Good Meromorphic Connections (Analytic Theory) and the Riemann-Hilbert Correspondence

This chapter is similar to Chap. 10, but we now assume that D is a divisor with normal crossings. We start by proving the many-variable version of the Hukuhara-Turrittin theorem, that we have already encountered in the case of a smooth divisor. It will be instrumental for making the link between formal and holomorphic aspects of the theory. The new point in the proof of the Riemann-Hilbert correspondence is the presence of non-Hausdorff eale spaces, and we need to use the level structure to prove the local essential surjectivity of the Riemann-Hilbert functor. As an application of the Riemann-Hilbert correspondence in the good case and of the fundamental results of K. Kedlaya and T. Mochizuki on the elimination of turning points by complex blowing-ups, we prove a conjecture of M. Kashiwara asserting that the Hermitian dual of a holonomic D-module is holonomic, generalizing the original result of M. Kashiwara for regular holonomic D-modules to possibly irregular holonomic D-modules and the result of Chap. 6 to higher dimensions.