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${}$\vskip1cm
\centerline{\hugebf On the Mathematical Heritage of Henri Cartan}
\vskip7mm
\centerline{\bigbf Jean-Pierre Demailly}
\vskip1mm
\centerline{\bf Universit\'e de Grenoble I, Institut Fourier}
\centerline{\bf demailly@fourier.ujf-grenoble.fr}
\bigskip

Henri Cartan left us on August 13, 2008 at the age of 104. His influence
on generations of mathematicians worldwide has been considerable. 
In France especially, his role as a professor at 
\'Ecole Normale Sup\'erieure in Paris between 1948 and 1965 led him to
supervise the PhD theses of Jean Pierre Serre (Fields Medal  1954),
Ren\'e Thom (Fields Medal 1958), and of many other prominent mathematicians 
such as Pierre Cartier, Jean Cerf, Adrien Douady, 
Roger Godement, Max Karoubi and Jean-Louis Koszul.
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\hbox{\vbox{\noindent \bf Henri Cartan during the 50's,\\ 
while he was Professor at the\\
Faculty of Sciences of Paris}\kern-6cm
\vbox{\noindent\bf Henri Cartan at Oberwolfach,\\
September 3, 1981}}
\vskip5mm


However, rather than rewriting history which is well known to many
people, I would like here to share lesser known facts about his career
and work, especially those related to parts I have been involved
with. It is actually quite surprising, in spite of the fact that I was
born more than half a century later, how present Henri Cartan still was
during my studies. My first mathematical encounter with Cartan
was when I was about 12, in 1969. In the earlier years, my father had
been a primary school teacher and had decided to go back to Lille
University to try to become a math teacher in secondary education;
there was a strong national effort in France to recruit teachers, due
to the much increased access of pupils and students to higher
education, along with a strong research effort in technology and
science. I remember quite well that my father had a book with a
mysterious title ``Th\'eorie \'el\'ementaire des fonctions analytiques
d'une ou plusieurs complexes'' (Hermann, 4th edition from 1961)'' [Ca3], by
Henri Cartan, which contained magical stuff like contour integrals and
residues. I could then of course not understand much of it, but my
father was quite absorbed with the book; I was equally impressed by
the photograph of Cartan on the cover pages and by the style of the
contents which brought obvious similarity with the ``New Math'' we
started being taught at school -- namely set theory and symbols like
$\cup$, $\cap$, $\in$, $\subset$~... My father explained to me that
Henri Cartan was one of the leading French mathematicians, and that he
was one of the founding members of the somewhat secretive Bourbaki group
which had been the source of inspiration for the new symbolism
and for the reform of education. In France, the leader of the reform
commission was A.~Lichnerowicz, at least as far as mathematics are
concerned, and I got myself involved with the new curriculum in grade
ten in 1970. Although overly zealous promoters of the ``New Math''
made the reform fail less than 15 years later, for instance by pushing
abstract set theory even down to kindergarten -- a failure which
resulted into very bad counter reforms around 1985 -- I would like to
testify that in spite of harsh criticism sometimes geared towards the
reform, what we were taught appeared well thought, quite rigorous and
even very exciting. In the rather modest high school I was frequenting
at the time, the large majority of my fellows in the science class was
certainly enjoying the menu and taking a large benefit. The disaster
came only later, from the great excess of reforms applied at earlier
stages of education.

In any case, my father left me from that period three books by Henri
Cartan, namely the one already described and two other texbooks
``Differential calculus'' and ``Differential forms'' [Ca4] (also by
Hermann, Paris) which I never ceased using. These books are still
widely used and are certainly among the primary references for the
courses I have been delivering at the University of Grenoble since
1983. I find it actually quite remarkable that French secondary
school teachers of the 1960-1990 era could be taught mathematics in
the profound textbooks by mathematicians as Cartan, Dieudonn\'e or
Serre, especially in comparison with the general evolution of
education in the last two or three decades in France, and other
Western countries as well, about which it seems that one cannot be so
optimistic~...

In 1975, I entered \'Ecole Normale Sup\'erieure in Paris and although
Henri Cartan had left the \'Ecole 10 years earlier, he was still very
much in the backyard when I began learning holomorphic functions of
one variable. His role was eminently stressed in the course proposed
to first year students by Michel Herv\'e, who made great efforts to
introduce sheaves to us, e.g.\ as a means to explain analytic
continuation and the maximal domain of existence of a germ of
holomorphic function.

Two years later I started a PhD thesis under the supervision of Henri
Skoda in Paris, and it is only at this period that I began realizing
the full extent of Cartan's contributions to mathematics, in particular
those on the theory of coherent analytic sheaves, and his fundamental
work in homological algebra and in algebraic topology [CE, CS1]. Taking part 
of its inspiration from J.~Leray's ideas and from the important work of
K.~Oka in Japan, the celebrated Cartan seminar [Ca2] ran from 1948 to 1964,
and as an outcome of the work by its participants, especially
H.~Cartan, J.-P.~Serre and A.~Grothendieck, many
results concerning topology and holomorphic functions of several
variables received their final modern formulation. One should
mention especially the proof of the coherence of the ring of
holomorphic functions ${\Cal O}_X$ in an arbitrary number of variables,
after ideas of Oka, and the coherence of the ideal sheaf of an
analytic set proved by Cartan in 1950.  Another important
result is the coherence of the sheaf of weakly holomorphic
meromorphic functions, which leads to Oka's theorem on the existence
of the normalization of any complex space. In this area of complex
analysis, Henri Cartan had a long record of collaboration with German
mathematicians, in particular H.~Behnke and P.~Thullen [CT] already before 
World War~II, and after the dramatic events of the war during which Cartan's
brother was beheaded, a new era of collaboration started with the
younger German generation represented by K.~Stein, H.~Grauert and
R.~Remmert. These events were probably among the main reasons for
Cartan's strong engagement in politics, especially towards human
rights and the construction of Europe; at age 80, Henri Cartan even
stood unsuccessfully for election to the European Parliament in 1984,
as head of list for a party called ``Pour les \'Etats-Unis d'Europe'',
declaring himself to be a European Federalist.

In 1960, pursuing ideas and suggestions of Cartan, Serre [CS2, Se] and 
Grothendieck [Gt], H.~Grauert proved the coherence of direct images of
coherent analytic sheaves under proper
holomorphic morphisms [Gr].  Actually, a further important coherence
theorem was to be discovered more than three decades later as the
culmination of work on $L^2$ techniques by L.~H\"ormander,
E.~Bombieri, H.~Skoda, Y.T.~Siu, A.~Nadel and myself~: if $\varphi$ is
a plurisubharmonic function, for instance a function of the form
$\varphi(z)=c\log|\sum g_j(z)|^2$ where $c>0$ and the $g_j$ are
holomorphic on an open set $\Omega$ in ${\Bbb C}^n$, then the sheaf
${\Cal I}(\varphi)\subset{\Cal O}_\Omega$ of germs of holomorphic 
functions $f$ such that $|f|^2e^{-\varphi}$ is locally integrable is a
coherent ideal sheaf~[Na]. The sheaf ${\Cal I}(\varphi)$ is now called
the Nadel mutiplier ideal sheaf associated with~$\varphi$; its algebraic
counterpart plays a fundamental role in modern algebraic geometry.
The main philosophical reason is probably that $L^2$ theory is a
natural framework for duality and vanishing theorems. It turns out
that I got the privilege of explaining this material to young students
of \'Ecole Normale Sup\'erieure around 1992. It was therefore a
considerable honor to me that Henri Cartan came to listen to this lecture
along with the younger members of the audience. Although he was close
to being 90 years old at that time, it was a rare experience for me to
have somebody there not missing a word of what I~was saying -- and
sometimes raising embarrassing questions about insufficiently
explained points~!  I remember that the lecture actually had to be
expanded at least half an hour beyond schedule, just to satisfy
Cartan's pressing demands~...

During the nineties, my mathematical interests went to the study of
entire curves drawn on projective algebraic varieties, especially in
the direction of the work of Green-Griffiths [GG] on the ``Bloch theorem''
-- for which they had provided a new proof in 1979. Henri Cartan had also
taken an eminent role in this area, which is actually the subject of 
his PhD thesis [Ca1] under the supervision of Paul Montel, although these
achievements are perhaps not as widely known as his later work on
sheaves. In any case, Cartan proved after A.~Bloch [Bl] several
important results in the then nascent Nevanlinna theory, which, in his
own terms, can be stated by saying that sequences of entire curves
contained in the complex projective $n$-space minus $(n+2)$
hyperplanes in general position form an ``almost normal family'':
namely, they either have a subsequence which has a limit contained in
the complement, or a subsequence which approaches more and more
closely a certain union of the ``diagonal'' hyperplanes.  These
results were put much later in geometric form by Kobayashi and Kiernan [KK]
in terms of the concepts of taut and hyperbolically embedded
domains. Very recently, M.~Ru-P.M.~Wong [RW], E.~Nochka and P.~Vojta
[Vo] found various generalizations and improvements with a more
arithmetic flavor. It is remarkable that Cartan's early work already
contains many important ingredients such as the use of Nevanlinna
estimates for Wronskians, which are still at the heart of contemporary
research on the subject, e.g.\ in the form of the study of the
geometry of jet bundles [De1,~De2]. I~had once again the privilege of
explaining some of these modern developments in front of 
Henri Cartan in 1997, still as vigilant as ever, on the occasion of a 
celebration of his work by the French Mathematical Society.
\bigskip

{\startbiblio

\bibitem{[Bl]}{Andr\'e {\sc Bloch}, {\it Sur les syst\`emes de fonctions 
holomorphes \`a vari\'et\'es lin\'eaires lacunaires}, Ann.\ Scient.\ 
\'E.N.S.\ 3e s\'erie, tome 43 (1926), 309-262.}

\bibitem{[Ca1]}{Henri {\sc Cartan}, {\it Sur les syst\`emes de fonctions 
holomorphes \`a vari\'et\'es lin\'eaires lacunaires et leurs applications},
Ann.\ Scient.\ \'E.N.S.\ 3e s\'erie, tome 45 (1928), 255-346.}
                                   
\bibitem{[Ca2]}{Henri {\sc Cartan}, {\it S\'eminaire de l'\'Ecole 
Normale Sup\'erieure}, W.A.\ Benjamin, New York (1967).}

\bibitem{[Ca3]}{Henri {\sc Cartan}, {\it Th\'eorie \'el\'ementaire des
fonctions analytiques d'une ou plusieurs variables complexes},
Hermann, Paris (1961), 232~pp.}

\bibitem{[Ca4]}{Henri {\sc Cartan}, {\it Calcul diff\'erentiel,
formes diff\'erentielles}, Hermann, Paris (1967).}

\bibitem{[CE]}{Henri {\sc Cartan}, Samuel {\sc Eilenberg}, 
{\it Homological Algebra}, with an
appendix by David {\sc A.~Buchsbaum}. Reprint of the 1956
original, Princeton Landmarks in Mathematics. Princeton University
Press, Princeton, NJ, 1999. xvi+390 pp.}

\bibitem{[CS1]}{Henri {\sc Cartan}, Jean-Pierre {\sc Serre}, {\it 
Espaces fibr\'es et groupes d'homotopie, I.\ Constructions g\'en\'erales,
II.\ Applications}, C.~R.\ Acad.\ Sci.\ Paris 234 (1952), 288--290
and 393--395.}

\bibitem{[CS2]}{Henri {\sc Cartan}, Jean-Pierre {\sc Serre}, {\it Un 
th\'eor\`eme de finitude concernant les vari\'et\'es analytiques compactes}, 
C.~R.\ Acad.\ Sci.\ Paris 237 (1953), 128\--130.}

\bibitem{[CT]}{Henri {\sc Cartan} and Peter {\sc Thullen}, {\it Zur
Theorie der Singularit\"aten der Funktionen mehrerer komplexen
Ver\"anderlichen}, Math.\ Ann.\ 106 (1932), 617--647.}

\bibitem{[De1]}{Jean-Pierre {\sc Demailly}, {\it Algebraic criteria for 
Kobayashi hyperbolic projective varieties and jet differentials}, 
Proceedings of Symposia in
Pure Math., vol.~62.2, AMS Summer Institute on Algebraic Geometry held
at Santa Cruz, 1995, ed.\ J.~Koll\'ar, R.~Lazarsfeld (1997), 285--360.}

\bibitem{[De2]}{Jean-Pierre {\sc Demailly}, {\it Vari\'et\'es projectives 
hyperboliques et \'equations diff\'erentielles alg\'ebriques}, dedicated
to Henri Cartan, Gazette Math.\ 73 (juillet 1997), 3\--23.}

\bibitem{[Gr]}{Hans {\sc Grauert}, {\it Ein Theorem der analytischen
Garbentheorie und die Modulr\"aume komplexer Strukturen}, 
Publ.\ Math.\ I.H.E.S. 5 (1960), 233--292.}

\bibitem{[Gt]}{Alexander {\sc Grothendieck}, {\it
Sur les faisceaux alg\'ebriques et les faisceaux analytiques coh\'erents},
S\'eminaire Henri Cartan; 9e ann\'ee: 1956/57, Quelques questions de topologie, 
Expos\'e no. 2, 16 pp. Secr\'etariat math\'ematique, Paris 1958 73 pp.}

\bibitem{[GG]}{Mark {\sc Green}, Phillip {\sc Griffiths}, {\it Two 
applications of algebraic geometry to entire holomorphic mappings}, 
The Chern Symposium 1979, Proc.\ Internal.\ Sympos.\ Berkeley, CA, 1979, 
Springer-Verlag, New York (1980), 41--74.}

\bibitem{[KK]}{Peter {\sc Kiernan} and Shoshichi {\sc Kobayashi}, {\it
Holomorphic mappings into projective space with lacunary hyperplanes},
Nagoya Math.\ J.\ 50 (1973), 199--216.}

\bibitem{[Na]}{Alan M.\ {\sc Nadel}, {\it Multiplier ideal sheaves and
K\"ahler-Einstein metrics of positive scalar curvature},
Proc.\ Nat.\ Acad.\ Sci.\ U.S.A., 86 (1989), 7299--7300 and Annals of
Math., 132 (1990), 549--596.}

\bibitem{[RW]}{Min {\sc Ru} and Pitt-Man {\sc Wong}, {\it Integral points 
of ${\Bbb P}^n$ minus $2n+1$ hyperplanes in general position}. Invent.\ Math.\ 
106 (1991), 195--216.}

\bibitem{[Se]}{Jean-Pierre {\sc Serre}, {\it Faisceaux alg\'ebriques 
coh\'erents}, Ann.\ of Math., 61 (1955), 197--278.}

\bibitem{[Vo]}{Paul {\sc Vojta}, {\it On the Nochka-Chen-Ru-Wong proof of
Cartan's conjecture}, Journal of Number Theory 125 (2007), 229--234.}


}
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