The Riemann - Hilbert Problem for Holonomic Systems

The purpose of this paper is to give a proof to the equivalence of the derived category of holonomic systems and that of constructible sheaves. Let X be a paracompact complex manifold and let ® x and 0 x be the sheaf of differential operators and holomorphic functions, respectively. We denote by Mod(^z) the abelian category of left ^^-Modules and by D(^) its derived category. Let ~D^(^x) denote the full sub-category of D(^z) consisting of bounded complexes whose cohomology groups are regular holonomic ([KK]). By replacing @x with @*x, the sheaf of differential operators of infinite order, and "regular holonomic" with "holonomic," we similarly define Mod(^J), D(SJ) and DU^*)Let us denote by Mod(X) the category of sheaves of C-vector spaces on X and by D(X) its derived category. We denote by Dc(-X) the full sub -category of D(X) consisting of bounded complexes whose cohomology groups are constructible. Let us define