Luc Illusie

Thesis

In May 1971, Illusie defended a state doctorate ( Thèse d’État) 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 (in 1971 & 1972 ). The main results of the thesis are summarized in a paper in English (entitled "Cotangent complex and Deformations of torsors and group schemes") presented in Halifax, at Dalhousie University, in January 1971 as part of a colloquium on algebraic geometry. also exists in a slightly extended version. 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

Complexe cotangent et déformations, Lecture Notes in Mathematics 239 et 283, Berlin and New York, Springer, 1971–1972. (ed.) Cohomologie ℓ-adique et fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie 1965–66, SGA 5, dir. A. Grothendieck, Lecture Notes in Mathematics 589, Berlin and New York, Springer, 1977. (with Pierre Berthelot and Alexander Grothendieck), Théorie des intersections et théorème de Riemann–Roch, Séminaire de Géométrie Algébrique du Bois Marie 1966–67, SGA 6, Lecture Notes in Mathematics 225, Berlin and New York, Springer, 1971. "Complexe de de Rham–Witt et cohomologie cristalline", Annales Scientifiques de l'École Normale Supérieure, 1979, ser. 4, vol. 12, 4, pp. 501–661, url=http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1979_4_12_4/ASENS_1979_4_12_4_501_0/ASENS_1979_4_12_4_501_0.pdf. (coed. with Jean Giraud and Michel Raynaud), Surfaces algébriques, Séminaire de géométrie algébrique d'Orsay 1976–78, Lecture Notes in Mathematics 868, Berlin and New York, Springer, 1981. (with Michel Raynaud), "Les suites spectrales ssociées au complexe de De Rham–Witt", Publ. Math. IHÉS, vol. 57, 1983, pp. 73–212. (with Pierre Deligne),"Relèvements modulo p2 et décomposition du complexe de de Rham", Inv. math. (1987), vol. 89, pp. 247–270. "Sur la formule de Picard–Lefschetz", in Algebraic Geometry 2000, ed. Azumino (Hotaka), Advanced Studies in Pure Mathematics 36, 2002, pp. 249–268, Mathematical Society of Japan, Tokyo.