Constructible nabla-modules on curves

Abstract

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 ${\mathcal{D}}_{\hat{X}\mathbf{Q}}^{†}$-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 $F$-${\mathcal{D}}_{\hat{X}\mathbf{Q}}^{†}$-modules.

39 pages