# TCC course: D-modules

Wednesdays, 2pm-4pm, starting 16/01/2019.

Lectures take place in whichever room your department uses for video lectures. In Oxford, this is VC.

There is no lecture on 27/02. Instead we'll have an extra lecture on 13/03.

## Lecture notes

Will be posted here.
## Course syllabus

D-modules are modules over the sheaf of differential operators on an algebraic variety (or scheme, or manifold, ...).

They give one of the most fundamental instances of non-commutative algebra entering algebraic geometry.

We will develop the basic language of D-modules and see how it connects with areas like differential equations and in particular representation theory.

The final goal of the course is the Riemann-Hilbert correspondence, which shows that a category of D-modules encodes topological information about the underlying space: the category of regular holonomic D-modules on X is equivalent to the categories of perverse sheaves on X.
- Definitions and examples. The Weyl algebra and the sheaf of differential operators. Integrable connections. Links to PDEs.
- Operations. Tensor product, sideswitching. Direct and inverse image. Kashiwara's equivalence.
- Coherent and holonomic D-modules. Characteristic varieties and Bernstein's inequality. Inverse images under smooth maps, direct images under proper maps. Duality.
- The six functor formalism. Minimal extensions and simple holonomic modules.
- Constructible sheaves and perverse sheaves. De Rham and solution complexes.
- Worked example: D-modules on P1 and representation theory of SL_2.
- The Riemann-Hilbert correspondence.

## Preliminaries

Good knowledge of basic algebraic geometry (complex algebraic varieties, basic operations on O-modules).

Familiarity with the language of derived categories is useful, but all notions will be recalled in lectures.

## References

- Bernstein: Lecture notes on D-modules.
- Borel et al: Algebraic D-modules. Academic Press, 1987.
- Hotta, Takeuchi, Tanisaki: D-modules, perverse sheaves and representation theory. Birkhauser, 2008.

The course essentially covers the first half (chapters 1-8) of HTT.
## Exercises

## Assessment

If you need any form of assessment/credit for this course, please contact me.

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