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Constructible \(\nabla \)-modules on curves. (English) Zbl 1320.14035

The goal of this paper is to construct a category of constructible convergent \(F\)-\(\nabla\)-modules over a proper smooth curve \(X\) over a \(p\)-adic field with good reduction, which under the specialization map, is equivalent to the category of perverse holonomic \(F\)-\(\mathcal D^\dagger_{\hat X, \mathbb Q}\)-modules. One may view this as an “overconvergent Deligne-Kashiwara correspondence”.
Let \(\mathcal V\) be a complete discrete valuation ring of mixed characteristic with perfect residue field \(k\) and fraction field \(K\), and let \(X\) denote a geometrically connected smooth proper curve over \(\mathcal V\). Let \(X_K^{\mathrm{an}}\) denote the Berkovich space associated to \(X\). The author defines a constructible overconvergent \(\nabla\)-module to be an \(\mathcal O_{X_K^{\mathrm{an}}}\)-module \(E\) with a convergent connection, such that there exists a finite covering of the special fiber \(X_k\) by locally closed subsets \(Y\) with the property that if \(i_Y:]Y[ \hookrightarrow X_K^{\mathrm{an}}\) denotes the inclusion map, then \(i_Y^{-1}E\) is a coherent \(i_Y^{-1}\mathcal O_{X_K^{\mathrm{an}}}\)-module. (Note that the pullbacks are taken for the Berkovich topology, so when \(Y\) is an open subspace of \(X_k\), \(i_Y^{-1}\mathcal O_{X_K^{\mathrm{an}}}\) is the limit of functions on strict neighborhoods of \(]Y[\).) The first main result of this paper is to describe the direct image under the specialization map \(\mathrm{Rsp}_* E_0\), where \(E_0\) is the corresponding module for the rigid topology. Explicitly, if \(D\) is a smooth relative divisor on \(X\) with affine open complement \(\mathrm{Spec}\; A\). Let \(A_K^\dagger\) denote the generic fiber of the weak completion of \(A\). For each \(a \in D_K\), write \(\mathcal R_a\) for the corresponding Robba ring at \(a\) over the residue field \(K(a)\). Then a constructible convergent \(\nabla\)-module is equivalent to the following data (for sufficiently large divisor \(D\)): (1) an overconvergent \(\nabla\)-module \(M\) over \(A_K^\dagger\), (2) for each \(a \in D_K\), a finite dimensional \(K(a)\)-vector space \(H_a\), and (3) a horizontal \(A_K^\dagger\)-linear map \[ M \to \bigoplus_{a \in D_K}\mathcal R_a \otimes_{K(a)} H_a. \] The image of \(E\) under the specialization, namely \(\mathrm{Rsp}_*E_0\) is roughly just (the sheaf version of) the complex \(M \to \bigoplus_{a \in D_K}\delta_a \otimes_{K(a)} H_a\), where \(\delta_a = \mathcal R_a / \mathcal O_a^{\mathrm{an}}\) is the arithmetic analogue of the delta \(D\)-module. This is a perverse complex of \(\mathcal D_{\hat X, \mathbb Q}^\dagger\)-modules in the sense that it has \(\mathcal O_{\hat X, \mathbb Q}\)-cohomology in degree \(0\), finite support in degree \(1\), and no other cohomology.
Now, if one starts with a constructible overconvergent \(F\)-\(\nabla\)-module \(E\), by a theorem of D. Caro [Bull. Soc. Math. Fr. 137, No. 4, 453–543 (2009; Zbl 1300.14021)], \(\mathrm{Rsp}_*E_0\) is a holonomic \(F\)-\(\mathcal D_{\hat X, \mathbb Q}^\dagger\)-module. The main result of this paper says that this construction in fact gives an equivalence of categories between the category of constructible overconvergent \(F\)-\(\nabla\)-modules and the category of perverse holonomic \(F\)-\(\mathcal D_{\hat X, \mathbb Q}^\dagger\)-modules.
The paper is elegantly written, mostly self-contained.

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F30 \(p\)-adic cohomology, crystalline cohomology

Citations:

Zbl 1300.14021
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References:

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