Constructible $$\nabla $$ ∇ -modules on curves

Abstract

Let $$\mathcal{V }$$ be a complete 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 fiber 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 specialization induces an equivalence between constructible $$F$$ - $$\nabla $$ -modules and perverse holonomic $$F$$ - $$\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$$ -modules.