In this talk we will explain some recent results we obtained on the constructibility of tempered solutions of holonomic D-modules on complex manifolds conjectured by M. Kashiwara and P. Schapira. We will start by recalling the importance of the constructibility of complexes of holomorphic solutions inside the classical Riemann-Hilbert correspondence. Then we will discuss the one dimensional case which we established some years ago. In the end we will discuss the higher dimensional case and the use we made of recent results of T. Mochizuki and K. Kedlaya on formal and asymptotic decomposition of holonomic D-modules.
Seminar on Analysis at University of Tsukuba