Beijing Institute of Mathematical Sciences and Applications

The algebraic approach to complex analysis, the D-module language, was pioneered by the Japanese school (Kashivara, Sato,...) and found applications in a wide variety of subjects. One of the analytic results in the theory is due to P. Schapira and J.-P. Schneiders, (Index theorem for elliptic pairs, Elliptic pairs II, Asterisque vol. 224, 1994.) As the title suggests, they gave conditions for ellipticity of holonomic modules, and formulated a conjecture giving the value of the index of such elliptic system. This should be understood as a very general reformation of the classical Riemann-Roch theorem which computes the Euler characteristic of a holomorphic vector bundle on a compact complex manifold. In this series of lectures we will review basic elements of the theory of D-modules and develop it sufficiently to end up with the proof of the Schapira-Schneider conjecture..
It is in some way a continuation of the previous lectures on non-commutative calculus, since the final formulas use the same type of arguments as those in the proofs of algebraic index theorem.
The course will be based on the lecture notes of Bernstein "Algebraic theory of D-modules", "Lectures on algebraic D-modules" by Alexander Braverman and Tatyana Chmutova" and two older papers on the Riemann-Roch theorem for D-modules, Bresler, Nest, Tsygan, "RiemannRoch Theorems via Deformation Quantization, I, II", Advances in Mathematics 167, Issue 1 (2002) pp. 1-25 and pp. 26-73.