Lecturer: Clemens Koppensteiner, clemens.koppensteiner@maths.ox.ac.uk

D-modules are quasi-coherent sheaves with an action of vector fields, or, in other words, integrable connections which are not required to be vector bundles. Thus they link the geometry, analysis and topology of complex varieties or manifolds. They are central objects of study in geometry, but have also found crucial application in other areas, such as representation theory.

In this course we will cover the theory of D-modules on complex algebraic varieties, working towards (a sketch of) the proof of the Riemann-Hilbert correspondence. The goal is to give good foundations of the theory in this setting, which can then be expanded upon. Due to time constraints we will not be able to go into applications of the theory, though some may be touched upon in exercise sheets.

**Prerequisites:** Complex varieties. Sheaf theory. A basic understanding of homological algebra and, ideally, derived categories.

There will be exercise sheets. For pass/fail grades I want to see reasonable progress on at least two of the sheets. Numerical grades will depend on the quality of the solutions. Please contact me with details of your grade requirements.

- [Exercise sheet 1]: direct and inverse image functor
- [Exercise sheet 2]: duality and characteristic varieties
- [Exercise sheet 3]: Gabber and Sato-Kashiwara filtrations
- [Exercise sheet 4]: Rees construction and Kashiwara's estimate

- Hotta, Takeuchi, Tanisaki,
*D-modules, perverse sheaves, and representation theory.*Progress in Mathematics 236, BirkhĂ¤user, 2008

Date | Topics | Class notes |
---|---|---|

Oct 15 | D-modules: Basic properties and examples; pullback functor | [PDF] |

Oct 22 | Pushforward functor; Kashiwara's Equivalence | [PDF] |

Oct 29 | Base change; coherent D-modules and duality; filtered rings | [PDF] |

Nov 5 | Characteristic varieties; holonomic D-modules | [PDF] |

Nov 12 | Lemma on b-functions and preservation of holonomic subcategory | [PDF] |

Nov 19 | Minimal extensions; analytic D-modules and the de Rham functor; constructible sheaves | [PDF] |

Nov 26 | de Rham complexes are constructible; regular connections | [PDF] |

Dec 3 | regular holonomic D-modules; Riemann-Hilbert correspondence; perverse sheaves | [PDF] |