## ** MATH757 (Spring 2020): D-modules.**

**Instructor: **Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

**Course web page: **https://gauss.math.yale.edu/~il282/Dmod

**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):

- 1) Algebras and sheaves of algebras of differential operators.
- 2) Modules over sheaves of differential operators. Support and singular support.
The Bernstein inequality and the Gabber theorem. Holonomic D-modules.
- 3) Left and right D-modules, pullback and pushforward functors. Kashiwara's lemma.
- 4) Verdier duality. Classification of simple holonomic modules. Preservation of
holonomicity under pullbacks and pushforrwards.
- 5) Variations: twisted differential operators, equivariant D-modules.
- 6*) Connections with representation theory: the differential operators on flag varieties
and the Beilinson-Bernstein localization theorem.
- 7*) The Riemann-Hilbert correspondence.

** 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.

**References**:

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.