# arXiv reviews 2: Algebraic foliations I

This post is on the recent paper [2] 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 [2], 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 connective^{1} 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,
- $\gr^1\dR_\Fscr[1]$ 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 $\L_\Fscr:=\gr^1\dR_\Fscr[1]$ 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

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 [2] 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

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 [3] and in [1]. In particular, a theory of Weyl algebras and differential operators is developed in [3].

**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

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

**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

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 [2]. 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

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 [2] 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

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
$D$.

# References

[1] Toën, *Classes caractéristiques des schémas feuilletés*,
arXiv:2008.10489.

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

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