Constructible nabla-modules on curves

Bernard Le Stum (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $\mathcal V$ be a discrete valuation ring of mixed characteristic with perfect residue field. Let $X$ be a geometrically connected smooth proper curve over $\mathcal V$. We introduce the notion of constructible convergent $\nabla$-module on the analytification $X_{K}^{\mathrm{an}}$ of the generic fibre of $X$. A constructible module is an $\mathcal O_{X_{K}^{\mathrm{an}}}$-module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $X_{k}$ of $X$. The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $\nabla$-modules to the category of $\mathcal D^\dagger_{\hat X \mathbf Q}$-modules. We show that if $X$ is endowed with a lifting of the absolute Frobenius of $X$, then specialization induces an equivalence between constructible $F$-$\nabla$-modules and perverse holonomic $F$-$\mathcal D^\dagger_{\hat X \mathbf Q}$-modules.

Mots clés: Overconvergent; Constructible; D-module; Frobenius

Domaines: Géométrie algébrique [math.AG]

Bernard Le Stum. Constructible nabla-modules on curves. Selecta Mathematica (New Series), 2014, 20 (2), pp.627-674. ⟨10.1007/s00029-013-0130-x⟩. ⟨hal-00546514v2⟩