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\centerline{\twelvebf Pierre Lelong~: a foundational work}
\smallskip
\centerline{\twelvebf in complex analysis and in analytic geometry}
\medskip
\centerline{\bf Jean-Pierre demailly}
\centerline{Universit\'e de Grenoble I, Institut Fourier}
\centerline{and Acad\'emie des Sciences de Paris}
\bigskip

My first encounter with Pierre Lelong goes back to 1977, a year during
which I started to attend his ``S\'eminaire d'Analyse'' in Paris,
coorganized in collaboration with Henri Skoda since 1976. During the
same year, I~also benefitted from a series of lectures that Pierre Lelong
gave on the theory of plurisubharmonic functions and positive currents,
following a PhD course presented a few months earlier by Henri
Skoda at Universit\'e Paris VI. These early contacts have had a strong
and lasting influence on my scientific career, that's the very least
I~can say. In fact, my later scientific investigations almost never
departed from the fundamental theories initiated by Pierre Lelong
during the decades 1940--50 and 1950--60 ([Lel42], [Lel57]): these
theories have wonderful applications to vast subdomains of
mathematics, e.g.\ in Number Theory or Algebraic Geometry. Even though
many mathematicians throughout the world have continued exploring these
directions in the following decades, most experts would certainly
agree that a lot remains to be done today.

A few years later, when I defended my ``Th\`ese de Doctorat d'\'Etat''
in 1982, I had the privilege of being invited at Pierre Lelong's
private dwelling in Paris. This was the occasion for me to realize
another important aspect of his past activities, namely, his deep
commitment to politics, and the guidance he exerted in 1959--1961
during a major reform of higher education and research, as one of
the scientific advisors of G\'eneral de Gaulle, then the President
of the French Republic. Especially impressive were Pierre Lelong's
private library and the unusual number of books and documents 
dealing with politics and political science. At present, the French 
system of higher education is faced with severe difficulties, and 
I~cannot refrain from thinking what a benefit my country had
during the 60's, a period of course economically much more 
favorable, when the scientific policy was guided by such enlightened 
minds as Pierre Lelong. In fact, France enjoyed at that time 
a sustained scientific and technological development, as well as 
a very strong increase of the number of students at universities. One
would like to see clearer signs today that the European governments
are ready to invest in science and to give it again a prominent role
in the evolution of society -- and, as a consequence, to rely
extensively on the expertise of the scientific community rather than
on technocrats$\,$!

Even though this is probably not the most central part of Pierre Lelong's
scientific work, I~would like to discuss here one contribution
that has in my view shed light on several important problems. This is
a paper entitled {\it ``\'El\'ements extr\'emaux sur le c\^one des
  courants positifs ferm\'es'' $($``Extremal elements in the cone of closed
positive currents''$)$}, published in the ``S\'eminaire d'Analyse'' in
1971/1972~[Lel71]. The first main statement of the paper is:

\noindent {\bf Theorem 1.} {\it Let $M$ be an analytic set of complex
  codimension~$p$, assumed to be irreducible, in a connected complex analytic
  analytic~$X$ $($countable at infinity$)$. Then the current of
  integration $[M]$ is extremal in the cone of closed positive currents
  of bidegree $(p,p)$ in~$X$.}

After stating this result, Pierre Lelong observed that many other examples
of closed positive currents that had then been investigated were
not extremal, especially those arising from smooth convex functions
or smooth plurisubharmonic exhaustion functions like $\log|z|$,
and he concluded: ``{\it It is likely that Theorem~$1$ does not produce
all extremal elements in the cone of closed positive currents; this seems
to be an important unsolved question of complex geometry$\,$}''. I~still
remember a private discussion we had on this issue at Jussieu. Strongly 
stimulated by these observations and by a further exchange with 
Jean-Louis Verdier, I~realized a couple of weeks later, around end of 1981,
that such a restrictive property concerning extremal elements could not hold.
In fact, it would have implied via Choquet's represention theorem a 
formulation of Hodge conjecture that was much too strong to be true.
Therefore, some of the extremal elements must be more complicated
than currents of integration on analytic sets, and I~found shortly
afterwards an explicit example of such an extremal closed positive
current of bidegree $(1,1)$ in the complex projective plane [De82].
As first noticed by Eric Bedford, further examples appear in a
natural way in complex dynamics of several variables; many invariant
closed positive currents produced by complex dynamical systems are
actually extremal currents, although their support is in general
a fractal set, and therefore is not analytic. This is for instance
what happens for a current of the form $\lim_{k\to+\infty}d^{-k}{i\over\pi}
\partial\overline\partial\log|P^{k}(z)|$, where $P^k$ is the $k$-th
iteration of a polynomial endomorphism of degree $d>1$ on
projective space, the support of such a current being a Julia set~[Sib99].
The dynamical study of ``hyperbolic''  endomorphisms of certain
algebraic surfaces, e.g.\ K3 surfaces, also leads to such extremal
invariant currents~[Ca03].

Another fundamental statement contained in the above cited article 
[Lel71] is the following.

\noindent {\bf Theorem 2.} {\it if $G$ is a pseudoconvex domain 
in~$\bC^n$, the positive cone generated over rational coefficients by
functions of the form $\log|f|$, where $f$ is holomorphic in~$G$, is
dense in the cone of plurisubharmonic functions on~$G$.}

\noindent {\bf Corollary.} {\it If moreover $H^2(G,\bR)=0$, the positive
cone generated over rational coefficients by currents of integration $[D]$ 
on irreducible divisors of $G$ is dense in the cone of closed positive
currents of type $(1,1)$ on~$G$.}

The original proof of Lelong rests upon a use of complex function
theory on Hartogs domains of the type $|w|<e^{-\varphi(z)}$,
$(z,w)\in\bC^n\times\bC$.  If $\varphi$ is plurisubharmonic and if
$z$ is taken in a pseudoconvex domain~$G$, it is known that the
corresponding Hartogs domain in $G\times\bC$ is again
pseudoconvex$\,$; as a consequence, there exists a holomorphic
function $F(z,w)$, the domain of existence of which is precisely the
Hartogs domain $|w|<e^{-\varphi(z)}$. The approximation of the
function $\varphi$ by logarithms of holomorphic functions $f_j(z)$ is
then obtained by applying Hadamard's formula to compute the radius of
convergence of the power series $\sum_{k\in\bN}a_k(z)w^k$
of~$F(z,w)$. The corollary is then derived by means of the fundamental
``Lelong-Poincar\'e'' equation, stating that for every holomorphic
function $F$ the current ${i\over\pi}\partial\overline\partial\log|F|$
coincides with the current of integration $[Z_F]$ on the zero divisor
of~$F$.

These approximation results for currents are now a central ingredient
of modern analytic geometry. By replacing the qualitative existence theorem of
defining holomorphic functions and Hadamard's formula with deeper
results such as the Ohsawa-Takegoshi $L^2$ extension theorem ([OT87]), 
one can obtain more precise statements in which the multiplicities of
the approximating $\bQ$-divisors converge uniformly to the
``Lelong numbers'' of the given closed positive $(1,1)$-current. 
In that way, one gets a very strong analytic tool that allows in particular
to prove numerous geometric results -- for instance,
Siu's theorem on the analyticity of level sets associated with Lelong
numbers of closed positive currents~[Siu74]. Another consequence of
such techniques in algebraic geometry is the proof of the conjecture
on the invariance of plurigenera for deformations of arbitrary non singular
projective algebraic varieties ([Siu00], [Pa07])$\,$; the latter result
relies again on the Ohsawa-Takegoshi theorem and on a compactness
argument for closed positive currents of type $(1,1)\;$; it comes as
a surprise that no algebraic proof is known at this point in time,
although the statement involves only algebraic objects$\,$!

Finally, among applications to number theory, one should mention
Bombieri's theorem on algebraic values of meromorphic
functions of several variables satisfying algebraic differential
equations [Bo70], [Sk76], [Lel79]. The proof, here again, 
exploits in an essential way the compactness properties of closed
positive currents of type $(1,1)$ in classes of currents of finite order,
in conjunction with H\"ormander's $L^2$ estimates for the
$\overline\partial$ operator.

Pierre Lelong's clever use of ``flexible objects''\footnote{${}^
{(*)}$}{\sevenrm \baselineskip8.4pt ``Objets souples'', according to
the terminology employed by Pierre Lelong himself in his notice of
scientific work sent to Acad\'emie des Sciences [Lel73].\par},
such as plurisubharmonic functions and positive currents, has permitted
the emergence of various important techniques that have led to strong
effective formulations of many results in algebraic, analytic or
arithmetic geometry, especially in areas where previously known
techniques could only produce qualitative results. Pierre Lelong
was perfectly aware of the philosophical dimension of the contributions
he made, and he has very early set up their most fundamental consequences.

\vskip8pt

\noindent
{\bf References}
\vskip3pt
{\baselineskip 8.4pt \parskip 1.5pt plus 1pt minus 0.5pt
\let\rm=\sevenrm \let\bf=\sevenbf \let\it=\sevenit

\bibitem[Bo70]&Bombieri, E.&Algebraic values of meromorphic maps&Invent.\ Math.\ {\bf 10} (1970) 267--287$\,$; Addendum, Invent.\ Math.\ {\bf 11} (1970) 163--166&

\bibitem[Ca03]&Cantat, S.&Dynamique des automorphismes des surfaces 
{\rm K3}&Acta Math.\ {\bf 187} (2001) 1--57&

\bibitem[De82]&Demailly, J.-P.&Courants positifs extr\^emaux et conjecture de
Hodge&Invent.\ Math.\ {\bf 69} (1982) 347--374&

\bibitem[Lel42]&Lelong, P.&D\'efinition des fonctions plurisousharmoniques&C.~R.\ Acad.\ Sci.\ Paris {\bf 215} (1942) p.~398 et p.~454&

\bibitem[Lel57]&Lelong, P.&Int\'egration sur un ensemble analytique complexe&Bull.\ Soc.\ Math.\ France {\bf 85} (1957) 239--262&

\bibitem[Lel68]&Lelong, P.&Fonctions plurisousharmoniques et formes diff\'erentielles positives&Dunod, Paris, Gordon \&\ Breach, New York (1968)&

\bibitem[Lel71]&Lelong, P.&\'El\'ements extr\'emaux sur le c\^one des
courants positifs ferm\'es&Springer Lecture Notes in Math.\ {\bf 332} (1973) 112--130&

\bibitem[Lel73]&Lelong, P.&Notice sur les titres et travaux scientifiques&site
de l'Acad\'emie des Sciences,\hfil\break
{\seventt http://www.academie-sciences.fr/academie/membre/Lelong\_{\kern0.5pt}notice\_{\kern0.5pt}1973.pdf}&

\bibitem[Lel79]&Lelong, P.&Sur les cycles holomorphes \`a coefficients
positifs dans $\scriptstyle\bC^n$ et un compl\'ement au th\'eor\`eme de
E.~Bombieri&C.~R.\ Math.\ Acad.\ Sc.\ Canada, vol.~1,
n$\scriptstyle{}^\circ$4 (1979) 211--213&

\bibitem[OT87]&Ohsawa, T., Takegoshi, K.&On the extension of $\scriptstyle L^2$
holomorphic functions&Math.\ Zeitschrift {\bf 195} (1987) 197--204&

\bibitem[Pa07]&P\u{a}un, M.&Siu's invariance of plurigenera: a one-tower
proof&J.~Differential Geom.\ {\bf 76} (2007) 485--493&

\bibitem[Sib99]&Sibony, N.&Dynamique des applications rationnelles de
$\scriptstyle\bP^k$&Dynamique et g\'eom\'etrie complexes (Lyon, 1997), ix--x, 
xi--xii, 97--185, Panor.\ Synth\`eses {\bf 8}, Soc.\ Math.\
France, Paris (1999)&

\bibitem[Siu74]&Siu, Y.T.&Analyticity of sets associated to Lelong numbers and the extension of closed positive currents&Invent.\ Math.\ {\bf 27} (1974) 53--156&

\bibitem[Siu00]&Siu, Y.T.&Extension of twisted pluricanonical sections 
with plurisubharmonic weight and invariance of semipositively twisted 
plurigenera for manifolds not necessarily of general type&Complex 
Geometry (G\"ottingen, 2000), Springer, Berlin (2002) 223--277&

\bibitem[Sko76]&Skoda, H.&Estimations $\scriptstyle L^2$ pour 
l'op\'erateur $\scriptstyle\overline \partial $ et applications arithm\'etiques&S\'eminaire P.~Lelong (Analyse), ann\'ee 1975/76, Lecture Notes in Math.\ no~{\bf 538}, Springer-Verlag, Berlin (1977) 314--323&

\par}

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