This post is on the recent paper  of Toën and Vezzosi on the Riemann–Hilbert correspondence for derived foliations.

There are two versions of the Riemann–Hilbert correspondence. The first, more classical form, deals with algebraic differential equations with regular singularities on a smooth complex manifold $U$ and relates these to local systems of finite dimensional complex vector spaces on $U$, i.e., complex representations of $\pi_1(U)$. The second, strictly more general, treats regular holonomic $D$-modules and relates these to constructible sheaves on $U$. In , Toën and Vezzosi prove an analogue of the classical Riemann–Hilbert correspondence for local systems for a large class of their derived foliations.

# Derived foliations

Definition. Let $R$ be a connective1 cdga over $\bC$. A derived foliation $\Fscr$ over $R$ is a complete filtered cdga $\F^\star\dR_\Fscr$ with a map to $R$ such that

• $\gr^s\dR_\Fscr\we 0$ for $s<0$,
• $\gr^0\dR_\Fscr\we R$ via the fixed map above,
• $\gr^1\dR_\Fscr$ is a perfect connective $R$-module, and
• the natural map $\Sym_R(\gr^1\dR_\Fscr)\rightarrow\gr^\star\dR_\Fscr$ is an equivalence.

The complex $\L_\Fscr:=\gr^1\dR_\Fscr$ is called the cotangent complex of $\Fscr$.

One wants derived foliations to form a geometric kind of $\infty$-category, so let $Fol(R)$ be the opposite of the $\infty$-category of derived foliations over $R$. The functions space correspondence is then $\dR_\Fscr\leftrightarrow\Fscr.$ This notion globalizes and if $X$ is a scheme or later a complex analytic space, one obtains $Fol(X)$, which is opposite to the $\infty$-category of sheaves of quasicoherent derived foliations in the obvious sense.

I will test each idea in this exposition on the following four basic examples.

Case A. The $\mathbf{0}_X$ foliation is the unique foliation with $\L_\Fscr\we 0$. It is the initial object of $Fol(X)$ and corresponds to the case where the leaves’ are the points of $X$.

Case B. If $X$ is locally of finite presentation over $\bC$ as a derived scheme,2 then the cotangent complex $\L_X$ is perfect so that the Hodge-complete derived de Rham cohomology $\dR_{X/\bC}$ is a derived foliation with cotangent complex the cotangent complex of $X$. This is the final object of $Fol(X)$ by the universal property of derived de Rham cohomology. It corresponds to the foliation with a single leaf, $X$ itself. I will further restrict attention below to the case where $X$ is an ordinary smooth scheme over $\bC$.

Case C. If $X\xrightarrow{f} Y$ is a morphism of smooth $\bC$-schemes, then the Hodge-complete relative derived de Rham cohomology $\dR_{f}$ is a derived foliation with cotangent complex $\L_f$. The leaves are the fibers of $f.$

Case D. If $\gfrak$ is a finite dimensional coconnective dg Lie algebra over $\bC$, then the Chevalley–Eilenberg complex $\mathrm{CE}(\gfrak)$ is naturally a derived foliation over $\bC$ with cotangent complex $\gfrak^\ast$, the $\bC$-linear dual of $\gfrak$. At least if $\gfrak$ is an ordinary Lie algebra (not dg), this complex computes the Lie algebra cohomology of $\bC$ over $\gfrak$ and the filtration arises as the Koszul dual of the Poincaré–Birkhoff–Witt filtration on the universal enveloping algebra. There is only a single leaf of this foliation and if $\gfrak$ is the tangent Lie algebra to an algebraic group $G$ over $\bC$, then the leaf is equivalent to $G$ itself. Indeed, in this case

$\mathrm{CE}(\gfrak)\we\dR_{\ast/\B G},$

the Hodge-complete derived de Rham cohomology of a point $\ast\we\Spec\bC$ over the classifying stack $\B G$.

Case D illustrates well that the leaves’ of a foliation are only well-defined formally as different algebraic groups can share the same Lie algebra. It would be better here to say that the leaf of $\mathrm{CE}(\gfrak)$ is the formal group of $G$ at the origin. I will not go into depth on the formal definition of the leaves of a foliation. These are certain formal moduli problems and are indeed studied in the paper  of Toën–Vezzosi, but they are not needed for the main story today.

The main theorem is about crystals over foliations.

# Crystals

Let $\Fscr$ be a derived foliation over $X$.

Definition. A perfect crystal over $\Fscr$ is a $\F^\star\dR_\Fscr$-module $\F^\star M$ in sheaves of complete filtered complexes such $\gr^0M$ is a perfect complex of $\Oscr_X$-modules and such that the natural map

$\gr^\star\dR_\Fscr\otimes_{\Oscr_X}\gr^0M\rightarrow\gr^\star M$

is a graded equivalence. Let $\Perf(\Fscr)$ denote the stable $\infty$-category of perfect crystals over $\Fscr$. This is a rigid symmetric monoidal stable $\infty$-category with a symmetric monoidal exact functor $\gr^0\colon\Perf(\Fscr)\rightarrow\Perf(X)$.

The theory of crystals over foliations is further developed in  and in . In particular, a theory of Weyl algebras and differential operators is developed in .

Case A. The functor $\gr^0\colon\Perf(\mathbf{0}_X)\rightarrow\Perf(\Oscr_X)$ is an equivalence.

Case B. If $X$ is a smooth $\bC$-scheme, the stable $\infty$-category $\Perf(\dR_{X/\bC})$ is equivalent to the stable $\infty$-category of perfect complexes with integrable connection, $\D^b(\mathrm{MIC}(X/\bC))$.

Case C. In this case, $\Perf(\dR_{X/Y})$ consists of integrable connections along $f$.

Case D. If $\gfrak$ is a finite-dimensional coconnective dg Lie algebra, then $\Perf(\mathrm{CE}(\gfrak))\we\mathrm{Rep}_\gfrak(\Perf(\bC))$, the $\infty$-category of representations of of $\gfrak$ in perfect complexes over $\bC$.

# Homotopy-coherent chain complexes

To a complete filtration $\F^\star M$ one can associate a homotopy-coherent chain complex

$\cdots\rightarrow\gr^{-1}M[-1]\rightarrow\gr^0M\rightarrow\gr^1M\rightarrow\cdots,$

which gives the $\E_1$-page of the spectral sequence upon taking cohomology. From this perspective, a derived foliation over $R$ is a homotopy-coherent cdga of the form

$0\rightarrow R\rightarrow\L_\Fscr\rightarrow\Lambda^2\L_{\Fscr}\rightarrow\cdots.$

If $\F^\star M$ is a perfect crystal over $\F^\star\dR_\Fscr$, then the associated homotopy-coherent chain complex looks like

$0\rightarrow\gr^0M\xrightarrow{\nabla}\L_\Fscr\otimes_R\gr^0M\xrightarrow{\nabla}\Lambda^2\L_\Fscr\otimes_R\gr^0M\rightarrow\cdots.$

Here, the fact that we have a homotopy-coherent chain complex means that there is a canonical nullhomotopy $\nabla^2\we 0$, expressing integrability as well as higher coherence data, necessary since $\gr^0M$ and $\L_\Fscr$ are typically not discrete.

Case A. Nothing interesting to say here.

Case B. If $\F^\star M$ is a perfect crystal over $\F^\star_\H\dR_{X/\bC}$ corresponding to a perfect $\Oscr_X$-module with integrable connection $E$, then the homotopy-coherent chain complex is the de Rham complex of the connection:

$0\rightarrow E\xrightarrow{\nabla}\L_X\otimes_{\Oscr_X}E\xrightarrow{\nabla}\Lambda^2\L_X\otimes_{\Oscr_X}E\rightarrow\cdots.$

Case C. This is a relative version of Case B, clearly. However, it makes sense even when the fibers of the map are not smooth.

Case D. For $\F^\star M\in\Perf(\mathrm{CE}(\gfrak))$, the associated homotopy-coherent chain complex is

$\gr^0M\xrightarrow{\nabla}\gfrak^\ast\otimes_\bC\gr^0M\xrightarrow{\nabla}\Lambda^2(\gfrak^\ast)\otimes_\bC\gr^0M\rightarrow\cdots,$

which computes the Lie algebra cohomology $\R\Gamma(\gfrak,\gr^0M)$ of the $\gfrak$-representation $\gr^0M$. Here, $\nabla$ is adjoint to the action map $\gfrak\otimes_\bC\gr^0M\rightarrow\gr^0M$ of the corresponding $\gfrak$-representation and the fact that $\nabla^2\we 0$ expresses the Jacobi identity.

# The Riemann–Hilbert correspondence

Now, I can state the main theorem of . First, note that if $X$ is a smooth $\bC$-scheme, then for any foliation $\Fscr$ on $X$ there is a corresponding holomorphic foliation $\Fscr^h$ on $X^h$. Let $\Oscr_{\Fscr^h}$ denote the sheaf of cdgas on $X^h$ obtained by taking the underlying object of the filtration defined by the foliation. In other words, $\Oscr_{\Fscr^h}=\F^0\Fscr^h$.

Let $\Perf^\nil(\Fscr)\subseteq\Perf(\Fscr)$ be the full subcategory of perfect crystals over $\Fscr$ which are locally in the analytic topology in the stable subcategory generated by the unit $\Fscr$ itself. Similarly for $\Perf^\nil(\Fscr^h)$. Finally, let $\Perf(\Oscr_{\Fscr^h})$ denote the stable $\infty$-category of sheaves of $\Oscr_{\Fscr^h}$-modules which are analytic-locally in the stable subcategory generated by $\Oscr_{\Fscr^h}$.

Theorem (The Riemann–Hilbert correspondence). If $X$ is a smooth and proper $\bC$-scheme and $\Fscr$ is a derived foliation on $X$ such that $\L_\Fscr$ has Tor-amplitude in cohomological degrees $[-1,0]$, then the natural functor

$\Perf^\nil(\Fscr)\rightarrow\Perf(\Oscr_{\Fscr^h})$

is an equivalence.

The condition on the cotangent complex $\L_\Fscr$ in the theorem, called quasi-smoothness, guarantees that $\H^n(\Oscr_{\Fscr^h})=0$ for $n<0$.

Sketch of proof. By GAGA (proved in the paper for this setting), the natural map $\Perf(\Fscr)\rightarrow\Perf(\Fscr^h)$ is an equivalence of stable $\infty$-categories, which induces an equivalence $\Perf^\nil(\Fscr)\we\Perf^\nil(\Fscr^h)$ on subcategories of nilpotent crystals. Now, $\Perf^\nil(\Fscr^h)\rightarrow\Perf(\Oscr_{\Fscr^h})$ is the functor on global sections of a map of stacks $\Perfscr^\nil(\Fscr^h)\rightarrow\Perfscr^\nil(\Oscr_{\Fscr^h})$. This map of stacks is a local equivalence. Indeed, both sides are locally generated by the unit object and by definition the sheaf of endomorphisms of the unit is $\Oscr_{\Fscr^h}$. The theorem follows now by taking global sections. $\square$

I really like this fundamentally Morita-theoretic proof. To get finer information, especially about which crystals are nilpotent, one needs a foliated analogue of the Cauchy–Kovalevskaya theorem on existence and uniqueness of solutions of linear partial differential equations. This is proved by Toën and Vezzosi as Theorem 3.2.3 in  for $\Fscr$-crystal structures on vector bundles on $X$ when $X$ is smooth, $\Fscr$ is quasi-smooth and rigid ($\H^0(\L_X)\rightarrow\H^0(\L_\Fscr)$ is surjective), and the singular locus of $\Fscr$ has codimension at least $2$.

Corollary. Suppose that $X$ is a smooth and proper $\bC$-scheme and that $\Fscr$ is a rigid quasi-smooth foliation on $X$ with singular locus of codimension at least $2$. The Riemann–Hilbert correspondence restricts to an equivalence

$\Perf^\mathrm{v}(\Fscr)\we\Perf^\mathrm{v}(\Oscr_{\Fscr^h}),$

where $\Perf^\mathrm{v}(\Fscr)$ denotes the stable subcategory of $\Perf(\Fscr)$ generated by foliations of vector bundles and $\Perf^\mathrm{v}(\Oscr_{\Fscr^h})$ denotes the full subcategory of $\Perf(\Oscr_{\Fscr^h})$ generated by those which are locally free over $\Oscr_{\Fscr^h}$, or equivalently vector bundles on $U$ with integrable connection and regular singularities.

Case A. This is the classical GAGA theorem. Here, $\Oscr_{\mathbf{0}_X}\we\Oscr_X^h$.

Case B. This is the classical Riemann–Hilbert correspondence of Deligne applied to the smooth proper scheme $X$. In this case, $\Oscr_{\dR_{X/\bC}}$ is the constant sheaf $\bC$ on $X^h$.

Case C. This is a relative Riemann–Hilbert correspondence.

Case D. The Riemann–Hilbert correspondence recovers the equivalence between nilpotent crystals over $\mathrm{CE}(\gfrak)$ and perfect complexes over the underlying cdga $\F^0\mathrm{CE}(\gfrak)$.

The beauty of this RH correspondence is that it applies not only in the familiar Cases A-D, but to many other cases, which might not be globally integrable. One example is when $X$ is a smooth proper compactification of a smooth $\bC$-scheme $U$ with a simple normal crossing divisor $D$ as complement. Let $\Fscr$ denote the foliation corresponding to the log de Rham complex. In this case, $\Oscr_{\Fscr^h}\we j_*\bC,$ which is a good sign, but the corollary does not apply directly, because $\Fscr$ is not rigid (unless $D$ is empty). Nevertheless, one can interpret the theorem as giving an equivalence between vector bundles on $X$ with nilpotent residues along $D$ and local systems on $U$ with unipotent local monodromy around $D$.

 Toën, Classes caractéristiques des schémas feuilletés, arXiv:2008.10489.

 Toën and Vezzosi, Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence, arXiv:2001.05450.

 Toën and Vezzosi, Algebraic foliations and derived geometry II: the Grothendieck-Riemann-Roch theorem, arXiv:2007.09251.

# Footnotes

1. A cdga $R$ is connective if $\H^n(R)=0$ for $n>0$.

2. This includes the case of ordinary lci schemes over $\bC$, but not general finite type $\bC$-schemes.