An existence theorem for tempered solutions of D-modules on complex curves

Abstract

Let X be a complex curve, $X_{sa}$ the subanalytic site associated to X, M a holonomic $D_X$-module. Let $O^t$ be the sheaf on $X_{sa}$ of tempered holomorphic functions, Sol(M) (resp. $Sol^t$(M)) the complex of holomorphic (resp. tempered holomorphic) solutions of M. We prove that the natural morphism from $H^1 Sol^t$(M) to $H^1$Sol(M) is an isomorphism of sheaves on $X_{sa}$. As a consequence, we prove that $Sol^t$(M) is R-constructible in the sense of sheaves on $X_{sa}$. Such a result is conjectured by M. Kashiwara and P. Schapira in Ast\'erisque 284 in any dimension.