Sur une caractérisation des $\mathcal{D}$-modules holonomes réguliers

Let $X$ be a smooth complex manifold. Let $\mathbf{S}$ denote the solution functor for $\mathcal{D}$-modules on $X$. Traditionally, the fully-faithfulness of Riemann–Hilbert correspondance is proved by showing that if $\mathcal{M}_1$ and $\mathcal{M}_2$ are regular holonomic $\mathcal{D}_X$-modules, then the canonical morphism of complexes of sheaves\[RH_{\mathcal{M}_1, \mathcal{M}_2} : R\mathcal{Hom}_{\mathcal{D}_X} ({\mathcal{M}_1, \mathcal{M}_2}) \longrightarrow R\mathcal{Hom}_{\mathrm{Sh}_{\mathbb{C}}(X)} (\mathbf{S}(\mathcal{M}_2), \mathbf{S}(\mathcal{M}_1)))\]is an isomorphism, in a derived sense. This paper has to do with the converse statement. We prove that if $\mathcal{M}$ is an holonomic $\mathcal{D}_X$ module for which $RH_{\mathcal{M}, \mathcal{M}}$ is an isomorphism, then $\mathcal{M}$ is regular.

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Accepted 13 January 2015

Published 25 May 2016