Course 18.735, Topics in Algebra “Introduction to algebraic D-modules” Instructor: Roman Bezrukavnikov MW 9:30–11

The formalism of algebraic D-modules is a tool to account for ODE phenomena in the context of algebraic geometry.

The course will cover (some of) the following topics.

Definitions and basic properties of the ring of differential operators. Weyl algebra, formal Fourier transform.

Singular support, Bernstein inequality, Gabber’s involutivity and holonomicity.

Application: analytic continuation of distribution P^{λ} and b-function. Left and right D-modules. Pull-back and push-forward functors, Kashiwara’s Lemma. Intermediate extension and classification of irreducible holonomic D-modules. Application: D-affineness of generalized flag varieties. Derivation of

Kazhdan-Lusztig conjectures from Hodge D-module theory.

Time permitting the following ingredients of the theory of Hodge Dmodules will be discussed:

Verdier specialization and gluing. Malgrange-Kashiwara filtration (V - filtration).

Nearby cycles and b-function. V -filtration and multiplier ideals (after Budur, Saito et.al.)

The course will have an intersection of about 30–40% with the course given by me in 2006.

Prerequisites amount to basic algebraic geometry (including sheaves). The application to Kazhdan-Lusztig conjectures (about 1 or 2 lectures) relies on basic semi-simple Lie algebras theory, the rest of the course is independent of this material.

Students taking the course for credit will be asked to submit a written work, either solutions to a set of exercises or a paper with an exposition of a result from the literature.