[Paper Review] Logarithmic Riemann-Hilbert correspondences for rigid varieties

Hansheng Diao, Kai‐Wen Lan+2 more|arXiv (Cornell University)|Mar 15, 2018

Advanced Algebra and Geometry4 citations

TL;DR

This paper constructs a p-adic Riemann-Hilbert correspondence for smooth rigid varieties over p-adic local fields, establishing a tensor functor from de Rham p-adic étale local systems to filtered algebraic vector bundles with integrable connections satisfying Griffiths transversality. The key contribution is a canonical extension of connections to compactifications with logarithmic poles, characterized by residue eigenvalues, and a compatibility with classical Riemann-Hilbert correspondence on Shimura varieties.

ABSTRACT

On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a $p$-adic analogue of Deligne's classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this $p$-adic Riemann--Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.

Motivation & Objective

• To establish a p-adic analogue of Deligne’s classical Riemann-Hilbert correspondence for smooth algebraic varieties over p-adic local fields.
• To construct a tensor functor from de Rham p-adic étale local systems to filtered algebraic vector bundles with integrable connections satisfying Griffiths transversality.
• To prove that this functor is compatible with the classical Riemann-Hilbert correspondence on Shimura varieties via Galois-equivariant comparison isomorphisms.
• To develop a canonical extension theory for connections on compactifications with logarithmic poles, characterized by residue eigenvalues.
• To provide a framework for comparing p-adic étale cohomology with algebraic de Rham cohomology via Fontaine’s period ring $B_{ ext{dR}}$.

Proposed method

• Construct a tensor functor $D_{ ext{dR}}^{ ext{alg}}$ from de Rham p-adic étale local systems to filtered algebraic vector bundles with integrable connections satisfying Griffiths transversality.
• Use log adic spaces and period sheaves, particularly $\mathcal{O}\mathbb{B}_{\text{dR},\log}$, to define the de Rham condition in the p-adic setting.
• Establish canonical extensions of connections to compactifications with logarithmic poles, where the residue eigenvalues determine the extension uniquely.
• Apply a formalism of decompletion to handle inverse limits of cohomology groups and prove cohomological compatibility via the $\Gamma$-action on period rings.
• Utilize the theory of filtered log connections 'relative to $B_{\text{dR}}$' to define the target category and ensure compatibility with filtrations.
• Prove the comparison isomorphism $H^i_{\text{ét}}(X_{\overline{k}},\mathbb{L})\otimes_{\mathbb{Q}_p} B_{\text{dR}} \cong H^i_{\text{dR}}(X,D_{\text{dR}}^{\text{alg}}(\mathbb{L}))\otimes_k B_{\text{dR}}$ as Galois modules with compatible filtrations.

Experimental results

Research questions

• RQ1 How can one construct a p-adic Riemann-Hilbert correspondence that generalizes Deligne’s classical correspondence to varieties over p-adic fields?
• RQ2 What is the correct category of filtered connections to which de Rham p-adic local systems should be mapped in the p-adic setting?
• RQ3 How can connections be extended canonically to compactifications with logarithmic poles, and what role do residue eigenvalues play in this extension?
• RQ4 Is the proposed p-adic Riemann-Hilbert functor compatible with the classical correspondence on Shimura varieties?
• RQ5 Can a Galois-equivariant comparison isomorphism be established between étale and de Rham cohomology via $B_{\text{dR}}$?

Key findings

• The paper constructs a tensor functor $D_{\text{dR}}^{\text{alg}}$ from de Rham p-adic étale local systems to filtered algebraic vector bundles with integrable connections satisfying Griffiths transversality.
• The functor is compatible with the classical Riemann-Hilbert correspondence on Shimura varieties, as shown by a Galois-equivariant comparison isomorphism involving $B_{\text{dR}}$.
• The connections in the image of $D_{\text{dR}}^{\text{alg}}$ admit canonical extensions to compactifications with logarithmic poles, uniquely determined by the eigenvalues of their residues.
• The comparison isomorphism $H^i_{\text{ét}}(X_{\overline{k}},\mathbb{L})\otimes_{\mathbb{Q}_p} B_{\text{dR}} \cong H^i_{\text{dR}}(X,D_{\text{dR}}^{\text{alg}}(\mathbb{L}))\otimes_k B_{\text{dR}}$ is shown to be compatible with the canonical filtrations and Galois actions.
• The formalism of decompletion is used to prove that the system $(\{\mathbb{B}_{r,m}\}_{m\geq 1},\widehat{\mathbb{B}}_{r,\infty},\Gamma)$ is a decompletion system, ensuring the existence of good models for finite projective modules over $\widehat{\mathbb{B}}_{r,\infty}$.
• The method relies on lifting cochains in cohomology via a recursive argument using the nilpotency of $\xi$, with bounds on norms controlled by $|\varpi|^{-1}$ and constants depending on the cohomological degree.