Lectures on $\mathscr{D}$-Modules
HU Berlin, Summer 2019
Problems and Notes
Here are the lecture notes and problem sheets: Notes / Problems 01 02 03 04 05 06
Topics
The theory of algebraic $\mathscr{D}$-modules provides a bridge from algebra to analysis and topology. It can be regarded as ``mildly noncommutative algebra´´ in the sense that it is obtained by extending the methods of commutative algebra to noncommutative rings (or rather sheaves of rings) of algebraic differential operators on complex varieties. As such it gives a substitute in algebraic geometry for the theory of linear partial differential equations. For instance, the ring of differential operators in one variable is the Weyl algebra \[ \mathscr{D}_{\mathbb{A}^1} \;=\; \Bigl\{ \, f_n(x)\, \partial^n + f_{n-1}(x)\, \partial^{n-1} + \cdots + f_0(x) \, \mid \, n\in \mathbb{N_0}, f_0, \dots, f_n \in \mathbb{C}[x] \, \Bigr\}, \] the free algebra on non-commuting variables $x,\partial$ modulo the commutator relation $[\partial, x] = 1$. Finitely presented modules over this algebra correspond to systems of ordinary linear differential equations with polynomial coefficients. Many interesting transcendental functions satisfy such algebraic differential equations, which makes them accessible with tools from algebraic geometry! Another viewpoint comes from differential geometry: $\mathscr{D}$-modules generalize holomorphic vector bundles with flat connections, allowing in particular for connections with singularities. In this context the classical equivalence of categories between flat vector bundles and representations of the fundamental group generalizes to the celebrated Riemann-Hilbert correspondence between regular holonomic $\mathscr{D}$-modules and perverse sheaves, one of the most fundamental links between algebraic/analytic geometry and topology.
The lecture will give a general introduction to the theory of $\mathscr{D}$-modules on algebraic varieties, with a view towards the Riemann-Hilbert correspondence. On the way we will become familiar with powerful tools from homological algebra such as derived categories.
Prerequisites
Some basic knowledge in commutative algebra and algebraic geometry will be helpful.
Literature
- J. Bernstein, Algebraic theory of D-modules, unpublished lecture notes [www]
- A. Borel et al., Algebraic D-modules, Academic Press (1987) [Primus]
- S. Coutinho, A primer of algebraic D-modules, Cambridge Univ. Press (1995) [Primus]
- R. Hotta et al., D-modules, perverse sheaves and representation theory, Birkhäuser (2008) [Primus]
- M. Kashiwara, D-modules and microlocal calculus, AMS (2003) [Primus]