nLab Riemann-Hilbert correspondence




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.


Let XX be a smooth algebraic variety over \mathbb{C}. Let D(X Zar,𝒟 X)D(X^\Zar,\mathcal{D}_{X}) be the derived category of algebraic D-modules on XX (i.e. quasi-coherent 𝒪 X\mathcal{O}_{X}-modules with flat connection), and let D rh b(X Zar,𝒟 X)D_{rh}^{b}(X^\Zar,\mathcal{D}_{X}) be the full subcategory of D b(X Zar,𝒟 X)D^{b}(X^\Zar,\mathcal{D}_{X}) consisting of objects whose cohomology sheaves are regular holonomic algebraic D-modules. Let D(X an,)D(X^{an},\mathbb{C}) be the derived category of sheaves of \mathbb{C}-vector spaces on X anX^{an}, and let D c b(X an,)D_{c}^{b}(X^{an},\mathbb{C}) be the full subcategory of D b(X an,)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 hol=𝒪 X hol 𝒪 X\mathcal{E}^{\hol}=\mathcal{O}_{X}^{\hol}\otimes_{\mathcal{O}_{X}}\mathcal{E}. The assignment Ω *( hol)\mathcal{E}\to\Omega^{*}(\mathcal{E}^{\mathrm{hol}}) determines a functor

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


(Mebkhout, Kashiwara, Beilinson-Bernstein)

The restriction of the functor dRdR gives an equivalence of categories

D rh b(X Zar,𝒟 X)D c b(X an,)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 GBun_{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:


(Bhatt21, Theorem 5.1)

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

RH:D cons b(X C,𝔽 p)D qc b(X 𝒪 C𝒪 C/p).\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).


(Bhatt21, Theorem 5.4)

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

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

where D wHT b(X C, p)D_{wHT}^{b}(X_{C},\mathbb{Q}_{p}) is a full subcategory of D cons b(X C, p)D_{cons}^{b}(X_{C},\mathbb{Q}_{p}) spanned by “weakly Hodge-Tate sheaves” and DF coh(𝒟 X)DF_{coh}(\mathcal{D}_{X}) is a suitable derived category of 𝒟 X \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 KK be a discretely-valued p-adic field and let XX be a smooth projective variety over KK with 𝒳\mathcal{X} its associated rigid analytic space. Let \mathcal{F} be a constructible sheaf over XX. 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, p)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 ν:𝒳 proet𝒳 et\nu:\mathcal{X}_{proet}\to \mathcal{X}_{et} be the projection and let dR\mathcal{B}_{dR} be the de Rham period sheaf on 𝒳 proet\mathcal{X}_{proet}. If \mathcal{F} is weakly de Rham, we define

RH():=Rν *(ν 1^ p dRu)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 φ\varphi-modules, which in his context are simultaneously solid modules and almost modules over 𝒪 X +/π\mathcal{O}_{X}^{+}/\pi together with an action of the Frobenius φ\varphi, and overconvergent etale 𝔽 p\mathbb{F}_{p}-sheaves (the corresponding derived \infty-categories are denoted by 𝒟 a(𝒪 X +/π) φ\mathcal{D}_{\square}^{a}(\mathcal{O}_{X}^{+}/\pi)^{\varphi} and 𝒟 et(X,𝔽 p) oc\mathcal{D}_{et}(X,\mathbb{F}_{p})^{oc} respectively).


(Mann22, Theorem 1.2.7, Theorem 3.9.23)

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

𝒪 X +a/π:𝒟 et(X,𝔽 p) oc𝒟 a(𝒪 X +/π) φ-\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.


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

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

Last revised on June 15, 2023 at 19:34:02. See the history of this page for a list of all contributions to it.