D-modules in characteristic p and applications
Wednesdays 12:15-13:45 (CEST)
Room 1.007 (at the University of Bonn)
Next (and the last) class on 13th July!
D-modules in characteristic p and applications
Wednesdays 12:15-13:45 (CEST)
Room 1.007 (at the University of Bonn)
Next (and the last) class on 13th July!
Description: In this course I am going to discuss the theory of differential operators in characteristic p and their application to various areas of mathematics. We will start with the general structure theory of the algebra D_X of (crystalline) differential operators, covering most importantly the "Azumaya property" and p-curvature map, through which we are going to explain several classical phenomena like Frobenius-descent, Milne's short exact sequence, et.c. Here "the Azumaya property" is the observation that D_X is naturally an Azumaya algebra over the cotangent bundle T^*X' on the Frobenius-twist of X. We will then discuss in great detail the nonabelian Hodge theory of Ogus-Vologodsky, the main idea being that a Simpson-like correspondence in characteristic p can be obtained by splitting the above Azumaya algebra. Though the splitting almost never exists globally, the theorem of Ogus-Vologodsky states that there is one in the PD-neighborhood of the zero section once we fix a lifting of X to W_2. We will consider some examples where the splitting can be extended to the formal neighborhood (e.g. abelian varieties) and discuss some applications like a Deligne-Illusie-style proof of the Barannikov-Kontsevich theorem and Drinfeld's splitting of the de Rham complex via the action mu_p. Then we will also discuss another important example given by the flag variety G/B, where the splitting of D_X along the fibers of the Springer map T^*G/B --> N to the nilpotent cone leads to the derived equivalence of the categories of U(g)-modules with a given generalized central character and coherent sheaves with support on the corresponding Springer fiber. If time permits we will also briefly discuss the work of Bezrukavnikov-Braverman on the geometric Langlands correspondence in this crystalline D-modules context.