nLab Riemann-Hilbert correspondence

Idea

The most basic form of the Riemann–Hilbert correspondence states that the category of flat vector bundles on a suitable space is equivalent to the category of local systems.

While Hilbert's 21st problem has a negative solution, there is a generalized sheaf-theoretical formulation which leads to an equivalence of categories discovered by Mebkhout and a bit later also by Kashiwara.

Statement

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Let $D(X^\Zar,\mathcal{D}_{X})$ be the derived category of algebraic D-modules on $X$ (i.e. quasi-coherent $\mathcal{O}_{X}$-modules with flat connection), and let $D_{rh}^{b}(X^\Zar,\mathcal{D}_{X})$ be the full subcategory of $D^{b}(X^\Zar,\mathcal{D}_{X})$ consisting of objects whose cohomology sheaves are regular holonomic algebraic D-modules. Let $D(X^{an},\mathbb{C})$ be the derived category of sheaves of $\mathbb{C}$-vector spaces on $X^{an}$, and let $D_{c}^{b}(X^{an},\mathbb{C})$ be the full subcategory of $D^{b}(X^{an},\mathbb{C})$ consisting of objects whose cohomology sheaves are Zariski constructible.

Let $\mathcal{E}$ be an algebraic D-module and let $\mathcal{E}^{\hol}=\mathcal{O}_{X}^{\hol}\otimes_{\mathcal{O}_{X}}\mathcal{E}$. The assignment $\mathcal{E}\to\Omega^{*}(\mathcal{E}^{\mathrm{hol}})$ determines a functor

$$dR: D(X^\Zar,\mathcal{D}_{X})\to D(X^{an},\mathbb{C}).$$

Proposition

(Mebkhout, Kashiwara, Beilinson-Bernstein)

The restriction of the functor $dR$ gives an equivalence of categories

$$D_{rh}^{b}(X^\Zar,\mathcal{D}_{X})\xrightarrow{\sim} D_{c}^{b}(X^{an},\mathbb{C})$$

Use in the geometric Langlands correspondence

The Riemann-Hilbert correspondence allows one to translate the automorphic side of the geometric Langlands correspondence from the language of perverse sheaves (which in turn is motivated by functions on $Bun_{G}$, via Grothendieck’s functions-to-sheaves dictionary) to the language of D-modules (see Frenkel05, 3.4 to 3.6).

p-torsion and p-adic Riemann-Hilbert correspondence

Bhargav Bhatt and Jacob Lurie have formulated a version of the Riemann-Hilbert correspondence for finite type schemes over a complete and algebraically closed extension of the p-adic numbers:

Proposition

(Bhatt21, Theorem 5.1)

Let $C$ be a complete and algebraically closed extension of $\mathbb{Q}_{p}$ and let $X$ be of finite type over $\mathcal{O}_{C}$. Then there is a natural exact functor

$$\RH: D_{\cons}^{b}(X_{C},\mathbb{F}_{p})\to D_{\qc}^{b}(X\otimes_{\mathcal{O}_{C}} \mathcal{O}_{C}/p).$$

This functor commutes with proper pushforward, intertwines Verdier duality and Grothendieck duality in the almost category, and interacts well with the perverse t-structure.

Bhatt and Lurie also have a version with p-adic, instead of p-torsion, coefficients (this time the schemes have to be smooth proper varieties over a finite extension of the p-adic numbers).

Proposition

(Bhatt21, Theorem 5.4)

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $X$ be a smooth proper variety over $K$. Then there is a natural exact functor

$$\RH_{\mathcal{D}}: D_{wHT}^{b}(X_{C},\mathbb{Q}_{p})\to DF_{coh}(\mathcal{D}_{X})$$

where $D_{wHT}^{b}(X_{C},\mathbb{Q}_{p})$ is a full subcategory of $D_{cons}^{b}(X_{C},\mathbb{Q}_{p})$ spanned by “weakly Hodge-Tate sheaves” and $DF_{coh}(\mathcal{D}_{X})$ is a suitable derived category of $\mathcal{D}_{X}$-modules equipped with a “good” filtration. This functor commutes with proper pushforward, intertwines Verdier duality and Grothendieck duality in the almost category, and interacts well with the perverse t-structure.

The construction of Bhatt and Lurie is as follows. Let $K$ be a discretely-valued p-adic field and let $X$ be a smooth projective variety over $K$ with $\mathcal{X}$ its associated rigid analytic space. Let $\mathcal{F}$ be a constructible sheaf over $X$. We say that $\mathcal{F}$ is weakly de Rham if the stalk $\mathcal{F}$ is a weakly de Rham Galois representation. More generally if $\mathcal{F}$ is an object of the derived category $D_{c}^{b}(X,\mathbb{Q}_{p})$ we say that $\mathcal{F}$ is weakly de Rham if its cohomology sheaves are weakly de Rham.

Let $\nu:\mathcal{X}_{proet}\to \mathcal{X}_{et}$ be the projection and let $\mathcal{B}_{dR}$ be the de Rham period sheaf on $\mathcal{X}_{proet}$. If $\mathcal{F}$ is weakly de Rham, we define

$$RH(\mathcal{F}):=R\nu_{*}(\nu^{-1}\mathcal{F}\widehat{\otimes}_{\mathbb{Q}_{p}}\mathcal{B}_{dR}\langle u\rangle)$$

Lucas Mann has also formulated a version of the p-torsion Riemann-Hilbert correspondence for small v-stacks as part of his proof of mod p Poincare duality for rigid analytic spaces. This version is stated in terms of $\varphi$-modules, which in his context are simultaneously solid modules and almost modules over $\mathcal{O}_{X}^{+}/\pi$ together with an action of the Frobenius $\varphi$, and overconvergent etale $\mathbb{F}_{p}$-sheaves (the corresponding derived $\infty$-categories are denoted by $\mathcal{D}_{\square}^{a}(\mathcal{O}_{X}^{+}/\pi)^{\varphi}$ and $\mathcal{D}_{et}(X,\mathbb{F}_{p})^{oc}$ respectively).

Proposition

(Mann22, Theorem 1.2.7, Theorem 3.9.23)

Let $X$ be a small v-stack over $\mathbb{Z}_{p}$ with pseudouniformizer $\pi$ such that $\pi\vert p$. Then the functor

$$-\otimes\mathcal{O}_{X}^{+a}/\pi:\mathcal{D}_{et}(X,\mathbb{F}_{p})^{oc}\to\mathcal{D}_{\square}^{a}(\mathcal{O}_{X}^{+}/\pi)^{\varphi}$$

is fully faithful and induces an equivalence of categories of perfect objects on both sides.

Related entries

• Hilbert's problems

• flat infinity-connection

• infinity-local system

• D-modules

References

• Alain Connes, Matilde Marcolli Noncommutative Geometry, Quantum Fields and Motives

• Wikipedia, Riemann-Hilbert correspondence

• Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les $\mathcal{D}_X$-modules cohérents, Travaux en Cours 35. Hermann, Paris, 1989. x+254 pp. MR90m:32026

• Andrea D'Agnolo?, Masaki Kashiwara, Riemann-Hilbert correspondence for holonomic D-modules, arxiv/1311.2374

The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic D-modules and that of constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin’s work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.

Generalization to flat infinity-connections via the dg-nerve and iterated integrals is discussed in

• Jonathan Block, Aaron Smith, A Riemann–Hilbert correspondence for infinity local systems, Adv. Math. 252 (2014) 382–405 arXiv:0908.2843 doi

Another approach to higher Riemann-Hilbert

• J. Chuang, J. Holstein, A. Lazarev, Maurer-Cartan moduli and theorems of Riemann–Hilbert type, Appl Categor Struct 29 (2021) 685-728, doi

The relation to the geometric Langlands correspondence is discussed in

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory, in Frontiers in number theory, physics, and geometry II, Springer Berlin Heidelberg, 2007. 387-533. (arXiv:hep-th/0512172)

An analogue in p-adic geometry is discussed in

• Jacob Lurie, A Riemann–Hilbert Correspondence in p-adic Geometry, Felix Klein Lectures 2022.

• Bhargav Bhatt, Algebraic geometry in mixed characteristic, arXiv:2112.12010

• Lucas Mann, A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry [arXiv:2206.02022]

Extension to the irregular case

• Andrea D'Agnolo?, Masaki Kashiwara, On the irregular Riemann-Hilbert correspondence, 2408.04260