An Introduction to Modules

Abstract

Introduction This text gathers notes of a five hours course on D-modules given for the Winter School on Derived Categories, Weyl Algebras and Hodge Theory organized by D. Rumynin and T. Stafford on March 16-20 in Warwick. During this week, a course on Hodge theory and algebraic geometry was given by L. Migliorini and a course on Derived categories and constructible sheaves was given by G. Williamson. The primary goal of the school was to explain to which extend topology, Hodge theory and D-modules interact. The audience was not supposed to be familiar with D-modules. The following theorem [Kas75] was used as a guideline for this course Theorem 1. — Let M be a holonomic D-module on a complex manifold X. Then the de Rham complex DR M of M is a perverse sheaf. A full proof p1q of the following theorem was given Theorem 2. — Let X be a complex manifold and let M be a complex of D X -modules with bounded and holonomic cohomology. Then DR M has bounded and constructible cohomology. This theorem is a superb application of the machinery of derived categories and functorialities: trying to prove it for a single holonomic D X -module sticking to X does not lead anywhere whereas push-forward allows to argue by induction on the dimension of X. Since push-forward is not an exact functor, we are naturally led to use derived push-forward, thus producing complexes even if the input M is concentrated in degree 0. Hence, derived category is the right setting for both the statement and the proof of theorem 2. Let us explain the content of each section of these notes. The first section introduces the notion of D-modules on a complex manifold, DR and Sol for D-modules and state p1q Note that we don't claim originality in the proofs given in these notes. 2 J.-B. TEYSSIER the Riemann-Hilbert correspondence. Section 2 has to do with functorialities for D-modules. We give a proof of Kashiwara theorem on direct images by a closed immersion and a proof of the commutativity of DR with push-forward. In section 3, we explain why the characteristic variety is of fundamental importance in the theory of D-module and how it can be used. Section 4 gives a full proof of Kashiwara constructibility theorem, following [LM93]. Section 5 has to do with regularity. For meromorphic connections, we insist on the necessity of a meromorphic structure in the analytic setting already to state a definition, and then give some fundamental theorems leading to the algebraic Riemann-Hilbert correspondence for algebraic flat connections. I thank the organizers of the Winter School Derived Categories, Weyl Algebras and Hodge Theory for giving me the opportunity to teach this course, as well as L. Migliorini and G. Williamson for advices on what should be and what should not be in a first course on D-modules, saving the audience from a certain number of unpleasant computations. I also thank the students who attended the school for providing a very pleasant and stimulating work atmosphere during the week.