MATH757 (Spring 2020): D-modules.

Instructor: Prof. Ivan Loseu (email:

Course web page:

Lectures: MW 1:00-2:15pm, LOM 214.

Office hours: by appointment, DL 416.

Goals: This course is an introduction to the theory of D-modules, i.e., modules over (sheaves of) algebras of linear differential operators.

Topics to be covered (6 and 7 time permitting):

Schedule : the class will start on 1/22. It will be cancelled for 1/27. Otherwise it will run as scheduled.
  • Jan 22, Lecture 1: Differential operators on the affine space.
  • Jan 29, Lecture 2 (by Sasha Tsymbaliuk): Differential operators on smooth affine varieties.
  • Feb 3, Lecture 3: Sheaves of differential operators. Quasi-coherent and coherent D-modules. Some examples.
  • Feb 5, Lecture 4: Further examples, O-coherent D-modules. Support of a D-module and why it fails to measure the size.
  • Feb 10, Lecture 5: Good filtrations and singular supports.
  • Feb 12, Lecture 6: Properties of singular support, holonomic modules. Characteristic cycles.
  • Feb 17, Lecture 7: Poisson structures from deformations. Gabber's theorem. Notes (that, in particular, clarify deformation orders needed for inducing a Poisson structure).
  • Feb 19, Lecture 8: Overview. Pullback of D-modules.
  • Feb 24, Lecture 9: Pullback continued. Discussion of pushforward. Left vs right D-modules. Pushforward for affine morphisms.
  • Feb 26, Lecture 10: Pushforward continued. Kashiwara's lemma.
  • Mar 2, Lecture 11: D-modules on singular varieties. Ext's of D-modules
  • Mar 4, Lecture 12: Duality for D-modules.
  • Week of Mar 23: Lectures 13 and 14 (updated 3/25). Duality, cont'd. Classification of simple holonomic D-modules.
  • Week of Mar 30: Lectures 15 and 16 (updated 4/1). Preservation of holonomicity. O-coherent D-modules vs representations of the fundamental group.
  • Week of Apr 6: Lectures 17 and 18. Equivariant coherent sheaves and D-modules.
  • Week of Apr 13: Lectures 19 and 20 (updated 4/13). Equivariant D-modules in the case of finitely many orbits. Applications to Representation theory.
  • Week of Apr 20: Lectures 21 to 23 (the last ones!). Twisted equivariant D-modules, sheaves of twisted differential operators..

    Homeworks : there will be optional homeworks.

    Homework 1 concentrating on the algebras of differential operators.

    Homework 2 concentrating on singular supports and characteristic cycles.

    Homework 3 (updated 4/1 and it's not a joke!) mostly on pullbacks, pushforward and related questions.

    Homework 4 on equivariant D-modules and twisted differential operators.

    Prerequisites: Algebraic geometry (for 1-3 the first two chapters of Hartshorne, as well as the 3rd chapter starting 4). Derived categories in 4, the first chapter in the Kashiwara-Shapira's "Sheaves on manifolds" may serve as a reasonable introduction. For 6, we assume the knowledge of complex semisimple Lie groups, Lie algebras and their representations.


    J. Bernstein, Algebraic theory of D-modules.

    V. Ginzburg, Lectures on D-modules.

    R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves and representation theory. Progress in Mathematics, 236. Birkhauser, 2008.