$c_{\phi}=\begin{pmatrix}F[1-i]&0\\\ \phi[1-i]&G\end{pmatrix}$

######  Remark 16.14.

Note that $(C_{\phi},c_{\phi})$ will not be an $\infty$-local system in
general unless $\phi$ is closed.

######  Definition 16.15.

Let
$\phi\in\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})(F,G)$
be a degree 0 closed morphism, we say $\phi$ is a homotopy equivalence if it
induces an isomorphism in
$\mathsf{Ho}\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})$.

Next, we will give an easy criterion to determine whether a map is a homotopy
equivalence. Consider the following decreasing filtration

$F^{k}\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})(F,G)=\\{\phi\in\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})(F,G)|\phi^{i}=0\text{for}i<k\\}$

######  Lemma 16.16.

There exists a spectral sequence

$E^{pq}_{0}\Longrightarrow
H^{p+q}(\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})(F,G))$

where

$E^{pq}_{0}=\operatorname{gr}\big{(}\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})(F,G)\big{)}=\\{\phi:K_{p}\to{\mathsf{C}}^{q}\phi(\sigma)\in{\mathsf{C}}^{q}(F(\sigma_{(p)}),G(\sigma_{(0)})\\}$

with differential $d_{0}(\phi^{p})=d_{G}\circ\phi^{p}-(-1)^{p+q}\phi^{p}\circ
d_{F}$.

######  Corollary 16.17.

The $E_{1}$-page of the above spectral sequence is a local system valued in
graded vector space in the usual sense.

Now we can give the criterion we want.

######  Proposition 16.18.

For ${\mathsf{C}}=\mathsf{Ch}_{k}$, a closed morphism
$\phi\in\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})^{0}(F,G)$
is a homotopy equivalence if and only if
$\phi^{0}:(F_{x},d_{F})\to(G_{x},d_{G})$ is a quasi-isomorphism of complexes
for all $x\in K_{0}$.

###### Proof.

Follows from [Blo05, Proposition 2.9]. ∎

###  17\. Cohomology of derived Lie $\infty$-groupoids

####  17.1. Actions of derived Lie $\infty$-groupoids

Let’s look at the action of derived Lie $\infty$-groupoids on a general space.
First, let’s recall the ordinary Lie groupoid action on a manifold. Let
$G_{\bullet}$ be a Lie groupoid acting on a manifold $M$. The data of this
groupoid action is encoded in an action groupoid $A_{\bullet}$ and a groupoid
morphism $\pi:A_{\bullet}\to G_{\bullet}$ over a ${\mathcal{C}}^{\infty}$ map
$M\to G_{0}$, where $A_{0}=M$ and

$A_{1}=M\times_{G_{0},t}G_{1}=\\{(x,g):t(g)=\epsilon(y)$

with structure maps $s(x,g)=xg,t(x,g)=x$. In fact we have a double pullback
square

${M\times_{G_{0},t}G_{1}}$${G_{1}}$${M}$${G_{0}}$$\scriptstyle{{\operatorname{pr}}_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}t}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}t}$$\scriptstyle{\epsilon}$

######  Lemma 17.1.

Kan fibrations between Lie groupoids are equivalent to the data of Lie
groupoid actions.

###### Proof.

$(\Leftarrow)$ Given a Lie groupoid action $G_{\bullet}$ on $M$, we get a
groupoid morphism $\pi:A_{\bullet}\to G_{\bullet}$. It suffices to show $\pi$
is a Kan fibration. $\operatorname{Kan}(1,0)$ is equivalent to

$A_{1}\to M\times_{\epsilon,G_{0},t}G_{1}$

which is an isomorphism by construction. By applying the inverse map,
$\operatorname{Kan}(1,0)$ is also satisfied. Higher Kan conditions follows
from degree 1 case.

$(\Rightarrow)$ Given a Kan fibration $\pi:A_{\bullet}\to G_{\bullet}$, we
want to show there is an action of $G_{\bullet}$ on $A_{0}$. By the
$\operatorname{Kan}(1,0)$ condition and unique Kan conditions for $n>1$, we
see that

$A_{1}\to A_{0}\times_{\epsilon,G_{0},t}G_{1}$

Hence we can define $s(x,g)=xg,t(x,g)=x$, which gives us the desired data for
action. ∎

Therefore, this inspires us to define a higher groupoid action using Kan
fibrations.

######  Definition 17.2.

Let $G_{\bullet}$ be a Lie $\infty$-groupoid, then an $\infty$-action of
$G_{\bullet}$ is a Kan fibration $\pi:A_{\bullet}\to G_{\bullet}$.

If $G_{\bullet}$ be a Lie $n$-groupoid, an $n$-action is a $n$-Kan fibration
$\pi:A_{\bullet}\to G_{\bullet}$ of Lie $n$-groupoids.

In [Li15] the 2-groupoid case is shown to be the correct definition of
actions.

####  17.2. Derived Lie $\infty$-groupoid Cohomology

######  Definition 17.3.

Let $G_{\bullet}$ be a derived Lie $\infty$-groupoid over a dga $A$. We denote
by $C^{\bullet}(G_{\bullet})$ the smooth cochain complex on $G_{\bullet}$,
where $C^{k}(G_{\bullet})$ consists of smooth functions on $G_{k}$, i.e.

$C^{k}(G_{\bullet})={\mathcal{O}}_{G_{k}}$

The differential $d=\sum_{i}(-1)^{i}d^{*}_{i}$.

Consider a derived Lie $\infty$-groupoid $G_{\bullet}$ over a cdga $A$, and
$E_{\bullet}\in\mathsf{Mod}_{A}^{{\operatorname{dg}}}$. We form a
dg-$C^{\bullet}(G_{\bullet})$-module $C^{\bullet}(G_{\bullet};E_{\bullet})$
whose degree $k$ part is

(17.1)
$C^{k}(G_{\bullet};E_{\bullet})=\bigoplus_{i+j=k}\Gamma(G_{i};Q_{0}^{*}E_{j})$

where $Q_{0}$ is defined as

$Q_{i}:=d_{1}\circ\cdots\circ d_{i}:G_{i}\to G_{0}$

is the projection on the last vertex.

######  Definition 17.4.

We define a ${\mathbb{Z}}$-connection

${\mathbb{E}}:C^{\bullet}(G_{\bullet};E_{\bullet})\to
C^{\bullet+1}(G_{\bullet};E_{\bullet})$

on $C^{\bullet}(G_{\bullet};E_{\bullet})$ to be a $k$-linear map of total
degree one which satisfies the grade Leibniz rule

$\mathbb{E}(e\omega)=(\mathbb{E}(e\otimes 1))\omega+(-1)^{|e|}ed\omega,$

for $e\in C^{\bullet}(G_{\bullet};E_{\bullet}),\omega\in
C^{\bullet}(G_{\bullet})$.

######  Definition 17.5.

We define an $\infty$-representation of $G_{\bullet}$ to be a dg-$A$-module
$E_{\bullet}$ together with a flat ${\mathbb{Z}}$-connection on
$C^{\bullet}(G_{\bullet};E_{\bullet})$.

We denote the resulting

We equip the category of $\infty$-representation a dg structure by defining
the morphism complex

$\mathsf{Rep}(G_{\bullet})(F,G)^{k}=\bigoplus_{i+j=k}\Gamma\big{(}G_{i},\underline{\operatorname{Hom}}^{j}(F,G)\big{)}$

######  Proposition 17.6.

Let ${\mathsf{C}}=\mathsf{Mod}_{A}^{{\operatorname{dg}}}$, then there exists
an dg equivalence between $\mathsf{Loc}_{\infty}^{A}(G_{\bullet})$ and
$\mathsf{Rep}^{A}_{\infty}(G_{\bullet})$.

###### Proof.

It’s easy to see that $C^{\bullet}(G_{\bullet};E_{\bullet})$ is a
$C^{\bullet}(G_{\bullet})$-module generated by

$\Gamma(G_{\bullet},E_{\bullet})=\bigoplus_{i+j=k}\Gamma(G_{k},Q_{1}^{*}E_{j})$

The Leibniz rule for ${\mathbb{E}}$ implies that we have a decomposition

${\mathbb{E}}={\mathbb{E}}^{0}+{\mathbb{E}}^{1}+{\mathbb{E}}^{2}+\cdots$

where
${\mathbb{E}}^{i}\in\operatorname{Hom}\big{(}\Gamma(G_{\bullet},E_{\bullet}),\Gamma(G_{\bullet+i},E_{\bullet+1-i})\big{)}$.
For $i\not=1$, since $D_{i}$ is $C^{\bullet}(G_{\bullet})$-linear, we can
identify them as an element of
$\Gamma\Big{(}G_{i},\operatorname{Hom}^{1-i}\big{(}P^{*}_{i}(E_{\bullet}),Q^{*}_{0}(E_{\bullet})\big{)}\Big{)}$,
which is exactly the ${\mathsf{C}}^{i,1-i}_{F}$ we defined in $\infty$-local
systems. For $i=1$, ${\mathbb{E}}^{1}$ is a derivation, which can be
identified as ${\mathbb{E}}^{1}=\hat{\delta}+\omega$ for some
$\omega\in\Gamma\Big{(}G_{1},\operatorname{Hom}^{0}\big{(}P^{*}_{1}(E_{\bullet}),Q^{*}_{0}(E_{\bullet})\big{)}\Big{)}$
and $\hat{\delta}$ is dual to the face map. Now the conditions for $E$ to be
an $\infty$-local system are

  1. (1)

$d_{F(\sigma_{(i)})}=f^{0}(\sigma_{(i)})$, which means
$\mathbb{E_{0}}\in\Gamma\Big{(}G_{0},\operatorname{Hom}^{1}\big{(}P^{*}_{0}(E_{\bullet}),Q^{*}_{0}(E_{\bullet})\big{)}\Big{)}$
is exactly the differential for $E_{\bullet}$.

  2. (2)

$\hat{\delta}f+f\cup f=0$ means ${\mathbb{E}}\circ{\mathbb{E}}=0$.

which are both satisfied by the construction. The dg structures on
$\mathsf{Loc}_{\infty}^{A}(G_{\bullet})$ and
$\mathsf{Rep}^{A}_{\infty}(G_{\bullet})$ are exactly the same. ∎

######  Definition 17.7.

We define the differentiable cohomology of a derived Lie $\infty$-groupoid
valued in $E$ to be

$H^{\bullet}_{\operatorname{diff}}(G_{\bullet};E)=H^{\bullet}\big{(}C^{\bullet}(G_{\bullet};E_{\bullet}),{\mathbb{E}}\big{)}$

###  18\. Chern-Weil theory for perfect dg modules

####  18.1. Chern-Weil theory for perfect dg modules

Let $(E^{\bullet},\mathbb{E})$ be a dg-$A^{0}$ module over a dga $A^{0}$ with
a ${\mathbb{Z}}$-connection $\mathbb{E}$. In this section, we will develop a
general theory of constructing characteristic classes valued in $A$. Later we
shall apply it to singular foliations.

The curvature of the ${\mathbb{Z}}$-connection $\mathbb{E}$ is defined by the
usual formula

$R_{\mathbb{E}}=\mathbb{E}^{2}=\frac{1}{2}[\mathbb{E},\mathbb{E}]\in
A({\mathcal{F}},\underline{\operatorname{End}}(E))$

In order to define the characteristic forms, we need to define a
${\mathbb{Z}}$-graded supertrace map
$\operatorname{Str}:A(M,\underline{\operatorname{End}}(E))\to\mathsf{A}$. For
each $\phi_{i}\in\Gamma(\operatorname{End}(E^{i}))$, define
$\operatorname{Str}(\phi_{i})=(-1)^{i}\operatorname{Tr}(\phi_{i})$. Extend
$\operatorname{Str}$ to a (${\mathbb{Z}}$-graded) $A$-linear map we get
$\operatorname{Str}:A({\mathcal{F}},\underline{\operatorname{End}}(E))\to
A({\mathcal{F}})$. Note that by construction $\operatorname{Str}$ vanishes on
$A({\mathcal{F}},\underline{\operatorname{End}}_{i}(E))$ for all $i\not=0$.

######  Proposition 18.1 (Bianchi identity).

$\mathbb{E}R_{\mathbb{E}}^{i}=0$ for all $i\geq 1$.

###### Proof.

It follows from $\mathbb{E}R_{\mathbb{E}}^{i}=[\mathbb{E},\mathbb{E}^{2i}]=0$.
∎

######  Lemma 18.2.

$d_{\mathsf{A}}\operatorname{Str}(R_{\mathbb{E}}^{i})=0$.

###### Proof.

$d_{\mathsf{A}}\operatorname{Str}(R_{\mathbb{E}}^{i})=\sum_{i}\operatorname{Str}(R_{\mathbb{E}}^{j-1}[\mathbb{E},R_{\mathbb{E}}]R_{\mathbb{E}}^{i-j})$

where each summand is zero by the Bianchi identity. Hence,
$\operatorname{Str}(R_{\mathbb{E}}^{i})$ is closed. ∎

Let $f(z)$ be a convergent formal power series in $z$, then
$f(R_{\mathbb{E}})$ is an element of
$A^{even}(M,\underline{\operatorname{End}}(E))$, defined by

$f(R_{\mathbb{E}})=\sum\frac{f^{(k)}(0)}{k!}(\mathbb{E}^{2})^{k}$

Applying supertrace map to $f(R_{\mathbb{E}})$ we get an element in
$\mathsf{A}$ which is a combination of even elements. We will call this
element the $\mathsf{A}$-characteristic form, or simply characteristic form if
there are no confusions, of $\mathbb{E}$ corresponding to $f(z)$.

######  Proposition 18.3.

Given a perfect $A^{0}$-module with ${\mathbb{Z}}$-connection $(E,\mathbb{E})$
over a dga $\mathsf{A}$. Then

  1. (1)

The characteristic form $\operatorname{Str}(f(R_{\mathbb{E}}))$ is a closed
element of even degree.

  2. (2)

(Transgression formaula) If $\mathbb{E}_{t}$ is a smooth 1-parameter family of
${\mathbb{Z}}$-connection on $E$, then

(18.1)
$\frac{d}{dt}\operatorname{Str}(f(R_{\mathbb{E}_{t}}))=d\operatorname{Str}\Big{(}\frac{d\mathbb{E}_{t}}{dt}f^{\prime}(R_{\mathbb{E}_{t}})\Big{)}$

  3. (3)

The cohomology class of $\operatorname{Str}(f(R_{\mathbb{E}}))$ in
$H^{\bullet}(\mathsf{A})$ is independent of the choice of $\mathbb{E}$.

###### Proof.

We need the following lemma

######  Lemma 18.4.

For any $\alpha\in A(M,\underline{\operatorname{End}}(E))$, we have
$d_{\mathsf{A}}(\operatorname{Str}\alpha)=\operatorname{Str}([\mathbb{E},\alpha])$.

###### Proof.

Locally, we can write $\mathbb{E}=d+\omega$, so
$\operatorname{Str}([\mathbb{E},\alpha])=\operatorname{Str}([d,\alpha])+\operatorname{Str}([\omega,\alpha])$.
The second term vanishes by the definition of $\operatorname{Str}$, and the
first terms equals $\operatorname{Str}(d\alpha)$. ∎

By this lemma, we have

$d\operatorname{Str}(f(R_{\mathbb{E}}))=\operatorname{Str}([\mathbb{E},f(\mathbb{E}^{2})])=0.$

Hence, $\operatorname{Str}(f(R_{\mathbb{E}}))$ is closed. The degree of it is
clearly even. We show prove (2) and (3) together. Let $\mathbb{E}_{1}$ and
$\mathbb{E}_{2}$ be two ${\mathbb{Z}}$-connection over $\mathsf{A}$ on
$E^{\bullet}$. $\mathbb{E}_{1}-\mathbb{E}_{2}$ is $\mathsf{A}$-linear hence we
can write $\mathbb{E}_{1}-\mathbb{E}_{2}=\mathbb{E}^{\prime}$ for some
$\mathbb{E}^{\prime}\in A(M,\underline{\operatorname{End}}(E))_{1}$. Set
$\mathbb{E}_{t}=\mathbb{E}_{2}+t\mathbb{E}^{\prime}$. Note that
$\mathbb{E}_{t}$ is a ${\mathbb{Z}}$-connection for any $t\in[0,1]$. Let
$\mathbb{E}_{t}$ be a smooth 1-parameter family of ${\mathbb{Z}}$-connections
on $E$. The curvature of $\mathbb{E}_{t}$ is

$R_{\mathbb{E}_{t}}=(\mathbb{E}_{2}+t\mathbb{E}^{\prime})^{2}=\mathbb{E}_{2}^{2}+t[\mathbb{E}_{2},\mathbb{E}^{\prime}]+\frac{1}{2}t^{2}[\mathbb{E}^{\prime},\mathbb{E}^{\prime}]$

The deformation of curvature is

$\displaystyle\frac{d}{dt}R_{\mathbb{E}_{t}}=[\mathbb{E}_{2},\mathbb{E}^{\prime}]+t[\mathbb{E}^{\prime},\mathbb{E}^{\prime}]=[\mathbb{E}_{t},\mathbb{E}^{\prime}]$

We will prove a more general lemma first.

######  Lemma 18.5.

Let $\alpha_{t}\in A^{even}(M,\underline{\operatorname{End}}(E))$ be a smooth
family of forms of even total degree, then

$\frac{d}{dt}\operatorname{Str}(f(\alpha_{t}))=\operatorname{Str}(\frac{d}{dt}(\alpha_{t})f^{\prime}(\alpha_{t}))$

###### Proof.

It suffices to consider $f$ being monomials. Consider $f(z)=z^{n}$. Then

$\frac{d}{dt}\operatorname{Str}(\alpha_{t}^{n})=\operatorname{Str}\Big{(}\sum_{i=0}^{n-1}\alpha_{t}^{i}\Big{(}\frac{d}{dt}\alpha_{t}\Big{)}\alpha_{t}^{n-i-1}\Big{)}=n\operatorname{Str}\Big{(}\Big{(}\frac{d}{dt}\alpha_{t}\Big{)}\alpha_{t}^{n-1}\Big{)}$

∎

Apply this lemma to $\alpha_{t}=\mathbb{E}_{t}^{2}$,

$\displaystyle\frac{d}{dt}\operatorname{Str}(f(\mathbb{E}_{t}^{2}))=$
$\displaystyle\operatorname{Str}\bigg{(}\frac{d\mathbb{E}_{t}^{2}}{dt}f^{\prime}(\mathbb{E}_{t}^{2})\bigg{)}$
$\displaystyle=$
$\displaystyle\operatorname{Str}\bigg{(}\Big{[}\mathbb{E}_{t},\Big{(}\frac{d}{dt}\mathbb{E}_{t}^{2}\Big{)}f^{\prime}(\mathbb{E}_{t}^{2})\Big{]}\bigg{)}$
$\displaystyle=$ $\displaystyle
d\operatorname{Str}\bigg{(}\frac{d\mathbb{E}_{t}}{dt}f^{\prime}(\mathbb{E}_{t}^{2})\bigg{)}$

This proves (2).

Now we integrate the transgression formula with respect to $t$, then we get

(18.2)
$\operatorname{Str}(f(R_{\mathbb{E}_{1}}))-\operatorname{Str}(f(R_{\mathbb{E}_{2}}))=d\int_{0}^{1}\operatorname{Str}\big{(}\mathbb{E}^{\prime}f^{\prime}(\mathbb{E}_{t}^{2})\big{)}dt$

Note that $\frac{d\mathbb{E}_{t}}{dt}=\mathbb{E}^{\prime}$. Hence, the
cohomology class of $\operatorname{Str}(f(R_{\mathbb{E}_{1}}))$ and
$\operatorname{Str}(f(R_{\mathbb{E}_{0}}))$ in $H^{\bullet}(\mathsf{A})$ are
the same.

∎

######  Definition 18.6.

Given a dg-$A^{0}$ module over a dga $A^{0}$ with a ${\mathbb{Z}}$-connection
$\mathbb{E}$ $(E,\mathbb{E})$, we define the $\mathsf{A}$-Pontryagin algebra
of $E$

$\operatorname{Pont}^{\bullet}_{\mathsf{A}}\subset H^{\bullet}(\mathsf{A})$

to be the subalgebra generated by

$\sigma^{i}_{\mathsf{A}}(E)=[\operatorname{Str}(R_{\mathbb{E}}^{i})]\in
H^{2i}(\mathsf{A})$

and we call $\sigma^{i}_{\mathsf{A}}(E)$ the $\mathsf{A}$-Pontryagin character
of $E$.

####  18.2. $L_{\infty}$-pairs over a dga

In this section, we will define $L_{\infty}$-pairs over a dga. As a special
case, given a regular foliation $F$, then $(T_{M},{\mathcal{F}})$ is an
$L_{\infty}$-pair over ${\mathcal{C}}^{\infty}(M)$.

######  Definition 18.7.

Let $\mathfrak{g}$ be a $L_{\infty}$-algebroid over a dga $\mathsf{A}$, and
$\mathfrak{h}\subset\mathfrak{g}$ a subalgebroid. Note the brackets
$\\{\lambda_{i}\\}_{i}$ of $\mathfrak{h}$ is the restriction of the brackets
$\\{\lambda_{i}\\}_{i}$. We call $(\mathfrak{g},\mathfrak{h})$ an
$L_{\infty}$-pair.

Note that the inclusion map $\iota:\mathfrak{h}\to\mathfrak{g}$ gives
$\iota^{\vee}:\mathfrak{g}^{\vee}\to\mathfrak{h}^{\vee}$, consequently we have
a surjective morphism of dga
$\iota^{\vee}:\operatorname{Sym}\mathfrak{g}^{\vee}[-1]\to\operatorname{Sym}\mathfrak{h}^{\vee}[-1]$

######  Example 18.8.

For a regular foliation $(M,{\mathcal{F}})$, $(T_{M},{\mathcal{F}})$ is an
$L_{\infty}$-pair.

######  Example 18.9.

Let $\mathfrak{g}$ be a Lie algebroid and $\mathfrak{h}\subset\mathfrak{g}$ a
subalgebroid, then $(\mathfrak{g},\mathfrak{h})$ is an $L_{\infty}$-pair,
which is called a Lie pair. As a special case, if $\mathfrak{g}$ is a Lie
algebra and $\mathfrak{h}\subset\mathfrak{g}$ a Lie subalgebra, then
$(\mathfrak{g},\mathfrak{h})$ is an $L_{\infty}$-pair over a point.

Now, for a Lie pair $(\mathfrak{g},\mathfrak{h})$, we denote the quotient
$\mathfrak{g}/\mathfrak{h}$ by $N$.

######  Lemma 18.10.

There is an $\infty$-representation over $\mathfrak{h}$ which gives $N$ an
$\mathfrak{h}$-module structure.

###### Proof.

There is an exact sequence of $dg$-$\mathsf{A}$-modules

$0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\mathfrak{g}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}N\to 0$

The $\mathfrak{h}$-module structure

$[x_{1},\cdots,x_{n-1},y]:\operatorname{Sym}^{n-1}\mathfrak{g}\otimes N\to
N,n\geq 1$

is given by

$[x_{1},\cdots,x_{n-1},y]=p\circ l_{k}(x_{1},\cdots,x_{n-1},y^{\prime})$

for any $y^{\prime}\in\mathfrak{g}$ such that $p(y^{\prime})=y$. These
brackets are well-defined, since $\mathfrak{h}$ is a subalgebroid. By
construction, $[\cdots]$ is an $\infty$-representation of $\mathfrak{h}$ on
$N$. ∎

######  Lemma 18.11.

$N^{\vee}$ is also an $\mathfrak{h}$-module.

######  Remark 18.12.

Note that
$N^{\vee}=(\mathfrak{g}/\mathfrak{h})^{\vee}\simeq\mathfrak{h}^{\perp}=\ker(\iota^{\vee}:\mathfrak{g}^{\vee}\to\mathfrak{h}^{\vee})$.
We denote the $\mathfrak{h}$-module structure of $N^{\vee}$ by
$(\mathfrak{h}^{\vee},\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}})$,
where
$\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}}=d_{{\mathcal{O}}(\mathfrak{h})}+D^{\mathfrak{h}^{\perp}}$.

Let $d^{\mathsf{dR}}_{\mathfrak{g}}$ and $d^{\mathsf{dR}}_{\mathfrak{h}}$ be
the algebraic de Rham operator on ${\mathcal{O}}(\mathfrak{g})$ and
${\mathcal{O}}(\mathfrak{h})$ respectively. Define an operator
$J:{\mathcal{O}}(\mathfrak{g})\to A(\mathfrak{h},\mathfrak{g}^{\vee})$ by
$J=(\iota^{\vee}\otimes 1\circ d^{\mathsf{dR}}_{\mathfrak{g}}$, i.e. the
following diagram commutes

${{{\mathcal{O}}(\mathfrak{g})}}$${{A(\mathfrak{g},\mathfrak{g}^{\vee})}}$${{A(\mathfrak{h},\mathfrak{g}^{\vee})}}$$\scriptstyle{d_{\mathfrak{g}}^{\mathsf{dR}}}$$\scriptstyle{J}$$\scriptstyle{\iota^{\vee}\otimes
1}$

Hence $(1\otimes\iota^{\vee})\circ J=\iota^{\vee}\circ
d_{\mathfrak{g}}^{\mathsf{dR}}=d_{\mathfrak{h}}^{\mathsf{dR}}\circ\iota^{\vee}$.
It is obvious that $J$ is a derivation on the
${\mathcal{A}}(\mathfrak{g})$-bimodule
${\mathcal{A}}(\mathfrak{h},\mathfrak{g}^{\vee})$, i.e. for all
$\omega,\omega^{\prime}\in{\mathcal{A}}(\mathfrak{g})$,
$J(\omega\odot\omega^{\prime})=\iota^{\vee}(\omega)\odot
J(\omega^{\prime})+J(\omega)\odot\iota^{\vee}(\omega^{\prime})=\iota^{\vee}(\omega)\odot
J(\omega^{\prime})+(-1)^{|\omega||\omega^{\prime}|}\iota^{\vee}(\omega^{\prime})\odot
J(\omega)$.

Now we get a map

(18.3) $J\otimes
1:{\mathcal{A}}(\mathfrak{g},\underline{\operatorname{End}}(E))\to{\mathcal{A}}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))$

by setting

(18.4) $(J\otimes 1)(\phi\circ\psi)=(\iota^{\vee}\otimes
1)(\phi)\circ(J\otimes 1)(\psi)+(J\otimes 1)(\phi)\circ(\iota^{\vee}\otimes
1)(\psi)$

for all
$\phi,\psi\in{\mathcal{A}}(\mathfrak{g},\underline{\operatorname{End}}(E))$.

######  Lemma 18.13.

Let
$\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}}=d_{{\mathcal{A}}(\mathfrak{h})}+D^{\mathfrak{h},\mathfrak{h}^{\perp}}$
be the $\mathfrak{h}$-module structure on
$N^{\vee}\simeq\mathfrak{h}^{\perp}$, then for $\omega\in\ker(\iota^{\vee})$,
we have $J(\omega)\in{\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\perp})$ and

(18.5)
$\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}}(J(\omega))=J(d_{{\mathcal{A}}(\mathfrak{g})}(\omega))$

###### Proof.

First, we will show that

$D^{\mathfrak{h},\mathfrak{h}^{\perp}}(\xi)=J(d_{{\mathcal{A}}(\mathfrak{g})}\xi)$

for all $\mathfrak{h}^{\perp}$. For $k=2$ For $k>=3$.

$\displaystyle<D^{\mathfrak{h},\mathfrak{h}^{\perp}}(\xi),p(l)>(a_{1}\odot\cdots\odot
a_{k})=$
$\displaystyle(-1)^{|\xi|+1}<\xi,D^{\mathfrak{h},\mathfrak{h}}(p(l))(a_{1}\odot\cdots\odot
a_{k})>$ $\displaystyle=$
$\displaystyle(-1)^{|\xi|+|l|_{*_{k}}+k}<\xi,m_{k+1}(a_{1}\odot\cdots\odot
a_{k},p(l))>$ $\displaystyle=$
$\displaystyle(-1)^{|\xi|+|l|_{*_{k}}+k}<\xi,p\circ
l_{k+1}(a_{1}\odot\cdots\odot a_{k},p(l))>$ $\displaystyle=$
$\displaystyle(-1)^{k+1}<(\iota^{\vee}\circ
d^{\mathsf{dR}}_{\mathfrak{g}}\circ
l_{k+1}^{\vee})(\xi),p(l))>(a_{1}\odot\cdots\odot a_{k})$ $\displaystyle=$
$\displaystyle<(J\circ
d_{{\mathcal{A}}(\mathfrak{g})}(\xi),p(l)>(a_{1}\odot\cdots\odot a_{k})$

where $*_{k}$ denotes $\sum_{i=1}^{k}|a_{i}|$.

Now let us prove the proposition. It suffices to consider the elements of the
form $\omega\odot\xi\in{\mathcal{A}}(\mathfrak{g})\odot\mathfrak{h}^{\perp}$.
Applying the previous equation, we get

$\displaystyle J\circ d_{{\mathcal{A}}(\mathfrak{g})}(\omega\odot\xi)=$
$\displaystyle
J\big{(}d_{{\mathcal{A}}(\mathfrak{g})}(\omega)\odot\xi+(-1)^{|\omega|}\omega\odot
d_{{\mathcal{A}}(\mathfrak{g})}\xi\big{)}$ $\displaystyle=$
$\displaystyle\iota^{\vee}(d_{{\mathcal{A}}(\mathfrak{g})}(\omega)\odot\xi+J(d_{{\mathcal{A}}(\mathfrak{g})}(\omega))\odot\iota^{\vee}(\xi)+$
$\displaystyle(-1)^{|\omega|}\big{(}J(\omega)\odot\iota^{\vee}(d_{{\mathcal{A}}(\mathfrak{g})}\xi)+\iota^{\vee}(\omega)\odot
J(d_{{\mathcal{A}}(\mathfrak{g})}\xi)\big{)}$ $\displaystyle=$ $\displaystyle
d_{{\mathcal{A}}(\mathfrak{g})}(\iota^{\vee}(\omega))\odot\xi+(-1)^{|\omega|}\iota^{\vee}(\omega)\odot
D^{\mathfrak{h},\mathfrak{h}^{\perp}}(\xi)=\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}}(J(\omega\odot\xi))$

∎

####  18.3. Characteristic classes of singular foliations

Let $(M,{\mathcal{F}})$ be a perfect singular foliation, i.e. ${\mathcal{F}}$
is a perfect module. Let $(E_{\bullet},\mathbb{E})$ be a cohesive module
resolves ${\mathcal{F}}$, i.e

$0\to E_{-n}\stackrel{{\scriptstyle
d_{n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
d_{2}}}{{\longrightarrow}}E_{-1}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}E_{0}\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}{\mathcal{F}}\longrightarrow
0$

By similar method for holomorphic singular foliation, we can construct an
$L_{\infty}$-algebroid structure on $E_{\bullet}$. Let
$\operatorname{CE}(E_{\bullet})=\operatorname{Sym}^{\bullet}E^{\vee}_{\bullet}[-1]$
be Chevalley-Eilenberg algebra of the $L_{\infty}$-algebroid $E_{\bullet}$.

######  Proposition 18.14.

Let $N{\mathcal{F}}=T_{M}/{\mathcal{F}}$ be the normal sheaf of the singular
foliation ${\mathcal{F}}$ which is perfect, then $N{\mathcal{F}}$ is also
perfect, and we have a normal complex

$0\to E_{-n}\stackrel{{\scriptstyle
d_{n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
d_{2}}}{{\longrightarrow}}E_{-1}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}E_{0}\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}T_{M}\longrightarrow
N{\mathcal{F}}\longrightarrow 0$

which resolves $N{\mathcal{F}}$ and carries a Bott ${\mathbb{Z}}$-connection
$\mathbb{B}$.

###### Proof.

Directly follows from the $L_{\infty}$-pair $(T_{M},E_{\bullet})$. ∎

######  Corollary 18.15.

Let ${\mathcal{F}}$ be a perfect singular foliation and let $E_{\bullet}$ be
an $L_{\infty}$-algebroid which resolves ${\mathcal{F}}$. Then there exists an
$L_{\infty}$-algebroid structure on $E_{\bullet}\oplus(E_{\bullet}[1]\to
T_{M})$ which is quasi-isomorphic to the tangent module $T_{M}$.

####  18.4. Atiyah class for $L_{\infty}$-algebroids

In this section, we will construct the Atiyah class for an $L_{\infty}$-pair
$(\mathfrak{g},\mathfrak{h})$. Let $(E,\mathbb{E}_{\mathfrak{h}}^{E})$ be a
cohesive module over $\mathfrak{h}$. Here
$\mathbb{E}_{\mathfrak{h}}^{E}=d_{{\mathcal{O}}(\mathfrak{h})}+D^{\mathfrak{h},E}$
where $D^{\mathfrak{h},E}$ corresponds to the $\mathfrak{h}$-module structure
on $E$ which is an ${\mathcal{O}}(\mathfrak{h})$-linear map
${\mathcal{O}}(\mathfrak{h})\otimes E\to{\mathcal{O}}(\mathfrak{h})\otimes E$.

Recall that $\mathfrak{h}^{\perp}=N^{\vee}$ is also an $\mathfrak{h}$-module,
hence $\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E)$ inherits
a $\infty$-representation over $\mathfrak{h}$, with the
${\mathbb{Z}}$-connection defined by

(18.6)
$\mathbb{E}^{\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E)}_{\mathfrak{h}}=d_{{\mathcal{O}}(\mathfrak{h})}+D^{\mathfrak{h}^{\perp}}+[D^{\mathfrak{h},E},-]$

Denote the cohomology of the complex
$(\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E),\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E)})$
by
$H^{\bullet}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))$.

By the surjectivity of the map $\iota^{\vee}$, we can lift
$D^{\mathfrak{h},E}\in({\mathcal{O}}(\mathfrak{h})\otimes\underline{\operatorname{End}}(E))_{1}$
to an element
$D^{\mathfrak{g},E}\in({\mathcal{O}}(\mathfrak{g})\otimes\underline{\operatorname{End}}(E))_{1}$.
We get a ${\mathbb{Z}}$-connection
$\mathbb{E}_{\mathfrak{g}}^{E}=d_{{\mathcal{O}}(\mathfrak{g})}+D^{\mathfrak{g},E}$.

Note that $(E,\mathbb{E}_{\mathfrak{g}}^{E})$ is not necessarily a cohesive
module, i.e. the curvature $R_{\mathbb{E}_{\mathfrak{g}}^{E}}$ might not
vanish.

We can easily calculate
$R_{\mathbb{E}_{\mathfrak{g}}^{E}}=d_{{\mathcal{O}}(\mathfrak{h})}\circ
D^{\mathfrak{g},E}+(D^{\mathfrak{g},E})^{2}$.

We have the following commutative diagram

${E}$${{{\mathcal{O}}(\mathfrak{g})\otimes
E}}$${{{\mathcal{O}}(\mathfrak{h})\otimes\mathfrak{g}^{\vee}\otimes
E}}$${E}$${{{\mathcal{O}}(\mathfrak{h})\otimes
E}}$${{{\mathcal{O}}(\mathfrak{h})\otimes\mathfrak{h}^{\vee}\otimes
E}}$$\scriptstyle{R_{\mathbb{E}_{\mathfrak{g}}^{E}}}$$\scriptstyle{J\otimes
1}$$\scriptstyle{=}$$\scriptstyle{R_{\mathbb{E}_{\mathfrak{h}}^{E}}}$$\scriptstyle{\iota^{\vee}\otimes
1}$$\scriptstyle{1\otimes\iota^{\vee}\otimes
1}$$\scriptstyle{d^{\mathsf{dR}}_{\mathfrak{h}}\otimes 1}$

which implies that

$(1\otimes\iota^{\vee}\otimes 1)\circ(J\otimes 1)\circ
R_{\mathbb{E}_{\mathfrak{g}}^{E}}=0$

Therefore, we get an element $\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}$ of total
degree 2.

######  Proposition 18.16.

  1. (1)

$d_{{\mathcal{O}}(h)}\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}=0$, hence we get a
cocycle in the Chevalley-Eilenberg complex of
$(\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E),\mathbb{E}^{\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E)})$.

  2. (2)

The cohomology class $[\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}]$ in the
$L_{\infty}$-algebroid cohomology
$H^{\bullet}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))$
is independent of the extension $\mathbb{E}_{\mathfrak{g}}^{E}$. We call
$[\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}]$ the Atiyah class of the
$L_{\infty}$-pair $(\mathfrak{g},\mathfrak{h})$ with respect to $E$.

  3. (3)

For the canonical $\mathfrak{h}$-module $(\mathfrak{g}/\mathfrak{h})$, there
is a canonical Atiyah class

$[\alpha^{\mathfrak{g}/\mathfrak{h}}]\in
H^{2}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(\mathfrak{g}/\mathfrak{h}))=H^{2}(\mathfrak{h},\underline{\operatorname{Hom}}(\mathfrak{g}/\mathfrak{h}\otimes\mathfrak{g}/\mathfrak{h},\mathfrak{g}/\mathfrak{h})).$

###### Proof.

First we need a lemma

######  Lemma 18.17.

Let $x\in{\mathcal{A}}(\mathfrak{g},\underline{\operatorname{End}}(E))$
satisfy $(\iota^{\vee}\otimes 1)(x)=0$, then

(18.7) $[D^{\mathfrak{h},E},(J\otimes 1)(x)]=(J\otimes
1)[D^{\mathfrak{g},E},x].$

###### Proof.

It suffices to prove for $x$ homogeneous. We have

$\displaystyle(J\otimes 1)[D^{\mathfrak{g},E},x]=$ $\displaystyle(J\otimes
1)(D^{\mathfrak{g},E}\circ x-(-1)^{|x|}x\circ D^{\mathfrak{g},E})$
$\displaystyle=$ $\displaystyle(\iota^{\vee}\otimes
1)(D^{\mathfrak{g},E})\circ(J\otimes 1)(x)+(J\otimes
1)(D^{\mathfrak{g},E})\circ(\iota^{\vee}\otimes 1)(x)$
$\displaystyle-(-1)^{|x|}\big{(}(\iota^{\vee}\otimes 1)(x)\circ(J\otimes
1)(D^{\mathfrak{g},E})+(J\otimes 1)(x)\circ(\iota^{\vee}\otimes
1)(D^{\mathfrak{g},E})\big{)}$ $\displaystyle=$ $\displaystyle
D^{\mathfrak{h},E}\circ(J\otimes 1)(x)-(-1)^{|x|}(J\otimes 1)(x)\circ
D^{\mathfrak{h},E}$ $\displaystyle=$
$\displaystyle[D^{\mathfrak{h},E},(J\otimes 1)(x)].$

∎

Now, let us prove the proposition. By the previous commutative diagram
$(\iota^{\vee}\otimes 1)\circ
R_{\mathbb{E}_{\mathfrak{g}}^{E}}=R_{\mathbb{E}_{\mathfrak{h}}^{E}}=0$, hence
$(J\otimes
1)(R_{\mathbb{E}_{\mathfrak{g}}^{E}})\in{\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E))$.
We have

$\displaystyle\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E)}(\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}})=$
$\displaystyle(d_{{\mathcal{O}}(\mathfrak{h})}+D^{\mathfrak{h}^{\perp}}+[D^{\mathfrak{h},E},-])((J\otimes
1)(R_{\mathbb{E}_{\mathfrak{g}}^{E}}))$ $\displaystyle=$
$\displaystyle(d_{{\mathcal{O}}(\mathfrak{h})}+D^{\mathfrak{h}^{\perp}})((J\otimes
1)(R_{\mathbb{E}_{\mathfrak{g}}^{E}}))+[D^{\mathfrak{h},E},(J\otimes
1)(R_{\mathbb{E}_{\mathfrak{g}}^{E}})]$ $\displaystyle=$
$\displaystyle(J\otimes
1)(d_{{\mathcal{O}}(\mathfrak{g})}R_{\mathbb{E}_{\mathfrak{g}}^{E}}+[D^{\mathfrak{g},E},(R_{\mathbb{E}_{\mathfrak{g}}^{E}})])$

where the last step follows from the Bianchi identity.

Next, let us look at (2). Consider another ${\mathbb{Z}}$-connection
$\mathbb{E}^{\prime}=d_{{\mathcal{O}}(\mathfrak{g})}+{D^{\mathfrak{g},E}}^{\prime}$
lifts the flat ${\mathbb{Z}}$-connection
$\mathbb{E}^{\mathfrak{h},E}_{\mathfrak{h}}$. Let
$\omega=\mathbb{E}^{\mathfrak{g},E}_{\mathfrak{g}}-\mathbb{E}^{\prime}=D^{\mathfrak{g},E}-{D^{\mathfrak{g},E}}^{\prime}\in{\mathcal{A}}(\mathfrak{g},\underline{\operatorname{End}}(E))^{1}$,
and we have $(\iota^{\vee}\otimes 1)(\omega)=0$ and $(J\otimes
1)(\omega)\in{\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))$,
which implies $(J\otimes 1)(\omega^{2})=0$. Now we have

$\displaystyle\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}-\alpha_{\mathbb{E}^{\prime}}=$
$\displaystyle(J\otimes
1)(R_{\mathbb{E}_{\mathfrak{g}}^{E}}-R_{\mathbb{E}^{\prime}})$
$\displaystyle=$ $\displaystyle(J\otimes
1)(d_{{\mathcal{A}}(\mathfrak{g})}(D^{\mathfrak{g},E})+(D^{\mathfrak{g},E})^{2}-d_{{\mathcal{A}}(\mathfrak{g})}({D^{\mathfrak{g},E}}^{\prime})-({D^{\mathfrak{g},E}}^{\prime})^{2})$
$\displaystyle=$ $\displaystyle(J\otimes
1)(d_{{\mathcal{A}}(\mathfrak{g})}\omega+\omega^{2}+[{D^{\mathfrak{g},E}}^{\prime},\omega])$
$\displaystyle=$
$\displaystyle(d_{{\mathcal{A}}(\mathfrak{h})}+D^{\mathfrak{h},\mathfrak{h}^{\perp}})((J\otimes
1)(\omega)+[D^{\mathfrak{g},E},(J\otimes 1)(\omega)]$ $\displaystyle=$
$\displaystyle(d_{{\mathcal{A}}(\mathfrak{h})}+D^{\mathfrak{h},\mathfrak{h}^{\perp}}+[D^{\mathfrak{g},E},-])((J\otimes
1)(\omega))=\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E)}((J\otimes
1)(\omega))$

which implies that the cohomology classes of
$\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}$ and $\alpha_{\mathbb{E}^{\prime}}$
are the same.

Finally, (3) follows from the standard identification
$\mathfrak{h}^{\perp}\simeq(\mathfrak{g}/\mathfrak{h})^{\vee}$.

∎

Next, we shall construct the Atiyah classes from another way. Again, let
$(\mathfrak{g},\mathfrak{h})$ be an $L_{\infty}$-pair. On
$\mathfrak{h}^{\vee}$, there exists a coadjoint $\mathfrak{h}$-module
structure
$\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\vee}}=d_{{\mathcal{A}}(\mathfrak{h})}+D^{\mathfrak{h},\mathfrak{h}^{\vee}}$
which is dual to the adjoint $\infty$-representation of $\mathfrak{h}$ on
itself. There is a natural $\mathfrak{h}$-module structure on $\mathfrak{g}$,
and similarly on $\mathfrak{g}^{\vee}$. We have a short exact sequence of
$\mathfrak{h}$-modules

$0\to\mathfrak{h}^{\perp}\to\mathfrak{g}^{\vee}\stackrel{{\scriptstyle\iota^{\vee}}}{{\to}}\mathfrak{h}^{\vee}\to
0$

Now consider $(E,\mathbb{E})$ an $\mathfrak{h}$-module, then we have

$0\to\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E)\to\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E)\stackrel{{\scriptstyle\iota^{\vee}\otimes
1}}{{\to}}\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E)\to 0$

which induces a short exact sequence of dga’s

$0\to{\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))\to{\mathcal{A}}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))\stackrel{{\scriptstyle
1\otimes\iota^{\vee}\otimes
1}}{{\to}}{\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E))\to
0$

Hence we have a long exact sequence of $L_{\infty}$-algebroid cohomology

$\displaystyle\cdots
H^{1}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))\to
H^{1}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))\stackrel{{\scriptstyle
1\otimes\iota^{\vee}\otimes
1}}{{\to}}H^{1}(\mathfrak{h},\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E))$
$\displaystyle\stackrel{{\scriptstyle\delta}}{{\to}}H^{2}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))\to
H^{2}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))\stackrel{{\scriptstyle
1\otimes\iota^{\vee}\otimes
1}}{{\to}}H^{2}(\mathfrak{h},\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E))\cdots$

######  Lemma 18.18.

The element $(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes
1)(D^{\mathfrak{h},E})\in{\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E))$
is a degree 1 cocycle.

###### Proof.

Since $d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}$ is a derivation on
${\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\vee})$,
$d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes 1$ is a derivation on
${\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E))$.
Hence,

$\displaystyle(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)({(D^{\mathfrak{h},E})}^{2})=$
$\displaystyle(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(D^{\mathfrak{h},E})\circ
D^{\mathfrak{h},E}+D^{\mathfrak{h},E}\circ(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(D^{\mathfrak{h},E})$ $\displaystyle=$
$\displaystyle[D^{\mathfrak{h},E},(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)D^{\mathfrak{h},E}]$

It is easy to verify that

$(\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\vee}}\otimes
1)(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(D^{\mathfrak{h},E})=(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(d_{A(\mathfrak{h})}D^{\mathfrak{h},E})$

Now

$\displaystyle\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E)}((d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(D^{\mathfrak{h},E}))$
$\displaystyle=(\mathbb{E}_{\mathfrak{h}}^{\mathfrak{h}^{\vee}}+[D^{\mathfrak{h},E},-])((d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(D^{\mathfrak{h},E}))$
$\displaystyle=(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(d_{A(\mathfrak{h})}D^{\mathfrak{h},E})+(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)({(D^{\mathfrak{h},E})}^{2})$
$\displaystyle=(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(d_{A(\mathfrak{h})}D^{\mathfrak{h},E}+{(D^{\mathfrak{h},E})}^{2})=0$

since $d_{A(\mathfrak{h})}D^{\mathfrak{h},E}+{(D^{\mathfrak{h},E})}^{2}=0$
which is the Maurer-Cartan equation. ∎

######  Proposition 18.19.

The cohomology class $\delta[(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes
1)(D^{\mathfrak{h},E})]\in
H^{2}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))$
is the same as the Atiyah class $\alpha^{E}_{\mathfrak{g},\mathfrak{h}}$.

###### Proof.

First we choose an element
$\beta\in{\mathcal{A}}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))$
of degree 1 such that $(1\otimes\iota^{\vee}\otimes
1)(\beta)=(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes 1)(D^{\mathfrak{h},E})$. Then
we get
$\alpha\in{\mathcal{A}}^{2}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))$
by

$\alpha=\mathbb{E}_{\mathfrak{h}}^{\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E)}(\beta)=(d_{{\mathcal{A}}(\mathfrak{h})}+D^{\mathfrak{g}^{\vee}}+[D^{\mathfrak{h},E},-])(\beta)$

whose cohomology class is the one given by
$\delta[(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes 1)(D^{\mathfrak{h},E})]$. We
want to show that $[\alpha]$ actually agrees with the Atiyah class
$[\alpha^{E}_{\mathfrak{g},\mathfrak{h}}]$. We will do this by extending the
${\mathbb{Z}}$-connection $\mathbb{E}_{\mathfrak{h}}^{E}$ given by the
$\mathfrak{h}$-module structure of $E$ and extend it to a
${\mathbb{Z}}$-connection $\mathbb{E}^{E}_{\mathfrak{g}}$ over $\mathfrak{g}$,
and show that the resulting Atiyah cocycle
$\alpha^{E}_{\mathbb{E}^{E}_{\mathfrak{g}}}$ coincides with $\alpha$.

First, we want to find an element
$D^{\mathfrak{g},E}\in{\mathcal{A}}(\mathfrak{g},\underline{\operatorname{End}}(E))$
with $(J\otimes 1)(D^{\mathfrak{g},E})=\beta$ and $(\iota^{\vee}\otimes
1)(D^{\mathfrak{g},E})=D^{\mathfrak{h},E}$. By surjectivity of $J$, we can
find some
$K^{\mathfrak{g},E}\in{\mathcal{A}}(\mathfrak{g},\underline{\operatorname{End}}(E))$
such that $(J\otimes 1)(K^{\mathfrak{g},E})=\beta$. Now

$\displaystyle(d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\otimes
1)(\iota^{\vee}\otimes 1)(K^{\mathfrak{g},E})=$
$\displaystyle(1\otimes\iota^{\vee}\otimes 1)(J\otimes 1)(K^{\mathfrak{g},E})$
$\displaystyle=$ $\displaystyle(1\otimes\iota^{\vee}\otimes 1)(\beta)$
$\displaystyle=$ $\displaystyle(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes
1)(D^{\mathfrak{h},E})$

e Now $(\iota^{\vee}\otimes 1)(K^{\mathfrak{g},E})-D^{\mathfrak{h},E}=\phi$
for some $\phi\in\Gamma(\underline{\operatorname{End}}(E))$ as $\ker
d^{\mathsf{dR}}_{{\mathcal{A}}(\mathfrak{h})}\simeq{\mathcal{C}}^{\infty}(M)$.
Hence, we could let $D^{\mathfrak{g},E}=K^{\mathfrak{h},E}-\phi$.

Now

$\displaystyle\alpha^{E}_{\mathbb{E}^{E}_{\mathfrak{g}}}=$
$\displaystyle(J\otimes
1)(d_{{\mathcal{A}}(\mathfrak{g})}D^{\mathfrak{g},E}+(D^{\mathfrak{g},E})^{2})$
$\displaystyle=$
$\displaystyle(\mathbb{E}_{\mathfrak{h}}^{\mathfrak{g}^{\vee}}\circ(J\otimes
1)(D^{\mathfrak{g},E})+[(\iota^{\vee}\otimes 1)(D^{\mathfrak{g},E}),(J\otimes
1)(D^{\mathfrak{h},E})]$ $\displaystyle=$
$\displaystyle(d_{{\mathcal{A}}(\mathfrak{h})}+D^{\mathfrak{g}^{\vee}})(J\otimes
1)(D^{\mathfrak{g},E})+[D^{\mathfrak{h},E},(J\otimes 1)(D^{\mathfrak{h},E})]$
$\displaystyle=$
$\displaystyle(d_{{\mathcal{A}}(\mathfrak{h})}+D^{\mathfrak{g}^{\vee}}+[D^{\mathfrak{h},E},-])(\beta)=\alpha$

Hence, $[\alpha]$ agrees with $[\alpha^{E}_{\mathbb{E}^{E}_{\mathfrak{g}}}]$.
∎

Let $\mathfrak{h}$ be an $L_{\infty}$-algebroid and $(E,\mathbb{E})$ a
$\mathfrak{h}$-module, we have the canonical Abelian extension
$(\mathfrak{g}=\mathfrak{h}\oplus E,\mathbb{E})$ of $\mathfrak{h}$ along $E$,
which induces an $L_{\infty}$-pair $(\mathfrak{g},\mathfrak{h})$. The Atiyah
class $\alpha_{\mathbb{E}}$ is trivial in this case. Hence, we see that the
Atiyah class measures the nontriviality of the extension of $\mathfrak{h}$ to
$\mathfrak{g}$.

Next, we are going to see some simple situation when does Atiyah classes
vanish.

######  Proposition 18.20.

Let $(\mathfrak{g},\mathfrak{h})$ be an $L_{\infty}$-pair and
$(E,\mathbb{E}=d_{{\mathcal{A}}(h)}+D^{\mathfrak{h},E})$ be an
$\mathfrak{h}$-module, then the Atiyah class
$[\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}]\in
H^{\bullet}(\mathfrak{h},\mathfrak{h}^{\perp}\otimes\underline{\operatorname{End}}(E))$
vanishes if any of the following equivalent condition is met:

  1. (1)

There exists a ${\mathbb{Z}}$-connection $\mathbb{E}^{\prime}$ over
$\mathfrak{g}$ on $E$ extending $\mathbb{E}$ such that the Atyiah cocycle
$\alpha_{\mathbb{E}_{\mathfrak{g}}^{E}}$ relative to
$\alpha_{\mathbb{E^{\prime}}}$ vanishes.

  2. (2)

There exists a degree 1 cocycle
$\phi\in{\mathcal{A}}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))$
such that $(1\otimes\iota^{\vee}\otimes
1)(\phi)=(d_{\mathfrak{h}}^{\mathsf{dR}}\otimes 1)(D^{\mathfrak{h},E})$.

  3. (3)

There exists an $\mathfrak{h}$-module morphism
$\\{\phi_{k}:\operatorname{Sym}\mathfrak{h}\otimes\mathfrak{g}\to\underline{\operatorname{End}}(E)[1]\\}_{k\geq
0}$ from $\mathfrak{g}$ to $\underline{\operatorname{End}}(E)[1]$ extending
the canonical $\mathfrak{h}$ module morphism
$\\{\phi_{k}^{\mathfrak{h},E}\\}_{k\geq 0}$ from $\mathfrak{h}$ to
$\underline{\operatorname{End}}(E)[1]$, i.e.

$(\phi_{k})\circ(1\otimes
j)=\phi_{k}^{\mathfrak{h},E}:\operatorname{Sym}\mathfrak{h}\otimes\mathfrak{g}\to\underline{\operatorname{End}}(E)[1]$

###### Proof.

(1) $\Rightarrow$ (2): Clearly (3) implies that the Atiyah class of $E$
vanishes. Hence $\delta[(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes
1)(D^{\mathfrak{h},E})]=0$. We can find a degree 1 element
$\tilde{\phi}\in{\mathcal{A}}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))$
such that $[(1\otimes\iota^{\vee}\otimes
1)(\tilde{\phi}]=[(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes
1)(D^{\mathfrak{h},E})]$. Then we can find an element
$\beta\in{\mathcal{A}}(\mathfrak{h},\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E))$
of degree 0 such that
$\mathbb{E}^{\mathfrak{h}^{\vee}\otimes\underline{\operatorname{End}}(E)}_{\mathfrak{h}}(\beta)=(1\otimes\iota^{\vee}\otimes
1)(\tilde{\phi})-(d^{\mathsf{dR}}_{\mathfrak{h}}\otimes
1)(D^{\mathfrak{h},E})$, and therefore a
$\gamma\in{\mathcal{A}}^{1}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))$
maps to $\mathfrak{h}$, i.e. $(1\otimes\iota^{\vee}\otimes 1)(\gamma)=\beta$.

Now we let $\phi=\tilde{\phi}-\beta$, by an easy calculation, we have
$(1\otimes\iota^{\vee}\otimes 1)(\phi)=(d_{\mathfrak{h}}^{\mathsf{dR}}\otimes
1)(D^{\mathfrak{h},E})$.

(2) $\Rightarrow$ (1): Given
$\phi\in{\mathcal{A}}^{1}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))$,
we can find a
$D^{\mathfrak{g},E}\in{\mathcal{A}}(L,\underline{\operatorname{End}}(E)))$
such that $(J\otimes 1)(D^{\mathfrak{g},E})=\beta$ and $(\iota^{\vee}\otimes
1)(D^{\mathfrak{g},E})=D^{\mathfrak{h},E}$. Now
$\mathbb{E}^{\prime}=d_{{\mathcal{A}}(\mathfrak{g})}+D^{\mathfrak{g},E}$ is a
${\mathbb{Z}}$-connection extending $\mathbb{E}$, and the associated Atiyah
cocycle

$\displaystyle\alpha^{E}_{\mathbb{E}^{\prime}}=$ $\displaystyle(J\otimes
1)(R_{\mathbb{E}}^{\prime})$ $\displaystyle=$ $\displaystyle(J\otimes
1)(d_{{\mathcal{A}}(\mathfrak{g})}D^{\mathfrak{g},E}+(D^{\mathfrak{g},E})^{2})$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mathfrak{h}}^{\mathfrak{g}^{\vee}}((J\otimes
1)(D^{\mathfrak{g},E}))+[(\iota^{\vee}\otimes 1)(D^{\mathfrak{g},E}),(J\otimes
1)(D^{\mathfrak{g},E})]$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\mathfrak{h}}^{\mathfrak{g}^{\vee}}(\phi)+[D^{\mathfrak{h},E},\phi]=0$

(2) $\Leftrightarrow$ (3): Note that
$\phi\in{\mathcal{A}}^{1}(\mathfrak{h},\mathfrak{g}^{\vee}\otimes\underline{\operatorname{End}}(E))$
consists of a family of map
$\phi_{k}:\operatorname{Sym}\mathfrak{h}\otimes\mathfrak{g}\to\underline{\operatorname{End}}(E)[1]$.

∎

####  18.5. Scalar Atiyah classes and Todd classes

Let $(E,\mathbb{E})$ be an $\mathfrak{h}$-module and
$(\mathfrak{g},\mathfrak{h})$ an $L_{\infty}$-pair. We define the scalar
Atiyah classes of the $L_{\infty}$-pair to be

$c_{k}(\mathfrak{g},\mathfrak{h})=\frac{1}{k!}\bigg{(}\frac{i}{2\pi}\bigg{)}^{k}\operatorname{Str}(\alpha_{\mathfrak{g},\mathfrak{h}})\in
H^{k}(\mathfrak{h},\mathfrak{h}^{\perp})$

Let
$\operatorname{Ber}:\Gamma(\underline{\operatorname{End}}(E))\to{\mathcal{C}}^{\infty}(M)$
be the Berezinian map (superdeterminant), then we define the Todd class of an
$L_{\infty}$-pair $(\mathfrak{g},\mathfrak{h})$ to be

$\operatorname{Td}_{\mathfrak{g},\mathfrak{h}}=\operatorname{Ber}\Bigg{(}\frac{\alpha_{(\mathfrak{g},\mathfrak{h})}}{1-e^{-\alpha_{(\mathfrak{g},\mathfrak{h})}}}\Bigg{)}\in\bigoplus_{k\geq
0}H^{k}(\mathfrak{h},\mathfrak{h}^{\perp})$

######  Example 18.21.

Let $X$ be a compact Kahler manifold. Consider the $L_{\infty}$-pair
$(T_{{\mathbb{C}}}X,T^{0,1}_{X})$, then the natural map of sheaf cohomology
$\bigoplus_{k}H^{k}(X,\Omega_{X}^{k})\to\bigoplus_{k}H^{2k}(X,{\mathbb{C}})$
sends the scalar Atiyah classes $c_{k}(T_{X}^{1,0})$ and the Todd class
$\operatorname{Td}_{T_{X}^{1,0}}$ of the $L_{\infty}$-pair
$(T_{{\mathbb{C}}}X,T^{0,1}_{X})$ to the $k$-th Chern characters
$\operatorname{ch}_{k}(X)$ and the Todd class $\operatorname{Td}_{X}$ of $X$
respectively.

####  18.6. Infinitesimal ideal system of $L_{\infty}$-algebroids

In this section, we will define infinitesimal ideal systems associated to an
$L_{\infty}$-pair, and show that there is a natural infinitesimal ideal
systems associated to an $L_{\infty}$-algebroid fibration. The infinitesimal
ideal system structure is actually related to the vanishing to Atiyah classes.

######  Definition 18.22.

Consider an $L_{\infty}$-pair $(\mathfrak{g},\mathfrak{h})$, we define an
infinitesimal ideal system to be a triple
$({\mathcal{F}}_{M},\mathfrak{h},\mathbb{E})$ such that
${\mathcal{F}}_{M}\subset T_{M}$ is an involutive locally free subsheaf of the
tangent sheaf $T_{M}$, $\rho(\mathfrak{h})\subset{\mathcal{F}}_{M}$, and
$\mathbb{E}$ is a flat ${\mathbb{Z}}$-connection over ${\mathcal{F}}_{M}$ on
$\mathfrak{g}/\mathfrak{h}$ which satisfies

  1. (1)

If $g\in\mathfrak{g}$ is $\mathbb{E}$-flat, then
$[g,h_{1},\cdots,h_{i-1}]\in\mathfrak{h}$ for all $h_{i}\in h$ and all
$i$-brackets for $i\geq 2$.

  2. (2)

If $g_{1},\cdots,g_{i}\in\mathfrak{g}$ are $\mathbb{E}$-flat, then
$[g_{1},\cdots,g_{i}]$ is also $\mathbb{E}$-flat.

  3. (3)

If $g\in\mathfrak{g}$ is $\mathbb{E}$-flat, then $\rho(g)$ is
$\nabla^{F_{M}}$-flat, where $\nabla^{F_{M}}$ is the Bott connection on
$T_{M}/{\mathcal{F}}_{M}$.

This is a direct generalization of infinitesimal ideal systems in Lie
algebroids.

######  Proposition 18.23.

For any $L_{\infty}$-algebroid fibration $\phi:\mathfrak{g}\to\mathfrak{h}$
over a ${\mathcal{C}}^{\infty}$-map $f:M\to N$, there exist an infinitesimal
ideal system $({\mathcal{F}},\mathfrak{h},\mathbb{E})$ associated to it.

######  Definition 18.24.

We define the Atiyah class of an infinitesimal ideal system
$({\mathcal{F}}_{M},\mathfrak{h}.\mathbb{E})$ in an $L_{\infty}$-algebroid
$\mathfrak{g}$ to be the Atiyah class of the flat ${\mathbb{Z}}$-connection
$\mathbb{E}$.

######  Proposition 18.25.

Let $(\mathfrak{g},\mathfrak{h})$ be an $L_{\infty}$-pair on $M$. If there
exists an infinitesimal ideal system
$({\mathcal{F}}_{M},\mathfrak{h}.\mathbb{E})$ in $\mathfrak{g}$, such that the
quotient $(\mathfrak{g}/\mathfrak{h})/\mathbb{E}\to M/{\mathcal{F}}_{M}$
exists and is smooth, then the Atiyah class of the infinitesimal ideal system
vanishes.

######  Example 18.26 (Simple foliations).

For example, if ${\mathcal{F}}$ is a simple foliation, i.e. the leaf space of
${\mathcal{F}}$ is a manifold, then the Atiyah class associated to
$L_{\infty}$-pair $(TM,{\mathcal{F}})$ vanishes.

##  Part V Singular foliations and $L_{\infty}$-algebroids

###  19\. Singular foliation and their homotopy theory

####  19.1. Foliations

A foliation is a partition of a manifold into immersed submanifolds.

######  Definition 19.1 ([MM03]).

Let $M$ be a smooth manifold. A (regular) foliation ${\mathcal{F}}$ of
codimension $q$ on $M$ can be described in the following equivalent data:

  1. (1)

A foliation atlas
$\\{\phi_{i}:U_{i}\to{\mathbb{R}}^{n-q}\times{\mathbb{R}}^{q}\\}$ of $M$ for
which the change-of-coordinates diffeomorphisms $\phi_{ij}$’s are globally of
the form

$\phi_{ij}(x,y)=\big{(}g_{ij})(x,y),h_{ij}(y)\big{)}$

with respect to the decomposition
${\mathbb{R}}^{n}={\mathbb{R}}^{n-q}\times{\mathbb{R}}^{q}$, where $n=\dim M$.
Note that each leaf is partitioned into plaques, which are connected
components of the submanifolds
$\phi^{-1}_{i}({\mathbb{R}}^{n-1}\times\\{y\\}),y\in{\mathbb{R}}^{q}$. The
plaques globally glue to leaves, which are immersed submanifolds of $M$. We
call the first $n-q$ directions in the decomposition the leaf directions, and
the last $q$ directions the transversal directions.

  2. (2)

An open cover $\\{U_{i}\\}$ of $M$ with submersions
$s_{i}:U_{i}\to{\mathbb{R}}^{q}$ such that there are diffeomorphisms

$\gamma_{ij}:s_{j}(U_{i}\cap U_{j})\to s_{i}(U_{i}\cap U_{j})$

with $\gamma_{ij}\circ s_{j}|_{U_{i}\cap U_{j}}=s_{i}|_{U_{i}\cap U_{j}}$,
which is necessarily unique. $\gamma_{ij}$’s satisfy the cocycle condition
$\gamma_{ij}\circ\gamma_{jk}=\gamma_{ik}$, which is called the Haefliger
cocycle representing ${\mathcal{F}}$.

  3. (3)

An integrable sub-bundle $F$ of $TM$ of rank $n-q$, i.e. for any
$X,Y\in\Gamma(F)$, $[X,Y]\in\Gamma(F)$. We usually denote $\Gamma(F)$ by
${\mathcal{F}}$ and $F$ by $T{\mathcal{F}}$.

  4. (4)

A locally trivial differential ideal
$\mathcal{J}=\oplus_{k=1}^{n}\mathcal{J}^{k}$ of rank $q$ in the de Rham dga
$\Omega^{\bullet}(M)$.

When there is a foliation on a manifold $M$, we denote by $(M,{\mathcal{F}})$
a foliated manifold.

The first two conditions are descriptions of ${\mathcal{F}}$ by local charts,
which tells us that locally a foliation is decomposed into two distinction
directions: the leaf direction and the transversal direction. We denote
$M/{\mathcal{F}}$ the leaf space by quotienting equivalence relations such
that $x\sim y$ if $x$ and $y$ lie on the same leaf.

######  Definition 19.2.

Let $M$ be a smooth manifold. We define a complex foliation to be an
involutive sub-bundle of the complexified tangent bundle
$T_{{\mathbb{C}}}M=TM\otimes_{{\mathbb{R}}}{\mathbb{C}}$.

A complex foliation ${\mathcal{F}}$ is called real if and
$\overline{T{\mathcal{F}}}=T{\mathcal{F}}$. In this case, we can define a real
foliation $T{\mathcal{F}}_{{\mathbb{R}}}=T{\mathcal{F}}\cap TM$.

######  Definition 19.3.

A map between two foliated manifolds
$f:(M_{1},{\mathcal{F}}_{1})\to(M_{2},{\mathcal{F}}_{2})$ is called foliated
if it preserves the foliated structure, i.e. $f$ maps leaves of
$(M_{1},{\mathcal{F}}_{1})$ into leaves of $(M_{2},{\mathcal{F}}_{2})$.

We denote $\mathsf{Mfd}^{{\operatorname{Fol}}}$ the category of foliated
manifolds where the morphisms are foliated maps, and
$\mathsf{Mfd}^{{\operatorname{Fol}}}_{{\mathbb{C}}}$ the category of complex-
foliated manifolds.

######  Example 19.4.

Let $M$ be a smooth manifold. The two most simple foliation on $M$ is given by
(1) foliation by whole manifold, i.e. ${\mathcal{F}}=T_{M}$, and (2) foliation
by points , i.e. ${\mathcal{F}}=M\times\\{0\\}$. For the first case, the
transversal direction is trivial, i.e. 0-dimensional. For the second case, the
transversal direction is the whole manifold, which is again foliated by
points. Hence, we see that the transversal direction always locally admits a
foliation by points.

######  Example 19.5 (Product foliations).

Given two foliations $(M_{1},{\mathcal{F}}_{1})$ and
$(M_{2},{\mathcal{F}}_{2})$, we can form their product $(M_{1}\times
M_{2},{\mathcal{F}}_{1}\times{\mathcal{F}}_{2})$, where
${\mathcal{F}}_{1}\times{\mathcal{F}}_{2}$ is given by $F_{1}\times
F_{2}\subset TM_{1}\times TM_{2}=T(M_{1}\times M_{2})$.

We say $f$ is transversal to ${\mathcal{F}}$ if $f$ is transverse to all
leaves of ${\mathcal{F}}$ in the image of $f$, i.e. for any $x\in N$,

$(df)_{x}(T_{x}N)+T_{f(x)}({\mathcal{F}})=T_{f(x)}M$

######  Example 19.6 (Pullback foliations).

Let $(M,{\mathcal{F}})$ be a manifold with foliation ${\mathcal{F}}$. Consider
a smooth map $f:N\to M$ transversal to ${\mathcal{F}}$. It is not hard to show
that the pull-back $f^{*}{\mathcal{F}}$ is a foliation on $N$. This example
will be important when we consider the ’derived’ counterpart of the foliation.

######  Example 19.7 (Flat bundles).

Let $G$ be a group acting freely and properly discontinuesly on a connected
manifold $\tilde{M}$ and $\tilde{M}/G=M$. For example, we can take $\tilde{M}$
be the universal cover of $M$ and $G=\pi_{1}(M,x)$ for some $x\in M$. Here we
let $G$ be a right action on $\tilde{M}$. Suppose $G$ also acts on the left on
some manifold $F$, then we can form a quotient space $E=\tilde{M}\times_{G}F$
from the product space $\tilde{M}\times F$ by identifying $(yg,z)\sim(y,gz)$.
It is easy to show that $E$ is a manifold, and we have the following
commutative diagram

${\tilde{M}\times
F\par}$${E}$${\tilde{M}}$${M}$$\scriptstyle{{\operatorname{pr}}_{1}}$$\scriptstyle{\pi}$

The projection ${\operatorname{pr}}_{1}$ induces a submersion $\pi$, which
gives $E$ a fiber bundle structure over $M$ with fiber $F$.

This construction also produces a foliation
${\mathcal{F}}({\operatorname{pr}}_{2})$ on $\tilde{M}\times F$, which is
given by the submersion ${\operatorname{pr}}_{2}:\tilde{M}\times F\to p$. By
the construction, ${\mathcal{F}}({\operatorname{pr}}_{2})$ is $G$-invariant,
hence we can form a quotient foliation
${\mathcal{F}}={\mathcal{F}}({\operatorname{pr}}_{2})/G$ on $E$.

We can look into more detail about leaves of ${\mathcal{F}}$. Let $x\in F$,
and $G_{x}\subset G$ the isotropy group of the $G$-action at $x$, then the
leaf of the foliation $F$ associated to $\tilde{M}\times\\{z\\}$ is
diffeomorphic $\tilde{M}/G_{x}$.

####  19.2. Singular foliation

We also often see foliated structures with singularities, i.e. the dimensions
of leaves are not constant.

######  Definition 19.8 ([AC09], [LLS20]).

Let $M$ be a smooth manifold. A singular foliation ${\mathcal{F}}$ on $M$ is a
locally finitely generated subsheaf of ${\mathcal{O}}_{M}$-modules of the
tangent sheaf $T_{M}$ which is involutive, i.e. closed under Lie brackets.

Equivalently, we can characterize a singular foliation ${\mathcal{F}}$ as a
locally finitely generated $O_{M}$-submodule of $\Gamma(TM)$. Clearly, regular
foliations are singular foliations, since sub-bundles of $TM$ are finitely
generated. By a result of Hermann [Her60], a singular foliation on $M$ induces
a partition of $M$ into leaves.

A singular sub-foliation ${\mathcal{F}}^{\prime}$ of a singular foliation
${\mathcal{F}}$ is a singular foliation such that, for all open sets $U\subset
M$, we have ${\mathcal{F}}^{\prime}(U)\subset{\mathcal{F}}(U)$.

######  Definition 19.9.

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold and $x\in M$. The
tangent space of the leaf at $x\in M$ is the image $F_{x}$ of ${\mathcal{F}}$
in $T_{x}M$. The fiber of ${\mathcal{F}}$ at $x$ is
${\mathcal{F}}_{x}={\mathcal{F}}/I_{x}{\mathcal{F}}$,
where$I_{x}=\\{f\in{\mathcal{O}}_{M}:f(x)=0\\}$.

Let $\operatorname{ev}_{x}:{\mathcal{F}}\to T_{x}M$ be the evaluation map of
${\mathcal{F}}$ at $x$. Clearly $\operatorname{ev}_{x}$ vanishes on
$I_{x}{\mathcal{F}}$, therefore it descends to a map
$\tilde{\operatorname{ev}_{x}}:{\mathcal{F}}_{x}\to F_{x}\subset T_{x}M$.
$\ker{\operatorname{ev}_{x}}$ is a Lie subalgebra of ${\mathcal{F}}$ and
$I_{x}{\mathcal{F}}$ is an ideal in this Lie algebra. It follows that
$\ker\tilde{\operatorname{ev}_{x}}=\ker{\operatorname{ev}_{x}}/I_{x}{\mathcal{F}}$
is a Lie algebra, and we call this Lie algebra the isotropy Lie algebra.

Below are some basic results for fibers and tangent spaces of the leaves of
singular foliations:

######  Proposition 19.10 ([AS06]).

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold, and $x\in M$. We have

  1. (1)

Let $X_{1},\cdots,X_{k}\in{\mathcal{F}}$ whose images in ${\mathcal{F}}_{x}$
form a basis of ${\mathcal{F}}_{x}$, then there is a neighborhood $U$ of $x$
such that ${\mathcal{F}}_{U}$ is generated by $X_{1},\cdots,X_{k}$.

  2. (2)

$\dim F_{x}$ is lower semi-continuous and $\dim{\mathcal{F}}_{x}$ is upper
semi-continuous.

  3. (3)

The set

$U=\\{x\in M:\tilde{\operatorname{ev}}_{x}:{\mathcal{F}}_{x}\to F_{x}\\}\text{
is an isomorphism}$

is the set of continuity of $x\mapsto\dim F_{x}$. $U$ is open and dense.
$F|_{U}\subset TM|_{U}$ is a sub-bundle, hence ${\mathcal{F}}_{U}$ is a
regular foliation.

###### Proof.

See [AS06, Proposition 1.5]. ∎

####  19.3. Holonomy and monodromy

Holonomy of a regular foliation is defined as germs of local diffeomorphisms
of transversals along a path on a leaf. It turns out to be one of the most
import concepts related to foliations, for example, we can construct the
holonomy groupoid of a foliation. A related notion is the monodromy, which
describes the leafwise homotopy classes of paths.

Let $\operatorname{Diff}_{x}(M)$ denote the group of diffeomorphisms of a
manifold fixing $M$.

######  Definition 19.11 ([MM03]).

Let $({\mathfrak{M}},{\mathcal{F}})$ be a foliated manifold. Let $x,y$ be some
points on some leaf $L$ of ${\mathcal{F}}$. Let $S,T$ be transversal sections
(or transversal) at $x$ and $y$, then for any path $\alpha:x\to y$ we can
associate a germ of a diffeomorphism

$\operatorname{Hol}^{S,T}(\alpha):(S,x)\to(T,y)$

which is called the holonomy of $\alpha$ with respect to the transversal
sections $S$ and $T$. For details about the construction of
$\operatorname{Hol}^{S,T}(\alpha)$, see [MM03, Section 2.1].

Two easy but import properties of the holonomy are

  1. (1)

Homotopic paths induces the same holonomy;

  2. (2)

Holonomy is independent of the choice of transversals by identifying the
holonomy of different transversals along the constant path.

Let $x\in L$ and $S$ be a transversal at $x$. By the above properties, we have
a group homomorphism

$\operatorname{Hol}:\pi_{1}(L,x)\to\operatorname{Diff}_{0}({\mathbb{R}}^{q})$

by the independence of the choice of $T$, we get the holonomy homomorphism
$\operatorname{Hol}$

$\operatorname{Hol}:\pi_{1}(L,x)\to\operatorname{Diff}_{x}(T)\simeq\operatorname{Diff}_{0}({\mathbb{R}}^{q})$

which is defined up to conjugations in
$\operatorname{Diff}_{0}({\mathbb{R}}^{q})$, where
$q=\operatorname{codim}{\mathcal{F}}$. We call the image of
$\operatorname{Hol}$ the holonomy group of $L$ at $x$, which is determined up
to an inner automorphism of $\operatorname{Diff}_{0}({\mathbb{R}}^{q})$. As a
direct consequence, we have a short exact sequence

$1\to
K\hookrightarrow\pi_{1}(L,x)\stackrel{{\scriptstyle\operatorname{Hol}}}{{\to}}\operatorname{Hol}(L,x)\to
1$

We will look at this sequence again when we generalize to higher holonomies.
Similar to the case of homotopy, we say two path $\alpha,\beta:x\to y$ lying
in the same leaf $L$ are in the same holonomy class if
$\operatorname{Hol}(\alpha^{-1}\beta)=\operatorname{Id}$.

######  Definition 19.12 ([MM03]).

Taking the differential at 0 gives a homomorphism
$d_{0}:\operatorname{Diff}_{0}({\mathbb{R}}^{q})\to\operatorname{GL}(q,{\mathbb{R}})$.
We call the composition

$d\operatorname{Hol}=d_{0}\circ\operatorname{Hol}:\pi_{1}(L,x)\to\operatorname{GL}(q,{\mathbb{R}})$

the linear holonomy homomorphism of $L$ at x, and we call the image the linear
holonomy group.

An important construction in foliations is associating various groupoids to a
foliation. The most important two are holonomy groupoids and monodromy
groupoids.

######  Definition 19.13 ([MM03]).

Let $({\mathfrak{M}},{\mathcal{F}})$ be a foliated manifold. Define the
monodromy groupoid $\operatorname{Mon}({\mathcal{F}})$ of ${\mathcal{F}}$ to
be a groupoid over $M$ whose arrows are homotopy classes of paths along leaves
of ${\mathcal{F}}$. Similarly, define the holonomy groupoid
$\operatorname{Hol}({\mathcal{F}})$ of ${\mathcal{F}}$ to be a groupoid over
$M$ whose arrows are holonomy classes of paths along leaves of ${\mathcal{F}}$

Both holonomy groupoids and monodromy groupoids are Lie groupoids, hence they
are powerful tools in studying the geometry and topology of foliations.

Next, we move to singular foliations.

######  Definition 19.14.

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold. A slice $\mathcal{T}$
at $x$ is an embedded submanifold $\mathcal{T}\subset M$ such that
$x\in\mathcal{T}$ and $T_{x}\mathcal{T}\oplus F_{x}=T_{x}M$.

This is similar to the definition of local transversals in regular foliations.

Consider a path $\gamma:[0,1]\to M$ from $x$ to $y$ which lie in a single leaf
$L$ of ${\mathcal{F}}$. Fixed two slices $\mathcal{T}_{x}$ and
$\mathcal{T}_{y}$ at $x$ and $y$ respectively. For each time $t$, we lift
$\dot{\gamma}(t)$ to a vector field $X_{t}$ lying in ${\mathcal{F}}$, such
that the flow of the time-dependent vector field $\\{X_{t}\\}$ maps
$\mathcal{T}_{x}$ to $\mathcal{T}_{y}$. However, if ${\mathcal{F}}$ is not a
regular foliation, the map
$f\in\operatorname{Hom}(\mathcal{T}_{x},\mathcal{T}_{y})$ will dependent on
the extension. Hence, we want to modify this such that we are not affected by
the choice of extensions.

First, we want to make some notations. Let
$\operatorname{Aut}_{{\mathcal{F}}}(M)$ be the subgroup of local
diffeomorphisms of $M$ preserving ${\mathcal{F}}$. Let $\exp$ denote the space
of time-one flows of time-dependent vector fields in ${\mathcal{F}}$. Recall
we have the following exact sequence

$0\to\mathfrak{g}_{x}\to{\mathcal{F}}_{x}\stackrel{{\scriptstyle\operatorname{ev}_{x}}}{{\to}}F_{x}\to
0$

where $\mathfrak{g}_{x}={\mathcal{F}}(x)/I_{x}{\mathcal{F}}$ is the isotropy
Lie algebra. We have that both $\exp(I_{x}{\mathcal{F}})$ and
$\exp({\mathcal{F}}(x))$ are subgroups of
$\operatorname{Aut}_{{\mathcal{F}}}(M)$. Denote the restriction of
${\mathcal{F}}$ to the slice $\mathcal{T}_{x}$ by
${\mathcal{F}}_{\mathcal{T}_{x}}$, i.e.
${\mathcal{F}}_{\mathcal{T}_{x}}={\mathcal{F}}|_{{\mathcal{F}}_{\mathcal{T}_{x}}}\cap
T\mathcal{T}_{x}$. Finally, we denote
$\operatorname{Germ}\operatorname{Aut}_{{\mathcal{F}}}(\mathcal{T}_{x},\mathcal{T}_{y})$
the space of germs of local diffeomorphisms from
$(\mathcal{T}_{x},{\mathcal{F}}_{\mathcal{T}_{x}})$ to
$(\mathcal{T}_{y},{\mathcal{F}}_{\mathcal{T}_{x}})$.

######  Theorem 19.15.

The class $\Gamma(-,1):\mathcal{T}_{x}\to\mathcal{T}_{y}$ in the quotient

(19.1)
$\frac{\operatorname{Germ}\operatorname{Aut}_{{\mathcal{F}}}(\mathcal{T}_{x},\mathcal{T}_{y})}{\exp({\mathcal{F}}_{\mathcal{T}_{x}})}$

is independent of the choice of extension $\Gamma$.

###### Proof.

See [AZ12, Proposition 2.3]. ∎

This seems to be a good candidate to define the holonomy transformation.
However, as pointed out in [AZ12], $\exp({\mathcal{F}}(x))$ is too large to do
linearization. Hence, we replace ${\mathcal{F}}(x)$ by $I_{x}{\mathcal{F}}$
and get the following definition:

######  Definition 19.16 ([AZ12]).

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold, and $x,y\in M$ lying
in some leaf $L$. We fix slices $\mathcal{T}_{x}$ and $\mathcal{T}_{y}$ at $x$
and $y$ respectively. We define a holonomy transformation from $x$ to $y$ to
be an element of

(19.2)
$\frac{\operatorname{Germ}\operatorname{Aut}_{{\mathcal{F}}}(\mathcal{T}_{x},\mathcal{T}_{y})}{\exp(I_{x}{\mathcal{F}}_{\mathcal{T}_{x}})}$

######  Lemma 19.17.

Let $x$ be a point on a regular leaf, then
$\exp(I_{x}{\mathcal{F}}_{\mathcal{T}_{x}})$ is trivial.

###### Proof.

In this case, $\mathcal{T}_{x}$ is equipped with a trivial singular foliation,
i.e. foliation by points. Hence, the flow is trivial. ∎

Therefore, we see that for regular foliation, the holonomy transformations
reduce to the ordinary holonomy transformations
$\operatorname{Germ}\operatorname{Diff}(\mathcal{T}_{x},\mathcal{T}_{y})$. It
is easy to see that we can form a topological groupoid
$\operatorname{HolTrans}({\mathcal{F}})$ over $M$ with morphisms being

$\bigcup_{x,y}\frac{\operatorname{Germ}\operatorname{Aut}_{{\mathcal{F}}}(\mathcal{T}_{x},\mathcal{T}_{y})}{\exp(I_{x}{\mathcal{F}}_{\mathcal{T}_{x}})}$

There is a natural map from the holonomy groupoid in the sense of
Androulidakis and Skandalis in [AS06], which justifies the correctness of the
definition of holonomy transformations. Let’s first review basics about
holonomy groupoids defined by Androulidakis and Skandalis.

######  Definition 19.18 ([AS06]).

Let $(M,{\mathcal{F}}_{M}),(N,{\mathcal{F}}_{N})$ be two singular foliated
manifolds. A bisubmersion is a manifold $P$ with two surjective submersions
$s:P\to M$ and $t:P\to N$ such that

$s^{-1}{\mathcal{F}}_{M}=t^{-1}{\mathcal{F}}_{N}=\Gamma(\ker(s)_{*})+\Gamma(\ker(t)_{*})$

We have the following diagram

${{(P,{\mathcal{F}})}}$${{(M,{\mathcal{F}}_{M})}}$${{(N,{\mathcal{F}}_{N})}}$$\scriptstyle{s}$$\scriptstyle{t}$

where ${\mathcal{F}}=s^{-1}{\mathcal{F}}_{M}=t^{-1}{\mathcal{F}}_{N}$ is the
pullback singular foliation on $P$.

######  Definition 19.19.

A morphism between bisubmersions $(U,s_{U},t_{U}),(V,s_{V},t_{V})$ is a smooth
map $f:U\to V$ such that for all $u\in U$, we have $s_{V}(f(u))=s_{U}(u)$ and
$t_{V}(f(u))=t_{U}(u)$.

A local morphism between bisubmersions $(U,s_{U},t_{U}),(V,s_{V},t_{V})$ is a
smooth map $f:U^{\prime}\to V$ for some $U^{\prime}\subset U$ such that for
all $u\in U^{\prime}$, we have $s_{V}(f(u))=s_{U}(u)$ and
$t_{V}(f(u))=t_{U}(u)$.

######  Definition 19.20.

Let $\mathcal{U}=(U_{i},s_{i},t_{i})_{i\in I}$ be a family of bisubmersions.

  1. (1)

A bisubmersion $(U,s,t)$ is said to be adapted to $\mathcal{U}$ at $u\in U$ if
there exists an open subset $U^{\prime}\subset U$ containing $u$ and a
morphism of bisubmersion $U^{\prime}\to U$.

  2. (2)

A bisubmersion $(U,s,t)$ is said to be adapted to $\mathcal{U}$ if $(U,s,t)$
is adapted to $\mathcal{U}$ at $u\in U$ for all $u\in U$.

  3. (3)

We call the family $\mathcal{U}=(U_{i},s_{i},t_{i})_{i\in I}$ an atlas if

    1. (a)

$\cup_{i\in I}s_{i}(U_{i})=M$.

    2. (b)

Any elements of $\mathcal{U}$ is still adapted to $\mathcal{U}$ under taking
inverses and compositions.

Recall that an atlas of a manifold allows us to reconstruct the manifold,
similarly an atlas of a singular foliation $(M,{\mathcal{F}})$ allows us to
reconstruct the foliated structure by a groupoid over $M$, and this is the
first step in constructing the holonomy groupoid.

######  Theorem 19.21 ([AS06]).

Let $\mathcal{U}=(U_{i},s_{i},t_{i})_{i\in I}$ be an atlas of a singular
foliated manifold $(M,{\mathcal{F}})$.

  1. (1)

Let $G=\coprod_{i\in I}U_{i}/\sim$ where $\sim$ is the equivalence relation
generated by local morphisms, i.e. $u\in U_{i}$ is equivalent to $v\in U_{j}$
if there exists a local morphism from $U_{i}\to U_{j}$ which takes $u$ to $v$.
There are maps $s,t:G\to M$ such that $s\circ q_{i}=s_{i}$ and $t\circ
q_{i}=t_{i}$, where $Q=(q_{i})_{i\in I}:\coprod_{i\in I}\to G$ is the quotient
map.

  2. (2)

For any $(U,s_{U},t_{U})$ adapted to $\mathcal{U}$, there exists a map
$q_{U}:U\to G$ such that for every local morphism $f:U^{\prime}\subset U\to
U_{i}$ and every $u\in U^{\prime}$, we have $q_{U}(u)=q_{i}(f(u))$.

  3. (3)

There exists a (topological) groupoid structure on $G$ over $M$ with source
and target maps $s$ and $t$ defined before, and
$q_{i}(u)q_{j}(v)=q_{U_{i}\circ U_{j}}(u,v)$ .

Hence, given any atlas, we can construct a groupoid which encodes information
about the singular foliation. Apparently, we can simply take all possible
bisubmersions to be an atlas, which is called the full holonomy atlas.
However, this atlas is obviously too big which will make the arrow space of
the groupoid to be incredibly large and nasty. The second choice is to take
all leaf-preserving bisubmersions, i.e. bisubmersions $(U,s,t)$ that $s(u)$
and $t(u)$ lying in the same leaf for all $u\in U$. This is called the leaf-
preserving atlas, which is much smaller than the full holonomy atlas. [AS06]
constructs an atlas $\mathcal{W}$ which is as minimal as possible.

######  Proposition 19.22 ([AS06]).

Let $x\in M$ and $X_{1},\cdots,X_{n}\in{\mathcal{F}}$ be vector fields whose
images at $x$ form a basis of ${\mathcal{F}}_{x}$ (the fiber of
${\mathcal{F}}$ at $x$). For $y=(y_{1},\cdots,y_{n})\in{\mathbb{R}}^{n}$, set
$\phi_{y}=\exp(\sum y_{i}X_{i})\in\exp{\mathcal{F}}$, i.e. the image of $y$
under the time-1 flow of the vector field $\sum y_{i}X_{i}$. Let
$W_{0}={\mathbb{R}}^{n}\times M,s_{0}(y,x)=x,s_{0}(y,x)=\phi_{y}(x)$. Then

  * •

There exists a neighborhood $W$ of $(0,x)$ in $W_{0}$ such that
$(W,s|_{W},t|_{W})$ is a bisubmersion.

  * •

For any bisubmersions $(V,s_{V},t_{v})$ carries the identity of $M$ at some
$v\in V$, then there exists a local morphism from $(V,s_{V},t_{v})$ to some
$(W,s_{W},t_{W})\in\mathcal{W}$ at $v$ which sends $v$ to $(0,x)$.

######  Definition 19.23 ([AS06]).

We define the path holonomy atlas of a singular foliated manifold
$(M,{\mathcal{F}})$ to be the maximal atlas generated by a cover of $M$ by
$s$-connected bisubmersions of the form in Proposition 19.22.

The corresponding groupoid of the path holonomy atlas is the smallest due to
(2) in Proposition 19.22, which implies that this atlas is adapted to any
other atlas.

######  Definition 19.24.

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold. We define the
holonomy groupoid of ${\mathcal{F}}$ to be the (topological) groupoid
associated to the path holonomy atlas of ${\mathcal{F}}$. We denote the
holonomy groupoid of ${\mathcal{F}}$ by $\operatorname{Hol}({\mathcal{F}})$ or
$\operatorname{Hol}^{AS}({\mathcal{F}})$.

The topology of holonomy groupoids in this definition is usually pretty bad.
For example, [AS06, Example 3.7] consider the singular foliation on
${\mathbb{R}}^{2}$ generated by the action of
$\operatorname{SL}(2,{\mathbb{R}})$, whose holonomy groupoid has a highly non-
Hausdorff arrow space, for example, for any $x$ outside the origin, the
sequence $\\{(x/n,x/n)\\}$ will converge to all $(g,0)$ for stabilizers $g$ of
$x$. Therefore, in general we don’t expect holonomy groupoids to be Lie
groupoids. However, we do have the following local smoothness results.

######  Theorem 19.25 ([Deb13]).

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold. The $s$-fibers of
$\operatorname{Hol}({\mathcal{F}})$ are smooth manifolds.

######  Corollary 19.26 ([AZ11]).

The transitive groupoid $\operatorname{Hol}_{L}({\mathcal{F}})$ is smooth and
integrates the Lie algebroid $A_{L}=\cup_{x\in L}{\mathcal{F}}_{x}$, where
$\operatorname{Hol}_{L}({\mathcal{F}})=\operatorname{Hol}({\mathcal{F}})|_{s^{-1}L}=\operatorname{Hol}({\mathcal{F}})|_{t^{-1}L}$

Hence a natural question is can we find a higher categorical geometric object
such that the 1-truncation equals the holonomy groupoid?

We will answer this question later, and let’s return to the holonomy
transformation first.

######  Theorem 19.27 ([AZ12]).

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold, and $x,y\in M$ lying
in some leaf $L$. We fix slices $\mathcal{T}_{x}$ and $\mathcal{T}_{y}$ at $x$
and $y$ respectively. There is a natural injective map

$\Phi^{y}_{x}:\operatorname{Hol}^{AS}({\mathcal{F}})^{y}_{x}\to\operatorname{HolTrans}({\mathcal{F}})^{y}_{x}=\frac{\operatorname{Germ}\operatorname{Aut}_{{\mathcal{F}}}(\mathcal{T}_{x},\mathcal{T}_{y})}{\exp(I_{x}{\mathcal{F}}_{\mathcal{T}_{x}})}$

where $\operatorname{Hol}^{AS}({\mathcal{F}})$ denotes the holonomy groupoid
in the sense of Androulidakis and Skandalis [AS06]. Moreover, $\Phi^{y}_{x}$
assembles to a global groupoid morphism

$\Phi:\operatorname{Hol}({\mathcal{F}})\to\operatorname{HolTrans}({\mathcal{F}})$

######  Definition 19.28.

Let $f:\mathcal{T}_{x}\to\mathcal{T}_{y}$ be a holonomy transformation.
Suppose $f$ is also an embedding, then we call $f$ a holonomy embedding.

####  19.4. Hausdorff Morita equivalences

There are various notions of two (singular) foliations to be equivalent.
Garmendia and Zambon [GZ19] proposed a notion called Hausdorff Morita
equivalence of singular foliations, which is constructed to be compatible to
Androulidakis and Skandalis’s construction of holonomy groupoids in [AS06].

######  Definition 19.29.

[AS06]

######  Definition 19.30 ([GZ19]).

Let $(M,{\mathcal{F}}_{M}),(N,{\mathcal{F}}_{N})$ be two singular foliated
manifolds. We say $(M,{\mathcal{F}}_{M})$ and $(N,{\mathcal{F}}_{N})$ are
Hausdorff Morita equivalent if there exists a manifold $P$ and two surjective
submersions with connected fibers $\pi_{M}:P\to M$ and $\pi_{N}:P\to N$ such
that $\pi_{M}^{-1}{\mathcal{F}}_{M}=\pi_{N}^{-1}{\mathcal{F}}_{N}$. We have
the following diagram

${{(P,{\mathcal{F}})}}$${{(M,{\mathcal{F}}_{M})}}$${{(N,{\mathcal{F}}_{N})}}$$\scriptstyle{\pi_{M}}$$\scriptstyle{\pi_{N}}$

where
${\mathcal{F}}=\pi_{M}^{-1}{\mathcal{F}}_{M}=\pi_{N}^{-1}{\mathcal{F}}_{N}$ is
the pullback singular foliation on $P$.

This definition is almost the same as the bisubmersion except that we do not
require
$\pi_{M}^{-1}{\mathcal{F}}_{M}=\pi_{N}^{-1}{\mathcal{F}}_{N}=\Gamma(\ker(\pi_{M})_{*})+\Gamma(\ker(\pi_{N})_{*})$.
Hence, Hausdorff Morita equivalences are weaker notions than bisubmersions.

The following theorem gives basic properties of Hausdorff Morita equivalences.

######  Theorem 19.31.

[GZ19] Let two singular foliated manifolds
$(M,{\mathcal{F}}_{M}),(N,{\mathcal{F}}_{N})$ be Hausdorff Morita equivalent.
We have

  1. (1)

There is a homeomorphism between the leaf spaces of $(M,{\mathcal{F}}_{M})$
and the leaf space of $(N,{\mathcal{F}}_{N})$, which maps the leaf though some
$x\in M$ to the leaf of ${\mathcal{F}}_{N}$ containing $\pi_{N}(\pi^{-1}(x))$,
and preserves the codimension of leaves and the property of being an embedded
leaf.

  2. (2)

Consider $x\in M,y\in N$. Pick slices $S_{x}$ at $x$ and $S_{y}$ at $y$, then
the singular foliated manifolds $(S_{x},\iota_{S_{x}}^{-1})$ and
$(S_{y},\iota_{S_{y}}^{-1})$ are diffeomorphic.

  3. (3)

Consider $x\in M,y\in N$. The isotropy Lie algebras
$\mathfrak{g}_{x}^{{\mathcal{F}}_{M}}$ and
$\mathfrak{g}_{y}^{{\mathcal{F}}_{N}}$ are isomorphic.

The next theorem justifies Hausdorff Morita equivalence is the correct notion
which preserves holonomy groupoids.

######  Theorem 19.32 ([GZ19]).

If two singular foliated manifolds
$(M,{\mathcal{F}}_{M}),(N,{\mathcal{F}}_{N})$ are Hausdorff Morita equivalent,
then their holonomy groupoids (in the sense of Androulidakis and Skandalis in
[AS06]) are Morita equivalent as open topological groupoids.

Next, let’s look at the behavior of Hausdorff Morita equivalences under
pullbacks.

######  Proposition 19.33.

Pullbacks of Hausdorff Morita equivalences are Hausdorff Morita equivalences.

###### Proof.

Let $f:(M,{\mathcal{F}}_{M})\to(N,{\mathcal{F}}_{N})$,
$g:(L,{\mathcal{F}}_{L})\to(N,{\mathcal{F}}_{N})$ be morphisms in
$\mathsf{Mfd}^{{\operatorname{SFol}}}$. Suppose the pullback singular foliated
manifold
$(M\times_{N}L,{\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})$
exists and $f$ is a Hausdorff Morita equivalence, then the induces map
$f^{\prime}:(M\times_{N}L,{\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})\to(L,{\mathcal{F}}_{L})$
is also a Hausdorff Morita equivalence. ∎

###### Proof.

Let
$(M,{\mathcal{F}}_{M})\stackrel{{\scriptstyle\pi_{M}}}{{\leftarrow}}(P,{\mathcal{F}})\stackrel{{\scriptstyle\pi_{N}}}{{\rightarrow}}(N,{\mathcal{F}})$
be a morphism representing the Hausdorff Morita equivalence. We have the
following commutative diagram

${{(P^{\prime},{\mathcal{F}}^{\prime})}}$${{(P,{\mathcal{F}})}}$${{(M\times_{N}L,{\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})}}$${{(M,{\mathcal{F}}_{M})}}$${{(L,{\mathcal{F}}_{L})}}$${{(N,{\mathcal{F}})}}$$\scriptstyle{\pi_{M}^{\prime}}$$\scriptstyle{\pi_{N}^{\prime}}$$\scriptstyle{\pi_{M}}$$\scriptstyle{\pi_{N}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{g}$

Since $\pi_{M},\pi_{N}$ are submersions, we see their pullbacks exist, and the
left triangle in the diagram is actually a composition of pullbacks, which
implies that all three squares are pullbacks. In particular.
$(\pi_{M}^{\prime})^{-1}({\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})=(\pi_{N}^{\prime})^{-1}({\mathcal{F}}_{L})$
follows from the composition pullback. Clearly the fiber of $\pi_{M}^{\prime}$
and $\pi_{N}^{\prime}$ since $\pi_{M}$ and $\pi_{N}$’s are. ∎

####  19.5. Homotopy theory of singular foliations

######  Definition 19.34.

Let $f:(M,{\mathcal{F}}_{M})\to(N,{\mathcal{F}}_{N})$,
$g:(L,{\mathcal{F}}_{L})\to(N,{\mathcal{F}}_{N})$ be morphisms in
$\mathsf{Mfd}^{{\operatorname{SFol}}}$. We say $f$ is foliated transverse to
$g$ if the natural map $(d_{f}\times
d_{g})({\mathcal{F}}_{M}\times{\mathcal{F}}_{L})\to
f^{*}{\mathcal{F}}_{N}\times_{N}g^{*}{\mathcal{F}}_{N}$ is surjective. Here we
use $f^{*}{\mathcal{F}}_{N}\times_{N}g^{*}{\mathcal{F}}_{N}$ to denote the
pullback of ${\mathcal{F}}_{N}$ on $M\times_{N}L$.

######  Proposition 19.35.

Let $f:(M,{\mathcal{F}}_{M})\to(N,{\mathcal{F}}_{N})$,
$g:(L,{\mathcal{F}}_{L})\to(N,{\mathcal{F}}_{N})$ be morphisms in
$\mathsf{Mfd}^{{\operatorname{SFol}}}$. Suppose the $f$ is foliated transverse
to $g$. If the pullback $(M\times_{N}L)$ exists, then the pullback
$({\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})$ is a singular
foliation on $(M\times_{N}L)$.

${(M\times_{N}L,{\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})}$${(M,{\mathcal{F}}_{M})}$${(L,{\mathcal{F}}_{L})}$${(N,{\mathcal{F}}_{N})}$$\scriptstyle{g^{\prime}}$${\ulcorner}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$

###### Proof.

First let’s look at the involutivity. Denote
${\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L}$ by
${\mathcal{F}}$. Obviously, $g^{\prime}$ is foliated, i.e.
$d_{g^{\prime}}({\mathcal{F}})\subset{\mathcal{F}}_{M}$. Let
$X,X^{\prime}\in{\mathcal{F}}_{M}$. Write $d_{g^{\prime}}(X)=\sum
f_{i}Y_{i}\circ g^{\prime}$ and $d_{g^{\prime}}(X^{\prime})=\sum
f_{i}^{\prime}Y_{i}^{\prime}\circ g^{\prime}$, then

$d_{g^{\prime}}([X,X^{\prime}])=\sum
f_{i}f_{j}^{\prime}[Y_{i},Y_{j}^{\prime}]\circ g^{\prime}+\sum
X(f_{j}^{\prime})Y_{j}^{\prime}\circ g^{\prime}-\sum
X^{\prime}(f_{i})Y_{i}\circ g^{\prime}$

hence we see the pullback is closed under brackets.

Next, we want to show ${\mathcal{F}}$ is locally finitely generated. By
restricting to sufficient small open subsets of $M$, $L$ and $N$, we can
assume ${\mathcal{F}}_{M}$, ${\mathcal{F}}_{L}$, and ${\mathcal{F}}_{N}$ are
finitely generated, and tangent bundles of $M$, $L$, and $N$ are trivial.

${\mathcal{F}}={\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L}={\mathcal{F}}_{M}\tensor[_{f^{-1}({\mathcal{F}}_{N})}]{\times}{{}_{g^{-1}({\mathcal{F}}_{N})}}{\mathcal{F}}_{L}$

Note that

$\displaystyle f^{-1}({\mathcal{F}}_{N})=$ $\displaystyle
f^{*}({\mathcal{F}}_{N})\times_{\Gamma(M,f^{*}(TN))}\Gamma(M,TM)$
$\displaystyle g^{-1}({\mathcal{F}}_{N})=$ $\displaystyle
g^{*}({\mathcal{F}}_{N})\times_{\Gamma(L,g^{*}(TN))}\Gamma(L,TL)$

By foliated transversality, we see there exists a section

$s:{\mathcal{F}}_{N}\to f^{-1}({\mathcal{F}}_{N})\times
g^{-1}({\mathcal{F}}_{N})\subset{\mathcal{F}}_{M}\times{\mathcal{F}}_{L}$

It follows that
${\mathcal{F}}_{M}\tensor[_{f^{-1}({\mathcal{F}}_{N})}]{\times}{{}_{g^{-1}({\mathcal{F}}_{N})}}{\mathcal{F}}_{L}$
is finitely generated. ∎

Similarly, we can define foliated submersions.

######  Definition 19.36.

Let $f:(M,{\mathcal{F}}_{M})\to(N,{\mathcal{F}}_{N})$ be morphisms in
$\mathsf{Mfd}^{{\operatorname{SFol}}}$. We say $f$ is a foliated submersion if
the natural map $d_{f}{\mathcal{F}}_{M}\to f^{*}{\mathcal{F}}_{N}$ is
surjective.

By this definition, a foliated submersion is then foliated transversal to any
foliated maps. Hence, a direct corollary is:

######  Corollary 19.37.

Let $f:(M,{\mathcal{F}}_{M})\to(N,{\mathcal{F}}_{N})$,
$g:(L,{\mathcal{F}}_{L})\to(N,{\mathcal{F}}_{N})$ be morphisms in
$\mathsf{Mfd}^{{\operatorname{SFol}}}$. Suppose $f$ is a surjective foliated
submersion. If the pullback $(M\times_{N}L)$ exists, then the pullback
$({\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})$ is a singular
foliation on $(M\times_{N}L)$. Moreover, the induced map
$f^{\prime}:(M\times_{N}L,{\mathcal{F}}_{M}\times_{{\mathcal{F}}_{N}}{\mathcal{F}}_{L})\to(N,{\mathcal{F}}_{N})$
is a surjective foliated submersion.

Next, we will construct a finite dimensional model for the path object of
singular foliations. Recall that, in general, ${\mathcal{C}}^{\infty}$ path
spaces for finite dimensional manifolds are infinite dimensional, even if we
restrict to $C^{1}$ paths. One way to remedy this is to consider only those
’short paths’. We will follow the construction in [BLX21, Section 2.1].

First, we can some connection $\nabla$ on $M$. Let $\exp^{\nabla}$ denote the
exponential map with respect to $\nabla$. Let $I=(a,b)\supset[0,1]$.

######  Proposition 19.38 ([BLX21]).

There exists a manifold $P_{g}M$, which is called the manifold of short
geodesic paths in $M$, which parametrizes a family of geodesic paths, such
that

${{M}}$${{TM}}$${{M\times
M}}$${{{P_{g}M}}}$$\scriptstyle{0}$$\scriptstyle{\Delta}$$\scriptstyle{const}$$\scriptstyle{\gamma(0)\times\gamma^{\prime}(0)}$$\scriptstyle{\gamma(0)\times\gamma(1)}$

where the two lower diagonal maps are open embeddings.

###### Proof.

Consider $U\subset TM$ an open neighborhood of the zero section, where the
$\exp^{\nabla}(tv)$ is defined for any $t\in I,v\in U$. $U$ parametrize a
family of geodesic paths with domain $I$:

$\displaystyle U\times I$ $\displaystyle\to M$
$\displaystyle(x,v,t)\mapsto\gamma_{x,v}(t)$
$\displaystyle=\exp^{\nabla}_{x}(tv)$

We can restrict to a smaller open neighborhood $V\subset V$ such that the both
maps $U\to TM$ and $V\to M\times M$ are open embeddings. Now take $P_{g}M=V$.
∎

Hence, $P_{g}M$ is diffeomorphic to an open neighborhood of the zero section
of $TM$. The evaluation map $\operatorname{ev}_{0}:P_{g}M\to M$ is just the
restriction of the projection $TM\to M$, and $\operatorname{ev}_{1}:P_{g}M\to
M$ is given by the exponential map

$v_{x}\mapsto\exp^{\nabla}_{x}v_{x}$

######  Lemma 19.39.

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold. Then there exists a
associated foliation ${\mathcal{F}}_{g}$ on $P_{g}M$ which projects to
${\mathcal{F}}$ along the projection $P_{g}M\to M$.

###### Proof.

It suffices to construct $(M,{\mathcal{F}})$ locally. The connection gives a
splitting $TTM\simeq TM\oplus vTM$, where $vTM$ denotes the vertical tangent
bundle of $TM$. Hence, locally, $P_{g}M\simeq U\times{\mathbb{R}}^{n}$ for
some $U\subset M$ and $n=\dim M$. Pick $X_{1},\cdots,X_{k}$ to be generators
of ${\mathcal{F}}|_{U}$, and $e_{1},\cdots,e_{n}$ be the local coordinate
sections of $vTM|_{U}$. Now we define ${\mathcal{F}}_{g}|_{U}$ to be the
module generated by $X_{1},\cdots,X_{k},e_{1},\cdots,e_{n}$. The involutivity
follows directly from our construction. Note the projection $P_{g}M\to M$ is
clearly a foliated map which just kills $e_{i}$’s, which then maps
${\mathcal{F}}_{g}$ to ${\mathcal{F}}$. ∎

######  Lemma 19.40.

Both $\operatorname{ev}_{0}$ and $\operatorname{ev}_{1}$ are foliated
submersions.

###### Proof.

Follows from the construction that

$(X_{1},\cdots,X_{k},e_{1},\cdots,e_{n})\mapsto(X_{1},\cdots,X_{k})$

is surjective. ∎

######  Proposition 19.41.

There exists an incomplete category of fibrant objects structure on
$\mathsf{Mfd}^{{\operatorname{SFol}}}$, where

  1. (1)

Fibrations are surjective foliated submersions.

  2. (2)

Weak equivalences are Hausdorff Morita equivalences.

  3. (3)

Path objects are foliated short geodesic path space.

###### Proof.

(2) follows from Corollary 19.40. (3) then follows from (4) and Proposition
19.33.

For (4), clearly isomorphisms are Hausdorff Morita equivalences, let’s prove
2-out-of-3 property. By construction, the composition of Hausdorff Morita
equivalences is the fiber product which is again a Hausdorff Morita
equivalences, this directly implies that, if
$(M,{\mathcal{F}}_{M})\simeq(N,{\mathcal{F}}_{N})$ and
$(L,{\mathcal{F}}_{L})\simeq(N,{\mathcal{F}}_{N})$, then we have
$(M,{\mathcal{F}}_{M})\simeq(L,{\mathcal{F}}_{L})$. By symmetry of the
Hausdorff Morita equivalences, this will generate all cases of 2-out-of-3.

For (5), clearly isomorphisms are surjective foliated submersions, and
composition of foliated submersions are again foliated submersions by
definition.

For (6), given a singular foliated manifold $(M,{\mathcal{F}})$, we construct
its path object $(M,{\mathcal{F}})^{\Delta[1]}$ to be
$(P_{g}M,{\mathcal{F}}_{g})$. By Theorem 19.38 and Lemma 19.39, we have the
following factorization

$(M,{\mathcal{F}})\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}(P_{g}M,{\mathcal{F}}_{g})\stackrel{{\scriptstyle(\gamma(0),\gamma(1))}}{{\longrightarrow}}(M,{\mathcal{F}})\times(M,{\mathcal{F}})$

which composes to the diagonal map. $(\gamma(0),\gamma(1))$ are fibrations by
Lemma 19.40. To see $\iota$ is a Hausdorff Morita equivalence, notice that the
$\iota$ is an embedding, and $p:P_{g}M\to\iota(M)$ is a submersion. By
construction $p_{-1}(\iota({\mathcal{F}}))$ is exactly ${\mathcal{F}}_{g}$.
Hence, $(P_{g}M,{\mathcal{F}})$ itself gives a Hausdorff Morita equivalence by
$(\operatorname{Id},p)$.

Finally, the trivial map $(M,{\mathcal{F}})\to*$ is clearly a surjective
foliated submersion, hence a fibration. ∎

We denote the $\infty$-category of singular foliated manifolds presented by
this iCFO by
$\mathtt{Mfd}^{{\operatorname{SFol}}}=\mathsf{Mfd}^{{\operatorname{SFol}}}[W^{-1}]$.
We will use this later in constructing algebraic $K$-theory of singular
foliations.

####  19.6. Algebraic $K$-theory of singular foliations

In this section, we will construct the algebraic $K$ theory sheaves
$\mathsf{K}$ on $\mathsf{Mfd}^{{\operatorname{SFol}}}$ following [Bun18] for
regular foliations. Then we can calculate Algebraic $K$-theory of
$(M,{\mathcal{F}})$ by taking homotopy groups of
$\mathsf{K}(M,{\mathcal{F}})$:

$K^{\bullet}(M,{\mathcal{F}})=\pi_{-\bullet}(\big{(}\mathsf{K}(M,{\mathcal{F}})\big{)}$

We consider $\mathsf{Cat}$ with its Cartesian symmetric monoidal structure.
Let $W$ denote the class of categorical equivalence, then we get a symmetric
monoidal category $\mathsf{Cat}[W^{-1}]$. We denote the category of
commutative algebras in $\mathsf{Cat}[W^{-1}]$ by
$\mathsf{CAlg}(\mathsf{Cat}[W^{-1}])$.

###### Construction of algebraic $K$ theory sheaves

  1. (1)

Let $M$ be a manifold, we denote $\mathsf{Vect}(M)$ the category of vector
bundles over $M$. Any map $f:M\to N$ induces a functor
$f^{*}:\mathsf{Vect}(N)\to\mathsf{Vect}(M)$. Hence, we get a stack
$\mathsf{Vect}$ on the site $\mathsf{Mfd}$ with open covering topology.
Similarly, we get a stack
$\mathsf{Vect}^{{{\mathbb{C}}}{\operatorname{SFol}}}$ by pullback along the
forgetful functor

$F:\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}}\to\mathsf{Mfd}$

We will write $\mathsf{Vect}$ for
$\mathsf{Vect}^{{{\mathbb{C}}}{\operatorname{SFol}}}$ for simplicity.

  2. (2)

Similarly, we can consider the category of pairs $(V,\nabla)$ of a vector
bundle $V\to M$. Denote the resulting symmetric monoidal stack (with the
Cartesian symmetric monoidal structure) by $\mathsf{Vect}^{\nabla}$.

  3. (3)

Let $(M,{\mathcal{F}})\in\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}}$.
Denote $\mathsf{Vect}^{{\operatorname{flat}}}(M,{\mathcal{F}})$ the category
of pairs $(V,\nabla_{I})$ of a vector bundle $V\to M$ and a flat partial
connection $\nabla_{I}$ on $(M,{\mathcal{F}})$. A foliated map
$f:(M,{\mathcal{F}})\to(M^{\prime},{\mathcal{F}}^{\prime})$ induces a functor
$f^{\prime}:\mathsf{Vect}^{{\operatorname{flat}}}(M^{\prime},{\mathcal{F}}^{\prime})\to\mathsf{Vect}^{{\operatorname{flat}}}(M,{\mathcal{F}})$.
We get a stack $\mathsf{Vect}^{{\operatorname{flat}}}$ on the site
$\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}}$.

  4. (4)

Finally, let
$(M,{\mathcal{F}})\in\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}}$.
Denote $\mathsf{Vect}^{{\operatorname{flat}},\nabla}(M,{\mathcal{F}})$ the
category of pairs $(V,\nabla_{{\mathcal{F}}})$ of a vector bundle $V\to M$ and
a flat ${\mathcal{F}}$-connection $\nabla_{{\mathcal{F}}}$ on
$(M,{\mathcal{F}})$. A foliated map
$f:(M,{\mathcal{F}})\to(M^{\prime},{\mathcal{F}}^{\prime})$ induces a functor
$f^{\prime}:\mathsf{Vect}^{{\operatorname{flat}},\nabla}(M^{\prime},{\mathcal{F}}^{\prime})\to\mathsf{Vect}^{{\operatorname{flat}},\nabla}(M,{\mathcal{F}})$.
We get a symmetric monoidal stack
$\mathsf{Vect}^{{\operatorname{flat}},\nabla}$ on the site
$\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}}$.

We have the following commutative diagrams of stacks by forgetful maps

${{\mathsf{Vect}^{{\operatorname{flat}},\nabla}}}$${{\mathsf{Vect}^{{\operatorname{flat}}}}}$${{\mathsf{Vect}^{\nabla}}}$${{\mathsf{Vect}}}$

in
$\mathsf{Sh}_{\mathsf{CAlg}(\mathsf{Cat}(W^{-1}))}(\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}})$.

  5. (5)

Now we can apply the K-theory machine developed in [BNV13], we get a
commutative diagram of presheaves of spectra

${{\mathsf{K}(\mathsf{Vect}^{{\operatorname{flat}},\nabla})}}$${{\mathsf{K}(\mathsf{Vect}^{{\operatorname{flat}}})}}$${{\mathsf{K}(\mathsf{Vect}^{\nabla})}}$${{\mathsf{K}(\mathsf{Vect})}}$${{\widehat{\mathsf{ku}}^{\nabla}=s\big{(}\mathsf{K}(\mathsf{Vect}^{\nabla})\big{)}}}$${{\widehat{\mathsf{ku}}=s\big{(}\mathsf{K}(\mathsf{Vect})\big{)}}}$

Here $s:\mathsf{PSh}\to\mathsf{Sh}$ denotes the sheafification functor.

The K-theory machine developed in [BNV13] is basically a composition,

$\displaystyle\mathsf{CAlg}(\mathsf{Cat}(W^{-1}))\to\mathsf{CAlg}(\mathsf{Grpd}(W^{-1}))\to\mathsf{ComMon}(s\mathsf{Set}[W^{-1})$
$\displaystyle\to\mathsf{ComGrp}(s\mathsf{Set}[W^{-1})\simeq\mathsf{Sp}_{\geq
0}\to\mathsf{Sp}$

where we

  1. (1)

First take the groupoid underlying $\mathsf{Cat}$.

  2. (2)

Applying nerve to get a commutative monoid in the category of spaces
$s\mathsf{Set}[W^{-1}]$, i.e. an $E_{\infty}$-space.

  3. (3)

Then applying the group completion, we get a commutative group in the category
of spaces $s\mathsf{Set}[W^{-1}]$, i.e. a grouplike $E_{\infty}$-space.

  4. (4)

Finally, we apply the functor which maps a commutative group in spaces to the
corresponding connective spectrum whose $\infty$-group is this group.

For more details about the K-theory machine $\mathsf{K}$, see [BNV13,
Definition 6.1, Remark 6.4]. Let $L$ and $\mathcal{H}^{{\operatorname{flat}}}$
denote the sheafification and homotopification functors.

######  Definition 19.42.

We define the following sheaves of spectra

$\displaystyle\mathbf{K}=$
$\displaystyle\mathcal{H}^{{\operatorname{flat}}}(L(\mathsf{K}(\mathsf{Vect}^{{\operatorname{flat}}})))\in\mathsf{Sh}_{{\mathcal{S}}}^{h}(\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}})$
$\displaystyle\mathbf{K}^{\nabla}=$ $\displaystyle
L(\mathsf{K}(\mathsf{Vect}^{{\operatorname{flat}}},\nabla))\in\mathsf{Sh}_{{\mathcal{S}}}(\mathsf{Mfd}^{{{\mathbb{C}}}{\operatorname{SFol}}})$

and for $i\in{\mathbb{Z}}$, we define the algebraic $K$-theory of a singular
foliation $(M,{\mathcal{F}})$ by

$K^{i}(M,{\mathcal{F}})=\pi_{-i}(\mathbf{K}(M,{\mathcal{F}}))$

Note that $\mathsf{K}(\mathsf{Vect}^{{\operatorname{flat}}})$ is a homotopy
invariant, hence we expect that homotopification will preserve this
invariance. Therefore, the homotopification might not be necessary.

###  20\. Higher groupoids arised in singular foliations

####  20.1. Leaf spaces of singular foliations and Čech $\infty$-groupoids

Recall that for a foliated manifold $(M,{\mathcal{F}})$, the leaf space of
${\mathcal{F}}$ is a space ${\mathcal{F}}$ which is a quotient of $M$ by
identifying points within the same leaves. We want to construct a smooth model
for the leaf space of a singular foliation.

######  Definition 20.1.

We define a transversal basis for $(M,{\mathcal{F}})$ as a family
$\mathcal{U}$ of slices $U$ such that given any slice $V$ at $x$, we can find
a $U$ at $y$ and $x,y$ lying in the same leaf, and there exists a holonomy
embedding $h:V\hookrightarrow U$.

Given a point $x$, we can take a

######  Definition 20.2.

Let $(M,{\mathcal{F}})$ be a singular foliated manifold and $\mathcal{U}$ a
transversal basis of for $(M,{\mathcal{F}})$. We define the Čech
$\infty$-groupoid $\operatorname{\check{C}ech}_{\bullet}({\mathcal{F}})$ whose
$k$-simplices are

$\operatorname{\check{C}ech}({\mathcal{F}})_{k}=\coprod_{U_{0}\stackrel{{\scriptstyle
h_{1}}}{{\to}}\cdots\stackrel{{\scriptstyle h_{k}}}{{\to}}U_{k}}U_{0}$

where $h_{i}:U_{i-1}\to U_{i}$ are holonomy embeddings and
$U_{i}\in\mathcal{U}$ for all $i$.

There structure maps $d_{i}$’s and $s_{i}$’s are defined as

######  Definition 20.3.

For a regular foliation, the standard model for the leaf space is the
classifying space of the holonomy groupoid. If ${\mathcal{F}}$ is regular,
then our construction reduces to [CM00], and we have the following isomorphism

######  Theorem 20.4 ([CM00]).

For a regular foliated-manifold $(M,{\mathcal{F}})$. There is a natural
isomorphism

$\check{H}^{\bullet}_{\mathcal{U}}(M/{\mathcal{F}})\simeq
H^{\bullet}(B\operatorname{Hol}(M{\mathcal{F}});{\mathbb{R}})$

between the Čech-de Rham cohomology and the cohomology of the classifying
space of the holonomy groupoid. The left hand side is independent of the
choice of $\mathcal{U}$.

Hence, we can regard $\operatorname{\check{C}ech}({\mathcal{F}})_{\bullet}$ as
a model for the leaf space of a singular foliated manifold
$(M,{\mathcal{F}})$.

####  20.2. Holonomy $\infty$-groupoids

In this section, we are going to construct a higher groupoid model which
extends the holonomy groupoids in the sense of Androulidakis and Skandalis.

Recall that, when we construct the groupoid associated to an atlas of
bisubmersions, we take the quotient of equivalence relations induced by local
morphisms of bisubmersions. This is the reason we get a crappy arrow space for
the holonomy groupoid. One natural idea is that, instead of brutally quotient
the (local) equivalence, we keep the gluing data, and take a ’nerve’ similar
to the case when we construct classifying spaces.

Let $(M,{\mathcal{F}})$ be a singular-foliated manifold. In this section, we
fix the atlas $\mathcal{U}=(U_{i},s_{i},t_{i})_{i\in I}$ to be the path
holonomy atlas of $(M,{\mathcal{F}})$.

######  Definition 20.5.

We say a morphism between bisubmersions $f:(U,s_{U},t_{U})\to(V,s_{V},t_{V})$
is an equivalence if there exists a morphism
$g:(V,s_{V},t_{V})\to(U,s_{U},t_{U})$. We say $f$ is an isomorphism if $f\circ
g=\operatorname{Id},g\circ f=id$.

We say a morphism between bisubmersions $f:(U,s_{U},t_{U})\to(V,s_{V},t_{V})$
is a local equivalence (isomorphism) if there exists a morphism
$f^{\prime}:U^{\prime}\to V^{\prime}$ where $U^{\prime}\subset
U,V^{\prime}\subset V$ which is an equivalence (isomorphism).

First, let’s recall the following lemma about the local morphisms.

######  Lemma 20.6 ([AS06]).

Let $(U,s_{U},t_{U}),(V,s_{V},t_{V})$ be bisubmersions and let $u\in U,v\in V$
with $s_{U}(u)=s_{V}(v)$. Then:

  1. (1)

If the identity local diffeomorphism is carried by $U$ at $u$ and by $V$ at
$v$, then there exists an open neighborhood $U^{\prime}$ of $u$ in $U$, and a
morphism $f:U^{\prime}\to V$ such that $f(u)=v$.

  2. (2)

If there exists a local diffeomorphism is carried by $U$ at $u$ and by $V$ at
$v$, then there exists an open neighborhood $U^{\prime}$ of $u$ in $U$, and a
morphism $f:U^{\prime}\to V$ such that $f(u)=v$.

  3. (3)

If there exists a morphism of bisubmersions $f:U\to V$ such that $f(u)=v$,
then there exists an open neighborhood $V^{\prime}$ of $v$ in $V$, and a
morphism $g:V^{\prime}\to U$ such that $g(v)=u$.

######  Lemma 20.7.

Let $(U,s_{U},t_{U}),(V,s_{V},t_{V})$ be bisubmersions and let $u\in U,v\in V$
with $s_{U}(u)=s_{V}(v)$, then any local morphisms around $u$ are local
equivalences.

###### Proof.

Let $f^{\prime}:U^{\prime}\to V$ denote the morphism induced by $f$. Applying
the previous lemma, we can restrict to an open neighborhood $V^{\prime}\subset
V$ with a morphism $g:V^{\prime}\to U^{\prime}$ such that $g(v)=u$, which
realizes a local equivalence. ∎

######  Corollary 20.8.

Local morphisms are local equivalences.

###### Proof.

Let $f:(U,s_{U},t_{U})\to(V,s_{V},t_{V})$ be a local morphism. Pick some $u\in
U^{\prime}$ where $f:U^{\prime}\to V$ is the morphism defined by the local
morphism $f$, and let $v=f(u)$. Then clearly $s_{U}(u)=s_{V}(v)$ and we can
apply the previous lemma. ∎

######  Proposition 20.9.

Let $f:(U,s_{U},t_{U})\to(V,s_{V},t_{V})$ be a local morphism of bisubmersions
which send $u\to v$, then there exists a fiber product

######  Lemma 20.10.

Let $f:(U,s_{U},t_{U})\to(V,s_{V},t_{V})$ be a local morphism of
bisubmersions. Then $(V,t_{V},s_{V})$ is locally isomorphic to
$(U,s_{U},t_{U})$.

######  Definition 20.11.

We say two bisubmersions $(U,s_{U},t_{U}),(V,s_{V},t_{V})$ of an atlas
$\mathcal{U}$ are $s$-sufficiently close, if there exists some
$(W,s_{W},t_{W})\in\mathcal{U}$ such that $s_{W}(U\times_{s_{U},t_{W}}W)\cap
s_{V}(V)$ is not empty. Similarly, we can define the notion of
$t$-sufficiently close.

######  Lemma 20.12.

Let $\mathcal{U}=(U_{i},s_{i},t_{i})_{i\in I}$ an atlas of
$(M,{\mathcal{F}})$. Let $f:(U,s_{U},t_{U})\to(V,s_{V},t_{V})$ be a local
morphism of bisubmersions of elements in $\mathcal{U}$. Then there exists a
fiber product $(U\times_{s_{U},t_{W}c}W,s_{W},t_{U})$ locally equivalent to
$(V,s_{V},s_{V})$ which consists of identity diffeomorphisms on $s_{V}(V)$ for
some $(W,s_{W},t_{W})\in\mathcal{U}$.

###### Proof.

Pick $u\in U^{\prime}$ in the domain of local morphism, and $v=f(u)$. We have
the following commutative diagram

${u}$${s_{U}(u)}$${v}$${t_{U}(u)}$$\scriptstyle{s_{U}}$$\scriptstyle{t_{U}}$$\scriptstyle{f}$$\scriptstyle{s_{V}}$$\scriptstyle{t_{V}}$

Since the inverse of $(V,s_{V},t_{V})$ is adapted to $\mathcal{U}$. Without
loss of generality, we can let
$(V,s_{V},t_{V})^{-1}=(V,t_{V},s_{V})\in\mathcal{U}$. Now let
$(W,s_{W},t_{W})=(V,t_{V},s_{V})$. Let’s consider the fiber product

$(W\times_{{s_{W},t_{U}}}U,s_{U},t_{W})$

We want to show this fiber product is locally equivalent to
$(U,\operatorname{Id},\operatorname{Id})$. First, let $V^{\prime}$ denote the
domain of the induced morphism $g:(V,s_{V},t_{V})\to(U,s_{U},t_{U})$. It
suffices to show there exists a local morphism from
$(U,\operatorname{Id},\operatorname{Id})$ to
$(W^{\prime}\times_{{s_{W},t_{U}}}U^{\prime},s_{U},t_{W})$ around $u\in
U^{\prime}$. Define $\phi:W\times_{{s_{W},t_{U}}}U\to V$ by

$\phi\big{(}(w,u)\big{)}=f(u)$

Then

$s_{V}\Big{(}\phi\big{(}(w,u)\big{)}\Big{)}=s_{V}(f(u))=s_{V}(v)=s_{U}\big{(}(w,u)\big{)}$
$s_{V}(v)=t_{W}(v)=t_{W}\big{(}(w,u)\big{)}$

Therefore, we see that there exists a local morphism from the fiber product
$(W\times_{{s_{W},t_{U}}}U,s_{U},t_{W})$ to $(V,s_{V},s_{V})$. Therefore, we
have a local equivalence.

Note that, by assumption, we only have $h(w)=v$ at $v=f(u)$. But if we
restrict to a small enough neighborhood of $V$, since the exponential map is
uniquely determined locally, if we have an identity Now we pick a
$(W,s_{W},t_{W})\in\mathcal{U}$ such that $t_{W}(w)=s_{U}(u)$ and
$s_{W}(w)=s_{V}(v)$ for some $w\in W$. This is possible since
$(U,s_{U},t_{U})$ and $(V,s_{V},t_{V})$ are sufficiently close. Now we have
$s_{U}\times_{s_{U},t_{W}}W((u,w))=s_{V}(v)$. Hence, we can apply previous
lemmas to get $(U\times_{s_{U},t_{W}}W,s_{W},t_{U})$ are locally equivalent to
$(V,s_{V},t_{V})$ by the local morphism $g:V\to W$ defined by

$g(v)=f(u)$

Clearly we have $S_{W}(F((u,w)))=s_{W}(w)=S_{V}(v)$

For general case, we can connect $s_{U}(U)$ and $s_{V}(V)$ by taking fiber
products with a series of bisubmersions $W_{1},\cdots,W_{k}$.

$Z=U\times_{s_{U},t_{W_{1}}}W_{1}\cdots\times_{s_{W_{k-1}},t_{W_{k}}}W_{k}$

Denote $(Z,s_{Z},t_{Z})$ the resulting fiber product. Notices that all fiber
products are adapted by the atlas. Let $z\in Z$ such that $s_{Z}(z)=s_{V}(v)$,
then by adeptness, there exists a $(U_{i},s_{i},s_{j})\in\mathcal{U}$ with a
local morphism $Z\to U_{i}$ at $z$. By previous lemmas, we see $U_{i}$ is
locally equivalent to $Z$ at $z$. By construction, $U_{i}$ ∎

###### Proof.

Pick $U^{\prime}\subset U$ such that $f:U^{\prime}\to V$ is a morphism, and
pick $u\in U^{\prime},v=f(u)$. Let’s take two bisubmersion $(W,s_{W},t_{w})$
and $(X,s_{X},t_{X})$ such that

$\displaystyle s$

By definition $U\times_{s_{U},s_{V}}V=\\{(u,v)\in U\times
V|s_{u}(u)=s_{v}(v)\\}$. Since $f$ is a morphism between $(U,s_{U},t_{U})$ and
$(V,s_{V},t_{V})$, we have

$\displaystyle s_{V}(f(u))$
$\displaystyle=s_{U}(u)=s_{V}(v)t_{V}(f(u))=t_{U}(u)$

hence

$s_{W}(u,v)=t_{v}(v)$

there exists a fiber product
$(W=U\times_{s_{U},s_{V}}V,s_{W}=t_{V},t_{W}=t_{U})$ of $(U,s_{U},t_{U})$ and
$(V,s_{V},t_{V})^{-1}$ ∎

Now let’s construct the holonomy $\infty$-groupoids which enhances the
holonomy groupoids of singular foliations.

######  Definition 20.13.

Let $(M,{\mathcal{F}})$ be a singular foliation manifold with an atlas
$\mathcal{U}=(U_{i},s_{i},t_{i})_{i\in I}$. We define a simplicial manifold
$\operatorname{Hol}^{\infty}_{\bullet}({\mathcal{F}})$ by

$\displaystyle\operatorname{Hol}^{\infty}_{k}({\mathcal{F}})$
$\displaystyle=\coprod_{i_{1},\cdots,i_{k}\in
I}U_{i_{1}}\times_{s_{U_{i_{1}}},t_{U_{i_{2}}}}U_{i_{2}}\times\cdots\times
U_{i_{k-1}}\times_{s_{U_{i_{k-1}}},t_{U_{i_{k}}}}U_{i_{k}}$ $\displaystyle
k\geq 1$ $\displaystyle\operatorname{Hol}^{\infty}_{1}({\mathcal{F}})$
$\displaystyle=M$

with face maps generated by

$\displaystyle d_{l}(u_{1},u_{2}\cdots,u_{k})$
$\displaystyle=(u_{2}\cdots,u_{k})$ $\displaystyle l=0$ $\displaystyle
d_{l}(u_{1},u_{2}\cdots,u_{k})$
$\displaystyle=(u_{1},u_{2}\cdots,u_{l-1},(u_{l},u_{l+1}),\cdots,u_{k})$
$\displaystyle 1\leq l\leq k-1$ $\displaystyle d_{l}(u_{1},u_{2}\cdots,u_{k})$
$\displaystyle=(u_{1},u_{2}\cdots,u_{k-1})$ $\displaystyle l=k$

where $(u_{l},u_{l+1})\in
U_{i_{l}}\times_{{s_{U_{i_{l}}}},t_{U_{i_{l+1}}}}U_{i_{l+1}}$.

######  Proposition 20.14.

$\operatorname{Hol}^{\infty}_{\bullet}({\mathcal{F}})$ is a Lie
$\infty$-groupoid.

###### Proof.

Note that given any length $k$ fiber product

$U_{i_{1}}\times\cdots\times U_{i_{k}}$

we can get its inverse simply by

$U_{i_{k}}\times\cdots\times U_{i_{1}}$

with all $s_{i_{l}}$ and $t_{i_{l}}$ switched, i.e. we take the inverse of
each $(U_{i},s_{i},s_{j})$ and then take the fiber product in the reverse
order. This implies that all $k$-simplices are invertible. In particular,
given a horn

$\Lambda^{k}[n]\to\operatorname{Hol}^{\infty}_{\bullet}({\mathcal{F}})$

we get a map $f:u_{1}\to u_{k}$ as an element of $U_{i_{1}}\times\cdots
U_{i_{k}}$ , a map $g:u_{k}\to u_{n}$ as an element of
$U_{i_{k+1}}\times\cdots U_{i_{n}}$, and a map $h:u_{n}\to u_{1}$ as an
element of $U_{i_{n}}\times U_{i_{1}}$. Now we can simply take
$\phi=g^{-1}\circ h\circ f^{-1}$, which gives the desired horn filling. Hence,
the induced map

$\operatorname{Hol}^{\infty}_{n}({\mathcal{F}})\to
M_{\Lambda^{k}[n]}\operatorname{Hol}^{\infty}_{\bullet}({\mathcal{F}})$

is clearly a surjective submersion. ∎

We call $\operatorname{Hol}^{\infty}_{\bullet}({\mathcal{F}})$ the holonomy
$\infty$-groupoid of the singular foliation ${\mathcal{F}}$. Next, we shall
justify the correctness of our construction.

######  Proposition 20.15.

The 1-truncation $\tau^{\leq
1}\operatorname{Hol}^{\infty}_{\bullet}({\mathcal{F}})$ of the holonomy
$\infty$-groupoid is equivalent to the holonomy groupoid
$\operatorname{Hol}({\mathcal{F}})$ in the sense of Androulidakis and
Skandalis.

###### Proof.

Recall that the arrow space of the holonomy groupoid is the quotient of path
holonomy atlas by the relation such that $u\in U_{i}$ is equivalent to $v\in
U_{j}$ if there exists a local morphism from $U_{i}\to U_{j}$ which takes $u$
to $v$. Now suppose $u\sim v$ for $u\in U_{i},v\in U_{j}$, and there exists a
local morphism $f:U_{i}\to U_{j}$ send $u$ to $v$. Then by Lemma 20.12,
$(v,u)\in U_{j}^{-1}\times_{t_{U_{j}},t_{U_{i}}}U_{i}=W$ and there exists a
local morphism near $(v,u)$ which maps to identity diffeomorphisms near
$s_{U_{j}}(v)$. In addition, at $(v,u)$,

$s_{W}\big{(}(v,u)\big{)}=t_{U_{j}}(v)=t_{U_{i}}(u)=t_{W}\big{(}(v,u)\big{)}$

∎

###  21\. Holomorphic singular foliations

####  21.1. $L_{\infty}$-algebroids for holomorphic singular foliations

Let $(X,{\mathcal{F}})$ be a holomorphic singular foliation, i.e.
${\mathcal{F}}$ is a locally finitely generated involutive coherent
${\mathcal{O}}_{X}$-module. Since $T_{X}$ has a natural $L_{\infty}$ structure
(indeed Lie structure), ${\mathcal{F}}$ is closed with this $L_{\infty}$
structure. Denote the Dolbeault dga $(A^{0,\bullet},\overline{\partial},0)$ by
$A$. Recall that a sheaf ${\mathcal{S}}$ of
${\mathcal{C}}^{\infty}(X)$-modules is called $\overline{\partial}$-analytic
coherent if ${\mathcal{S}}$ locally admits a resolution by finitely generated
free modules and equips with a flat $\overline{\partial}$-connection.

By a simple application of Hilbert’s Syzygy theorem, we have that
${\mathcal{F}}$ admits a local resolution.

######  Proposition 21.1 ([LLS20]).

Any holomorphic singular foliations on a complex manifold $X$ of dimension $n$
locally admits a finite resolution by finitely generated free
${\mathcal{O}}_{X}$-modules of length $\leq n$.

Equivalently, we can regard $X$ has local resolution by trivial vector
bundles.

The $\overline{\partial}$-connection is directly inherited the flat
$\overline{\partial}$-connection on the $T_{X}$. Therefore, ${\mathcal{F}}$ is
$\overline{\partial}$-analytic coherent . In (3), the bundle $F$ is formed by
taking all tangent vectors tangent to leaves, whereas in (4) a $k$-forms is in
$\mathcal{J}$ if it vanishes on any $k$-tuples of vectors tangent to leaves.
Laurent-Gengoux, Lavau, Strob in [LLS20] considered the case when a singular
foliation on a smooth manifold admits a (global) resolution by vector bundles,
and they proved that we can always construct an $L_{\infty}$-algebroid
associated to that singular foliation:

######  Theorem 21.2 ([LLS20]).

Give a foliation ${\mathcal{F}}$ which admits a resolution by vector bundles
${\mathcal{F}}_{\bullet}$, there exists a universal $L_{\infty}$-algebroid
$\mathfrak{g}\in\mathsf{L_{\infty}Algd}^{\operatorname{dg}}_{{\mathcal{C}}^{\infty}{M}}$
whose linear part is the given resolution ${\mathcal{F}}_{\bullet}$. Here
universal means that $\mathfrak{g}$ is the terminal object in the category of
$\mathsf{L_{\infty}Algd}^{\operatorname{dg}}_{{\mathcal{C}}^{\infty}{M}}/{\mathcal{F}}$
which consists of $L_{\infty}$-algebroids resolving ${\mathcal{F}}$.

We would like to generalize this to holomorphic singular foliations. Though we
don’t have global resolutions for holomorphic singular foliations, we can glue
the local resolutions by higher homotopical information
[TT76][TT78][Blo05][Wei16]. As natural to coherent sheaves on complex
manifolds, we will use the Dolbeault enhancement introduced by Block in
[Blo05]:

######  Theorem 21.3 ([Blo05]).

Let $X$ be a complex manifold, and $\mathfrak{g}=T^{0,1}X$ be the Dolbeault
Lie algebroid. The homotopy category of the dg-category
$\mathsf{Mod}^{Coh}_{\operatorname{CE}(\mathfrak{g})}=\mathtt{Rep}_{\mathfrak{g},A}$
is equivalent to the bounded derived category of chain complexes of sheaves of
${\mathcal{O}}_{X}$-modules with coherent cohomology on $X$

As a corollary, for any coherent analytic sheaf ${\mathcal{F}}$, there exists
a cohesive module $(E,\mathbb{E})\in\mathsf{Mod}^{Coh}_{A}$, which is unique
up to quasi-isomorphism, corresponds to ${\mathcal{F}}$. In fact, take
${\mathcal{F}}^{\infty}={\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{C}}^{\infty}(X)$
which is the $\overline{\partial}$-coherent sheaf associated with
${\mathcal{F}}$. we have a projective resolution

$0\to E_{-n}\stackrel{{\scriptstyle
d_{n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
d_{2}}}{{\longrightarrow}}E_{-1}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}E_{0}\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}{\mathcal{F}}^{\infty}\longrightarrow
0$

Tensoring the above sequence with the Dolbeault dga gives a resolution
$E_{\bullet}\otimes_{{\mathcal{C}}^{\infty}(X)}A\to{\mathcal{F}}^{\infty}\otimes_{{\mathcal{C}}^{\infty}(X)}A$.
Denote ${\mathcal{F}}^{\infty}\otimes_{{\mathcal{C}}^{\infty}(X)}A$ by $F$. We
can equip the dg-$A$-module
${\mathcal{E}}_{\bullet}=E_{\bullet}\otimes_{{\mathcal{C}}^{\infty}(X)}A$ with
a ${\mathbb{Z}}$-connection $\nabla$ and then get a cohesive module. Denote
the associated map $E_{0}\to 0$ by $d_{0}$.

####  21.2. Lifting dg-$A$-Module structure to $L_{\infty}$-algebroid
structure

We shall construct $L_{\infty}$-structure on ${\mathcal{E}}_{\bullet}$
following similar methods in [Bar+97] and [LLS20].

For simplicity, sometimes we will denote the $n$-ary bracket
$[-,\cdots,-]_{n}$ by $l_{n}$. We regard
$F\in\mathsf{Mod}^{\operatorname{dg}}_{A}$ concentrated in degree zero, then
$\rho$ naturally extends to a chain map. Let $C_{\bullet}$ and $B_{\bullet}$
denote the $d$-cycles and $d$-boundaries, respectively. We have $F\simeq
H_{0}({\mathcal{E}}_{\bullet})$. The existence of contracting homotopy
$s:{\mathcal{E}}_{\bullet}\to{\mathcal{E}}_{\bullet-1}$ specifies a homotopy
inverse $\delta:F\to{\mathcal{E}}_{\bullet}$.

#####  21.2.1. Construction of $l_{2}$

First, we want to construct $l_{2}$ on ${\mathcal{E}}_{0}$.

######  Lemma 21.4.

There exists a skew-symmetric $A$-linear map
$\tilde{l}_{2}:{\mathcal{E}}_{0}\otimes{\mathcal{E}}_{0}\to{\mathcal{E}}_{0}$
satisfying

  1. (1)

$\tilde{l}_{2}(c_{1},b_{1})=0$

  2. (2)

$[c_{1},a\cdot c_{2}]=a[c_{1},c_{2}]+\rho(c_{1})(a)c_{2}$

  3. (3)

$\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\tilde{l}_{2}(\tilde{l}_{2}(c_{\sigma(1)},c_{\sigma(2)}),c_{\sigma(3)})\in
B_{0}$

where $c_{i}\in{\mathcal{E}}_{0}$, $b_{i}\in B_{0}$, $a\in A$.

###### Proof.

Define $\tilde{l}_{2}=\delta\circ[-,-]\circ(\rho\otimes\rho)$. It is clearly
skew-symmetric. Property (1) is satisfied since $\rho(b_{1})=0$. We claim that
$[-,-]=\rho\circ\tilde{l}_{2}\circ(\delta\otimes\delta)$. In fact,

$d_{0}\circ\tilde{l}_{2}\circ(\delta\otimes\delta)=d_{0}\circ\delta\circ[-,-]\circ(\rho\otimes\rho)\circ(\delta\otimes\delta)$

since $d_{0}\circ\delta=\operatorname{Id}$ the result follows.

For (2), we want to construct $\delta$ explicitly. In fact, pick an open
neighborhood $U\subset X$ such that $F|_{U}$ is finitely generated by
$\\{f_{i}\otimes a_{j}\\}$. Pick $\\{e_{i}\otimes a_{j}\\}$ such that
$\rho{(e_{i}\otimes a_{j})}=f_{i}\otimes a_{j}$. Then we can define
$[e_{i},e_{j}]=\sum_{l=1}^{k}c_{ij}^{l}e_{k}$ where $c_{ij}^{l}$ comes from
$[f_{i},f_{j}]=\sum_{l=1}^{k}c_{ij}^{l}f_{k}$. Extend the brackets to all
$E_{0}$ by Leibniz rules, i.e.

$[e_{i},a\cdot e_{j}]=a[e_{i},e_{j}]+\rho(e_{i})(a)e_{j}$

. Note that here we regard $\rho(e_{i}\otimes a_{j})\in
F={\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}A$ sits inside
$T_{X}\otimes_{{\mathcal{O}}_{X}}A\subset\operatorname{Der}_{k}(A)$. Finally,
we glue all the local brackets by partition of unity.

Next, we want to show (3) holds. On ${\mathcal{F}}$, the Jacobi identity
implies that, for $f_{i}\in{\mathcal{F}}$

$\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}[[f_{\sigma(1)},f_{\sigma(2)}],f_{\sigma(3)}]=0$

Since $[-,-]=\rho\circ\tilde{l}_{2}\circ(\delta\otimes\delta)$, the left hand
side becomes

$\displaystyle\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\big{(}\rho\circ\tilde{l}_{2}\circ(\delta\otimes\delta)\big{)}\circ\big{(}\rho\circ(\tilde{l}_{2}\otimes{\mathbb{1}})\circ(\delta\otimes\delta\otimes{\mathbb{1}})\big{)}(f_{\sigma(1)},f_{\sigma(2)},f_{\sigma(3)})$
$\displaystyle=$
$\displaystyle\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\big{(}\rho\circ\tilde{l}_{2}\circ(\delta\otimes\delta)\big{)}\circ\big{(}\rho\circ(\tilde{l}_{2}(\delta
f_{\sigma(1)}\otimes\delta f_{\sigma(2)}))\otimes f_{\sigma(3)}\big{)}$
$\displaystyle=$
$\displaystyle\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\rho\circ\tilde{l}_{2}\Big{(}\delta\circ\rho\circ(\tilde{l}_{2}(\delta
f_{\sigma(1)}\otimes\delta f_{\sigma(2)})),\delta f_{\sigma(3)}\Big{)}$

Note that $d_{0}\circ\delta={\mathbb{1}}_{{\mathcal{F}}}$ and there exists a
chain homotopy $s:{\mathcal{E}}_{\bullet}\to{\mathcal{E}}_{\bullet-1}$ with
$\delta\circ d-{\mathbb{1}}_{{\mathcal{E}}_{\bullet}}=s\circ d+d\circ s$.
Hence, we get,

$\displaystyle\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\rho\circ\tilde{l}_{2}\Big{(}({\mathbb{1}}_{{\mathcal{E}}_{0}}+d_{1}\circ
s)\circ(\tilde{l}_{2}(\delta f_{\sigma(1)}\otimes\delta f_{\sigma(2)})),\delta
f_{\sigma(3)}\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\rho\circ\tilde{l}_{2}\Big{(}(\tilde{l}_{2}(\delta
f_{\sigma(1)}\otimes\delta f_{\sigma(2)})),\delta f_{\sigma(3)}\Big{)}$
$\displaystyle+\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\rho\circ\tilde{l}_{2}\Big{(}d_{1}\circ
s\circ(\tilde{l}_{2}(\delta f_{\sigma(1)}\otimes\delta f_{\sigma(2)})),\delta
f_{\sigma(3)}\Big{)}$

By property (1), the second term is 0. Therefore, we have

$\rho\Big{(}\sum_{\sigma\in\operatorname{UnSh}(2,1)}(-1)^{\sigma}\circ\tilde{l}_{2}\Big{(}(\tilde{l}_{2}(\delta
f_{\sigma(1)}\otimes\delta f_{\sigma(2)})),\delta
f_{\sigma(3)}\Big{)}\Big{)}=0$

which implies the term inside belongs to $B_{0}$. ∎

Next, we extend $\tilde{l}_{2}$ to a chain map $l_{2}:E_{\bullet}\otimes
E_{\bullet}\to E_{\bullet}$. Consider $e_{1}\otimes e_{0}\in E_{1}\otimes
E_{0}$, then $d(_{1}\otimes e_{0})=de_{1}\otimes e_{0}$. We define
$l_{2}\big{(}d(x_{1}\otimes x_{0})\big{)}=\tilde{l}_{2}\big{(}d(x_{1}\otimes
x_{0})\big{)}=\tilde{l}_{2}\big{(}(dx_{1}\otimes x_{0})\big{)}=0$. Pick some
$x_{1}\in B_{1}$ such that $dx_{1}=0$, then we define $l_{2}(e_{1}\otimes
e_{0})=x_{1}$. Extend this to $E_{0}\otimes E_{1}$ skew-symmetrically.
Finally, by induction, we can extend $l_{2}$ to all $E_{i}\otimes E_{j}$.

It turns out that $l_{1}=d$ and $l_{2}$ satisfy a higher homotopy identity by
introducing a new map $l_{3}$.

######  Proposition 21.5.

There exists an almost Lie algebroid structure on ${\mathcal{E}}_{\bullet}$.

######  Lemma 21.6.

There exists a degree one skew-symmetric map
$l_{3}:\bigotimes_{3}E_{\bullet}\to E_{\bullet}$ such that
$l_{1}l_{3}+l_{3}l_{1}+l_{2}l_{2}=0$.

###### Proof.

First let $e=e_{1}\otimes e_{2}\otimes e_{3}$ in degree 0, then by previous
lemma, $l_{2}l_{2}e=b\in B_{0}$, hence we can find a $z\in E_{1}$ with $dz=b$.
Now we define $l_{3}(e)=-z$, then $l_{1}l_{3}+l_{3}l_{1}+l_{2}l_{2}=0$ since
$l_{3}l_{1}(e)=0$.

Next we proceed by induction, suppose the $l_{3}$ is constructed up to degree
$k$, then $l_{2}l_{2}+l_{3}l_{1}$ is then defined for degree $k+1$ elements.
Compose with $l_{1}$ we have
$l_{1}(l_{2}l_{2}+l_{3}l_{1})=l_{2}l_{2}l_{1}+l_{1}l_{3}l_{1}$ used the fact
that $l_{1}$ and $l_{2}$ commutes. Using the equality on degree $k$, we get
$l_{2}l_{2}l_{1}+l_{1}l_{3}l_{1}=-l_{3}l_{1}l_{1}=0$, hence we see
$(l_{2}l_{2}+l_{3}l_{1})$ on degree $k+1$ element is a boundary $b$, then we
can define the image of $l_{3}$ to be a preimage of $b$ under $l_{1}$ just as
before. ∎

Next, we want to construct $l_{i}$ for all $i\geq 3$. The idea is still the
same as before. We first consider the degree 0 elements. Suppose we have
construct $l_{i}$ for $1\leq i<n$ which satisfy strong homotopy Jacobi
identities. For simplicity, we denote $\sum_{i+j=n+1}(-1)^{i(j-1)}l_{j}l_{i}$
which is already summed under appropriate unshuffles. We will use the
following lemma:

######  Lemma 21.7.

Let $\\{l_{i}\\}$ define an $L_{\infty}$ structure, then we have

$l_{1}\sum_{i+j=n+1,i,j>1}(-1)^{i(j-1)}l_{j}l_{i}=(-1)^{(n-1)}\sum_{i+j=n+1,i,j,>1}(-1)^{i(j-1)}l_{j}l_{i}l_{1}$

Hence on $\bigotimes^{n}X_{0}$, both $l_{1}l_{1}l_{n}$ and $l_{1}l_{n}l_{1}$
vanishes, therefore we know $l_{1}l_{n}$ is a cycle, so we can define $l_{n}$
by acyclicity of $E_{\bullet}$.

Next, suppose $l_{n}$ has been constructed on all degree $<k$ elements.

$l_{1}\sum_{i,j>1,i+j=n+1}(-1)^{i(j-1)}l_{j}l_{i}=(-1)^{(n-1)}\sum_{i,j>1,i+j=n+1}(-1)^{i(j-1)}l_{j}l_{i}l_{1}=0$

Hence, $\sum_{i,j>1,i+j=n+1}(-1)^{i(j-1)}l_{j}l_{i}$ is a cycle $b$ in
$E_{n-3}$. Let $z\in E_{n-2}$ such that $l_{1}z=b$, then with appropriate care
of signs from unshuffles, we can define $l_{n}=z$.

######  Theorem 21.8.

Given a holomorphic foliation ${\mathcal{F}}$ on a compact complex manifold
${\mathcal{F}}$ , there exist an $L_{\infty}$-algebroid $\mathfrak{g}$ over
$A$, where the linear part of $\mathfrak{g}$ corresponds to the cohesive
module $E^{\bullet}$ associated to
${\mathcal{F}}^{\infty}={\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{C}}^{\infty}{X}$

####  21.3. Cofibrant replacement

Since $F\in\mathsf{Mod}^{\operatorname{dg}}_{A}$, $({\mathcal{E}},\nabla)$ is
essentially a cofibrant replacement in the model category
$\mathsf{Mod}^{\operatorname{dg}}_{A}$. Note that we have a Quillen adjunction
$\mathsf{L_{\infty}Algd}^{\operatorname{dg}}_{A}\leftrightarrow\mathsf{Mod}^{\operatorname{dg}}_{A}$,
a natural question is whether we can lift $(E,\mathbb{E})$ to a cofibrant
object in $\mathsf{L_{\infty}Algd}^{\operatorname{dg}}_{A}$.

Since every strict morphism between $L_{\infty}$-morphism is an
$\infty$-morphism, we have a function
$\iota:\mathsf{L_{\infty}Algd}^{\operatorname{dg}}_{A}\to\mathtt{L_{\infty}Algd}_{A}$.

####  21.4. Perfect singular foliations

Following the idea of our construction of $L_{\infty}$-algebroid, we refine
the notion of singular foliation as follows.

######  Definition 21.9.

Let $M$ be a ${\mathcal{C}}^{\infty}$-manifold, we define a perfect singular
foliation ${\mathcal{F}}$ to be a subsheaf ${\mathcal{F}}$ of
${\mathcal{O}}_{M}$-module of the tangent sheaf $T_{M}$ such that:

  1. (1)

(Perfectness)${\mathcal{F}}$ is a (strict) perfect ${\mathcal{O}}_{M}$-module,
i.e. there exists a (global) local resolution by finite projective
${\mathcal{O}}_{M}$-modules

$0\to E_{-d}\to E_{d-1}\to\cdots\to E_{-1}\to E_{0}\to{\mathcal{F}}\to 0$

  2. (2)

(Involutivity) ${\mathcal{F}}$ is closed under brackets.

In this definition, we replace the local finite generativity by local finite
presentivity, which allows us to do many operations in homological algebras
and homotopical operations as we did in the case of holomorphic singular
foliations previously. For perfect singular foliations, we can always endow
$L_{\infty}$-algebroid structures by [LLS20]. Note that what we defined here
is using the existing of a global resolution, which we might weaken to exist
local resolutions. In that case, we call ${\mathcal{F}}$ a weakly perfect
singular foliation. In the case of compact manifolds, these two definition
agrees. For weakly perfect foliations, we cannot construct
$L_{\infty}$-structure directly, but we can follow the similar idea of Theorem
21.8, and use twisted perfect complex (c.f. [Wei16][TT76]) resolving
${\mathcal{F}}$, and then use similar freely generating method as Theorem 21.8
to construct an $L_{\infty}$-algebroid structure.

We can easily define perfect complex singular foliation to be a subsheaf
${\mathcal{F}}$ of the complexified tangent sheaf $T^{{\mathbb{C}}}_{M}$ which
satisfies perfectness and involutivity.

We can also generalize this to derived manifolds:

######  Definition 21.10.

Let $(X,{\mathcal{O}}_{X})$ be a derived manifold, we define a perfect
singular foliation ${\mathcal{F}}$ to be a complex of subsheaf of
${\mathcal{F}}$ of ${\mathcal{O}}_{X}$-module of the tangent sheaf $T_{X}$
complex such that:

  1. (1)

(Perfectness)${\mathcal{F}}$ is a perfect ${\mathcal{O}}_{X}$-module, i.e.
there exists a resolution by a double complex of finite projective
${\mathcal{O}}_{X}$-modules

$0\to E_{-d}\to E_{d-1}\to\cdots\to E_{-1}\to E_{0}\to{\mathcal{F}}\to 0$

  2. (2)

(Involutivity) ${\mathcal{F}}$ is closed under brackets.

For example, in coisotropic reduction (c.f. Example 15.4), if ${\mathcal{F}}$
has constant dimension in each stratum, then ${\mathcal{F}}$ is a perfect
singular foliation on the derived critical locus.

Following the ideas for perfect singular foliations, we can generalize the
elliptic involutive structures

######  Definition 21.11.

Let $M$ be a ${\mathcal{C}}^{\infty}$-manifold, and ${\mathcal{F}}\subset
T^{{\mathbb{C}}}_{M}$ a complex singular foliation. We say ${\mathcal{F}}$ is
an elliptic singular foliation if

  1. (1)

${\mathcal{F}}$ is a complex perfect singular foliation.

  2. (2)

${\mathcal{F}}+\overline{{\mathcal{F}}}=T^{{\mathbb{C}}}_{M}$.

Clear, elliptic involutive structures are elliptic singular foliations, which
we can also call elliptic regular foliations.

###  22\. Elliptic involutive structures

####  22.1. Elliptic involutive structures and foliations

Let $M$ be a compact manifold.

######  Definition 22.1.

A complex Lie algebroid $A$ is elliptic if its associated dga
$\operatorname{Sym}A^{\vee}[-1]$ is an elliptic complex.

Note that here the ellipticity is equivalent to require
$\rho(A)+\overline{\rho(A)}=T_{{\mathbb{C}}}M$.

######  Definition 22.2.

Let $M$ be a smooth manifold. An elliptic involutive structure (EIS) consists
of the following data:

  1. (1)

An involutive sub-bundle $V$ of the complexified tangent bundle
$T_{{\mathbb{C}}}M$.

  2. (2)

$V$ is an elliptic Lie algebroid.

######  Example 22.3.

The complexified tangent Lie algebroid is clearly a trivial EIS.

######  Example 22.4.

Take $V=T^{0,1}M$ the anti-holomorphic tangent Lie algebroid, then an $EIS$ on
$V$ corresponds to a complex structure on $M$.

######  Theorem 22.5 (Newlander-Nirenberg).

Let $V$ be an elliptic involutive structure on $M$. Then, locally, there exist
on $M$ real coordinates $(t_{1},\cdots,t_{d})$ and complex coordinates
$(z_{1},\cdots,z_{n})$ such that

(22.1) $V=\operatorname{Span}\big{\\{}\frac{\partial}{\partial
t_{1}},\cdots,\frac{\partial}{\partial
t_{d}},\frac{\partial}{\partial\overline{z}_{1}},\cdots,\frac{\partial}{\partial\overline{z}_{n}}\big{\\}}=\operatorname{Span}\big{\\{}\frac{\partial}{\partial
z_{1}},\cdots,\frac{\partial}{\partial z_{n}}^{\perp}\big{\\}}$

Thus, locally, V looks like the product distribution ${\mathbb{R}}^{d}\oplus
T^{0,1}{\mathbb{C}}^{n}$, where ${\mathbb{C}}^{n}$ has its standard complex
structure.

The real part of an EIS is always a foliation on $M$, hence we also call an
EIS an elliptic (regular) foliation.

######  Proposition 22.6.

Let ${\mathcal{O}}_{V}$ denote the structure sheaf of an $EIS$ on $M$, then
there exists an equivalence of categories

(22.2) $\\{\text{locally free sheaves of ${\mathcal{O}}_{V}$
modules}\\}\simeq\\{\text{ finitely generated projective $V$-modules}\\}$

###### Proof.

First let $\mathcal{E}$ be a locally free sheaf of ${\mathcal{O}}_{V}$-module.
Note that this condition is equivalent to that there exists a trivializing
cover $\\{U_{i}\\}$ such that the transition functions of $E$ corresponding to
this cover take value in ${\mathcal{O}}_{V}$, where
$E=\Gamma(E)\otimes_{{\mathcal{O}}_{V}}{\mathcal{C}}^{\infty}(M)$ is the
vector bundle corresponding to $\mathcal{E}$. Hence, in order to construct a
flat $V$-connection on $E$, we just need to let the frame on
$U_{i}\times{\mathbb{C}}^{r}$ to be parallel.

Let $E$ be finitely generated projective $V$-modules, i.e. a vector bundle
with a flat $V$-connection. We want to show that, for any $x\in M$, there
exists a parallel local frame on some neighborhood of $x$. Let $\nabla$ be the
flat connection on $E$, and $\\{e_{i}\\}$ a local frame on some neighborhood
$U$ of $x$, then $\nabla e_{i}=\omega^{j}_{i}e_{j}$ where $\omega$ is the
connection 1-form. By Newlander-Nirenberg, we can let $U$ be small such that
$V|_{U}=\operatorname{Span}_{{\mathbb{C}}}\\{dz^{1},\cdots,dz^{m}\\}^{\perp}$,
where $z^{1},\cdots,z^{m}$ are some complex coordinates on $E|_{U}$.

Now we have $E|_{U}\simeq U\times{\mathbb{C}}^{r}$. Let $u^{1},\cdots,u^{r}$
be complex coordinates on ${\mathbb{C}}^{r}$. Consider a distribution

$V^{\prime}=\operatorname{Span}_{{\mathbb{C}}}\bigg{\\{}\frac{\partial}{\partial\overline{u}^{k}},v-\omega^{j}_{i}(v)u^{i}\frac{\partial}{\partial
u^{k}}|v\in V\subset
T_{{\mathbb{C}}}(U\times{\mathbb{C}}^{r})\bigg{\\}}\subset
T_{{\mathbb{C}}}(U\times{\mathbb{C}}^{r})$

By flatness of $\nabla$, we can show that this distribution is involutive. By
our construction,
$V^{\prime}+\overline{V^{\prime}}=T_{{\mathbb{C}}}U\times{\mathbb{C}}^{r}$,
therefore, we get an EIS on $U\times{\mathbb{C}}^{r}$. According to our
construction, the space of 1-form annihilate this distribution is

$\displaystyle(V^{\prime})^{\perp}$
$\displaystyle=V^{\perp}+\operatorname{Span}_{{\mathbb{C}}}\\{du^{j}+\tilde{\omega}^{j}_{i}u_{i}:j=1,\cdots,r\\}$

Note that $V^{\perp}=\\{dz^{1},\cdots,dz^{m}\\}$ by Newlander-Nirenberg, and
by same reason we also have
${V^{\prime}}^{\perp}=\\{d\tilde{z}^{1},\cdots,d\tilde{z}^{m}\\}$ for some
coordinates $\\{\tilde{z}^{i}\\}$. Hence, we have
$d\tilde{z}^{j}=F^{j}_{i}dz^{i}+G^{j}_{i}(du^{i}+\tilde{\omega}^{i}_{k}u^{k})$.
With possibly rearranging indices, we have $G^{j}_{i}\in GL_{r}({\mathbb{C}})$
in some neighborhood of zero-section. Differentiating previous equation and
set $u^{k}=0$ for all $k$, we get

$0=dF^{j}_{i}\wedge dz_{i}+dG^{j}_{i}\wedge
du^{i}-G^{j}_{i}\tilde{\omega}^{i}_{k}\wedge du^{k}$

on $U\times\\{0\\}$. Pulling back to
$V^{\vee}\otimes(^{(}T^{1,0}{\mathbb{C}}^{r})^{\vee}$, we get

$d_{V}G^{j}_{i}-G^{j}_{k}\tilde{\omega}^{k}_{i}=0$

Now let $\tilde{\sigma}_{i}=(G^{i}_{j})^{-1}\sigma_{j}$ on $U\times\\{0\\}$.
Next, we want to show that $\tilde{\sigma}_{k}$’s are all parallel.

First, note that

$\displaystyle\nabla\sigma_{j}=$
$\displaystyle\omega^{k}_{j}\otimes\sigma_{k}=d_{V}G^{k}_{j}\otimes\tilde{\sigma}_{k}+G^{k}_{j}\nabla\tilde{\sigma}_{k}$
$\displaystyle=$ $\displaystyle
G^{k}_{i}\omega^{i}_{j}\otimes\tilde{\sigma}_{k}+G^{k}_{j}\nabla\tilde{\sigma}_{k}$
$\displaystyle=$
$\displaystyle\omega^{i}_{j}\otimes{\sigma}_{i}+G^{k}_{j}\nabla\tilde{\sigma}_{k}$

Hence $G^{k}_{j}\nabla\tilde{\sigma}_{k}=0$, which implies
$\nabla\tilde{\sigma}_{k}=0$ for all $k$. Therefore,
$\\{\tilde{\sigma}_{k}\\}$ is a parallel local frame for $E$. ∎

####  22.2. Sheaf of ${\mathcal{O}}_{V}$-modules

######  Definition 22.7.

Let $(X,V)$ be a compact manifold with an elliptic involutive structure $V$.
An $V$-analytic sheaf on $X$ is a sheaf of ${\mathcal{O}}_{V}$-modules. An
$V$-analytic sheaf ${\mathcal{F}}$ is called a coherent $V$-analytic sheaf if
each point of $X$ is contained in a neighborhood $U$ such that
${\mathcal{F}}|_{U}$ is the cokernel of a morphism ${\mathcal{O}}_{V}^{\oplus
m}\to{\mathcal{O}}_{V}^{\oplus n}$ between free finite rank $V$-analytic
sheaves.

######  Lemma 22.8.

Let $k\in{\mathbb{N}}$. If for every open set $U\subset M$ and every positive
integer $m$, every morphism ${\mathcal{O}}_{V}^{\oplus
m}|_{U}\to{\mathcal{O}}_{V}|_{U}$ has locally finitely generated kernel, then
every morphism ${\mathcal{O}}_{V}^{\oplus
m}|_{U}\to{\mathcal{O}}_{V}^{k}|_{U}$ also has locally finitely generated
kernel.

###### Proof.

We proceed by induction on $k$. The case $k=1$ is just the assumption. Now
assume $k>1$ and we have already proved the case for all $j<k$, i.e. for every
$U\subset M$, every morphism ${\mathcal{O}}_{V}^{\oplus
m}|_{U}\to{\mathcal{O}}_{V}^{j}|_{U}$ has finitely generated kernel. Now we
fix a $U$, and consider a morphism $f:{\mathcal{O}}_{V}^{\oplus
m}|_{U}\to{\mathcal{O}}_{V}^{k}|_{U}$. We can regard $f=(g,h)$, where
$g:{\mathcal{O}}_{V}^{\oplus m}|_{U}\to{\mathcal{O}}_{V}^{k-1}|_{U}$ and
$h:{\mathcal{O}}_{V}^{\oplus m}|_{U}\to{\mathcal{O}}_{V}^{1}|_{U}$ is
constructed by composing $f$ with projection on the first $k-1$ entries and
the last entry respectively. By induction hypothesis, $\ker g$ is locally
finitely generated. Hence, for any $x\in U$, we can find a neighborhood
$V\subset U$ such that there exists some $p\in{\mathbb{N}}$ and a morphism
$\phi:{\mathcal{O}}_{V}^{\oplus p}|_{V}\to{\mathcal{O}}_{V}^{m}|_{V}$, and
$g\circ\phi$ is an exact sequence. Note that
$h\circ\phi:{\mathcal{O}}_{V}^{\oplus p}|_{V}\to{\mathcal{O}}_{V}|_{V}$ also
has a locally finitely generated kernel, by shrinking $V$ is necessary, there