots\wedge dd^c\varphi_k^\nu\wedge dd^c\psi\cr &\leq\int_\Omega V^\nu\,dd^c\varphi_1^\nu\wedge\ldots\wedge dd^c\varphi_k^\nu\wedge dd^c\psi -\varepsilon^{-2k}\int_{\partial\Omega} V\,d^c\rho\wedge(dd^c\rho) ^{k-1} \wedge dd^c\psi\cr &=\int_\Omega \varphi_k^\nu\,dd^cV^\nu\wedge dd^c\varphi_1^\nu\wedge\ldots \wedge dd^c\varphi_k^\nu\wedge dd^c\psi\cr &\kern40pt{}- +\varepsilon^{-2k}\int_{\partial\Omega} (V^\nu-V)\, d^c\rho\wedge(dd^c\rho) ^{k-1}\wedge dd^c\psi.\cr} $$ The latter integral extends to $0$ by convergence monotone, those that finishes the test of 2.6~(a). \medskip {\it Test of} 2.7~(a). Immediate consequence of 2.4~(b) and 2.6~(a). \medskip {\it Test of} 2.6~(b). The inequality 2.2~(b) trains that the continuation $V^\nu\,dd^c\varphi_1^\nu\wedge\ldots\wedge dd^c\varphi_k^\nu$ is of mass locally uniformément limited on $X$. Besides all courage of adhérence $T$ of that continuation is tel that $$ T\leq V\,dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k, $$ with equality on $\Omega\ssm\Omega_\varepsilon$ (there where $\varphi_j^\nu = \varphi_j = \varepsilon^{-2}\rho$ ). As 2.6~(a) and 2.7~(a) has on on the other hand $$ dd^cT= dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k= dd^c\big(V\,dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k\big). $$ Distinguish now deux cases tracking the courage of the entire~$k$. \medskip If $k\leq n-1$, the lemme 2.5 applied with $v = \rho(dd^c\rho)^{n-k-1}$ trains that the positive current $$ u = V\,dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k - T $$ is any on~$\Omega$. If $k = n$, pouvoir on consider $V$, $\varphi_1$, $\ldots$~, $\varphi_k$ comme of the functions on $X\times\bC$ no dépendant of the latter variable, and apply the outcome 2.6~(b) already known at $X\times\bC$. The démonstration of 2.6~(c) is identical.\hfil\square \medskip {\statement Observes 2.9.\pointir} If $\varphi_j$ is${}\geq 0$, pouvoir on type $$ d\varphi_j\wedge d^c\varphi_j = {1\over 2} dd^c\varphi_j^2 - \varphi_j\,dd^c\varphi_j~; $$ therefore, the théorème of convergence faible 2.6 valid rest for all produce of $V$ or $dd^cV$ by of the $(1,1)$ -forms of the type $dd^c\varphi_j$, $d\varphi_j\wedge d^c\varphi_j$, or still (by polarisation) $d\varphi_i\wedge d^c\varphi_j+d\varphi_j\wedge d^c\varphi_i$. \medskip {\statement Observes 2.10.\pointir} The reader will find an interesting argument on the problem of the definition and of the continuity of the operator of Monge-Ampère on [Ki] and [That]. At particular, is possible of stretch some of the précédent outcomes au cas où the functions $\varphi_j$ be not any more necessarily limited, at condition of take a hypothesis of compacité on the poles of the $\varphi_j$. Assume Prpers who il there is a compact $K\subset X$ as $\varphi_1,\ldots,\varphi_k$ have limited locally on $X\ssm K$. Then the definition (2.1), the lemme 2.4~(a) and the théorème 2.6~(a) remain valid. \medskip For see, observes on that the problem states se only at the voisinage of~$K$. Be $\rho$ a function $\cC^\infty$ strictly psh and $\omega$ an open tel that $K\subset\omega\compact\Omega=\{\rho<0\}\compact X$. Quitte à replace $\varphi_j$ by $$ \cases{ \varphi_j-{2\over\varepsilon}&sur $\omega$,\cr \noalign{\vskip8pt} \varepsilon^{-2}\rho&sur $X\ssm\Omega$,\cr \noalign{\vskip6pt} \max\big(\varphi_j-{2\over\varepsilon},\varepsilon^{-2}\rho\big)&sur $\Omega\ssm\omega$,\cr} $$ pouvoir on assume $\varphi_j = \varepsilon^{-2}\rho$ at the voisinage of~$\partial\Omega$. Show firstly by récurrence that $\varphi_k\,dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_{k-1}$ (and so also $dd^cV\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_k$ ) is of mass locally ended if~$k\leq n-1$. For all $a < 0$ has on at effect, with the notation $\varphi_{k,a}=\max(\varphi_k,a)$~:\eject $$ \eqalign{ \int_\Omega|\varphi_{k,a}|\,dd^cV&\wedge dd^c\varphi_1\wedge\ldots\wedge dd^c\varphi_{k-1}\wedge (dd^c\rho)^{n-k}\cr &=\int_\Omega|\rho|\,dd^c\big(\varphi_{k,a} dd^cV\wedge dd^c\varphi_1 \wedge\ldots\wedge dd^c\varphi_{k-1}\big)\wedge (dd^c\rho)^{n-k-1}\cr &\le C\int_\Omega dd^c\big(\varphi_{k,a} dd^cV\wedge dd^c\varphi_1 \wedge\ldots\wedge dd^c\varphi_{k-1}\big)\wedge (dd^c\rho)^{n-k-1}\cr &=C\varepsilon^{-2k}\int_\Omega dd^cV\wedge(dd^c\rho)^{n-1}<+\infty,\cr} $$ the latter equality pertinent of the théorème of Stokes. The démonstration of 2.4~(a) and 2.6~(a) takes se then without any modification. \bigskip \section{3}{Mesures of Monge-Ampère and formula of Jensen.} HAS all function $\varphi$ psh continuous and exhaustive on a space of Stein, go prpers associate of way canonique a family of positive measures scopes by the groups of level of$\varphi$. That measures appear naturally when looks for on stretch the formula of Jensen at plusieurs variables. The principal ideas of that paragraph rest on the taken calculations by P.~Lelong [The1] for display the existence of the numbers of Lelong of a positive current closed. Restart Prpers for the essential the notations of [Of4], [Of5] (see also the article of H.~Skoda [Sk1]). Considers On a space of Stein $X$ of pure dimension $n$, reduced, catered of a function psh continuous $\varphi:X\to[-\infty,R[\,$, where $R\in{} ]-\infty,+\infty]$. For all $r-\infty$, and if $r=-\infty$, $(dd^c\varphi_r)^n = (dd^c\varphi)^n$ exist as observes prpers 2.10. \medskip {\statement Lemme 3.1.\pointir}{\it The application $r\mapsto (dd^c\varphi_r)^n$ is continuous of $[-\infty,R[$ on the space of the measures on $X$ catered of the topologie faible.} \medskip {\it Démonstration}. The continuity at right results of the théorème 2.6~(a), while the continuity at left obtains se typing $$ (dd^c\varphi_r)^n = \big(dd^c\max(\varphi-r,0)\big)^n.\eqno\square $$ Comme $(dd^c\varphi_r)^n$ is any on $B(r)$ and coincides with $(dd^c\varphi)^n$ on $X\ssm\overline B(r)$, the continuity at left train $$ (dd^c\varphi_r)^n\geq \bOne_{X\ssm B(r)}(dd^c\varphi)^n, $$ where $\bOne_A$ designates the characteristic function of a part~$A\subset X$. The outcomes here-below découlent forthwith of that observe and justify the following definition. \medskip {\statement Théorème And definition 3.2.\pointir}{\it Urge On measures of Monge-Ampère associated at $\varphi$ the families of positive measures $(\mu_r)$, $(\overline\mu_r)$ scopes by $S(r)$, $r\in[-\infty,R[\,$, defined by $$ \eqalign{ \mu_r&=(dd^c\varphi_r)^n-\bOne_{X\ssm B(r)}(dd^c\varphi)^n,\cr \overline\mu_r&=(dd^c\varphi_r)^n-\bOne_{X\ssm\overline B(r)}(dd^c\varphi)^n.\cr} $$ The family $\mu_r$ $($ resp.\ $\overline\mu_r)$ Is faiblement continues at left $($ resp.\ At right$)$, and has on the accounts $$ \eqalign{ \overline\mu_r&=\lim_{\rho\to r+0}\mu_\rho,\qquad \mu_r=\lim_{\rho\to r-0}\overline\mu_\rho,\cr \overline\mu_r&=\bOne_{S(r)}(dd^c\varphi_r)^n=\mu_r+\bOne_{S(r)} (dd^c\varphi)^n.\cr} $$ Be $D_\varphi\subset [-\infty,R[$ the ensemble at the best dénombrable of the real $r$ tel that $S(r)$ be négli\-geable for the measure $(dd^c\varphi)^n$ on~$X$. Then $\overline\mu_r =\mu_r$ for all $r\notin D_\varphi$, and the applications $r\mapsto\mu_r$, $r\mapsto\overline\mu_r$ are continuous entirely $r\notin D_\varphi$.\hfil\square} \medskip Appreciate Prpers vivement E.~Bedford Of have suggested that definition, who simplifies those that prpers have used on a former version of that work. Entirely where $\varphi$ is regular, $\mu_r$ and $\overline\mu_r$ pouvoir depict by a form différentielle simple on the hypersurface $S(r)$. \medskip {\statement Proposal 3.3.\pointir}{\it Be $x\in X$ a regular dot at the voisinage duquel $\varphi$ is of class $\cC^2$ and as $d\varphi(x)\neq 0$. Orienter On $S(r)$ comme edge of $B(r)$. Then the measures $\mu_r$ and $\overline\mu_r$ have defined at the voisinage of $x$ by the $(2n-1)$ -forms volume $(dd^c\varphi)^{n-1}\wedge\varphi_{|S(r)}$.} \medskip {\it Démonstration}. Be $\Omega$ a voisinage of $x$ on which $d\varphi \neq 0$, and $h$ a function $\cC^\infty$ at compact support on~$\Omega$. Type $$ \max(r,t) = \lim_{\nu\to+\infty}\chi_\nu(t) $$ where $\chi_\nu$ is a continuation of régulariser by convolution of $t\mapsto \max(r,t)$. Pouvoir On take at type that $(\chi_\nu)$ be a continuation décroissante of functions convexes~$\cC^\infty$, tel that $0\leq\chi'_\nu\leq 1$, with $\lim\chi'_\nu(t)$ equal at $0$ for $t r$. The théorème 2.6~(a) trains so $$ \eqalign{ \int_\Omega h\,(dd^c\varphi_r)^n &=\lim_{\nu\to+\infty}\int_\Omega h\,(dd^c\chi_\nu\circ\varphi)^n\cr &=\lim_{\nu\to+\infty}-\int_\Omega dh\wedge(dd^c\chi_\nu\circ\varphi)^{n-1} \wedge d^c(\chi_\nu\circ\varphi)\cr &=\lim_{\nu\to+\infty}-\int_\Omega \chi_\nu'(\varphi)^n\,dh\wedge (dd^c\varphi)^{n-1}\wedge d^c\varphi\cr &=-\int_{\Omega\ssm B(r)}dh\wedge(dd^c\varphi)^{n-1}\wedge d^c\varphi\cr &=\int_{\Omega\cap S(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi+ \int_{\Omega\ssm B(r)}h\,(dd^c\varphi)^n\cr} $$ as the formula of Stokes. Deduces so on $\Omega$ the equality of measures $$ (dd^c\varphi_r)^n=(dd^c\varphi)^{n-1}\wedge d^c\varphi_{|S(r)}+ \bOne_{\Omega\ssm B(r)}(dd^c\varphi)^n. \eqno\square $$ pouvoir Prpers now show the formula of Jensen-Lelong that have prpers at view. \medskip {\statement Théorème 3.4.\pointir}{\it Be $V$ a function psh on $X$. Then $V$ is $\mu_r$ -intégrable for all $r\in{}]-\infty,R[\,$. Besides $$ \int_{-\infty}^rdt\int_{B(t)}dd^cV\wedge\alpha^{n-1}= \mu_r(V)-\int_{B(r)}V\,\alpha^n, $$ where $\alpha=dd^c\varphi$. Both members have ended if $\inf_X\varphi> -\infty$ or if $\inf_{B(r)}V>-\infty$.} \medskip {\it Démonstration}. The intégrabilité of $V$ for $\mu_r$ (and for $\overline\mu_r$ ) results since $V$ is intégrable for $(dd^c\varphi_r)^n$ as the théorème 2.2~(b). Note also that the integral $\int_{B(t)}dd^cV\wedge(dd^c\varphi)^{n-1}\geq 0$ and $\int_{B(r)}V\,(dd^c\varphi)^n$ have very a sense en vertu de observes prpers 2.10, the first being of elsewhere always convergente. The deuxième converge if \hbox{$\inf_X\varphi>-\infty$} thanks to 2.2~(b), or if $\inf_{B(r)}V>-\infty$ thanks to 2.10. For show the formula 3.4, assumes on firstly $$ \inf_X\varphi>-\infty\quad\hbox{et}\quad\inf_{B(r)}V>-\infty, $$ and on se gives $c > r$. The théorème of Fubini involve $$ \eqalign{ \int_{-\infty}^cdt\int_{B(t)}dd^cV\wedge(dd^c\varphi)^{n-1} &=\int_{B(c)}\bigg[\int_{\{\varphi a$, so obtains on $$ \int_a^r dt\int_{B(t)} dd^cV\wedge(dd^c\varphi)^{n-1}=\mu_r(V)- \int_{B(r)}V\,(dd^c\varphi_a)^n. $$ That formula trains that the continuous functions $a\mapsto\int_{B(r)}V_\nu\,(dd^c\varphi_a)^n$ are increasing on $[-\infty,r[\,$. Leur limit décroissante $a\mapsto\int_{B(r)}V\,(dd^c\varphi_a)^n$ is so continues at right, those that allows spend at the limit at $a=-\infty$.\hfil\square \medskip Of the formula 3.4 deduces on forthwith the analogous formula for the measures $\overline\mu_r$~: $$ \int_{-\infty}^rdt\int_{B(t)}dd^cV\wedge(dd^c\varphi)^{n-1}= \overline\mu_r(V)-\int_{\overline B(r)}V\,(dd^c\varphi)^n. \leqno(3.6) $$ At particular for $V = 1$ prpers comes~: \medskip {\statement Corollaire 3.7.\pointir}{\it The total masses of $\mu_r$ and $\overline\mu_r$ have given by $$ \Vert\mu_r\Vert=\int_{B(r)}(dd^c\varphi)^n=\int_{B(r)}\alpha^n,\qquad \Vert\overline\mu_r\Vert=\int_{\overline B(r)}(dd^c\varphi)^n =\int_{\overline B(r)}\alpha^n. $$} On all the continuation, allow prpers at the reader the cure of translate the obtained outcomes on the cases of the measures~$\overline\mu_r$. Study Prpers now the continuity of the measures $\mu_r$ en fonction of the exhaustion~$\varphi$. \medskip {\statement Proposal 3.8.\pointir}{\it Be $(\varphi^\nu)_{\nu\in\bN}$ a continuation décroissante of functions psh continuous convergeant to $\varphi$ on~$X$, and $\mu_r^\nu$ the measures of Monge-Ampère associated at $\varphi^\nu$. Then $\mu_r^\nu$ converge faiblement to $\mu_r$ for all $r \in{}]-\infty,R[{}\ssm D_\varphi$.} \medskip {\it Démonstration}. Prpers suffice of apply the definition 3.2, who gives $$ \mu_r^\nu=(dd^c\varphi_r^\nu)^n-\bOne_{X\ssm B(r)}(dd^c\varphi^\nu)^n $$ with $$ \varphi^\nu_r=\max(\varphi^\nu,r),\qquad B^\nu_r=\big\{|z\in X\,;\;\varphi^\nu(z) 1$. Fix $r < R_2$ and state $\varphi=\max(\varphi_1-A,\varphi_2)$ where $A$ has chosen enough big so that $\varphi$ coincide with $\varphi_2$ at the voisinage of $S_2(r)$. Are $\mu_r$ the associé measures at~$\varphi$. The hypothesis $\liminf \varphi_2/\varphi_1 > 1$ train what il there is $t < r$ as $\varphi$ coincide with $\varphi_1-A$ on $B_1(t) =\{\varphi_1 < t\}$. Obtains On so $$ \overline\mu_{-\infty,1}(V)=\lim_{t\to-\infty}\overline\mu_{t,1}(V)= \lim_{t\to-\infty}\overline\mu_{t}(V)\leq \mu_r(V)=\mu_{r,2}(V), $$ of where $$ \overline\mu_{-\infty,1}(V)\leq\lim_{r\to-\infty}\overline\mu_{r,2}(V)= \overline\mu_{-\infty,2}(V).\eqno\square $$ Under the précédent hypotheses, pouvoir on conjecturer that the inequality amid measures\break $\overline\mu_{-\infty,2}\geq\ell^n\,\overline\mu_{-\infty,1}$ has always place, but the conclusions of the théorème 4.2 no prpers have allowed show. Have Prpers however of the particular outcome tracking~: \medskip {\statement Corollaire 4.3.\pointir}{\it With the notations of the théorème~$4.2$, be $A\subset S_1(-\infty)$ a part borélienne who is meeting of composantes connexes of $S_1(-\infty)$ and $\bOne_A$ the characteristic function of~$A$. Then for all function psh $V\geq 0$ $$ \overline\mu_{-\infty,2}(\bOne_A V)\geq \ell^n\,\overline\mu_{-\infty,1}(\bOne_A V). $$ At particular, if $S_1(-\infty)$ is totally discontinu, has on $\overline\mu_{-\infty,2}\geq\ell^n\,\overline\mu_{-\infty,1}$.} {\it Démonstration}. Il there is an increasing continuation of compact $K_\nu\subset A$ tel that have on $\overline\mu_{-\infty,j}(A\ssm K_\nu) < 2^{-\nu}$, $j = 1,2$. The account of équivalence whose class are the composantes connexes of $S_1(-\infty)$ is of graphe closed ($S_1(-\infty)$ being compact). The saturer $\tilde K_\nu$ of $K_\nu$ is so a compact part of~$A$~; besides $\tilde K_\nu$ is intersection of a continuation décroissante of parts at a time opened and closed on $S_1(-\infty)$ (cf.\ Bourbaki [Bo], chap.~II,\S4, n${}^\circ$ 4). Pouvoir On so assume that $A$ has opened and closed on $S_1(-\infty)$. Il there is then an open $U\compact X$ as $A = U\cap S_1(-\infty)$, $\partial U\cap S_1(-\infty) = \emptyset$. Be $r_0 = \inf_{\partial U}\varphi_1 > -\infty$ and $$ \Omega = \big\{z\in U\,;\;\varphi_1(z) < r_0\big\}. $$ The open $\Omega$ is meeting of composantes connexes of $B_1(r_0)$, so $\Omega$ is of Stein~; besides $\varphi_1:\Omega\to [-\infty,r_0[$ is exhaustive. State $$ \varphi_\nu = \max\big(\varphi_2,\nu(\varphi_1-r_0+1)\big). $$ For $\nu > \sup_\Omega\varphi_2$, the application $\varphi_\nu:\Omega\to [-\infty,\nu[$ is exhaustive, while for $\nu\geq\ell$ has on $$ \liminf_{\varphi_1(z)\to-\infty}{\varphi_\nu(z)\over\varphi_1(z)} =\liminf{\varphi_2\over\varphi_1}=\ell. $$ As the théorème 4.2 applied at $\varphi_1$ and $\varphi_\nu$ on~$\Omega$, prpers comes $$ \overline\mu_{-\infty,\nu}(\bOne_\Omega V) \geq \ell^n\,\overline\mu_{-\infty,1}(\bOne_\Omega V). $$ If $\ell$ is${}> 0$~ (alone case interesting at consider) has on $S_2(-\infty)\supset S_1(-\infty)$, so\break $S_\nu(-\infty) = S_1(-\infty)$ and $\Omega\cap S_1(-\infty) = U\cap S_1(-\infty)= A$. Therefore $$ (dd^c\varphi_\nu)^n(\bOne_A V) = \overline\mu_{-\infty,\nu}(\bOne_A V) \geq \ell^n\,\overline\mu_{-\infty,1}(\bOne_A V). $$ observes On maintenant que the continuation $\varphi_\nu$ décroît to $\varphi_2$ on the open $B_1(r_0-1)$ when $\nu\to+\infty$, so $(dd^c\varphi_\nu)^n$ extends faiblement to $(dd^c\varphi_2)^n$ on $B_1(r_0-1)$ as 2.6~(a) and 2.10. Comme $A$ is compact and that $A\subset S_1(-\infty)\subset B_1(r_0-1)$, and comme $\bOne_A V$ is semi-continuous supérieurement, deduces at the limit $$ \overline\mu_{-\infty,2}(\bOne_A V)= (dd^c\varphi_2)^n(\bOne_A V) \geq \ell^n\,\overline\mu_{-\infty,1}(\bOne_A V).\eqno\square $$ On the classical calculation who tracks, have prpers need of score the mass $\Vert\mu_{-\infty}\Vert$ at split of the function $\varphi^*=e^\varphi$.\`HAS that effect, observes on as the proposal~3.10 that $\mu^*_r=\mu_{\log r}$, of where $$ \overline\mu_{-\infty}(1) = \lim_{r\to 0}r^{-n}\mu^*_r(1) = \lim_{r\to 0}r^{-n}\int_{\{\varphi^* 0$ and $(\varphi^*)^{-1}(0) = 0$. Then $$ (dd^c\varphi)^n= (2\pi\ell)^n\delta_0 $$ where $\delta_0$ is the measure of Dirac at~$0$.} \medskip {\it Démonstration}. The homogeneity of $\varphi^*$ involve $(dd^c\varphi)^n = 0$ on $\bC^n\ssm\{0\}$. On the particular case $\varphi^*(z)=|z|^2$, finds on $(dd^c\varphi^*)^n= 4^n\,n!\,d\lambda$~($d\lambda = {}$ measure of Lebesgue), of where~$\overline\mu_{-\infty}(1) = (4\pi)^n$ and $(dd^c\varphi)^n =\overline\mu_{-\infty} = (4\pi)^n\delta_0$. The general case results of the théo\-rème~4.2.\hfil\square \medskip {\statement Example 4.6.\pointir} HAS title of illustration of those that precedes, watch the cases where $\varphi(z) = \log\max(|z_1|,\ldots,|z_n|)$ on~$\bC^n$. Comme $\varphi$ no depends that of $n-1$ variable at the voisinage of all dot of the complementary of the\lguil\?diagonale\?\rguil~ $\Delta = \{|z_1|= \cdots = |z_n|\}$, deduces by homogeneity of $e^\varphi$ that $(dd^c\varphi)^{n-1} = 0$ on $\bC^n\ssm\Delta$. The proposal 3.7 watch whereas the measure $\mu_r$ is at support on the distinguished edge $$ \Gamma(r) = \big\{|z_1| = \cdots = |z_n| = e^r\big\}=S(r)\cap\Delta $$ of the polydisque $B(r)$. Comme $\mu_r$ is invariante by the rotations preserving~$B(r)$, and comme $\Vert\mu_r\Vert = (2\pi)^n$ as~4.5, prpers results that $\mu_r= d\theta_1\wedge\ldots\wedge d\theta_n$ with $z_j=e^{r+i\theta_j}$, $1\leq j\leq n$. Obtains On besides : $$ \eqalignno{ &(dd^c\varphi)^n = (2\pi)^n\delta_0\,,\cr \noalign{\vskip5pt} &(dd^c\varphi)^{n-1}\wedge d\varphi\wedge d^c\varphi = dr\wedge d\theta_1\wedge\ldots\wedge d\theta_n\quad \hbox{sur $\Delta\ssm\{0\}$}.&\square\cr} $$ Go back now at the general case. Be $x\in X$ a dot quelconque and $w ={}$\break $(w_1, w_2,\ldots,w_N)$ the $N$ relative coordinated function at a plongement of a voisinage $U\subset X$ of $x$ on~$\bC^N$, as $w(x) = 0$. Il there is a voisinage $\Omega\compact U$ of $x$ and $r_0 < 0$ tel that the function $\varphi_1(z) = \log|w(z)|^2 :\Omega\to{} ]\infty,r_0[$ be exhaustive. The formula 4.4 gives then (cf.\ [Of5] ) : $$ \overline\mu_{-\infty,1}(1)= (4\pi)^n\,\nu([X],x) $$ where $\nu([X],x)$ is the number of Lelong $x$ of the current of integration of $X$ on~$\bC^N$, equal as P.~Thie [Th] At the multiplicity algébrique $m(X, x)$ of $X$ at the dot~$x$. Concludes that $$ \overline\mu_{-\infty,1} = (4\pi)^n\,m(X,x)\,\delta_x\,. \leqno(4.7) $$ For a function $\varphi$ quelconque, obtains on the following outcome, who is very known at least dans le cas où $X$ is lisse. \medskip {\statement Corollaire 4.8.\pointir}{\it Designate by $\nu(\varphi,x)$ the number of Lelong of $\varphi$ entirely $x\in X$. Then $$ (dd^c\varphi)^n\geq\overline\mu_{-\infty}\geq (2\pi)^n\sum_{x\in X}m(X,x)\,\nu(\varphi,x)^n\,\delta_x\,. $$} {\it Démonstration}. With the précédent notations, the an of the equivalent definitions of the numbers of Lelong is the following : $$ \nu(\varphi,x)=\liminf_{z\to x}{\varphi(z)\over\log|w(z)|}. $$ State then $\varphi_1(z) = \chi(z)\log |w(z)|^2 + A\psi(z)$ where $\chi$ is $\cC^\infty$ at compact support on~$\Omega$, $\chi\equiv 1$ at the voisinage of~$x$, $\psi$ strictly psh of class $\cC^2$ on $X$ and $A > 0$ enough big. As the corollaire 4.3 and the formula (4.7) prpers comes~: $$ \liminf_{z\to x} {\varphi(z)\over\varphi_1(z)}= {1\over 2}\nu(\varphi, x), $$ of where $$ \overline\mu_{-\infty}\geq \Big({1\over 2}\nu(\varphi,x)\Big) ^n\overline\mu_{-\infty,1}\geq (2\pi)^n\,m(X,x)\,\nu(\varphi,x)^n\,\delta_x. \eqno\square $$ \section{5}{Principe of the maximum.} Be $\varphi$ a function of exhaustion psh continues on a complex space $X$. Go Prpers see that the functions plurisousharmoniques on $X$ satisfy the principle of the maximum relatively at the measures of Monge-Ampère associated at $\varphi$. \medskip {\statement Théorème 5.1.\pointir}{\it If $B(r) = \{\varphi < r\}\neq \emptyset$, then $\Vert\mu_r\Vert > 0$ and for all function $V$ psh on~$X$ has on~$:$ $$ \sup_{B(r)}V = \hbox{sup essentiel de $V$ relativement à $\mu_r$.} $$}\vskip-\parskip The example 4.6 watch that the hypothesis of plurisousharmonicité of $V$ on the théorème 5.1 is pertinent. \medskip {\it Démonstration}. No is restrictif of assume $V\leq 0$. Go Prpers then display that $\sup_{B(r)}V = \Vert V\Vert_{L^\infty(\mu_r)}$ applying the formula of Jensen at a function of exhaustion $\varphi'$ very chosen. Be $\psi$ a function strictly psh of class $\cC^2$ on $X$, $z_0\in B(r)\cap X_\reg$ a regular dot and $U\compact B(r)\cap X_\reg$ a voisinage of~$z_0$. For $\varepsilon > 0$ enough small, the function $$ \varphi'(z) = \max\big(\varphi(z),\varphi(z_0), r-\sqrt{\varepsilon} + \varepsilon\psi(z)\big) $$ is equal at $\varepsilon\psi(z) +\hbox{Cte}$ on $U$ and coincides with $\varphi$ at the voisinage of~$S(r)$. The measure $\mu_r$ pouvoir so also very stand defined by $\varphi'$, those that gives $$ \eqalign{ \mu_r(V)&= \int_{-\infty}rdt\int_{B(r)\cap\{\varphi' 0$. Replace now $V$ by $V^p$ and take extend $p$ to~$+\infty$. Prpers comes~: $$ V(z_0)\leq \lim_{p\to+\infty}\bigg[\int_U V^p\,(dd^c\psi)^n\bigg]^{1/p}\leq \lim_{p\to+\infty}\Big[\varepsilon^{-n}\mu_r(V^p)\Big]^{1/p} =\Vert V\Vert_{L^\infty(\mu_r)}. $$ obtains On therefore $$ \sup_{B(r)}V = \sup_{B(r)\cap X_\reg}V \leq \Vert V\Vert_{L^\infty(\mu_r)}. $$ On the autre sense, the inequality $$ \Vert V\Vert_{L^\infty(\mu_r)}\leq \sup_{S(r)}V $$ is evident. If tests on the continuity at left of the function $r\mapsto \Vert V\Vert_{L^\infty(\mu_r)}$ have on so $$ \Vert V\Vert_{L^\infty(\mu_r)}\leq \lim_{tA\}$}. \leqno(6.1) $$ For all function psh $V$ on $X$ and all $r > A$ the théorème 3.4 watch whereas the derive at left $$ {d\over dr_-}\,\mu_r(V) = \int_{B(r)}dd^cV\wedge\alpha^{n-1} \leqno(6.2) $$ is positive increasing at $r$, of where the \medskip {\statement Théorème 6.3.\pointir}{\it The function half courage $r\mapsto M_V(r) = \mu_r(V)$ is convexe increasing on $]A, R[\,$.} \medskip The classical case evoked at first corresponds at the bowl of ray $e^R$ on $\bC^n$ with\hbox{$\varphi(z) = \log|z|$}, $A = -\infty$. Best generally, has on an outcome of convexité for the half at norm $L^p$ defined by $$ M_V^p(r) = \Big[\mu_r(V_+^p)\Big]^{1/p},\qquad p\in[1,+\infty[\,. $$ {\statement Théorème 6.4.\pointir}{\it The function $r\mapsto M_V^p(r)$ is convexe increasing on $]A,R[\,$.} \medskip {\it Démonstration}. By regularisation on se bring au cas où $V$ is psh${}> 0$ of class~$\cC^\infty$. Having given $\varepsilon>0$, consider the function $$ h_\varepsilon(r) = \int_{r-\varepsilon}^r \mu_t(V^p)\,dt =\int_{B(r)\ssm B(r-\varepsilon)}V^p\,\alpha^{n-1} d\varphi\wedge d^c\varphi\,dt,\qquad r\in{}]A+\varepsilon,R[ $$ (the latter equality results of the proposal 3.9~(a)) . Comme $\mu_r(V^p) = \lim_{\varepsilon\to 0} h_\varepsilon(r)$, prpers suffice of test that $h_\varepsilon^{1/p}$ is convexe for all~$\varepsilon > 0$. Owes On so check the inequality $$ h_\varepsilon h''_\varepsilon-\Big(1-{1\over p}\Big)h_\varepsilon^{\prime 2}\geq 0 $$ where the derive second $h''_\varepsilon$ is, say, reckoned at left. As the proposal 3.9~(b) and the hypothesis (6.1) prpers comes $$ \eqalign{ h'_\varepsilon(r) &= \mu_r(V^p) - \mu_{r-\varepsilon}(V^p)\cr &=\int_{B(r)\ssm B(r-\varepsilon)}d\big[V^p\,\alpha^{n-1}\wedge d^c\varphi\big]\cr &=\int_{B(r)\ssm B(r-\varepsilon)}p\,V^{p-1}\,dV\wedge\alpha^{n-1} \wedge d^c\varphi.\cr} $$ The formula (6.2) involves on the other hand $$ h''_\varepsilon(r) = \int_{B(r)\ssm B(r-\varepsilon)}dd^c(V^p)\wedge\alpha^{n-1}. $$ Grace at the inequality of Cauchy-Schwarz obtains on $$ h'_\varepsilon(r)^2\leq \int_{B(r)\ssm B(r-\varepsilon)}V^p\,\alpha^{n-1}\wedge d\varphi\wedge d^c\varphi~ \cdot\int_{B(r)\ssm B(r-\varepsilon)}p^2V^{p-2}\,dV\wedge d^cV\wedge\alpha^{n-1}, $$ and the account (6.5) searched découle of the inequality $$ dd^c(V^p)\geq 2 p(p-1)\,V^{p-2}dV\wedge d^cV. \eqno\square $$ {\statement Corollaire 6.6.\pointir}{\it The defined functions by {\parindent=6.5mm \vskip2pt \item{\rm (a)} $M^{\exp}_V(r) = \log \mu_r(e^V)$, \vskip2pt \item{\rm (b)} $M^\infty_V(r) = \sup_{B(r)}V$, \vskip2pt} are increasing convexes on $]A,R[$.} \medskip {\it Démonstration.} The ownership (a) results of the théorème 6.4 and of the equality $$ \log \mu_r(e^V)=\lim_{p\to+\infty}p\bigg\{\Big[\mu_r\Big(1+{V\over p}\Big)_+^p \Big]^{1/p}-1\bigg\}. $$ The principle of the maximum (théorème 5.1) trains on the other hand $$ \sup_{B(r)}V=\lim_{\lambda\to+\infty}{1\over\lambda}\log \mu_r(e^{\lambda V}) $$ by continuation (b) is consequence of (a).\hfil\square \medskip En vue de the applications at the studio of the spaces fibrés, show prpers now a version with parameter of the théorème 6.4. On se gives a morphisme $\pi : X\to Y$ of spaces analytique of pure dimensions $\dim X = m+n$, $\dim Y = m$ and of the functions $\varphi : X\to[-\infty, +\infty[$ psh continuous, $R : Y\to{}]-\infty,+\infty]$ ) (resp.\ $ A : Y\to [-\infty,+\infty[$ ) semi-Continuous inférieurement (resp.\ supérieurement) Checking the ownerships here-below. \medskip {\statement Hypothesis 6.7.\pointir}{\it {\parindent=6.5mm \vskip2pt \item{\rm (a)} $\pi$ is surjectif, and the fibres $\pi^{-1}(y)$, $y\in Y$ are of pure dimension~$n$. \vskip2pt \item{\rm (b)} $\pi$ is a morphisme of Stein, i.e.\ $Y$ Owns a recouvrement opened $(\Omega_j)_{j\in J}$ as $\pi^{-1}(\Omega_j)$ be of Stein for all $j\in J$. \vskip2pt \item{\rm (c)} $\varphi(x) < R(\pi(x))$ and $A(y) < R(y)$ quel that are $x\in X$, $y\in Y$.\vskip2pt \item{\rm (d)} For all $y\in Y$ and all $r < R(y)$, il there is a voisinage $U$ of $y$ on $Y$ as $\pi^{-1}(U)\cap B(r)\compact X$. \vskip2pt \item{\rm (e)} $(dd^c\varphi)^n\equiv 0$ on the open $\{x\in X\,;\;\varphi(x) > A(\pi(x))\}$.\medskip}} Notes On here still $B(r) = \{\varphi 1$ pouvoir on assume besides $V > 0$. Be $\varepsilon > 0$ arbitrary and $\chi :{}]r-\varepsilon, r[{}\to\bR$ a function $\cC^\infty\geq 0$ no any at compact support. By analogie with the théorème~6.4, enter the auxiliary function $$ h(y) = \int_{r-\varepsilon}^r \mu_{y,t}(V^p)\,\chi(t)\,dt = \int_{\pi^{-1}(y)}V^p\,\chi(\varphi)\,\alpha^{n-1}\wedge d\varphi \wedge d^c\varphi $$ defined on the open $$ U_\varepsilon = \big\{y\in Y\,;\; A(y) + \varepsilon < r < R(y)\big\}. $$ For conclude, prpers suffice of display that $h^{1/p}$ is faiblement psh on~$U_\varepsilon$. If $p > 1$, prpers acts se so of display that $$ h\,dd^ch - \Big(1-{1\over p}\Big)dh\wedge d^ch\geq 0. $$ Are $u$, $v$, $w$ of the real forms of class $\cC^\infty$ on $Y$ at compact support on~$U_\varepsilon$, of bidegrés respective $(m, m)$, $(m,m-1)\oplus (m-1,m)$ and $(m-1,m-1)$. As the théorème of Fubini, what apply on firstly assuming $\varphi$ of class $\cC^\infty$, prpers comes $$ \int_ Y hu = \int_X V^p\,\chi(\varphi)\,\alpha^{n-1}\wedge d\varphi \wedge d^c\varphi\wedge \pi^* u~; $$ the case where $\varphi$ is alone continues deduces by limit décroissante (théorème~2.6). Observes On maintenant que the intégrande is at support on $$ \pi^{-1}(\Supp u)\cap \big(B(r)\ssm B(r-\varepsilon)\big)\compact X $$ (hypothesis 6.7~(d)) and that run $\chi(\varphi)\,\alpha^{n-1}\wedge d\varphi\wedge d^c\varphi$ is $d$ -closed (hypothesis 6.7~(e)). At the half of an integration by parts on $Y$ and of other reverse on $X$ obtains on so successively $$ \leqalignno{ \int_Y dh\wedge v&=\int_X d(V^{p})\wedge\chi(\varphi)\,\alpha^{n-1} \wedge d\varphi\wedge d^c\varphi\wedge \pi^*v,&(6.9)\cr \int_Y dd^ch\wedge w&=\int_X dd^c(V^{p})\wedge\chi(\varphi)\,\alpha^{n-1} \wedge d\varphi\wedge d^c\varphi\wedge \pi^*w.&(6.10)\cr} $$ Assume that the $(m-1,m-1)$ -form $w$ be${}\geq 0$. The equality (6.10) ontre already that $\int_Y dd^ch\wedge w\geq 0$, so $dd^ch\geq 0$ on~$U_\varepsilon$, those that resolves the case~$p = 1$. On the general case $p > 1$, be $\gamma$ a $1$ -real form $\cC^\infty$ on $Y$ and $\gamma^c = i(\gamma^{0,1}-\gamma^{1,0})$. The equality (6.9) combined at the inequality of Cauchy-Schwarz train $$ \eqalign{ \int_Y dh&\wedge\gamma^c\wedge w =\int_X p\,V^{p-1}\wedge dV\wedge\pi^*\gamma^c \wedge\chi(\varphi)\,\alpha^{n-1}\wedge d\varphi\wedge d^c\varphi \wedge \pi^*w\cr &\leq~{1\over 2}\int_X\big(V^{p}\,\pi^*(\gamma\wedge\gamma^c)+ p^2\,V^{p-2}dV\wedge d^cV\big) \wedge\chi(\varphi)\,\alpha^{n-1}\wedge d\varphi\wedge d^c\varphi \wedge \pi^*w\cr &\leq~{1\over 2}\int_Y h\,\gamma\wedge\gamma^c+{p\over p-1}dd^ch\wedge w,\cr} $$ kept account that $dd^cV^p\geq p(p-1)\,dV\wedge d^cV$. Comme this is vrai for all form $w\geq 0$, deduces at the sense of the common the inequality $$ dh\wedge\gamma^c + \gamma\wedge d^ch\leq h\,\gamma\wedge\gamma^c+{p\over p-1}\, dd^ch. $$ Observe that $h$ is all around${}> 0$ on $U_\varepsilon$ as 4.1~; if take prpers extend now $\gamma$ to $dh/h$, prpers comes the awaited inequality $$ {1\over h}\,dh\wedge d^ch\leq {p\over p-1}dd^ch. $$ For see that $h$ is locally majorée on~$Y$, prpers suffice of watch the case where $V\equiv 1$. The equality (6.9) watch whereas $dh= 0$, so $h$ is locally constante on $X_\reg$.\hfil\square \medskip The proposal 6.8 contains in fact the best general outcome following, who be our principal objective. \medskip {\statement Théorème 6.11.\pointir}{\it The functions on $Y\times\bC$ defined by $$ (y,z)\mapsto M_V(y,\Re z),\quad M^p_V(y,\Re z), \quad M^{\exp}_V(V,\Re z),\quad M^\infty_V(y,\Re z) $$ are faiblement psh on the open $$ \big\{(y,z)\in Y\times\bC\,;\; A(y) < \Re z < R(y)\big\}. $$} {\it Démonstration}. Considers On the morphisme $$ \tilde\pi = \pi\times \Id : X \times\bC \to Y\times \bC $$ and caters on $X\times\bC$, $Y\times\bC$ of the functions $$ \tilde\varphi(x, z) = \varphi(x) - \Re z,\quad \tilde R(y,z) = R(y) - \Re z,\quad \tilde A(y, z) = A(y) - \Re z, $$ so that the hypotheses 6.7~(has-e) have checked relatively at that data. If $\tilde V(x, z) = V(x)$ has on by building $$ \tilde\mu_{(y,z),0}(\tilde V) = \mu_{y,\Re z}(V), $$ and the théorème 6.11 découle so of the proposal 6.8.\hfil\square \medskip {\statement Corollaire 6.12.\pointir}{\it Are $(X_j)_{1\leq j\leq k}$ of the spaces of Stein of pure dimension $n_j$ and $\varphi_j : X_j\to [-\infty,R_j[$ of the functions psh continuous exhaustive tel that $(dd^c\varphi_j)^{n_j}\equiv 0$ on the open $\{x\in X_j\,;\;\varphi_j(x) > A_j\}$. If $V$ is psh on $X_1\times\cdots\times X_k$, the functions $$ \eqalign{ M_V(r_1,\ldots,r_k)&=\mu_{1,r_1}\otimes\cdots\otimes\mu_{k,r_k}(V),\cr \noalign{\vskip7pt} M^p_V(r_1,\ldots,r_k)&=M_{V_+^p}(r_1,\ldots,r_k)^{1/p},\cr \noalign{\vskip7pt} M_V^{\exp}(r_1,\ldots,r_k)&=\log M_{e^V}(r_1,\ldots,r_k),\cr \noalign{\vskip7pt} M_V^\infty(r_1,\ldots,r_k)&=\sup_{B(r_1)\times\cdots\times B(r_k)}V\cr} $$ are convexes at the variable $(r_1,\ldots,r_k)\in\prod_{1\leq j\leq k}{} ]A_j,R_j[$ simultaneously, and increasing by report at each of the $r_j$. \vskip1pt{} Best generally, if $X_0$ is a space analytique of pure dimension $n$ and if $V$ is psh on $X_0\times X_1\times\cdots\times X_k$, the function $$ M^p_V(x_0,\Re z_1,\ldots,\Re z_k) = M^p_{V(x_0,\bullet)}(\Re z_1,\ldots,\Re z_k) $$ $($ resp.\ $p =\emptyset$, $\exp$, $\infty)$ is psh on the open $$ X_0\times \prod_{1\leq j\leq k}\big\{A_j<\Re z_j< R_j\big\}\subset X_0\times\bC^k. $$} {\it Démonstration}. Prpers suffice of test the latter affirmation. Reasons On by récurrence on~$k$. For $k = 1$, the théorème 6.11 applied at $\pi : X = X_0\times X_1\to X_0 = Y$ and $\varphi=\varphi_1$ displays that the function $$ (x_0,z_1)\mapsto M^p_V(x_0,\Re z_1) $$ is faiblement psh. If besides $V$ is continuous, that function is séparément continues at $x$ and convexe $\Re z$, so continues at $(x_0,\Re z_1)$. Thanks to the corollaire 1.12, $M^p_V(x_0,\Re z)$ is so psh. The case where $V$ is quelconque obtains se typing $V$ comme limit décroissante of functions psh continuous. For $k > 1$, the ownership results at the order $k$ of his validité at the orders $1$ and $k-1$ stating $$ W(x_0,x_1,\ldots,x_{k-1},z_k)=M^p_{V(x_0,\ldots,x_{k-1},\bullet)}(\Re z_k) $$ and observing that $$ M^p_V(x_0,\Re z_1,\ldots,\Re z_k) = M^p_{W(x_0,\bullet,z_k)}(\Re z_1,\ldots, \Re z_{k-1}).\eqno\square $$ Finish Prpers that paragraph at réexaminant at the light of the précédent outcomes the inequality of convexité of P. Lelong, who measures of manner needs the variations of growth of a function psh on a space fibré along the different fibres. That inequality has be used by H. Skoda [Sk2] for build a first against-example at the problem, stated by J.-P.\ Serre at 1953, of know if a fibré at base and at fibre of Stein has luire-even of Stein~; see also [Of1], [Of2] for d'autre against-examples and [Of3] for a simple and rapid building. Be $\Omega$ a space of Stein irréductible of dimension~$m$, who will act the role of base of the fibré, and $X$ a space of Stein of pure dimension~$n$, who will be the fibre. Assumes On who il there is of the functions $\psi:\Omega\to [-\infty,R[\,$, $\varphi:X\to[-\infty,+\infty[$ psh continuous exhaustive tel that $$ (dd^c\psi)^m = 0\quad\hbox{sur}\quad \{\psi> A\},\qquad (dd^c\varphi)^n = 0\quad\hbox{sur $\{\varphi > 0\}$}. $$ By example if $X$ is a variety algébrique affine of dimension~$n$, il there is a morphisme ended $F :X\to\bC^n$ (théorème of normalisation of Noether), and prpers suffice of seize $\varphi(z) = \log\Vert F(z)\Vert$ where $\Vert~~\Vert$ is a norm on $\bC^n$~; the same reasoning costs locally on $\Omega$ for the existence of~$\psi$. Are $V$ a function psh on $\Omega\times X$ and of the real $a,b,c,r$ tel that $A0$. The ownership of convexité of the corollaire~6.12 watch that $$ M_V^\infty(b,r)\leq M_V^\infty(a,\sigma r) + \Big(1-{1\over\sigma}\Big)\; \Big[M_V^\infty(c, 0) - M_V^\infty(a,\sigma r)\Big] $$ with $\sigma = {c-a\over c-b}$. II results of the théorème 7.5 showed at the paragraph as has on $M_V^\infty(a,r)\to+\infty$ when $r\to+\infty$, dès que $V$ is no constante on at least a fibre $\{z\}\times X$, $z\in B(a)$. Il there is then a constante $r_0$ dépendant of $a,b,c,V$ tel that $$ M_V^\infty(b,r) < M_V^\infty(a,\sigma r)\quad\hbox{pour $r > r_0$, où}~~ \sigma={c-a\over c-b}. \leqno(6.13) $$ If $\omega$ is an open relatively compact on~$\Omega$, states on now $$ M_V^\infty(\omega\,;\,r)=\sup_{\omega\times B(r)}V. $$ Thanks to an elementary reasoning of compacité and of connexité (cf.\ [The2], théorème 6.5.4) deduces on then of (6.13) the following outcome~: \medskip {\statement Corollaire 6.14{\rm(Inequality of P. Lelong)}.\pointir}{\it Are $\Omega$ a complex space irréductible, $\omega_1,\,\omega_2$ deux opened relatively compact on $\Omega$ and $V$ a function psh on $\Omega\times X$, assumed no constante on at least a fibre $\{z\}\times X$. Then il there is a constante $\sigma > 1$ no dépendant that of $\omega_1,\omega_2,\Omega$, and a constante $r_0$ dépendant outrer of $V$, tel that for all $r > r_0$ have on $$ M_V^\infty(\omega_2\,;\,r) < M_V^\infty(\omega_1\,;\,\sigma r). $$\vskip-\parskip} On the practical applications states se the problem of the explicit calculation of the constante~$\sigma$. The inequality (6.13) contributes a theoretical answer completes at that problem. If $\omega_1,\omega_2\compact\Omega$ are of the open of $\bC$, searches on a function harmonique $\psi$ on $\Omega\ssm\overline\omega_1$ who extends to $0$ on $\partial\omega_1$ and to $1$ on $\partial\Omega$ (resp.\ To $+\infty$ if $\partial\Omega$ is of capacity~$0$ )~; prolonger on $\psi$ by $0$ on $\omega_1$ and states on $b = \sup_{\omega_2}\psi$. All constante $\sigma > {1\over 1-b}$ (resp.\ $\sigma>1$ ) Attends then at the question. The case where the base $\Omega$ is of dimension $m > 1$ goes back at resolve a problem of Dirichlet analogous for the equation of Monge-Ampère $(dd^c\psi)^m=0$ on~$\Omega\ssm\overline\omega_1$. Prpers~est easy of see that the constante $\sigma$ thus obtained is the good possible. A calculation élé\-mentaire watch at effect that the function $$ \chi(t,r) = \exp\Big({r\over 1-t}+{1\over(1-t)^2}\Big) $$ is increasing convexe on $[0,1[{}\times[0,+\infty[\,$. The function psh $V =\chi(\psi_+,\varphi_+)$ contradict then (6.13) for all $\sigma\leq{1\over 1-b}$. \bigskip \section{7}{Croissance at the infinite of the functions psh.} Be $X$ a space of Stein irréductible of dimension $n$ and $\varphi : X\to [-\infty,+\infty[$ a function psh continuous exhaustive (has on so here $R = +\infty$ ). Notes On $$ \tau(r)=\Vert\mu_r\Vert=\int_{B(r)}\alpha^n $$ where $\alpha = dd^c\varphi$, the volume of the pseudoboule $B(r) = \{\varphi 0$ dépendant of $V,\varepsilon,r_0$ tel that for all $r\geq r_0$ have on $$ (l-\varepsilon)r\int_{B(r_0)}dd^cV\wedge\alpha^{n-1}\leq\mu_r(V_+)+C\,\tau(r). $$} {\it Démonstration}. State $V_\nu =\max(V,-\nu)$, $\nu\in\bN$. The measures $dd^cV_\nu\wedge\alpha^{n-1}$ convergent faiblement to $dd^cV\wedge \alpha^{n-1}$ when $\nu\to+\infty$, by continuation$$ \liminf_{\nu\to+\infty}\int_{B(r_0)}dd^cV_\nu\wedge\alpha^{n-1} \geq \int_{B(r_0)}dd^cV\wedge\alpha^{n-1}. $$ Il there is so $\nu\in\bN$ (dépendant of $V,\varepsilon,r_0$ ) as $$ \int_{B(r_0)}dd^cV_\nu\wedge\alpha^{n-1}\geq (1-\varepsilon)\int_{B(r_0)}dd^cV\wedge\alpha^{n-1}. $$ The formula 3.4 applied at $V_\nu$ gives on the other hand $$ (r-r_0)\int_{B(r_0)}dd^cV_\nu\wedge\alpha^{n-1}\geq \mu_r(V_\nu)-\int_{B(r)}V_\nu\,\alpha^n \leq\mu_r(V_+)+\nu\,\tau(r). $$ That deux inequalities combined train the proposal 7.1 with $$ C=\nu+{(1-\varepsilon)r_0\over\tau(r_0)}\int_{B(r_0)}dd^cV\wedge\alpha^{n-1}. \eqno\square $$ On all the continuation of that paragraph, take prpers a hypothesis of moderation on the growth of the volume of $X$. \medskip {\statement Hypothesis 7.2.\pointir}{\it $\displaystyle\lim_{r\to +\infty} {\tau(r)\over r}= 0$.} \medskip Obtain Prpers then comme immediate consequence of the proposal 7.1 the fundamental inequality following : \medskip {\statement Corollaire 7.3.\pointir}{\it Under the hypothesis $(7.2)$, has on for all function $V$ psh~$:$ $$ \int_{X}dd^cV\wedge\alpha^{n-1}\leq{\,}\liminf_{r\to+\infty}~{1\over r}\mu_r(V_+). $$} That inequality have exploited principally at the half of the lemme following : \medskip {\statement Lemme 7.4.\pointir}{\it Are $\psi$ a function strictly psh of class $\cC^2$ on $X$ and $r_1 < r_2$ with $B(r_2)\neq\emptyset$. Then il there is a constante $C(r_1,r_2) > 0$ tel that for all function psh $V$ have on$~:$ $$ \int_{B(r_1)}dd^cV\wedge(dd^c\psi)^{n-1}\leq C(r_1,r_2) \int_{B(r_2)}dd^cV\wedge\alpha^{n-1}. $$} {\it Démonstration}. State $\varphi' = \max(\varphi, r_1 + \varepsilon\psi + \sqrt{\varepsilon})$ where $\varepsilon > 0$ has chosen enough small so that $\varphi'=\varphi$ at the voisinage of $S(r_2)$ and $\varphi' = r_1 + \varepsilon\psi + \sqrt{\varepsilon}$ on~$B(r_1)$. As the théorème of Stokes prpers comes $$ \int_{B(r_2)}dd^cV\wedge(dd^c\varphi)^{n-1}= \int_{B(r_2)}dd^cV\wedge(dd^c\varphi')^{n-1}\geq \varepsilon^{n-1}\int_{B(r_2)}dd^cV\wedge(dd^c\psi)^{n-1}. \eqno\square $$ {\statement Théorème 7.5.\pointir}{\it All function psh $V$ on $X$ checking the an of the hypothesis of growth here-below is constante~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle\liminf_{r\to+\infty}~{1\over r}\mu_r(V_+)=0$. \vskip2pt \item{\rm(b)} $\displaystyle\sup_{B(r)}V = o\Big({r\over \tau(r)}\Big)$ when $r\to+\infty$.}} {\it Démonstration}. The théorème 5.1 gives $$ \mu_r(V_+)\leq\Vert\mu_r~\Vert\sup_{B(r)}V_+ = \tau(r)\sup_{B(r)}V_+, $$ so the hypothesis 7.5~(b) involves 7.5~(a). Under the hypothesis 7.5~(a), the corollaire 7.3 and the lemme 7.4 display that $dd^cV = 0$, i.e.\ $V$ Is pluriharmonique. For all $x\in X$ the function $z\mapsto \tilde V(z)=\max(V(z), V(x))$ checks still 7.5~(a), prpers is so pluriharmonique. As the principle of the maximum $-\tilde V$ is constante on~$X$ ($X$ having assumed irréductible), i.e.\ $V\leq V(x)$~; $V$~est So constante.\hfil\square \medskip On the situation usuelle of the space hermitien $\bC^n$ and of the function of exhaustion $\varphi(z) = \log|z|$, the théorème 7.5 redonner (with a démonstration a little best simple) an owed outcome at N.~Sibony And P.-M. Wong. Define the order logarithmique $\rho(V)$ of a function psh $V$ on $\bC^n$ (resp.\ Of an entire function $F$ ) by $$ 1 + \rho(V) = \limsup_{r\to +\infty} {\log\sup_{|z| 0$ has on $$ V_+(z)\leq\varphi(z)^{1+\rho+\varepsilon},\qquad\hbox{(resp.}~~ |F(z)|\leq\exp\big(\varphi(z)^{1+\rho+\varepsilon})\big) $$ when $|z|$ is enough big. At particular, all polynôme is of order logarithmique any, and all entire function of order logarithmique ended is of order any at the sense usuel. \medskip {\statement Corollaire 7.6{\rm([SW])}.\pointir}{\it Be $F$ an entire function no constante of order logarithmique $\rho < 1$, and $X$ a composante irréductible of the hypersurface $F^{-1}(0)$. Then all function psh $V$ on $X$ of order logarithmique${}< 1-\rho$ is constante. At particular, the functions psh and holomorphes limited on $X$ are constantes.} \medskip {\it Démonstration}. The théorème 7.5 brings prpers at price the volume of $X$ relatively at $\varphi$, those that is a classical problem. The current of integration on $X$ is at effect majoré by ${1\over2\pi}dd^c\log|F|$ en vertu de the equation of Lelong-Poincaré (cf.\ [The1])~; has on so $$ \tau(r) = \int_{X\cap\{\varphi 0$ tel that for all function $F\in A_\varphi(X)$ no any have on~$:$ $$ \ord\nolimits_a(F)\leq C(a) \,\delta_\varphi(F). $$} {\it Démonstration}. Be $(z_1,z_2,\ldots,z_n)$ a system of coordinated local on~$X$, centred at~$a$, as the bowl $|z|\leq\varepsilon$ be relatively compact on~$X$. The lemme 7.4 trains the existence of a constante $C_1 > 0$ tel that $$ \int_{|z|\leq\varepsilon}[Z_F]\wedge(dd^c|z|^2)^{n-1}\leq C_1\int_{X}[Z_F]\wedge\alpha^{n-1}. $$ The corollaire 8.4 results then of the proposal 8.3 and of the classical inequality of P.~Lelong [The1]~: $$ {1\over (4\pi\varepsilon^2)^{n-1}} \int_{|z|\leq\varepsilon}[Z_F]\wedge(dd^c|z|^2)^{n-1}\geq \ord\nolimits_a(F).\eqno\square $$ Using a classical reasoning remontant à Poincaré and developed by Siegel [If1], [If2], go prpers now at deduce a théorème of algébricité very general. \medskip {\statement Théorème 8.5.\pointir}{\it The grade of transcendence on $\bC$ of the body $K_\varphi(X)$ of the functions $\varphi$ -ra\-tion\-nelles is as~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $0\leq\deg \tr_\bC K_\varphi(X)\leq n = \dim X.$ \vskip2pt \item{\rm(b)} If $\deg\tr_\bC K_\varphi(X) = n$, then $K_\varphi(X)$ is an extension of ended type of~$\bC$.\medskip}} {\it Démonstration}. Are $F_1,\ldots,F_N$ of the functions $\varphi$ -polynomiales, $(k_1,\ldots,k_N)$ a\hbox{$N$ -uplet} of entire${}\geq 0$, and $P\in\bC[X_1,\ldots,X_N]$ a polynôme of $N$ variable as $\deg_{X_j}P\leq k_j$ and $P(F_1,\ldots,F_N)\not\equiv 0$. Has On then $$ \eqalign{ &\log_+|P(F_1,\ldots,F_N)|\leq \sum_{1\leq j\leq N}k_j\log_+|F_j|+\hbox{Cte},\cr &\delta_\varphi(P(F_1,\ldots,F_N))\leq \sum_{1\leq j\leq N} k_j\,\delta_\varphi(F_j).\cr} $$ The corollaire 8.4 gives so the inequality $$ \ord\nolimits_aP(F_1,\ldots,F_N)\leq C(a) \sum_{1\leq j\leq N}k_j\,\delta_\varphi(F_j).\leqno(8.6) $$ Assume $F_1,\ldots,F_N$ algébriquement independent. Then the dimension of the space vectoriel of the polynômes $P(F_1,\ldots,F_N)$ is equal at $(k_1+1)\ldots (k_N+1)$. For all entire $s\geq 0$ and all dot $a\in X_\reg$ given, the linear system homogène $$ {\partial^\nu\over \partial z^\nu}P(F_1,\ldots,F_N)_{|z=a}=0,\qquad \nu\in\bN^n,~~|\nu|\leq s, $$ admits a solution no any dès que that dimension exceeds the number of equations, equal at ${n+s\choose n}\leq {1\over n!}(n+s)^n$. The function $P(F_1,\ldots,F_N)$ cancels se then at least at the order $s+1$ at the dot~$a$, and the election of $s$ as $s\leq C(a)\sum k_j\,\delta_\varphi(F_j)0$ and $k\to+\infty$. Comes at the limit $$ q_1\ldots q_nd\leq{1\over n!}\bigg[C(a) \sum_{1\leq j\leq n}q_j\,\delta_\varphi(F_j)+(d-1)\,\delta_\varphi(F_{n+1})\bigg]^n, $$ and the election $q_j = 1/\delta_\varphi(F_j)$ gives the majoration explicit awaited of the grade~: $$ d\leq {(n C(a))^n\over n!}~\delta_\varphi(F_1)\cdots\delta_\varphi(F_n). \eqno\square $$ Signal that the théorème 8.5~(b) no say nothing as for the algèbre $A_\varphi(X)$ prpers-same~; comme prpers see prpers at the\S10 , prpers pouvoir se fort although the algèbre $A_\varphi(X)${\it no~soit~pas of ended type}. HAS title of application, consider the particular case where $X$ is an under-ensemble analytique (closed) of pure dimension $n$ on~$\bC^N$, catered of the function of exhaustion usuelle\hbox{$\varphi(z) = \log(1+|z|^2)$}. The associé métrique $\alpha = dd^c\varphi$ identifies se with the métrique of Fubini-Study of the projectif space~$\bP^N$, while the métrique $\beta = dd^ce^\varphi$ coincide with the métrique hermitienne flat of~$\bC^N$. The proposal 3.10 involves the accounts $$ \eqalign{ &\int_{X\cap\{|z| 0$~: $$ i\partial\overline\partial \log(\varepsilon + |\sigma|^2) = i\partial\bigg[{\langle\sigma,D\sigma\rangle\over\varepsilon+|\sigma|^2} \bigg]={\varepsilon\langle D\sigma,D\sigma\rangle\over (\varepsilon+|\sigma|^2) ^2}-{|\sigma|^2\over\varepsilon+|\sigma|^2}\,ic(L). $$ The formula of Jensen 3.4 applied at the function $V = {1\over 2} \log(\varepsilon+|\sigma|^2)$ give then, kept account that $V\geq \log \varepsilon^{1/2}$~: $$ \eqalign{ \int_0^rdt&\int_{B(t)}{\varepsilon\langle D\sigma,D\sigma\rangle \over(\varepsilon+|\sigma|^2) ^2}\wedge\alpha^{n-1}\cr &\leq\int_0^rdt\int_{B(t)}{|\sigma|^2\over\varepsilon+|\sigma|^2}\, ic(L)\wedge\alpha^{n-1}+\mu_r(V)-\mu_0(V)- \int_{B(r)\ssm B(0)}V\,\alpha^n\cr &\leq r\int_X\big[ic(L)\wedge\alpha^{n-1}\big]_++\mu_r(V) +\tau(r)\,\log\varepsilon^{-{1\over 2}},\cr} $$ where $[ic(L)\wedge\alpha^{n-1}]_+$ designates the positive part of the measure $c(L)\wedge\alpha^{n-1}$. Divide that inequality by $r$ and take extend $r$ to~$+\infty$. Comme $V\leq\log_+|\sigma|+{1\over 2}\log(1+\varepsilon)$, prpers comes $$ \int_X{\varepsilon\langle D\sigma,D\sigma\rangle\over (\varepsilon+|\sigma|^2)^2}\wedge\alpha^{n-1} \leq\limsup_{r\to+\infty}~{1\over r}\,\mu_r(\log_+|\sigma|) +\int_X\big[ic(L)\wedge\alpha^{n-1}\big]_+\,. $$ When $\varepsilon$ extends to~$0$, the term ${\varepsilon\langle D\sigma,D\sigma\rangle\over (\varepsilon+|\sigma|^2)^2}$ converge faiblement to the current of integration $2\pi[Z_\sigma]$ associé at the diviseur of the zero of~$\sigma$. Obtains On so the following generalisation of the inequality~8.3. \medskip {\statement Proposal 8.9.\pointir}{\it For all section holomorphe $\sigma\not\equiv 0$ of~$L$, has on $$ 2\pi\int_X [Z_\sigma]\wedge\alpha^{n-1} \leq \delta_\varphi(\sigma) + \delta_\varphi(L) $$ where $\delta_\varphi(\sigma)$, $\delta_\varphi(L)$ designate the \lguil\?Grades\?\rguil\ Respective of $\sigma$ and $L~:$ $$ \eqalign{ \delta_\varphi(\sigma) &= \limsup_{r\to+\infty}~{1\over r}\mu_r(\log_+|\sigma|),\cr \delta_\varphi(L) &= \int_X\big[ic(L)\wedge\alpha^{n-1}\big]_+\,.\cr} $$} The théorème of algébricité se énonce now comme tracks. \medskip {\statement Théorème 8.10.\pointir}{\it Designates On by $K_\varphi(X,L)$ the body of the functions méromorphes on $X$ of the form $\sigma_1/\sigma_2$ with $\sigma_j\in H^0(X,\cO(L^m))$, $m\in\bN$, $\delta_\varphi(\sigma_j) < +\infty$, $j =1,2$. If the fibré $L$ is of grade $\delta_\varphi(L)$ ended, then~$:$ {\parindent=6.5mm \vskip2pt \item{\rm (a)} $0\leq \deg\tr K_\varphi(X,L) \leq n = \dim X~;$ \vskip2pt \item{\rm (b)} If $\deg\tr K_\varphi(X,L) = n$, the body $K_\varphi(X,L)$ is of ended type.\medskip}} {\it Démonstration}. Are $F_1 = \sigma'_1/\sigma_1,\ldots, F_N = \sigma'_N/\sigma_N$ of the elements of $K_\varphi(X,L)$ with $\sigma_j,\sigma'_j\in\cO(L^{m_j})$, and $P\in\bC[X_1,\ldots,X_N]$ a polynôme as $\deg_{X_j} P \leq k_j$. State $$ \sigma=P\Big({\sigma'_1\over\sigma_1},\ldots,{\sigma'_N\over\sigma_N}\Big) \sigma_1^{k_1}\cdots\sigma_N^{k_N}\in H^0(X,\cO(L^m)),\qquad m=\sum k_jm_j. $$ The inequality (8.6) generalises se then under the following form : $$ \ord\nolimits_a(\sigma)\leq C(a)\sum_{1\leq j\leq N} k_j\Big[\max(\delta_\varphi(\sigma_j),\delta_\varphi(\sigma'_j))+ m_j\delta_\varphi(L)\Big], \leqno(8.11) $$ and the rest of the démonstration is identical at that of 8.5.\hfil\square \vskip1.5cm {\hugebf B. Geometrical characterisation\vskip0pt \strut\phantom{B.}des Varieties algébriques affines.\par} \vskip7mm\rm \section{9}{Énoncé of the criterion of algébricité.} The object of the paragraphs who track is of display that the varieties algébriques affines have characterised among the spaces of Stein by of the geometrical conditions simple, at know the finitude of the volume of Monge-Ampère and a minoration convenient of the courbure of Ricci. Recall who a variety algébrique affine is by definition an under-variety algébrique closed of a space $\bC^N$. On the case of a space $X$ at isolated singularities, obtain prpers the necessary and sufficient characterisations here-below. \medskip {\statement Théorème 9.1.\pointir}{\it Be $X$ a space complex analytique of dimension $n$, having at the plus an ended number of singular dots. Then $X$ is analytiquement isomorphe at a variety algébrique affine $X$ if and alone if $X$ owns a function of exhaustion $\varphi$ of class $\cC^\infty$ strictly psh having the ownerships{\rm(a), (b), (c)} here-below. {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle\Vol(X) = \int_X (dd^c\varphi)^n <+\infty~;$ \vskip2pt \item{\rm(b)} The courbure of Ricci of the métrique $\beta= dd^c(e^\varphi)$ admits a minoration of the form $$ \Ricci(\beta)\geq-{1\over 2}dd^c\psi, $$ with $\psi\in L^1_{\loc}(X,\bR)\cap\cC^0(X_\reg,\bR)$ and $\psi\leq A\varphi + B$, where $A$, $B$ are of the constantes${}\geq 0~;$ \vskip2pt \item{\rm(c)} $\varphi$ no has who an ended number of dot critics on $X_\reg$.\vskip2pt} If that conditions have checked, the ring $R_\varphi(X)=K_\varphi(X)\cap \cO(X)$ $($ cf. Definition~$8.2)$ is a $\bC$ -algèbre of ended type and of grade of transcendence~$n$. The structure algébrique $X_\alg$ is then defined comme the unique structure algébrique on $X$ whose ring of the regular functions is $R_\varphi(X)$.\medskip} The extension of that characterisation at the case of the spaces analytique having of the singularities quelconques present of the difficulties who have examined at the\S14. The role of the different hypotheses of the théorème 9.1 shares se grosso modo comme tracks. The existence of a function of exhaustion $\varphi$ strictly psh guarantees that $X$ is a variety of Stein, as the solution of the problem of Levi data by H. Grauert [Gr]. Under the hypothesis~(a), the théorème 8.5 involves on the other hand that the body of the functions $\varphi$ -rational is of grade of ended transcendence. The hypothesis~(b), regarding prpers, assures the existence of a sufficient number of functions $\varphi$ -polynomiales, thanks to the assessments $L^2$ of Hörmander-Nakano-Bombieri-Skoda for the operator $\overline\partial$. Signal here what on no pouvoir replace the condition (b) by a condition taking on the courbure of the métrique $dd^c\varphi$ (cf.\ Observes 10.2). Obtains On instead an equivalent condition replacing $\beta$ by a métrique $\gamma$ quelconque tel that $$ \exp(-A_1\varphi-B_1)\leq\gamma\leq\exp(A_2\varphi+B_2, $$ by example the métrique $\gamma = dd^c\log(1 + e^\varphi)$ or $\gamma = dd^c(\varphi^2)$. Lastly the hypothesis~(c) trains as the theory of Morse that $X$ has even characterises of homo\-topie who a complex cellular ended, and so that the cohomologie of $X$ is of ended type. Prpers no know in fact if the hypotheses~(c) is really indispensable, since what assume on $X$ irréductible. Without the hypothesis~(c), pouvoir on already display that $X$ is meeting of an increasing continuation of varieties X algébriques quasi-affines (${}={}$ opened of Zariski of varieties affines), cf.~proposition 13.1. Of that outcome découle the following improvement of the théorème 9.1. \medskip {\statement Théorème $\bf 9.1^{\prime}$.\pointir}{\it The théorème $9.1$ true rest if the hypothesis{\rm (has,b,c)} have weakened comme tracks~$:$ \vskip2pt $\hbox{\rm(a')} =\hbox{\rm(a)}\;:$~ $\Vol(X) = \int_X (dd^c\varphi)^n < +\infty~;$ {\parindent=7.5mm \vskip2pt \item{\rm (b')} $\Ricci(\beta)\geq -{1\over 2}dd^c\psi$, where $\psi\in L^1_\loc(X,\bR) \cap Cc^0(X_\reg,\bR)$ admits an assessment of the form $$ \int_X \exp(c\psi-A\varphi)\,\beta^n < +\infty,\qquad c > 0,\quad A > 0~; $$ \item{\rm (this )} the spaces of cohomologie of grade pair $H^{2q}(X_\reg;\bR)$ are of ended dimension.\medskip}} The hypothesis (has'), (b') train although $X = \bigcup_{k\in\bN}X_k$ with $X_k$ quasi-affine, $X_k\subset X_{k+1}$, and the hypotheses (this ) involves that the continuation $X_k$ is necessarily stationnaire~; therefore, $X$ is algébrique. Observe that the hypotheses (this ) has checked always if $n = 1$~; when $n = 2$ or $n = 3$, prpers équivaloir at assume alone $\dim H^2(X;\bR) < +\infty$, because the groups $H^q(X;\bR)$ are always nuls for $q>n$ when $X$ is of Stein. The vraisemblance of the théorèmes 9.1, 9.1' prpers has be suggested partly by the works of W.~Stoll And D.~Burns On the varieties paraboliques. Recall here the fundamental outcome of W.~Stoll (1980), who characterises the varieties strictly paraboliques of ray quelconque. \medskip {\statement Théorème 9.2{\rm(cf.\ [St2] and [Drunk])}.\pointir}{\it Be $M$ a variety complex analytique connexe of dimension~$n$. Assumes On who il there is a real $R\in{}]0, +\infty]$ and a function $\tau: M\to{}]0,R^2[$ strictly psh exhaustive of class~$\cC^\infty$, tel that $\log\tau$ be psh and checks $(dd^c\tau)^n\equiv 0$ on $M\ssm\tau^{-1}(0)$. Then il there is an application biholomorphe $F : B(R)\to M$ of the bowl of ray $R$ on $M$ tel that $F^*\tau(z) = |z|^2$.} \medskip If raises on the hypothesis of strict plurisousharmonicité of $\tau$, sees on easily that all variety algébrique affine $M$ checks still the condition of the théorème 9.2 (with $R = +\infty$ )~: prpers suffice of choose $\tau(z) = \log|\pi(z)|^2$ where $\pi : M \to \bC^n$ is a morphisme own ended. That observes has conducted D.~Burns At state the problem of the characterisation of tel varieties at terms of functions of exhaustion having of the particular ownerships. Signal at particular the opened problems following. \medskip {\statement Problem 9.3.\pointir}{\it Considers On a variety of Stein $M$ of dimension~$n$, having a function of exhaustion psh $\tau: M\to[0,+\infty[$ of class $\cC^\infty$ tel that $\log\tau$ be psh and checks $(dd^c\log\tau)^n\equiv 0$ on $M\ssm\tau^{-1}(0)$. The variety $M$ is- algébrique affine~$?$} \medskip {\statement Problem 9.4.\pointir}{\it Characterise the varieties $M$ admitting a function of exhaustion $\tau : M\to [0,+\infty[$ strictly psh tel that $\log\tau$ be psh and checks $(dd^c\log\tau)\equiv 0$ out of a compact.} \medskip D.~Burns Has displayed what il there is of the varieties algébriques affines no checking the condition 9.4~: tel is the case of $M = (\bC^*)^n$, $n\ge 2$. Néanmoins, the condition 9.4 has checked by a variety affine\lguil\?Generic\rguil, at know a submerged variety on $\bC^N$ whose the complétion projectif is lisse and transverse at the hyperplan at the infinite. Little after have showed the théorème 9.1, prpers have learnt on the other hand that N.~Mok Have obtained formerly a geometrical condition sufficient (no necessary at general) so that a variety be algébrique affine. \medskip {\statement Théorème 9.5{\rm([Mok 1,2,3])}.\pointir}{\it Be $X$ a variety kahlérienne completes of dimension~$n$, of courbure bisectionnelle positive, tel that {\parindent=6.5mm \vskip2pt \item{\rm(a)} volume$(B(x_0,r))\geq c\,r^{2n},$ \vskip2pt \item{\rm(b)} $0 < \hbox{courbure scalaire} \leq C/d(x_0,x)^2,$} \vskip2pt where $B(x_0,r)$ and $d(x_0,x)$ designate respectively the bowls and the distance géodésiques, and~$c,C > 0$. Then $X$ is biholomorphiquement isomorphe at a variety algébrique affine.} \medskip N. Mok Has deduced of that théorème that all surface $X$ of courbure riemannienne positive checking the hypothesis 9.5 (a), (b) is in fact isomorphe at~$\bC^2$. The analogous outcome at dimension $n > 2$ remains a conjecture. The théorème~9.5 rests essentially on a work [MSY] of Mok, Siu and Yau taking on the resolution of the equation of Poincaré-Lelong on the varieties kählériennes at courbure bisectionnelle${}> 0$. That laid outcome aside, the démonstration of N.~Mok Tracks on her big lines a noticeably parallel démarche at the nôtre. The hypothesis that the courbure bisectionnelle be positive appears cependant enough restrictive, and no allows of cover at general the case of the varieties algébriques affines (the courbure euclidienne of a tel variety is always negative, cf.~\S10). Cite cependant any known outcomes on the cases of the courbures no necessarily positive. Siu And Yau [SY] have showed who a variety kählérienne complete $X$ simply connexe whose the courbure sectionnelle check $$ - {C\over d(x_0,x)^{2+\varepsilon}}< \hbox{courbure sectionnelle} < 0 $$ is biholomorphe at~$\bC^n$. Prpers is likewise if the courbure check $$ \hbox{courbure sectionnelle} \leq {C_\varepsilon\over d(x_0,x)^{2+\varepsilon}} $$ with a constante $C_\varepsilon$ enough small (cf.\ [MSY]). \def\section#1#2{{\baselineskip=15pt\parindent=8.5mm\bigbf \item{#1.} #2\vskip2pt}} \bigskip \section{10}{Nécessité of the conditions on the volume and the courbure.} Go Prpers show here that the conditions (a), (b), (c) of the théorème 9.1 have checked for all under-variety algébrique irréductible $X\subset\bC^N$ of dimension~$n$. Chooses On on that case $\varphi(z) = \log(1+|z|^2)$, so that the métrique $\alpha = dd^c\varphi$ coincide with the métrique of Fubini-Study of the projectif space~$\bP^N$. Comme the adhérence $\overline X$ of $X$ on $\bP^N$ is an under-variety algébrique compact, obtains on $$ \int_X(dd^c\varphi)^n = \int_{\overline X} \alpha^n < +\infty, $$ therefore the condition (a) has checked. As the théorème of Bertini-Sard, the ensemble of the courages critics of $\varphi$ on $X_\reg$ has ended. By continuation the ensemble critical of $\varphi$ is compact. Quitte à disturb slightly $\varphi$ on $\cC^\infty(X,\bR)$ at the voisinage of that compact ([Mil], cor.~6.8), builds on a function $\varphi'$ whose dot critics are no dégénérés. The dots critics of $\varphi'$ are then at ended number [hypothesis (c)]. Prpers remains prpers now at display that $X$ satisfied the condition of courbure 9.1~(b) relatively at the métrique $$ \beta = dd^c(e^\varphi) = dd^c|z|^2 = 2i\sum_{j=1}^N dz_j\wedge d\overline z_j\,, $$ those that go prpers check by an explicit calculation of $\Ricci(\beta_{|X})$ and of~$\psi$. Be $(P_1,\ldots,P_m)$ a system of polynômes générateur for the ideal $I(X)$ of the under-variety $X$ on $\bC[z_1,\ldots,z_N]$, and be $s = \codim X = N-n$. For all couple of multi-rates $$ K = \{k_1<\ldots0$ and at convergence enough rapid so that $\varphi$ be on$\cC^\infty(\bC\ssm E)$. Comme $$ \log{|z-z_j|\over 1+|z_j|}\leq\log(1+|z|), $$ prpers comes $$ \varphi(z)\geq\log{1+|z|^2\over 1+|z|}~; $$ besides $\lim_{z\to z_j}\varphi(z) = +\infty$ for all $z_j\in E$. The function $\varphi$ is so exhaustive on~$X$. On the other hand, $dd^c\varphi = dd^c\log(1+|z|^2)$, so $\int_X dd^c\varphi = 4\pi < +\infty$. Cependant $X$ no is algébrique.\hfil\square \medskip That example displays incidemment what on no pouvoir neither replace the condition 9.1~(b) by a condition taking on the courbure of the métrique $dd^c\varphi$. Is easy of see on the other hand that the algèbres $A_\varphi(X)$ and $R_\varphi(X) = K_\varphi(X)\cap \cO(X)$ coïncident with the algèbre $\cA_E$ of the fractions $\varphi$ -rational of $\bC[z]$ whose pole appar\-keep at~$E$. At effect, all element $f\in\cA_E$ admits visibly a majoration\break\hbox{$\log|f|\leq C_1\varphi + C_2$}, so $\cA_E\subset A_\varphi(X)\subset R_\varphi(X)$~; inversement, all element of $K_\varphi(X)$ is algébrique on $\bC(z)$ by the théorème~8.5, so $$ R_\varphi(X)\subset \bC(z)\cap\cO(X)=\cA_E. $$ is clear that the algèbre $\cA_E$ no is of ended type. \bigskip \section{11}{Existence of a plongement on an open\\ of a variety algébrique.} The paragraphs 11-14 who track have consacrer the démonstration of the suffisance of the criterion of algébricité 9.1'. Assume Prpers here that $X$ is a variety lisse and connexe, and that the given functions $(\varphi,\psi)$ check the conditions 9.1'(has',b'). Takes On on the other hand the additional hypothesis no restrictive $\varphi\geq 0$. State Prpers comme précédemment $\alpha = dd^c\varphi\geq 0$, $\beta = dd^c(e^\varphi)$~; the measures $\mu_r$ have defined comme at the\S3. \medskip {\statement Definition 11.1.\pointir}{\it Be $p\in{}]0,+\infty]$. Notes On {\parindent=6.5mm \vskip2pt \item{\rm(a)} $L^p_\varphi$ (X) the space vectoriel of the measurable applications $f : X \to \bC$ tel what il there is a constante $C\geq 0$ tel that~$:$\vskip5pt $\displaystyle \eqalign{ &\int_X|f|^pe^{-C\varphi}\;\beta^n < +\infty, \kern54.5pt\hbox{si $ 00}L^p_\varphi(X)~;$ \vskip2pt \item{\rm(c)} $A^p_\varphi(X) = L^p_\varphi(X)\cap \cO(X),\kern15pt\hbox{si $ p\in[0,+\infty]$}$~.\medskip}} The interest of that definition appears on both technical outcomes here-below, who have used at of numerous recoveries on the continuation. \medskip {\statement Lemme 11.2.\pointir}{\it Has On the following ownerships~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $1\in A^p_\varphi(X)$ for all $p > 0~;$ \vskip2pt \item{\rm(b)} $L^p_\varphi(X)\subset L^q_\varphi(X)$ and $A^p_\varphi(X)\subset A^q_\varphi(X)$ for all $p\geq q > 0~;$ \vskip2pt \item{\rm(c)} $L^0_\varphi(X)$ is a $\bC$ -algèbre~$;$ \vskip2pt \item{\rm(d)} $A^0_\varphi(X)$ is an under-algèbre intégralement clos of $L^0_\varphi(X)$.\medskip}} {\statement Lemme 11.3.\pointir}{\it Has On the inclusion $A^0_\varphi(X) \subset A_\varphi(X)$.\'Etant Given $f\in A^0_\varphi(X)$ tel that $$ \int_X |f|^p\exp(-C\varphi)\;\beta^n<+\infty, $$ then~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle \delta_\varphi(f) \leq {C - n\over p}\Vol(X)~;$ \vskip2pt \item{\rm(b)} $\displaystyle \int_X |df|_\beta^p \exp\Big[\Big({p\over 2}-C\Big)\varphi\Big]\;\beta^n <+\infty\quad$ if $p\in{}]0,2]~,$\vskip5pt where the norm $|df|_\beta$ has reckoned relatively at the métrique~$\beta$. \medskip}} {\it Démonstration} of 11.2. The proposal 3.10 trains successively $$ \eqalign{ &v(r) := \int_{\{\varphi 0$. Obtains On so 1'inclusion $$ L^p_\varphi(X)\cdot L^q_\varphi(X) \subset L^{{pq\over p+q}}_\varphi(X), \leqno(11.4) $$ and the ownership (c) se ensuit. Check now the affirmation (d). Be $f$ a function méromorphe on $X$ checking an entire equation on $A^0_\varphi(X)$ of the form $$ f^d + a_1f^{d-1}+ \cdots + a_{d-1}f + a_d = 0,\qquad a_j\in A^0_\varphi(X). $$ Of that equation, deduces on the majoration $$ |f|\leq 2\,\max_{1\leq j\leq d}|a_j|^{1/j}, $$ sinon the equality $$ -1 = a_1f^{-1}+ \cdots + a_df^{-d} $$ would conduct at absolute courage at the absurd inequality $$ 1\le 2^{-1}+\cdots+2^{-d}. $$ Therefore, comme $X$ is lisse, $f$ prolonger se at a function holomorphe on $X$, and is clear that $f\in A^0_\varphi(X)$.\hfil\square \medskip {\it Démonstration} of 11.3. (a) has On $\beta^n\geq e^{n\varphi}(dd^c\varphi)^{n-1}\wedge d\varphi\wedge d^c\varphi$, therefore (proposal 3.8) the integral $$ \int_{-\infty}^{+\infty}e^{(n-C)r}\,\mu_r(|f|^p)\,dr =\int_X|f|^pe^{(n-C)\varphi}\,(dd^c\varphi)^{n-1}\wedge d\varphi\wedge d^c\varphi \leq\int_X|f|^pe^{-C\varphi}\,\beta^n $$ has ended. Comme the application $r\mapsto\mu_r(|f|^p)$ is increasing, deduces $$ \mu_r(|f|^p)\leq \exp\big((C-n)(r+1)\big) \int_{r}^{r+1}e^{(n-C)t}\,\mu_t(|f|^p)\,dt\leq C_1e^{(C-n)r} $$ with a constante $C_1\geq 0$. As the inequality of convexité of Jensen and the inequality $\Vert\mu_r\Vert\leq\Vol(X)$, prpers comes $$ \eqalign{ &{\mu_r\big(\log(1+|f|^p)\big)\over\Vert\mu\Vert_r}\leq \log\Bigg[{\mu_r\big(1+|f|^p\big)\over\Vert\mu\Vert_r}\Bigg]\leq (C-n)r+C_2\,,\cr &\delta_\varphi(f)= \limsup_{r\to+\infty}~{1\over r}\; \mu_r(\log_+|f|) \leq {C-n\over p}\,\Vol(X).\cr} $$ (b) For majorer $|dF|_\beta$, observes on that $$ dd^c(|F|^p)\wedge\beta^{n-1} ={p^2\over 2n}\,|F|^{p-2}|dF|^2_\beta\;\beta^n. $$ Thanks to the théorème of Stokes, that equality trains for all $r > 0$~: $$ \int_{B(r)}|F|^{p-2}|dF|^2_\beta\Big(1-{\varphi\over r}\Big)^2e^{(1-C)\varphi} \;\beta^n\leq {2n\over p^2}\int_{B(r)}|F|^p\,dd^c\Big[\Big(1-{\varphi\over r}\Big)^2 e^{(1-C)\varphi}\Big]\wedge\beta^{n-1}. $$ An easy calculation gives on the other hand on $B(r)=\{\varphi0. $$ After passage at the limit when $r\to+\infty$, obtains on so $$ \int_{X}|F|^{p-2}|dF|^2_\beta\;e^{(1-C)\varphi} \;\beta^n\leq C_3\int_X|F|^pe^{-C\varphi}\beta^n. $$ The ownership 11.3 (b) results now of the inequality of Hölder applied at the measure $|F|^pe^{-C\varphi}\beta^n$, at the couple of functions $\smash{(|F|^{-p}|dF|^p_\beta\exp({p\over 2}\varphi)~;~1)}$ and at the exposant conju\-gués~$({2\over p},{2\over 2-p})$.\hfil\square \medskip The existence of functions holomorphes no constantes on $\smash{L^p_\varphi(X)}$ goes to result of the classical assessments of L.~Hörmander [Hö1] for the operator~$\overline\partial$. \medskip {\statement Proposal 11.5.\pointir}{\it The following ownerships have satisfied. {\parindent=6.5mm \vskip2pt \item{\rm(a)} Be $\tau\in L^1_\loc(X)$ a tel function that $$ i\,\partial\overline\partial\tau+ \Ricci(\beta) \geq \lambda\beta $$ where $\lambda$ is a continuous function${}> 0$ on $X$. Be $u$ a $(0,1)$ -form at coefficients $L^2_\loc$ on $X$ tel that $\overline\partial u = 0$ and $$ \int_X \lambda^{-1}|u|^2e^{-\tau}\,\beta^n < +\infty. $$ Then, il there is a function $g\in L^2_\loc(X)$ tel that $\overline\partial g = u$ and $$ \int_X |g|^2e^{-\tau}\,\beta^n \leq \int_X \lambda^{-1}|u|^2e^{-\tau}\,\beta^n < +\infty. $$ \item{\rm(b)} Are $\psi,c,A$ the data of{\rm 9.1'(b')}. If $\rho$ is psh on $X$ and if $u$ check $\overline\partial u = 0$ and $$ \int_X|u|^2\,e^{-\rho-\psi}\,\beta^n<+\infty, $$ then il there is $g\in L^2_\loc(X)$ tel that $\overline\partial g = u$ and $$ \int_X |g|^2e^{-\rho-\psi-2\varphi}\,\beta^n \leq 4\int_X |u|^2e^{-\rho-\psi}\,\beta^n. $$ \item{\rm(c)} Be an ensemble ended $\{x_1,x_2,\ldots,x_m\}\subset X$ and $\rho$ a function psh on $X$ tel that $e^{-\rho}$ be sommable at the voisinage of $x_1,x_2,\ldots,x_m$. Then il there is a function holomorphe $f$ having a jet of order $s$ given at each dot $x_j$ and tel that $$ \int_X |f|^2e^{-\rho-\psi-C_1\varphi}\,\beta^n < +\infty,\qquad \hbox{où $C_1\geq 0$.} $$ At particular, if $\rho\equiv 0$, obtains on $f\in A^b_\varphi(X)$ with $b = {2c\over 1+c}$ and $c$ comme on~{\rm 9.1'(b')}. \vskip2pt \item{\rm(d)} $A^b_\varphi(X)$ is dense on $\cO(X)$ for the topologie of the convergence uniform on all compact.\medskip}} {\it Démonstration}. (a) is classical, see by example H.~Skoda [Sk3]. \medskip (b) Apply (a) with $\tau = \rho + \psi + 2\log(1+e^\varphi)$. Comme $\varphi\geq 0$ has on $\tau\leq \rho+\psi+2\varphi + \log 4$ and the hypothesis 9.1'(b') trains $$ i\,\partial\overline\partial\tau + \Ricci(\beta) \geq {e^\varphi\;dd^c\varphi\over 1+e^\varphi} + {e^\varphi\;d\varphi \wedge d^c\varphi\over (1+e^\varphi)^2} \geq\lambda\beta $$ with $\lambda = (1+e^\varphi)^{-2}$. The assessment (b) results. \medskip (c) Is consequence of (b) thanks to a classical reasoning owed at Bombieri and Skoda~[Sk1]. Are $U_1,\ldots,U_m$ of the opened voisinage deux at deux disjoints of $x_1,\ldots,x_m$ on which $e^{-\rho}$ is locally sommable. Assumes On $U_j$ catered of a system of coordinated local $z^{(j)} = (z^j_1,z^j_2,\ldots,z^j_n)$ centred at $x_j$ and states on $$ \rho_1=\rho+(n+s)\Bigg[\sum_{j=1}^m\chi_j\log|z^{(j)}|^2+C_j\varphi\Bigg] $$ where $\chi_j$ is a function${}\geq 0$ of class $\cC^\infty$ at compact support on~$U_j$, equalises at $1$ at the voisinage of~$x_j$, and $C_j$ a constante ${}\geq 0$ enough big so that $\rho_1$ be psh on~$X$. The constante $n+s$ has chosen here so that the jet of order $s$ of a function $g$ of class $\cC^\infty$, locally sommable for the measure $e^{-\rho_1}\,\beta^n$, be necessarily any at the dots~$x_j$. Be now $P_j(z^{(j)})$ a polynôme of grade${}\leq s$ having the jet imposed at~$x_j$. States On $$ h = \sum_{j=1}^m\chi_j\,P_j(z^{(j)}). $$ The $(0,1)$ -form $$ u:=\overline\partial h = \sum_{j=1}^m\overline\partial\chi_j\;P_j(z^{(j)}). $$ is of class $\cC^\infty$, any at the voisinage of $x_1,\ldots,x_m$, and by building $$ \int_X |u|^2e^{-\rho_1-\psi}\,\beta^n <+\infty~; $$ on has used here takes prpers that $\psi$ have limited locally. As (b), il there is $g\in\cC^\infty(X)$ tel that $\overline\partial g = u$ and $$ \int_X |g|^2e^{-\rho_1-\psi-2\varphi}\,\beta^n <+\infty~. $$ The function $f = h-g$ attends then at the question. If $\rho\equiv 0$, pouvoir on type $$ |f| = \Big[|f|\exp\Big(-{1\over 2}\psi\Big)\Big]\;\exp\Big({1\over 2}\psi\Big) $$ where $|f|\exp(-{1\over 2}\psi)\in L^2_\varphi(X)$ and $\exp({1\over 2}\psi)\in L^{2c}_\varphi(X)$. Prpers se ensuit thanks to (11.4) that $f\in L^b_\varphi(X)$. \medskip (D) shows se at split of (b) exactly comme the lemme 4.3.1 of~[Hö2].\hfil\square \medskip Uses On now the proposal 11.5 for build of numerous functions holomorphes on $X$, and obtain thus a plongement partial of $X$ on~$\bC^N$. Be $x_0 \in X$ a fixed dot. As 11.5~(c) applied with $\rho\equiv 0$, il there is of the functions $f_1,\ldots,f_n\in A^b_\varphi(X)$ tel that $$ df_1\wedge\ldots\wedge df_n(x_0) \neq 0. $$ At particular, $f_1,\ldots,f_n$ are algébriquement independent on~$A_\varphi(X)$. The théorème 8.5 (b) applies se so, those that gives~: \medskip {\statement Proposal 11.6.\pointir}{\it The body $K_\varphi(X)$ of the functions $\varphi$ -rational is an extension of ended type of~$\bC$, of grade of transcendence~$n$.\hfil\square} \medskip Comme go prpers see, is easy of deduce of the précédent outcomes the existence of a morphisme $F : X\to M$ of $X$ on a variety algébrique $M$ of dimension~$n$, who is, out of a hypersurface algébrique $S \subset X$, an isomorphisme of $X\ssm S$ on an open of~$M$. The principal difficulty who will remain at surpass will be of test that $F$ is quasi-surjectif, i.e.\ That $F(X\ssm S)$ is an open of Zariski of~$M$. \medskip {\statement Proposal 11.7.\pointir}{\it The ownerships of following existence have satisfied. {\parindent=6.5mm \vskip2pt \item{\rm(a)} Il there is a function $f_{n+1}\in A^0_\varphi(X)$ tel that $$ f_{n+1}(x_0)=1\quad\hbox{et}\quad \int_X|f_{n+1}|^2\,|df_1\wedge\ldots\wedge df_n|_\beta^{-2}\; \exp(-2\psi-C\varphi)\;\beta^n<+\infty. $$ At particular $\{x\in X\,;\;df_1\wedge \ldots\wedge df_n(x) = 0\} \subset f_{n+1}^{-1}(0)$. \vskip2pt \item{\rm(b)} Il there is of the functions $f_{n+2},\ldots,f_N\in A^0_\varphi(X)$ and an under-variety algébrique irréductible $M\subset\bC^N$ of dimension~$n$ tel that the morphisme $F = (f_1,\ldots,f_N)$ forward $X$ on $M$ and be an isomorphisme analytique of $X\ssm f_{n+1}^{-1}(0)$ on an open lisse of~$M$.\medskip}} {\it Démonstration}. (a) Prpers suffice of apply 11.5~(c) at the function psh $$ \rho = \psi + \log|df_1\wedge\ldots\wedge df_n|^2. $$ If $Z$ is the diviseur of the zero of the $n$ -form holomorphe $df_1\wedge\ldots\wedge df_n$ considered comme section of $\bigwedge^n T^*X$, has on very at effect as the hypothesis 9.1'(b')~: $$ dd^c\rho = dd^c\psi + 2\Ricci(\beta) + 4\pi\,[Z] \geq 0, $$ and $\rho$ is continuous at the voisinage of~$x_0$. Il there is so $f_{n+1}\in\cO(X)$ tel that $f_{n+1}(x_0) = 1$, checking the assessment $L^2$~annoncée. That assessment trains that $f_{n+1}$ cancels se on the support of~$Z$, and as the lemme 11.3~(b) has on $|df_j|\in L^b_\varphi(X)$, of where $$ |f_0|\leq \Big(|f_0|\;|df_1\wedge\ldots\wedge df_n|^{-1}\,e^{-\psi}\Big)\; |df_1|\cdots|df_n|\;e^\psi\in L ^0_\varphi(X). $$ (b) Builds On firstly by récurrence on $j$ of the functions $f_1,\ldots,f_{N_j}\in A^0_\varphi(X)$ with $N_0 = n + 1 < N_1 < N_2 \ldots$ (has taken for $j = 0$ ). As 11.6 the image of the morphisme $$ F_j=(f_1,\ldots,f_{N_j}):X\to \bC^{N_j} $$ has contained on a variety algébrique irréductible $M_j\subset\bC^{N_j}$ of dimension~$n$. Be $\tilde M_j$ the normalisation of~$M_j$. Il there is a diagramme commutatif $$ \matrix{ ~\tilde M_j&\hookrightarrow &\bC^{\tilde N_j}\ni{}&\kern-8pt(z_1,\ldots,z_{\tilde N_j})\cr \noalign{\vskip5pt} \big\downarrow&\phantom{{}\longrightarrow{}}&\big\downarrow~~~~~ &\mathop{\big\downarrow}\kern-7.5pt\raise6pt\hbox{--}~~~~~\cr \noalign{\vskip5pt} ~M_j&\hookrightarrow&\bC^{N_j}\ni{}&\kern-8pt(z_1,\ldots,z_{N_j}) \hbox{\rlap{{}$\;$.}}\cr} $$ Comme the variety $X$ is lisse (and so normal), the morphisme $F_j: X \to M_j$ raises se at a morphisme $$ \tilde F_j=(f_1,\ldots,f_{\tilde N_j}):X\to \tilde M_j. $$ By building, the coordinated functions $z_{N_j+1},\ldots,z_{\tilde N_j}$ (resp.\ $f_{N_j+1},\ldots,f_{\tilde N_j}$ ) Are of the entire algébriques on the ring $\bC[z_1,\ldots,z_{N_j}]/I(M_j)$ (resp.\ On $\bC[f_1,\ldots,f_{N_j}]$ ), so $f_{N_j+1},\ldots,f_{\tilde N_j}\in A^0_\varphi(X)$ as 11.2~(d). Besides, the restriction $$ \tilde F_j : X\ssm f_{n+1}^{-1}(0)\to M_j $$ is extends, because $df_1\wedge\ldots\wedge df_n\neq 0$ on $X\ssm f_{n+1}^{-1}(0)$ as~(a). Comme $\tilde M_j$ is locally irréductible, the image $\tilde F_j(X\ssm f_{n+1}^{-1}(0))$ has contained necessarily on the ensemble of the dots lisses of~$\tilde M_j$. If $\tilde F_j$ is injective on $X\ssm f_{n+1}^{-1}(0)$, the building have finished with $F = \tilde F_j$, $M = \tilde M_j$, $N = \tilde N_j$. Sinon, are deux dots $z_1 \neq z_2$ on $X\ssm f_{n+1}^{-1}(0)$ tel that $\tilde F_j(z_1) = \tilde F_j(z_2)$. The proposal 11.5~(c) watch what il there is a function $g\in A^b_\varphi(X)$ tel that $g(z_1) \neq g(z_2)$. States On $N_{j+1} = \tilde N_j+1$, $f_{\tilde N_j+1} = g$. As 11.6, $g$~est algébrique on $\bC[f_1,\ldots,f_{\tilde N_j}]$, $g$~vérifie so an equation irréductible of the form $$ \sum_{k=0}^da_k(\tilde F_j)\,g^k = 0,\qquad a_k \in \bC[f_1,\ldots,f_{\tilde N_j}],\quad a_d(\tilde F_j)\not\equiv 0. \leqno(11.8) $$ Comme $\tilde F_j$ is extends at the voisinage of $z_1$ and $z_2$, il there is of the dots $w_1$ neighbouring of $z_1$, $w_2$ neighbouring of $z_2$ tel that $\tilde F_j(w_1) = \tilde F_j(w_2) \notin a_d^{-1}(0)$ and $g(w_1)\neq g(w_2)$. This trains that the equation (11.8) is of grade $d\geq 2$. Comme $K_\varphi(X)$ is an extension of ended grade of $\bC(f_1,\ldots,f_n)$ the proceed arrests se necessarily au bout de an ended number of stages.\hfil\square \bigskip \section{12}{Quasi-surjectivité of the plongement.} Restart Prpers the notations of the proposal 11.7. The objective of that paragraph is of show that the image of the morphisme $F : X\ssm f_{n+1}^{-1}(0)\to M$ is an open of Zariski of~$M$. Be $Q\in\bC[z_1,\ldots,z_N]$ a polynôme no any on~$M$, divisible by~$z_{n+1}$, as the hypersurface $Q^{-1}(0)$ contain the singular place~$M_\sing$. States On $$ \check M = M\ssm Q^{-1}(0),\qquad \check X = X\ssm Q(F)^{-1}(0) \subset X\ssm f_{n+1}^{-1}(0), $$ so that $\check M$ is lisse and that the morphisme of restriction $$ \check F : \check X \to \check M $$ is an isomorphisme of $\check X$ on the open $\Omega = \check F(\check X)$. The variety $\check M$ peut être (and will be) identified at an under-variety algébrique affine of $\bC^{N+1}$ via the application $\check M\to \bC^{N+1}$ defined by $$ (z_1,\ldots,z_N)\mapsto\big(z_1,\ldots,z_N,z_{N+1}=Q(z_1,\ldots,z_N)^{-1}\big)~; $$ the morphisme $\check F : \check X \to \check M \subset \bC^{N+1}$ is then given by $\check F = (F,Q(F)^{-1})$. An of the crucial dots of the reasoning is of display that the positive current closed $\check F_*dd^c\varphi$ prolonger se the open $\Omega = \check F(\check X)$ at the variety $M$ all entire. Has On need for that of precise assessments of the mass, who have afforded by the lemme following. \medskip {\statement Lemme 12.1.\pointir}{\it Be $G = (g_1,\ldots,g_m) \in \big[A^0_\varphi(X)\big]^m$ and $\gamma = dd^c \log(1 + |G|^2)$. Then for all entire $k \geq 0$, has on~$:$ {\parindent=6.5mm \vskip4pt \item{\rm(a)} $\displaystyle \int_X (dd^c\varphi)^{n-k}\wedge \gamma^k < +\infty,\qquad 0\leq k\leq n,$ \vskip4pt \item{\rm(b)} $\displaystyle \int_{B(r)}d\varphi\wedge d^c\varphi\wedge (dd^c\varphi)^{n-k-1}\wedge \gamma^k \leq Cr,\qquad 0\leq k\leq n-1,$\vskip4pt where $C$ is a constante${}\geq 0$.\medskip}} {\it Démonstration}. Apply the théorème 2.2 (c) with $\rho = \varphi-r$, $\Omega = \{\rho<0\} = B(r)$ and $V_1 = \ldots = V_k = \log(1+|G|^2) \geq 0$. Prpers comes $$ \int_{B(r)}\beta_k\wedge\gamma^k\leq C_3\int_{B(r)}\big(\log(1+|G|^2)\big)^k\; \beta_0 $$ where, for $a > 0$ and $k \geq 0$, has be stated~: $$ \beta_k=(r-\varphi)^{k+a}\,(dd^c\varphi)^{n-k}+(k+a) (r-\varphi)^{k-1+a}\,d\varphi\wedge d^c\varphi\wedge (dd^c\varphi)^{n-k-1}. $$ has On~: $$ \beta_0 = 2(r-\varphi)^a\,(dd^c\varphi)^n+{1\over 2(1+a)}\,dd^c\beta_1. $$ The théorème of Stokes train so for all $r > 0$~: $$ \int_{B(r)}\beta_0 = 2 \int_{B(r)}(r-\varphi)^a\,(dd^c\varphi)^n \leq 2\,r^a\,\Vol(X). $$ On the other hand, the function $t\mapsto (\log (e^k+t))^k$ is concave on~$[0,+\infty[\,$. Obtains On so for all $p > 0$ the inequality of convexité~: $$ {\int_{B(r)}\big(\log (e^k+|G|^p)\big)^k\;\beta_0\over \int_{B(r)}\beta_0} \leq\Bigg\{\log\Bigg[e^k+ {\int_{B(r)}|G|^p\;\beta_0\over \int_{B(r)}\beta_0} \Bigg]\Bigg\}^k. $$ Comme $g_j\in A^0_\varphi(X)$ [cf.\ Definition 11.1], il there is $p > 0$ enough small and $C_4,C_5\geq 0$ enough big tel that $$ \int_{B(r)}|G|^p\;\beta_0 \leq \exp(C_4r+C_5). $$ The précédent inequalities train then $$ \int_{B(r)}\beta_k\wedge\gamma^k \leq C_6 \int_{B(r)}\big(\log (e^k+|G|^p)\big)^k\;\beta_0 \leq C_7\,r^{k+a}. $$ kept-Account of the definition of $\beta_k$, this involves the lemme 12.1 after replacement of $2r$~à~$r$.\hfil\square \medskip Cater $\bC^{N+1}$ and $\check M \subset \bC^{N+1}$ of the métrique of Fubini-Study $\omega = dd^c\log(1+|z|^2)$. Has On then the théorème of following elongation, whose démonstration have inspired of H.~Skoda [Sk5] and of H.~El Mir [EM]~; See also the article of synthesis of N.~Sibony [Sib]. \medskip {\statement Proposal 12.2.\pointir}{\it Be $T$ the simple extension at $\check M$ of the current $\check F_*dd^c\varphi$, defined by $$ T = \check F_*dd^c\varphi\quad\hbox{sur}~\check F(\check X),\qquad T = 0\quad\hbox{sur}~ \check M\ssm \check F(\check X). $$ Then $T$ is a positive current closed on $\check M$ of total mass $\int_{\check M}T\wedge\omega^{n-1}$ ended.} \medskip {\it Démonstration}. Reckon firstly the mass of~$T$~: $$ \int_{\check M}T\wedge\omega^{n-1}=\int_{\check F(\check X)} \check F_*dd^c\varphi\wedge\omega^{n-1}=\int_{\check X}dd^c\varphi\wedge (\check F^*\omega)^{n-1}. $$ The $(1,1)$ -form $\check F^*\omega$ has given here by $$ \eqalign{ \check F^*\omega &=dd^c\log(1+|\check F|^2)=dd^c\log(1+|F|^2+|Q(F)|^{-2})\cr &=dd^c\log(1+|Q(F)|^2+|F|^2|Q(F)|^2).\cr} $$ The finitude of the mass results then of the lemme 12.1~(a). For all $1$ -real form $v$ of class $\cC^\infty$ at compact support on $M$ and for all multi-rates $J,K\subset \{1,\ldots,N+1\}$ tel that $|J| = |K| = n-2$, displays on now the nullity of the integral $$ I = \int_{\check M} dv \wedge T \wedge dz_J \wedge d\overline z_K, $$ those that will test that $dT = 0$. Be $\chi$ a function of class $\cC^\infty$ on $\bR$ tel that $0 \leq \chi \leq 1$, $\chi(t) = 1$ if $t < 0$, $\chi(t) = 0$ if $t > 1$ and $0 \leq \chi' \leq 2$. By definition of $T$, prpers comes $$ \eqalign{ I &= \int_{\check X} \check F^*(dv) \wedge dd^c\varphi \wedge d\check F_J\wedge \overline{d\check F_K}\cr &= \lim_{r\to +\infty} \int_{\check X}\chi\Big({\varphi\over r}\Big)\; d(\check F^*v) \wedge dd^c\varphi \wedge d\check F_J\wedge \overline{d\check F_K}\;.\cr} $$ The form $\chi({\varphi\over r})\;d(\check F^*v)$ is at support on $\check F^{-1}(\Supp v)\cap\overline{B(r)}\compact \check X$. An integration by parts gives so $$ I= \lim_{r\to +\infty} \pm\int_{\check X} \check F^*v \wedge \chi'\Big({\varphi\over r}\Big)\;{d\varphi\over r}\wedge dd^c\varphi \wedge d\check F_J\wedge \overline{d\check F_K}\;. $$ Thanks to the inequality of Cauchy-Schwarz, that latter integral is majorée by ${2\over r}\sqrt{I_1I_2(r)}$ with $$ \eqalign{ I_1&=\int_{\check X} dd^c\varphi \wedge \check F^*\big(v\wedge \overline v^c\wedge dz_J\wedge d\overline z_J\big)~,\cr I_2(r)&= \int_{\check F^{-1}(\Supp v)\cap B(r)} d\varphi\wedge d^c\varphi\wedge dd^c\varphi \wedge d\check F_K\wedge \overline{d\check F_K}\;.\cr} $$ Comme $v$ is at compact support on~$\check M$, il there is of the constantes $C_1,C_2\geq 0$ tel that $$ \eqalign{ &v\wedge \overline v^c\wedge dz_J\wedge d\overline z_J\leq C_1(\check F^*\omega)^{n-1}~,\quad\cr &d\check F_K\wedge \overline{d\check F_K}=\check F^*(dz_K\wedge d\overline z_K) \leq C_2(\check F^*\omega)^{n-2}\quad \hbox{sur $\check F^{-1}(\Supp v)$}.\cr} $$ The lemme 12.1 (a) and (b) trains then $$ \eqalign{ I_1&\leq C_1\int_X dd^c\varphi \wedge (\check F^*\omega)^{n-1} < +\infty~,\cr I_2(r)&\leq C_2\int_{B(r)} d\varphi\wedge d^c\varphi\wedge dd^c\varphi \wedge (\check F^*\omega)^{n-2}\leq C\,C_2\,r~,\cr} $$ of where $$ |I|\leq \lim_{r\to+\infty} {2\over r}\sqrt{I_1I_2(r)} = 0.\eqno\square $$ As the théorème 15.3 of the appendix, il there is a function psh $V$ and a $(1,0)$ -form $u$ of class $\cC^\infty$ on $\check M$ having the following ownerships, for of the constantes $C_1,C_2,C_3\geq 0$ convenient. \medskip {\statement Ownerships 12.3.\pointir}{\it Has On the $($ in$)$ equalities {\parindent=6.5mm \vskip2pt \item{\rm(a)} $dd^cV \geq T~;$ \vskip2pt \item{\rm(b)} $V(z) \leq C_1\log(1 + |z|^2)~;$ \vskip2pt \item{\rm(c)} $dd^cV - T = \overline\partial u~;$ \vskip2pt \item{\rm(d)} $|u|_\omega \leq C_2(1 + |z|^2)^{C_3}\;$.\medskip}} Consider then the function $\tau = V - \check F_*\varphi$ defined on the open $\Omega = \check F(\check X) \subset \check M$. As 12.3~(a), $\tau$ is psh on~$\Omega$, and besides $\tau \leq V$. Comme $\check F_*\varphi$ extends to $+\infty$ at the voisinage of~$\partial\Omega$, $\tau$ extends to $-\infty$ entirely of~$\partial\Omega$. Therefore, $\tau$ prolonger se at a function psh on~$\check M$, still noted $\tau$, tel that $\tau = -\infty$ on $\check M\ssm\Omega$. \medskip {\statement Corollaire 12.4.\pointir}{\it $\check M\ssm\Omega$ Is a fermé part pluripolaire of~$\check M$.\hfil\square} \medskip The next stage is of display that $\check M\ssm\Omega$ is in fact a hypersurface algébrique of~$\check M$. As 12.3~(c) and the definition of $T$ has on~: $$ 2i\,\partial\overline\partial(V-\check F_*\varphi) = \overline\partial u \quad\hbox{sur}~\Omega, $$ so the $(1,0)$ -form $h$ defined by $$ h =\partial(V - \check F_*\varphi) + {u\over 2i} = \partial\tau + {u\over 2i} \leqno(12.5) $$ is holomorphe on~$\Omega$~; comme $u$ is of class $\cC^\infty$ on~$\check M$, this show at the passage that $\tau$ is of class $\cC^\infty$ on~$\Omega$. Go Prpers now test that $h$ prolonger se at a $1$ -form méromorphe rational on~$\check M$. This goes to result essentially of the assessments 12.3 (b,d) and of the théorème of algébricité 8.5. By building of $F$ and $\check X$, the forms $(df_1,\ldots,df_n)$ define a repérer global of~$T^*\check X$. The forms $(dz_1,\ldots,dz_n)$ constitute so also a repérer of $T^*\check M$ at the-above of the open $\Omega = \check F(\check X)$, and pouvoir on type $$ h = \sum_{j=1}^n h_jdz_j $$ with of the functions $h_j\in\cO(\Omega)$. The principle of the reasoning consister check that the functions $h_j\circ\check F$ are at growth $\varphi$ -polynomiale, at split of the majoration of $\tau = V - \check F_*\varphi$ afforded by 12.3~(b). Takes prpers that prpers no have of minoration of $\tau$ entered an additional difficulty that go prpers short-circuiter researching alone an assessment of the functions $\exp({1\over 2}\tau\circ\check F)\;|h_j(\check F)|$. \medskip {\statement Lemme 12.6.\pointir}{\it Considers On on $X$ the function of exhaustion $$ \check\varphi = \log(1+e^\varphi) + \log(1+|\check F|^2) $$ and the associé métrique $$ \check\alpha = dd^c\check\varphi = \log(1+e^\varphi) + \check F^*\omega. $$ The following ownerships are then checked for of the constantes $p > 0$ enough small and $C_4,C_5$ enough big. {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\displaystyle \int_{\check X}\check\alpha^n < + \infty~;$ \vskip2pt \item{\rm(b)} $\displaystyle \int_{\check X} e^{\tau\circ\check F-C_4\check\varphi}\;\check F^*(ih\wedge\overline h)\wedge \check\alpha^{n-1} < + \infty~;$ \vskip2pt \item{\rm(c)} $\displaystyle \int_{\check X}\Big[ \exp\Big({1\over 2}\tau\circ\check F\Big)\;|Q(F)|^{C_4+1}\; |h_j(\check F)|\Big]^p\;e^{-C_5\varphi}\;\beta^n < + \infty~.$ \medskip}} {\it Démonstration}. (a) is immediate consequence of the lemme 12.1 if observes on that $$ dd^c\log(1+e^\varphi)= {e^\varphi\;dd^c\varphi\over 1+e^\varphi} + {e^\varphi\;d\varphi\wedge d^c\varphi\over (1+e^\varphi)^2} \leq dd^c\varphi+e^{-\varphi}\;d\varphi\wedge d^c\varphi. $$ (b) The assessment 12.3 (b) involves $$ \tau = V - \check F^*\varphi \leq V \leq C_1 \log(1+|z|^2), \leqno(12.7) $$ so the function $\theta = \log(1+e^{\tau\circ\check F})$ satisfied the majoration $$ \theta \leq \log\big(1 + (1+|\check F|^2)^{C_1} \big) \leq C_1\check\varphi+\log 2. $$ The corollaire 7.3 applied at $(\check X,\check\varphi)$ train then $$ \int_{\check X} i\partial\overline\partial\theta \wedge \check\alpha^{n-1} < +\infty. $$ An immediate calculation gives on the other hand $$ i\partial\overline\partial\theta =\check F^*\bigg( {i\partial\overline\partial\tau\over 1+e^{-\tau}}+ {ie^\tau\;\partial\tau\wedge\overline\partial\tau\over (1+e^\tau)^2}\bigg) \geq {1\over 2}\; \check F^*\big(ie^\tau\;\partial\tau\wedge\overline\partial\tau\big)\; e^{-2C_1\check\varphi}, $$ and prpers se ensuit $$ \int_{\check X}e^{\tau\circ\check F-2C_1\check\varphi}\; \check F^*(\partial\tau\wedge\overline\partial\tau)\wedge\check\alpha^{n-1} < +\infty. $$ By definition of $\check\alpha$ has on on the other hand $\check\alpha\geq \check F^*\omega$. The assessment 12.3 (d) trains so $$ \eqalign{ &|\check F^*u|_{\check\alpha}\leq\big(|u|_\omega\big)\circ\check F\leq C_2(1+|\check F|^2)^{C_3}\le C_2\;e^{C_3\check \varphi},\cr \noalign{\vskip5pt} &\int_{\check X}|\check F^*u|_{\check\alpha}^2\; e^{\tau\circ\check F-(C_1+2C_3)\check\varphi}\;\check\alpha^n\leq C_2^2\int_{\check X}\check\alpha^n<+\infty.\cr} $$ The ownership (b) results now of the definition of $h = \partial\tau + {u\over 2i}$ and of the equality $$ |\check F^*u|_{\check\alpha}^2\;\check\alpha^n=n\; \check F^*(iu\wedge\overline u)\wedge\check\alpha^{n-1}. $$ (c) The trivial inequality $$ dd^c\log(1+e^\varphi) \geq {1\over 2}\,dd^c\varphi+{1\over 4}\, e^{-\varphi}\;d\varphi\wedge d^c\varphi $$ trains successively $$ \eqalign{ \check\alpha^{n-1}&\geq {1\over 2^{n-1}}\,(dd^c\varphi)^{n-1}+ {(n-1)\,e^{-\varphi}\over 2^n}\,(dd^c\varphi)^{n-2}\wedge d\varphi \wedge d^c\varphi\cr &\geq 2^{-n}e^{-n\varphi}\;(dd^ce^\varphi)^{n-1} = 2^{-n}e^{-n\varphi}\;\beta^{n-1},\cr \noalign{\vskip7pt} \check F^*(ih\wedge\overline h)\wedge\check\alpha^{n-1}&\geq 2^{-n}e^{-n\varphi}\;|\check F^*h|_\beta^2\;\beta^n.\cr} $$ By definition of $\check\varphi$, has on on the other hand $$ \eqalign{ \check\varphi &= \log(1+e^\varphi)-2\log|Q(F)| + \log\big(1+|Q(F)|^2+|Q(F)|^2|F|^2\big)\cr \noalign{\vskip5pt} &\leq \varphi - 2\log|Q(F)| + C_6\log(1+|F|^2) + C_7 ,\cr} $$ where $C_6 = 1 + \deg(Q)$. The inequality (b) gives prpers so $$ \int_{\check X} e^{\tau\circ\check F-C_4\varphi}\;|Q(F)|^{2C_4}\; (1+|F|^2)^{-C_4C_6}e^{-n\varphi}\;|\check F^*h|_\beta^2\;\beta^n < + \infty $$ and comme $|F|\in L^0_\varphi(X)$, prpers se ensuit $$ \exp\Big({1\over 2}\tau\circ\check F\Big)\;|Q(F)|^{C_4}\; |\check F^*h|_\beta \in L^0_\varphi(X). $$ On the other hand, the function $h_j$ pouvoir type under the form $$ h_j = (-1)^{j-1}\; {h\wedge dz_1\wedge \ldots\wedge \widehat{dz_j}\wedge\ldots \wedge dz_n\over dz_1\wedge\ldots\wedge dz_n}. $$ has On so $$ |h_j(\check F)| \leq |\check F^*h|_\beta\;|df_1|_\beta\ldots\widehat{|df_j|_\beta} \ldots|df_n|_\beta\;|df_1\wedge\ldots\wedge df_n|_\beta^{-1}, $$ and comme $$ \eqalign{ &|df_n|_\beta\,,\,\ldots\,,\,|df_n|_\beta \in L^0_\varphi(X)\kern57pt \hbox{[lemme 11.3 (b)], et}\cr \noalign{\vskip5pt} &|f_{n+1}|\;|df_1\wedge\ldots\wedge df_n|_\beta^{-1} \in L^0_\varphi(X)\kern30pt\hbox{[inégalité 11.7 (a)],}\cr} $$ prpers comes $$ \exp\Big({1\over 2}\tau\circ\check F\Big)\;|Q(F)|^{C_4}\; |f_{n+1}|\;|h_j\circ\check F| \in L^0_\varphi(X). $$ By hypothesis $Q$ is divisible by~$z_{n+1}$, i.e.\ $Q =z_{n+1}R$. The ownership (c) obtains se then multiplying the function here-above by $|R(F)|\in L^0_\varphi(X)$.\hfil\square \medskip Afin de pouvoir work on $X$ plutôt que on~$\check X$, have prpers need of the lemme elementary of elongation here-below. \medskip {\statement Lemme 12.8.\pointir}{\it Be $S = g^{-1}(0)$ a hypersurface of~$X$, and $\theta$ a function psh on $X\ssm S$ tel that $e^\theta\in L^1_\loc(X)$. Then $\theta + \log|g|^2$ prolonger se at a function psh on~$X$.} \medskip {\it Démonstration}. Prpers suffice of display that $\theta + \log|g|^2$ is majorée at the voisinage of all regular dot of~$S$. Pouvoir On so assume that $X$ is an open of $\bC^n$ containing the polydisque closed unity $\overline\Delta^n$, and that $S = \{z_1 = 0\}$. The inequality of half applied at the polydisque $$ (z_1 + |z_1|\,\Delta) \times \Delta^{n-1} \subset X\ssm S $$ for all dot $z\in\Delta^n$, $0 < |z_1| < {1\over 2}$, involves $$ e^{\theta(z)}\leq{1\over\pi^n|z_1|^2}\int_{\Delta^n}e^\theta\;d\lambda. $$ The function $\theta + \log |z_1|^2$ is by continuation majorée at the voisinage of~$S$.\hfil\square \medskip {\statement Proposal 12.9.\pointir}{\it The $1$ -form $h = \sum_{1\leq j \leq n} h_jdz_j$ prolonger se at a $1$ -form méromorphe rational on~$M$.} \medskip {\it Démonstration}. Comme $\check X = X\ssm Q(F)^{-1}(0)$, the lemmes 12.6~(c) and 12.8 display that $$ p\;\log\Big[\exp\Big({1\over 2}\tau\circ\check F\Big)\;|Q(F)|^{C_4+1}\; |h_j\circ\check F|\;\Big] + \log|Q(F)|^2 $$ stretches se at a function psh on $X$. Il there is so an entire $s > 0$ enough big and $\varepsilon > 0$ enough small tel that, if $g$ designates the function holomorphe on $X$ defined by $$ g = Q(F)^sh_j(F), $$ then the function ${1\over 2}\tau\circ F + \log|g|$ is psh on $X$, and $$ \int_X \exp\Big[\varepsilon\Big({1\over 2}\tau\circ\check F+\log|g|\Big) -C_8\varphi\Big] \;\beta^n<+\infty. $$ Reasoning comme on the lemme 11.3 (a), obtains on therefore $$ \limsup_{r\to+\infty}~{1\over r}\;\mu_r\Big[\Big({1\over 2}\tau\circ F + \log|g|\Big)_+\Big] \leq {C_8 - n \over \varepsilon}\Vol(X) < +\infty. $$ Be $P\in\bC[X_0,X_1,\ldots,X_n]$ a polynôme as $\deg_{X_\ell} P \leq k_\ell$ and be $\theta$ the defined function by $$ \theta=\log|P(g,f_1,\ldots,f_n)| + k_0\Big({1\over 2}\tau\circ F + C_1\log|Q(F)|\Big). $$ As the assessment (12.7) and the outcomes here-above, $\theta$ is psh on $X$ and checks an assessment $$ \theta_+\leq \sum_{j=1}^m k_j\log_+|f_j| + k_0 \Big({1\over 2}\tau\circ F + C_9\log|Q(F)|\Big) + C_{10}. $$ Thanks to the corollaire 7.3, obtains on the majoration $$ \int_X dd^c\theta\wedge\alpha^{n-1}\leq C_{11}k_0+\sum_{j=1}^n k_j\delta_\varphi(f_j) $$ with a constante $C_{11} \geq 0$. If $a \in X$, prpers results the inequality $$ \ord\nolimits_a P(g,f_1,\ldots,f_n) \leq C_{12}(k_0+k_1+\cdots+k_n),\qquad C_{12}\geq 0. $$ The reasoning of the théorème 8.5 watch whereas $g$ is algébrique on $\bC(f_1,\ldots,f_n)$, and prpers is so likewise for the function $h_j(\check F) = Q(F)^{-s}g$. By continuation $h_j$ is algébrique on $\bC(z_1,\ldots,z_n)$, i.e.\ $h_j$ Checks an equation $$ \sum_{\ell=0}^d a_\ell(z_1,\ldots,z_ n)\;h_j^\ell = 0,\qquad a_\ell \in \bC[z_1,\ldots,z_n],\quad a_d\not\equiv 0. $$ The element $a_dh_j$ is so entire algébrique on $\bC[z_1,\ldots,z_n]$~; deduces a majoration $$ |a_d(z)h_j(z)| \leq C_{13}(1+|z|)^{C_{14}}. $$ Comme $h_j$ is holomorphe on the open $\Omega\subset \check M$ and that the complementary $\check M\ssm\Omega$ is pluripolaire, $a_dh_j$ prolonger se at a polynôme on~$\check M$. Therefore $h = \sum h_jdz_j$ prolonger se at a $1$ -form méromorphe rational on~$\check M$.\hfil\square\medskip {\statement Proposal 12.10.\pointir}{\it Be $\Omega_1$ the big plus opened of Zariski of $\check M$ on which $h$ is holomorphe. Then $\Omega = \Omega_1$.} \medskip {\it Démonstration}. Has On evidently $\Omega \subset \Omega_1$. For obtain the reciprocal inclusion, display firstly that $\tau$ is of class $\cC^\infty$ on~$\Omega_1$. Knows On that the equation (12.5) $$ \partial\tau = h - {u \over 2i} $$ has place on~$\Omega$, and that $v : = h - {u\over 2i} \in \cC^\infty_{1,0}(\Omega_1)$, since $u \in \cC^\infty_{1,0}(\check M)$. Prpers comes $v + \overline v = d\tau$ on~$\Omega$, of where $d(v+\overline v) = 0$ on $\Omega_1$ by continuity. Be $(\Omega_{1,j})_{j\in J}$ a recouvrement of $\Omega_1$ by of the open simply connexes. Il there is of the functions $\tau_j \in \cC^\infty(\Omega_{1,j})$ tel that $d\tau_j = v + \overline v$ on~$\Omega_{1,j}$. The function $\tau - \tau_j$ is then locally constante on $\Omega_{1,j}\cap \Omega$, so constante, because $\Omega_{1,j}\cap \Omega = \Omega_{1,j}\ssm( \check M\ssm\Omega)$ is connexe. By continuation $\tau \in \cC^\infty(\Omega_1)$, and comme $\tau=-\infty$ on $\check M\ssm\Omega$, prpers se ensuit that $\Omega_1\ssm \Omega = \emptyset$.\hfil\square \medskip {\statement Corollaire 12.11.\pointir}{\it $F(X\ssm f_{n+1}^{-1}(0))$ Is an open of Zariski of~$M$.} \medskip {\it Démonstration}. If $x$ is a dot quelconque of the open $X\ssm f_{n+1}^{-1}(0)$, then $F(x)\notin M_\sing \cup \{z_{n+1} = 0\}$, so il there is a polynôme $Q_x$ divisible by $z_{n+1}$ and cancel on~$M_\sing$, as $Q_x(F(x)) \neq 0$. As the corollaire 12.10, $\Omega_x := F(X\ssm Q_x(F)^{-1}(0))$ is an open of Zariski of~$M$, so also the meeting $$ \bigcup_{x\in X\ssm f_{n+1}^{-1}(0)}\Omega_x = F(X\ssm f_{n+1}^{-1}(0)).\eqno\square $$ Comme $X\ssm f_{n+1}^{-1}(0)$ is of Stein as well as his image biholomorphe by~$F$, sees on in fact that the complementary $M\ssm F(X\ssm f_{n+1}^{-1}(0))$ is necessarily a hypersurface algébrique of~$M$. \bigskip \section{13}{Démonstration of the criterion of algébricité (case lisse).} As the proposal 11.7~(a), at all dot $x_0 \in X$ pouvoir on associate a morphisme $$ F^{(0)} = (f^{(0)}_1,\ldots,f^{(0)}_{N_0}) \in \big[A^0_\varphi(X)\big]^{N_0} $$ and a function $g_0 = f_{n+1}^{(0)}$ tel that the open $X\ssm g_0^{-1}(0) \ni x_0$ be biholomorphe by $F^{(0)}$ at an open of Zariski of a variety algébrique on $\bC^{N_0}$. Il there is so a recouvrement dénombrable of $X$ by of tel opened $X\ssm g_k^{-1}(0)$, associé at of the morphismes \hbox{$F^{(k)} : X \to \bC^{N_k}$}. Consider the morphisme produced $$ F_k = F^{(0)} \times F^{(1)} \times\cdots \times F^{(k)}: X \to \bC^{N_0+\cdots+N_k}. $$ As the proposal 8.5, the image $F_k(X)$ has contained on a variety algébrique ir\-ré\-duke\-tible $M_k\subset \bC^{N_0\;+\;\cdots\;+\;N_k}$ of dimension~$n$, and the corollaire 12.11 watch that \hbox{$F_k(X\ssm g_j^{-1}(0))$} is an open of Zariski of $M_k$ if~$j\leq k$. State $$ Y_k = \bigcap_{j\leq k} g_j^{-1}(0),\qquad X_k = X\ssm Y_k = \bigcup_{j\leq k} \big(X \ssm g_j^{-1}(0)\big). $$ By building $F_k:X_k\to F_k(X_k)\subset M_k$ is an isomorphisme, and $F_k(X_k)$ is an open of Zariski of~$M_k$. Pouvoir On so énoncer~: \medskip {\statement Propostion 13.1.\pointir}{\it If $X$ checks the hypothesis{\rm 9.1'(has',b')}, then $X$ is meeting of an increasing continuation of varieties quasi-affines $X_k$, where each $X_k$ identifies se at an open of Zariski of $X_{k+1}$ with the structure algébrique induced.\hfil\square} \medskip En d'autres termes, $X$ has very a structure of space annelé who is\lguil\?Locally\?\rguil\ That of a variety algébrique, but the \lguil\?topologie Of Zariski\?\rguil\ Pouvoir no stand quasi-compact. Signal who il there is sure enough of tel varieties. Prpers suffice of seize for $X$ the surface (lisse) of equation $\sin x = yz$ on~$\bC^3$, and for $Y_k$ the meeting dénombrable of right $$ (\{j\pi\}\times\{0\}\times\bC)~ \cup ~(\{j\pi\}\times\bC\times\{0\}), $$ $j\in\bZ$, $|j|>k$. The open $X_k= X\ssm Y_k$ identifies se then at the variety algébrique $$ V_k=\Big\{(x,y,z)\in\bC^3\,;\; x\Big(1-{x^2\over\pi^2}\Big)\cdots\Big(1-{x^2\over k^2\pi^2}\Big)=yz \Big\} $$ via the application $V_k\hookrightarrow X$ defined by $$ (x,y,z) \mapsto (x,y,z')\quad\hbox{où}\quad z' = z \prod_{|j|>k}\Big(1-{x^2\over j^2\pi^2}\Big), $$ with of the inclusions $V_k\hookrightarrow V_{k+1}$ data by the morphismes algébriques $$ (x,y,z) \mapsto (x,y,z')\quad\hbox{où}\quad z' = z \Big(1-{x^2\over (k+1)^2\pi^2}\Big).\eqno\square $$ go Prpers now display that the continuation $(X_k)$ is necessarily stationnaire if the spaces of cohomologie $H^{2q}(X\,;\,\bR)$ are of ended dimension [hypothesis 9.1'(this )]. \medskip {\statement Lemme 13.2.\pointir}{\it Be $X$ a variety complex analytique of dimension~$n$, $Y$ an ensemble analytique of dimension${}\leq p$ on $X$, and $d = n - p = \codim_\bC Y$. Then the space of cohomologie relative $H^q(X,X\ssm Y\,;\,\bR)$ is any if $q < 2d$ and $$ H^{2d}(X, X\ssm Y\,;\,\bR) \simeq \bR^J , $$ where $(Y_j)_{j\in J}$ is the family of the composantes irréductibles of dimension $p$ of~$Y$.} \medskip {\it Démonstration}. Prpers renvoyons by example at E.~Spanier [Sp] For the elementary arguments of topologie algébrique who go to stand used. Reasons On by récurrence on~$p$, the outcome being trivial for~$p = 0$. If $p \geq 1$, be $Z$ the meeting of the singular place $Y_\sing$ and of the composantes irréductibles of $Y$ of dimension${} < p$, so that $\dim Z \leq p-1$. The~suite exact of the triplet type se $$ H^q(X,X\ssm Z) \to H^q(X, X\ssm Y) \to H^q(X, X\ssm Y) \to H^{q+1}(X,X\ssm Z). $$ By hypothesis of récurrence $H^q(X,X\ssm Z) = H^{q+1}(X,X\ssm Z) = 0$ for $q\leq 2d$, so $H^q(X, X\ssm Y) \simeq H^q(X\ssm Z,X\ssm Y)$. Quitte à replace $(X, Y)$ by $(X\ssm Z,Y\ssm Z)$, pouvoir on assume $Y$ lisse of dimension~$p$. $Y$ Own then a voisinage tubulaire $U$ homéomorphe at the fibré normal~$NY$. Thanks to the théorème of excision, obtains on so $$ H^q(X,X\ssm Y) \simeq H^q(U, U\ssm Y) \simeq H^q(NY,N^\bullet Y) $$ where $N^\bullet Y$ is the complementary of the section any of~$NY$. Comme the fibré $NY$ is of real rank~$2d$, the théorème of isomorphisme of Thom-Gysin involve $$ H^q(NY,N^\bullet Y) \simeq H^{q-2d}(Y), $$ and for $q = 2d$, $H^0(Y) \simeq \bR^J$.\hfil\square \medskip Go back now at the situation of the proposal 13.1, where $X_k = X\ssm Y_k$, and state $\dim Y_k = p_k$, $d_k = n-p_k$. The~suite exact of the pair $(X, X\ssm Y_k)$ give $$ H^{2d_k-1}(X\ssm Y_k) \to H^{2d_k}(X,X\ssm Y_k) \to H^{2d_k}(X). $$ Since $X\ssm Y_k$ is isomorphe at a variety algébrique, $H^{2d_k-1}(X\ssm Y_k)$ is of ended dimension, and is likewise by hypothesis for $H^{2d_k}(X)$. The lemme 13.2 watch so that $Y_k$ no has who an ended number of composantes irréductibles of maximum dimension $p_k$. Comme $Y_k$ is a continuation décroissante of empty intersection, sees on who il there is $\ell > k$ as $\dim Y_\ell < p_k$. Au bout de an ended number of stages have on so $Y_\ell = \emptyset$, be $X = X_\ell$. State $$ F = F_\ell = F^{(0)}\times \cdots \times F^{(\ell)},\qquad M = M_\ell,\qquad N= N_0 +\cdots+ N_\ell. $$ The morphisme $F : X \to M \subset \bC^N$ is then an isomorphisme analytique of $X$ on an open of Zariski $\Omega \subset M$. Quitte à replace $M$ by his normalisation comme on the démonstration 11.7~(b), pouvoir on assume $M$ normal. Since $\Omega$ is of Stein, the complementary $H = M\ssm \Omega$ is necessarily a hypersurface of~$M$. Designate by $K(\Omega)\simeq K(M)$ the bodies of the rational functions on~$\Omega$, and by $R(\Omega)$ the ring of the functions algébriques regular on~$\Omega$. Type $$ F = (f_1,\ldots,f_N) \in \big[A^0_\varphi(X)\big]^N. $$ The co-morphisme $F^*$ forward $K(\Omega)$ on the body $\bC(f_1,\ldots,f_N)\subset K_\varphi(X)$. The proposal here-below displays that the structures algébriques $(X,K_\varphi(X)\cap\cO(X))$ and $(\Omega,R(\Omega))$ are isomorphes. \medskip {\statement Proposal 13.3.\pointir}{\it Has On the following ownerships~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $F^* : K(\Omega) \to K_\varphi(X)$ is an isomorphisme. \vskip2pt \item{\rm(b)} $F^*R(\Omega) = K_\varphi(X)\cap \cO(X)$.\medskip}} {\it Démonstration}. (a) Prpers suffice of show the surjectivité of $F^*$. Or, if $g\in K_\varphi(X)$, the function $g$ is algébrique on $\bC(f_1,\ldots,f_N)$ as 11.6. By continuation $g\circ F^{-1}$ is méromorphe on $\Omega$ and algébrique on $K(\Omega)$. Prpers results that $g\circ F^{-1}\in K(\Omega) = K(M)$, reasoning by example comme à la fin de the démonstration~12.9. (b) Deduces se forthwith of (a) , at condition of check the equality $R(\Omega) = K(\Omega)\cap\cO_\anal(\Omega)$. The inclusion $R(\Omega)\subset\ldots$ is clear. Inversement, having given $g \in K(\Omega)$ and $x \in \Omega$, be $g = u/v$, where $u,v\in \cO_{x,\alg}$, a writing irréductible of $g$ at the dot $x$ (who is lisse by hypothesis). That writing is also irréductible on $\cO_{x,\anal}$. Comme $g \in \cO_{x,\anal}$, has on so $v(x)\ne 0$, by continuation $g\in\cO_{x,\alg}$ and $g\in R(\Omega)$.\hfil\square \medskip For finish the test of the théorème 9.1', prpers remains now at display that $\Omega$ is algébriquement isomorphe at a variety algébrique affine, autrement dit prpers falloir test the existence of a plongement algébrique own $\Omega=M\ssm H\to\bC^{N'}$. Is easy if $M$ is lisse, but when $M$ is singular prpers pouvoir se that the hypersurface algébrique $H$ no be locally complete intersection, and on that situation Goodman [Go] has given of the examples for which the algèbre $R(M\ssm H)$ no is of ended type. I Appreciate N.~Mok Of have signalled that difficulty, who return caduque my initial démonstration. The reasoning of [Mok2] consister observe that $\Omega$ is {\it rationnellement convexe} on the following sense~: for all compact $K\subset\Omega$ the enveloppe $$ \hat K= \Big\{x\in\Omega\,;\;|g(x)| \leq \sup_K|g|~\hbox{pour tout $g\in R(\Omega)$}\Big\} $$ is compact. This result at effect of 11.5~(d) and since $\Omega\simeq X$ is of Stein. Applies On then the part (b) of the théorème here-below. \medskip {\statement Théorème 13.4.\pointir}{\it Be $M\subset\bC^N$ a variety algébrique affine $($ éventuellement singu\-lière$)$ of pure dimension~$n$, and $H$ a hypersurface algébrique of~$M$. Then $M\ssm H$ is isomorphe at a variety algébrique affine under the one quelconque of the deux hypotheses following~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} $H$ is $($ comme under-reduced schéma$)$ locally complete intersection on~$M$. \vskip2pt \item{\rm(b)} $M\ssm H$ is rationnellement convexe{\rm([Mok2])}. \medskip}} {\it Démonstration} under the hypothesis (a). For all $x \in H$, il there is by hypothesis a polynôme $P\in\bC[z_1,\ldots,z_n]$ and a voisinage of Zariski $V(x) \subset M$ tel that $H\cap V(x) = P^{-1}(0)\cap V(x)$. Be $H'$ the meeting of the composantes irréductibles of $P^{-1}(0)$ no contained on~$H$. Comme $x\notin H'$, il there is a polynôme $Q$ cancel on~$H'$, as $Q(x) = 1$. The théorème of the zero of Hilbert entraine the existence of an entire $s\in\bN$ as $Q^s/P \in R(M\ssm H)$. Comme the topologie of Zariski is quasi-compact, pouvoir on extract a recouvrement ended$V(x_1),\ldots, V(x_m)$ of~$H$ and an ended family of polynômes $P_j$, $Q_j^{s_j}$ associé at the dots~$x_j$. Our building displays whereas the morphisme $$ \big(z_1,\ldots,z_N,Q_1^{s_1}/P_1,\ldots,Q_m^{s_m}/P_m\big) : M\ssm H\to\bC^{N+m} $$ is a plongement own. \medskip {\it Démonstration} under the hypothesis (b). Builds On firstly by récurrence downward on $k$ a continuation of under-varieties algébriques $M_k\subset M$ fermé, of pure dimension~$k$, tel that $M_k\cap H$ be a hypersurface of~$M_k$. States On $M_n = M$~; if $M_k$ has be built, chooses on a polynôme $P_k\in R(M)$ cancel on $H$ but no cancel identiquement on any composante irréductible of~$M_k$~; notes on then $M_{k-1}$ the meeting of the composantes irréductibles of $M_k\cap P_k^{-1}(0)$ no contained on~$H$. Shows On now by récurrence increasing on $k$ the existence of rational fractions $g_1,\ldots,g_{m_k}\in R(M\ssm H)$ tel that the morphisme $$ \Phi_k:(z_1,\ldots,z_N)\mapsto (z_1,\ldots,z_N,g_1(z),\ldots,g_{m_k}(z)) : M\ssm H\to\bC^{N+m_k} $$ be a plongement own at restriction at $M_k\ssm H$ (for $k = n$ the théorème have showed thus). If $k = \dim M_k = 1$, that ownership is clear, because the hypothesis (b) and the principle of the maximum train the existence of rational fractions $g_1,\ldots,g_{m_1} \in R(M\ssm H)$ whose restriction at $M_1$ have of the poles at the different dots $x_1,\ldots,x_{m_1}$ of~$M_1\cap H$. Assume now $\Phi_k$ built. Be $\pi_k : \bC^{N+m_k} \to \bC^{N}$ the projection, $\overline M$ the adhérence of $\Phi_k(M\ssm H)$ on $\bC^{N+m_k}$ and $\overline H = \overline M \cap \pi_k^{-1}(H)$, so that $$ \Phi_k : M\ssm H \to \overline M\ssm \overline H $$ is an isomorphisme (of reverse $\Phi_k^{-1} = \pi_{k|\overline M\ssm\overline H}$ ). By hypothesis of récurrence $\overline M_k =\Phi_k(M_k\ssm H)$ is an under-variety algébrique closed of $\overline M_{k+1} = \overline{\Phi_k(M_{k+1}\ssm H)}$. Comme $\overline M_{k+1} \cap (P_k\circ \pi_k)^{-1}(0)$ is the meeting disjointe $(\overline M_{k+1}\cap\overline H) \cup \overline M_k$, sees on that $\overline M_{k+1}\cap \overline H$ is locally complete intersection on $\overline M_{k+1}$ (and has defined locally by $P_k\circ\pi_k$ ). For all $x \in \overline M_{k+1}\cap \overline H$, il there is so a polynôme $Q \in R(\overline M)$ as $Q(x) = 1$, who cancel se on all the composantes irréductibles of $(P_k\circ \pi_k)^{-1}(0)$ no finding ~$\overline M_{k+1}\cap \overline H$. Pouvoir On so complete $\Phi_k$ at a morphisme $\Phi_{k+1}$ comme on the case~(a), adjoignant at $\Phi_k$ of the functions $g_j=(Q_j\circ\Phi_k)^{s_j}/P_k$, $m_k \log_2 r$ on the other hand, obtains on easily for $|z| = r$ the assessments $$ \leqalignno{ &\qquad\quad\varphi(z) = 2\log(1 + |z|^2) + O\Big({\log r\over r}\Big)\qquad \hbox{si $\forall j$, $|z-j| > {1\over 2}$},&(13.7) \cr &\qquad\quad\varphi(z) = 2\log(1 + |z|^2) + 2^{-j}\log(\varepsilon_j+|z-j|^2) + O\Big({\log r\over r}\Big)\qquad \hbox{si $|z-j| \leq {1\over 2}$}.&(13.8) \cr} $$ Choose $\varepsilon_j$ so that $$ 2\log(1+j^2) + 2^{-j}\log\varepsilon_j = \log(1 + \log j),\quad j\geq 1,\quad \hbox{i.e.}~~\varepsilon_j=\Big[{1+\log j\over(1+j^2)^2}\Big]^{2^j}. \leqno(13.9) $$ has On then $\varphi(j) \sim \log\log j$ when $j\to+\infty$, so that $\varphi$ is exhaustive. The functions $f\in A_\varphi^\infty(\bC)$ are at growth polynomiale and have to check besides $|f(j)| \leq (\log j)^{\rm Cte}$ when $j\to +\infty$. By continuation$A_\varphi^\infty(\bC)$ reduces se at the constantes. The conditions 9.1~(a) and (b) are néanmoins checked. An immediate calculation gives at effect $$ dd^c\varphi = 2i\,dz\wedge d\overline z~ \sum_{j\in\bN}{2^{-j}\varepsilon_j\over(\varepsilon_j+|z-j|^2)^2}, $$ so that $\int_\bC dd^c\varphi = 8\pi$. The majoration 9.1~(b) has place with the function $$ \psi(z) = -\log\Bigg(\sum_{j\in\bN}{2^{-j}\varepsilon_j\over (\varepsilon_j+|z-j|^2)^2}\Bigg). $$ Considering the alone term $j = 0$, obtains on the majoration $\psi(z) \leq 2\log(1 + |z|^2)$. For $\varepsilon_j^{1/3}\leq |z-j| \leq {1\over 2}$, has on on the other hand thanks to (13.8) and $(13.9)\times {2\over 3}$~: $$ \varphi(z) \geq 2\log(1+|z|^2) + 2^{-j}{2\over 3}\log\varepsilon_j + O(1) \geq {2\over 3}\log(1 + |z|^2) + O(1), $$ while for $|z-j| \leq \varepsilon_j^{1/3}$ prpers comes $$ {2^{-j}\varepsilon_j\over(\varepsilon_j+|z-j|^2)^2}\geq {2^{-j}\varepsilon_j\over4\,\varepsilon_j^{4/3}}=2^{-j-2}\varepsilon_j^{-1/3}, $$ so that $\psi(z)\leq 0$ if $j$ is enough big. Sees On so what il there is a constante $B$ tel that $\psi\leq 3\varphi + B$.\hfil\square \bigskip \section{14}{Algébricité of the complex spaces singular.} Be $X$ a space analytique of dimension~$n$. If $X$ is an ensemble algébrique on~$\bC^N$, the calculations of the\S10 display that the geometrical conditions 9.1~(has,b,c) have checked. Inversement, for show the suffisance of the geometrical conditions, on se heurter at deux principal difficulty. On the one hand the assessments $L^2$ of Hörmander no are a priori valid that on an open of Stein lisse of the form $X\ssm H$, where $H$ is a hypersurface of $X$ containing the singular place $X_\sing$. For pouvoir apply a lemme of elongation, owes on so assume that $X$ is normal. \medskip {\statement Lemme 14.1.\pointir}{\it Be $f$ a function holomorphe on $X\ssm H$ tel that $f \in L^2_\loc(X_\reg)$. Then, if $X$ is normal, $f$ prolonger se at a function holomorphe on~$X$.} \medskip {\it Démonstration.} Under the hypothesis $f \in L^2_\loc(X_\reg)$, is classical that $f$ prolonger se $X_\reg\ssm H$ at $X_\reg$, and all function holomorphe on $X_\reg$ prolonger se at $X$ if $X$ is normal (cf.\ [Nar], proposal~VI.4). Other difficulty comes since the weight $e^{-\psi}$ pouvoir no stand locally sommable at some singular dots, by report at a métrique ambiante lisse. Consider by example the case of the variety conique $X$ of equation $z_0^p+\cdots+z_n^p=0$ on $\bC^{n+1}$, $p\in\bN^*$. The courbure of Ricci of $X$ is then given thanks to the proposal 10.1~(a) by the formula $$ \Ricci(\beta_{|X})=-{1\over 2}dd^c\psi $$ with $\psi(z)=\log(|z_0|^{2p-2}+\cdots+|z_n|^{2p-2})$. Sees On so that the function $e^{-\psi}$ no is locally sommable $0$ that if $p \leq n$. On the case of a space $X$ for which $e^{-\psi}$ is no sommable at the voisinage of all dot of a bend, the proposal 11.5~(c) no applies se best. Is On so brought à supposer que the singularities of $X$ have isolated. \medskip {\it Démonstration of the théorème} 9.1' (suffisance of the conditions on the cases of isolated singularities). The hypothesis (this ) trains that the composantes irréductibles of $X$ are at ended number. Be $$ \pi : \tilde X \to X $$ the normalisation of $X$. The function $\varphi\circ\pi$ no is at general strictly psh at the voisinage of $\pi^{-1}(X_\sing)$, but quitte à modify $\varphi\circ\pi$ and $\psi\circ\pi$ at the voisinage of the ensemble ended $\pi^{-1}(X_\sing)$, sees on that the hypothesis have satisfied by $\tilde X$. At definite, pouvoir on assume $X$ normal and irréductible. The démonstration is now all at alike fact at those that has be given during the\S11, 12, 13, also cheer prpers of indicate the big lines and the changes at contribute. The lemmes 11.2 and 11.3 are true without any modification, as well as the ownerships 11.5~(has,b,d). The énoncé 11.5~(c) valid rest if $\{x_1,\ldots,x_m\}\subset X_\reg$, and if some of the dots $x_j$ are singular, has on the partial outcome following (who corresponds at the case~$\rho = 0$ ). \medskip {\statement Lemme 14.2.\pointir}{\it Be an ensemble ended $\{x_1,\ldots,x_m\}\subset X$. Then il there is a function $f \in A_\varphi^b(X)$, $b = {2c\over 1+c}$, having a jet of order $s$ given at each dot $x_1,x_2,\ldots,x_m$.} \medskip {\it Démonstration}. Restarts On the same arguments that on 11.5~(c), replacing the system of coordinated local $z^{(j)}$ by a générateur system $(z_1^{(j)},\ldots,z_{N_j}^{(j)})$ of the ideal maximum $\gm_{X,x_j}$ of $\cO_{X,x_j}$ and $\rho_1$ by $$ \rho_1 = s(n+2)\Bigg[\sum_{j=1}^m \chi_j \log|z^{(j)}|^2 + C_1\varphi\Bigg]. $$ At the voisinage of $x_j$, the building gives then $f = P_j(z^{(j)}) - g$ with $g$ holomorphe tel that $$ |g|^2\;|z^{(j)}|^{-2s(n+2)}\;e^{-\psi}\in L^2_\loc(X). $$ As the assessments $L^2$ of H.~Skoda [Sk4], this trains that $g \in \gm_{X,x_j}^s$.\hfil\square \medskip The proposal 11.7 rest so applicable if $x_0\in X_\reg\;$, and restarting the argu\-ments of the\S12, 13, builds on a variety algébrique normal $M$ and a morphisme $F = (f_1,\ldots,f_N) : X \to M$ whose restriction at $X_\reg$ is an isomorphisme of $X_\reg$ on an open of Zariski of~$M$. Thanks to the lemme 14.2, pouvoir on (quitte à complete $F$ by an ended number of functions $f_j$ ) assume that $F$ defines a plongement of $X$ at the voisinage of each singular dot. The morphisme $F$ is then an isomorphisme of $X$ on the open of Zariski $F(X) \subset M$. The end of the test is identical at that data at the~\S13.\hfil\square \medskip The reasoning who acabar of stand esquissé gives on the other hand the interesting outcome here-below. \medskip {\statement Théorème 14.3.\pointir}{\it Be $X$ a space normal analytique of dimension~$n$, checking the hypothesis{\rm 9.1'(has',b',this )}. Then $X_\reg$ is analytiquement isomorphe at a variety algébrique quasi-affine, the isomorphisme having given by a morphisme $\varphi$ -polynomial $F$ of $X$ on a variety algébrique affine normal $M \subset \bC^N$ of dimension~$n$.\hfil\square} \bigskip \section{15}{Appendice : common and functions plurisousharmoniques\\ at minimum growth on a variety algébrique affine.} Be $M$ an under-variety algébrique affine lisse of dimension~$n$. Caters On $M$ of the métrique kählériennes $$ \beta = dd^c|z|^2,\qquad \omega = dd^c\log(1+|z|^2) $$ induced respectively by the flat métrique of $\bC^N$ and by the métrique of Fubini-Study of the projectif space~$\bP^N$. \medskip {\statement Definition 15.1.\pointir}{\it Functions psh and common at minimum growth~$:$ {\parindent=6.5mm \vskip2pt \item{\rm(a)} A function psh $V$ on $M$ has said at minimum growth se il there is of the constantes $C_0,C_1\geq 0$ tel that $$ V(z) \leq C_1\log_+|z| + C_0. $$ \item{\rm(b)} A positive current closed $T$ of bidegré $(1,1)$ on $M$ has said at minimum growth if $$ \int_M T\wedge\omega^{n-1} < +\infty. $$\vskip-\parskip}} Of the corollaire 7.3 results forthwith the \medskip {\statement Proposal 15.2.\pointir}{\it If $V$ is psh of minimum growth on~$M$, then $T = dd^cV$ is at minimum growth. \hfil\square} \medskip Réciproquement, having given a current $T \geq 0$ closed at minimum growth, on no will be able to find of solution at the equation $dd^cV = T$ that if the class of cohomologie of $T$ is any. The objective of that paragraph is of show the general outcome following, who is a reciprocal partial of the proposal~15.2. \medskip {\statement Théorème 15.3.\pointir}{\it Be $T$ a $(1,1)$ -running positive closed on $M$ as $$ \int_M T\wedge\omega^{n-1} < +\infty. $$ Then il there is a function psh $V$ and a $(1,0)$ -form $u$ of class $\cC^\infty$ on $M$ having the ownerships here-below, where $C_1,C_2,C_3$ are of the constantes${}\geq 0$. {\parindent=6.5mm \vskip2pt \item{\rm(a)} $dd^cV \geq T~;$ \vskip2pt \item{\rm(b)} $V(z)\leq C_1\log_+|z|~;$ \vskip2pt \item{\rm(c)} $dd^cV - T = \overline\partial u~;$ \vskip2pt \item{\rm(d)} $|u|_\omega\leq C_2(1+|z|)^{C_3}\;$.\medskip}} The démonstration take se at plusieurs stages. Observe firstly that the condition 15.1~(b) is equivalent at the following : $$ \sigma(r) = \int_{|\zeta|2|z|\}$, of uniform way when $z$ depicted a compact of~$M$. The ownership $V_T\in L^1_\loc(M)$ result then by the théorème of Fubini since $|L_n(z,\zeta)|$ is, locally on $M\times M$, intégrable at $z$ uniformément by report at~$\zeta$.\hfil\square \medskip {\statement Lemme 15.7.\pointir}{\it For all dot $z\in M$, il there is of the bowls $B'_z\subset T_zM$, $B''_z\subset (T_zM)^\perp$ of centre $0$ and of ray $r(z)=C_6(1+z|)^{-C_7}$ where $C_6,C_7 > 0$, and an application holomorphe $g_z : B'_z\to B''_z$ tel that $M\cap (z+B'_z+B''_z)$ be the graphe of~$g$, i.e.\ If $\zeta-z = \zeta'+\zeta''$ is the writing of a dot $\zeta\in M$ tracking the decomposition $\bC^N = (T_zM) \oplus (T_zM)^\perp$, then $$ M\cap (z+B'_z+B''_z) = \big\{ \zeta\in\bC^N\,;\;\zeta''=g_z(\zeta'),\; \zeta'\in B'_z\big\}. $$\vskip-\parskip} {\it Démonstration}. Be $(P_1,\ldots,P_m)$ a system of polynômes générateur for the ideal of the variety $M$ on $\bC[X_1,\ldots,X_N]$. Since $M$ is lisse, the jacobiens partial $J_{K,L}$ of order $N-n$ (cf.\\S10) no cancel se all simultaneously on~$M$. As the théorème of the zero of Hilbert, the polynômes $J_{K,L}$ engendrent the ideal unity on~$M$~; il there is so of the constantes $C_8,C_9 > 0$ tel that $$ \max_{K,L}|J_{K,L}(z)| \geq C_8(1+|z|)^{-C_9},\qquad z\in M. $$ The lemme result then of the théorème of the functions implicites (on his version quanti\-tative).\hfil\square \medskip . Observes On maintenant que the formula (15.5) pouvoir se récrire under the form $$ V_T(z)=\int_M T(\zeta)\wedge\big[K_n(z,\zeta)-H_n(\zeta)\big] \leqno(15.8) $$ with $$ \eqalign{ K_n(z,\zeta) & = -{1\over(n-1)(4\pi)^n}\; \bigg[{dd^c|z-\zeta|^2\over|z-\zeta|^2}\bigg]^{n-1},\cr \noalign{\vskip4pt} H_n(\zeta) &= -{1\over(n-1)(4\pi)^n}\;{\beta(\zeta)^{n-1}\over(1+|\zeta|^2)^{n-1}}.\cr} $$ The ownerships of the core $K_n$ go to allow reckon easily $dd^cV_T$ en fonction of~$T$. \medskip {\statement Lemme 15.9.\pointir}{\it $dd^c K_n = [\Delta] + R_n$, where $[\Delta]$ is the current of integration on the diagonale of $M \times M$ and where $R_n$ is a $(n,n)$ -running${}\geq 0$ at coefficients locally intégrables on $M\times M$, checking the assessment $$ \Vert R_n(z,\zeta)\Vert_{\beta\oplus\beta}\leq C_{10}\min\bigg[ {1\over|z-\zeta|^{2n}}\;,\;{(1+|z|)^{C_{11}}\over|z-\zeta|^{2n-1}}\bigg]. $$\vskip-\parskip} \medskip {\it Démonstration}. Outside the diagonale $\Delta$, a classical calculation (whose vérification have allowed at the reader) gives $$ \leqalignno{ dd^cK_n&={(dd^c|z-\zeta|^2)^n-n\;|z-\zeta|^{-2}\;d|z-\zeta|^2\wedge d^c|z-\zeta|^2\wedge (dd^c|z-\zeta|^2)^{n-1}\over (4\pi)^n\;|z-\zeta|^{2n}}\cr &=\Big({1\over 4\pi}\;dd^c\log|z-\zeta|^2\Big)^n,&(15.10)\cr} $$ at see on best far that $\bOne_{\Delta}\;dd^cK_n=[\Delta]$. At particular, has on $$ R_n = \bOne_{M\times M\ssm\Delta}\;dd^cK_n \geq 0\quad\hbox{et}\quad \Vert R_n(z,\zeta)\Vert \leq C|z-\zeta|^{-2n}. $$ For obtain the deuxième part of the majoration, place at a dot $z \in M$ and use the lemme~15.7. At restriction at~$M$, has on at the dot~$z$ $$ dz = dz' = \hbox{composante de $dz$ sur $T_zM$}, $$ while at a neighbouring dot $\zeta\in z + (B'_z + B''_z)$ has on~: $$ d\zeta = d\zeta' +d(g_z(\zeta')). $$ As (15.10) prpers comes so : $$ \eqalign{ R_n(z,\zeta)&=\Bigg[{dd^c\big(|z'-\zeta'|^2+|g_z(\zeta')|^2\big) \over (4\pi)\;(|\zeta'|^2+|g_z(\zeta')|^2)}\cr &\kern50.5pt{}-{d\big(|z'-\zeta'|^2 +|g_z(\zeta')|^2\big)\wedge d^c\big(|z'-\zeta'|^2 +|g_z(\zeta')|^2\big) \over (4\pi)\;(|\zeta'|^2+|g_z(\zeta')|^2)^2}\Bigg]^n,\cr} $$ where the différentiation of $g_z(\zeta')$ takes only on $\zeta'$. The lemme~15.7 gives by con\-trick\-tion $g_z(0) = D_0g_z = 0$~; the lemme of Schwarz involve then the inequalities $$ \eqalign{ &|g_z(\zeta')| \leq |\zeta'|,\kern67.5pt\zeta'\in B'_z~;\cr \noalign{\vskip5pt} &\Vert D_{\zeta'}g_z(\zeta')\Vert \leq C(\lambda){|\zeta'|\over r(z)},\qquad \zeta'\in\lambda B'_z,\quad 0<\lambda<1.\cr} $$ observes On maintenant que $R_n(z,\zeta) \equiv 0$ if $g_z \equiv 0$. Prpers se ensuit for $\zeta'\in {1\over 2}B'_z$ the inequality $$ \Vert R_n(z,\zeta)\Vert\leq {C_{12}\;|\zeta'|\;r(z)^{-1}\over\big(|\zeta'|^2 +|g_z(\zeta')|^2\big)^n} \leq {C_{12}\;r(z)^{-1}\over |z-\zeta|^{2n-1}}, $$ who completes the assessment of the lemme 15.9. The classical formula of Bochner-Martinelli on $\bC^n$ gives on the other hand $$ dd^cK_n(z',\zeta') = [\Delta]. $$ By an analogous calculation at that here-above, obtains on the inequality $$ \Vert K_n(z,\zeta)-K_n(z',\zeta')\Vert\leq {C_{13}\;r(z)^{-1}\over |z-\zeta|^{2n-3}}, $$ and for each différentiation of $K_n$ the exposant of $|z-\zeta|$ accroître se of a unity. Sees On so that $dd^cK_n-[\Delta]$ is at coefficients $L^1_\loc$ on $M\times M$, and en conséquence no takes of mass on $\Delta$. The~preuve has finished.\hfil\square \medskip {\statement Proposal 15.11.\pointir}{\it If $T$ has closed, then $$ dd^cV_T = T + \Theta_T\qquad\hbox{où}\quad \Theta_T(z) = \int_M R_n(z,\zeta)\wedge T(\zeta) \geq 0. $$ At particular, $V_T$ is psh.} \medskip {\it Démonstration}. Be $\chi:\bR\to[0,1]$ a function of class $\cC^\infty$ tel that $\chi(t) = 1$ for $t<1$, $\chi(t) = 0$ for $t > 2$, and be $w$ a $(n-1,n-1)$ form $\cC^\infty$ at compact support on~$M$. The writing (15.8) gives prpers $$ \eqalign{ &\int_MV_T\;dd^cw = \lim_{r\to+\infty}~I(r),\cr \noalign{\vskip5pt} &I(r)=\int_{M\times M}\chi\Big({|\zeta|\over r}\Big)\;T(\zeta)\wedge \big(K_n(z,\zeta)-H_n(\zeta)\big)\wedge dd^cw(z).\cr} $$ The théorème of Stokes and the lemme 15.9 involve $$ \eqalign{I(r) &= \int_{M\times M}dd^c\Big[\chi\Big({|\zeta|\over r}\Big)\;T(\zeta)\wedge K_n(z,\zeta)\Big]\wedge w(z)\cr &= \int_{M\times M}\chi\Big({|\zeta|\over r}\Big)\;T(\zeta)\wedge \big([\Delta]+R_n(z,\zeta)\big)\wedge w(z)\cr &\qquad{}+ \int_{M\times M}d\Big[\chi\Big({|\zeta|\over r}\Big)\Big]\wedge T(\zeta)\wedge d^c K_n(z,\zeta)\wedge w(z)\cr &\qquad{}- \int_{M\times M}d^c\Big[\chi\Big({|\zeta|\over r}\Big)\Big]\wedge T(\zeta)\wedge dK_n(z,\zeta)\wedge w(z)\cr &\qquad{}+ \int_{M\times M}dd^c\Big[\chi\Big({|\zeta|\over r}\Big)\Big]\wedge T(\zeta)\wedge K_n(z,\zeta)\wedge w(z)\cr} $$ because $dT = d^cT = 0$. For justify that calculation, pouvoir on firstly assume that $T$ is of class~$\cC^\infty$, quitte à régulariser afterwards $T$ at the voisinage of the support of $\chi\big({|\zeta|\over r}\big)\subset \{|\zeta|\leq 2r\}$. Uses On now (15.4) and the majorations evident $$ \eqalign{ &\Big\Vert d\chi\Big({|\zeta|\over r}\Big)\Big\Vert = O\Big({1\over r}\Big), \kern70.5pt \Big\Vert dd^c\chi\Big({|\zeta|\over r}\Big)\Big\Vert = O\Big({1\over r^2}\Big), \cr &\Vert d^cK_n(z,\zeta)\Vert = O\Big({1\over |z-\zeta|^{2n-1}}\Big), \qquad \Vert dd^cK_n(z,\zeta)\Vert = O\Big({1\over |z-\zeta|^{2n-2}}\Big),\quad n\neq 1\cr} $$ for see that both latter integral on the calculation of $I(r)$ admit a majoration of the form $O(r^{-2})$. Has On so the awaited formula $$ \lim_{r\to+\infty} I(r) = \int_M T(\zeta)\wedge w(\zeta) + \int_{M\times M} T(\zeta)\wedge R_n(z,\zeta)\wedge w(z).\eqno\square $$ {\it Démonstration} of the théorème 15.3. As the proposal 15.2 and the lemme 15.6, run $\Theta_T$ is positive closed at minimum growth. Pouvoir On so build by récurrence on $k$ of the functions psh $V_k$ and of the common $T_k$ positive closed at tel minimum growth that $$ \eqalign{ &T_0=T,\qquad V_k = V_{T_{k-1}},\qquad T_k = \Theta_{T_{k-1}},\cr \noalign{\vskip6pt} & dd^cV_k = T_{k-1}+T_k.\cr} $$ Effect the alterné sum of that identities. For the rates impair prpers comes~: $$ dd^c\big(V_1-V_2+\cdots-V_{2k}+V_{2k+1}\big) = T+T_{2k+1} \geq 0, $$ and the lemme 15.15 here-below involves that the function psh $V = V_1 - V_2 +\cdots+ V_{2k+1}$ is at minimum growth. As the proposal 15.11, has on the account of récurrence $$ T_{k+1}(z)=\int_M R_n(z,\zeta)\wedge T_k(\zeta). $$ exploits On now takes prpers that $R_n$ is a core régulariser of type convolution. \medskip {\statement Lemme 15.12.\pointir}{\it Has On the following ownerships. {\parindent=6.5mm \vskip2pt \item{\rm(a)} For all entire $k$, $1\leq k<2n$, il there is of the constantes $A_k,B_k\geq 0$ tel that for all $\varepsilon\in{}]0,1[$ have on $$ \Vert T_k(z)\Vert\leq A_k(1+|z|)^{B_k}\bigg[\varepsilon^{-2}+ \int_{|\zeta-z|<\varepsilon\,r(z)} {T(\zeta)\wedge\beta(\zeta)^{n-1} \over|\zeta-z|^{2n-k}}\bigg]. $$ where $r(z) = C_6(1 +|z|)^{-C_7}$ $[\,$ cf.\ lemme{\rm 15.7}$\,]$. \vskip2pt \item{\rm(b)} For $k \geq 3$ run $T_k$ is at continuous coefficients and $$ \Vert T_k(z)\Vert = O\big((1+|z|)^{B_k}\big). $$\vskip-\parskip}} {\it Démonstration}. (a) reasons On by récurrence on~$k$. State $$ \sigma_k(z,r) = \int_{|\zeta-z|0, $$ On the open $\Omega_k$ prpers comes $$ dd^cW - \Theta = dd^c(W - \tau_k) = dd^c \Bigg[ \sum_{j\in\bN} \psi_j(\tau_j-\tau_k-\tau_j^{\varepsilon_j})\Bigg], $$ and since $\tau_j-\tau_k \in \cC^\infty(\Omega_j\cap\Omega_k)$, sees on that $dd^cW-\Theta \in \cC^\infty_{1,1}(M)$. Comme run $\Theta$ is at continuous coefficients, $\tau_j$ and the $1$ -form $d\tau_j$, $d^c\tau_j$ are continuous. When the $\varepsilon_j$ have chosen enough small, obtains on so $|W|\leq 1$ and $-\omega \leq dd^cW \leq \omega$, with $\omega = dd^c\log(1+|z|^2)$. The function $V' = V - W + \log(1+|z|^2)$ is then psh at minimum growth, and checks $dd^cV' = T+\Theta'$ where $$ \Theta' = \Theta - dd^cW + \omega $$ is a positive current closed of class $\cC^\infty$, as $\Vert\Theta'(z)\Vert = O\big((1 + |z|)^{C_{20}}\big)$. Assumes On so désormais that $\Theta$ is of class $\cC^\infty$. Applies On then the assessments $L^2$ of Hörmander-Nakano-Skoda [Nak], [Sk4] au courant $\Theta$, considered comme a $(n,1)$ -fermé form at courages on the fibré holomorphe $E = T^*M \otimes \bigwedge^n TM$. The fibré cotangent $T^*M$ is semi-positive at the sense of Griffiths for the métrique $\beta$ (is a quotient of the fibré flat $T^*\bC^N_{|M}$ ), so as [DS] the fibré $T^*M\otimes\bigwedge^n T^*M$ is semi-positive at the sense of Nakano. The proposal 10.1~(b) displays that the fibré $E$ has luire-same semi-positive at the sense of Nakano for the métrique $\beta\,e^{-2\psi}$, where $\psi = \log\big(\sum_{K,L}|J_{K,L}|^2\big)$. As the assessments of [Sk4] applied at the fibré hermitien $$ \Big(E, \beta\,\exp\big(-2\psi-C_{21}\log(1+|z|^2)\big)\Big), $$ obtains on the existence of a form $u\in\cC^\infty_{1,0}(M)$ tel that $\overline\partial u = \Theta$ and $$ \int_M|u|_\beta^2\;(1+|z|^2)^{-C_{22}}\;\beta^n<+\infty. $$ For finish the test of the théorème 15.3 and at particular of 15.3~(d), prpers suffice of convert that assessment $L^2$ at an assessment $L^\infty$ $$ |u|_\beta^2\leq C_{23}(1+|z|)^{-C_{24}}. $$ kept-Account that $\overline\partial u = \Theta$ admits a majoration at norm $L^\infty$, prpers suffice of use the inéga\-lité here-below, place on the bowls $|\zeta-z| < {1\over 2}\,r(z)$ of the lemme~15.7.\hfil\square \medskip {\statement Lemme 15.14.\pointir}{\it Be $v$ a function of class $\cC^1$ on the bowl $B(r)\subset\bC^n$. Then $$ |v(0)| \leq \bigg[{n!\over \pi^nr^{2n}}\int_{B(r)}|v(z)|^2\;d\lambda(z) \bigg]^{1/2}+{4n\over 2n+1}\;r\cdot\sup_{B(r)}|\overline\partial v|. $$\vskip-\parskip} {\it Démonstration}. Apply the formula of Cauchy with rest at the function $t\mapsto v(tz)$, $z \in B(r)$, $t\in\bC$, $|t| < 1$. Prpers comes $$ \eqalign{ v(0) &= {1\over 2\pi} \int_0^{2\pi}v(e^{i\theta}z)\,d\theta-{1\over\pi} \int_{|t|<1}{\overline\partial v(tz)\cdot z \over t}\;d\lambda(t),\cr |v(0)| &\leq {1\over 2\pi} \int_0^{2\pi}|v(e^{i\theta}z)|\,d\theta +2|z|\; \sup_{B(r)}|\overline\partial v|.\cr} $$ After calculation of the half courage (VM) for $z \in B(r)$, obtains on $$ |v(0)|\leq \VM\big[\,|v|\;;\;B(r)\big] + {4n\over 2n+1}\;r\cdot \sup_{B(r)}|\overline\partial v|, $$ and $$ \VM\big[\,|v|\;;\;B(r)\big] \leq \VM\big[\,|v|^2\;;\;B(r)\big]^{1/2} $$ thanks to the inequality of Cauchy-Schwarz.\hfil\square \medskip No remains prpers best what at check the elementary outcome following, who has be used during the démonstration. \medskip {\statement Lemme 15.15.\pointir}{\it Are $V_1,V_2$ deux functions psh at minimum growth on~$M$. Assumes On that $V = V_1 - V_2$ is psh. Then $V$ is at minimum growth.} \medskip {\it Démonstration}. As the théorème of normalisation of Noether, il there is of the functions polynomiales $f_1,\ldots,f_n$ on $M$ tel that $R(M)=\bC[z_1,\ldots,z_N]/I(M)$ be an algèbre entire on $\bC[f_1,\ldots,f_n]$. The morphisme $F=(f_1,\ldots,f_n) : M \to \bC^n$ is so own and ended, and has on an encadrement $$ C_{25}(1+|z|)^{C_{26}}\leq |F(z)| \leq C_{27}(1+|z|)^{C_{28}} $$ with $C_{25},\ldots,C_{28} > 0$. Thanks to the evident inequality $$ V \leq V_+ \leq F^*(F_*V_+) $$ prpers suffice of display that $F_*V_+$ is at minimum growth on~$\bC^n$. Comme $$ V_+ \leq (V_1)_+ + (V_2)_+ - V_2, $$ deduces for the half courage of $F_*V_+$ on the bowl $B(r)\subset\bC^n$ the majoration $$ \VM\big[F_*V_+\,;\,B(r)\big] \leq \VM\big[F_*(V_1)_++F_*(V_2)_+\,;\,B(r)\big] - \VM\big[F_*V_2\,;\,B(r)\big]. $$ The functions $F_*(V_1)_+$, $F_*(V_2)_+$ are psh at minimum growth, while the function $r \mapsto \VM[F_*V_2\,;\,;B(r)]$ is increasing. 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Math.\ Ann.\ {\bf 172} (1967), pp.~269--312. \bigskip} Jean-Pierre Demailly\\ Institut Fourier\\ Laboratoire de Mathématiques associé au C.N.R.S.\\ BP 74\\ 38402 St Martin d'Hères Cedex \vfill\eject \strut\vskip1cm \centerline{\bigbf Mémoires de la Société Mathématique de France} \medskip \centerline{\bigbf Nouvelle série} \bigskip {\parindent=5mm \noindent{\bf 1980} \item{1.} J.~Briançon, A.~Galligo, M.~Granger -- Déformations équisingulières des germes de courbes \item{2.} D.~Bertrand, M.~Waldschmidt -- Fonctions abéliennes et nombres transcendants \item{3.} Y.~Félix -- Dénombrement des types de $K$-homotopie \item{4.} L.~Bégueri -- Dualité sur un corps local \medskip \noindent{\bf 1981} \item{5.} S.~Ochanine -- Signature modulo-16, invariants de Kervaire généralisés \item{6.} Nguyen Tien Dai, Nguyen Huu Duc, F.~Pham -- Singularités non dégénérées des systèmes de Gauss-Manin réticulés \medskip \noindent{\bf 1982} \item{7.} P.~Ellia -- Sur les fibrés uniformes de rang $(n + 1)$ sur $\bP^n$ \medskip \noindent{\bf 1983} \item{8.} M.~Granger -- Géométrie des schémas de Hilbert ponctuels {\parindent=11.5mm \item{9/10.~} S.~Halperin -- Lectures on Minimal Models \item{11/12.} G.~Henniart -- La conjecture de Langlands locale pour GL(3) \medskip} \noindent{\bf 1984} {\parindent=6.3mm \item{13.} D.~Bertrand, M.~Emsalem, F.~Gramain, M.~Huttner, M.~Langevin, M.~Laurent, M.~Mignotte, J.-C.~Moreau, P.~Philippon, E.~Reyssat, M.~Waldschmidt -- Les nombres transcendants \item{14.} G.~Dloussky -- Structure des surfaces de Kato \item{15.} M.~Duflo, P.~Eymard, G.~Schiffmann -- Analyse harmonique sur les groupes de Lie et les espaces symétriques \item{16.} F.~Delon, D.~Lascar, M.~Parigot, G.~Sabbagh (Editeurs), Logique, octobre 1983, Paris \item{17.} Bernadette Perrin-Riou -- Arithmétique des courbes elliptiques et théorie d'Iwasawa \medskip \noindent{\bf 1985} \item{18.} Corinne Blondel -- Les représentations supercuspidales des groupes métaplectiques sur GL(2) et leurs caractères \item{19.} J.-P.~Demailly -- Mesures de Monge-Ampère et caractérisation géométrique des\break variétés algébriques affines\par}} \end % Local Variables: % TeX-command-default: "TeX" % End: