Luc Illusie


Luc Illusie is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences.

Biography

Luc Illusie entered the École Normale Supérieure in 1959. At first a student of the mathematician Henri Cartan, he participated in the Cartan–Schwartz seminar of 1963–1964. In 1964, following Cartan’s advice, he began to work with Alexandre Grothendieck, collaborating with him on two volumes of the latter’s Séminaire de Géométrie Algébrique du Bois Marie. In 1970, Illusie introduced the concept of the cotangent complex.
A researcher in the Centre national de la recherche scientifique from 1964 to 1976, Illusie then became a professor at the University of Paris-Sud, retiring as emeritus professor in 2005. Between 1984 and 1995, he was the director of the arithmetic and algebraic geometry group in the department of mathematics of that university. and Gérard Laumon are among his students.

Thesis

In May 1971, Illusie defended a state doctorate entitled "Cotangent complex; application to the theory of deformations" at the University of Paris-Sud, in front of a jury composed of
Alexander Grothendieck, Michel Demazure and Jean-Pierre Serre and presided by Henri Cartan.
The thesis was published in French by Springer-Verlag as a two-volume book. The main results of the thesis are summarized in a paper in English presented in Halifax, at Dalhousie University, on January 1971 as part of a colloquium on algebraic geometry. This paper, originally published by Springer-Verlag in 1972, also exists in a slightly extended version.
Illusie's construction of the cotangent complex generalizes that of Michel André and Daniel Quillen to morphisms of ringed topoi. The generality of the framework makes it possible to apply the formalism to
various first-order deformation problems: schemes, morphisms of schemes,
group schemes and torsors under group schemes. Results concerning commutative
group schemes in particular were the key tool in Grothendieck's proof of his
existence and structure theorem for infinitesimal deformations of Barsotti–Tate groups, an ingredient in Gerd Faltings' proof of the Mordell conjecture. In Chapter VIII of the second volume of the thesis, Illusie introduces
and studies derived de Rham complexes.

Awards

Illusie has received the Langevin Prize of the French Academy of Sciences in 1977 and, in 2012, the Émile Picard Medal of the French Academy of Sciences for "his fundamental work on the cotangent complex, the Picard–Lefschetz formula, Hodge theory and logarithmic geometry".

Selected works