**4.3.1. Cotangent stacks of derived foliations.** Let $X$ be a derived scheme and $E$ be a perfect complex on $X$ of tor-amplitude contained in $[0, n]$ for some non-negative integer $n$ (see [TV07] for the notion of tor-amplitude of perfect complexes). Recall from §2.5.1 that the linear stack $\mathcal{V}(E)$ associated to $E$ comes equipped with its natural $\mathbb{G}_m$-action, covering the projection $\pi : \mathcal{V}(E) \to X$. We pass to the corresponding morphism on quotient stacks

$$ \pi^{\text{gr}} : [\mathcal{V}(E)/\mathbb{G}_m] \to X \times B\mathbb{G}_m. $$

The direct image along this morphism is a symmetric lax monoidal $\infty$-functor

$$ (19) \quad \pi_*^{\text{gr}} : \mathcal{QCoh}^{\text{gr}}(\mathcal{V}(E)) := \mathcal{QCoh}([\mathcal{V}(E)/\mathbb{G}_m]) \to \mathcal{QCoh}(X \times B\mathbb{G}_m) =: \mathcal{QCoh}^{\text{gr}}(X). $$

Since the structure sheaf $\mathcal{O} := \mathcal{O}[\mathcal{V}(E)/\mathbb{G}_m]$ is the monoidal unit in $\mathcal{QCoh}^{\text{gr}}(\mathcal{V}(E))$, the lax-monoidal the $\infty$-functor $\pi_*^{\text{gr}}$ in (19) factors via a $\infty$-functor (denoted by the same symbol)

$$ \pi_*^{\text{gr}} : \mathcal{QCoh}^{\text{gr}}(\mathcal{V}(E)) \to \pi_*^{\text{gr}}(\mathcal{O}) - \mathbf{dg}_{qcoh}^{\text{gr}}, $$

from graded quasi-coherent complexes on $\mathcal{V}(E)$ to graded $\pi_*^{\text{gr}}(\mathcal{O})$-modules over $X$ which are quasi-coherent as $\mathcal{O}_X$-modules.

**PROPOSITON 4.3.1.1.** There exists a fully faithful $\infty$-functor

$$ \mathrm{Sym}_{\mathcal{O}_X}(E) - \mathbf{dg}_{\mathrm{perf}}^{\mathrm{gr}} \to \mathcal{QCoh}^{\mathrm{gr}}(\mathcal{V}(E)) $$

from perfect graded $\mathrm{Sym}_{\mathcal{O}_X}(E)$-modules over $X$, sending the $i$-th weight twist $\mathrm{Sym}_{\mathcal{O}_X}(E)(i)$ of the tautological graded module $\mathrm{Sym}_{\mathcal{O}_X}(E)$, to the $i$-th weight twist $\mathcal{O}_{\mathcal{V}(E)}(i)$ of the structure sheaf of $\mathcal{V}(E)$.

*Proof.* We start by assuming that $X = \mathrm{Spec} A$ is affine, with $A$ a connective cdga, and by [Mon21] we known that there is a natural equivalence graded cdga's over $X$

$$ \pi_*^{\text{gr}}(\mathcal{O}) \simeq \mathrm{Sym}_{\mathcal{O}_X}(E). $$

The direct image $\infty$-functor

$$ \pi_*^{\text{gr}} : \mathcal{QCoh}^{\text{gr}}(\mathcal{V}(E)) \to \pi_*^{\text{gr}}(\mathcal{O}) - \mathbf{dg}_{qcoh,X}^{\text{gr}}, $$

from graded quasi-coherent complexes on $\mathcal{V}(E)$ to graded $\pi_*(\mathcal{O})$-modules over $X$ which are quasi-coherent as $\mathcal{O}_X$-modules possesses a left adjoint $\pi_{gr}^*$

$$ \pi_{gr}^* : \pi_*^{\text{gr}}(\mathcal{O}) - \mathbf{dg}_{qcoh}^{\text{gr}} \simeq \mathrm{Sym}_{\mathcal{O}_X}(E) - \mathbf{dg}_{\mathrm{perf}}^{\text{gr}} \to \mathcal{QCoh}^{\text{gr}}(\mathcal{V}(E)). $$

As the unit of this adjunction

$$ \mathrm{Sym}_{\mathcal{O}_X}(E) \to \pi_*^{\text{gr}}(\pi_{gr}^*(\mathrm{Sym}_{\mathcal{O}_X}(E))) = \pi_*^{\text{gr}}(\mathcal{O}) $$

is an equivalence, the $\infty$-functor $\pi_{gr}^*$, when restricted to the thick triangulated sub-$\infty$-category generated by the objects $\mathcal{O}(i)$, induces an equivalences with perfect graded $\mathrm{Sym}_{\mathcal{O}_X}(E)$-modules over $X$. This proves the proposition when $X$ is affine.
The case where $X$ is a general derived Artin stack easily follows by descent from the affine case thanks to the commutative diagram

$$ \begin{array}{ccc} \mathrm{Sym}_{\mathcal{O}_X}(E) - \mathrm{dg}_{\mathrm{perf}}^{\mathrm{gr}} & \xrightarrow{\sim} & \mathrm{QCoh}^{\mathrm{gr}}(\mathbb{V}(E)) \\ \llap{$\simeq$}\downarrow & & \downarrow\llap{$\simeq$} \\ \lim_{u:\mathrm{Spec} A \to X} (\mathrm{Sym}_A(u^*(E)) - \mathrm{dg}_{\mathrm{perf}}^{\mathrm{gr}}) & \longrightarrow & \lim_{u:\mathrm{Spec} A \to X} \mathrm{QCoh}^{\mathrm{rm}}(\mathbb{V}(u^*(E))). \end{array} $$

□

The proposition 4.3.1.1 will be applied later to the particular case where $E = \mathbb{T}_F$, the tangent complex of a derived foliation $\mathcal{F}$ on $X$. The derived stack $V(\mathbb{T}_F)$ will be denoted by $T^*\mathcal{F}$, and will be called the *global derived cotangent stack of $\mathcal{F}$*.

**DEFINITION 4.3.1.2.** For a derived scheme $X$ and a perfect derived foliation $\mathcal{F} \in \mathrm{Fol}^p(X)$, the (derived) cotangent stack of $\mathcal{F}$ is defined by

$$ T^*\mathcal{F} := V(\mathbb{T}_F). $$

equipped with is natural $\mathbb{G}_m$-action.

In the next section, we will use the functor of the proposition 4.3.1.1 in order to define *characteristic cycles* of crystals on $\mathcal{F}$ admitting a good filtration.

**4.3.2. Coherent crystals and good filtrations.** Let $X$ be a separated and quasi-compact derived Deligne-Mumford stack and $\mathcal{F} \in \mathrm{Fol}^p(X)$ be a perfect derived foliation on $X$. We let $E \in \mathrm{QCoh}(\mathcal{F})$ be a quasi-coherent crystal. Via the equivalence of our theorem 4.1.2.1, we will freely identify $E$ with a $\mathcal{D}_F$-module quasi-coherent over $X$.

**DEFINITION 4.3.2.1.** Let $X$, $\mathcal{F}$ and $E$ as above.

(1) The crystal $E$ is called coherent if it is a compact object in $\mathrm{QCoh}(\mathcal{F})$. The full sub- $\infty$-category of coherent crystals is denoted by $\mathrm{Coh}(\mathcal{F}) \subset \mathrm{QCoh}(\mathcal{F})$.

(2) For $E \in \mathrm{Coh}(\mathcal{F})$, a good filtration on $E$ is the data of a compact object $E^{\mathrm{fil}} \in \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$ together with an equivalence in $\mathrm{QCoh}(\mathcal{F})$

$$ (E^{\mathrm{fil}})^u \simeq E. $$

Compact objects in $\mathrm{QCoh}(\mathcal{F})$ and in $\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$ can be easily characterized, either through the induction $\infty$-functors or as sheaves of $\mathcal{D}_F$-modules.

**PROPOSITON 4.3.2.2.** Let $X$ and $\mathcal{F}$ be as above. An object $E \in \mathrm{QCoh}(\mathcal{F})$ (resp. $E^{\mathrm{fil}} \in \mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$) is compact if and only if it satisfies one of the following two equivalent conditions.

(1) The object $E$ belongs to the thick triangulated sub-category generated by objects of the form $\mathrm{Ind}_{\mathcal{F}}(E_0)$ (resp. $\mathrm{Ind}_{\mathcal{F}}^{\mathrm{fil}}(E_0^{\mathrm{fil}})$) for a perfect complex of $\mathcal{O}_X$-modules $E_0$ (resp. a filtered perfect complex of $\mathcal{O}_X$-modules $E_0^{\mathrm{fil}}$).
(2) $E$ (resp. $E^{\text{fil}}$) is a perfect $\mathcal{D}_{\mathcal{F}}$-module (resp. a perfect $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$-module) i.e. locally on $X_{et}$, $E$ (resp. $E^{\text{fil}}$) is a retract of a finite cell $\mathcal{D}_{\mathcal{F}}$-module (resp. of a finite cell filtered $\mathcal{D}_{\mathcal{F}}^{\text{fil}}$-module).

*Proof.* Condition (1) have been already considered in the proof of Corollary 4.1.3.3. Clearly, condition (1) implies condition (2). Finally condition (2) clearly implies compactness when $X$ is affine. The general case follows from the fact that $X$ is assumed to be separated and quasi-compact and therefore 4.1.3.4 holds and from the fact that any object satisfying condition (2) is a compact object (see our remark 4.1.3.5). $\square$

Suppose that $E$ is a coherent crystal along $\mathcal{F}$, equipped with a good filtration $E^{\text{fil}}$ in the sense of definition 4.3.2.1 above. By proposition 4.3.2.2, $\text{Gr}(E^{\text{fil}})$ is a perfect graded $\text{Gr}(\mathcal{D}_{\mathcal{F}}^{\text{fil}}) = \text{Sym}_{\mathcal{O}_X}(\mathbb{T}_{\mathcal{F}})$-module over $X$. Now, proposition 4.3.1.1 implies that $\text{Gr}(E^{\text{fil}})$ defines a graded perfect complex on the stack $T^*\mathcal{F}$, and by forgetting the $\mathbb{G}_m$-action we get a perfect complex on $T^*\mathcal{F}$. We consider the $K$-theory spectrum $K(T^*\mathcal{F})$, defined as the $K$-theory of the $\infty$-category of perfect (not graded) modules over $T^*\mathcal{F}$. The perfect complex $\text{Gr}(E^{\text{fil}})$ on $T^*\mathcal{F}$ thus defines a class

$$ \mathrm{Ch}(E^{\text{fil}}) \in K_0(T^*\mathcal{F}). $$

**DEFINITION 4.3.2.3.** The characteristic cycle of $E^{\text{fil}}$ is the element

$$ \mathrm{Ch}(E^{\text{fil}}) \in K_0(T^*\mathcal{F}) $$

defined above.

Note that the above definition depends a priori on the choice of $E^{\text{fil}}$. We will see later that, in fact, modulo a class of *phantoms objects*, it does not (see §4.3.2.2, and proposition 4.3.2.7). When $X$ and the derived foliation $\mathcal{F}$ are both smooth these phantoms are all trivial, but in general we have not been able to show that $\mathrm{Ch}(E^{\text{fil}})$ only depends on $E$ and not of the choice of the good filtration.

4.3.2.1. *Existence of good filtrations.* The existence of good filtrations in general seems to be a complicated question, and the authors do not know if good filtrations always exist for arbitrary coherent crystals, as it is the case for usual $\mathcal{D}$-modules on smooth varieties. It can be shown that they do exist for smooth foliations on smooth varieties, but the fact that, for a general foliation $\mathcal{F}$, its ring of differential operators $\mathcal{D}_{\mathcal{F}}$ is not concentrated in degree 0, creates complications in constructing good filtrations. The following result is therefore very useful in practice.

**PROPOSITON 4.3.2.4.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism of separated and quasi-compact derivbed DM-stacks endowed with perfect derived foliations and assume that $g$ is proper and quasi-smooth. If $E^{\text{fil}}$ is a good filtration on a coherent crystal $E \in \mathbf{QCoh}(\mathcal{F})$, then $f!(E^{\text{fil}})$ is a good filtration on $f!(E)$.
*Proof.* This follows from Corollary 4.2.1.7 together with the fact that direct images commutes with taking the underlying object (see diagram (16)). □

We can also single out a large class of coherent crystals for which existence of good filtrations is guaranteed: the *finite cell* crystals. For a derived scheme $X$ and $\mathcal{F} \in \mathcal{F}ol(X)$, we define a non-thick triangulated sub-∞-category $\mathrm{Coh}^{\mathrm{cell}}(\mathcal{F}) \subset \mathrm{Coh}(\mathcal{F})$, as being generated (by finite limits and shifts) by the objects of the form $\mathrm{Ind}_{\mathcal{F}}(E)$ for $E$ a perfect complex on $X$. An object of $\mathrm{Coh}^{\mathrm{cell}}(\mathcal{F})$ will be called a *finite cell crystal*. There is an obvious filtered version, too. Finite cell crystals are obviously coherent, but the converse has no reason to be true in general. Finite cell crystals are however useful because of the following result.

**PROPOSITON 4.3.2.5.** With the notation above, any object $E \in \mathrm{Coh}^{\mathrm{cell}}(\mathcal{F})$ admits a good filtration. Moreover, a good filtration $E^{\mathrm{fil}}$ can be chosen to be a finite cell object in $\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$.

*Proof.* Let $E \in \mathrm{Coh}^{\mathrm{cell}}(\mathcal{F})$. By definition of a finite cell object, there is a finite sequence of morphisms in $\mathrm{Coh}(\mathcal{F})$

$$E_{-1} = 0 \longrightarrow E_0 \longrightarrow \dots \longrightarrow E_i \longrightarrow E_{i+1} \longrightarrow \dots \longrightarrow E_n = E,$$

with the following property: for all $i$ there is a perfect complex $K_i$ on $X$ and a cartesian square in $\mathrm{Coh}(\mathcal{F})$

<figure><img src="image_27.png" /></figure>

We will show, by induction, that each $E_i$ has a good filtration. For this, assume that $E_i$ has a good filtration $E_i^{\mathrm{fil}}$. The morphism $u_i$ is given, by adjunction, by a morphism $v_i : K_i \to E_i = (E_i^{\mathrm{fil}})^u$ in $\mathrm{QCoh}(\mathcal{F})$. The quasi-coherent sheaf $(E_i^{\mathrm{fil}})^u$ is the filtered colimit $\mathrm{colim}_k F^k(E_i^{\mathrm{fil}})$, and as $K_i$ is a compact object in $\mathrm{QCoh}(X)$, $v_i$ can be factored as $K_i \xrightarrow{w_i} F^k(E_i^{\mathrm{fil}}) \longrightarrow E_i$, for some index $k$. By using the left adjoint $\mathrm{Ind}_{\mathcal{F}}^{\mathrm{fil}}$, the morphism $w_i$ corresponds to a morphism of filtered $\mathcal{D}_{\mathcal{F}}^{\mathrm{fil}}$-modules

$$\alpha_i : \mathrm{Ind}_{\mathcal{F}}^{\mathrm{fil}}(K_i)(-k) \longrightarrow E_i,$$

where $(-k)$ denotes the endofunctor of $\mathrm{QCoh}^{\mathrm{fil}}(\mathcal{F})$ that shifts by $-k$ the filtration. The cone of $\alpha_i$ clearly defines a good filtration on $E_{i+1}$. □

**4.3.2.2. Independence of the good filtration.** We already noticed that the characteristic cycle of definition 4.3.2.3 depends, a priori, on the good filtration $E^{\mathrm{fil}}$. In order to solve this problem, we introduce a reduced $K$-group, and prove that the image of $\mathrm{Ch}(E^{\mathrm{fil}})$ in this reduced $K$-group, only depends on the object $E$.
Let $\mathcal{F}$ be a derived foliation on a derived scheme $X$. We consider $\mathbf{Coh}^{\text{fil}}(\mathcal{F})$, the $\infty$-category of compact objects $\mathbf{QCoh}^{\text{fil}}(\mathcal{F})$, and the underlying object $\infty$-functor $(-)^u : \mathbf{Coh}^{\text{fil}}(\mathcal{F}) \to \mathbf{Coh}(\mathcal{F})$. An object $E \in \mathbf{Coh}^{\text{fil}}(\mathcal{F})$ will be called a phantom if $E^u \simeq 0$. We then give the following definition.

**DEFINITION 4.3.2.6.** With the notations above, the reduced $K$-group $K_0^{\text{red}}(T^*\mathcal{F})$ is the quotient of $K_0(T^*\mathcal{F})$ by the subgroup generated by the classes of $\text{Gr}(E)$ for $E \in \mathbf{Coh}^{\text{fil}}(\mathcal{F})$ a phantom.

We have the following independence result.

**PROPOSITON 4.3.2.7.** Let $\mathcal{F}$ be a derived foliation on a derived scheme $X$, and $E \in \mathbf{QCoh}(\mathcal{F})$ be a coherent crystal. Let $E_1^{\text{fil}}$ and $E_2^{\text{fil}}$ be two good filtrations on $E$ in the sense of Definition 4.3.2.1. Then we have $\text{Ch}(E_1^{\text{fil}}) = \text{Ch}(E_2^{\text{fil}})$ in $K_0^{\text{red}}(T^*\mathcal{F})$.

*Proof.* We start by the following lifting lemma.

**LEMMA 4.3.2.8.** Let $M, N \in \mathbf{QCoh}^{\text{fil}}(\mathcal{F})$ with $M$ compact. Then any morphism $u : M^u \to N^u$ in $\mathbf{QCoh}(\mathcal{F})$ can be lifted, via the $\infty$-functor $(-)^u$, to a morphism $v : M \to N(k)$ in $\mathbf{QCoh}^{\text{fil}}(\mathcal{F})$, where $N(k)$ the $k$-weight shift of $N$, for some integer $k$.

*Proof of the lemma.* Recall that $N(k)$ denotes the filtered crystal with the filtration shifted by $k$, i.e.

$$F^i(N(k)) := F^{i+k}N.$$

As $M$ is compact, we know by the proof of Corollary 4.1.3.3, that $M$ is a retract of a finite cell object in $\mathbf{QCoh}^{\text{fil}}(\mathcal{F})$ in the sense of 4.3.2.5. Clearly, if the lemma is true for $M$ (and any $N$) it is also true for any of its retracts. We may therefore assume that $M$ is a finite cell object. By induction on the number of cells, we reduce ourselves to the following statement. Assume that the lemma is true for $M$ (and any $N$), and let us consider a push-out

<figure><img src="image_28.png" /></figure>

with $K$ compact in $\mathbf{QCoh}(X)$. We must prove that the lemma remains true for $M'$ (and any $N$). Let $v : (M')^u \to N^u$ be a morphism. It consists of the data of a morphism $u : M^u \to N^u$ and a homotopy to zero of $(u \circ \alpha) : \text{Ind}_\mathcal{F}(K) \to N^u$, i.e., by adjunction, a homotopy to zero $h$ of the induced morphism $\beta : K \to N^u$. By assumption on $u$ we can lift $u$ to $w : M \to N(k)$ for some $k$. Moreover, $h$ defines a homotopy to zero of $v = (w)^u : M^u \to N(k)^u \simeq N^u$. As $K$ is compact and $N^u = \text{colim}_k F^i N$, the homotopy $h$ factors as a homotopy to zero $h'$ of $K \to F^{k'} N \to N^u$ for some $k' \ge k$. This pair $(w, h')$ then defines the required lift $M' \to N(k')$. $\square$
Let us go back to the proof of Proposition 4.3.2.7. As $E_i^{\text{fil}}$, $i = 1, 2$ are a good filtrations on the same crystal $E$, we have a natural equivalence in $\mathbf{QCoh}(F)$

$$u : (E_1^{\text{fil}})^u \simeq (E_2^{\text{fil}})^u.$$

By Lemma 4.3.2.8, $u$ can be lifted to a morphism of filtered crystals $v : E_1^{\text{fil}} \to E_2^{\text{fil}}$. We set $M$ to be the cone of $v$ in $\mathbf{QCoh}^{\text{fil}}(\mathcal{F})$. This is a compact object which is obviously a phantom, i.e. $M^u \simeq 0$. We thus have an $\text{Ch}(E_1^{\text{fil}}) = \text{Ch}(E_2^{\text{fil}})$ in $K_0^{\text{red}}(T^*\mathcal{F})$. $\square$

Proposition 4.3.2.7 shows that the following notion is well defined.

**DEFINITION 4.3.2.9.** Let $X$ be a quasi-compact and separated derived DM-stack and $\mathcal{F} \in \mathcal{Fol}^p(X)$ a perfect derived foliation. If $E \in \mathbf{Coh}(\mathcal{F})$ admits a good filtration $E^{\text{fil}}$, then its characteristic cycle is defined as

$$\text{Ch}(E) = \text{Ch}(E^{\text{fil}}) \in K_0^{\text{red}}(T^*\mathcal{F}).$$

It is possible to show that when $X$ and $\mathcal{F}$ are both smooth, then the natural projection $K_0(T^*\mathcal{F}) \to K_0^{\text{red}}(T^*\mathcal{F})$ is bijective, or, in other words, that the class of $\text{Gr}(E)$ is trivial in $K_0(T^*\mathcal{F})$ for any phantom $E \in \mathbf{Coh}^{\text{fil}}(\mathcal{F})$. This relies on using regularity and Quillen's devissage techniques, that do not work in our general setting. We will not need this isomorphism $K_0(T^*\mathcal{F}) \simeq K_0^{\text{red}}(T^*\mathcal{F})$ in the rest of the book, so we omit its proof. However, the following very simple particular case will be useful later, in order to get numerical formulas out of our general Grothendieck-Riemann-Roch statement (Section 4.4).

**PROPOSITON 4.3.2.10.** Let $X$ be a quasi-compact and separated derived DM-stacks and $\mathcal{F} = 0_X$ be the initial foliation, so that $T^*\mathcal{F} \simeq X$. Then, the natural projection

$$K_0(X) \to K_0^{\text{red}}(X)$$

is bijective.

*Proof.* In this case $\mathbf{QCoh}^{\text{fil}}(\mathcal{F}) \simeq \mathbf{QCoh}^{\text{fil}}(X)$ is the $\infty$-category of filtered quasi-coherent complexes. The compact objects in $\mathbf{QCoh}^{\text{fil}}(X)$ are therefore finite filtered perfect complexes, and thus $\text{Gr}(E)$ has only a finite number of non-trivial summand for any compact object $E \in \mathbf{QCoh}^{\text{fil}}(X)$. As a result, if $E$ is a phantom, we have $0 = [\text{Gr}(E)]$ in $K_0(X)$. $\square$

**REMARK 4.3.2.11.** A particularly important case of proposition 4.3.2.10 is when $X = \text{Spec } k$, for which we find $K_0^{\text{red}}(\text{Spec } k) \simeq \mathbb{Z}$.

## 4.4. Grothendieck-Riemann-Roch for derived foliations

In this Section we will state and prove the Grothendieck-Riemann-Roch (GRR) formula for proper maps between derived schemes endowed with derived foliations.
**4.4.1. The GRR formula.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism of derived stack endowed with derived foliaions, with $g$ proper and quasi-smooth. We will make the standard assumption that $Y$ is either a qcqs derived scheme, or a separated and quasi-compact derived DM-stack. Associated to $f$ is the so-called “Japanese correspondence”

$$T^*\mathcal{F} \xleftarrow[q]{\mathcal{G}} T^*\mathcal{G} \times_Y X \xrightarrow{p} T^*\mathcal{G}.$$

Here $q$ is induced by the morphism of $g^*(\mathbb{L}_{\mathcal{G}}) \to \mathbb{L}_{\mathcal{F}}$ perfect complexes on $X$ induced by $u$ while the morphism $p$ is the first projection.

Define the push-forward on K-groups

$$f_! := p_!(q^*(-)) : K_0(T^*\mathcal{F}) \to K_0(T^*\mathcal{G}).$$

where $p_!$ is the left adjoint of the pull-back of quasi-coherent sheaves $p^* : \mathbf{QCoh}(T^*\mathcal{G} \times_Y X) \to \mathbf{QCoh}(T^*\mathcal{G})$. This left adjoint exists because $p$ is representable, proper and quasi-smooth, and it is given explicitly by

$$p_!(E) = p_*(E \otimes \omega_p[d])$$

for $E \in \mathrm{Perf}(T^*\mathcal{G} \times_Y X)$, where the integer $d$ is the relative dimension of $X$ over $Y$, and $\omega_p$ is the relative canonical sheaf of the morphism $p$. Note that $\omega_p$ coincides with the pull-back of $\omega_{X/Y}$ along the projection $T^*\mathcal{G} \times_Y X \to X$.

We start by noticing that $f_!$ is compatible with the quotient defining reduced K-groups. Indeed, if $E \in \mathrm{Coh}^{\mathrm{fil}}(\mathcal{F})$ is a phantom, then so is $f_!(E)$, and the image by $f_!$ of $\mathrm{Gr}(E)$ is $\mathrm{Gr}(f_!E)$ (see Corollary 4.2.2.1). Therefore, $f_!$ induces a well defined map

$$f_! : K_0^{\mathrm{red}}(T^*\mathcal{F}) \to K_0^{\mathrm{red}}(T^*\mathcal{G}).$$

**THEOREM 4.4.1.1.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism of derived DM-stack schemes endowed with perfect derived foliations with $g$ proper and quasi-smooth. We assume that either $Y$ is separated and quasi-compact, or a qcqs derived scheme. For any coherent crystal $E \in \mathrm{Coh}(\mathcal{F})$ along $\mathcal{F}$ admitting a good filtration (e.g. a finite cell object by proposition 4.3.2.5), we have

$$\mathrm{Ch}(f_!(E)) = f_!(\mathrm{Ch}(E))$$

in the group $K_0^{\mathrm{red}}(T^*\mathcal{G})$.

*Proof.* This is a direct consequence of the fact that direct images commute with taking associated graded for good filtrations (corollary 4.2.2.1). □

The following consequence of Theorem 4.4.1.1 is obtained when $Y = \mathrm{Spec} k$ and $\mathcal{G} = 0_Y$. Let $s : X \to T^*\mathcal{F}$ denotes the zero section of the cotangent stack of $\mathcal{F}$. When $X$ is a proper and quasi-smooth derived DM-stack, we denote by $p : X \to \mathrm{Spec} k$ the canonical map, and by

$$p_* := \int_X : K_0(X) \to K_0(k) \simeq \mathbb{Z}$$

the induced push-forward on perfect complexes.
**COROLLARY 4.4.1.2.** Let $(X, \mathcal{F})$ be a quasi-smooth and proper derived DM-stack endowed with a perfect derived foliation $\mathcal{F} \in \mathcal{Fol}^p(X)$. Let $f : (X, \mathcal{F}) \to (\text{Spec } k, \mathcal{O}_{\text{Spec } k})$ be the canonical morphism. For any coherent crystal $E$ that admits a good filtration we have

$$ \chi(f_!(E)) = \int_X s^*(\mathrm{Ch}(E)) \otimes \omega_X. $$

This corollary is a *Hirzeburch-Riemann-Roch (HRR) formula* for crystals along the foliation $\mathcal{F}$. The complex $f_!(E)$ is what should be called the *foliated cohomology of $(X, \mathcal{F})$ with coefficients in $E$*. If we denote this cohomology by $H_{\mathcal{F}}^*(X, E)$, the HRR formula reads

$$ \chi(H_{\mathcal{F}}^*(X, E)) = \int_X s^*(\mathrm{Ch}(E)) \otimes \omega_X. $$

**4.4.2. The non-proper case: Fredholm crystals.** We explain here briefly how to extend theorem 4.4.1.1 to *non-proper maps*, by introducing the notion of *Fredholm crystal*.

We start by extending direct images to the case of *compactifiable morphisms*. Assume that $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ is a morphism of derived DM-stack endowed with perfect derived foliations, such that $g : X \to Y$ is quasi-smooth. A *quasi-smooth compactification* of $f$ is the datum of a factorization

$$ (20) \qquad f : (X, \mathcal{F}) \xrightarrow{(j, v)} (\bar{X}, \bar{\mathcal{F}}) \xrightarrow{\bar{f}} (Y, \mathcal{G}) $$

where $j$ is an open embedding, $v$ is an equivalence $\mathcal{F} \simeq j^*(\bar{\mathcal{F}})$, and $\bar{f}$ is a proper and quasi-smooth morphism. For such a compactification, we set

$$ j_* : \mathrm{QCoh}(\mathcal{F}) \to \mathrm{QCoh}(\bar{\mathcal{F}}) $$

to be the right adjoint to the $\infty$-functor $j^!$. We note here that $j_*$ exists because $j^!$ commutes with colimits. Moreover, $j_*$ is compatible with the usual push-forward of quasi-coherent sheaves through the forceful $\infty$-functors, i.e. the following diagram naturally commutes

$$ \begin{tikzcd} \mathrm{QCoh}(\mathcal{F}) \arrow[r, "j_*"] & \mathrm{QCoh}(\bar{\mathcal{F}}) \arrow[d] \\ \mathrm{QCoh}(X) \arrow[r, "j_*"] & \mathrm{QCoh}(\bar{X}) \arrow[d] \end{tikzcd} $$

where the vertical $\infty$-functors are the natural forgetful functors. Since $j$ is an open immersion and $v$ is an equivalence, we have a natural equivalence $j^{-1}\mathcal{D}_{\bar{\mathcal{F}}} \simeq \mathcal{D}_{\mathcal{F}}$ of sheaves of dg-algebras on $X$, and a canonical adjunction morphism on $\bar{X}$

$$ a : \mathcal{D}_{\bar{\mathcal{F}}} \to j_*(\mathcal{D}_{\mathcal{F}}). $$

The $\infty$-functor $j_* : \mathrm{QCoh}(\mathcal{F}) \to \mathrm{QCoh}(\bar{\mathcal{F}})$ is then simply induced by the usual push-forward of sheaves along $j : X \hookrightarrow \bar{X}$: it sends a $\mathcal{D}_{\mathcal{F}}$-module $E$ to $j_*(E)$, viewed as a $\mathcal{D}_{\bar{\mathcal{F}}}$-module via the
map a above. This description shows that we also have a commutative square involving the induction $\infty$-functors

$$ \begin{array}{ccc} \mathbf{QCoh}(\mathcal{F}) & \xrightarrow{j_*} & \mathbf{QCoh}(\bar{\mathcal{F}}) \\ \multicolumn{1}{c}{\updownarrow} & & \multicolumn{1}{c}{\updownarrow} \\ \mathbf{Ind}_{\bar{\mathcal{F}}} & & \mathbf{Ind}_{\bar{\mathcal{F}}} \\ \mathbf{QCoh}(X) & \xrightarrow{j_*} & \mathbf{QCoh}(\bar{X}). \end{array} $$

We define the *-direct image along f as

$$ f_* = \bar{f}_! \circ j_* : \mathbf{QCoh}(\mathcal{F}) \to \mathbf{QCoh}(\mathcal{G}). $$

A standard argument, by using the product embedding, proves that $f_*$, defined as above, does not depend on the choice of the factorization (20) (see e.g. [AGV73, Exp XVII] or [FK88, §8]). Indeed, $f_*$ clearly commutes with colimits, so the independence of the factorization in its definition can be checked on compact $\mathcal{D}_{\bar{\mathcal{F}}}$-modules of the form $\mathbf{Ind}_{\bar{\mathcal{F}}}(E)$, where it is easily deduced from the compatibility of push-forwards for quasi-coherent sheaves, and from the explicit formula of proposition 4.2.1.6.

**DEFINITION 4.4.2.1.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism of derived DM-stack endowed with perfect derived foliations. Suppose that $f$ admits a quasi-smooth compactification. An object $E \in \mathbf{Coh}(\mathcal{F})$ is called $f$-Fredholm if it admits a good filtration $E^{\text{fil}} \in \mathbf{Coh}^{\text{fil}}(\mathcal{F})$ such that $j_*(E^{\text{fil}})$ is a compact object in $\mathbf{QCoh}^{\text{fil}}(\mathcal{F})$ for some quasi-smooth compactification $j : (X, \mathcal{F}) \to (\bar{X}, \bar{\mathcal{F}})$ of the morphism $f$.

Suppose $E \in \mathbf{Coh}(\mathcal{F})$ is $f$-Fredholm; pick a quasi-smooth compactification $(\bar{X}, \bar{\mathcal{F}})$ and a filtration $E^{\text{fil}}$ as in the definition above. We have the associated graded $Gr(E^{\text{fil}}) \in \text{Perf}(T^*\mathcal{F})$, and its direct image by $j$ is by again a perfect complex on $T^*\bar{\mathcal{F}}$, so that, in particular, the support of $Gr(E^{\text{fil}})$, $SS(E) \subset T^*\mathcal{F}$ is closed in $T^*\bar{\mathcal{F}}$. If we pull-back this support by the canonical map $T^*\mathcal{G} \times_Y X \to \mathbb{T}^*\mathcal{F}$, we get a closed subset in $T^*\mathcal{G} \times_Y X$ which is proper over $T^*\mathcal{G}$. In other words, the morphism $f$ is proper when restricted to the support of $Gr(E^{\text{fil}})$. In our situation, the notion of Fredholm is a priori stronger, it is unclear that properness of $f$ on the support of $Gr(E^{\text{fil}})$ is enough to recover the fact that $j_*(E^{\text{fil}})$ remains a compact object. Therefore, our definition of being $f$-Fredholm might be hard to check in practice, already because it involves the existence of a compactification of $f$ which seems a delicate question even when $X$ and $Y$ are both smooth quasi-projective varieties.

We can now state the Grothendieck-Riemann-Roch formula for possibly non-proper maps, and Fredholm coefficients.

**THEOREM 4.4.2.2.** Let $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ be a morphism of derived DM-stacks endowed with perfect derived foliations. We assume as usual that either $Y$ is a $\text{qcqs}$ derived schemes, or a separated and quasi-compact DM-stack. If $f$ admits a quasi-smooth
compactification, and $E \in \mathbf{Coh}(\mathcal{F})$ is an $f$-Fredholm crystal, then we have an equality

$$ \mathrm{Ch}(f_*(E)) = f_*(\mathrm{Ch}(E)) $$

of elements in $K_0^{\mathrm{red}}(T^*\mathcal{G})$.

*Proof.* Simply apply Theorem 4.4.1.1 to the morphism $\bar{f}$ and to the object $j_*(E)$ over $(\bar{X}, \bar{\mathcal{F}})$, which has a good filtration given by $j_*(E^{\mathrm{fil}})$ for $E^{\mathrm{fil}}$ given by Definition 4.4.2.1. $\square$

## 4.5. Examples and applications

We finish this Chapter by giving a sample of examples and applications of the general Grothendieck-Riemann-Roch formula of Theorem 4.4.1.1.

**4.5.1. Grothendieck-Riemann-Roch for derived $\mathcal{D}$-modules.** The very first special case of Theorem 4.4.1.1 is when $\mathcal{F} = *_X$ and $\mathcal{G} = *_Y$ are the final derived foliations on $X$ and $Y$ respectively. The $\infty$-categories $\mathbf{QCoh}(*_X)$ and $\mathbf{QCoh}(*_Y)$ are then called the $\infty$-categories of derived $\mathcal{D}$-modules on $X$ and $Y$, and denoted by $\mathcal{D}_X - \mathrm{dg}_{\mathrm{qcoh}}$ and $\mathcal{D}_Y - \mathrm{dg}_{\mathrm{qcoh}}$, respectively. We think they coincide, for all $X$, with the derived $\mathcal{D}$-modules introduced and studied in [Ber21], where they are denoted by $\mathcal{D}^{\mathrm{der}}(X)$ and $\mathcal{D}^{\mathrm{der}}(Y)$. We refer the reader to [Buc] for precise comparison statements. However, we remark that, contrary to the notion of crystals introduced in [GR14], these $\infty$-categories of derived $\mathcal{D}$-modules are sensitive to derived structures (see remark 4.1.1.7).

In this situation, $T^*\mathcal{F} = T^*X$ and $T^*\mathcal{G} = T^*Y$ are the global derived cotangent stacks of $X$ and $Y$, and the GRR formula is the equality

$$ (21) \qquad \mathrm{Ch}(f_!(E)) = f_!\mathrm{Ch}(E) \quad \text{in } K_0^{\mathrm{red}}(T^*Y) $$

for a compact object $E \in \mathcal{D}_X - \mathrm{dg}_{\mathrm{qcoh}}$ admitting a good filtration, and $f: X \to Y$ a proper and quasi-smooth morphism. When $X$ and $Y$ are smooth varieties, this recovers the well known GRR formula for $\mathcal{D}$-modules of [Lau83] (see also [Sab12]). However, already when $X$ and $Y$ are just underived $k$-schemes, formula (21) is new and provides a Grothendieck-Riemann-Roch formula for $\mathcal{D}$-modules on possibly singular schemes.

An interesting feature of this situation is that any morphism $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ fits in a commutative diagram

$$ \begin{tikzcd} (X, \mathcal{F}) \arrow[r, "f"] \arrow[d, "u"] & (Y, \mathcal{G}) \arrow[d, "v"'] \\ (X, *_X) \arrow[r, "g"] & (Y, *_Y). \end{tikzcd} $$

As explained at the end of §4.1.3, the push-fowards $u_!$ and $v_!$ define induction $\infty$-functors

$$ u_! : \mathbf{QCoh}(\mathcal{F}) \to \mathcal{D}_X - \mathrm{dg}_{\mathrm{qcoh}} \qquad v_! : \mathbf{QCoh}(\mathcal{G}) \to \mathcal{D}_Y - \mathrm{dg}_{\mathrm{qcoh}}. $$
The GRR formula for $f$, which is an equality in $K^{\text{red}}(T^*\mathcal{G})$, can be pushed forward along $v$ to a formula in $K_0^{\text{red}}(T^*Y)$:

$$v_!Ch(f_!(E)) = g_!Ch(u_!E))$$

where $E$ is a coherent crystal along $\mathcal{F}$ admitting a good filtration, $u_!(E)$ the induced coherent $\mathcal{D}_X$-module, and $v_! : K_0^{\text{red}}(T^*\mathcal{G}) \to K_0^{\text{red}}(T^*Y)$ the direct image along the canonical morphism of foliations $\mathcal{G} \to *_Y$. This gives a formula for the push-forward of the characteristic cycle of a $\mathcal{D}_X$-module induced from the derived foliation $\mathcal{F}$. When $f = id$ the formula reads

$$v_!Ch(E) = Ch(u_!E)$$

and should be understood as a formula for the characteristic cycle on a $\mathcal{D}_X$-module induced from the derived foliation $\mathcal{F}$.

**4.5.2. Smooth Lie algebroids.** Let $X$ be a smooth variety and $\mathcal{F} \in \mathcal{Fol}(X)$ a smooth foliation on $X$ (i.e. $\mathbb{L}_{\mathcal{F}}$ is a vector bundle on $X$). As explained in theorem 2.2.0.7, $\mathcal{F}$ is determined by a Lie algebroid $\mathbb{T}_{\mathcal{F}} \to \mathbb{T}_X$. The sheaf $\mathcal{D}_{\mathbb{T}_{\mathcal{F}}}^{\text{fil}}$ is then isomorphic to the universal enveloping algebra $U(\mathbb{T}_{\mathcal{F}})$ of $\mathbb{T}_{\mathcal{F}}$, equipped with its natural PBW filtration. As already noticed $K_0^{\text{red}}(T^*\mathcal{F}) \simeq K_0(T^*\mathcal{F})$, in this case (of a smooth foliation on a smooth variety). It can also be shown that any coherent crystal along $\mathcal{F}$ admits a good filtration. Moreover, $\mathcal{D}_{\mathbb{T}_{\mathcal{F}}}$ satisfies all the conditions of Quillen theorem (see [Qui73]) and we thus have natural isomorphisms of K-groups

$$\tau_X : K_0(\mathbf{Coh}(\mathcal{F})) \simeq K_0(T^*\mathcal{F}) \simeq K_0(X).$$

The first of this isomorphisms is precisely given by $E \mapsto \text{Gr}(E^{\text{fil}})$ for $E^{\text{fil}}$ a good filtration on $E$, and the second isomorphism is the pull-back along the zero section $s : X \to T^*\mathcal{F}$.

The GRR formula 4.4.1.1 tells us that the isomorphism $\tau_X : K_0(\mathbf{Coh}(F)) \simeq K_0(X)$ is covariantly functorial in $(X, \mathcal{F})$ in the following sense: given a morphism $f = (g, u) : (X, \mathcal{F}) \to (Y, \mathcal{G})$ of smooth varieties endowed with smooth foliations, with $g$ proper, the following diagram

<figure><img src="image_29.png" /></figure>

commutes. This recovers the well known GRR formula for $\mathcal{D}$-modules on smooth varieties (see [Lau83]), and its natural extension to Lie algebroids. This extension to Lie algebroids is probably a folklore result (as its proof is word by word the same as for $\mathcal{D}$-modules), but we have not been able to find an appropriate reference in the literature.
**4.5.3. Shifted Poisson structures.** An important class of derived foliations are provided by shifted Poisson structures in the sense of [CPT$^{+17}$]. Indeed, let $X$ be a derived scheme endowed with a shifted Poisson structure of degree $n$. The Poisson bracket defines a morphism of perfect complexes on $X$

$$a : \mathbb{L}_X[-n] \to \mathbb{T}_X$$

making $\mathbb{L}_X[-n]$ into a dg-Lie algebroid over $X$. This defines (§ 2.2) a derived foliation $\mathcal{F}$ whose underlying graded mixed cdga is $\mathrm{Sym}_{\mathcal{O}_X}(\mathbb{T}_X[n])$, where the mixed structure is induced the bracket $[-, p]$, $p$ being the bivector defining the Poisson structure. We refer to [Tom25] where this construction is explained in details, as several technical complications related to homotopy coherences have to be overcome.

This derived foliation associated to a shifted Poisson structure is the derived analogue of the foliation by symplectic leaves of a classical Poisson strcuture on a smooth variety. Its leaves, in the sense of Chapter 6, are by definition the symplectic leaves of the shifted Poisson structure. When the Poisson structure is non-degenerate, then the foliation $\mathcal{F}$ is the final foliation. In general $\mathcal{F}$ is a very interesting derived foliation containing information about the Poisson structure. For instance, our notion $\mathbf{QCoh}(\mathcal{F})$ of quasi-coherent crystals provides a useful setting to study various versions of Poisson cohomology. As an example, we may define derived Poisson cohomology to be

$$H_{der}^{\mathrm{Poiss}}(X) := \mathbb{R} \mathrm{Hom}_{\mathbf{QCoh}(\mathcal{F})}(\mathcal{O}_X, \mathcal{O}_X),$$

which coincides with derived de Rham cohomology when the Poisson structure is a symplectic structure. It is also possible to consider the induced $\mathcal{D}_X$-module $u_!(\mathcal{O}_X) \in \mathcal{D}_X - \mathrm{dg}_{X,qcoh}$, where $u : \mathcal{F} \to *_X$ is the unique morphism to the final foliation. Then, we may use the $\mathcal{D}_X$-module $u_!(\mathcal{O}_X)$ in order to define a version of Poisson homology by

$$H_{\mathrm{Poiss}}(X) := p_*(u_!(\mathcal{O}_X)) \in \mathrm{dg}$$

where $p : X \to \mathrm{Spec} k$ is the structure map (assuming here that $X$ is quasi-smooth and $p$ admits a quasi-smooth compactification as defined in §4.4.2). This offers a criterion for finiteness of Poisson homology by requiring $u_!(\mathcal{O}_X)$ to be Fredholm as a $\mathcal{D}_X$-module on $X$. When $X$ is smooth and the Poisson structure is a classical Poisson structure of degree 0, the $\mathcal{D}_X$-module $u_!(\mathcal{O}_X)$ has been considered in [PS18], where it was used to define the notion of holonomic Poisson varieties and to get finiteness results for Poisson cohomology. The GRR formula 4.4.2.2, and more generally the general formalism of crystals along derived foliations, provides a way to extend these notions and results to shifted Poisson structures.

**4.5.4. A foliated index formula.** As a last application of Theorem 4.4.2.2 we propose an index formula for a weakly Fredholm differential operator along the leaves of a derived foliation on a quasi-smooth derived scheme. We like to think of it as an algebraic version of the longitudinal index theorem of [CS84], possibly valid outside the smooth setting, i.e. for non-smooth derived schemes and non-smooth derived foliation over them.
Let $(X, \mathcal{F})$, where $X$ is a derived DM-stack and $\mathcal{F} \in \mathcal{F}ol^p(X)$ a perfect derived foliation. We assume that $X$ has a quasi-smooth compactification $j : X \hookrightarrow \bar{X}$ (i.e. the projection $X \to \text{Speck}$ admits a quasi-smooth compactification as in §4.4.2), which implies in particular that $X$ is separated and quasi-compact. Note that we only assume the existence of $\bar{X}$, and we do not assume that $\mathcal{F}$ can be extended to $\bar{X}$. Let $E$ and $E'$ be two perfect complexes on $X$. A differential operator $P$ along $\mathcal{F}$ from $E$ to $E'$ is by definition a morphism of quasi-coherent sheaves on $X$

$$ P : E \to \mathrm{Ind}_{\mathcal{F}}(E') = \mathcal{D}_{\mathcal{F}} \otimes_{\mathcal{O}_X} E', $$

or, equivalently, a morphism of coherent crystals along $\mathcal{F}$

$$ a_P : \mathrm{Ind}_{\mathcal{F}}(E) \to \mathrm{Ind}_{\mathcal{F}}(E'). $$

As $E$ is compact in $\mathbf{QCoh}(X)$, we note that $P$ must factor through a morphism $E \to \mathcal{D}_{\bar{\mathcal{F}}}^{\le i} \otimes_{\mathcal{O}_X} E'$ for some integer $i$. In this case, we say that $P$ is of order $\le i$.

We let $\mathcal{D}(P)$ be the cone of the morphism $a_P$ inside $\mathbf{Coh}(\mathcal{F})$, and we want to apply Theorem 4.4.2.2 to the coherent crystal $\mathcal{D}_X \otimes_{\mathcal{D}_F} \mathcal{D}(P)$. Note that this is also the cone of the morphism $\mathcal{D}_X \otimes_{\mathcal{D}_F} E \to \mathcal{D}_X \otimes_{\mathcal{D}_F} E'$, induced by the composition

$$ P : E \to \mathcal{D}_{\mathcal{F}} \otimes_{\mathcal{O}_X} E' \to \mathcal{D}_X \otimes_{\mathcal{O}_X} E'. $$

In order to apply Theorem 4.4.2.2, we need to impose a condition on $P$ ensuring that $\mathcal{D}_X \otimes_{\mathcal{D}_F} \mathcal{D}(P)$ is a Fredholm object (Definition 4.4.2.1). We consider the least integer $i$ such that $P$ factors as

$$ E \to \mathcal{D}_{\bar{\mathcal{F}}}^{\le i} \otimes_{\mathcal{O}_X} E'. $$

Associated to this, we have a morphism of filtered $\mathcal{D}_{\bar{\mathcal{F}}}^{\text{fil}}$-modules

$$ \mathrm{Ind}_{\mathcal{F}}^{\text{fil}}(E)(-i) \to \mathrm{Ind}_{\mathcal{F}}^{\text{fil}}(E), $$

whose cone defines a good filtration $\mathcal{D}^{\text{fil}}(P)$ on $\mathcal{D}(P)$, and thus, by base change, a good filtration $\mathcal{D}_X^{\text{fil}} \otimes_{\mathcal{D}_{\bar{\mathcal{F}}}^{\text{fil}}} \mathcal{D}^{\text{fil}}(P)$ on $\mathcal{D}_X \otimes_{\mathcal{D}_F} D(P)$.

**DEFINITION 4.5.4.1.** The operator $P$ along $\mathcal{F}$ is weakly Fredholm if $j_*(\mathcal{D}_X^{\text{fil}} \otimes_{\mathcal{D}_{\bar{\mathcal{F}}}^{\text{fil}}} \mathcal{D}^{\text{fil}}(P))$ is a compact object in $\mathbf{QCoh}^{\text{fil}}(\bar{X})$, for a quasi-smooth compactification $j : X \to \bar{X}$.

Assuming that $P$ is weakly Fredholm, we are thus able to apply Theorem 4.4.2.2 to get our index formula, that has values in $K_0(k) \simeq \mathbb{Z}$, and thus is an equality of two numbers we are now going to describe.

Let $f : X \to \text{Speck}$ the projection, that we factor as $X \xrightarrow{j} \bar{X} \xrightarrow{\bar{f}} \text{Speck}$. We first describe $f_*(\mathcal{D}(P)) \simeq \bar{f}_!(j_*(\mathcal{D}(P))) \simeq \bar{f}_!(\mathcal{D}_X \otimes_{\mathcal{D}_F} D(P))$. This is a perfect complex of $k$-modules, which by definition, is the cone of the morphism induced by $P$

$$ \Gamma(P) : \Gamma(X, E \otimes \omega_X)[d] \to \Gamma(X, E' \otimes \omega_X)[d]. $$

Note that none of the two complexes $\Gamma(X, E \otimes \omega_X)$ or $\Gamma(X, E' \otimes \omega_X)$ is perfect, and only the cone of $\Gamma(P)$ is so. This is the effect of the weakly Fredholm property, implying that $\Gamma(P)$ is
indeed a Fredlhom operator. Therefore, we have a first well defined number, called the *algebraic index of P* Ind(P), which is the Euler characteristic of the cone of Γ(P)

$$Ind(P) := (-1)^d \chi(\text{cone } (\Gamma(P) : \Gamma(X, E \otimes \omega_X) \to \Gamma(X, E' \otimes \omega_X))).$$

On the other hand, the object $\mathcal{D}_X \otimes_{\mathcal{D}_F} \mathcal{D}(P)$ is endowed with the good filtration $\mathcal{D}_X^{\text{fil}} \otimes_{\mathcal{D}_F^{\text{fil}}} \mathcal{D}^{\text{fil}}(P)$, and by assumption its associated graded defines a perfect complex of $T^*X$ which remains perfect on $T^*\bar{X}$. This associated graded can be described as follows. Recall that $i$ is the least integer such that $P$ defines a morphism in $\text{QCoh}(X)$ $P : E \to \mathcal{D}_{\bar{X}}^{\le i} \otimes_{\mathcal{O}_X} E'$. By projection $\mathcal{D}_{\bar{X}}^{\le i} \to \text{Sym}_X^i(\mathbb{T}_X)$, we find

$$\sigma(P) : E \to \text{Sym}_X^i(\mathbb{T}_X) \otimes_{\mathcal{O}_X} E'$$

which is called the *symbol of P*. This extends to

$$\text{Sym}_X(\mathbb{T}_X) \otimes_{\mathcal{O}_X} E \to \text{Sym}_X(\mathbb{T}_X) \otimes_{\mathcal{O}_X} E$$

as a morphism of graded $\text{Sym}_{\mathcal{O}_X}(\mathbb{T}_X)$-modules

The cone of this morphism defines a perfect complex, still denoted by $\sigma(P)$, on $T^*X$, which remains perfect on $T^*\bar{X}$ by the weakly Fredholm assumption. If we denote by $s : X \to T^*X$ the zero section map, we thus get a well defined perfect complex of $k$-modules $\Gamma(X, s^*(\sigma(P)) \otimes \omega_X)[d]$. The Euler characteristic of this complex is called the *K-theoretical index of the operator P*, and denoted by

$$\text{Ind}_{\sigma}(P) := (-1)^d \chi(\Gamma(X, s^*(\sigma(P)) \otimes \omega_X)).$$

Theorem 4.4.2.2 then implies the following

COROLLARY 4.5.4.2. With the notations above, we have

$$\text{Ind}(P) = \text{Ind}_{\sigma}(P).$$

In plain words, Corollary 4.5.4.2, says that the *algebraic index of P*, computing the Fredholm index of P acting on global sections of E and E', equals the *K-theoretic index of P*, which should be understood as an intersection number between X and the cycle defined by the symbol of P inside the total cotangent stack $T^*X$.
CHAPTER 5

Analytic aspects

This chapter is devoted to introduce the notion of derived foliation in the holomorphic context. We remind some basic definitions on holomorphic rings and holomorphic cdga, and the corresponding notion of derived analytic stacks. We then introduce sheaves of holomorphic graded mixed cdga and use them to define holomorphic derived foliations on general derived analytic stacks in a very similar fashion than the in the algebraic setting. We study analytic integrability of quasi-smooth and rigid derived foliations, locally in the analytic topology. Finally we construct an analytification functor and prove a version of GAGA relating derived algebraic foliations and derived analytic foliations on a proper derived Deligne-Mumford algebraic stack.

Throughout this chapter the base ring $k$ is the field of complex numbers $\mathbb{C}$.

5.1. Quick reminders on derived analytic geometry

We recall the definition of an algebraic theory *hol*, of holomorphic rings as follows (see [Pir15, Pri20, Por19] and [CR13, Nui12] for the $\mathbb{C}^\infty$-version). The algebraic theory *hol* is by definition the full sub-category of $\mathbb{C}Man$, of complex manifolds and holomorphic maps, consisting of all the objects $\mathbb{C}^n$ for $n \ge 0$. We will denote by $hol(n)$ the set of holomorphic maps $\mathbb{C}^n \to \mathbb{C}$, so that in the category *hol* the maps from $\mathbb{C}^n$ to $\mathbb{C}^m$ are in natural bijection with $hol(n)^m$.

By definition, a *holomorphic ring A* is an algebra over the algebraic theory *hol*, that is a product preserving functor $A : hol \to Sets$. In more concrete terms, such a holomorphic ring consists of a commutative $\mathbb{C}$-algebra $A$, endowed with extra operations: for any holomorphic function $f : \mathbb{C}^n \to \mathbb{C}$ we have an application

$$ \gamma_f : \mathbb{A}^n \to \mathbb{A}, $$

satisfying some natural properties with respect to compositions of holomorphic maps. Morally, $\gamma_f(a_1, \dots, a_n)$ stands for $f(a_1, \dots, a_n)$, the "evaluation of $f$ at the elements $a_i$". When $f$ is restricted to polynomials maps, the operations $\gamma_f$ determines a commutative $\mathbb{C}$-algebra structure on $A$ (the sum and multiplication being induced by the two holomorphic maps $\mathbb{C}^2 \to \mathbb{C}$, $(x, y) \mapsto x+y$ and $(x, y) \mapsto xy$). The typical example of a holomorphic ring is the set $\mathcal{O}_X(X)$ of (globally defined) holomorphic functions on a given complex analytic space $X$, where the operations $\gamma_f$ are actually defined by $\gamma_f(a_1, \dots, a_n) = f(a_1, \dots, a_n)$, for $a_i \in \mathcal{O}_X(X)$.
Morphisms between holomorphic rings are maps commuting with all the operations $\gamma_f$ (for all $f$ as above). Holomorphic rings thus form a category denoted by $\mathbf{CR}^h$, which comes equipped with a forgetful functor $\mathbf{CR}^h \to \mathbb{C} - \mathbf{CR}$ to commutative $\mathbb{C}$-algebras. We will often use the same notations for a holomorphic ring $A$ and its underlying $\mathbb{C}$-algebra, but in situations where the difference is important the latter will be denoted by $N(A)$. The forgetful functor $\mathbf{CR}^h \to \mathbb{C} - \mathbf{CR}$ commutes with limits and filtered colimits, and thus possesses a left adjoint, denoted by $A \mapsto A^h$, sending a $\mathbb{C}$-algebra to its "holomorphication". This left adjoint $A \mapsto A^h$ can be understood as a left Kan extension as follows. We start by the functor sending the polynomial algebra $\mathbb{C}[x_1, \dots, x_n]$ to the ring $hol(n)$ of holomorphic functions on $\mathbb{C}^n$ (with its natural holomorphic structure), and any $\mathbb{C}$-algebra morphism $\mathbb{C}[X_1, \dots, X_n] \to \mathbb{C}[Y_1, \dots, Y_m]$, defined by polynomials $P_1(Y), \dots, P_n(Y)$, to the map $hol(n) \to hol(m)$ obtained by composition with $(P_1, \dots, P_n) : \mathbb{C}^m \to \mathbb{C}^n$. We then obtain $(-)^h$ as the left Kan extension of this functor along the canonical inclusion of the category of polynomial algebras into the whole category of commutative $\mathbb{C}$-algebras. For any $\mathbb{C}$-algebra $A$ the holomorphic ring $A^h$ can thus be described by choosing a presentation of $A$ by polynomial algebras, whose image by $(-)^h$ provides a presentation by holomorphic rings $hol(n)$. More explicitly, if $A$ is finite generated, we can write $A$ as a co-equalizer

$$ (\mathbb{C}[X_1, \dots, X_n] \rightrightarrows \mathbb{C}[Y_1, \dots, Y_m]) \longrightarrow A $$

and obtain $A^h$ as the corresponding coequalizer in $\mathbf{CR}^h$

$$ (hol(n) \rightrightarrows hol(m)) \longrightarrow A^h. $$

When $A$ is not finitely presented we write $A = \text{colim}_i A_i$ as a filtered colimit of finite presented $\mathbb{C}$-algebras and we have $A^h = \text{colim}_i (A_i)^h$.

The forgetful functor $\mathbf{CR}^h \to \mathbf{CR}$ commutes with filtered colimits (and even sifted colimits), but does not commute with push-outs in general. In order to avoid confusions we will then use $\otimes^h$ to denote the categorical sum in $\mathbf{CR}^h$. For instance the push-out $hol(n) \otimes^h hol(m)$ is naturally isomorphic to $hol(n+m)$, simply because $hol(n)$ turns out to be the free holomorphic ring over $n$ generators

$$ \operatorname{Hom}_{\mathbf{CR}^h}(hol(n), A) \simeq N(A)^n. $$

However, the $\mathbb{C}$-algebras $hol(n+m)$ is certainly not isomorphic, by the canonical map of $\mathbb{C}$-algebras, to the algebraic tensor product $hol(n) \otimes hol(m)$. However, as $(-)^h$ is left adjoint, we always have a canonical isomorphism of holomorphic rings

$$ A^h \otimes^h B^h \simeq (A \otimes_{\mathbb{C}} B)^h $$

for any two commutative $\mathbb{C}$-algebras $A$ and $B$.

Any holomorphic ring $A$ possesses a module of (holomorphic) differential forms $\Omega_A^1$, which is an $A$-module with the following universal property: $A$-modules maps $\Omega_A^1 \to M$ are in one-to-one correspondence with holomorphic derivations $\delta : A \to M$. Recall here that a derivation
$\delta : A \to M$ is holomorphic if for all $f \in hol(n)$ we have the chain rule

$$ \delta(\gamma_f(a_1, \dots, a_n)) = \sum_i \frac{\partial f}{\partial z_i}(a_1, \dots, a_n) \cdot \delta(a_i). $$

It is important here to make a difference between a holomorphic ring $A$ and its underlying $\mathbb{C}$-algebra $N(A)$. Indeed, $\Omega_A^1$ is different from the usual model of Kahler differentials $\Omega_{N(A)/\mathbb{C}}^1$ of $N(A)$. There is an obvious morphism $\Omega_{N(A)/\mathbb{C}}^1 \to \Omega_A^1$ coming from the fact that the universal holomorphic derivation $A \to \Omega_A^1$ is a derivation in the usual algebraic sense, but this is not an isomorphism in general. On the other hand, if we start with a commutative $\mathbb{C}$-algebra $R$, then there exists a natural isomorphism of $R^h$-modules (which follows from the fact that the forgetful functor $\mathbf{CR}^h \to \mathbb{C} - \mathbf{CR}$ preserves trivial square zero extensions)

$$ R^h \otimes_R \Omega_{R/\mathbb{C}}^1 \simeq \Omega_{R^h}^1. $$

When $A$ is a holomorphic ring $\Omega_A^1$ will always mean the holomorphic differentials, while we will use the notation $\Omega_{N(A)}^1$ for the (non-holomorphic) Kahler differentials.

**5.1.1. The $\infty$-category of holomorphic cdga's.** The next definition is essentially similar to the notions used in [Pir15, Pri20]. Note however that in the *non-connective* case, our notion of holomorphic cdga differs slightly, as the holomorphic structure will exist on the co-homological degree 0 subalgebra (and not merely on the algebra of 0-cocyles). Note also the extra condition on the differential which does not appear in the above mentioned references. However, for connective cdga's it coincides with the notions of [Pir15, Pri20] (note that for connective cdga's, a holomorphic structure, as in Definition 5.1.1.1 below, is just a holomorphic ring structure on its degree 0 part). We will non-connective holomorphic cdga's in an essential manner, for instance in order to consider the holomorphic *de Rham complex* as a holomorphic cdga.

**DEFINITION 5.1.1.1.** (1) Let $A \in \mathbf{cdga}_{\mathbb{C}}$ be a possibly non-connective $\mathbb{C}$-linear cdga. A holomorphic structure on $A$ consists of a holomorphic ring structure on the $\mathbb{C}$-algebra $A^0$ of elements of cohomological degree 0, such that the cohomological differential $d : A^0 \to A^1$ is a holomorphic derivation. A holomorphic cdga is a $\mathbb{C}$-linear cdga endowed with a holomorphic structure.

(2) A morphism $A \to B$, between two holomorphic cdga's, is a morphism of (underlying) cdga's for which the induced morphism $A^0 \to B^0$ is a morphism of holomorphic rings.

Holomorphic cdga's form a category $cdga^h$, endowed with a forgetful functor $cdga^h \to cdga_{\mathbb{C}}$, to the category of $\mathbb{C}$-linear cdga's. This functor is the right adjoint of an adjunction, whose left adjoint is denoted by $A \to A^h$. Explicitly, for $A \in cdga_{\mathbb{C}}$, $A^h$ is defined as follows. We start by the graded $\mathbb{C}$-algebra $(A^0)^h \otimes_{A^0} A$, by extending $A^0$ to its holomorphication. The cohomological differential $d : A^0 \to A^1$ extends uniquely into a holomorphic derivation
$d^h : (A^0)^h \to (A^0)^h \otimes_{A^0} A^1$. We then define the cohomological differential $d^h$ on the whole graded algebra $(A^0)^h \otimes_{A^0} A$ by the formula

$$d^h(a \otimes x) = d^h(a).x + a.d(x),$$

for any homogenous $x \in A$ and $a \in (A^0)^h$. This defines a structure of cdga on $A^h$. Moreover, $(A^h)^0 = (A^0)^h$ comes equipped with its natural holomorphic ring structure making $A^h$ into a holomorphic cdga in the sense of the above definition.

More generally, it is possible to describe the coproducts in the category $cdga^h$ as follows. For this we start by understanding the case of $\mathbb{Z}$-graded holomorphic rings $(\mathbf{CR}^h)^{\mathbb{Z}}$. These are, by definition, graded commutative $\mathcal{C}$-algebras $A$, with the usual sign rules $xy = (-1)^{|x||y|} = yx$, together with a holomorphic structure on $A^0$. We have a corresponding notion of graded holomorphic derivations and a graded module of holomorphic differential forms $\Omega_A^1$ for $A \in (\mathbf{CR}^h)^{\mathbb{Z}}$. In particular, any holomorphic cdga $A$ can be understood via its underlying graded holomorphic algebra $A^{gr} \in (\mathbf{CR}^h)^{\mathbb{Z}}$, together with its differential given by a morphism of graded $A^{gr}$-modules

$$d : \Omega_{A^{gr}}^1 \to A^{gr}(-1),$$

where $A^{gr}(-1)$ is the weight twist by 1 of the unit graded module $A^{gr}$. With these notations coproducts in $(\mathbf{CR}^h)^h$ are easily identified by the formula

$$(A \otimes^h B)^n = \bigoplus_{p+q=n} (A^p \otimes B^q) \otimes_{A^0 \otimes B^0} (A^0 \otimes^h B^0),$$

with the usual commutative multiplicative structure. Note that by construction $(A \otimes^h B)^0 = A^0 \otimes^h B^0$ is the coproduct of holomorphic rings in weight 0.

Suppose now that $A$ and $B$ are two objects in $cdga^h$, which we represent as graded holomorphic rings $A^{gr}$ and $B^{gr}$ together with graded holomorphic derivations

$$d_A : \Omega_{A^{gr}}^1 \to A^{gr}(-1) \qquad B_A : \Omega_{B^{gr}}^1 \to B^{gr}(-1).$$

We can form the coproduct in graded rings $A^{gr} \otimes^h B^{gr}$, and the two differentials $d_A$ and $d_B$ provide for us a canonical morphism of graded $A^{gr} \otimes^h B^{gr}$-modules

$$d_A \otimes id + id \otimes d_B : \Omega_{A^{gr} \otimes^h B^{gr}}^1 \simeq (\Omega_{A^{gr} \otimes A^{gr}}^1 A^{gr} \otimes^h B^{gr}) \oplus (A^{gr} \otimes^h B^{gr} \otimes_{B^{gr}} \Omega_{B^{gr}}^1) \to (A^{gr} \otimes^h B^{gr})(-1).$$

This defines a differential on the graded holomorphic ring $A^{gr} \otimes^h B^{gr}$ which is holomorphic in degree 0, and thus defines a holomorphic cdga. This holomorphic cdga is the coproduct of $A$ and $B$ computed in the category $cdga^h$. The important point here is that the forgetful functor $cdga^h \to (\mathbf{CR}^h)^{\mathbb{Z}}$, from holomorphic cdga's to graded holomorphic rings, preserves coproducts.

The adjunction $cdga^h \stackrel{\sim}{\to} cdga_{\mathbb{C}}$ can be used in order to lift the model structure on $cdga_{\mathbb{C}}$ to a model structure on $cdga^h$, for which fibrations and equivalences are defined via the forgetful functor, as shown in the proposition below.

**PROPOSITON 5.1.1.2.** *The above notions of equivalences and fibrations make $cdga^h$ into a model category for which the forgetful functor $cdga^h \to cdga$ is a right Quillen functor.*
*Proof.* We apply the criterion of transferring model structures along a right adjoint using the path object argument (see [BM03, 2.6]). The only non-trivial statement is to prove that any object $A \in \mathrm{cdga}^h$ admits a path object. We construct this path object explicitly has follows.

We introduce the object $\mathbb{C}[\Delta^1]^h \in \mathrm{cdga}^h$ which is the image by $(-)^h$ of the **cdga** freely generated by the acyclic complex $(\mathbb{C} \xrightarrow{id} \mathbb{C})$ concentrated in cohomological degrees $[0, 1]$. It can be explicitly described as the holomorphic cdga with only two non-trivial degrees

$$ (\mathbb{C}[\Delta^1]^h)^0 = (\mathbb{C}[\Delta^1]^h)^1 = \mathrm{hol}(1) \qquad (\mathbb{C}[\Delta^1]^h)^i = 0 \quad \text{for } i \neq 0, 1 $$

and whose differential in degree 0 is given by

$$ \mathrm{hol}(1) \to \mathrm{hol}(1) $$

sending a holomorphic function $f$ to its derivative $f' = \frac{\partial f}{\partial z}$. The holomorphic cdga $\mathbb{C}[\Delta^1]^h$ admits two augmentation to $\mathbb{C}$, by simply evaluating a function $f$ at 0 and at 1

$$ ev_0, ev_1 : \mathbb{C}[\Delta^1]^h \to \mathbb{C}. $$

We claim that for any $A \in \mathrm{cdga}^h$ the induced diagram

$$ A \xrightarrow{j} A \otimes^h \mathbb{C}[\Delta^1]^h \xrightarrow{\substack{ev_0 \\ ev_1}} A $$

is a path object for $A$: $j$ is a weak equivalence, $ev_0 \times ev_1 : A \otimes^h \mathbb{C}[\Delta^1]^h \to A \times A$ is a fibration and $ev_i \circ j = id$. The last equality $ev_i \circ j = id$ is obvious and we focus on the two other properties.

As a start, we want to describe $A \otimes^h \mathbb{C}[\Delta^1]^h$ explicitly. We know that coproducts in $\mathrm{cdga}^h$ can be described on the level of the underlying graded holomorphic rings by forgetting the differential. Therefore, in degree $i$ we have

$$ (A \otimes^h \mathbb{C}[\Delta^1]^h)^i \simeq (A \otimes^h \mathrm{hol}(1))^i \oplus (A \otimes^h \mathrm{hol}(1))^{i-1}, $$

and the differential is given by

$$ \begin{pmatrix} d & \partial \\ 0 & d \end{pmatrix} : (A \otimes^h \mathrm{hol}(1))^i \oplus (A \otimes^h \mathrm{hol}(1))^{i-1} \to (A \otimes^h \mathrm{hol}(1))^{i+1} \oplus (A \otimes^h \mathrm{hol}(1))^i, $$

where $d$ the differential of $A \otimes^h \mathrm{hol}(1)$, and $\partial : (A \otimes^h \mathrm{hol}(1))^{i-1} \to (A \otimes^h \mathrm{hol}(1))^i$ is the natural morphism induced by the holomorphic derivation $\partial/\partial z : \mathrm{hol}(1) \to \mathrm{hol}(1)$ by base change to $A \otimes^h \mathrm{hol}(1)$. In other words, the underlying complex of $\mathbb{C}$-modules of $A \otimes^h \mathbb{C}[\Delta^1]^h$ is the cocone of the morphism

$$ \partial : A \otimes^h \mathrm{hol}(1) \to A \otimes^h \mathrm{hol}(1). $$

In order to prove that $j$ is a weak equivalence, it is therefore enough to show that the natural inclusion $A \to A \otimes^h \mathrm{hol}(1)$ induces a short exact sequence of complexes

$$ (22) \qquad 0 \longrightarrow A \longrightarrow A \otimes^h \mathrm{hol}(1) \stackrel{\partial}{\longrightarrow} A \otimes^h \mathrm{hol}(1) \longrightarrow 0. $$
The statement we want to prove here is idenpendant of the differential of A and we can thus work entirely with the underlying graded holomorphic rings. We start by the degree 0 part. The holomorphic ring $A^0$ can be written as a filtered colimit in $\mathbf{CR}^h$ of finitely presented holomorphic rings $A^0 = \mathrm{colim}_i B_i$. Each $B_i$ is thus the holomorphic ring of closed analytic space $X_i \subset \mathbb{C}^n$ (for some $n$ depending on $i$), given by a finite number of global holomorphic equations. For each $i$, we can identify $B_i \otimes^h \mathrm{hol}(1) \simeq \mathcal{O}(X_i \times C)$. The diagram above is then isomorphic to the following

$$0 \longrightarrow \mathrm{colim}_i \mathcal{O}(X_i) \longrightarrow \mathrm{colim}_i \mathcal{O}(X_i \times C) \xrightarrow{\partial} \mathrm{colim}_i \mathcal{O}(X_i \times C) \longrightarrow 0.$$

Moreover, $\partial : \mathcal{O}(X_i \times C) \to \mathcal{O}(X_i \times C)$ turns out to be the partial derivative along the canonical coordinate on $C$. It is then clear that the diagram above is a filtered colimit of short exact sequences and thus is an exact sequence. This shows that 22 is indeed exact in degree 0. Exactness in non-zero degrees is proven similarly. For $i$ fixed, let $M = A^i$ considered as an $A^0$-module, and we can replace $A$ by the graded holomorphic ring $A^0 \oplus M[-i]$ in order to prove that 22 is exact in degree $i$. We first reduce to the case where $A^0$ is of the form $\mathcal{O}(X)$ for some $X \subset \mathbb{C}^n$ as above. We then write $M$ as a filtered colimit of finite presented $\mathcal{O}(X)$-modules, and thus reduce to the case where $M$ is finite presented. In this case it corresponds to a well defined coherent $\mathcal{M}$ sheaf on the Stein space $X$ with a global presentation $\mathcal{O}_X^q \to \mathcal{O}_X^p \to \mathcal{M} \to 0$. The diagram 22 in degree $i$ then can be identified with

$$0 \longrightarrow \Gamma(X, \mathcal{M}) \longrightarrow \Gamma(X \times \mathbb{C}, \pi^*(\mathcal{M})) \xrightarrow{\partial} \Gamma(X \times \mathbb{C}, \pi^*(\mathcal{M})) \longrightarrow 0,$$

where $\pi : X \times \mathbb{C} \to X$ is the first projection and $\partial$ is again the partial derivative along the coordinate on $C$, which exists canonically on $\pi^*(\mathcal{M})$. The above diagram is again a short exact sequence.

We have thus shown that the sequence 22 is exact, and thus that the morphism $j : A \to A \otimes^h \mathbb{C}[\Delta^1]^h$ is indeed a weak equivalence of holomorphic cdga's. It remains to show that

$$ev_0 \times ev_1 : A \otimes^h \mathbb{C}[\Delta^1]^h \to A \times A$$

is a fibration, or equivalently that for any $i$ the induced map $(A \otimes^h \mathbb{C}[\Delta^1]^h)^i \to A^i \times A^i$ is surjective. This last map is the composition

$$(A \otimes^h \mathrm{hol}(1))^i \oplus (A \otimes^h \mathrm{hol}(1))^{i-1} \to (A \otimes^h \mathrm{hol}(1))^i \to A^i \times A^i,$$

where the first morphism is the canonical projection on the first factor, and the second map is induced by $ev_{0,1} : \mathrm{hol}(1) \to \mathbb{C} \times \mathbb{C}$ given by evaluating a function $f$ at the subset $\{0, 1\} \subset \mathbb{C}$. As the projection is obviously surjective it remains to show that

$$(A \otimes^h \mathrm{hol}(1))^i \to A^i \times A^i,$$

is always a surjective morphism for any $A \in \mathrm{cdga}^h$. Again, this is a statement idenpendant of the differential of A and thus can be proven more generally for any graded holomorphic ring A. As above, when $i = 0$ we reduce the question to the case where $A^0 = \mathcal{O}(X)$ for $X \subset \mathbb{C}^n$ globally
defined by a finite number of holomorphic equations. Then, the map under consideration is the evaluation at $X \times \{0, 1\}$

$$ \mathcal{O}(X \times \mathbb{C}) \to \mathcal{O}(X) \times \mathcal{O}(X). $$

This is surjective as a couple of functions $(f, g)$ will be the image of $z.f+(1-z).g$ where $z$ is the global coordinate on $\mathbb{C}$. The case where $i \neq 0$ is treated similarly as before, by reducing to the case where $A^0 = \mathcal{O}(X)$ and furthermore $A^i$ is a finitely presented $A^0$-module corresponding to a coherent sheaf $\mathcal{M}$ on $X$ which admits a global free resolution. The map under consideration in degree $i$ is then

$$ \Gamma(X \times \mathbb{C}, \mathcal{M}) \to \Gamma(X, \mathcal{M}) \times \Gamma(X, \mathcal{M}). $$

To prove the surjectivity of this last map we can write $\mathcal{M}$ as a quotient $\mathcal{O}_X^p \to \mathcal{M}$. As $X$ is Stein, the induced morphism $\Gamma(X, \mathcal{O}_X^p) \to \Gamma(X, \mathcal{M})$ is surjective, and thus we are reduced to the case where $\mathcal{M}$ is free which follows from the case already treated $\mathcal{M} = \mathcal{O}_X$.

This finishes the proof that $A \otimes^h \mathbb{C}[\Delta^1]^h$ is a path object in $cdga^h$, and thus by [BM03, 2.6] of the proposition. $\square$

The $\infty$-category obtained from $cdga^h$ by inverting the weak equivalences (the morphisms inducing quasi-isomorphisms on the underlying cdga's) will be denoted by $cdga^h$. The Quillen adjunction of the above proposition produces an adjunction of $\infty$-categories

$$ \mathrm{cdga}_{\mathbb{C}} \stackrel{\sim}{\Rightarrow} \mathrm{cdga}^h, $$

whose left adjoint will be denoted by $A \mapsto A^h$. The forgetful right adjoint, if necessary, will be denoted by $A \mapsto N(A)$, but most of the time it will not be written explicitly (so that we'll simply write again $A$ for the underlying cdga).

### 5.1.2. Cotangent complexes.
As a first comment, for a holomorphic cdga $A \in \mathrm{cdga}^h$ an $A$-dg-module will be, by definition, a dg-module over $N(A)$ the underlying cdga of $A$. The $\infty$-category is denoted as usual

$$ \mathrm{dg}_A := \mathrm{dg}_{N(A)}. $$

Any holomorphic cdga $A$ possesses a module of holomorphic differential forms $\Omega_A^1$. This is an $A$-dg-module such that morphisms of dg-modules $\Omega_A^1 \to M$ are in one-to-one correspondence with holomorphic derivations $A \to M$. In order to be more precise, we introduce the holomorphic trivial square zero extension $A \oplus M$. This is a holomorphic cdga whose underlying cdga is the trivial square zero extension of $N(A)$ by $M$. The holomorphic structure on $A^0 \oplus M^0$ is defined by formula

$$ \gamma_f(a_1 + m_1, \dots, a_n + m_n) = \gamma_f(a_1, \dots, a_n) + \sum_i \frac{\partial f}{\partial z_i}(a_1, \dots, a_n).m_i. $$

The holomorphic derivations from $A$ to $M$ are then defined as sections of the natural projection $A \oplus M \to A$. They can be naturally identified with derivations $d: N(A) \to M$ on the underlying
cdga which satisfies furthermore the condition that $d^0 : A^0 \to M^0$ is a holomorphic derivation from the holomorphic ring $A^0$ to the $A^0$-module $M^0$.

As in the algebraic case, the construction $A \mapsto \Omega_A^1$ can be left derived in order to produce a cotangent complex $\mathbb{L}_A$ for any $A \in \mathrm{cdga}^h$. This cotangent complex satisfies the usual properties of stability by base change and functoriality inside $\mathrm{cdga}^h$. For $A \in \mathrm{cdga}_\mathbb{C}$ we have a natural equivalence

$$A^h \otimes_A \mathbb{L}_A \simeq \mathbb{L}_{A^h},$$

as this follows by adjunction from the straight forward observation that $N(A \oplus M) \simeq N(A) \oplus M$ (i.e. trivial square zero extensions are preserved by the forgetful $\infty$-functor $\mathrm{cdga}^h \to \mathrm{cdga}_\mathbb{C}$).

Any object $A \in \mathrm{cdga}^h$ possesses a connective cover $\tau_{\le 0}A$. This is the right adjoint of the inclusion $\infty$-functor $\mathrm{cdga}_c^h \hookrightarrow \mathrm{cdga}^h$, where $\mathrm{cdga}_c^h$ stands for the full sub-$\infty$-category spanned by holomorphic cdga's whose underlying cdga are connective. The connective cover construction $A \mapsto \tau_{\le 0}A$ is moreover compatible with the forgetful $\infty$-functor to $\mathrm{cdga}$. Indeed, we simply have to notice that if $A$ is a holomorphic cdga, the kernel $\mathrm{Ker}(d : A^0 \to A^{-1}) \subset A^0$ is a sub-holomorphic ring of $A^0$ (because $d$ is a holomorphic derivation).

A connective holomorphic cdga $A \in \mathrm{cdga}_c^h$ is called a *finite cell object* if there is a finite sequence of morphisms in $\mathrm{cdga}_c^h$

$$0 = A_0 \longrightarrow A_1 \longrightarrow \dots \longrightarrow A_{n-1} \longrightarrow A_n = A,$$

where each $A_{i+1}$ is obtained from $A_i$ by choosing some elements $a_i \in A_i^{d_i}$ of degree $n_i$, and freely adding a finite number of variables $y_i$ of degrees $n_{i-1}$ with $d(y_i) = a_i$. In a slightly weaker manner, $A \in \mathrm{cdga}_c^h$ is an *almost finite cell object* if there is a countable sequence of morphisms

$$0 = A_0 \longrightarrow A_1 \longrightarrow \dots \longrightarrow A_{n-1} \longrightarrow A_n \longrightarrow \dots,$$

with $A = \mathrm{colim}_i A_i$ and where each $A_i \to A_{i+1}$ is as above but with the condition that the sequence of integers $n_i$ tends to $\infty$. In other words, an almost finite cell object is a cell object with a finite number of cell in each dimension.

**DEFINITION 5.1.2.1.** (1) A connective holomorphic cdga $A$ is of finite presentation if it is a retract, in the $\infty$-category $\mathrm{cdga}_c^h$ of a finite cell object.

(2) A connective holomorphic cdga $A$ is almost of finite presentation if it is a retract of an almost finite cell object.

Using [TV07, Prop. 2.2], it can be shown that finitely presented holomorphic cdga's are the compact objects in $\mathrm{cdga}_c^h$. Similarly, almost finitely presented holomorphic cdga's are the object $A$ for which $\mathrm{Map}(A, -)$ commutes with filtered colimits of uniformly truncated objects: for any filtered system $\{A_i\}$, with $H^j(A_i) = 0$ for all $j \le n$ (for some fixed integer $n$), we have $\mathrm{colim}_i \mathrm{Map}(A, A_i) \simeq \mathrm{Map}(A, \mathrm{colim}_i A_i)$.
**5.1.3. Derived complex analytic spaces and stacks.**

**DEFINITION 5.1.3.1.** The $\infty$-category of affine derived analytic spaces is the full sub-$\infty$-category of the opposite $\infty$-category of cdga$_c^h$ formed by holomorphic cdga's almost of finite presentation. It is denoted by $\mathrm{dAff}^h$. The object in $\mathrm{dAff}^h$ corresponding to an object $A \in \mathrm{cdga}_c^h$ will be denoted symbolically by $\mathrm{Spec}^h A$.

We note that for an affine derived analytic space $X = \mathrm{Spec}^h A$, the holomorphic ring $H^0(A)$ can be canonically identified with the ring of holomorphic functions on a closed analytic subspace $X$ in $\mathbb{C}^n$. More precisely, we can write $A$ as a connective cell object with finitely many cells in each dimension. For such a cell object, $A^0$ is a free holomorphic ring on $n$ generators (for some $n$), and thus $H^0(A)$ becomes a quotient of the holomorphic ring $hol(n) = \mathcal{O}(\mathbb{C}^n)$ by a finitely number of relations. These relations are given by elements $f_i \in hol(n)$ and thus they define a complex analytic subspace $\tau_0(X)$ inside $\mathbb{C}^n$. The holomorphic ring $H^0(A)$ is then naturally isomorphic to the holomorphic ring $\mathcal{O}_X(X)$ of holomorphic functions on $\tau_0(X)$. The complex space $\tau_0(X)$ is called the truncation of $X$. In the same manner, the cohomology groups $H^i(A)$ are finitely generated $H^0(A)$-modules, and therefore they correspond to globally generated coherent sheaves on $\tau_0(X)$ (see [Pri20, Prop. 1.24, Lem. 1.25]).

The natural transcendent topology on $\mathbb{C}$ defines a Grothendieck topology on the $\infty$-categories $\mathrm{dAff}^h$. If $i : \mathrm{Spec}^h A \to \mathrm{Spec}^h B$ is a morphism of affine derived analytic spaces, we say that $i$ is an open immersion if the induced morphism $i_0 : \mathrm{Spec}^h H^0(A) \to \mathrm{Spec}^h H^0(B)$ is an open immersion of analytic spaces, and if furthermore the natural morphisms $H^i(A) \otimes_{H^0(A)} H^0(B) \to H^i(B)$ are isomorphisms (i.e. the coherent sheaves $H^i(A)$ on $\mathrm{Spec}^h H^0(A)$ restrict to $H^i(B)$ along the open immersion $i_0$). An open covering of affine derived analytic spaces is then a family of open immersion $\{U_i \to X\}_i$ such that $\coprod_i \tau_0(U_i) \to \tau_0(X)$ is a surjective map of complex spaces.

We can also define smooth and étale morphisms, simply by using the holomorphic cotangent complexes. We say that $\mathrm{Spec}^h A \to \mathrm{Spec}^h B$ is smooth (resp. étale) if the relative cotangent complex $\mathcal{L}_{B/A}$ is a projective $B$-module of finite rank (resp. $\mathcal{L}_{B/A} \simeq 0$). We refer to [Pir15, Pri20, Por19] for more details on these notions.

We are now in a situation in which we have a Grothendieck $\infty$-site $\mathrm{dAff}^h$ together with a notion of smooth morphisms. We can thus define Artin stacks by using the formal geometricity procedure (see [TV08, Ch. 1.3], [Toe14, 3.3] and [PV21, §4]). In particular, we have the $\infty$-topos of stacks on this $\infty$-site, which will be denoted by $\mathrm{dSt}^h$ and whose objects are called *derived analytic stacks*. We also have the full sub-$\infty$-category of Artin (aka geometric) stacks inside $\mathrm{dSt}^h$.

**REMARK 5.1.3.2.** Note that, by definition, the site $\mathrm{dAff}^h$ consists only of almost finitely presented objects, and thus Artin stacks in these settings will be automatically locally almost finitely presented. In particular, for any Artin derived analytic stack $X$, its truncation $\tau_0 X$
is an (underived) Artin analytic stack locally of finite presentation, and the homotopy groups sheaves $H^i(\mathcal{O}_X)$ always define coherent sheaves on $\tau_0(X)$.

## 5.2. De Rham algebras in the holomorphic context

We introduce here a category of holomorphic $\mathbb{C}$-linear graded mixed cdga's. Its objects are graded mixed cdga's $B$ (see §1.4) together with a holomorphic structure on the weight 0 cdga $B^{(0)}$, and such that the mixed structure $\epsilon: B^{(0)} \to B^{(1)}[-1]$ is a holomorphic derivation on the holomorphic cdga $B^{(0)}$. We can endow this category with a model category structure simply by defining equivalences and fibrations on the underlying complexes $B^{(i)}$ (by forgetting the holomorphic structures).

**PROPOSITON 5.2.0.1.** The category $(\epsilon - CAlg(k)^{gr})^h$ of holomorphic graded mixed cdgas, with the above notions of fibrations and equivalences, defines a model category structure such that the forgetful functor to graded mixed cdga's is right Quillen.

*Proof.* This is similar to the proof of Proposition 5.1.1.2 (with model structure on $\epsilon - CAlg(k)^{gr}$ as in Definition 1.4.0.1), and in fact easier, since the forgetful functor

$$ (\epsilon - CAlg(k)^{gr})^h \to \epsilon - CAlg(k)^{gr} $$

commutes with push-outs. We leave the details to the reader. □

The $\infty$-category obtained from the model category of holomorphic graded mixed cdga's will be denoted by $(\epsilon - \mathbf{cdga}^{gr})^h$, and will be called the $\infty$-category of holomorphic graded mixed cdga's. For any holomorphic cdga $A$ its holomorphic de Rham algebra is defined using the holomorphic cotangent complex

$$ \mathbf{DR}(A) = \mathrm{Sym}_A(\mathbb{L}_A[1]). $$

It is first a graded mixed cdga where the mixed structure is the de Rham differential. But, the de Rham differential being obviously holomorphic, $\mathbf{DR}(A)$ is an object in $(\epsilon - \mathbf{cdga}^{gr})^h$. We can state things differently, by considering the $\infty$-functor $(\epsilon - \mathbf{cdga}^{gr})^h \to \mathbf{cdga}^h$, sending $B$ to $B^{(0)}$, and observe that it has a left adjoint given by $B \to \mathbf{DR}(B)$. This adjunction of $\infty$-categories is moreover induced by a Quillen adjunction on the level of model categories. Of this adjunction we retain the following property

**PROPOSITON 5.2.0.2.** Let $A$ be a holomorphic cdga, and $\mathbf{DR}(A)$ its holomorphic de Rham complex as an object in $(\epsilon - \mathbf{cdga}^{gr})^h$. For any $B \in (\epsilon - \mathbf{cdga}^{gr})^h$, the morphism induced on mapping spaces

$$ \mathbf{Map}_{(\epsilon - \mathbf{cdga}^{gr})^h}(\mathbf{DR}(A), B) \to \mathbf{Map}_{\mathbf{cdga}^h}(A, B^{(0)}) $$

is an equivalence.

**REMARK 5.2.0.3.** There is of course a big difference between $\mathbf{DR}(A)$ and $\mathbf{DR}(N(A))$, as already the holomorphic cotangent complex $\mathbb{L}_A$ differs from the algebraic cotangent complex $\mathbb{L}_{N(A)}$.
We will also make use of a second adjunction between holomorphic graded mixed cdga's and holomorphic cdga's. Namely, we consider the $\infty$-functor $A \mapsto A(0)$, sending a holomorphic cdga $A$ to the holomorphic graded mixed cdga $A(0)$ consisting of $A$ sitting purely in weight 0 and endowed with the trivial mixed structure. This defines an $\infty$-functor $\mathrm{cdga}^h \to (\epsilon - \mathrm{cdga}^{gr})^h$ which commutes with colimits, and so it admits a right adjoint. This right adjoint is called the *holomorphic realization* and is denoted by

$$|-| : (\epsilon - \mathrm{cdga}^{gr})^h \to \mathrm{cdga}^h.$$

**PROPOSITON 5.2.0.4.** For any $B \in (\epsilon - \mathrm{cdga}^{gr})^h$, the underlying cdga $N(|B|)$ associated to $|B|$ is canonically equivalent to $|N(B)|$, the realization (as in Remark 1.3.4.5) of the underlying graded mixed cdga $N(B)$ associated to $B$.

*Proof.* This is clear by adjunction and from the fact that $A(0)^h$ obviously identifies with $A^h(0)$. For $A \in \mathrm{cdga}_c$, we have a chain of natural equivalences

$$ \begin{align*} \mathrm{Map}_{\mathrm{cdga}_c}(A, N(|B|)) &\simeq \mathrm{Map}_{\mathrm{cdga}^h}(A^h, |B|) \simeq \mathrm{Map}_{(\epsilon - \mathrm{cdga}^{gr})^h}(A(0)^h, B) \\ &\simeq \mathrm{Map}_{\epsilon - \mathrm{cdga}^{gr}}(A(0), N(B)) \simeq \mathrm{Map}_{\mathrm{cdga}_c}(A, |N(B)|). \end{align*} $$

□

**DEFINITION 5.2.0.5.** Let $A$ be a (connective) holomorphic cdga and $\mathbf{DR}(A)$ its de Rham holomorphic graded mixed cdga. The holomorphic derived de Rham complex of $A$ is defined by $\hat{C}_{\mathbf{DR}}^*(A) := |\mathbf{DR}(A)| \in \mathrm{cdga}^h$.

**REMARK 5.2.0.6.** (1) There is discrepancy between the notation of Definition 5.2.0.5 and Definition 1.4.2.1, as the former involved also a filtration. To be perfectly coherent we should have used instead the notation $\hat{C}_{\mathbf{DR}}^*(A)^u$. However, we will not use the Hodge filtration in the holomorphic context, so will keep the notation $\hat{C}_{\mathbf{DR}}^*(A)$ as above. It is clear that Definition 5.2.0.5 also has a refined filtered version, using the natural notion of a *filtered holomorphic cdga*. We will not pursue is that direction and leave these details to the interested readers.

(2) In most cases we will only consider the underlying cdga of $\hat{C}_{\mathbf{DR}}^*(A)$ and will not use often its natural holomorphic structure. However, this structure will be useful for some specific arguments (see for instance the proof of Theorem 5.4.0.1). Because of this, we will still denote by $\hat{C}_{\mathbf{DR}}^*(A)$ the underlying cdga and simply call it the derived de Rham complex of $A$.

## 5.3. Analytic derived foliations

For any affine derived analytic space $\mathrm{Spec}^h A$, we remind that we have a holomorphic graded mixed cdga $\mathbf{DR}(A)$ whose underlying graded cdga is $\mathrm{Sym}_A(\mathbb{L}_A[1])$, and for which the
mixed structure in given by the de Rham differential. The construction $A \mapsto \mathbf{DR}(A)$ defines an $\infty$-functor

$$ \mathbf{DR}(-) : (\mathbf{dAff}^h)^{op} \to (\epsilon - \mathbf{cdga}^{gr})^h. $$

We then consider the $\infty$-functor

$$ \mathcal{Fol}^{pr} : (\mathbf{dAff}^h)^{op} \to \mathbf{Cat}_{\infty}, $$

sending $X = \mathbf{Spec}^h A \in \mathbf{dAff}^h$ to the $\infty$-category of holomorphic graded mixed $\mathbf{DR}(A)$-cdga's $B$ such that the following two conditions are satisfied

(1) the natural morphism $A \to B^{(0)}$ is a quasi-isomorphism

(2) $B^{(1)}$ is an almost perfect complex of $A$-dg-modules

(3) the natural morphism of graded cdga's

$$ \mathrm{Sym}_A(B^{(1)}) \to B $$

is a graded quasi-isomorphism.

As opposed to the algebraic situation, the $\infty$-functor $\mathcal{Fol}^{pr} : (\mathbf{dAff}^h)^{op} \to \mathbf{Cat}_{\infty}$ is only a prestack i.e. it does not satisfy descent. Indeed, if $X \subset \mathbb{C}^n$ is a closed analytic subspace, globally defined by a finite number of equations, and with holomorphic ring of functions $A$, then almost perfect $A$-modules correspond to almost perfect complexes on $X$ than can be written as bounded complex of globally generated vector bundles on $X$. Of course, these are not all the almost perfect complexes on $X$. The associated stack $\mathcal{Fol}$ to $\mathcal{Fol}^{pr}$ can however be computed easily. For this, we sheafify everything on $X$. We first consider the small site of affine opens $U \subset X$. The holomorphic ring $A$ sheafifies to a sheaf of holomorphic cdga's $\mathcal{O}_X$. We next consider $\mathbf{DR}_X$ the sheaf $(U = \mathbf{Spec}^h B \subset X) \mapsto \mathbf{DR}(B)$ of holomorphic graded mixed cdga's on the underlying topological space of $X$. The objects in $\mathcal{Fol}(X)$ can then be described as sheaves of holomorphic graded mixed cdga's $B$, together with a morphism $\mathbf{DR}_X \to B$ in $(\epsilon - \mathbf{cdga}^{gr})^h(X)$ such that

(a) the natural morphism $\mathcal{O}_X \to B^{(0)}$ is a quasi-isomorphism of sheaves of cdga's on $X$

(b) $B^{(1)}$ is an almost perfect complex of $\mathcal{O}_X$-dg-modules

(c) the natural morphism of sheaves of graded cdga's on $X$

$$ \mathrm{Sym}_{\mathcal{O}_X}(B^{(1)}) \to B $$

is a quasi-isomorphisms of sheaves of graded complexes.

By definition $\mathcal{Fol}(X)$ is a full sub-$\infty$-category of the opposite of $\mathbf{DR}_X/(\epsilon - \mathbf{cdga}^{gr})^h(X)$ of sheaves of holomorphic graded mixed cdga's under $\mathbf{DR}_X$. We thus get an $\infty$-functor $\mathcal{Fol} : (\mathbf{dAff}^h)^{op} \to \mathbf{Cat}_{\infty}$ satisfying the descent property can be left Kan extended to a colimit preserving $\infty$-functor on the $\infty$-category of all derived analytic stacks

$$ \mathcal{Fol} : (\mathbf{dSt}^h)^{op} \to \mathbf{Cat}_{\infty}. $$
**DEFINITION 5.3.0.1.** For a derived analytic stack $X \in \mathrm{dSt}^h$, the $\infty$-category of derived foliations on $X$ is $\mathcal{F}ol(X)$. For a morphism of derived analytic stacks $f : X \to Y$ the induced $\infty$-functor $\mathcal{F}ol(Y) \to \mathcal{F}ol(X)$ is called the pull-back of derived foliations and is denoted by $f^*$.

**REMARK 5.3.0.2.** By definition derived foliations in the analytic settings are always almost perfect by assumptions. This is only because we want to avoid technical complications related to the notion of quasi-coherent sheaves in the analytic setting, and moreover all the examples treated in this book will be almost perfect.

We finish this section by mentioning that most of the general results and notions we have seen for derived foliations on derived stacks remains valid in the analytic setting, possibly with mild modifications. As a start, if $\mathcal{F} \in \mathcal{F}ol(X)$ is a derived foliation on a derived analytic stack $X \in \mathrm{dSt}^h$, we can define its fake (or big) cotangent complex $\mathbb{L}_{\mathcal{F}}$, which is an $\mathcal{O}_X$-module, simply by sending $u : S := \mathrm{Spec}^h A \to X$ to the $\mathcal{O}_S$-module $\mathbb{L}_{u^*(\mathcal{F})}$. Here, $u^*(\mathcal{F}) \in \mathcal{F}ol(\mathrm{Spec}^h A)$ and can thus be represented by a sheaf of graded mixed $\mathrm{DR}_S$-algebras $\mathrm{DR}_{u^*(\mathcal{F})}$ satisfying conditions (a), (b), (c) above. The $\mathcal{O}_S$-module $\mathbb{L}_{u^*(\mathcal{F})}$ is then by definition $\mathrm{DR}^{(1)}_{u^*(\mathcal{F})}[-1]$, the (shifted) weight one piece of $\mathrm{DR}_{u^*(\mathcal{F})}$, which by assumption is an almost perfect $\mathcal{O}_S$-module. We get this way a family of $\mathcal{O}_S$-modules, functorial in $S \to X$, or in other words an $\mathcal{O}_X$-module denoted by $\mathbb{L}_{\mathcal{F}}^{\mathrm{big}}$. It comes equipped with a canonical morphism $\mathbb{L}_X^{\mathrm{big}} \to \mathbb{L}_{\mathcal{F}}^{\mathrm{big}}$.

A first observation is that the cone of this morphism is an almost perfect $\mathcal{O}_X$-modules. Indeed, this is due to the fact that conormal complexes of derived foliations are stable by pull-backs. We denote this cone by $\mathcal{N}_{\mathcal{F}}^*$. Suppose now that $X$ is a derived analytic Artin stack, and let us consider its cotangent complex $\mathbb{L}_X$, which comes equipped with its canonical morphism $\mathbb{L}_X^{\mathrm{big}} \to \mathbb{L}_X$.

**DEFINITION 5.3.0.3.** Let $X$ be an analytic derived Artin stack and $\mathcal{F} \in \mathcal{F}ol(X)$ be a derived foliation on $X$. The cotangent complex of $\mathcal{F}$, $\mathbb{L}_{\mathcal{F}}$, is defined by the cartesian square of $\mathcal{O}_X$-modules

<figure><img src="image_30.png" /></figure>

We note that the cone of $\mathbb{L}_X \to \mathbb{L}_{\mathcal{F}}$ is again $\mathcal{N}_{\mathcal{F}}^*$, that we have seen to be almost perfect. Therefore, $\mathbb{L}_{\mathcal{F}}$ is itself an almost perfect $\mathcal{O}_X$-module. Also, from the definition it is easy to deduce the functorial nature of cotangent complexes. For a morphism $f : X \to Y$ of analytic derived Artin stacks, and $\mathcal{F} \in \mathcal{F}ol(Y)$, we have a Cartesian square of almost perfect $\mathcal{O}_Y$-modules

<figure><img src="image_31.png" /></figure>
Finally, we write $\mathbf{DR}_F$ for the sheaf of holomorphic graded mixed cdga's associated to $\mathcal{F} \in \mathcal{F}ol(X)$, which is defined on the big site $\mathbf{dAff}^h/X$ of affine derived analytic spaces over $X$. We denote by $\hat{C}_{DR}^*(\mathcal{F}) = |\mathbf{DR}_F|$ its realization in the sense of Definition 5.2.0.5, which is a sheaf of holomorphic cdga's on $X$.

### 5.4. Analytic integrability and existence of derived enhancements

The following local integrability result is specific to the analytic setting, and is a version of the classical Frobenius theorem for transversely smooth and rigid derived foliations. It is a direct consequence of a result of Malgrange combined with the analytic version of Corollary 2.3.4.6. However, we still display the result as a theorem to insist on its importance.

As in the algebraic setting, we will say that a derived foliation $\mathcal{F} \in \mathcal{F}ol(X)$ on a derived analytic space $X$ is analytically integrable, if there exists a morphism $f: X \to Y$ to another derived analytic space $Y$ such that $\mathcal{F} \simeq f^*(0_Y)$, where $0_Y \in \mathcal{F}ol(Y)$ is the initial foliation on $Y$. When such an $f$ only exists locally on $X$ we say that $\mathcal{F}$ locally analytically integrable.

**THEOREM 5.4.0.1.** Let $X$ be smooth Deligne-Mumford analytic stack, and $\mathcal{F} \in \mathcal{F}ol(X)$ be derived foliation. We suppose that the following two conditions are satisfied.

(1) The derived foliation $\mathcal{F}$ is transversely smooth and rigid.

(2) The derived foliation is smooth in codimension 2.

The, locally on $X$, $\mathcal{F}$ is integrable.

**Proof.** The result being local on $X$ we can assume that $X$ is affine and we will allow to shrink it furthermore when necessary. For any point $x \in X$, we consider the formal completion $\hat{X}_x$ of $X$ at $x$, and the restriction $\hat{\mathcal{F}}_x$ of $\mathcal{F}$ on $\hat{X}_x$. We know by our results on formal integrability that $\hat{\mathcal{F}}_x$ is formally integrable (see Corollary 2.3.4.6).

We now consider the truncation $K$ of $\mathcal{F}$ in the sense of Definition 2.4.1.2. This truncation is a sheaf of holomorphic graded mixed cdga of the form $\text{Sym}_{\mathcal{O}_X}^{\text{naive}}((\Omega_X^1/K)[1])$, where $K \subset \Omega_X^1$ is the differential ideal defined as the kernel of the morphism $\Omega_X^1 \to \text{H}^0(\mathbb{L}_{\mathcal{F}})$. The sub-sheaf $K$ is also the image of $\mathcal{N}_{\mathcal{F}}^* \to \Omega_X^1$. By generic smoothness the morphism between vector bundles $\mathcal{N}_{\mathcal{F}}^* \to \Omega_X^1$ must be a monomorphism as it is a monomorphism on the dense open $U$. Therefore $K$ is, as an abstract $\mathcal{O}_X$-module isomorphic to $\mathcal{N}_{\mathcal{F}}^*$. Shrinking $X$ is necessary we can assume that $\mathcal{N}_{\mathcal{F}}^*$ is a free vector bundle, and thus that $K \simeq \mathcal{O}_X^d$. The image of the basis elements of $\mathcal{O}_X^d$ in $\Omega_X^1$ defines differential forms $(\omega_1, \dots, \omega_d)$ on $X$ which are linearly independent over $\mathcal{O}_X$. We can thus apply [Mal77, Thm. 3.1, B] to the forms $(\omega_1, \dots, \omega_d)$ and get that (possibly after localizing on $X$), there exists a holomorphic map $f: X \to S$, where $S$ is another complex manifold (that can be chosen to be an open in $\mathbb{C}^d$), such that $f$ integrates $K: f^*(\Omega_S^1) = K \subset \Omega_X^1$. We now claim that $f$ also integrates $\mathcal{F}$ in the sense that $\mathcal{F}$ is isomorphic to $f^*(0_S)$ as derived foliations on $X$.

For this, we consider $\mathcal{O}_{\mathcal{F}} \subset \mathcal{O}_X$ the subring of functions which are killed by the relative de Rham differential $d: \mathcal{O}_X \to \Omega_X^1/K$. This is also the 0-th naive de Rham cohomology group of the differential ideal $K$, denoted by $\text{H}_{\text{dR,naive}}^0(K)$. By our assumptions on the codimension of
the singularities of $\mathcal{F}$, we know that the natural morphism from derived de Rham cohomology along $\mathcal{F}$ to naive de Rham cohomology of $K$ induces an isomorphism of sheaves of holomorphic rings (see 2.4.2.5)

$$H_{dR}^{0}(\mathcal{F}) \to H_{dR,naive}^{0}(K) = \mathcal{O}_{\mathcal{F}}.$$

In particular, as $\hat{C}_{DR}^{*}(\mathcal{F}) = |\mathbf{DR}_{\mathcal{F}}|$ is the realization of $\mathbf{DR}_{\mathcal{F}}$, and because quasi-smoothness implies that $H^{i}(|\mathbf{DR}_{\mathcal{F}}|) \simeq 0$ for all $i < 0$, we have a canonical morphism of sheaves of holomorphic cdga's

$$\mathcal{O}_{\mathcal{F}} \simeq H_{dR}^{0}(\mathcal{F}) \to |\mathbf{DR}_{\mathcal{F}}|,$$

which we can compose with the adjunction morphism $|\mathbf{DR}_{\mathcal{F}}| \to \mathbf{DR}_{\mathcal{F}}$ where the sheaf of holomorphic cdga $|\mathbf{DR}_{\mathcal{F}}|$ is considered with its trivial graded mixed structure (remind that $|-|$ is right adjoint to the $\infty$-functor remembering only the weight 0 piece, see after Proposition 5.2.0.2).

**LEMMA 5.4.0.2.** There exists a commutative square of sheaves of holomorphic graded mixed cdga's on $X$

<figure><img src="image_32.png" /></figure>

where $\mathcal{O}_{\mathcal{F}} \to \mathbf{DR}_{\mathcal{F}}$ is the natural morphism described above, and $\mathbf{DR}_{\mathcal{O}_{\mathcal{F}}} \to \mathbf{DR}_{X}$ the morphism induced from the inclusion of sheaves of holomorphic rings $\mathcal{O}_{\mathcal{F}} \to \mathcal{O}_{X}$.

*Proof of the lemma.* The square of the lemma provides two possible composed morphisms $\mathbf{DR}_{\mathcal{O}_{\mathcal{F}}} \to \mathbf{DR}_{\mathcal{F}}$. These are morphisms of holomorphic graded mixed cdga's. By the universal property of $\mathbf{DR}_{\mathcal{O}_{\mathcal{F}}}$ (see Proposition 5.2.0.2), in order construct a homotopy between these two morphisms it is enough to see that they induce the homotopic morphisms of holomorphic rings in weight 0. But, by construction, these two morphisms in weight 0 both coincide with the canonical inclusion $\mathcal{O}_{\mathcal{F}} \to \mathcal{O}_{X}$ and thus are homotopic in a uniquely possible manner (the space of self homotopies of this inclusion is contractible). $\square$

By the previous lemma, we have constructed a natural morphism of sheaves of holomorphic graded mixed cdga's on $X$

$$\mathbf{DR}_{\mathcal{O}_{X}/\mathcal{O}_{\mathcal{F}}} = \mathbf{DR}_{X} \otimes_{\mathbf{DR}(\mathcal{O}_{\mathcal{F}})} \mathcal{O}_{\mathcal{F}} \to \mathbf{DR}_{\mathcal{F}}.$$

We claim that this morphism is an equivalence. Indeed, it is enough to check that it induces an equivalence on the associated graded, or equivalently (by multiplicativity) on weight 1. Now, in weight 1 the above morphism is the canonical morphism $\mathbb{L}_{\mathcal{O}_{X}/\mathcal{O}_{\mathcal{F}}} \to \mathbb{L}_{\mathcal{F}}$, where $\mathbb{L}_{\mathcal{O}_{X}/\mathcal{O}_{\mathcal{F}}}$ is the holomorphic relative cotangent complex of $\mathcal{O}_{\mathcal{F}} \subset \mathcal{O}_{X}$, and the morphism above is induced by the holomorphic derivation $\mathcal{O}_{X} \to \mathbf{DR}_{\mathcal{F}}^{(1)}$ which vanishes on $\mathcal{O}_{\mathcal{F}}$. To see that this morphism $\mathbb{L}_{\mathcal{O}_{X}/\mathcal{O}_{\mathcal{F}}} \to \mathbb{L}_{\mathcal{F}}$ is an equivalence we can work locally on $X$, and thus assume that the differential ideal $K$ is integrable by a morphism $f: X \to S$ with $S$ a smooth manifold. In this case, [Mal77,
Thm. 2.1.1] shows that $\mathcal{O}_F \simeq f^{-1}(\mathcal{O}_S)$, and thus $\mathbb{L}_{\mathcal{O}_X/\mathcal{O}_F}$ is identified with the coherent sheaf $\Omega_f^1$ of relative 1-forms on $X$ over $S$. The fact that $\Omega_f^1 \simeq \mathbb{L}_F$ then follows from the very definition of the fact that $f$ integrates the differential ideal $K$.

To summarize, we have seen that under the hypothesis of the theorem $\mathbf{DR}_F$ is of the form $\mathbf{DR}_{\mathcal{O}_X/\mathcal{O}_F}$. Moreover, locally on $X$, $\mathbf{DR}_{\mathcal{O}_X/\mathcal{O}_F}$ is the relative de Rham algebra $\mathbf{DR}_{X/S}$ for a morphism $f: X \to S$ integrating the differential ideal $K$. This shows, in particular, that $f$ also integrates $\mathcal{F}$ as claimed. $\square$

Also the following two existence and uniqueness results are specific to the analytic setting and does not have an algebraic counterpart (see however Corollary 5.5.0.8 in the proper case). They are very close to Theorem 5.4.0.1, and they are again consequences of the results of [Mal77].

**PROPOSITON 5.4.0.3.** Let $X$ be a smooth complex manifold and $K \subset \Omega_X^1$ be a differential ideal such that $K$ is a vector bundle.

(1) If the coherent sheaf $\Omega_X^1/K$ is a vector bundle in codimension 2, then there exists at most one, up to equivalence, transversely smooth and rigid derived enhancement for $K$.

(2) If the coherent sheaf $\Omega_X^1/K$ is a vector bundle in codimension 3, then there exists a unique, up to equivalence, transversely smooth and rigid derived enhancement for $K$.

**Proof.** (1) This is a consequence of the proof of Theorem 5.4.0.1. Indeed, in that proof we have shown that if $\mathcal{F} \in \mathcal{Fol}(X)$ is a transversely smooth and rigid derived enhancement of $K$, then we must have $\mathbf{DR}_F \simeq \mathbf{DR}_{\mathcal{O}_X/\mathcal{O}_F}$, for $\mathcal{O}_F \subset \mathcal{O}_X$ the holomorphic subring of functions killed by the de Rham differential $d: \mathcal{O}_X \to \Omega_X^1/K$. This shows indeed that $K$ determines $\mathcal{F}$ and thus implies the uniqueness of $\mathcal{F}$.

(2) The point (1) already insures unicity, so it remains to prove existence. By unicity it is enven enough to prove existence locally on $X$. The argument is then the same as in the proof of Theorem 5.4.0.1. We let $\mathcal{O}_F \subset \mathcal{O}_X$ be the holomorphic subring of functions killed by the de Rham differential $d: \mathcal{O}_X \to \Omega_X^1/K$. We set $\mathbf{DR}_F := \mathbf{DR}_{\mathcal{O}_X/\mathcal{O}_F}$ be the relative holomorphic de Rham algebra of the inclusion $\mathcal{O}_F \subset \mathcal{O}_X$. It is endowed with the canonical morphism $\mathbf{DR}_X \to \mathbf{DR}_F$, and we claim that as such it is a transversely smooth and rigid derived enhancement of $K$. For this we use case (A) of [Mal77, Thm. 3.1] which insures that $K$ is locally integrable. During the proof of Theorem 5.4.0.1 we have seen that $\mathbf{DR}_F$ is locally of the form $\mathbf{DR}_{X/S}$ for some morphism $f: X \to S$ with $S$ smooth. This clearly implies that the sheaf of graded mixed $\mathbf{DR}_X$-cdga $\mathbf{DR}_F$ satisfies the conditions of being a transversely smooth and rigid derived foliation and that the corresponding differential ideal is equal to $K$. $\square$
## 5.5. Analytification of derived foliations: the GAGA theorem

We consider the full sub-∞-category $dSt_C^{aft} \subset dSt_C$, consisting of derived stacks which are locally almost of finite presentation over $\mathbb{C}$. These are the derived stacks that can be obtained as colimits of objects of the form Spec $A$ with $A$ a connective cdga which is almost of finite presentation over $\mathbb{C}$. The canonical embedding $dSt_C^{aft} \subset dSt_C$ can also be identified with the left Kan extension of the Yoneda embedding restricted to derived affine schemes locally almost of finite presentation $dAff_C^{aft} \subset dAff_C \subset dSt_C$. In particular, an alternative description of $dSt_C^{aft}$ is as the ∞-category of (hypercomplete) stacks on the étale ∞-site $dAff_C^{aft}$.

The analytification $\infty$-functor $cdga_{c,\mathbb{C}}^{aft} \to cdga_c^h$ of §5.1 induces an analytification $\infty$-functor

$$(-)^h : dSt_C^{aft} \to dSt^h$$

from the $\infty$-category of derived stacks locally almost of finite presentation to the $\infty$-category of derived analytic stacks. It is obtained as the left Kan extension of the analytification $\infty$-functor followed by the Yoneda embedding

$$dAff_C^{aft} \to dAff^h \to dSt^h$$

where the leftmost $\infty$-functor is the (opposite of the) analytification functor of §5.1, and it sends $\mathbf{Spec} A$ to $\mathbf{Spec}^h A^h$, where $A^h$ is the holomorphic cdga generated by $A$. The analytification $\infty$-functor $(-)^h : dSt_C^{aft} \to dSt^h$ is the left adjoint of the restriction $\infty$-functor, sending $X \in dSt^h$ to the derived stack defined by $A \mapsto X(A^h)$. This restriction will be denoted by $r$ so that we have an adjunction

$$(-)^h : dSt_C^{aft} \stackrel{\sim}{\Rightarrow} dSt^h : r.$$

Moreover, the right adjoint $r$ is a geometric morphism of $\infty$-topoi, or equivalently $(-)^h$ commutes with finite limits. To avoid confusions with the various $(-)^h$ notations involved (for instance for graded mixed cdga's), we will denote this adjunction by

$$u^{-1} : dSt_C^{aft} \stackrel{\sim}{\Rightarrow} dSt^h : u_*.$$

Remind that to both $\infty$-topoi $dSt_C^{aft}$ and $dSt^h$ are associated canonical stacks (in $\infty$-categories) of almost perfect derived foliations, denoted by $\mathcal{F}ol^{ap}$ in the algebraic setting (see Definition 2.1.4.5), and simply by $\mathcal{F}ol$ in the holomorphic setting. These can be compared by means of the analytification $\infty$-functor as shown in the following proposition.

**PROPOSITON 5.5.0.1.** There exists a canonical morphism of derived analytic stacks

$$u^{-1}(\mathcal{F}ol^{ap}) \to \mathcal{F}ol.$$

*Proof.* By adjunction it is enough to construct a morphism $\mathcal{F}ol^{ap} \to u_*(\mathcal{F}ol)$. For an object $\mathbf{Spec} A \in dAff_C^{aft}$, we consider the analytification construction $(-)^h : \mathcal{F}ol^{ap}(X) \to \mathcal{F}ol(X^h)$, sending a graded mixed cdga $B$ with $B^{(0)} \simeq A$ to $B^h$, which is a holomorphic graded mixed cdga with $(B^h)^{(0)} \simeq A^h$ satisfying the conditions to define a holomorphic derived foliations on
$X^h$. This is clearly functorial in $A$, and thus defines a morphism $\mathcal{F}ol^{ap} \to u_*(\mathcal{F}ol)$ as required. $\square$

The above proposition is needed in order to construct the analytification $\infty$-functor for derived foliations. Indeed, for $X \in \mathbf{dSt}_C^{aft}$, we define an $\infty$-functor

$$(-)^h : \mathcal{F}ol^{ap}(X) \to \mathcal{F}ol(X^h)$$

by means of the analytification $\infty$-functor followed by the morphism of proposition

$$\mathcal{F}ol^{ap}(X) \simeq \mathrm{Map}(X, \mathcal{F}ol^{ap}) \to \mathrm{Map}(X^h, u^{-1}(\mathcal{F}ol)) \to \mathrm{Map}(X^h, \mathcal{F}ol) \simeq \mathcal{F}ol(X^h).$$

**DEFINITION 5.5.0.2.** For $X \in \mathbf{dSt}_C^{aft}$, the $\infty$-functor defined above

$$(-)^h : \mathcal{F}ol^{ap}(X) \to \mathcal{F}ol(X^h)$$

is called the analytification $\infty$-functor for derived foliations.

It is easy to check that $(-)^h$ extends to a morphism of stacks $\mathcal{F}ol^{ap}(-) \to \mathcal{F}ol(-) \circ (-)^h$. i.e. it is functorial for pullbacks of derived foliations: for any $f : X \to Y$ in $\mathbf{dSt}_C^{aft}$, there is an induced $\infty$-functor $f^* : \mathcal{F}ol(Y^h) \to \mathcal{F}ol(X^h)$, compatible with composition.

For the next proposition, remind from [Por19] the existence of an analytification $\infty$-functor for almost perfect complexes

$$(-)^h : \mathbf{APerf}(X) \to \mathbf{APerf}(X^h),$$

functorially in $X \in \mathbf{dSt}_k^{aft}$.

**PROPOSITON 5.5.0.3.** For $X \in \mathbf{dSt}_C^{aft}$ a derived Artin stack locally of finite presentation, the analytification functor $(-)^h : \mathbf{APerf}(X) \to \mathbf{APerf}(X^h)$ preserves cotangent complexes of derived foliations, i.e. for any $\mathcal{F} \in \mathcal{F}ol^{ap}(X)$, we have a canonical isomorphism

$$(\mathbb{L}_{\mathcal{F}})^h \simeq \mathbb{L}_{\mathcal{F}^h}$$

in $\mathbf{APerf}(X^h)$.

*Proof.* This is an easy consequence of [PY20, Thm. 5.21]. $\square$

The following result is our version of GAGA theorem for (almost perfect) derived foliations.

**THEOREM 5.5.0.4.** Let $X$ be a proper algebraic Deligne-Mumford stack. Then the $\infty$-functor

$$(-)^h : \mathcal{F}ol^{ap}(X) \to \mathcal{F}ol(X^h)$$

from almost perfect algebraic derived foliations on $X$ to almost perfect holomorphic derived foliations on $X^h$, is an equivalence of $\infty$-categories.

*Proof.* We start by enlarging quite a bit the $\infty$-categories $\mathcal{F}ol^{ap}(X)$ and $\mathcal{F}ol(X^h)$. For this, let $C_X$ be the $\infty$-category of sheaves of graded mixed cdga's $B$ on $X_{\acute{e}t}$ together with an augmentation $B \to \mathcal{O}_X$ (of graded mixed cdga's for the trivial graded mixed structure on $\mathcal{O}_X$) and satisfying the following conditions.
(1) The augmentation $B \to \mathcal{O}_X$ induces an equivalence on weight zero parts $B^{(0)} \simeq \mathcal{O}_X$.

(2) The negative weight pieces of $B$ vanish: $B^{(i)} \simeq 0$ for all $i < 0$.

(3) For all $i$, the $\mathcal{O}_X$-module $B^{(i)}$ is almost perfect.

Similarly, we define $C_{X^h}$ the $\infty$-category of sheaves of holomorphic graded mixed cdga's $B$ on $X_{et}^h$ together with an augmentation $B \to \mathcal{O}_{X^h}$ (of holomorphic graded mixed cdga's for the trivial graded mixed structure on $\mathcal{O}_{X^h}$) and satisfying the following conditions.

(1) The augmentation $B \to \mathcal{O}_{X^h}$ induces an equivalence on weight zero parts $B^{(0)} \simeq \mathcal{O}_{X^h}$.

(2) The negative weight pieces of $B$ vanish: $B^{(i)} \simeq 0$ for all $i < 0$.

(3) For all $i$, the $\mathcal{O}_{X^h}$-module $B^{(i)}$ is almost perfect.

As a first step, we observe the following finiteness result for cotangent complexes.

**LEMMA 5.5.0.5.** Let $B \in C_X$ (resp. $B \in C_{X^h}$), then the graded cotangent complex $\mathbb{L}_B$ is such that each individual weight piece $\mathbb{L}_B^{(i)}$ is almost perfect over $\mathcal{O}_X$ and is zero for $i < 0$.

*Proof of Lemma 5.5.0.5.* We give the proof for objects in $C_X$, the holomorphic case can be treated similarly.

First of all, this is a statement about the underlying graded cdga so we can forget the mixed structure. We construct a "graded cell decomposition" of $B$ in the usual way, for which cells in a given dimension will be parametrized by an almost perfect $\mathcal{O}_X$-modules. We construct, by induction, a sequence of sheaves of graded $\mathcal{O}_X$-cdga's

$$C(0) = \mathcal{O}_X \longrightarrow C(1) \longrightarrow \dots \longrightarrow C(n) \longrightarrow C(n+1) \longrightarrow \dots \longrightarrow B$$

such that $C(n) \to B$ induces an equivalence in weights less or equal to $n$, and for each $n$ there exists a push-out square of sheaves of graded cdga's

$$\begin{tikzcd}
(23) \arrow[r] \arrow[d] & C(n) \arrow[r] \arrow[d] \arrow[ul] & C(n+1) \arrow[ul] \\
& \operatorname{Sym}_{\mathcal{O}_X}(E(n)) \arrow[r] \arrow[ul] & \mathcal{O}_X
\end{tikzcd}$$

with $E(n)$ an almost perfect $\mathcal{O}_X$-module pure of weight $n+1$. Assuming that $C(n) \to B$ has been constructed, we consider the induced morphism $C(n)^{(n+1)} \to B^{(n+1)}$ on weight $n+1$ pieces and we denote by $E(n)$ is homotopy fiber, which is a graded $\mathcal{O}_X$-module pure of weight $(n+1)$. The graded $\mathcal{O}_X$-cdga $C(n+1)$ is then defined by the push-out above. Because the morphism $K(n) \to B$ is canonically homotopic to zero, the morphism $C(n) \to B$ factors canonically through $C(n+1) \to B$. Finally, $C(n)$ and $C(n+1)$ do coincide in weight less than $n$, so $C(n+1) \to B$ induces an equivalence in weights less than $n$. Moreover, the weight $n$ part of $C(n+1)$ is by definition equivalent to the cone of $E(n) \to C(n)^{(n+1)}$, and thus is sent by an equivalence to $B^{(n+1)}$ as required.

The existence of the sequence $C$ does imply the lemma, as $\mathbb{L}_B$ will coincide with $\mathbb{L}_{C(n)}$ in weight less than $n$. Moreover, by induction on $n$ and from the push-out square (23), $\mathbb{L}_{C(n)}$ is seen
to be a graded perfect $C(n)$-module for all $n$, and therefore is graded almost perfect over $\mathcal{O}_X$. $\square$

A second observation is that $C_X$ (resp. $C_{X^h}$) contains $\mathcal{F}ol^{ap}(X)$ (resp. $\mathcal{F}ol(X^h)$) as a full sub-$\infty$-category.

LEMMA 5.5.0.6. The natural forgetful $\infty$-functors

$$ \mathcal{F}ol^{ap}(X) \to C_X \qquad \mathcal{F}ol(X^h) \to C_{X^h} $$

are fully faithful. Their essential images consists of $B \in C_X$ (resp. $B \in C_{X^h}$) such that $\mathrm{Sym}_{\mathcal{O}_X}(B^{(1)}) \simeq B$ (resp. $\mathrm{Sym}_{\mathcal{O}_{X^h}}(B^{(1)}) \simeq B$) as graded cdga's.

Proof of Lemma 5.5.0.6. This follows by simply unraveling the various definitions, and the universal property of the de Rham algebras. We give the argument for $C_X$, the case of $C_{X^h}$ being completely analogous.

For two derived foliations $\mathcal{F}$ and $\mathcal{F}'$ in $\mathcal{F}ol^{ap}(X)$, corresponding to sheaves of graded mixed $\mathbf{DR}_X$-cdga's $\mathbf{DR}_\mathcal{F}$ and $\mathbf{DR}_{\mathcal{F}'}$, the mapping space $\mathrm{Map}(\mathcal{F}, \mathcal{F}')$ is by definition the homotopy fiber, taken at the identity of $\mathcal{O}_X$, of the natural morphism

$$ \begin{aligned} \mathrm{Map}_{\epsilon-\mathrm{cdga}^{gr}(X)}(\mathbf{DR}_{\mathcal{F}'}, \mathbf{DR}_{\mathcal{F}}) &\to \mathrm{Map}_{\epsilon-\mathrm{cdga}^{gr}(X)}(\mathbf{DR}_X, \mathbf{DR}_{\mathcal{F}}) \simeq \\ &\simeq \mathrm{Map}_{\mathrm{cdga}(X)}(\mathcal{O}_X, \mathbf{DR}_{\mathcal{F}}^{(0)}) \simeq \mathrm{Map}_{\mathrm{cdga}(X)}(\mathcal{O}_X, \mathcal{O}_X) \end{aligned} $$

(where we have denoted $\epsilon - \mathrm{cdga}_X^{gr}$ and $\mathrm{cdga}_X$ the $\infty$-categories of sheaves of graded mixed cdga's and of cdga's on $X_{\epsilon t}$). Analogously, when $\mathbf{DR}_\mathcal{F}$ are considered as augmented towards $\mathcal{O}_X$ via their natural augmentation to $\mathbf{DR}_\mathcal{F}^{(0)} \simeq \mathcal{O}_X \simeq \mathbf{DR}_{\mathcal{F}'}^{(0)}$, their mapping space in $C_X$ is the homotopy fiber, taken at the identity, of the natural morphism

$$ \begin{aligned} \mathrm{Map}_{\epsilon-\mathrm{cdga}^{gr}(X)}(\mathbf{DR}_{\mathcal{F}'}, \mathbf{DR}_{\mathcal{F}}) &\to \mathrm{Map}_{\epsilon-\mathrm{cdga}^{gr}(X)}(\mathbf{DR}_{\mathcal{F}'}, \mathbf{DR}_{\mathcal{F}}^{(0)}) \simeq \\ &\simeq \mathrm{Map}_{\mathrm{cdga}(X)}(\mathbf{DR}_{\mathcal{F}'}^{(0)}, \mathbf{DR}_{\mathcal{F}}^{(0)}) \simeq \mathrm{Map}_{\mathrm{cdga}(X)}(\mathcal{O}_X, \mathcal{O}_X). \end{aligned} $$

These two morphisms are easily seen to be equivalent, and thus so are their homotopy fibers. This shows that the $\infty$-functors of the lemma are fully faithful. Similarly, if $B \to \mathcal{O}_X$ is an object in $C_X$, choosing an inverse of the equivalence $B^{(0)} \simeq \mathcal{O}_X$ provides a morphism $\mathcal{O}_X \to B^{(0)}$, and thus a morphism $\mathbf{DR}_X \to B$ by the universal property of de Rham algebras. The graded mixed cdga $B$, together with this morphism from $\mathbf{DR}_X$ clearly defines an object in $\mathcal{F}ol^{ap}(X)$, whose image by the forgetful $\infty$-functor is equivalent to $B$, showing the essential surjectivity. $\square$

The analytification $\infty$-functor $(-)^h : \mathcal{F}ol^{ap}(X) \to \mathcal{F}ol(X^h)$ extends to $(-)^h : C_X \to C_{X^h}$, simply by sending a graded mixed cdga to the corresponding holomorphic graded mixed cdga by the analytification construction. Concretely, this sends a sheaf of graded mixed cdga $B$ on $X_{\epsilon t}$ to the sheaf (on $X_{\epsilon t}^h$) of graded holomorphic cdga $\mathcal{O}_{X^h} \otimes_{u^{-1}(\mathcal{O}_X)} u^{-1}(B)$ endowed with its canonical mixed structure together with its natural augmentation to $\mathcal{O}_{X^h}$. To prove the theorem. It is therefore enough to prove that this extended $\infty$-functor $(-)^h : C_X \to C_{X^h}$ is an equivalence of $\infty$-categories.
In order to show this, we consider the forgetful $\infty$-functor to graded cdga's by forgetting the mixed structures. We let $C_X^o$ be the $\infty$-category of sheaves of graded cdga's $B$ on $X_{\hat{e}t}$ together with an augmentation $B \to \mathcal{O}_X$ and satisfying the graded analogues of the previous three conditions: $B^{(0)} \simeq \mathcal{O}_X$, $B^{(i)}$ are zero for $i < 0$ and are almost perfect for $i > 0$. Similarly we have $C_{X^h}^o$ consisting of sheaves of graded cdga's $B$ on $X_{\hat{e}t}^h$ together with an augmentation $B \to \mathcal{O}_{X^h}$ and satisfying the graded analogues of the previous three conditions: $B^{(0)} \simeq \mathcal{O}_{X^h}$, $B^{(i)}$ are zero for $i < 0$ and are almost perfect for $i > 0$. We have forgetful $\infty$-functors $C_X \to C_X^o$ and $C_{X^h} \to C_{X^h}^o$ which commute with analytification functors

$$ (24) \qquad \begin{tikzcd} C_X \arrow[r, "(-)^h"] \arrow[d] & C_{X^h} \arrow[d] \\ C_X^o \arrow[r, "(-)^h"] & C_{X^h}^o. \end{tikzcd} $$

**LEMMA 5.5.0.7.** *The analytification $\infty$-functor*

$$ (-)^h : C_X^o \to C_{X^h}^o $$

is an equivalence of $\infty$-categories.

*Proof of Lemma 5.5.0.7.* This is a simple application of the GAGA theorem for almost perfect complexes of proper derived Artin stack of [Por19]. Indeed, GAGA implies that the analytification $(-)^h : \mathbf{APerf}(X) \to \mathbf{APerf}(X^h)$ is an equivalence (note that $\mathbf{APerf}$ is denoted by $\mathrm{Coh}^-$ in [Por19]). We deduce that it also induces an equivalence on the $\infty$-categories of $\mathbb{N}$-graded objects $\mathbf{APerf}^{\mathbb{N}}(X) \to \mathbf{APerf}^{\mathbb{N}}(X^h)$. As this equivalence is moreover an equivalence of symmetric monoidal $\infty$-categories, we get an induced equivalence on commutative algebra objects. Finally, considering commutative algebra augmented towards $\mathcal{O}_X$ and $\mathcal{O}_{X^h}$ we get the statement of the lemma. $\square$

In order to deduce the theorem from the lemma we use the Barr-Beck theorem [Lur22a, Theorem 4.7.3.5]. Indeed, the forgetful $\infty$-functors $C_X \to C_X^o$ and $C_X \to C_{X^h}$ are both monadics. The left adjoint to $C_X \to C_X^o$ sends an augmented sheaf of graded algebras $B \to \mathcal{O}_X$ to $\mathrm{Sym}_B(\mathbb{L}_B[1])$, the graded de Rham algebra of $B$ endowed with its total graduation and its natural mixed structure induced by the de Rham differential. The weight 0 part of $\mathrm{Sym}_B(\mathbb{L}_B[1])$ is clearly $\mathcal{O}_X$, and thus we do get an object in $C_X$. Similarly, the forgetful $\infty$-functor $C_{X^h} \to C_{X^h}^o$ has a left adjoint given by the graded holomorphic de Rham algebra construction. Because cotangent complexes commutes with analytification (see Proposition 5.5.0.3), we have that the
commutative square 24 is adjointable and the adjoint square

<figure><img src="image_33.png" /></figure>

naturally commutes (the canonical natural transformation between the two composition is an equivalence). Therefore, the equivalence $(-)^h : C_X^o \simeq C_{X^h}^o$ preserves the monads defining $C_X$ and $C_{X^h}$, and we thus have that the induced $\infty$-functor $(-)^h : C_X \to C_{X^h}$ is an equivalence as required. $\square$

Combining Proposition 5.4.0.3 and Theorem 5.5.0.4 we get the following important corollary.

**COROLLARY 5.5.0.8.** Let $X$ be a smooth and proper algebraic complex variety and $K \subset \Omega_X^1$ be an algebraic differential ideal such that $K$ is a vector bundle. If the coherent sheaf $\Omega_X^1/K$ is a vector bundle in codimension 3, then there exists a unique, up to equivalence, transversely smooth and rigid derived enhancement for $K$ in the sense of Definition 2.4.1.2.

The theorem 5.5.0.4 also has a linear version, for crystals which are almost perfect as $\mathcal{O}_X$-modules. The proof of the theorem below can be formally deduced from Theorem 5.5.0.4 by interpreting graded mixed $\mathbf{DR}_F$-modules (and in particular thus crystals) as trivial square zero extensions of $\mathbf{DR}_F$. Its proof will thus be only sketched.

**THEOREM 5.5.0.9.** Let $X$ be a proper algebraic Deligne-Mumford stack, $\mathcal{F} \in \mathcal{F}\mathcal{O}\mathcal{L}^{ap}(X)$ and almost perfect derived foliation on $X$ and $\mathcal{F}^h \in \mathcal{F}\mathcal{O}\mathcal{L}(X^h)$ its analytification. The analytification $\infty$-functor

$$(-)^h : \mathbf{DR}_F - \epsilon - \mathbf{d}g^{gr}(X) \to \mathbf{DR}_{F^h} - \epsilon - \mathbf{d}g^{gr}(X^h)$$

induces an equivalence on the full sub-$\infty$-category of crystals which are almost perfect as $\mathcal{O}_X$-modules and $\mathcal{O}_{X^h}$-modules respectively.

*Proof.* We leave to the reader the precise construction of the analytification $\infty$-functor

$$(-)^h : \mathbf{QCoh}(\mathcal{F}) \to \mathbf{QCoh}(\mathcal{F}^h)$$

along the same lines as the construction $\mathcal{F} \mapsto \mathcal{F}^h$ explained at the beginning of this section. It is such that the following diagram of $\infty$-categories commutes

$$\begin{tikzcd}
\mathbf{DR}_{\mathcal{F}} - \epsilon - \mathbf{d}g^{gr}(X) \arrow[r] & \mathbf{DR}_{\mathcal{F}^h} - \epsilon - \mathbf{d}g^{gr}(X) \arrow[r] \\
\mathcal{O}_X - \mathbf{d}g^{gr}(X) \arrow[u] \arrow[r] & \mathcal{O}_{X^h} - \mathbf{d}g^{gr}(X^h),
\end{tikzcd}$$

where the vertical $\infty$-functor forget the module structures and only remember the graded $\mathcal{O}$-module structures. The commutative square above is adjointable, the monads being here given
by $\mathbf{DR}_F \otimes_{\mathcal{O}_X}$ — and $\mathbf{DR}_F \otimes_{\mathcal{O}_{X^h}}$ — respectively. Using the GAGA theorem for almost perfect $\mathcal{O}$-modules (see [Por19]) we easily deduce the theorem. $\square$
<figure><img src="image_34.png" /></figure>
CHAPTER 6

Leaf spaces and holonomy

In this chapter we pursue our study of holomorphic derived foliations focusing now of the notion of analytic leaf and analytic leaf spaces. We start by discussing the notion of leaves in a very general setting. In the specific case of quasi-smooth and rigid derived foliation we state natural conditions, principally related to the singularities of the derived foliation, insuring that a leaf space exists in the analytic category. The existence of this leaf space is used to define the holonomy, as well as the relative homotopy type describing the change of topology of the leaves. We provide two main applications of the existence of the leaf space, a Riemann-Hilbert correspondence relating algebraic crystals and relative local systems, and an algebraic integrability result based on notion of Reed stability.

We set $k = \mathbb{C}$ throughout the whole chapter.

6.1. Formal, algebraic and analytic leaves

In Definition 2.3.3.4 we have seen the notion of formal leaves at a given global point. We remind that, for any derived Artin stack $X$ locally of finite presentation, any perfect derived foliation $\mathcal{F} \in \mathcal{F}ol^p(X/\mathbb{C})$, and any $x \in X(\mathbb{C})$, the formal completion $\hat{\mathcal{F}}_x$ of $\mathcal{F}$ at $x$ is determined by a morphism of perfect dg-Lie algebras $p : \ell_{X,x} \to \ell_{\mathcal{F}_x}$. The formal leaf of $\mathcal{F}$ at $x$ is defined to be the formal moduli problem determined by the $\mathbb{C}$-dg-Lie algebra which is the fiber of $p$. We denote by $\hat{\mathcal{L}}_x$ this formal moduli problem (or the corresponding derived stack by the full embedding of Proposition 2.3.1.2). In this section we will extend this formal notion to more global algebraic and analytic versions.

As a start, we introduce the $\infty$-category $\mathbf{dSt}_{\mathbb{C}}^{\text{fol}}$, of foliated derived stacks. Its objects are pairs $(X, \mathcal{F})$, where $X$ is a derived stack and $\mathcal{F}$ is a derived foliation on $X$. A morphism $(X, \mathcal{F}_X) \to (Y, \mathcal{F}_Y)$ is defined to be a pair $(f, u)$, where $f : X \to Y$ is a morphism of derived stacks, and $u : \mathcal{F}_X \to f^*(\mathcal{F}_Y)$ is a morphism in $\mathcal{F}ol(X/\mathbb{C})$. Formally, $\mathbf{dSt}_{\mathbb{C}}^{\text{fol}}$ is obtained as the Grothendieck construction for the $\infty$-functor of (7)

$$ \mathcal{F}ol(-/\mathbb{C}) : \mathbf{dSt}_{\mathbb{C}}^{\text{op}} \to \mathbf{Cat}_{\infty}. $$

We let $X$ be a derived Artin stack locally of finite presentation over $\mathbb{C}$, and $\mathcal{F} \in \mathcal{F}ol^p(X/\mathbb{C})$ a perfect derived foliation over $X$, and consider $(X, \mathcal{F})$ as an object in $\mathbf{dSt}_{\mathbb{C}}^{\text{fol}}$. For any derived
Artin stack $L$, we consider a morphism of foliated derived stacks

$$ (f, u) : (L, *_{L}) \to (X, \mathcal{F}). $$

Such a morphism can be referred to as a *preleaf of $\mathcal{F}$*. Considering the induced morphisms on tangent complexes, we find the following cartesian square of perfect complexes on $L$ (see Proposition 2.1.4.3)

<figure><img src="image_35.png" /></figure>

The morphism $u$ determines a section of $T_{f^*(\mathcal{F})} \to T_L$, and thus a morphism of perfect complexes $\theta_u : T_L \to f^*(T_{\mathcal{F}})$.

**DEFINITION 6.1.0.1.** Let $X$ be a derived Artin stack locally of finite presentation over $\mathbb{C}$ and $\mathcal{F} \in \mathcal{Fol}^p(X/\mathbb{C})$ a perfect derived foliation over $X$. With the notations and assumptions as above, a preleaf $(f, u) : (L, *_{L}) \to (X, \mathcal{F})$ is called an (algebraic) leaf of $\mathcal{F}$, if $L$ is non-empty and the induced morphism $\theta_u : T_L \to f^*(T_{\mathcal{F}})$ is an equivalence.

We can organise the leaves of $\mathcal{F}$ into an $\infty$-category, by considering the full sub-$\infty$-category in $\mathrm{dSt}_{\mathbb{C}}^{\mathrm{fol}}/(X, \mathcal{F})$ of objects of the forms $(L, *_{L})$, with $L$ a derived Artin stack, and such that the structure morphism $(L, *_{L}) \to (X, \mathcal{F})$ satisfies the condition of the previous definition. A first important observation is that a morphism $(L, *_{L}) \to (L', *_{L'})$ between two leaves of $\mathcal{F}$ induces automatically an étale morphism $L \to L'$. As opposed to the formal notion of leaves of Definition 2.3.3.4, the leaf passing through a given point $x \in X(\mathbb{C})$ is not a well defined notion, as we could always modify $L$ by an étale cover to obtain another, non-equivalent, leaf. Under some extra conditions (e.g. for transversely smooth and rigid derived foliations), there is a canonical choice, namely the “maximal leaf” passing through a point, which will be studied later with the introduction of the *leaf space*. We have however the following comparison between the formal and global versions, showing that formally at each point $L$ is equivalent to the formal leaf defined previously.

**PROPOSITON 6.1.0.2.** We let $X$ be a derived Artin stack locally of finite presentation over $\mathbb{C}$ and $\mathcal{F} \in \mathcal{Fol}(X/\mathbb{C})$ a perfect derived foliation over $X$. Let $(L, *_{L}) \to (X, \mathcal{F})$ be a leaf of $\mathcal{F}$, and $x \in L(\mathbb{C})$ be a global point. Then, the formal completion $(L, x)^{\wedge}$ of $L$ at $x$ is, as a formal moduli problem, naturally equivalent to the formal leaf $\hat{\mathcal{L}}_x$ as in Definition 2.3.3.4.

*Proof.* First of all, any morphism $f : L \to X$, induces a morphism of derived foliations over $\mathrm{Spec}(\mathbb{k})$ (or of dg-Lie algebras according to Corollary 2.3.3.3) $x^*(0_L) \to x^*(0_X)$. The derived foliation $\mathcal{F}$ itself determines a second morphism $x^*(0_X) \to x^*(\mathcal{F})$. Now, a lift of $f$ to a morphism
of foliated derived stacks $(L, *_{L}) \to (X, \mathcal{F})$ defines a commutative diagram in $\mathcal{F}ol(*)$

<figure><img src="image_36.png" /></figure>

Finally, if we require the lift to be a leaf in the sense of Definition 7.1.0.1, then the square above is moreover cartesian. This implies that $x^*(0_L)$ is equivalent to the fiber of $x^*(0_X) \to x^*(\mathcal{F})$. This proves the Proposition, as via the equivalence of Corollary 2.3.3.3, $x^*(0_L)$ corresponds to the formal completion of $L$ at $x$, and the fiber of $x^*(0_X) \to x^*(\mathcal{F})$ is by definition the formal leaf at $x$. $\square$

Analytic leaves are defined in exactly the same way.

**DEFINITION 6.1.0.3.** We let $X$ be a derived analytic Artin stack locally of finite presentation over $\mathbb{C}$ and $\mathcal{F} \in \mathcal{F}ol(X/\mathbb{C})$ a perfect derived foliation over $X$. A morphism $(f, u) : (L, *_{L}) \to (X, \mathcal{F})$ of foliated derived analytic Artin stacks is called an analytic leaf of $\mathcal{F}$, if $L$ is non-empty and the induced morphism $\theta_u : \mathbb{T}_L \to f^*(\mathbb{T}_\mathcal{F})$ is an equivalence of perfect complexes on $L$

Of course, by analytification, an algebraic leaf $(L, *_{L}) \to (X, \mathcal{F})$ produces an analytic leaf $(L^h, *_{L^h}) \to (X^h, \mathcal{F}^h)$. Algebraic leaves exist very rarely, as opposed to analytic leaves.

**EXAMPLE 6.1.0.4.** Let $f : X \to Y$ be a morphism of derived Artin stacks locally of finite presentation. Then, for any global point $y \in Y(\mathbb{C})$, the canonical morphism $(f^{-1}(y), *_{f^{-1}(y)}) \to (X, f^*(0_Y) \simeq *_{X/Y})$ identifies the fibers of $f$ with leaves of the foliation $\mathcal{F} = f^*(0_Y) \in \mathcal{F}ol(X)$. The same argument applies in the analytic setting. In other words, when the derived foliation is integrable by the morphisms $f$, then the fibers of $f$ are naturally leaves of $\mathcal{F}$, as expected.

## 6.2. Existence of leaf space and holonomy

In this section we will restrict ourselves to the following specific setting. We let $X$ be a separated and connected (smooth) complex manifold. Let $\mathcal{F} \in \mathcal{F}ol(X/\mathbb{C})$ be a derived foliation on $X$ which is transversely smooth and rigid (i.e. satisfies Definition 2.3.4.1 in the analytic setting). We will study the existence of an analytic leaf space for $\mathcal{F}$, namely we would like to contract all the leaves to points and produce the quotient space. This quotient does not exist in general, and we start by studying extra conditions on $\mathcal{F}$ that will ensure its existence.

**6.2.1. Some conditions.** We start by discussing some conditions on $\mathcal{F}$ that will later on ensure the existence of a leaf space.

**Codimension 2.** Because $\mathcal{F}$ is assumed transversely smooth and rigid, the cotangent complex $\mathbb{L}_{\mathcal{F}}$ is a length two complex of vector bundles on $X$ of the form $\mathcal{N}_{\mathcal{F}}^* \to \Omega_{X/\mathbb{C}}^1$, where
$\mathcal{N}_{\mathcal{F}}^*$ is the conormal bundle of $\mathcal{F}$. The codimension 2 condition for $\mathcal{F}$ states that there exists a closed analytic subset $Z \subset X$, of codimension 2 or higher, such that $\mathbb{L}_{\mathcal{F}}$ restricts on $U = X - Z$ to a vector bundle. Equivalently, $\mathcal{N}_{\mathcal{F}}^* \to \Omega_{X/\mathbb{C}}^1$ is injective, and is a sub-bundle when restricted to $U$. In particular, the perfect complex $\mathbb{L}_{\mathcal{F}}$ is equivalent to a single coherent sheaf in degree zero, namely $\Omega_{X/\mathbb{C}}^1/\mathcal{N}_{\mathcal{F}}^*$, which is a vector bundle outside of $Z$. The closed subset $Z$ is called the singular set of $\mathcal{F}$.

When the codimension 2 condition is satisfied for $\mathcal{F}$, we will also say that $\mathcal{F}$ is smooth in codimension 2.

**Flatness.** For any point $x \in X$, we have seen that there exists a formal leaf $\hat{\mathcal{L}}_x(\mathcal{F})$ passing through $x$ (Definition 2.3.3.4). This formal leaf is a formal moduli problem given by a dg-Lie algebra whose underlying complex is $\mathbb{T}_{\mathcal{F},x}[-1]$, the fiber of the tangent complex of $\mathcal{F}$ at $x$, shifted by $-1$. As $\mathcal{F}$ is transversely smooth and rigid, $\mathbb{T}_{\mathcal{F},x}[-1]$ is cohomologically concentrated in degrees $[1, 2]$, and therefore corresponds to a pro-representable formal moduli problem (see [Lur22b, Prop. 13.3.3.1]): there exists a pro-Artinian connective local cdga $\underline{\mathcal{A}} = \text{``lim''} A_i$ such that $\hat{\mathcal{L}}_x(\mathcal{F}) \simeq \text{Spf} \underline{\mathcal{A}}$, where $\text{Spf} \underline{\mathcal{A}}$ is the formal spectrum of $\mathcal{A}$.

The flatness condition states that, if $A = \lim_i A_i$ is the realization of $\underline{\mathcal{A}}$, then $H^i(A) = 0$ for all $i < 0$. Equivalently, this means that $\hat{\mathcal{L}}_x(\mathcal{F})$ is flat over $\mathbb{C}$. As $A$ is naturally quasi-isomorphic to the Chevalley complex of the dg-Lie algebra $\mathbb{T}_{\mathcal{F},x}[-1]$, the flatness condition can also be stated as the following vanishing for dg-Lie algebra cohomology

$$H^i(\mathbb{T}_{\mathcal{F},x}[-1], \mathbb{C}) \simeq 0 \quad \forall i < 0.$$

**Local connectedness.** The local connectedness condition only makes sense under the codimension 2 condition. Indeed, we have already seen that when $\mathcal{F}$ is smooth in codimension 2, it is locally (analytically) integrable on $X$ (Theorem 5.4.0.1). Therefore, there exists a basis $\mathcal{B}$ for the topology of $X$, and for all open $U \in \mathcal{B}$, a holomorphic map $f: U \to \mathbb{C}^d$, such that $\mathcal{F}|_U \simeq f^*(0)$. The local connectedness coinditon states that $\mathcal{B}$ and the holomorphic maps $f$ above can be moreover chosen so that the non-empty fibers of $f$ are all connected. This condition is of topological nature and is not always satisfied. When $d = 1$, it is always satisfied as $f$ is smooth in codimension 2 and thus it is well known that the Milnor fibers of $f$ are always connected (by [KM75] the Milnor fiber is $(n-2-k)$ connected, where $n$ is the dimension of $X$ and $k$ the dimension of the singular locus of $\mathcal{F}$). When $d > 1$, there are examples of holomorphic maps $f$ such that the fibers close to a singular fibers are connected or not depending on the direction we approach the singular values. A simple example is given in [Sab83], explicitly given by $f: \mathbb{C}^3 \to \mathbb{C}^2$ given by $f(x, y, z) = (x^2 - y^2z, y)$.

Our existence theorem below requires all three conditions, codimension 2, flatness and local connectedness. Before going further we would like to make some comments on these conditions.

**REMARK 6.2.1.1.** (1) As already reminded before the codimension 2 condition implies that $\mathcal{F}$ is locally integrable (Theorem 5.4.0.1). So, for any given point $x \in X$, there is
an open $x \in U$ and a holomorphic map $f: U \to \mathbb{C}^d$ such that $f^*(0) \simeq \mathcal{F}|_U$. This implies that the formal leaf $\mathcal{L}_x(\mathcal{F})$ is the formal completion of $f^{-1}(f(x))$ at the point $x$, where $f^{-1}(f(x))$ denotes here the derived fiber of $f$ passing through $x$. As formal completion are faithfully flat (see Corollary A.3.0.6), the flatness of the formal moduli space $\mathcal{L}_x(\mathcal{F})$ implies that $f^{-1}(f(x))$ is flat over $\text{Spec } \mathbb{C}$ locally at $x$. Therefore, flatness implies that the derived fibers are all flat over $\mathbb{C}$, and thus that $f$ must be a flat morphism. The converse is obviously true. Therefore, we see that under the codimension 2 condition, flatness is equivalent to the fact that all the local holomorphic maps integrating $\mathcal{F}$ are actually flat holomorphic maps.

(2) If $\mathcal{F}$ is moreover smooth, and thus is given by an integrable subbundle of $\mathbb{T}_X$, then all three conditions above are all automatic. Indeed, the local holomorphic maps $f$ must be smooth and thus their fibers will be themselves smooth. In particular, the local connectedness is true because the maps $f$ are submersions. Finally, the formal leaves are the formal completions of smooth sub-varieties (the fibers of $f$), and are thus all equivalent to formal affine spaces $\mathbb{A}^{n-d}$ (where $n = \dim X$), and so they are flat.

**6.2.2. Existence theorem.** The following result establishes, under appropriate hypotheses, the existence of a leaf space.

**THEOREM 6.2.2.1.** Let $X$ be a smooth, connected and separated complex manifold and $\mathcal{F} \in \text{Fol}(X/\mathbb{C})$ be a derived foliation which is flat, smooth in codimension 2 and locally connected. There exists a smooth and 1-truncated Deligne-Mumford analytic stack $X \parallel \mathcal{F}$, together with a holomorphic morphism $\pi: X \to X \parallel \mathcal{F}$ with the following properties.

(1) The morphism $\pi$ is flat, surjective, and we have $\pi^*(0) \simeq \mathcal{F}$.

(2) The Deligne-Mumford stack $X \parallel \mathcal{F}$ is quasi-separated, effective and is of dimension $d$ the codimension of $\mathcal{F}$.

(3) For any global point $x: * \to X \parallel \mathcal{F}$ the fiber of $\pi$ at $x$

$$ \pi^{-1}(x) := * \times_{X \parallel \mathcal{F}} X $$

is a connected and separated analytic space.

(4) For any global point $x: * \to X \parallel \mathcal{F}$, the isotropy group $H_x$ of $X \parallel \mathcal{F}$ at $x$ acts freely and properly on $\pi^{-1}(x)$.

(5) The morphism $\pi$ is relatively connected in the sense of topos theory: the pull-back $\infty$-functor $\pi^{-1}: \text{St}(X \parallel \mathcal{F}) \to \text{St}(X)$ is fully faithful when restricted to 0-truncated objects (i.e. sheaves of sets).

Before giving a proof of Theorem 6.2.2.1 some of the terms in (2) require some explanations. First of all, *quasi-separated* here means that the diagonal of $X \parallel \mathcal{F}$ is a separated morphism. In more concrete terms, if the stack $X \parallel \mathcal{F}$ is presented as an étale groupoid object $G \twoheadrightarrow U$, with $U$ a disjoint union of Stein manifolds, then the smooth analytic space $G$ must be separated. *Effectiveness of $X \parallel \mathcal{F}$* means that the action of $G$ on $U$, by germs of biholomorphic maps, is faithful (see [Car19, Def. 2.4]). Equivalently, the groupoid $G$ is a sub-groupoid of the
Haefliger's groupoid of germs of biholomorphic isomorphisms. As shown in [Car19, Def. 2.9], any Deligne-Mumford stack $Y$ possesses a universal effective quotient $Y^{\text{eff}}$. We will see later than $X \parallel \mathcal{F}$ is itself the effective quotient of a bigger stack which is pro-étale over it, given as the relative pro-homotopy type of $X$ over $X \parallel \mathcal{F}$ (see §6.3).

Finally, Theorem 6.2.2.1 allows to set the following useful terminology.

**DEFINITION 6.2.2.2.** Let $X$ be a smooth complex manifold and $\mathcal{F}$ be a derived foliation on $X$, and assume the conditions of Theorem 6.2.2.1 hold.

(1) The stack $X \parallel \mathcal{F}$ is called the leaf space of the derived foliation $\mathcal{F}$.

(2) For any $x \in X$, the quotient analytic space $\pi^{-1}(\pi(x))/H_{\pi(x)}$ is called the maximal leaf passing through $x$.

(3) The $H_{\pi(x)}$-covering $\pi^{-1}(\pi(x)) \to \pi^{-1}(\pi(x))/H_{\pi(x)}$ is called the holonomy covering, and the group $H_{\pi(x)}$ the holonomy group of $\mathcal{F}$ at $x$.

The maximal leaf will be denoted by $\mathcal{L}_x^{\text{max}}(\mathcal{F})$, and the holonomy group $\text{Hol}_{\mathcal{F}}(x)$. The holonomy covering of $\mathcal{L}_x^{\text{max}}(\mathcal{F})$ will be denoted by $\tilde{\mathcal{L}}_x^{\text{max}}(\mathcal{F})$.

We now prove Theorem 6.2.2.1, except for (3): the connectedness of $\pi^{-1}(x)$ will be proved at the very end of the section, as a consequence of an explicit description of the underlying groupoid of $X \parallel \mathcal{F}$.

**Construction of $X \parallel \mathcal{F}$**. We let $\mathcal{B}$ be the set of all open subsets $U \subset X$ such that there exists a flat holomorphic function $f_U : U \to \mathbb{C}^d$ with connected fibers, and with $f_U^*(0) \simeq \mathcal{F}|_U$. As $f_U$ is flat, it has an open image $V \subset \mathbb{C}^d$, and thus factors $U \to V \subset \mathbb{C}^d$. For each $U \in \mathcal{B}$ we fix once for all $f_U : U \to V \subset \mathbb{C}^d$, where $V$ is open, $f_U$ is flat, surjective with connected fibers, and such that $f_U^*(0) \simeq \mathcal{F}|_U$. Our conditions on $\mathcal{F}$ ensure that the set $\mathcal{B}$ forms a basis for the topology of $X$.

Before going further, we note that $\mathcal{B}$ consists of all opens $U \subset X$ with the required properties, so the only choices that have been made here are the choices of the functions $f_U$. In order to understand the impact of these choices we will need the following lemma. By the codimension 2 condition we know that the perfect complex $\mathbb{L}_{\mathcal{F}}$ is a coherent sheaf, denoted by $\Omega_{\mathcal{F}}^1 \simeq H^0(\mathbb{L}_{\mathcal{F}})$. The de Rham differential for $\mathcal{F}$ therefore induces a morphism $dR_{\mathcal{F}} : \mathcal{O}_U \to \Omega_{\mathcal{F}}^1$ of sheaves on $U$. Because $f_U^*(0) \simeq \mathcal{F}$, this de Rham differential can be identified with the relative de Rham differential $dR_{U/V} : \mathcal{O}_U \to \Omega_{U/V}^1$. In particular, the composite morphism $f^{-1}(\mathcal{O}_V) \to \mathcal{O}_U \to \Omega_{\mathcal{F}}^1$ is zero.

**LEMMA 6.2.2.3.** *The natural morphism of sheaves of rings on $U$*

$$f^{-1}(\mathcal{O}_V) \to \text{Ker}(dR_{\mathcal{F}})$$

is an isomorphism.

*Proof of Lemma 6.2.2.3.* Because $f$ is flat and surjective the induced morphism $f^{-1}(\mathcal{O}_V) \to \mathcal{O}_U$ is a monomorphism of sheaves. Therefore, $f^{-1}(\mathcal{O}_V) \to \text{Ker}(dR_{\mathcal{F}})$ is also mono. It remains
to show that a local section of $Ker(dR_{\mathcal{F}})$, locally descend to a local holomorphic function on $V$.

**SUB-LEMMA 6.2.2.4.** Let $f : X \to Y$ be a flat surjective morphism of complex manifolds which is smooth in codimension 2. If an application $u : Y \to \mathbb{C}$ is such that $f \circ u : X \to \mathbb{C}$ is holomorphic then $u$ is holomorphic.

**Proof of sublemma 6.2.2.4.** Let $U_0 \subset X$ be the open on which $f$ is smooth, so that $Z = X - U_0$ is a closed analytic subset of codimension 2 or higher. The image of $U_0$ by $f$ is an open $V_0 \subset Y$, because $f$ is flat, which is dense by Sard's theorem. As $f \circ u : U_0 \to V_0$ is a holomorphic submersion, it has local sections, and thus we deduce that $u$ is holomorphic when restricted to $V_0$. Moreover, $f$ being flat and surjective implies that the topology of $Y$ is the quotient of the topology on $X$ induced by the map $f$. This shows that $u$ is also continuous on $Y$. The application $u$ is continuous on $Y$ and holomorphic on a dense open, and so is holomorphic on the whole $Y$. $\square$

Now, let $s$ be a local section of $Ker(dR_{\mathcal{F}})$ defined on an open $U' \subset U$. By possibly shrinking $U'$ and using the local connectedness condition, we can assume that the restriction of $f_U$ on $U'$ has connected fibers. The codimension 2 condition implies that the canonical map from derived de Rham cohomology of $U$ over $V$ coincides with the naive de Rham cohomology in degree 0 (see Corollary 2.4.2.5). In particular, $dR_{\mathcal{F}}(s) = 0$ implies that $s$ determines a natural element in $H_{dR}^0(U'/V)$, the derived de Rham cohomology of $U'$ relative to $V$. We can write this as $s \in |\mathbf{DR}(U/V)|(U')$, where $\mathbf{DR}(U/V)$ is the sheaf on $U$ of graded mixed cdga's of relative de Rham theory of $U$ over $V$. The sheaf $|\mathbf{DR}(U/V)|$ is naturally a module over $f^{-1}(\mathcal{O}_V)$. Moreover, if $y \in V$ is a point, corresponding to a morphism of sheaves of rings $f^{-1}(\mathcal{O}_V) \to \mathbb{C}$, then we have an equivalence of sheaves of cdga's

$$|\mathbf{DR}(U/V)| \otimes_{f^{-1}(\mathcal{O}_V)} \mathbb{C} \simeq |\mathbf{DR}(U/V) \otimes_{f^{-1}(\mathcal{O}_V)} \mathbb{C}| \simeq j_*(|\mathbf{DR}(U_y/\mathbb{C})|),$$

where $j : U_y \hookrightarrow U$ is the inclusion of the fiber $U_y = f^{-1}(y)$, and where the first equivalence follows from the fact that $\mathbb{C}$ is a perfect $f^{-1}(\mathcal{O}_V)$-module (so that the tensor operation $\otimes_{f^{-1}(\mathcal{O}_V)} \mathbb{C}$ commutes with limits and with the realization). To summarize, the restriction of $s$ on $U'_y$, lies in derived de Rham cohomology $H_{dR}^0(U'_y/\mathbb{C})$. Now $U'_y$ is connected, so that by the comparison between derived de Rham cohomology and Betti cohomology (see Proposition 1.4.2.2 which also holds in the complex analytic setting), we have that $H_{dR}^0(U'_y/\mathbb{C}) \simeq \mathbb{C}$, and thus the restriction of $s$ on $U'_y$ is a constant function. As $f : U' \to V$ is flat with connected fibers, the function $s$ descends to an application defined on the open $V' = f(U') \subset V$, which is moreover holomorphic by the sublemma 6.2.2.4. This shows that $s$ comes from a local section of $f^{-1}(\mathcal{O}_V)$ as required. $\square$

**Lemma 6.2.2.3** has the following important corollary, showing that the choices of the local morphisms $f : U \to V$ are essentially unique.
**COROLLARY 6.2.2.5.** Let $f: U \to V$ and $g: U \to V'$ be two flat and surjective holomorphic morphisms of smooth manifolds with $f^*(0) \simeq g^*(0)$. Assume that $V'$ is isomorphic to an open in $\mathbb{C}^d$. If the fibers of $f$ are connected Then, there exists a unique étale holomorphic morphism $\alpha: V \to V'$ such that $\alpha f = g$.

*Proof of Corollary 6.2.2.5.* We apply Lemma 6.2.2.3 to local coordinate functions of $\mathbb{C}^d$ restricted on $V'$, and get a unique factorization $\alpha: V \to V'$ such that $\alpha f = g$. Because $f$ is surjective, $\alpha$ must be unique. Finally, as $f^*(0) \simeq g^*(0)$, comparing cotangent complexes show that $\alpha$ must be étale. $\square$

We now come back to the base $\mathcal{B}$ of the topology. The set $\mathcal{B}$ is ordered by the inclusions and as such will be considered as a category. We define a functor

$$ \mathcal{B} \to \mathbb{C}Man_{et}^d, $$

from $\mathcal{B}$ to the category of complex manifolds of dimension $d$ and étale morphisms between them. On objects, the functor $\phi$ sends $U$ to the space $V$, the base of the morphism $f_U: U \to V$. If $U' \subset U$ is an inclusion of opens in $\mathcal{B}$, with morphisms $f_U: U \to V$ and $f_{U'}: V \to U'$. Corollary 6.2.2.5 implies that there exists a unique étale factorization $\alpha: V' \to V$ such that $f_{U'}\alpha = (f_U)_{|U'}$. This provides for each open inclusion $U' \subset U$ in $\mathcal{B}$, a natural morphism $\phi(U') = V' \to \phi(U) = V$ in $\mathbb{C}Man_{et}^d$. This defines the functor $\phi$.

Smooth Deligne-Mumford stacks are stable arbitrary colimits of étale morphisms. We therefore consider the colimit of the functor $\phi$, and take its universal effective quotient defined in [Car19, Def. 2.9]

$$ X \llcorner \mathcal{F} := (\mathrm{colim}_{U \in \mathcal{B}} V)^{\mathrm{eff}}. $$

By construction $X \llcorner \mathcal{F}$ is an effective smooth Deligne-Mumford stack of dimension $d$. Moreover, as $\mathcal{B}$ is a basis for the topology of $X$, we have that $X \simeq \mathrm{colim}_{U \in \mathcal{B}} U$. The local morphisms $f_U: U \to V$ therefore define the projection

$$ \pi: X \simeq \mathrm{colim}_{U \in \mathcal{B}} U \to \mathrm{colim}_{U \in \mathcal{B}} V \to (\mathrm{colim}_{U \in \mathcal{B}} V)^{\mathrm{eff}} = X \llcorner \mathcal{F}. $$

**The morphism $\pi$ is flat surjective and $\pi^*(0_{X \llcorner \mathcal{F}}) \simeq \mathcal{F}$**. By construction, the morphism $\pi: X \to X \llcorner \mathcal{F}$, restricted to an open $U \in \mathcal{B}$ factorizes as $\pi|_U: U \xrightarrow{f_U} V \xrightarrow{p_V} X \llcorner \mathcal{F}$, where $p_V$ is the composition of the natural morphism $V \to \mathrm{colim}_{U \in \mathcal{B}} V$, with the canonical projection to the effective quotient $\mathrm{colim}_{U \in \mathcal{B}} V \to X \llcorner \mathcal{F}$. The morphism $p_V$ is étale, and $f_U$ is flat, so we see that $\pi|_U$ is a flat morphism for all $U \in \mathcal{B}$. As $\mathcal{B}$ is a basis for the topology of $X$, this shows that $\pi$ is flat.

To see that $\pi$ is surjective, it is enough to notice that the universal effective quotient map is always surjective (because it does not change the space of objects), and moreover that $\coprod_{U \in \mathcal{B}} V \to \mathrm{colim}_{U \in \mathcal{B}} V$ is surjective. As a result, we see that the composition $\coprod_{U \in \mathcal{B}} U \to X \to X \llcorner \mathcal{F}$ is a surjective morphism, and thus $\pi$ is surjective.
Finally, let us consider the pullback foliation $\pi^*(0_{X\parallel\mathcal{F}})$. We thus have two derived foliations on $X$, $\pi^*(0_{X\parallel\mathcal{F}})$ and $\mathcal{F}$. For all $U \in \mathcal{B}$, we know that $f_U^*(0) \simeq \mathcal{F}|_U$, but as $p_V: V \to X \parallel \mathcal{F}$ is étale, we have that $f_U^*(0) \simeq \pi^*(0)|_U$. Therefore, the two derived foliations $\pi^*(0)$ and $\mathcal{F}$ coincide on each open $U$. This implies in particular that the corresponding differential ideals $K_{\mathcal{F}}$ and $K_{\pi^*(0)}$, obtained as their truncations (Definition 2.4.1.2), agree on each $U$, and thus globally agree on $X$. This means that $\mathcal{F}$ and $\pi^*(0)$ are two transversely smooth and rigid derived enhancements of the same differential ideal $K_{\mathcal{F}}$. It follows from the uniqueness statement of derived enhancements in codimension 2 (Proposition 5.4.0.3) that $\mathcal{F} \simeq \pi^*(0)$ as required.

**Quasi-separatedness.** The quasi-separatedness property for $X \parallel \mathcal{F}$ follows from the general lemma below.

**LEMMA 6.2.2.6.** *Any smooth effective Deligne-Mumford stack is quasi-separated.*

*Proof.* Let $Y$ be an effective Deligne-Mumford stack and $U \to Y$ an étale atlas with $U$ a disjoint union of Stein manifolds (hence separated). The nerve of $U \to Y$ defines an étale groupoid $G = U \times_Y U$ acting on $U$. The quasi-separatedness of $Y$ is equivalent to the fact that the analytic space $G$ is separated.

Let $Haf(U)$ be the *Haefliger groupoid* on $U$. As a set, $Haf(U)$ consists of triplets $(x, y, u)$, with $x$ and $y$ points in $U$ and $u$ is a germ of biholomorphic isomorphism from $x$ to $y$. A basis for the topology on $Haf(U)$ is defined as follows. Fix $V \subset U$ and $W \subset U$ two opens, and $u: V \simeq W$ and isomorphism. Associated to $V, W$ and $u$ we have a subset $Hal(U)_{V,W,u} \subset Haf(U)$ consisting of all points $(x, u(x), u_x)$, where $x \in V$, $u(x) \in W$ its image and $u_x$ the germ of isomorphism at $x$ from $x$ to $u(x)$. By definition, a basis for the topology consists of all $Hal(U)_{V,W,u}$ where $(V, W, u)$ varies as above. The groupoid structure on $Haf(U)$ is the obvious one: the source (resp. target) map sends $(x, y, u)$ to $x$ (resp. $y$), and the composition is given by composing germs of isomorphisms. Since we are working in the holomorphic context, the space $Haf(U)$ is automatically separated, as two holomorphic maps agreeing on an open agree globally (on the connected components meeting this open).

Finally, there is a canonical morphism of groupoids over $U$, $\phi: G \to Haf(U)$. To a point $u \in G$, with source $x$ and target $y$, we associate $\phi(u)$ the germs of isomorphisms defined by the diagram

$$U \leftarrow G \longrightarrow U,$$

which, $G$ being an étale groupoid, induces an isomorphism on germs

$$U_x \xleftarrow{\simeq} G_u \xrightarrow{\simeq} U_y.$$

The isomorphism $U_x \simeq U_y$ obtained this way is $\phi(u)$. Now, $Y$ being effective is equivalent to the fact that the morphism $\phi$ is a monomorphism (see [Car19, Def. 2.4]). So $G \hookrightarrow Haf(U)$ is here an injective holomorphic map. As $Haf(U)$ is separated, this implies that $G$ is also separated, as claimed. $\square$
**Actions of isotropy groups.** Let $x : * \to X \llcorner \mathcal{F}$ be a point in $X \llcorner \mathcal{F}$, and $H_x = \text{aut}(x)$ be the isotropy group of the DM-stack $X \llcorner \mathcal{F}$ at $x$. We have a commutative diagram with cartesian squares

<figure><img src="image_37.png" /></figure>

The canonical morphism $BH_x \to X \llcorner \mathcal{F}$ is a monomorphism, and thus $[\pi^{-1}(x)/H_x] \to X$ is also a monomorphism. This implies that the stack $[\pi^{-1}(x)/H_x]$ is homotopically 0-truncated and thus is an analytic space. Moreover, as $X$ is separated, so is $[\pi^{-1}(x)/H_x]$. Therefore, $H_x$ acts freely and properly on $\pi^{-1}(x)$.

**Relative connectedness.** Let us denote by $\text{Sh}(Z) \subset \text{St}(Z) := \lim_{S \to X} \text{St}(S)$ the full sub-category of 0-truncated objects (for some stack $Z$), i.e. the category of sheaves of sets. By construction, the functor $\pi^{-1} : \text{Sh}(X \llcorner \mathcal{F}) \to \text{Sh}(X)$ is obtained as a functor on limit categories

$$ \lim_{U \in \mathcal{B}} f_U^{-1} : \text{Sh}(X \llcorner \mathcal{F}) \simeq \lim_{U \in \mathcal{B}} \text{Sh}(V) \to \lim_{U \in \mathcal{B}} \text{Sh}(U) \simeq \text{Sh}(X). $$

Therefore, in order to prove that $\pi^{-1}$ is fully faithful, it is enough to prove that $f_U^{-1} : \text{Sh}(V) \to \text{Sh}(U)$ is fully faithful. But this follows easily from the fact that $f_U$ is a flat surjective morphism with connected fibers, as shown by the next lemma.

**LEMMA 6.2.2.7.** Let $f : U \to V$ be an open and surjective morphism such that the topology of $V$ is the quotient of the topology of $U$ (i.e. a subset $T \subset V$ is open if and only if $f^{-1}(T) \subset U$ is open). Then, if $f$ has connected fibers, for any sheaf of sets $E$ on $V$ the canonical morphism

$$ E \to f_*f^{-1}(E) $$

is an isomorphism.

**Proof.** Let us represent $E$ by its espace étalé $p : E \to V$. As the statement is local on $V$, it is enough to show that the natural map $E(V) \to f^{-1}(E)(U)$ is bijective. Now, $E(V)$ is the set of continuous sections of $E \to V$, and $f^{-1}(E)(U)$ is the set of continuous maps $s : U \to E$ such that $ps = f$. The map $E(V) \to f^{-1}(E)(U)$ simply sends a section $s : V \to E$ to $sf : U \to E$. As $f$ is surjective this map is clearly injective. Moreover, if $s : U \to E$ is an element $f^{-1}(E)(U)$, we can restrict $s$ to the fiber of $f$ at a point $v \in V$. This provides a continuous map $f^{-1}(v) \to E_v$, where $E_v$ is the fiber of $E$ at $v$ endowed with the discrete topology. As $f^{-1}(v)$ is connected we see that the restriction of $s$ along all the fibers of $f$ is a constant morphism. Because $f$ is a topological quotient, this implies that $s : U \to E$ descend to a continuous map $V \to E$, and thus $s$ is the image of an element in $E(V)$. $\square$

**An étale groupoid description.** It is possible to describe the stack $X \llcorner \mathcal{F}$ by a (more or less explicit) étale groupoid acting on $W := \coprod_{U \in \mathcal{B}} V$. Recall that we have chosen, for
any $U \in \mathcal{B}$, a smooth surjective holomoprhic map with connected fibers $f_U : U \to V$ which integrates $\mathcal{F}|_U$. We consider all pairs of embeddings

$$U_1 \supset U_0 \subset U_2$$

of opens $U_i \in \mathcal{B}$. Because of Corollary 6.2.2.5, there is a corresponding pair of étale morphisms

$$V_1 \xleftarrow{b} V_0 \xrightarrow{s} V_2.$$

Therefore, for any $v \in V_0$, we deduce a germ of isomorphism $\alpha(v)$ between $(V_1, s(v))$ and $(V_2, b(v))$. When $v$ runs in $V_0$, we get this way a family of elements $\{(s(v), b(v), \alpha(v))\}_{v \in V_0}$ in $Haf(W)$, the Haefliger groupoid on $W = \coprod_{U \in \mathcal{B}} V$ (see the proof of 6.2.2.6 for the definition of $Haf$).

**LEMMA 6.2.2.8.** Let $G \to W$ be the étale groupoid obtained as the nerve of the canonical morphism $W \to X \parallel \mathcal{F}$. Then, $G$ is isomorphic, as a groupoid over $W$, to the subgroupoid of $Haf(W)$ generated by the elements $(s(v), b(v), \alpha(v))$ for all possible choices of $U_1 \supset U_0 \subset U_2$ in $\mathcal{B}$ and of $v \in V_0$.

*Proof.* This follows from a more general formula for the effective part of a colimit of étale morphisms. Let $V_* : I \to \mathbb{C}Man_{et}^d$ be a diagram of smooth complex manifolds of dimension $d$ with étale transition map. Let $X = (\mathrm{colim}U_i)^{eff}$ and $W = \coprod V_i \to X$ be the canonical map. Then, the nerve $W \to X$ is isomorphic, as a groupoid over $W$, to the subgroupoid of $Haf(W)$ generated by all the germs of isomorphisms generated by diagrams of the form $V_j \to V_i \to V_k$, image of $j \to i \to k$ in $I$ by $V_*$. This last general fact is obvious according to the universal properties of colimits and of the effective part. $\square$

**Connectedness of the leaves and their holonomy coverings.** We prove here Theorem 6.2.2.1 (3). We first prove that the leaves are connected. For this, let $x$ and $y$ be two points in $X$ such that $\pi(x)$ and $\pi(y)$ are isomorphic in $X \parallel \mathcal{F}$. By Lemma 6.2.2.8 we know that there exists a finite sequence of open inclusions in $\mathcal{B}$

$$U_1 \supset U_{1,2} \subset U_2 \supset U_{2,3} \subset U_3 \dots U_{n-1} \supset U_{n-1,n} \subset U_n,$$

with points $x_{i,j} \in U_{i,j}$, and such that

$$(1) x \in U_1 \text{ and } y \in U_n$$

$$(2) f_1(x_{1,2}) = f_1(x) \text{ and } f_n(x_{n-1,n}) = f_n(y)$$

$$(3) f_i(x_{i-1,i}) = f_i(x_{i,i+1})$$

where $f_i : U_i \to V_i$ is our chosen map integrating $\mathcal{F}|_{U_i}$. As the fibers of $f_i$ are all connected, it is possible to find a path $\gamma_i$ joining $x_{i-1,i}$ with $x_{i,i+1}$ and such that $f_i(\gamma_i)$ is constant (for all $i \neq 1, n$). In the same manner, we can choose paths $\alpha$ from $x$ to $x_{1,2}$ with $f_1(\alpha)$ constant, and $\beta$ from $x_{n-1,n}$ to $y$ with $f_n(\beta)$ constant. The concatenation of the $\gamma_i$ with $\alpha$ and $\beta$ provides a path $\gamma : [0, 1] \to X$, from $x$ to $y$ and such that for all $t \in [0, 1]$ the object $\pi(\gamma(t))$ are all isomorphic. By definition, this means that $\gamma$ factors through the image of the monomorphism
$\mathcal{L}_x^{max}(\mathcal{F}) \hookrightarrow X$, and thus it is a continuous path in $\mathcal{L}_x^{max}(\mathcal{F})$ joining $x$ and $y$. This implies that $\mathcal{L}_x^{max}(\mathcal{F})$ is path connected.

It remains to show that the $H_x$-covering $\tilde{\mathcal{L}}_x^{max}(\mathcal{F}) \to \mathcal{L}_x^{max}(\mathcal{F})$ is connected or, equivalently, that the corresponding classifying morphism $\pi_1(\mathcal{L}_x^{max}(\mathcal{F}), x) \to H_x$ is a surjective morphism of groups. This is proven using the same argument as before with $x = y$. Indeed, an element $h \in H_x$ is determined by a finite sequence of open inclusions in $\mathcal{B}$

$$U_1 \supset U_{2,3} \subset U_2 \supset U_{2,3} \subset U_3 \dots U_{n-1} \supset U_{n-1,n} \subset U_n,$$

with points $x_{i,j} \in U_{i,j}$, and such that

$$ (1) \quad x \in U_1 \cap U_n $$

$$ (2) \quad f_1(x_{1,2}) = f_1(x) \text{ and } f_n(x_{n-1,n}) = f_n(x) $$

$$ (3) \quad f_i(x_{i-1,i}) = f_i(x_{i,i+1}). $$

We have seen that there exists a loop $\gamma$ in $X$, pointed at $x \in X$, and such that $\pi(\gamma(t))$ is independent of $t \in [0, 1]$, up to isomorphism, in $X \parallel \mathcal{F}$. The image of $\gamma$ by the morphism $\pi_1(\mathcal{L}_x^{max}(\mathcal{F}), x) \to H_x$ is $h$ by construction.

We conclude this § with the following statement, concerning the functoriality of the leaf space construction, and thus of its characterization by a universal property.

**PROPOSITON 6.2.2.9.** Let $f : X \to Y$ be a morphism between smooth, connected and separated complex manifolds and $\mathcal{F}_Y \in \mathcal{Fol}(Y)$ and $\mathcal{F}_X := f^*(\mathcal{F}_Y)$. We assume that both derived foliations $\mathcal{F}_Y$ and $\mathcal{F}_X$ are transversely smooth, flat, smooth in codimension 2, and locally connected. Then, there exists a morphism, unique up to a unique isomorphism, $u : X \parallel \mathcal{F}_X \to Y \parallel \mathcal{F}_Y$ such that the following diagram commutes up to isomorphism

<figure><img src="image_38.png" /></figure>

*Proof.* Left to the reader. $\square$

<figure><img src="image_39.png" /></figure>

## 6.3. Relative homotopy theory

Let $f : \mathcal{T} \to \mathcal{T}'$ be a geometric morphism of $\infty$-topoi. The inverse image $f^{-1} : \mathcal{T}' \to \mathcal{T}$ is exact by definition, and thus admits a pro-left adjoint<fnref n="1" />

$$ f_{\sharp} : \mathcal{T} \to \mathrm{Pro}(\mathcal{T}') $$

<footnote n="1">¹We recall that this means that $\mathrm{Pro}(f^{-1}) : \mathrm{Pro}(\mathcal{T}') \to \mathrm{Pro}(\mathcal{T})$ has a genuine left adjoint, and $f_{\sharp}$ is then the composition of $\mathcal{T} \to \mathrm{Pro}(\mathcal{T})$ with such a genuine left adjoint.</footnote>
**DEFINITION 6.3.0.1.** The pro-object $f_{\#}(*) \in \text{Pro}(\mathcal{T}')$ is called the relative homotopy type of $\mathcal{T}$ over $\mathcal{T}'$. It is denoted either by $\mathcal{H}_f$ or $\mathcal{H}_{\mathcal{T}/\mathcal{T}'}$.

By definition, the relative homotopy type satisfies the following universal property: for any object $X \in \mathcal{T}'$, we have $\text{Map}_{\text{Pro}(\mathcal{T}')}(\mathcal{H}_f, X) \simeq \text{Map}_{\mathcal{T}}(*, f^{-1}(X))$. This must be understood as the fact that $\mathcal{H}_f$ pro-represents non-abelian cohomology of $\mathcal{T}$ with coefficients coming from $\mathcal{T}'$. When $\mathcal{T}' = \mathcal{T} = \text{Top}$ is the punctual $\infty$-topos, the pro-object $\mathcal{H}_f$ has been introduced in [TV02] as the shape of $\mathcal{T}$ (see also [Lur09, §7.1.6], [Hoy18, §2] and [Vol21, §3]).

We will restrict ourselves to the case where $\mathcal{T}$ and $\mathcal{T}'$ are $\infty$-topoi of stacks on nice topological spaces. In particular, all these spaces will be locally CW complexes of finite dimension, and thus locally contractible and locally of finite homotopical dimension. Therefore, all our $\infty$-topoi appearing below will be hypercomplete, making it easier to check that morphisms are equivalences by just looking at fibers at each points.

Let $f: X \to Y$ be a holomorphic morphism between analytic Stein varieties. In this case, $X$ and $Y$ are CW complexes of finite dimension by Hironaka (see [Hir75]), and thus are locally contractible, and locally of finite homotopical dimension. We still denote by $f: \text{St}(X) \to \text{St}(Y)$ the induced geometric morphism between the corresponding $\infty$-topoi of stacks on $X$ and $Y$. The pro-object $\mathcal{H}_f \in \text{Pro}(\text{St}(Y))$ is in general non-constant, but has the following remarkable properties.

**PROPOSITON 6.3.0.2.** With the notation above, the following assertions hold.

(1) Let $y \in Y$ be a point, then the fiber of $\mathcal{H}_f$ at $y$ is pro-constant as a pro-object in $\text{Top}$, equivalent to $\text{Sing}(f^{-1}(y))$, the homotopy type of the fiber of $f$ at $y$.

(2) If $f$ is a smooth morphism, then the pro-object $\mathcal{H}_f$ is pro-constant.

(3) Suppose that $f$ is topologically a locally trivial fibration with fiber $F_0$, then $\mathcal{H}_f$ is pro-constant, and a locally constant stack stack with fiber the homotopy type $\text{Sing}(F_0)$.

*Proof.* (1) Let $i: \{y\} \hookrightarrow Y$ the inclusion map. Let $X_y := f^{-1}(y)$ be the fiber of $f$ at $y$, $j: X_y \to X$ the inclusion and $f_y: X_y \to *$ the structure map. For any space $F \in \text{St}(*) = \text{Top}$, we have

$$ \text{Map}_{\text{Pro}(\text{Top})}(i^*(\mathcal{H}_f), F) \simeq \text{Map}_{\text{Pro}(\text{St}(Y))}(\mathcal{H}_f, i_*(F)) \simeq \text{Map}_{\text{St}(X)}(*, f^{-1}i_*(F)). $$

But, by proper base change, we have a canonical equivalence $f^{-1}i_*(F) \simeq (f_y)_*j^{-1}(F)$. Therefore, we have

$$ \text{Map}_{\text{St}(X)}(*, f^{-1}i_*(F)) \simeq \text{Map}_{\text{St}(X_y)}(*, j^{-1}(F)) \simeq \text{Map}_{\text{Pro}(\text{Top})}(\mathcal{H}_{f_y}, F). $$

However, $X_y$ is Stein analytic space, and thus is a CW complex of finite dimension by [Hir75]. Using [Vol21, Cor. 3.2] combined with [Lur09, Cor. 7.2.3.7], we have that $\mathcal{H}_{f_y}$ is pro-constant and equivalent to the usual homotopy type $\text{Sing}(X_y)$. As a final result, we have functorial equivalences

$$ \text{Map}_{\text{Pro}(\text{Top})}(i^*(\mathcal{H}_f), F) \simeq \text{Map}_{\text{Top}}(\text{Sing}(X_y), F), $$
which shows (1) by Yoneda.

(2) This is proven in [Vol21, Lem. 3.2], keeping in mind, as in the previous point, that Stein analytic spaces are essential in the sense of [Vol21] (as they are of finite covering dimension and locally contractible).

(3) The statement being local on $Y$, we may assume that $X = Y \times F_0$ with $F_0$ another Stein space, and $f: Y \times F_0 \to Y$ the natural projection. Let $K = \text{Sing}(F_0)$ and $\underline{K}_Y \in \text{St}(Y)$ be the constant stack with fiber $K$ on $Y$. There exists a canonical morphism $\mathcal{H}_f \to \underline{K}_Y$ defined as follows. Such a morphism, by previous (2), is in fact a morphism of stacks on $Y$ and not of pro-stacks. By definition of $\mathcal{H}_f$, it is thus determined by a global section $\ast \to \underline{K}_X$ of the constant stack $\underline{K}_X$ with fiber $K$ on $X$. As $X$ is a CW complex, we know that the stack $\underline{K}_X$ simply sends an open $U \subset X$ to $\text{Map}_{\text{Top}}(\text{Sing}(U), K)$. Therefore, a global section of $\underline{K}_X$ is determined by a morphism of simplicial sets $\text{Sing}(X) \to \text{Sing}(F_0)$, which we take as being the natural projection. This defines our morphism $\mathcal{H}_f \to \underline{K}_Y$. To check this is an equivalence we use that $\text{St}(Y)$ is hypercomplete, and thus it is enough to check it is a fiberwise equivalence. The point (1) therefore implies the result. $\square$

The relative homotopy theory mentioned above as several interesting applications, and one of them is the definition of a *monodromy groupoid*, bigger than the leave space $X \parallel \mathcal{F}$ of Definition 6.2.2.2. It is gather in the following definition, and will be an essential object for our Riemann-Hilbert correspondence studied in the next section.

**DEFINITION 6.3.0.3.** Let $X$ be a smooth complex manifold and $\mathcal{F} \in \text{Fol}(X)$ a derived foliation satisfying the condition of Theorem 6.2.2.1. The monodromic leaf space of $X$ with respect to $\mathcal{F}$ is the relative homotopy type

$$ \mathcal{H}_{\pi} \in \text{Pro}(X \parallel \mathcal{F}), $$

where $\pi: X \to X \parallel \mathcal{F}$ is the natural projection. It is denoted by $\widetilde{X} \parallel \mathcal{F}$.

The monodromic leaf space $\widetilde{X} \parallel \mathcal{F}$ can be understood as a projective system of étale morphism of analytic stacks over $\widetilde{X} \parallel \mathcal{F}$, and thus as a smooth pro-Deligne-Mumford stack. The major difference between $\widetilde{X} \parallel \mathcal{F}$ and $X \parallel \mathcal{F}$ is that the former retains the variation of the homotopy types of the leaves of $\mathcal{F}$, whereas in $X \parallel \mathcal{F}$ these leaves are contracted to stack points $BH_x$. By the base change formula Proposition 6.3.0.2 (1) the fiber of $\widetilde{X} \parallel \mathcal{F} \to X \parallel \mathcal{F}$ at a point $x$ is pro-constant and equivalent to the usual homotopy type of the complex space $\widetilde{\mathcal{L}}_{x}^{\max}(\mathcal{F})$.

**REMARK 6.3.0.4.** The expression *monodromic leaf space* is not standard but refers to the classical notion of the *monodromy groupoid* of a genuine foliation, which is an extension of the
holonomy groupoids. If we were to recover these groupoid notions we would simply have to consider the nerve groupoids of the projections $\tilde{\pi} : X \to \tilde{X} \parallel \mathcal{F}$ and $\pi : X \to X \parallel \mathcal{F}$, which should be called the *monodromy groupoid and the holonomy groupoid of* $\mathcal{F}$. The new feature here is that the presence of possible singularities in $\mathcal{F}$ first implies that these groupoids are flat groupoids but are not smooth in general (except when $\mathcal{F}$ is a genuine smooth foliation). Moreover, the monodromy groupoid does not even exist as a genuine groupoid but only as a pro-object in groupoids.

## 6.4. The Riemann-Hilbert correspondence

In this section we propose a first application of the existence of the leaf spaces and monodromy leaf spaces discussed previously, by proving a Riemann-Hilbert correspondence relating on the one side analytic crystals along a derived foliation $\mathcal{F}$, and a certain $\infty$-category of relative local systems along $\mathcal{F}$ that we will define. For integrable smooth foliation this recovers Deligne's relative Riemann-Hilbert correspondence of [Del70], but our result also includes a relative correspondence for flat, possibly non-smooth, morphisms between complex manifolds, as well as its generalization to the case of derived foliations.

We fix $X$ a separated smooth complex manifold and $\mathcal{F}$ be rigid and quasi-smooth derived foliation on $X$. We assume that the assumptions of Theorem 6.2.2.1 are satisfied so we have a leaf space $X \parallel \mathcal{F}$ and a monodromic leaf space $\tilde{X} \parallel \tilde{\mathcal{F}}$, related by natural projections

<figure><img src="image_40.png" /></figure>

Recall that $\pi$ is flat surjective and integrates $\mathcal{F}$, $\pi^*(0) \simeq \mathcal{F}$, and similarly for $\tilde{\pi}$. The holonomy leaf space $\tilde{X} \parallel \tilde{\mathcal{F}}$ is only a pro-Deligne Mumford stack étale over $X \parallel \mathcal{F}$ and which is relatively connected.

**DEFINITION 6.4.0.1.** With the notations and assumptions above, the $\infty$-category of sheaves locally constant along $\mathcal{F}$ (also called relative local systems along $\mathcal{F}$ is defined by

$$ \mathcal{D}_{lc}(\mathcal{F}) := \Gamma(X, \pi^{-1}(\mathrm{Perf})) $$

where $\mathrm{Perf}$ is the stack of perfect complexes of $\mathcal{O}$-modules on $X \parallel \mathcal{F}$.

Suppose that $\mathcal{F} = *_X$ is the tautological foliation on $X$, then $X \parallel \mathcal{F} = *$ and therefore $\pi^{-1}(\mathrm{Perf}) = \pi^{-1}(\mathrm{Perf}(k))$ is the stack of locally constant complexes of sheaves of $k$-modules on $X$. Therefore, $\mathcal{D}_{lc}(\mathcal{F})$ can be identified in this case with the $\infty$-category of complexes of
sheaves on X with locally constant cohomology sheaves and perfect fibers.

Definition 6.4.0.1 possesses an interesting interpretation in terms of the monodromic leaf space. Indeed, by definition we have and equivalence of $\infty$-categories

$$ \mathcal{D}_{ct}(\mathcal{F}) \simeq \mathbf{Map}_{\mathrm{Pro}(\mathrm{St}(X\parallel\mathcal{F}))}(\widetilde{X}\parallel\mathcal{F}, \mathrm{Perf}). $$

Therefore, objects in $\mathcal{D}_{ct}(\mathcal{F})$ can be, and probably should be, understood as representations of the connected pro-stack $\widetilde{X}\parallel\mathcal{F}$ in the stack of perfect complexes $\mathrm{Perf}$. Morally speaking these are analytic families, parameterised by $X\parallel\mathcal{F}$, of locally constant perfect complexes on the maximal leaves $\mathcal{L}_x^{\max}(\mathcal{F})$, where $x$ varies in $X\parallel\mathcal{F}$. The possible pathologies related to singularities of $\mathcal{F}$ are taken into account in the pro-structure of the object $\widetilde{X}\parallel\mathcal{F}$.

We let $X$ and $\mathcal{F} \in \mathcal{Fol}(X)$ as in Theorem 6.2.2.1. We define a sheaf of cdga's on $X$ as

$$ \mathcal{A}_{\mathcal{F}} := |\mathbf{DR}_{\mathcal{F}}| \in \mathrm{cdga}(X), $$

where $|\cdot|$ is the non-Tate realization functor of 1.3.4.5. In other words $\mathcal{A}_{\mathcal{F}}$ is the sheaf sending $U \subset X$ to the Hodge completed derived de Rham cohomology along the leaves $\widehat{\mathcal{C}}_{DR}^*(\mathcal{F}_U)$. This is a non-negatively graded sheaf of cdga's (because $\mathcal{F}$ is quasi-smooth) and moreover we know that our conditions imply that $H^0(\mathcal{A}_{\mathcal{F}})$ is naturally isomorphic to $\mathcal{O}_{\mathcal{F}} = \pi^{-1}(\mathcal{O}_{X\parallel\mathcal{F}})$ (because of Corollary 2.4.2.5 and Lemma 6.2.2.3), the sheaf of locally constant holomorphic functions along $\mathcal{F}$.

**DEFINITION 6.4.0.2.** With the notations and assumptions above, the $\infty$-category of sheaves almost locally constant along $\mathcal{F}$ is defined by

$$ \mathcal{D}_{alc}(\mathcal{F}) := \mathrm{Perf}(\mathcal{A}_{\mathcal{F}}) $$

the $\infty$-category of sheaves of perfect $\mathcal{A}_{\mathcal{F}}$-modules on $X$.

Finally, we also have the perfect crystals along $\mathcal{F}$ which, by definition, are sheaves $E$ of graded mixed $\mathbf{DR}_{\mathcal{F}}$-modules on $X$ satisfying the following two conditions.

(1) As a sheaf of graded $\mathbf{DR}_{\mathcal{F}}$-module $E$ is free of weight 0

$$ \mathbf{DR}_{\mathcal{F}} \otimes_{\mathcal{O}_X} E^{(0)} \simeq E. $$

(2) The $\mathcal{O}_X$-module $E^{(0)}$ is perfect.

This $\infty$-category will be denoted by $\mathrm{Perf}(\mathcal{F})$ as in the algebraic situation.

We can now consider two natural $\infty$-functors

$$ \mathrm{Perf}(\mathcal{F}) \xrightarrow{\mathrm{Sol}} \mathcal{A}_{\mathcal{F}} - \mathrm{dg}(X) \xleftarrow{\mathcal{A}_{\mathcal{F}} \otimes_{\mathcal{O}_{\mathcal{F}}} \mathcal{O}_{\mathcal{F}}} \mathcal{D}_{lc}(\mathcal{F}). $$

The $\infty$-functor $\mathrm{Sol}$ (for solutions) sends a crystal $E$ to the sheaf of morphisms $\mathbb{R}\mathrm{Hom}_{\mathrm{Perf}(\mathcal{F})}(\mathcal{O}_X, E)$, where $\mathcal{O}_X$ is the usual unit crystal along $\mathcal{F}$. It is obviouly equipped with a left $\mathcal{A}_{\mathcal{F}}$-module
structure, as $\mathcal{A}_F \simeq \mathbb{R}\operatorname{Hom}_{\operatorname{Perf}(\mathcal{F})}(\mathcal{O}_X, \mathcal{O}_X)$. The $\infty$-functor on the right hand side simply is the base change from $\mathcal{O}_F$-module to $\mathcal{A}_F$ via the canonical morphism

$$ \mathcal{O}_F = H^0(\mathcal{A}_F) \to \mathcal{A}_F $$

whose existence is insured by the fact that $\mathcal{A}_F$ is cohomologically concentrated in non-negative degrees. By construction, this second $\infty$-functor factors through the full sub-$\infty$-category $\operatorname{Perf}(\mathcal{A}_F) \subset \mathcal{A}_F - \operatorname{dg}(X)$. However, we do not if $\operatorname{Sol}$ has its image contained in perfect $\mathcal{A}_F$-modules as well, and we are thus forced to introduce the notion of unipotent objects as follows.

**DEFINITION 6.4.0.3.** An object $E \in \operatorname{Perf}(\mathcal{F})$ is unipotent, if locally on $X$, $E$ belongs to the thick triangulated sub-$\infty$-category of $\operatorname{Perf}(\mathcal{F})$ generated by the unit crystal $\mathcal{O}_X$. The full sub-$\infty$-category of unipotent objects will be denoted by

$$ \operatorname{Perf}^{\text{uni}}(\mathcal{F}) \subset \operatorname{Perf}(\mathcal{F}). $$

By the definition above, it is now clear that if $E$ is unipotent, then $\operatorname{Sol}(E)$ is itself locally in the thick triangulated sub-$\infty$-category of $\mathcal{A}_F - \operatorname{dg}(X)$ generated by $\mathcal{A}_F$. In other words, $\operatorname{Sol}$ sends unipotent crystals to perfect $\mathcal{A}_F$-modules.

**THEOREM 6.4.0.4.** The $\infty$-functors constructed above

$$ \operatorname{Perf}^{\text{uni}}(\mathcal{F}) \xrightarrow{\text{Sol}} \operatorname{Perf}(\mathcal{A}_F) \xleftarrow{\mathcal{A}_F \otimes \mathcal{O}_F} \mathcal{D}_{lc}(\mathcal{F}) $$

satisfy the following assertions.

(1) The $\infty$-functor $\operatorname{Sol}$ induces an equivalence $\operatorname{Perf}^{\text{uni}}(\mathcal{F}) \simeq \operatorname{Perf}(\mathcal{A}_F)$.

(2) The $\infty$-functor $\mathcal{A}_F \otimes_{\mathcal{O}_F}$ induces an equivalence when restricted to the genuine sub-category $\operatorname{Vect}(\mathcal{O}_F) \subset \mathcal{D}_{lc}(\mathcal{F})$, consisting of locally free $\mathcal{O}_F$-modules of finite rank.

(3) Any object $E \in \operatorname{Perf}^{\text{uni}}(\mathcal{F})$ which is a vector bundle as an $\mathcal{O}_X$-module is unipotent.

*Proof.* (1) By the definition of unipotent objects $\operatorname{Sol}$ is clearly fully faithful. It is also locally essentially surjective, as perfect $\mathcal{A}_F$-modules are locally build out from $\mathcal{A}_F$. As both $\infty$-categories are global sections of stacks over $X$, this shows that in fact $\operatorname{Sol}$ is an equivalence of the $\infty$-categories as wanted.

(2) This is formally implied by the fact that $\mathcal{A}_F$ is cohomologically non-negatively graded, and the fact that the morphism $\mathcal{O}_F \to \mathcal{A}_F$ identifies $\mathcal{O}_F$ with $H^0(\mathcal{A}_F)$. The image of $\operatorname{Vect}(\mathcal{O}_F)$ by $\mathcal{A}_F \otimes_{\mathcal{O}_F}$ clearly consists of all locally free $\mathcal{A}_F$-modules, which we can denote by $\operatorname{Vect}(\mathcal{A}_F)$.

(3) This is a consequence of [Mal77] and is proved in [TV23b, Thm. 3.2.3]. We do not reproduce the proof here. $\square$

**REMARK 6.4.0.5.** Part (1) of the theorem above is true under much weaker assumptions on $\mathcal{F}$. In particular it also applies to some situations where $\mathcal{F}$ is not rigid. We refer to [TV23b] for more on this.
An important consequence of Theorem 6.4.0.4 is the following corollary, which is the most interesting version of our Riemann-Hilbert correspondence.

**COROLLARY 6.4.0.6.** *Under the same assumptions as in Theorem 6.4.0.6, the functor $\text{Sol}$ induces an equivalence of categories*

$$ \text{Vect}(\mathcal{F}) \simeq \text{Vect}(\mathcal{O}_{\mathcal{F}}) \simeq \Gamma(X, \pi^{-1}(\text{Vect})). $$

Combine with GAGA, this gives the following corollary.

**COROLLARY 6.4.0.7.** *Let $X$ be a smooth and proper algebraic complex variety and $\mathcal{F} \in \mathcal{F}ol(X)$. We assume that $\mathcal{F}^h \in \mathcal{F}ol(X^h)$ satisfies the conditions of Theorem 6.2.2.1. Then, $\text{Sol}$ induces an equivalences of categories*

$$ \text{Vect}(\mathcal{F}) \simeq \text{Vect}(\mathcal{O}_{\mathcal{F}^h}) \simeq \Gamma(X^h, \pi^{-1}(\text{Vect})). $$

Tracking back the various functors, it is easy to see that the equivalences of the above corollary are equivalence of tensor categories, and are further functorial in the pair $(X, \mathcal{F})$. In particular, fixing a point $x \in X(\mathbb{C})$ provides a fiber functor

$$ \omega_x : \text{Vect}(\mathcal{F}) \to \text{Vect}(\mathbb{C}) $$

as explained in 3.4. The abstract algebraic holonomy group $\widehat{\text{Hol}}_{\mathcal{F}}^{alg}(x)$ is defined in Definition 3.4.0.2 as the group of tensor automorphisms of $\omega_x$ restricted to the tensor sub-category generated by the conormal bundle $\mathcal{N}_{\mathcal{F}}^*$. By corollary 6.4.0.7, when $X$ is smooth proper and $\mathcal{F}$ satisfies the assumptions of 6.2.2.1, this can also be interpreted as the group of tensor automorphisms of the fiber functor

$$ \omega_x : \text{Vect}(\mathcal{O}_{\mathcal{F}^h}) \to \text{Vect}(\mathbb{C}) $$

now restricted to the object $\pi^{-1}(\Omega_{X\parallel\mathcal{F}}^1) \in \text{Vect}(\mathcal{O}_{\mathcal{F}^h})$. Now, the holonomy group $H_x$, in the sense of Definition 6.2.2.2, obviously acts on the fiber of $\Omega_{X\parallel\mathcal{F}}^1$ at $x$, and thus there is a natural morphism

$$ H_x \to \widehat{\text{Hol}}_{\mathcal{F}}^{alg}(x) $$

from the holonomy in the sense of 6.2.2.2 to the abstract algebraic holonomy in the sense of 3.4.0.2. Both of these groups acts faithfully on the formal completion of $X \parallel \mathcal{F}$ at $\pi(x)$ and thus the morphism $H_x \to \widehat{\text{Hol}}_{\mathcal{F}}^{alg}(x)$ is injective. We believe it can also be Zariski dense in many situations but we do not adress this question here. Finally, we can now compare the two notions of holonomy we have defined in this book.

**COROLLARY 6.4.0.8.** *With the notations and assumptions of Corollary 6.4.0.7, the holonomy group $H_x$ of 6.2.2.2 identifies with a sub-group of the algebraic holonomy $\text{Hol}_{\mathcal{F}}^{alg}(x)$ of 3.4.*

An important aspect of the corollary above is that the group $\text{Hol}_{\mathcal{F}}^{alg}(x)$ admits a purely algebraic description whereas the group $H_x$ is of transcendental nature.
## 6.5. Reeb stability and algebraic integrability

In this section we give a second application of the existence of a leaf space, in order to prove a fully algebraic integrability result. This theorem is on the one side an algebraic version of [Per01] provided in the Khaler situation and for smooth foliations, but also a generalization to the quasi-smooth and thus singular setting. Our proof is mainly inspired from [Per01] with the use of Gromov compacity being replaced by the properness of Hilbert schemes.

**THEOREM 6.5.0.1.** Let $X$ be a smooth, proper and connected complex algebraic variety and $\mathcal{F} \in \mathcal{F}ol(X/\mathbb{C})$ be a transversely smooth derived foliation on $X$. We suppose that $\mathcal{F}$ is flat, smooth in codimension 2, and locally connected. We denote by $X^h$ and $\mathcal{F}^h$ the analytifications of $X$ and $\mathcal{F}$. Then, the following three conditions are equivalent.

(1) The analytic Deligne-Mumford stack $X^h \parallel \mathcal{F}^h$ is algebraizable and proper.

(2) There exists a smooth and proper effective Deligne-Mumford stack $M$, and a flat surjective morphism $p: X \to M$ with connected fibers such that $p^*(0) \simeq \mathcal{F}$.

(3) There exists one point $x \in X^h$, such that $\mathcal{L}_x^{\text{max}}(\mathcal{F}^h)$, the maximal leaf passing through $x$, is compact, and its holonomy group $\text{Hol}_{\mathcal{F}^h}(x)$ is finite (or equivalently the holonomy covering $\mathcal{L}_x^{\text{max}}(\mathcal{F}^h)$ is compact).

**Proof.** (1) $\Rightarrow$ (2) Let $M$ be a smooth and proper Deligne-Mumford stack with $M^h \simeq X^h \parallel \mathcal{F}^h$. By GAGA, the morphism $\pi: X^h \to M^h$ algebraizes to a morphism $p: X \to M$. Moreover, the GAGA theorem for perfect derived foliations (Theorem 5.5.0.4), we know that the analytification $\infty$-functor $\mathcal{F}ol(X) \to \mathcal{F}ol(X^h)$ is an equivalence of categories. From $\pi^*(0) \simeq \mathcal{F}^h$, and from the compatibility of the analytification with pull-backs (see §5.5), we deduce that $p^*(0) \simeq \mathcal{F}$. Moreover, as $\pi = p^h$ has connected fibers, so does $p$.

(2) $\Rightarrow$ (3) By the universal property of the leaf spaces (Proposition 6.2.2.9) we have a canonical identification $M^h \simeq X^h \parallel \mathcal{F}^h$. In particular, the maximal leaves of $\mathcal{F}^h$ are automatically the analytification of the fibers of $p$, and thus, by hypothesis, they are compact. In the same way, the holonomy groups of $\mathcal{F}^h$ are the isotropy groups of $M$, and so they are finite because $M$ is proper.

(3) $\Rightarrow$ (1) This implication is the main content of the theorem, and its proof will be somehow long. Let $x \in X$ be a closed point such that $\mathcal{L}_x^{\text{max}}(\mathcal{F}^h)$ is compact. The monomorphism $\mathcal{L}_x^{\text{max}}(\mathcal{F}^h) \hookrightarrow X^h$ is thus a closed immersion, and by GAGA $\mathcal{L}_x^{\text{max}}(\mathcal{F}^h)$ arises as the analytification of a closed subscheme $X_x \subset X$. We denote by $\widetilde{X}_x$ the formal completion of $X$ along $X_x$.

In the same manner, the monomorphism $BH_x \hookrightarrow X^h \parallel \mathcal{F}^h$ is a closed immersion. Indeed, $X^h \to X^h \parallel \mathcal{F}^h$ being flat and surjective, it is enough (by flat descent) to show that the pullback $BH_x \times_{X^h \parallel \mathcal{F}^h} X^h \hookrightarrow X^h$ is a closed immersion. But $BH_x \times_{X^h \parallel \mathcal{F}^h} X^h \simeq X_x^h \simeq \mathcal{L}_x^{\text{max}}(\mathcal{F}^h)$, and we have seen that it is closed in $X^h$, by assumption. We can therefore consider $\widetilde{BH_x}$, the
around $X_x \subset Z$. Therefore, by shrinking $S$ if necessary, we can assume that they are quasi-isomorphic on $Z$. This implies that also $q^*(0)$ is smooth in codimension 2. Finally, we can use the same argument but now for the stack of quasi-isomorphisms between $\mathbb{L}_{Z/[S/H_x]}$ and $f^*(\mathbb{L}_\mathcal{F})$ compatible with the canonical morphism

$$ \mathbb{L}_{Z/[S/H_x]} \leftarrow \Omega_Z^1 \longrightarrow f^*(\mathbb{L}_\mathcal{F}). $$

This shows that the two quotients of coherent sheaves

$$ \Omega_Z^1 \to H^0(\mathbb{L}_{Z/[S/H_x]}) \quad \Omega_Z^1 \to H^0(f^*(\mathbb{L}_\mathcal{F})) $$

are isomorphic, and thus their kernel must be equal. These kernels being precisely the differential ideals $K_{q^*(0)}$ and $K_{f^*(\mathcal{F})}$, we see that these must coincide. Therefore, all the conditions are met to apply Proposition 5.4.0.3 and we deduce the lemma. $\square$

Since the morphism $f: Z \to X$ is étale, it is easy to see that $f^*(\mathcal{F})^h$ is not only smooth in codimension 2, but it is also flat and locally connected. We can thus apply functoriality of leaf spaces (Proposition 6.2.2.9) in order to get a commutative diagram in the analytic category

$$ \begin{tikzcd} & Z^h \arrow[r, "f"] \arrow[d, "p"] & X^h \arrow[d, "\pi"] \\ Z^h \arrow[r, "f^*(\mathcal{F})^h"] & [S^h/H_x] \arrow[r, "g"] & X^g \arrow[r, "\mathcal{F}^h"] & [S^h/H_x]. \end{tikzcd} $$

The morphism $r$ is here a finite étale cover, which can be identified with the relative connected components of $Z \to [S/H_x]$. Since $X_x$ is connected, we see that this finite étale cover must be trivial, and thus $r$ is an equivalence. We therefore conclude the existence of commutative diagram

$$ \begin{tikzcd} & Z^h \arrow[r, "f"] \arrow[d, "p"] & X^h \arrow[d, "\pi"] \\ [S^h/H_x] \arrow[r, "g"] & X^g \arrow[r, "\mathcal{F}^h"] & [S^h/H_x]. \end{tikzcd} $$

Because $f$ is étale, and $q^*(0) \simeq \pi^*(0)$, a comparison of the cotangent complexes shows that $g$ must be étale, too.

As a result, for any $y \in S^h$, we have an étale morphism of connected analytic spaces

$$ \{y\} \times_{[S^h/H_x]} Z^h \to \tilde{\mathcal{L}}_{f(y)}^{\max}(\mathcal{F}). $$

However, because $p$ is proper we deduce that the left hand side is a proper analytic space, and that the above morphism is a finite étale cover, and thus a proper surjective morphism.
In particular, $\tilde{\mathcal{L}}^{\max}_{f(y)}(\mathcal{F})$ must be compact. We conclude that for all $y \in X$ in the image of $f$, the maximal leaf $\mathcal{L}_{y}^{\max}(\mathcal{F}) \subset X^h$ is compact and its holonomy $H_y$ is finite. As $f$ is an étale morphism, its image is a dense Zariski open subset in $X$, and we denote it by $W \subset X$. Let $y \in X-W$ be a closed point, and let $y_i$ be a sequence of closed points in $X$ converging to $y$ in the analytic topology. We can moreover assume that the holonomy groups of the $y_i$ are all trivial, as these form a Zariski dense open substack in $[S/H_x]$. We then consider the corresponding sequence of closed subschemes $\mathcal{L}_{y_i}^{\max}(\mathcal{F}^h) \subset X$. By compactness of the Hilbert scheme of $X$, this sequence of subschemes has a limit $L \subset X$. Clearly $\pi$ sends $L$ to $BH_y \subset X^h \mathcal{F}^h$, and thus $L$ is contained in the maximal leaf $\mathcal{L}_{y}^{\max}(\mathcal{F}^h)$. Moreover, $y$ being the limit of $y_i$, we have $y \in L$. More generally, any point $y' \in \mathcal{L}_{y}^{\max}(\mathcal{F})$ is a limit of points $y'_i \in \mathcal{L}_{y_i}^{\max}(\mathcal{F}^h)$, so that $y' \in L$. Therefore $L$ and $\mathcal{L}_{y}^{\max}(\mathcal{F}^h)$ coincide set theoretically. This implies in particular that $\mathcal{L}_{y}^{\max}(\mathcal{F}^h)$ is compact, that the monomorphism $\mathcal{L}_{y}^{\max}(\mathcal{F}^h) \to X^h$ is a closed immersion, and by GAGA that $\mathcal{L}_{y}^{\max}(\mathcal{F}^h)$ is algebraizable. We thus conclude that the maximal leaves $\mathcal{L}_{y}^{\max}(\mathcal{F}^h)$ are compact and algebraizable for all $y \in X^h$.

It remains to show that the holonomy group $H_y = \text{Hol}_y(\mathcal{F}^h)$ is finite for all $y$. For this, we use the same techniques. We write the maximal leaf $\mathcal{L}_{y}^{\max}(\mathcal{F}^h)$ as the limit, in the Hilbert scheme of $X$, of a sequence $\mathcal{L}_{y_i}^{\max}(\mathcal{F}^h)$ of compact leaves with trivial holonomy. We can fit this limit in a diagram of analytic spaces

<figure><img src="image_41.png" /></figure>

where $S$ is a small holomorphic disk, $Z$ is flat and proper with connected fibers over $S$, $Z \to X \times S$ is a closed immersion, and $Z_0 = \mathcal{L}_{y}^{\max}(\mathcal{F}^h) \subset X$ is the central fiber at the point $0 \in S$. We let $Z_t \subset X$ denotes the "generic" or Milnor fiber. The nearby cycles in dimension 0 form a constructible sheaf on the central fiber $X_0$, and therefore any open cover $U_i$ of $X_0$ in $X$ can be refined to a cover such that each $U_i \cap X_t$ has a number of connected components which is uniformly bounded. By covering $X_0$ with $U_i$ belonging to our basis $\mathcal{B}$ for the topology (see the construction of $X \mathcal{F}$ in §6.2.2), we deduce the existence of an étale morphism $V \to X^h \mathcal{F}^h$ covering the point $y$, and such that $\pi^{-1}(V) \mathcal{L}_{y_i}^{\max}(\mathcal{F}^h)$ possesses at most $m$ connected components for some fixed integer $m$. Equivalently, let $G \to V$ be the étale groupoid induced on $V$, so that $[V/G]$ is an open substack in $X^h \mathcal{F}^h$. Then, all the orbits of the points $y_i$ are finite of order at most $m$ in $V$. As a result, if $h \in G_y$ is a point in the stabilizer of $y \in V$, since all the $y_i$'s have trivial holonomies, we must have that $h^m$ is the identity near $y_i$, for all $i$ (for all $i$ big enough such that $h^m$ is defined at $y_i$). By analytic continuation, we have that $h^m = \text{id}$. Now, let $H_y$ be the holonomy group at a point $y$. We have seen that $H_y$ is of finite exponent $m$. Moreover, as it is a subgroup of the group of germs of holomorphic isomorphisms of $\mathbb{C}^d$ at 0, it must be finite (see [Per01, Lem. 2]).
To summarize, we have seen that all the maximal leaves are compact and all the holonomy groups are finite. We still need to show that this implies that $X^h \parallel \mathcal{F}^h$ is proper and algebraizable. We start by proving that $X^h \parallel \mathcal{F}^h$ is a separated Deligne-Mumford stack, i.e. that its diagonal is a finite morphism. For this, we come back to the very beginning of the proof of (3) ⇒ (1). Now that we know that all the maximal leaves are compact and all holonomy groups are finite, we can run exactly the same argument starting with any point $x \in X$. We have seen that there exists a commutative diagram

<figure><img src="image_42.png" /></figure>

In this diagram $p$ is a proper morphism, and moreover the natural morphism $Z^h \to [S^h/H_x] \times_{X^h \parallel \mathcal{F}^h} X^h$ is étale. Properness of $p$ implies that this étale morphism is moreover a finite covering. As a consequence $[S^h/H_x] \times_{X^h \parallel \mathcal{F}^h} X^h \to [S^h/H_x]$ is proper. As the étale morphisms of the form $[S^h/H_x] \to X^h \parallel \mathcal{F}^h$ cover the whole stack $X^h \parallel \mathcal{F}^h$, we deduce that the morphism $\pi: X^h \to X^h \parallel \mathcal{F}^h$ is a proper morphism.

We now consider the nerve of $\pi$, which is a proper and flat groupoid $G \rightrightarrows X^h$. The diagonal morphism $G \to X^h \times X^h$ is thus a proper and unramified morphism, hence a finite morphism. As $X^h \to X^h \parallel \mathcal{F}^h$ is a flat covering, this implies that the diagonal of the stack $X^h \parallel \mathcal{F}^h$ is a finite morphism, and thus that $X^h \parallel \mathcal{F}^h$ is a separated Deligne-Mumford stack. Finally, as $X^h \to X^h \parallel \mathcal{F}^h$ is proper and surjective we deduce that $X^h \parallel \mathcal{F}^h$ is also proper.

To finish the proof, we need to show that $X^h \parallel \mathcal{F}^h$ is algebraizable. But this follows by considering the groupoid $G \rightrightarrows X^h$ above (i.e. the nerve of the morphism $\pi$). Since $G \to X^h \times X^h$ is finite, and $X^h$ and $G$ are both proper algebraic spaces, we know by GAGA that the groupoid $G \rightrightarrows X^h$ is the analytification of a proper flat algebraic groupoid $G' \rightrightarrows X$. We can then consider the quotient stack $[X/G]$ of this flat groupoid: it is an (algebraic) Deligne-Mumford stack whose analytification is $X^h \parallel \mathcal{F}^h$. $\square$

**REMARK 6.5.0.3.** We can combine the above theorem with Corollary 6.4.0.8 which shows that finiteness of $H_x$ is implied by the finiteness of the algebraic holonomy group $\text{Hol}_{\mathcal{F}}^{alg}(x)$, which can be described purely in algebraic terms with using the transcendent topology.
<figure><img src="image_43.png" /></figure>
CHAPTER 7

Derived foliations on schemes and stacks in non-zero characteristic

In this last chapter we introduce derived foliation over arbitrary ground ring. We present two different non-equivalent notion; derived foliations and infinitesimal derived foliations. We show in particular how derived foliations can be useful to study characteristic classes, and prove in that direction vanishing theorem in crystalline cohomology. Finally we show that infinitesimal derived foliations have remarkable formal integrability properties, and explain how they share relations with infinitesimal cohomology as opposed to de Rham cohomology.

All along this chapter we denote by $k$ an arbitrary commutative ring.

7.1. The graded circle and graded derived loop spaces

We start by defining a group stack called the *graded circle* that we will use in order to define derived foliations in non-zero characteristic. Let $k<x>$ the free commutative $k$-algebra with divided powers over one generator. It can be presented as the quotient of the free commutative $k$-algebra over generators $x^{(n)}$, for $n \ge 0$, with relations

$$x^{(n)} \cdot x^{(m)} = \binom{n+m}{n} x^{(n+m)}$$

for all $n \ge 0$ (in particular $x^{(0)} = 1$). The commutative ring $k<x>$ is endowed with a natural comultiplication

$$\Delta : k<x> \to k<x> \otimes_k k<x>$$

which is the morphism of $k$-algebras defined on generators by

$$\Delta(x^{(n)}) = \sum_{i+j=n} x^{(i)} \otimes x^{(j)}.$$

This makes $k<x>$ into a commutative and cocommutative Hopf $k$-algebra, whose corresponding commutative group scheme $\text{Spec } k<x>$ can be naturally identified with the PD-hull at 0 of the additive group $\mathbb{G}_a$. Using the notations from [Dri24], this group scheme will be denoted by $\mathbb{G}_a^\sharp$ (it was denoted by Ker in [MRT22]). As the group $\mathbb{G}_a^\sharp$ is commutative, its classifying stack $B\mathbb{G}_a^\sharp$ inherits a canonical group structure and will be considered as a group stack.

DEFINITION 7.1.0.1. The graded circle (over $k$) is the group stack

$$S_{gr}^1 := B\mathbb{G}_a^\sharp$$
REMARK 7.1.0.2. The terminology *graded circle* suggests that $S_{gr}^1$ is the associated graded of a filtered object. This *filtered circle* is introduced in [MRT22] where it is identified with the universal Hochschild-Kostant-Rosenberg filtration.

We can give several interpretations of the group $\mathbb{G}_a^\sharp$ and of the group stack $S_{gr}^1$. We will mainly use the following cohomological interpretation, showing that $S_{gr}^1$ behaves like a *formal circle*.

The natural morphism $\mathbb{G}_a^\sharp \to \mathbb{G}_a$ induces a canonical morphism on the corresponding classifying stacks $S_{gr}^1 \to B\mathbb{G}_a$. This defines a canonical degree 1 cohomology class

$$ \epsilon^{(1)} \in H^1(S_{gr}^1, \mathcal{O}). $$

This class can be shown to be a generator for the cohomology ring and thus provides a natural graded ring isomorphism

$$ H^*(S_{gr}^1, \mathcal{O}) \simeq k \oplus k.\epsilon^{(1)}. $$

We can say more by considering the following commutative diagram of stacks

<figure><img src="image_44.png" /></figure>

for which the natural isomorphism making this diagram commutative is given by the element $\epsilon$ in $\mathbb{G}_a^\sharp(k[\epsilon])$: this is the element given by the morphism $k<x> \to k[\epsilon]$ sending $x^{(1)}$ to $\epsilon$ and all the $x^{(n)}$ for $n > 1$ to zero. The commutative diagram above then provides a canonical morphism on $\mathcal{O}$-cohomology complexes

$$ C^*(S_{gr}^1, \mathcal{O}) \to k \times_{k[\epsilon]} k $$

where the fiber product on the right hand side is taken in the $\infty$-category $\mathbf{dg}_k$.

PROPOSITON 7.1.0.3. The morphism above induces an equivalence in $\mathbf{dg}_k$

$$ C^*(S_{gr}^1, \mathcal{O}) \to k \times_{k[\epsilon]} k \simeq k \oplus k[-1]. $$

We will not provide the proof of the Proposition above and will refer to (see [MRT22, Prop. 3.4.3] for details) for the details. This equivalence lives in $\mathbf{dg}_k$ but, by construction, can be lifted to include algebra structures. For us it is important to realize it as an equivalence of commutative cosimplicial algebras (see [MRT22, Thm. 3.4.17]).

REMARK 7.1.0.4. Proposition 7.1.0.3 is one of the main reason to introduce the group stack $S_{gr}^1$. We warn the reader that it badly fails if one tries to use $B\mathbb{G}_a$ as in the characteristic zero case, as already $H^2(B\mathbb{G}_a, \mathbb{G}_a) \neq 0$ (as length 2 Witt vectors provides a non-trivial extension of $\mathbb{G}_a$ by itself).
The multiplicative group $\mathbb{G}_m$ acts on the group scheme $\mathbb{G}_a^\sharp$ by dilations. Factorially, this action is described as follows. For a commutative $k$-algebra $A$, $\mathbb{G}_m(A)$ is the group of invertible elements $\lambda$ in $A$ and $\mathbb{G}_a^\sharp(A)$ is the additive group of sequences $\{a^{(n)}\}$ of elements in $A$ satisfying $a^{(n)} \cdot a^{(m)} = \binom{n+m}{n} a^{(n+m)}$. The action of $\lambda \in A^*$ is then given by

$$ \lambda\{a^{(n)}\} = \{\lambda^n a^{(n)}\} $$

This provides an action of $\mathbb{G}_m$ on the group stack $S_{gr}^1$. The multiplicative group $\mathbb{G}_m$ also acts naturally on $\mathbf{Spec} k[\epsilon]$ by dilations, and the equivalence of Proposition 7.1.0.3 is then naturally compatible with the $\mathbb{G}_m$-actions on both sides.

The importance of Proposition 7.1.0.3 (when taking into account the algebras' structures) is that the graded circle acts as the spectrum of a square zero trivial extension by $k[-1]$. An important consequence of this fact is the following corollary.

**COROLLARY 7.1.0.5.** For any derived affine $k$-scheme $\mathbf{Spec} A$, we have a canonical $\mathbb{G}_m$-equivariant identification of derived stacks

$$ \underline{\mathbf{Map}}(S_{gr}^1, X) \simeq \mathcal{V}(\mathbb{L}_X[1]) = \mathbf{Spec} (\mathbf{Sym}_A(\mathbb{L}_{A/k}[1])) $$

*Sketch of proof.* The commutative diagram

<figure><img src="image_45.png" /></figure>

defines a commutative diagram of derived stacks

$$ \underline{\mathbf{Map}}(S_{gr}^1, X) \to X \times_{\mathcal{V}(\mathbb{L}_X)} X \simeq \mathcal{V}(\mathbb{L}_X[1]). $$

The fact that this morphism is an equivalence can be easily proven by the Whitehead theorem: a morphism of derived Artin stacks is an equivalence if and only if it is formally étale and an equivalence on the classical truncations (see [TV08, 2.2.2]). The formally étale assumption can be checked using Proposition 7.1.0.3 as the tangent complex of the right hand side can be understood as a cohomology complex of $S_{gr}^1$. We refer to [MRT22, Thm. 5.1.3] for more details. $\square$

As a consequence of Corollary 7.1.0.5, we see that, for any derived affine $k$-scheme $X$, $\mathcal{V}(\mathbb{L}_X[1])$ carries a canonical action of the semi-direct product group stack

$$ \mathcal{H} := \mathbb{G}_m \times S_{gr}^1. $$

This semi-direct group stack $\mathcal{H}$ is the pertinent generalization of our group $\mathcal{H}_0 = \mathbb{G}_m \times B\mathbb{G}_a$ (Definition 1.5.0.1) outside of characteristic zero situations. The representations of $\mathcal{H}$ are indeed graded mixed complexes, as we will see in the next sections. For the moment we use $\mathcal{H}$ in order to introduce the notion of graded loop space.
**DEFINITION 7.1.0.6.** For a derived affine k-scheme $X$, the graded loop space is the derived affine scheme

$$ \mathcal{L}^{gr}(X/k) := \underline{\mathrm{Map}}(S_{gr}^{1}, X). $$

It comes naturally equipped with a canonical action of the group $\mathcal{H}$.

For non-affine derived stacks, we use the left Kan extension of the functor $X \mapsto \mathcal{L}^{gr}(X/k)$. To avoid any confusion we will denote by $\hat{\mathcal{L}}^{gr}(-/k)$ this extension.

**DEFINITION 7.1.0.7.** The derived graded loop space of a derived stack $X$ is $\hat{\mathcal{L}}^{gr}(X/k)$ defined above. It comes with a canonical action of the group $\mathcal{H}$.

Note that for general derived Artin stack $X$, Corollary 7.1.0.5 fails. There is always a natural descent morphism $\hat{\mathcal{L}}^{gr}(X) \to \mathcal{V}(\mathbb{L}_X[1])$ but this is not an equivalence in general. For a general derived Artin stack things get even more complicated and $\hat{\mathcal{L}}^{gr}(X)$ involves some form of formal completion of $\mathcal{V}(\mathbb{L}_X[1])$ along the zero section.

The induced morphism $\hat{\mathcal{L}}^{gr}(X/k) \to \mathcal{V}(\mathbb{L}_X[1])$ is however an equivalence when $X$ is more specifically a derived Deligne-Mumford stack.

**COROLLARY 7.1.0.8.** For a derived Deligne-Mumford stack $X$ the natural morphism

$$ \hat{\mathcal{L}}^{gr}(X/k) \to \mathcal{V}(\mathbb{L}_X[1]) $$

is an equivalence.

*Proof.* Indeed, both functors $\hat{\mathcal{L}}^{gr}(-/k)$ and $\mathcal{V}(\mathbb{L}_-[1])$ are stacks on the small étale site of $X$, and Corollary 7.1.0.5 implies that the morphism in consideration defines a local equivalence between these stacks. $\square$

**REMARK 7.1.0.9.** For a general Artin stack $X$ there is a close connection between $\hat{\mathcal{L}}^{gr}(X/k)$ and $\mathcal{V}(\mathbb{L}_X[1])$. It is for instance possible to prove that these two derived stacks have equivalent algebras of $\mathbb{G}_m$-equivariant functions. This can be proved using some standard descent property as well as extensions of the results of §A.5 to the case of arbitrary characteristics (replacing $\Lambda$-powers with their left derived versions). We will not pursue this direction.

The geometric interpretation of graded mixed complexes in characteristic zero mentioned in §1.5 remains valid in any characteristic using the group stack $\mathcal{H}$ defined above when working over general base rings. The construction of §1.5 remains similar, and produces a canonical $\infty$-functor

$$ \phi : \epsilon - \mathrm{dg}_k \to \mathrm{QCoh}(\mathcal{B}\mathcal{H}). $$

**PROPOSITON 7.1.0.10.** The $\infty$-functor $\phi$ induces an equivalence of symmetric monoidal $\infty$-categories

$$ \phi : \epsilon - \mathrm{dg}_k \simeq \mathrm{QCoh}(\mathcal{B}\mathcal{H}). $$

*Proof.* This is proven in [MRT22, Prop. 4.2.3 (ii)]. $\square$
## 7.2. Derived foliations via graded derived loop spaces

The geometric interpretation of derived foliations given in Section 2.5.2 (see Definition 2.5.2.1) can be generalized in arbitrary characteristic by replacing $\mathcal{H}_0$ with the group stack $\mathcal{H} = \mathbb{G}_m \times S_{gr}^1$.

**DEFINITION 7.2.0.1.** Let $X$ be a derived stack over $k$. A derived foliation $\mathcal{F}$ on $X$ (relative to $k$) consists of the data of a $\mathcal{H}$-equivariant derived stack $\mathcal{V}(\mathcal{F})$ over $\hat{\mathcal{L}}^{gr}(X/k)$, such the following condition is satisfied: for any derived affine scheme $S$ and morphism $S \to X$, the $\mathbb{G}_m$-equivariant morphism

$$ \mathcal{V}(\mathcal{F}) \times_{\hat{\mathcal{L}}^{gr}(X/k)} \hat{\mathcal{L}}^{gr}(S/k) \to S $$

makes $\mathcal{V}(\mathcal{F})$ into a perfect derived linear stack over $S$.

By definition, derived foliations $\mathcal{F}$ on $X$ form a full sub-∞-category of $\mathrm{dSt}_{\mathcal{F}/\hat{\mathcal{L}}^{gr}(X/k)}^H$, the ∞-category of $\mathcal{H}$-equivariant derived stacks over $\hat{\mathcal{L}}^{gr}(X/k)$.

**REMARK 7.2.0.2.** Outside of the case where $k$ is a $\mathbb{Q}$-algebra, the above definition does not have an easy purely algebraic description. Indeed graded mixed cdga's must be replaced by the more complex notion of graded mixed $LSym$-algebra, where $LSym$ is the $\infty$-monad given by the left derived symmetric powers. We refer to [Rak20] for such a purely algebraic approach, and warn the reader that the comparison with our definition above is not straightforward. In what follows we will thus avoid any tentative to provide an algebraic version of derived foliations over general rings and will stick to the geometric definition above.

The derived foliations defined above share the same formal properties that we have already seen in characteristic zero, such as pull-backs and existence of finite limits. For a morphism of derived stacks $X \to Y$ we have an $\infty$-category $\mathcal{Fol}(X/Y)$ of relative derived foliations over $X$ relative to $Y$. This $\infty$-category is functorial by pull-back for commutative diagrams of derived stacks

<figure><img src="image_46.png" /></figure>

Explicitly, an object $\mathcal{F} \in \mathcal{Fol}(X/Y)$ is represented by an $\mathcal{H}$-equivariant derived stack $\mathcal{V}(\mathcal{F}) \to \hat{\mathcal{L}}^{gr}(X/Y)$. Its pull-back along the commutative diagram above is given by

$$ (f, s)^*(\mathcal{V}(\mathcal{F})) = \mathcal{V}(\mathcal{F}) \times_{\hat{\mathcal{L}}^{gr}(X/Y)} \hat{\mathcal{L}}^{gr}(X'/Y'). $$

Cotangent complexes of derived foliations are defined as in §4.3.1. Namely, for $\mathcal{V}(\mathcal{F}) \to \hat{\mathcal{L}}^{gr}(X)$ a derived foliation over $X$, and a derived affine $U = \mathrm{Spec} A$ with a morphism $U \to X$, the $\mathbb{G}_m$-equivariant derived affine stack object $\mathcal{V}(\mathcal{F}) \times_{\hat{\mathcal{L}}^{gr}(X)} U \to U$ corresponds, by [Mon21], to a well defined $A$-dg-module $\mathbb{L}_F^{\mathrm{big}}(U)$ such that $\mathcal{V}(\mathcal{F}) \times_{\hat{\mathcal{L}}^{gr}(X)} U$ and $\mathcal{V}(\mathbb{L}_F^{\mathrm{big}}(U))$ are equivalent in $\mathrm{dSt}_k/U$. When $U$ varies in $\mathrm{dAff}_k/X$, these glue to a (non-quasi-coherent) $\mathcal{O}_X$-dg-module