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Cartan and Complex Analytic Geometry Cartan and Complex Analytic Geometry Jean-Pierre Demailly effort in France to recruit teachers, due to the much increased access of pupils and students to On the Mathematical Heritage of Henri higher education, along with a strong research ef- Cartan fort in technology and science. I remember quite Henri Cartan left us on August 13, 2008, at the well that my father had a book with a mysterious age of 104. His influence on generations of math- title: Théorie Élémentaire des Fonctions Analytiques ematicians worldwide has been considerable. In d’une ou Plusieurs Variables Complexes (Hermann, France especially, his role as a 4th edition from 1961) [Ca3], by Henri Cartan, professor at École Normale Su- which contained magical stuff such as contour périeure in Paris between 1940 integrals and residues. I could then, of course, and 1965 led him to supervise not understand much of it, but my father was the Ph.D. theses of Jean-Pierre quite absorbed with the book; I was equally im- Serre (Fields Medal 1954), René pressed by the photograph of Cartan on the cover Thom (Fields Medal 1958), and pages and by the style of the contents, which had many other prominent math- obvious similarity to the “New Math” we started ematicians such as Pierre Cart- being taught at school—namely set theory and ier, Jean Cerf, Adrien Douady, symbols like ∪, ∩, ∈, ⊂,... My father explained Roger Godement, Max Karoubi, to me that Henri Cartan was one of the leading and Jean-Louis Koszul. French mathematicians and that he was one of However, rather than rewrit- the founding members of the somewhat secre- ing history that is well known tive Bourbaki group, which had been the source to many people, I would like of inspiration for the new symbolism and for the here to share lesser known reform of education. In France, the leader of the facts about his career and work, reform commission was A. Lichnerowicz, at least especially those related to parts I have been involved with. It as far as mathematics was concerned, and I got Henri Cartan at Oberwolfach, is actually quite surprising, myself involved with the new curriculum in grade September 3, 1981. in spite of the fact that I was ten in 1970. Although overly zealous promoters born more than half a century of the “New Math” made the reform fail less than later, how present Henri Cartan still was during fifteen years later, for instance by pushing abstract my studies. My first mathematical encounter with set theory even down to kindergarten—a failure Cartan was when I was about twelve, in 1969. In which resulted in very bad counter-reforms around the earlier years, my father had been an elemen- 1985—I would like to testify that in spite of harsh tary school teacher and had decided to go back to criticism sometimes geared toward the reform, Lille University to try to become a math teacher in what we were taught appeared well thought out, secondary education; there was a strong national quite rigorous, and even very exciting. In the rather modest high school I was frequenting at the time, Jean-Pierre Demailly is professor of mathematics at the the large majority of my fellows in the science Université de Grenoble. His email address is demailly@ class were certainly enjoying the menu and tak- fourier.ujf-grenoble.fr. ing a large benefit. The disaster came only later, 952 NOTICES OF THE AMS VOLUME 57, NUMBER 8 from the great excess of reforms applied at earlier of collaboration started with the younger German stages of education. generation represented by K. Stein, H. Grauert, and In any case, my father left me from that period R. Remmert. These events were probably among three books by Henri Cartan, namely the one al- the main reasons for Cartan’s strong engagement ready described and two other textbooks: Differ- in politics, especially toward human rights and ential Calculus and Differential Forms [Ca4] (also the construction of Europe; at age eighty, Henri by Hermann, Paris), which I never ceased using. Cartan even stood unsuccessfully for election to These books are still widely used and are certainly the European Parliament in 1984, as head of list among the primary references for the courses I for a party called “Pour les États-Unis d’Europe”, have been delivering at the University of Grenoble declaring himself to be a European Federalist. since 1983. I find it actually quite remarkable that In 1960, pursuing ideas and suggestions of French secondary school teachers of the 1960–90 Cartan, Serre [CS2, Se], and Grothendieck [Gt], era could be taught mathematics in the profound H. Grauert proved the coherence of direct im- textbooks by such mathematicians as Cartan, ages of coherent analytic sheaves under proper Dieudonné, or Serre, especially in comparison holomorphic morphisms [Gr]. Actually, a further with the general evolution of education in the last important coherence theorem was to be discov- two or three decades in France, and other Western ered more than three decades later as the culmina- countries as well, about which it seems that one tion of work on L2 techniques by L. Hörmander, cannot be so optimistic…. E. Bombieri, H. Skoda, Y. T. Siu, A. Nadel, and myself: In 1975 I entered École Normale Supérieure in if ϕ is a plurisubharmonic function, for instance a 2 Paris, and although Henri Cartan had left the École function of the form ϕ(z) = c log | gj (z)| where ten years earlier, he was still very much in the c>0 and the gj are holomorphic on an open set n background when I began learning holomorphic Ω in C , then the sheaf I(ϕ) ⊂OΩ of germs of functions of one variable. His role was eminently holomorphic functions f such that |f |2e−ϕ is locally stressed in the course proposed to first-year stu- integrable is a coherent ideal sheaf [Na]. The sheaf dents by Michel Hervé, who made great efforts to I(ϕ) is now called the Nadel multiplier ideal sheaf introduce sheaves to us, for example, as a means associated with ϕ ; its algebraic counterpart plays to explain analytic continuation and the maximal a fundamental role in modern algebraic geometry. domain of existence of a germ of a holomorphic The main philosophical reason is probably that L2 function. theory is a natural framework for duality and van- Two years later I started a Ph.D. thesis under ishing theorems. It turns out that I got the privilege the supervision of Henri Skoda in Paris, and it is of explaining this material to young students of only at this period that I began realizing the full École Normale Supérieure around 1992. It was extent of Cartan’s contributions to mathemat- therefore a considerable honor to me that Henri ics, in particular those on the theory of coherent Cartan came to listen to this lecture along with analytic sheaves and his fundamental work in the younger members of the audience. Although homological algebra and in algebraic topology he was close to being ninety years old at that time, [CE, CS1]. Taking part of its inspiration from it was a rare experience for me to have somebody J. Leray’s ideas and from the important work of there not missing a word of what I was saying—and K. Oka in Japan, the celebrated Cartan seminar sometimes raising embarrassing questions about [Ca2] ran from 1948 to 1964, and as an outcome insufficiently explained points! I remember that of the work by its participants, especially H. Car- the lecture actually had to be expanded at least tan, J.-P. Serre, and A. Grothendieck, many results half an hour beyond schedule, just to satisfy Car- concerning topology and holomorphic functions tan’s pressing demands…. of several variables received their final modern During the 1990s, my mathematical interests formulation. One should mention especially the went to the study of entire curves drawn on projec- proof of the coherence of the ring of holomorphic tive algebraic varieties, especially in the direction functions OX in an arbitrary number of variables, of the work of Green-Griffiths [GG] on the “Bloch after ideas of Oka, and the coherence of the ideal theorem”—for which they had provided a new sheaf of an analytic set proved by Cartan in 1950. proof in 1979. Henri Cartan had also taken an emi- Another important result is the coherence of nent role in this area, which is actually the subject the sheaf of weakly holomorphic meromorphic of his Ph.D. thesis [Ca1] under the supervision functions, which leads to Oka’s theorem on the of Paul Montel, although these achievements are existence of the normalization of any complex perhaps not as widely known as his later work on space. In this area of complex analysis, Henri sheaves. In any case, Cartan proved after A. Bloch Cartan had a long record of collaboration with [Bl] several important results in the then nascent German mathematicians, in particular H. Behnke Nevanlinna theory, which, in his own terms, can and P. Thullen [CT] already before World War II, be stated by saying that sequences of entire curves and after the dramatic events of the war, during contained in the complex projective n-space minus which Cartan’s brother was beheaded, a new era (n+2) hyperplanes in general position form an SEPTEMBER 2010 NOTICES OF THE AMS 953 “almost normal family”: namely, they either have [Gt] Alexander Grothendieck, Sur les faisceaux a subsequence that has a limit contained in the algébriques et les faisceaux analytiques cohérents, complement or a subsequence that approaches Séminaire Henri Cartan; 9e année: 1956/57, Quelques more and more closely a certain union of the “di- questions de topologie, Exposé no.
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