2.1 Complex distributions

A (complex) distribution on $X$ is a sub-bundleFootnote ² of ${\mathcal{T}}_{X}$ .

A distribution ${\mathcal{P}}$ is called involutive if it is closed under the Lie bracket, i.e.  $[{\mathcal{P}},{\mathcal{P}}]\subseteq {\mathcal{P}}$ .

For a distribution ${\mathcal{P}}$ on $X$ we denote by ${\mathcal{P}}^{\bot }\subseteq \unicode[STIX]{x1D6FA}_{X}^{1}$ the annihilator of ${\mathcal{P}}$ (with respect to the canonical duality pairing).

A distribution ${\mathcal{P}}$ of rank $r$ on $X$ is called integrable if, locally on $X$ , there exist functions $f_{1},\ldots ,f_{r}\in {\mathcal{O}}_{X}$ such that $df_{1},\ldots ,df_{r}$ form a local frame for ${\mathcal{P}}^{\bot }$ .

It is easy to see that an integrable distribution is involutive. The converse is true when ${\mathcal{P}}$ is real, i.e.  $\overline{{\mathcal{P}}}={\mathcal{P}}$ (Frobenius), and when ${\mathcal{P}}$ is a complex structure, i.e.  $\overline{{\mathcal{P}}}\cap {\mathcal{P}}=0$ and $\overline{{\mathcal{P}}}\oplus {\mathcal{P}}={\mathcal{T}}_{X}$ (Newlander–Nirenberg). More generally, according to [Reference RawnsleyRaw77, Theorem 1], a sufficient condition for integrability of a complex distribution ${\mathcal{P}}$ is

(1) $$\begin{eqnarray}{\mathcal{P}}\cap \overline{{\mathcal{P}}}\text{is a sub-bundle and both }{\mathcal{P}}~\text{and}~{\mathcal{P}}+\overline{{\mathcal{P}}}~\text{are involutive}.\end{eqnarray}$$

2.2 The Hodge filtration

Suppose that ${\mathcal{P}}$ is an involutive distribution on $X$ .

Let $F_{\bullet }\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ denote the filtration by the powers of the differential ideal generated by ${\mathcal{P}}^{\bot }$ , i.e.  $F_{-i}\unicode[STIX]{x1D6FA}_{X}^{j}=\bigwedge ^{i}{\mathcal{P}}^{\bot }\wedge \unicode[STIX]{x1D6FA}_{X}^{j-i}\subseteq \unicode[STIX]{x1D6FA}_{X}^{j}$ . Let $\overline{\unicode[STIX]{x2202}}$ denote the differential in $\operatorname{Gr}_{\bullet }^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ . The wedge product of differential forms induces a structure of a commutative differential-graded algebra (DGA) on $(\operatorname{Gr}_{\bullet }^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet },\overline{\unicode[STIX]{x2202}})$ .

In particular, $\operatorname{Gr}_{0}^{F}{\mathcal{O}}_{X}={\mathcal{O}}_{X}$ , $\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{1}=\unicode[STIX]{x1D6FA}_{X}^{1}/{\mathcal{P}}^{\bot }$ and $\overline{\unicode[STIX]{x2202}}:{\mathcal{O}}_{X}\rightarrow \operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{1}$ is equal to the composition ${\mathcal{O}}_{X}\xrightarrow[]{d}\unicode[STIX]{x1D6FA}_{X}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{X}^{1}/{\mathcal{P}}^{\bot }$ . Let ${\mathcal{O}}_{X/{\mathcal{P}}}:=\ker ({\mathcal{O}}_{X}\xrightarrow[]{\overline{\unicode[STIX]{x2202}}}\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{1})$ . Equivalently, ${\mathcal{O}}_{X/{\mathcal{P}}}=({\mathcal{O}}_{X})^{{\mathcal{P}}}\subset {\mathcal{O}}_{X}$ , the subsheaf of functions locally constant along ${\mathcal{P}}$ . Note that $\overline{\unicode[STIX]{x2202}}$ is ${\mathcal{O}}_{X/{\mathcal{P}}}$ -linear.

Theorem 2 of [Reference RawnsleyRaw77] says that, if ${\mathcal{P}}$ satisfies the condition (1), the higher $\overline{\unicode[STIX]{x2202}}$ -cohomology of ${\mathcal{O}}_{X}$ vanishes, i.e.

(2) $$\begin{eqnarray}H^{i}(\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet },\overline{\unicode[STIX]{x2202}})=\left\{\begin{array}{@{}ll@{}}{\mathcal{O}}_{X/{\mathcal{P}}}\quad & \text{if }i=0,\\ 0\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

In what follows we will assume that the complex distribution ${\mathcal{P}}$ under consideration is integrable and satisfies (2). The latter is implied by the condition (1).

2.3 $\overline{\unicode[STIX]{x2202}}$ -operators

Suppose that ${\mathcal{E}}$ is a vector bundle on $X$ , i.e. a locally free ${\mathcal{O}}_{X}$ -module of finite rank. A connection along ${\mathcal{P}}$ on ${\mathcal{E}}$ is, by definition, a map $\unicode[STIX]{x1D6FB}^{{\mathcal{P}}}:{\mathcal{E}}\rightarrow \unicode[STIX]{x1D6FA}_{X}^{1}/{\mathcal{P}}^{\bot }\otimes _{{\mathcal{O}}_{X}}{\mathcal{E}}$ which satisfies the Leibniz rule $\unicode[STIX]{x1D6FB}^{{\mathcal{P}}}(fe)=f\unicode[STIX]{x1D6FB}^{{\mathcal{P}}}(e)+\overline{\unicode[STIX]{x2202}}f\cdot e$ for $e\in {\mathcal{E}}$ and $f\in {\mathcal{O}}_{X}$ . Equivalently, a connection along ${\mathcal{P}}$ is an ${\mathcal{O}}_{X}$ -linear map $\unicode[STIX]{x1D6FB}_{(\bullet )}^{{\mathcal{P}}}:{\mathcal{P}}\rightarrow \text{}\underline{\operatorname{End}}_{\mathbb{C}}({\mathcal{E}})$ which satisfies the Leibniz rule $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}^{{\mathcal{P}}}(fe)=f\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}^{{\mathcal{P}}}(e)+\overline{\unicode[STIX]{x2202}}f\cdot e$ for $e\in {\mathcal{E}}$ and $f\in {\mathcal{O}}_{X}$ . In particular, $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}^{{\mathcal{P}}}$ is ${\mathcal{O}}_{X/{\mathcal{P}}}$ -linear. The two avatars of a connection along ${\mathcal{P}}$ are related by $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}^{{\mathcal{P}}}(e)=\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D6FB}^{{\mathcal{P}}}(e)$ .

A connection along ${\mathcal{P}}$ on ${\mathcal{E}}$ is called flat if the corresponding map $\unicode[STIX]{x1D6FB}_{(\bullet )}^{{\mathcal{P}}}:{\mathcal{P}}\rightarrow \text{}\underline{\operatorname{End}}_{\mathbb{C}}({\mathcal{E}})$ is a morphism of Lie algebras. We will refer to a flat connection along ${\mathcal{P}}$ on ${\mathcal{E}}$ as a $\overline{\unicode[STIX]{x2202}}$ -operator on ${\mathcal{E}}$ .

A connection on ${\mathcal{E}}$ along ${\mathcal{P}}$ extends uniquely to a derivation $\overline{\unicode[STIX]{x2202}}_{{\mathcal{E}}}$ of the graded $\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ -module $\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet }\otimes _{{\mathcal{O}}_{X}}{\mathcal{E}}$ which is a $\overline{\unicode[STIX]{x2202}}$ -operator if and only if $\overline{\unicode[STIX]{x2202}}_{{\mathcal{E}}}^{2}=0$ . The complex $(\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet }\otimes _{{\mathcal{O}}_{X}}{\mathcal{E}},\overline{\unicode[STIX]{x2202}}_{{\mathcal{E}}})$ is referred to as the (corresponding) $\overline{\unicode[STIX]{x2202}}$ -complex. Since $\overline{\unicode[STIX]{x2202}}_{{\mathcal{E}}}$ is ${\mathcal{O}}_{X/{\mathcal{P}}}$ -linear, the sheaves $H^{i}(\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet }\otimes _{{\mathcal{O}}_{X}}{\mathcal{E}},\overline{\unicode[STIX]{x2202}}_{{\mathcal{E}}})$ are ${\mathcal{O}}_{X/{\mathcal{P}}}$ -modules. The vanishing of higher $\overline{\unicode[STIX]{x2202}}$ -cohomology of ${\mathcal{O}}_{X}$ (2) generalizes easily to vector bundles.

2.4 Calculus

The adjoint action of ${\mathcal{P}}$ on ${\mathcal{T}}_{X}$ preserves ${\mathcal{P}}$ , hence descends to an action on ${\mathcal{T}}_{X}/{\mathcal{P}}$ . The latter action defines a connection along ${\mathcal{P}}$ , i.e. a canonical $\overline{\unicode[STIX]{x2202}}$ -operator on ${\mathcal{T}}_{X}/{\mathcal{P}}$ which is easily seen to coincide with that induced via the duality pairingFootnote ³ between the latter and ${\mathcal{P}}^{\bot }$ . Let ${\mathcal{T}}_{X/{\mathcal{P}}}:=({\mathcal{T}}_{X}/{\mathcal{P}})^{{\mathcal{P}}}$ (the subsheaf of ${\mathcal{P}}$ invariant section, equivalently, the kernel of the $\overline{\unicode[STIX]{x2202}}$ -operator on ${\mathcal{T}}_{X}/{\mathcal{P}}$ ). The Lie bracket on ${\mathcal{T}}_{X}$ (respectively, the action of ${\mathcal{T}}_{X}$ on ${\mathcal{O}}_{X}$ ) induces a Lie bracket on  ${\mathcal{T}}_{X/{\mathcal{P}}}$ (respectively, an action of ${\mathcal{T}}_{X/{\mathcal{P}}}$ on ${\mathcal{O}}_{X/{\mathcal{P}}}$ ). The bracket and the action on ${\mathcal{O}}_{X/{\mathcal{P}}}$ endow ${\mathcal{T}}_{X/{\mathcal{P}}}$ with a structure of an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -Lie algebroid.

The action of ${\mathcal{P}}$ on $\unicode[STIX]{x1D6FA}_{X}^{1}$ by Lie derivative restricts to a flat connection along ${\mathcal{P}}$ , i.e. a canonical $\overline{\unicode[STIX]{x2202}}$ -operator on ${\mathcal{P}}^{\bot }$ and, therefore, on $\bigwedge ^{i}{\mathcal{P}}^{\bot }$ for all $i$ . It is easy to see that the multiplication map $\operatorname{Gr}_{0}^{F}\unicode[STIX]{x1D6FA}^{\bullet }\otimes \bigwedge ^{i}{\mathcal{P}}^{\bot }\rightarrow \operatorname{Gr}_{-i}^{F}\unicode[STIX]{x1D6FA}^{\bullet }[i]$ is an isomorphism which identifies the $\overline{\unicode[STIX]{x2202}}$ -complex of $\bigwedge ^{i}{\mathcal{P}}^{\bot }$ with $\operatorname{Gr}_{-i}^{F}\unicode[STIX]{x1D6FA}^{\bullet }[i]$ . Let $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{i}:=H^{i}(\operatorname{Gr}_{-i}^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet },\overline{\unicode[STIX]{x2202}})$ (so that ${\mathcal{O}}_{X/{\mathcal{P}}}:=\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{0}$ ). Then, $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{i}\subset \bigwedge ^{i}{\mathcal{P}}^{\bot }\subset \unicode[STIX]{x1D6FA}_{X}^{i}$ . The wedge product of differential forms induces a structure of a graded-commutative algebra on $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{\bullet }:=\bigoplus _{i}\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{i}[-i]=H^{\bullet }(\operatorname{Gr}^{F}\unicode[STIX]{x1D6FA}_{X}^{\bullet },\overline{\unicode[STIX]{x2202}})$ . The multiplication induces an isomorphism $\bigwedge _{{\mathcal{O}}_{X/{\mathcal{P}}}}^{i}\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{i}$ . The de Rham differential $d$ restricts to the map $d:\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{i}\rightarrow \unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{i+1}$ and the complex $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{\bullet }:=(\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{\bullet },d)$ is a commutative DGA.

The Hodge filtration $F_{\bullet }\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{\bullet }$ is defined by

$$\begin{eqnarray}F_{i}\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{\bullet }=\bigoplus _{j\geqslant -i}\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{j},\end{eqnarray}$$

so that the inclusion $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{\bullet }{\hookrightarrow}\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ is filtered with respect to the Hodge filtration. It follows from Lemma 2.1 that it is, in fact, a filtered quasi-isomorphism.

The duality pairing ${\mathcal{T}}_{X}/{\mathcal{P}}\otimes {\mathcal{P}}^{\bot }\rightarrow {\mathcal{O}}_{X}$ restricts to a non-degenerate pairing ${\mathcal{T}}_{X/{\mathcal{P}}}\otimes _{{\mathcal{O}}_{X/{\mathcal{P}}}}\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}$ . The action of ${\mathcal{T}}_{X}/{\mathcal{P}}$ on ${\mathcal{O}}_{X/{\mathcal{P}}}$ the pairing and the de Rham differential are related by the usual formula $\unicode[STIX]{x1D709}(f)=\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709}}df$ , for $\unicode[STIX]{x1D709}\in {\mathcal{T}}_{X/{\mathcal{P}}}$ and $f\in {\mathcal{O}}_{X/{\mathcal{P}}}$ .

3.1 Picard stacks

We recall the definitions from 1.4. Champs de Picard strictement commutatifs of [Reference DeligneDel73].

A (strictly commutative) Picard groupoid ${\mathcal{P}}$ is a non-empty groupoid equipped with a functor $+:{\mathcal{P}}\times {\mathcal{P}}\rightarrow {\mathcal{P}}$ and functorial isomorphisms:

1. (a) $\unicode[STIX]{x1D70E}_{x,y,z}:(x+y)+z\rightarrow x+(y+z)$ ;

2. (b) $\unicode[STIX]{x1D70F}_{x,y}:x+y\rightarrow y+x$ ;

rendering $+$ associative and strictly commutative, and such that for each object $x\in {\mathcal{P}}$ the functor $y\mapsto x+y$ is an equivalence.

A Picard stack on $X$ is a stack in groupoids ${\mathcal{P}}$ equipped with a functor $+:{\mathcal{P}}\times {\mathcal{P}}\rightarrow {\mathcal{P}}$ and functorial isomorphisms $\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D70F}$ as above, which, for each open subset $U\subseteq X$ , endow the category ${\mathcal{P}}(U)$ a structure of a Picard groupoid.

3.2 Torsors

Suppose that $A$ is a sheaf of abelian groups on $X$ . The stack of $A$ -torsors will be denoted by $A[1]$ ; it is a gerbe since all $A$ -torsors are locally trivial.

Suppose that $\unicode[STIX]{x1D719}:A\rightarrow B$ is a morphism of sheaves of abelian groups. The assignment $A[1]\ni T\mapsto \unicode[STIX]{x1D719}(T):=T\times _{A}B\in B[1]$ extends to a morphism $\unicode[STIX]{x1D719}:A[1]\rightarrow B[1]$ of stacks. There is a canonical map of sheaves of torsors $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{T}:T\rightarrow \unicode[STIX]{x1D719}(T)$ compatible with the map $\unicode[STIX]{x1D719}$ of abelian groups and respective actions.

Suppose that $A$ and $B$ are sheaves of abelian groups. The assignment $A[1]\times B[1]\ni (S,T)\mapsto S\times T\in (A\times B)[1]$ extends to a morphism of stacks $\times :A[1]\times B[1]\rightarrow (A\times B)[1]$ .

Suppose that $A$ is a sheaf of abelian groups with the group structure $+:A\times A\rightarrow A$ . The latter is a morphism of sheaves of groups since $A$ is abelian. The assignment $A[1]\times A[1]\ni (S,T)\mapsto S+T:=+(S\times T)$ defines a structure of a Picard stack on $A[1]$ . If $\unicode[STIX]{x1D719}:A\rightarrow B$ is a morphism of sheaves of abelian groups the corresponding morphism $\unicode[STIX]{x1D719}:A[1]\rightarrow B[1]$ is a morphism of Picard stacks.

As a consequence, the set $\unicode[STIX]{x1D70B}_{0}A[1](X)$ of isomorphism classes of $A$ -torsors is endowed with a canonical structure of an abelian group. There is a canonical isomorphism of groups $\unicode[STIX]{x1D70B}_{0}A[1](X)\cong H^{1}(X;A)$ .

3.3 Gerbes

An $A$ -gerbe is a stack in groupoids which is a twisted form of (i.e. locally equivalent to) $A[1]$ . The (2-)stack of $A$ -gerbes will be denoted $A[2]$ . Since all $A$ -gerbes are locally equivalent the (2-)stack $A[2]$ is a (2-)gerbe. Every $A$ -gerbe ${\mathcal{S}}$ (is equivalent to one which) admits a canonical action of the Picard stack $A[1]$ by autoequivalences denoted by $+:A[1]\times {\mathcal{S}}\rightarrow {\mathcal{S}}$ , $(T,L)\mapsto T+L$ endowing ${\mathcal{S}}$ with a structure of a (2-)torsor under $A[1]$ . We shall not make a distinction between $A$ -gerbes and (2-)torsors under $A[1]$ and use the notation $A[2]$ for both.

Suppose that $\unicode[STIX]{x1D719}:A\rightarrow B$ is a morphism of sheaves of abelian groups and ${\mathcal{S}}$ is an $A$ -gerbe. In particular, for any two (locally defined) objects $s_{1},s_{2}\in {\mathcal{S}}$ the sheaf $\text{}\underline{\operatorname{Hom}}_{{\mathcal{S}}}(s_{1},s_{2})$ is an $A$ -torsor. The stack $\unicode[STIX]{x1D719}{\mathcal{S}}$ is defined as the stack associated to the prestack with the same objects as ${\mathcal{S}}$ and $\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x1D719}{\mathcal{S}}}(s_{1},s_{2}):=\unicode[STIX]{x1D719}(\text{}\underline{\operatorname{Hom}}_{{\mathcal{S}}}(s_{1},s_{2}))$ . Then, $\unicode[STIX]{x1D719}{\mathcal{S}}$ is a $B$ -gerbe and the assignment ${\mathcal{S}}\mapsto \unicode[STIX]{x1D719}{\mathcal{S}}$ extends to a morphism $\unicode[STIX]{x1D719}:A[2]\rightarrow B[2]$ . There is a canonical morphism of stacks $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{{\mathcal{S}}}:{\mathcal{S}}\rightarrow \unicode[STIX]{x1D719}{\mathcal{S}}$ which induces the map $\unicode[STIX]{x1D719}:A\rightarrow B$ on groups of automorphisms.

The Picard structure on $A[1]$ gives rise to one on $A[2]$ defined in analogous fashion. As a consequence, the set $\unicode[STIX]{x1D70B}_{0}A[2](X)$ of equivalence classes of $A[1]$ -torsors is endowed with a canonical structure of an abelian group. There is a canonical isomorphism of groups $\unicode[STIX]{x1D70B}_{0}A[2](X)\cong H^{2}(X;A)$ .

Let $A^{0}\xrightarrow[]{d}A^{1}$ be a complex of sheaves of abelian groups on $X$ concentrated in degrees zero and one. Recall that a $(A^{0}\xrightarrow[]{d}A^{1})$ -torsor is a pair $(T,\unicode[STIX]{x1D70F})$ , where $T$ is a $A^{0}$ -torsor and $\unicode[STIX]{x1D70F}$ is a trivialization (i.e. a section) of the $A^{1}$ -torsor $d(T)=T\times _{A^{0}}A^{1}$ . A morphism of $(A^{0}\xrightarrow[]{d}A^{1})$ -torsors $\unicode[STIX]{x1D719}:(S,\unicode[STIX]{x1D70E})\rightarrow (T,\unicode[STIX]{x1D70F})$ is a morphism of $A^{0}$ -torsors $\unicode[STIX]{x1D719}:S\rightarrow T$ such that the induced morphism of $A^{1}$ -torsors $d(\unicode[STIX]{x1D719}):d(S)\rightarrow d(T)$ which commutes with respective trivializations, i.e.  $d(\unicode[STIX]{x1D719})(\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D70F}$ .

The monoidal structure on the category of $(A^{0}\xrightarrow[]{d}A^{1})$ -torsors is defined as follows. Suppose that $(S,\unicode[STIX]{x1D70E})$ and $(T,\unicode[STIX]{x1D70F})$ are $(A^{0}\xrightarrow[]{d}A^{1})$ -torsors. The sum $(S,\unicode[STIX]{x1D70E})+(T,\unicode[STIX]{x1D70F})$ is represented by $(S+T,\unicode[STIX]{x1D70E}+\unicode[STIX]{x1D70F})$ , where $\unicode[STIX]{x1D70E}+\unicode[STIX]{x1D70F}$ is the trivialization of $d(S)+d(T)=d(S+T)$ induced by $\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D70F}$ .

Locally defined $(A^{0}\xrightarrow[]{d}A^{1})$ -torsors form a Picard stack on $X$ which we will denote by $(A^{0}\xrightarrow[]{d}A^{1})[1]$ . By a result of Deligne [Reference DeligneDel73, Proposition 1.4.15] all Picard stacks arise in this way. The group (under the operation induced by the monoidal structure) of isomorphism classes of $(A^{0}\xrightarrow[]{d}A^{1})$ -torsors on $X$ , i.e.  $\unicode[STIX]{x1D70B}_{0}(A^{0}\xrightarrow[]{d}A^{1})[1](X)$ is canonically isomorphic to $H^{1}(X;A^{0}\xrightarrow[]{d}A^{1})$ .

3.4 2-torsors

A $(A^{0}\xrightarrow[]{d}A^{1})$ -gerbe is equivalent to the data $({\mathcal{S}},\unicode[STIX]{x1D70E})$ , where ${\mathcal{S}}$ is an $A^{0}$ -gerbe and $\unicode[STIX]{x1D70F}$ is a trivialization of the $A^{1}$ -gerbe $d{\mathcal{S}}$ , i.e. an equivalence $\unicode[STIX]{x1D70F}:d{\mathcal{S}}\rightarrow A^{1}[1]$ . The composition ${\mathcal{S}}\xrightarrow[]{d}d{\mathcal{S}}\xrightarrow[]{\unicode[STIX]{x1D70F}}A^{1}[1]$ is a functorial assignment of an $A^{1}$ -torsor $\unicode[STIX]{x1D70F}(s)$ to a (locally defined) object $s\in {\mathcal{S}}$ . The 2-stack of $(A^{0}\xrightarrow[]{d}A^{1})$ -gerbes will be denoted $(A^{0}\xrightarrow[]{d}A^{1})[2]$ . The Picard structure on $(A^{0}\xrightarrow[]{d}A^{1})[1]$ gives rise to one on $(A^{0}\xrightarrow[]{d}A^{1})[2]$ defined in an analogous fashion. As a consequence, the set $\unicode[STIX]{x1D70B}_{0}(A^{0}\xrightarrow[]{d}A^{1})[2](X)$ of equivalence classes of $(A^{0}\xrightarrow[]{d}A^{1})$ -gerbes is endowed with a canonical structure of an abelian group. There is a canonical isomorphism of groups $\unicode[STIX]{x1D70B}_{0}(A^{0}\xrightarrow[]{d}A^{1})[2](X)\cong H^{2}(X;A^{0}\xrightarrow[]{d}A^{1})$ .

3.5 Algebroids

Recall that a $k$ -algebroid, $k$ a commutative ring with unit, is a stack in $k$ -linear categories ${\mathcal{C}}$ such that the substack of isomorphisms $i{\mathcal{C}}$ (which is a stack in groupoids) is a gerbe.

Suppose that ${\mathcal{A}}$ is a sheaf of $k$ -algebras on $X$ . For $U$ open in $X$ let ${\mathcal{A}}^{+}(U)$ denote the category with one object whose endomorphism algebra is ${\mathcal{A}}^{\mathtt{op}}$ . The assignment $U\mapsto {\mathcal{A}}^{+}(U)$ defines a prestack on $X$ , which we denote ${\mathcal{A}}^{+}$ . Let $\widetilde{{\mathcal{A}}^{+}}$ denote the associated stack. Note that $\widetilde{{\mathcal{A}}^{+}}$ is equivalent to the stack of ${\mathcal{A}}$ -modules locally isomorphic to ${\mathcal{A}}$ . The category $\widetilde{{\mathcal{A}}^{+}}(X)$ is non-empty.

Conversely, let ${\mathcal{C}}$ be a $k$ -algebroid such that the category ${\mathcal{C}}(X)$ is non-empty. Let $L\in {\mathcal{C}}(X)$ , and let ${\mathcal{A}}:=\text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L)^{\mathtt{op}}$ . The morphism ${\mathcal{A}}^{+}\rightarrow {\mathcal{C}}$ which sends the unique object to $L$ induces and equivalence $\widetilde{{\mathcal{A}}^{+}}\rightarrow {\mathcal{C}}$ .

For a sheaf of $k$ -algebras ${\mathcal{A}}$ on $X$ a twisted form of ${\mathcal{A}}$ is a $k$ -algebroid locally $k$ -linearly equivalent to $\widetilde{{\mathcal{A}}^{+}}$ .

Let $X$ be a $C^{\infty }$ manifold equipped with an integrable complex distribution ${\mathcal{P}}$ . Twisted forms of the sheaf of $\mathbb{C}$ -algebras ${\mathcal{O}}_{X/{\mathcal{P}}}$ are in one-to-one correspondence with ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes via ${\mathcal{S}}\mapsto i{\mathcal{S}}$ . Equivalence classes of twisted forms of ${\mathcal{O}}_{X/{\mathcal{P}}}$ form an group canonically isomorphic to $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times })$ .

4.1 ${\mathcal{O}}_{X/{\mathcal{P}}}$ -Lie algebroids

An ${\mathcal{O}}_{X/{\mathcal{P}}}$ -Lie algebroid or, simply, a Lie algebroid is the datum of:

1. (a) an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -module ${\mathcal{B}}$ ;

2. (b) an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -linear map $\unicode[STIX]{x1D70E}:{\mathcal{B}}\rightarrow {\mathcal{T}}_{X/{\mathcal{P}}}$ (the anchor);

3. (c) a structure of a $\mathbb{C}$ -Lie algebra on ${\mathcal{B}}$ given by $[~,~]:{\mathcal{B}}\otimes _{\mathbb{C}}{\mathcal{B}}\rightarrow {\mathcal{B}}$ ;

such that the Leibniz rule

$$\begin{eqnarray}[b_{1},fb_{2}]=\unicode[STIX]{x1D70E}(b_{1})(f)b_{2}+f[b_{1},b_{2}],\end{eqnarray}$$

$b_{1},b_{2}\in {\mathcal{B}}$ , $f\in {\mathcal{O}}_{X/{\mathcal{P}}}$ , holds. As a consequence, the anchor map is a morphism of Lie algebras.

A morphism of Lie algebroids is an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -linear map $\unicode[STIX]{x1D719}:{\mathcal{B}}_{1}\rightarrow {\mathcal{B}}_{2}$ of Lie algebras which is commutes with respective anchors.

The sheaf ${\mathcal{T}}_{X/{\mathcal{P}}}$ is a Lie algebroid under the Lie bracket of vector fields with the anchor the identity map. It is immediate from the definitions that ${\mathcal{T}}_{X/{\mathcal{P}}}$ is the terminal object in the category of Lie algebroids.

Further examples of Lie algebroids are given below.

4.1.1 Atiyah algebras

For a locally free ${\mathcal{O}}_{X/{\mathcal{P}}}$ -module of finite rank ${\mathcal{E}}$ the Atiyah algebra of ${\mathcal{E}}$ , denoted by ${\mathcal{A}}_{{\mathcal{E}}}$ is defined by the pull-back diagram

where the bottom horizontal arrow is the order-one principal symbol map. It is a Lie algebroid under (the restriction of) the commutator bracket, the anchor given by the top horizontal map (the restriction of the principal symbol map). There is an exact sequence

$$\begin{eqnarray}0\rightarrow \text{}\underline{\operatorname{End}}_{{\mathcal{O}}_{X/{\mathcal{P}}}}({\mathcal{E}})\rightarrow {\mathcal{A}}_{{\mathcal{E}}}\xrightarrow[]{\unicode[STIX]{x1D70E}}{\mathcal{T}}_{X/{\mathcal{P}}}\rightarrow 0.\end{eqnarray}$$

4.1.2 Poisson structures

Let $\unicode[STIX]{x1D70B}\in \unicode[STIX]{x1D6E4}(X;\bigwedge ^{2}{\mathcal{T}}_{X/{\mathcal{P}}})$ be a bi-vector; we denote by $\widetilde{\unicode[STIX]{x1D70B}}:\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}\rightarrow {\mathcal{T}}_{X/{\mathcal{P}}}$ the adjoint map $\unicode[STIX]{x1D6FC}\mapsto \unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6FC},\cdot )$ . The bi-vector $\unicode[STIX]{x1D70B}$ is Poisson if the operation $\{~,~\}:{\mathcal{O}}_{X/{\mathcal{P}}}\otimes _{\mathbb{C}}{\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}$ defined by $\{f,g\}=\unicode[STIX]{x1D70B}(df,dg)$ satisfies the Jacobi identity, i.e. endows ${\mathcal{O}}_{X/{\mathcal{P}}}$ with a structure of a $\mathbb{C}$ -Lie algebra.

Let $[~,~]_{\unicode[STIX]{x1D70B}}:\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}\otimes _{\mathbb{C}}\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}$ denote the map given by

(3) $$\begin{eqnarray}[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}]_{\unicode[STIX]{x1D70B}}=L_{\widetilde{\unicode[STIX]{x1D70B}}(\unicode[STIX]{x1D6FC})}\unicode[STIX]{x1D6FD}-L_{\widetilde{\unicode[STIX]{x1D70B}}(\unicode[STIX]{x1D6FD})}\unicode[STIX]{x1D6FC}-d\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}).\end{eqnarray}$$

The bi-vector $\unicode[STIX]{x1D70B}$ is Poisson if and only if the bracket (3) on $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}$ satisfies the Jacobi identity; the map $\widetilde{\unicode[STIX]{x1D70B}}$ is compatible with respective brackets.

Thus, a Poisson bi-vector $\unicode[STIX]{x1D70B}$ gives rise to a structure, denoted by $\unicode[STIX]{x1D6F1}$ in what follows, of a Lie algebroid on $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}$ with bracket (3) and the anchor $\widetilde{\unicode[STIX]{x1D70B}}$ .

4.1.3 The de Rham complex

We shall assume for simplicity that the Lie algebroid ${\mathcal{B}}$ in question is locally free of finite rank over ${\mathcal{O}}_{X/{\mathcal{P}}}$ . Let $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{0}={\mathcal{O}}_{X/{\mathcal{P}}}$ , $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}={\mathcal{B}}^{\vee }$ , where, for an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -module ${\mathcal{E}}$ , we put ${\mathcal{E}}^{\vee }:=\text{}\underline{\operatorname{Hom}}_{{\mathcal{O}}_{X/{\mathcal{P}}}}({\mathcal{E}},{\mathcal{O}}_{X/{\mathcal{P}}})$ .

Let $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{i}=\bigwedge ^{i}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ . Then, $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{\bullet }:=\bigoplus _{i}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{i}[-i]$ is a graded commutative algebra. The composition

$$\begin{eqnarray}{\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{d}\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}\xrightarrow[]{\unicode[STIX]{x1D70E}^{\vee }}B^{\vee }=\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\end{eqnarray}$$

extends uniquely to a square-zero derivation $d_{{\mathcal{B}}}$ of degree one of the algebra $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{\bullet }$ . Let $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{i,\text{cl}}:=\ker (\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{i}\xrightarrow[]{d_{{\mathcal{B}}}}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{i+1})$ . In the case ${\mathcal{B}}=\unicode[STIX]{x1D6F1}$ of the example in § 4.1.2  $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{i}=\bigwedge ^{i}{\mathcal{T}}_{X/{\mathcal{P}}}$ and the de Rham differential is given by $d_{\unicode[STIX]{x1D6F1}}=[\unicode[STIX]{x1D70B},\cdot ]$ , i.e the Schouten bracket with the bi-vector $\unicode[STIX]{x1D70B}$ .

4.2 Modules over Lie algebroids

Suppose that ${\mathcal{B}}$ is a Lie algebroid.

4.2.1 ${\mathcal{B}}$ -modules

Suppose that ${\mathcal{M}}$ is an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -module. A ${\mathcal{B}}$ -module structure on ${\mathcal{M}}$ is given by an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -linear map ${\mathcal{B}}\rightarrow \text{}\underline{\operatorname{End}}_{\mathbb{C}}({\mathcal{M}})$ , $b\mapsto (m\mapsto b.m)$ , $b\in {\mathcal{B}}$ , $m\in {\mathcal{M}}$ , which is a map of Lie algebras (with respect to the commutator bracket of endomorphisms) and satisfies the Leibniz rule $b.fm=\unicode[STIX]{x1D70E}(b)(f)m+f\cdot b.m$ .

Suppose that ${\mathcal{E}}$ is a locally free ${\mathcal{O}}_{X/{\mathcal{P}}}$ -module of finite rank. It transpires from the above definition that a ${\mathcal{B}}$ -module structure on ${\mathcal{E}}$ amounts to a morphism of algebroids ${\mathcal{B}}\rightarrow {\mathcal{A}}_{{\mathcal{E}}}$ . Equivalently, it is a morphism of Lie algebroids ${\mathcal{B}}\rightarrow {\mathcal{B}}\times _{{\mathcal{T}}_{X/{\mathcal{P}}}}{\mathcal{A}}_{{\mathcal{E}}}$ splitting the exact sequence

$$\begin{eqnarray}0\rightarrow \text{}\underline{\operatorname{End}}_{{\mathcal{O}}_{X/{\mathcal{P}}}}({\mathcal{E}})\rightarrow {\mathcal{B}}\times _{{\mathcal{T}}_{X/{\mathcal{P}}}}{\mathcal{A}}_{{\mathcal{E}}}\xrightarrow[]{\operatorname{pr}_{{\mathcal{B}}}}{\mathcal{B}}\rightarrow 0.\end{eqnarray}$$

A morphism of ${\mathcal{B}}$ -modules is an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -linear map which commutes with respective actions of ${\mathcal{B}}$ . The ${\mathcal{B}}$ -modules form an Abelian category.

The structure sheaf ${\mathcal{O}}_{X/{\mathcal{P}}}$ is a ${\mathcal{T}}_{X/{\mathcal{P}}}$ -module hence a ${\mathcal{B}}$ -module in a canonical way for any Lie algebroid ${\mathcal{B}}$ .

4.2.3 Invertible ${\mathcal{B}}$ -modules

For the purposes of this note an invertible ${\mathcal{B}}$ -module is a line bundle (i.e. a locally free ${\mathcal{O}}_{X/{\mathcal{P}}}$ -module of rank one) equipped with a structure of a ${\mathcal{B}}$ -module. Invertible ${\mathcal{B}}$ -modules form a Picard category, denoted $\operatorname{Pic}_{{\mathcal{B}}}(X)$ , under the tensor product over ${\mathcal{O}}_{X/{\mathcal{P}}}$ . Let $\text{}\underline{\operatorname{Pic}}_{{\mathcal{B}}}$ denote the Picard stack of invertible ${\mathcal{B}}$ -modules, i.e.  $\text{}\underline{\operatorname{Pic}}_{{\mathcal{B}}}(U)=\operatorname{Pic}_{{\mathcal{B}}}(U)$ for $U$ open in $X$ .

Suppose that ${\mathcal{L}}$ is a line bundle; we shall denote by ${\mathcal{L}}^{\times }$ the corresponding ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -torsor (the subsheaf of nowhere vanishing sections). Recall that a ${\mathcal{B}}$ -module structure on ${\mathcal{L}}$ is a morphism of Lie algebroids $\unicode[STIX]{x1D6FB}:{\mathcal{B}}\rightarrow {\mathcal{B}}\times _{{\mathcal{T}}_{X/{\mathcal{P}}}}{\mathcal{A}}_{{\mathcal{L}}}$ splitting the short exact sequence

(4) $$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{i}{\mathcal{B}}\times _{{\mathcal{T}}_{X/{\mathcal{P}}}}{\mathcal{A}}_{{\mathcal{L}}}\xrightarrow[]{\operatorname{pr}_{{\mathcal{B}}}}{\mathcal{B}}\rightarrow 0.\end{eqnarray}$$

For any two such $\unicode[STIX]{x1D6FB}_{1},\unicode[STIX]{x1D6FB}_{2}:{\mathcal{B}}\rightarrow {\mathcal{B}}\times _{{\mathcal{T}}_{X/{\mathcal{P}}}}{\mathcal{A}}_{{\mathcal{L}}}$ , their difference $\unicode[STIX]{x1D6FB}_{2}-\unicode[STIX]{x1D6FB}_{1}$ satisfies $\operatorname{pr}_{{\mathcal{B}}}\circ (\unicode[STIX]{x1D6FB}_{2}-\unicode[STIX]{x1D6FB}_{1})=0$ and therefore factors through (the inclusion of) ${\mathcal{O}}_{X/{\mathcal{P}}}$ . Moreover, the resulting section $\unicode[STIX]{x1D6FB}_{2}-\unicode[STIX]{x1D6FB}_{1}\in \operatorname{Hom}_{{\mathcal{O}}_{X/{\mathcal{P}}}}({\mathcal{B}},{\mathcal{O}}_{X/{\mathcal{P}}})=\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ satisfies $d_{{\mathcal{B}}}(\unicode[STIX]{x1D6FB}_{2}-\unicode[STIX]{x1D6FB}_{1})=0$ , where $d_{{\mathcal{B}}}$ is the de Rham differential introduced in § 4.1.3.

The sheaf of (locally defined) splittings of (4) by Lie algebroid morphisms is a $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}$ -torsor and will be denoted by $\operatorname{Conn}_{{\mathcal{B}}}^{\flat }({\mathcal{L}})$ . A structure of a ${\mathcal{B}}$ -module on ${\mathcal{L}}$ is a trivialization of $\operatorname{Conn}_{{\mathcal{B}}}^{\flat }({\mathcal{L}})$ .

The morphism of sheaves of groups

$$\begin{eqnarray}d_{{\mathcal{B}}}\log :{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}\end{eqnarray}$$

is defined by $f\mapsto d_{{\mathcal{B}}}f/f$ . There is a canonical morphism of sheaves

(5) $$\begin{eqnarray}d_{{\mathcal{B}}}\log :{\mathcal{L}}^{\times }\rightarrow \operatorname{Conn}_{{\mathcal{B}}}^{\flat }({\mathcal{L}})\end{eqnarray}$$

defined as follows. Let $s\in {\mathcal{L}}^{\times }$ be a (locally defined) section. The section $s$ gives rise to a (locally defined) isomorphism ${\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{\cong }{\mathcal{L}}$ . Via this isomorphism the canonical ${\mathcal{B}}$ -module structure on ${\mathcal{O}}_{X/{\mathcal{P}}}$ gives rise to a ${\mathcal{B}}$ -module structure on ${\mathcal{L}}$ which corresponds to a (locally defined) section denoted by $d_{{\mathcal{B}}}\log (s)\in \operatorname{Conn}_{{\mathcal{B}}}^{\flat }({\mathcal{L}})$ . We leave it to the reader to check that the morphism (5) is compatible with the map of sheaves of groups $d_{{\mathcal{B}}}\log$ and respective actions. Hence, (5) induces a canonical isomorphism of $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}$ -torsors $\operatorname{Conn}_{{\mathcal{B}}}^{\flat }({\mathcal{L}})\cong d_{{\mathcal{B}}}\log ({\mathcal{L}}^{\times }):={\mathcal{L}}^{\times }\times _{{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}$ . In particular, a structure of a ${\mathcal{B}}$ -module on ${\mathcal{L}}$ amounts to a trivialization of $d_{{\mathcal{B}}}\log ({\mathcal{L}}^{\times })$ .

4.2.4 Classification of invertible ${\mathcal{B}}$ -modules

Assigning to an invertible ${\mathcal{B}}$ -module ${\mathcal{L}}$ the pair $({\mathcal{L}}^{\times },\unicode[STIX]{x1D6FB})$ , where $\unicode[STIX]{x1D6FB}$ is the trivialization of $d_{{\mathcal{B}}}\log ({\mathcal{L}}^{\times })$ corresponding to the ${\mathcal{B}}$ -module structure we obtain a morphism of Picard stacks

(6) $$\begin{eqnarray}\text{}\underline{\operatorname{Pic}}_{{\mathcal{B}}}\rightarrow ({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}})[1].\end{eqnarray}$$

Lemma 4.1. The morphism (6) is an equivalence.

Proof. A quasi-inverse to (6) is given by the following construction. An ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -torsor $T$ determines the line bundle ${\mathcal{L}}:=T\times _{{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }}{\mathcal{O}}_{X/{\mathcal{P}}}$ such that there is a canonical isomorphism ${\mathcal{L}}^{\times }\cong T$ , hence $d_{{\mathcal{B}}}\log (T)\cong d_{{\mathcal{B}}}\log ({\mathcal{L}}^{\times })\cong \operatorname{Conn}_{{\mathcal{B}}}^{\flat }({\mathcal{L}})$ . Thus, a trivialization of $d_{{\mathcal{B}}}\log (T)$ gives rise to a ${\mathcal{B}}$ -module structure on ${\mathcal{L}}$ .◻

Corollary 4.2. We have $\unicode[STIX]{x1D70B}_{0}\operatorname{Pic}_{{\mathcal{B}}}(X)\cong H^{1}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}})$ .

The above result reduces to the well-known classification of line bundles equipped with flat connections in the case ${\mathcal{B}}={\mathcal{T}}_{X/{\mathcal{P}}}$ (see, for example, [Reference BrylinskiBry93]). In the case ${\mathcal{B}}=\unicode[STIX]{x1D6F1}$ (see § 4.1.2), Lemma 4.1 is a particular case of [Reference BursztynBur01, Proposition 2.8].

4.3 ${\mathcal{O}}$ -extensions

Suppose that ${\mathcal{B}}$ is an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -Lie algebroid. An ${\mathcal{O}}$ -extension of ${\mathcal{B}}$ is a triple $(\widetilde{{\mathcal{B}}},\mathfrak{c},\unicode[STIX]{x1D70E})$ which consists of:

1. (a) a Lie algebroid $\widetilde{{\mathcal{B}}}$ ;

2. (b) a central element $\mathfrak{c}$ of the Lie algebra of sections $\unicode[STIX]{x1D6E4}(X;\widetilde{{\mathcal{B}}})$ ;

3. (c) a morphism of Lie algebroids $\unicode[STIX]{x1D70E}:\widetilde{{\mathcal{B}}}\rightarrow {\mathcal{B}}$ ,

such that these data give rise to the associated short exact sequence

(7) $$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}\displaystyle \xrightarrow[]{f\mapsto f\cdot \mathfrak{c}}\widetilde{{\mathcal{B}}}\xrightarrow[]{\unicode[STIX]{x1D70E}}{\mathcal{B}}\rightarrow 0.\end{eqnarray}$$

Locally defined ${\mathcal{O}}$ -extensions of ${\mathcal{B}}$ form a Picard stack under the operation of Baer sum of extensions which we denote by ${\mathcal{O}}{\mathcal{E}}\!{\mathcal{X}}{\mathcal{T}}({\mathcal{B}})$ .

Since $\mathfrak{c}$ is central, it follows from the Leibniz rule that, for $f\in {\mathcal{O}}_{X/{\mathcal{P}}}$ , $b\in \widetilde{{\mathcal{B}}}$ , $[b,f\cdot \mathfrak{c}]=\overline{b}(f)\cdot \mathfrak{c}$ , where $\overline{b}\in {\mathcal{T}}_{X/{\mathcal{P}}}$ denotes the image of $b$ under the anchor map. That is to say, the (adjoint) action of $\widetilde{{\mathcal{B}}}$ on ${\mathcal{O}}_{X/{\mathcal{P}}}\cdot \mathfrak{c}\cong {\mathcal{O}}_{X/{\mathcal{P}}}$ factors through ${\mathcal{T}}_{X/{\mathcal{P}}}$ and the latter action coincides with the Lie derivative action of vector fields on functions.

Suppose that ${\mathcal{B}}$ is locally free of finite rank over ${\mathcal{O}}_{X/{\mathcal{P}}}$ . Then there is a canonical equivalence of Picard stacks

(8) $$\begin{eqnarray}{\mathcal{O}}{\mathcal{E}}\!{\mathcal{X}}{\mathcal{T}}({\mathcal{B}})\displaystyle \xrightarrow[]{\cong }(\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}})[1].\end{eqnarray}$$

The functor (8) associates to $(\widetilde{{\mathcal{B}}},\mathfrak{c},\unicode[STIX]{x1D70E})$ the $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ -torsor $\operatorname{Conn}_{{\mathcal{B}}}(\widetilde{{\mathcal{B}}})$ of (locally defined) splittings of $\unicode[STIX]{x1D70E}:\widetilde{{\mathcal{B}}}\rightarrow {\mathcal{B}}$ . The ‘curvature’ map $c:\operatorname{Conn}_{{\mathcal{B}}}(\widetilde{{\mathcal{B}}})\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}}$ , $\operatorname{Conn}_{{\mathcal{B}}}(\widetilde{{\mathcal{B}}})\ni \unicode[STIX]{x1D6FB}\mapsto c(\unicode[STIX]{x1D6FB})\in \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}}\subset \operatorname{Hom}(\bigwedge ^{2}{\mathcal{B}},{\mathcal{O}}_{X/{\mathcal{P}}})$ is determined by $c(\unicode[STIX]{x1D6FB})(b_{1},b_{2})\cdot \mathfrak{c}=[\unicode[STIX]{x1D6FB}(b_{1}),\unicode[STIX]{x1D6FB}(b_{2})]-\unicode[STIX]{x1D6FB}([b_{1},b_{2}])$ . We leave it to the reader to check that the standard calculation shows that the above formula defines a closed 2-form.

5.1 ${\mathcal{B}}$ -connective structures

Let ${\mathcal{S}}$ be an ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbe. A ${\mathcal{B}}$ -connective structure on ${\mathcal{S}}$ is a trivialization of the $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ -gerbe $d_{{\mathcal{B}}}\log {\mathcal{S}}$ , i.e. a morphism, thus an equivalence, of $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ -gerbes $\unicode[STIX]{x1D6FB}:d_{{\mathcal{B}}}\log {\mathcal{S}}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}[1]$ . The composition ${\mathcal{S}}\rightarrow d_{{\mathcal{B}}}\log {\mathcal{S}}\xrightarrow[]{\unicode[STIX]{x1D6FB}}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}[1]$ is a functorial assignment of a $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ -torsor $\unicode[STIX]{x1D6FB}(L)$ to a (locally defined) object $L\in {\mathcal{S}}$ which induces the map $d_{{\mathcal{B}}}\log :{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ on respective sheaves of groups of automorphisms. For locally defined objects $L_{1},L_{2}\in {\mathcal{S}}$ the connective structure induces the isomorphism $\unicode[STIX]{x1D6FB}:\operatorname{Conn}_{{\mathcal{B}}}(\text{}\underline{\operatorname{Hom}}_{{\mathcal{S}}}(L_{1},L_{2}))=\text{}\underline{\operatorname{Hom}}_{d_{{\mathcal{B}}}\log {\mathcal{S}}}(d_{{\mathcal{B}}}\log (L_{1}),d_{{\mathcal{B}}}\log (L_{2}))\rightarrow \text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}[1]}(\unicode[STIX]{x1D6FB}(L_{1}),\unicode[STIX]{x1D6FB}(L_{2}))\cong \unicode[STIX]{x1D6FB}(L_{2})-\unicode[STIX]{x1D6FB}(L_{1})$ .

It transpires from the above description that locally defined ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with ${\mathcal{B}}$ -connective structure, i.e. pairs $({\mathcal{S}},\unicode[STIX]{x1D6FB})$ as above are objects of the Picard 2-stack $({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1})[2]$ . As a consequence, the group of equivalence classes of ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with ${\mathcal{B}}$ -connective structure is canonically isomorphic to $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1})$ .

Since the map $d_{{\mathcal{B}}}\log :{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ factors through the inclusion $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}{\hookrightarrow}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ the $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}$ -gerbe $d_{{\mathcal{B}}}\log {\mathcal{S}}$ is induced from the $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}$ -gerbe which will be denoted $d_{{\mathcal{B}}}\log {\mathcal{S}}^{\flat }$ . A connective structure $\unicode[STIX]{x1D6FB}:d_{{\mathcal{B}}}\log {\mathcal{S}}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}[1]$ is called flat if it is induced from the trivialization of $d_{{\mathcal{B}}}\log {\mathcal{S}}^{\flat }$ , i.e. from an equivalence $\unicode[STIX]{x1D6FB}:d_{{\mathcal{B}}}\log {\mathcal{S}}^{\flat }\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}[1]$ .

It transpires from the above description that locally defined ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with flat ${\mathcal{B}}$ -connective structure are objects of $({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}})[2]$ . As a consequence, the group of equivalence classes of ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with flat ${\mathcal{B}}$ -connective structure is canonically isomorphic to $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}})$ .

5.2 Curving

A curving $\unicode[STIX]{x1D705}$ on the a ${\mathcal{B}}$ -connective structure $\unicode[STIX]{x1D6FB}$ on an ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbe ${\mathcal{S}}$ is a lift of the functor ${\mathcal{S}}\xrightarrow[]{\unicode[STIX]{x1D6FB}}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}[1]$ to $(\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2})[1]$ , i.e. a factorization of $\unicode[STIX]{x1D6FB}$ as

$$\begin{eqnarray}{\mathcal{S}}\xrightarrow[]{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2})[1]\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}[1].\end{eqnarray}$$

We refer to the functor ${\mathcal{S}}\xrightarrow[]{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2})[1]$ as a connective structure with curving (given by $\unicode[STIX]{x1D705}$ ).

Locally defined ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with ${\mathcal{B}}$ -connective structure with curving, i.e. pairs $({\mathcal{S}},\unicode[STIX]{x1D705})$ as above are objects of the Picard 2-stack which we denote by $({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\xrightarrow[]{d_{{\mathcal{B}}}}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2})[2]$ . The group of equivalence classes of ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with connective structure with curving is canonically isomorphic to $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\xrightarrow[]{d_{{\mathcal{B}}}}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2})$ .

The curving ${\mathcal{S}}\rightarrow (\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2})[1]$ is flat if it factors through $(\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}})[1]$ . In this case we refer to the functor ${\mathcal{S}}\rightarrow (\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}})[1]$ as a ${\mathcal{B}}$ -connective structure with a flat curving. Locally defined ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with ${\mathcal{B}}$ -connective structure with flat curving are objects of the Picard 2-stack which we denote by $({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\xrightarrow[]{d_{{\mathcal{B}}}}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}})[2]$ . The group of equivalence classes of ${\mathcal{O}}_{X/{\mathcal{P}}}^{\times }$ -gerbes with connective structure with flat curving is canonically isomorphic to $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\xrightarrow[]{d_{{\mathcal{B}}}\log }\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\xrightarrow[]{d_{{\mathcal{B}}}}\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}})$ .

The morphism of complexes $\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1,\text{cl}}\rightarrow (\unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{B}}}^{2,\text{cl}})$ induces a canonical flat curving on a flat ${\mathcal{B}}$ -connective structure.

6.1 Star-products

A star-product on ${\mathcal{O}}_{X/{\mathcal{P}}}$ is a map

$$\begin{eqnarray}{\mathcal{O}}_{X/{\mathcal{P}}}\otimes _{\mathbb{C}}{\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}[[t]]\end{eqnarray}$$

of the form

(9) $$\begin{eqnarray}f\otimes g\mapsto f\star g=fg+\mathop{\sum }_{i=1}^{\infty }P_{i}(f,g)t^{i},\end{eqnarray}$$

where $P_{i}$ are bi-differential operators. Such a map admits a unique $\mathbb{C}[[t]]$ -bilinear extension

$$\begin{eqnarray}{\mathcal{O}}_{X/{\mathcal{P}}}[[t]]\otimes _{\mathbb{C}[[t]]}{\mathcal{O}}_{X/{\mathcal{P}}}[[t]]\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}[[t]]\end{eqnarray}$$

and the latter is required to define a structure of an associative unital $\mathbb{C}[[t]]$ -algebra on ${\mathcal{O}}_{X/{\mathcal{P}}}[[t]]$ .

For a star-product given by (9) the operation

(10) $$\begin{eqnarray}f\otimes g\mapsto P_{1}(f,g)-P_{1}(g,f)\end{eqnarray}$$

is a Poisson bracket on ${\mathcal{O}}_{X/{\mathcal{P}}}$ which we refer to as the associated Poisson bracket. It follows from Lemma 6.6 below that isomorphic star-products give rise to the same associated Poisson bracket.

The following lemma is well-known.

6.2 DQ-algebras

A DQ-algebra is a sheaf of $\mathbb{C}[[t]]$ -algebras locally isomorphic to a star-product. For a DQ-algebra $\mathbb{A}$ there is a canonical isomorphism $\mathbb{A}/t\cdot \mathbb{A}\cong {\mathcal{O}}_{X/{\mathcal{P}}}$ . Therefore, there is a canonical map (reduction modulo $t$ ) $\mathbb{A}\xrightarrow[]{\unicode[STIX]{x1D70E}}{\mathcal{O}}_{X/{\mathcal{P}}}$ of $\mathbb{C}[[t]]$ -algebras.

A morphism of DQ-algebras is a morphism of sheaves of $\mathbb{C}[[t]]$ -algebras.

For an open subset $U\subseteq X$ let $\text{DQ}\text{-}\text{alg}_{X/{\mathcal{P}}}(U)$ denote the category of DQ-algebras on $U$ , where $U$ is equipped with the restriction of ${\mathcal{P}}$ . The assignment $U\mapsto \text{DQ}\text{-}\text{alg}_{X/{\mathcal{P}}}(U)$ defines a stack on $X$ denoted $\text{DQ}\text{-}\text{alg}_{X/{\mathcal{P}}}$ .

In what follows, we shall denote $\text{}\underline{\operatorname{Hom}}_{\text{DQ}\text{-}\text{alg}_{X/{\mathcal{P}}}}$ simply by $\text{}\underline{\operatorname{Hom}}_{\text{DQ}\text{-}\text{alg}}$ .

6.3 The associated Poisson structure

Suppose that $\mathbb{A}$ is a DQ-algebra. The composition

$$\begin{eqnarray}\mathbb{A}\otimes \mathbb{A}\xrightarrow[]{[.,.]}\mathbb{A}\xrightarrow[]{\unicode[STIX]{x1D70E}}{\mathcal{O}}_{X/{\mathcal{P}}}\end{eqnarray}$$

is trivial. Therefore, the commutator $\mathbb{A}\otimes \mathbb{A}\xrightarrow[]{[.,.]}\mathbb{A}$ takes values in $t\mathbb{A}$ . The composition

$$\begin{eqnarray}\mathbb{A}\otimes \mathbb{A}\xrightarrow[]{[.,.]}t\mathbb{A}\xrightarrow[]{t^{-1}}\mathbb{A}\xrightarrow[]{\unicode[STIX]{x1D70E}}{\mathcal{O}}_{X/{\mathcal{P}}}\end{eqnarray}$$

factors uniquely as

$$\begin{eqnarray}\mathbb{A}\otimes \mathbb{A}\xrightarrow[]{\unicode[STIX]{x1D70E}\otimes \unicode[STIX]{x1D70E}}{\mathcal{O}}_{X/{\mathcal{P}}}\otimes {\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{\{.\,,\,.\}}{\mathcal{O}}_{X/{\mathcal{P}}}.\end{eqnarray}$$

The latter map, $\{.\,,.\}:{\mathcal{O}}_{X/{\mathcal{P}}}\otimes {\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}$ is a Poisson bracket on ${\mathcal{O}}_{X/{\mathcal{P}}}$ , hence corresponds to a bi-vector $\unicode[STIX]{x1D70B}\in \unicode[STIX]{x1D6E4}(X;\bigwedge ^{2}{\mathcal{T}}_{X/{\mathcal{P}}})$ . If $\mathbb{A}$ is a star-product we recover the (10).

Proof. Suppose that $\mathbb{A}_{i}$ , $i=1,2$ , is a DQ-algebras with associated Poisson bracket $\{.\,,.\}_{i}$ and $\unicode[STIX]{x1D6F7}:\mathbb{A}_{1}\rightarrow \mathbb{A}_{2}$ is a morphism of such. Let $f,g\in {\mathcal{O}}_{X/{\mathcal{P}}}$ , $\widetilde{f},\widetilde{g}\in \mathbb{A}_{1}$ with $f=\widetilde{f}+t\mathbb{A}_{1}$ , $g=\widetilde{g}+t\mathbb{A}_{1}$ .

Since $\unicode[STIX]{x1D6F7}$ induces the identity map modulo $t$ ,

$$\begin{eqnarray}t\cdot \{f,g\}_{2}+t^{2}\mathbb{A}_{2}=\unicode[STIX]{x1D6F7}(\widetilde{f})\unicode[STIX]{x1D6F7}(\widetilde{g})-\unicode[STIX]{x1D6F7}(\widetilde{g})\unicode[STIX]{x1D6F7}(\widetilde{f})+t^{2}\mathbb{A}_{2}=\unicode[STIX]{x1D6F7}(\widetilde{f}\widetilde{g}-\widetilde{g}\widetilde{f}+t^{2}\mathbb{A}_{1})=t\cdot \{f,g\}_{1}+t^{2}\mathbb{A}_{2}.\end{eqnarray}$$

Thus, if $\mathbb{A}_{i}$ , $i=1,2$ , are locally isomorphic, then the associated Poisson brackets are locally equal, hence equal.◻

6.4 Standard sections

Let $\mathbb{A}$ be a DQ-algebra. Recall [Reference Kashiwara and SchapiraKS12, Definition 2.2.6] that a $\mathbb{C}$ -linear section $\unicode[STIX]{x1D719}:{\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow \mathbb{A}$ of the map $\mathbb{A}\xrightarrow[]{\unicode[STIX]{x1D70E}}{\mathcal{O}}_{X/{\mathcal{P}}}$ is called standard if there exist bi-differential operators $P_{i}$ , $i=0,1,\ldots ,$ such that for any $f,g\in {\mathcal{O}}_{X/{\mathcal{P}}}$

(11) $$\begin{eqnarray}\unicode[STIX]{x1D719}(f)\unicode[STIX]{x1D719}(g)=\unicode[STIX]{x1D719}(fg)+\mathop{\sum }_{i\geqslant 1}\unicode[STIX]{x1D719}(P_{i}(f,g))t^{i},\end{eqnarray}$$

where the left-hand side product is computed in $\mathbb{A}$ . In this case

$$\begin{eqnarray}f\otimes g\mapsto f\star _{\unicode[STIX]{x1D719}}g:=fg+\mathop{\sum }_{i\geqslant 1}P_{i}(f,g)t^{i}\end{eqnarray}$$

defines a star-product.

A standard section $\unicode[STIX]{x1D719}:{\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow \mathbb{A}$ extends by $t$ -linearity to a morphism of DQ-algebras $\widetilde{\unicode[STIX]{x1D719}}:({\mathcal{O}}_{X/{\mathcal{P}}}[[t]],\star _{\unicode[STIX]{x1D719}})\rightarrow \mathbb{A}$ . Conversely, a morphism of DQ-algebras $\unicode[STIX]{x1D711}:({\mathcal{O}}_{X/{\mathcal{P}}}[[t]],\star )\rightarrow \mathbb{A}$ restricts to a standard section $\unicode[STIX]{x1D719}:=\unicode[STIX]{x1D711}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}$ such that $\star _{\unicode[STIX]{x1D719}}=\star$ and $\widetilde{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D711}$ .

We will call a standard section $\unicode[STIX]{x1D719}$ special if the corresponding star-product $\star _{\unicode[STIX]{x1D719}}$ is special.

For $U\subseteq X$ and open subset, $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}\in \unicode[STIX]{x1D6F4}(\mathbb{A})(U)$ and $k=1,2,\ldots$ we say that $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ are equivalent modulo $t^{k+1}$ if the compositions ${\mathcal{O}}_{X/{\mathcal{P}}}(U)\xrightarrow[]{\unicode[STIX]{x1D719}_{i}}\mathbb{A}(U)\rightarrow \mathbb{A}(U)/t^{k+1}\mathbb{A}(U)$ , $i=1,2$ , coincide. Clearly, ‘equivalence modulo $t^{k+1}$ ’ is an equivalence relation on $\unicode[STIX]{x1D6F4}(\mathbb{A})$ which we denote by ${\sim}_{k}$ . We denote by $\unicode[STIX]{x1D6F4}_{k}(\mathbb{A})$ the sheaf associated to the presheaf $U\mapsto \unicode[STIX]{x1D6F4}(\mathbb{A})(U)/{\sim}_{k}$ .

Proof. Since the question is local we may assume that $\mathbb{A}=({\mathcal{O}}_{X/{\mathcal{P}}}[[t]],\star )$ is a star-product given by (9).

(i) It follows from Lemma 6.5 that locally there exists an isomorphism $\unicode[STIX]{x1D711}:({\mathcal{O}}_{X/{\mathcal{P}}}[[t]],\star )\rightarrow \mathbb{A}$ with $\star$ a special star-product. Then $\unicode[STIX]{x1D711}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}$ is a special standard section.

(ii) Since the question is local we may assume that the map $q:\mathbb{A}\rightarrow \mathbb{A}/t^{k+1}\mathbb{A}$ admits a splitting $s:\mathbb{A}/t^{k+1}\mathbb{A}\rightarrow \mathbb{A}$ , $q\circ s=\mathtt{Id}$ . The composition $s\circ q:\mathbb{A}\rightarrow \mathbb{A}$ preserves equivalence modulo $t^{k+1}$ , hence induces the map $s\circ q:\unicode[STIX]{x1D6F4}_{k}(\mathbb{A})\rightarrow \unicode[STIX]{x1D6F4}(\mathbb{A})$ . Since the composition $\unicode[STIX]{x1D6F4}_{k}(\mathbb{A})\rightarrow \unicode[STIX]{x1D6F4}(\mathbb{A})\rightarrow \unicode[STIX]{x1D6F4}_{k}(\mathbb{A})$ is the identity map, the claim follows.

(iii) Since $\unicode[STIX]{x1D6F7}$ induces the identity map modulo $t$ the composition $\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719}$ is a section. By Proposition 6.1 the terms of

$$\begin{eqnarray}(\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})(f)(\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})(g)=(\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})(fg)+\mathop{\sum }_{i\geqslant 1}(\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})(P_{i}(f,g))t^{i},\end{eqnarray}$$

are bi-differential operators and therefore the section $\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719}$ is standard if $\unicode[STIX]{x1D719}$ is. The formula shows that, if $\unicode[STIX]{x1D719}$ is special, then so is $\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719}$ .◻

Thus, the assignment $\mathbb{A}\mapsto \unicode[STIX]{x1D6F4}(\mathbb{A})$ , $(\mathbb{A}_{0}\xrightarrow[]{\unicode[STIX]{x1D6F7}}\mathbb{A}_{1})\mapsto (\unicode[STIX]{x1D6F4}(\unicode[STIX]{x1D6F7}):\unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})$ defines a functor, denoted by $\unicode[STIX]{x1D6F4}$ , on the category of DQ-algebras.

It is clear that the map $\unicode[STIX]{x1D6F4}(\unicode[STIX]{x1D6F7}):\unicode[STIX]{x1D6F4}(\mathbb{A}_{0})\rightarrow \unicode[STIX]{x1D6F4}(\mathbb{A}_{1})$ induced by the morphism of DQ-algebras $\unicode[STIX]{x1D6F7}:\mathbb{A}_{0}\rightarrow \mathbb{A}_{1}$ preserves equivalence modulo $t^{k}$ . Thus, the functor $\unicode[STIX]{x1D6F4}$ gives rise to functors $\unicode[STIX]{x1D6F4}_{k}$ for all $k=1,2,\ldots \,$ .

6.5 The functor $\unicode[STIX]{x1D6F4}_{1}$

Suppose that $\mathbb{A}$ is a DQ-algebra with the associated Poisson tensor $\unicode[STIX]{x1D70B}$ and the corresponding Lie algebroid (structure on $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}$ ) denoted $\unicode[STIX]{x1D6F1}$ .

Lemma 6.8. Suppose that $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D719}_{1}$ are standard sections of $\mathbb{A}$ with the corresponding star-products $\star _{(j)}:=\star _{\unicode[STIX]{x1D719}_{i}}$ given by

$$\begin{eqnarray}f\star _{(j)}g=fg+P_{1}^{(j)}(f,g)t+P_{2}^{(j)}(f,g)t^{2}+\cdots\end{eqnarray}$$

for $j=0,1$ . Let $R=1+\sum _{i=1}^{\infty }R_{i}t^{i}$ , $R_{i}$ differential operators, denote the solution of $\widetilde{\unicode[STIX]{x1D719}_{1}}=\widetilde{\unicode[STIX]{x1D719}_{0}}\circ R$ uniquely determined by Proposition 6.1. Then, $P_{1}^{(0)}=P_{1}^{(1)}$ if and only if $R_{1}$ is a derivation, i.e. a section of ${\mathcal{T}}_{X/{\mathcal{P}}}$ .

Proof. The operator $R$ defines a morphism of algebras $({\mathcal{O}}_{X/{\mathcal{P}}}[[t]],\star _{(1)})\rightarrow ({\mathcal{O}}_{X/{\mathcal{P}}}[[t]],\star _{(0)})$ , i.e.

$$\begin{eqnarray}R(f)\star _{(0)}R(g)=R(f\star _{(1)}g).\end{eqnarray}$$

Comparing the calculations

$$\begin{eqnarray}R(f)\star _{(0)}R(g)=fg+(fR_{1}(g)+R_{1}(f)g+P_{1}^{(0)}(f,g))t+\cdots \,.\end{eqnarray}$$

and

$$\begin{eqnarray}R(f\star _{(1)}g)=fg+(R_{1}(fg)+P_{1}^{(1)}(f,g))t+\cdots \,.\end{eqnarray}$$

one concludes that

$$\begin{eqnarray}P_{1}^{(0)}(f,g)=P_{1}^{(1)}(f,g)\quad \text{if and only if}\quad fR_{1}(g)+R_{1}(f)g=R_{1}(fg),\end{eqnarray}$$

i.e.  $R_{1}$ is a derivation.◻

For $\unicode[STIX]{x1D709}\in {\mathcal{T}}_{X/{\mathcal{P}}}=\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}$ let $R_{\unicode[STIX]{x1D709}}:=1+\unicode[STIX]{x1D709}t$ . Corollary 6.9 implies that for $\unicode[STIX]{x1D719}\in \unicode[STIX]{x1D6F4}(\mathbb{A})$ the section $\widetilde{\unicode[STIX]{x1D719}}\circ R_{\unicode[STIX]{x1D709}}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}$ is special, i.e.  $\unicode[STIX]{x1D719}\mapsto \widetilde{\unicode[STIX]{x1D719}}\circ R_{\unicode[STIX]{x1D709}}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}$ gives rise to a map $\unicode[STIX]{x1D6F4}(\mathbb{A})\rightarrow \unicode[STIX]{x1D6F4}(\mathbb{A})$ .

Suppose that $\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1}\in \unicode[STIX]{x1D6F4}(\mathbb{A})$ are equivalent modulo $t^{2}$ , which is to say $\unicode[STIX]{x1D719}_{0}(f)-\unicode[STIX]{x1D719}_{1}(f)\in t^{2}\mathbb{A}$ for any $f\in {\mathcal{O}}_{X/{\mathcal{P}}}$ which implies that $\widetilde{\unicode[STIX]{x1D719}_{0}}(f)-\widetilde{\unicode[STIX]{x1D719}_{1}}(f)\in t^{2}\mathbb{A}$ for any $f\in {\mathcal{O}}_{X/{\mathcal{P}}}[[t]]$ . Then, for any $f\in {\mathcal{O}}_{X/{\mathcal{P}}}$ , $\widetilde{\unicode[STIX]{x1D719}_{0}}\circ R_{\unicode[STIX]{x1D709}}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}(f)-\widetilde{\unicode[STIX]{x1D719}_{1}}\circ R_{\unicode[STIX]{x1D709}}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}(f)=\widetilde{\unicode[STIX]{x1D719}_{0}}(R_{\unicode[STIX]{x1D709}}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}(f))-\widetilde{\unicode[STIX]{x1D719}_{1}}(R_{\unicode[STIX]{x1D709}}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}(f))\in t^{2}\mathbb{A}$ . Therefore, the map $\unicode[STIX]{x1D719}\mapsto \widetilde{\unicode[STIX]{x1D719}}\circ R_{\unicode[STIX]{x1D709}}|_{{\mathcal{O}}_{X/{\mathcal{P}}}}$ preserves equivalence modulo $t^{2}$ , hence descends to a map $\unicode[STIX]{x1D6F4}_{1}(\mathbb{A})\rightarrow \unicode[STIX]{x1D6F4}_{1}(\mathbb{A})$ denoted $\unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D719}+\unicode[STIX]{x1D709}$ .

Proof. (i) The sheaf $\unicode[STIX]{x1D6F4}_{1}(\mathbb{A})$ is locally non-empty by Proposition 6.7.

Note that, for $\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}\in {\mathcal{T}}_{X/{\mathcal{P}}}$ the equality $R_{\unicode[STIX]{x1D709}}\circ R_{\unicode[STIX]{x1D702}}=R_{\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702}}+t^{2}\unicode[STIX]{x1D709}\circ \unicode[STIX]{x1D702}$ holds. Therefore, the formula $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D709})\mapsto \unicode[STIX]{x1D719}+\unicode[STIX]{x1D709}$ does indeed define an action of the group $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}$ . Since $\widetilde{\unicode[STIX]{x1D719}}(R_{\unicode[STIX]{x1D709}}(f))-\widetilde{\unicode[STIX]{x1D719}}(f)=t\unicode[STIX]{x1D719}(\unicode[STIX]{x1D709}(f))+t^{2}\mathbb{A}$ , it follows that the action is free. Corollary 6.9 implies that the action is in fact transitive.

(ii) By associativity of composition

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}(\widetilde{\unicode[STIX]{x1D719}}\circ R_{\unicode[STIX]{x1D709}}(f))=\widetilde{(\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})}\circ R_{\unicode[STIX]{x1D709}}(f)\end{eqnarray}$$

for all $\unicode[STIX]{x1D719}\in \unicode[STIX]{x1D6F4}(\mathbb{A}_{0})$ , $\unicode[STIX]{x1D709}\in {\mathcal{T}}_{X/{\mathcal{P}}}$ and $f\in {\mathcal{O}}_{X/{\mathcal{P}}}$ . Therefore,

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{1}(\unicode[STIX]{x1D719}+\unicode[STIX]{x1D709})=\unicode[STIX]{x1D6F4}_{1}(\unicode[STIX]{x1D719})+\unicode[STIX]{x1D709}.\end{eqnarray}$$

(iii) Since the statement is local, we can assume that $a=\exp \unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FC}\in \mathbb{A}$ . Note that, for $b\in \mathbb{A}$ ,

$$\begin{eqnarray}\operatorname{Ad}(\exp \unicode[STIX]{x1D6FC})(b)=\mathop{\sum }_{i=0}^{\infty }\frac{(\operatorname{ad}\unicode[STIX]{x1D6FC})^{i}(b)}{i!}=b+t\{\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FC}),\unicode[STIX]{x1D70E}(b)\}+t^{2}\mathbb{A},\end{eqnarray}$$

where $(\operatorname{ad}\unicode[STIX]{x1D6FC})(b):=[\unicode[STIX]{x1D6FC},b]$ . Therefore, for $f\in {\mathcal{O}}_{X/{\mathcal{P}}}$ , since $\unicode[STIX]{x1D70E}\circ \unicode[STIX]{x1D719}=\mathtt{Id}$ ,

$$\begin{eqnarray}\displaystyle ((\operatorname{Ad}a)\circ \unicode[STIX]{x1D719})(f) & = & \displaystyle \unicode[STIX]{x1D719}(f)+t\{\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FC}),\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}(f))\}+t^{2}\mathbb{A}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D719}(f)+td_{\unicode[STIX]{x1D6F1}}\log (\unicode[STIX]{x1D70E}(a))(f)+t^{2}\mathbb{A}\nonumber\\ \displaystyle & = & \displaystyle \widetilde{\unicode[STIX]{x1D719}}\circ R_{d_{\unicode[STIX]{x1D6F1}}\log (\unicode[STIX]{x1D70E}(a))}(f)+t^{2}\mathbb{A}.\square\nonumber\end{eqnarray}$$

In view of Lemma 6.10, the assignment $\mathbb{A}\mapsto \unicode[STIX]{x1D6F4}_{1}(\mathbb{A})$ , $\unicode[STIX]{x1D6F7}\mapsto \unicode[STIX]{x1D6F4}_{1}(\unicode[STIX]{x1D6F7})$ , defines a morphism of stacks

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{1}:\text{DQ}\text{-}\text{alg}_{X/{\mathcal{P}}}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}[1].\end{eqnarray}$$

6.6 Subprincipal symbols

The construction presented below can be traced back to [Reference VeyVey75]. Subprincipal curvature defined below appears in the context of classification of star-products in [Reference Bertelson, Cahen and GuttBCG97] and in [Reference BursztynBur02], where it is called ‘semiclassical curvature’.

Multiplication by $t^{n}$ induces the isomorphism ${\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{\cong }t^{n}\mathbb{A}/t^{n+1}\mathbb{A}$ . In particular, there is a short exact sequence

(12) $$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{\cdot t}\mathbb{A}/t^{2}\mathbb{A}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow 0.\end{eqnarray}$$

Let $\widetilde{\{~,~\}}$ denote operation on $\mathbb{A}/t^{2}\mathbb{A}$ induced by $(1/t)[~,~]$ . The operation $\widetilde{\{~,~\}}$ endows $\mathbb{A}/t^{2}\mathbb{A}$ with a structure of a Lie algebra so that the exact sequence (12) exhibits $\mathbb{A}/t^{2}\mathbb{A}$ as an abelian extension of (the Lie algebra) ${\mathcal{O}}_{X/{\mathcal{P}}}$ equipped with the associated Poisson bracket.

Lemma 6.11. For $\unicode[STIX]{x1D719}\in \unicode[STIX]{x1D6F4}(\mathbb{A})$ and $f,g\in {\mathcal{O}}_{X/{\mathcal{P}}}$ ,

$$\begin{eqnarray}\widetilde{\{\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)\}}-\unicode[STIX]{x1D719}(\{f,g\})\in t\mathbb{A}.\end{eqnarray}$$

Proof. Since (in the notation of (11))

$$\begin{eqnarray}[\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)]-(t\unicode[STIX]{x1D719}(\{f,g\})+t^{2}\unicode[STIX]{x1D719}(P_{2}(f,g)-P_{2}(g,f)))\in t^{3}\mathbb{A},\end{eqnarray}$$

it follows that

$$\begin{eqnarray}\widetilde{\{\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)\}}-\unicode[STIX]{x1D719}(\{f,g\})=t\unicode[STIX]{x1D719}(P_{2}(f,g)-P_{2}(g,f))+t^{2}\mathbb{A}\in t\mathbb{A}.\square\end{eqnarray}$$

By Lemma 6.11

$$\begin{eqnarray}\widetilde{\{\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)\}}-\unicode[STIX]{x1D719}(\{f,g\})+t^{2}\mathbb{A}\in t\mathbb{A}/\mathbb{A}/t^{2}\mathbb{A}\xleftarrow[]{\cdot t}{\mathcal{O}}_{X/{\mathcal{P}}}.\end{eqnarray}$$

For $\unicode[STIX]{x1D719}\in \unicode[STIX]{x1D6F4}(\mathbb{A})$ we define the map

(13) $$\begin{eqnarray}c(\unicode[STIX]{x1D719}):{\mathcal{O}}_{X/{\mathcal{P}}}\otimes _{\mathbb{C}}{\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}\end{eqnarray}$$

by

$$\begin{eqnarray}tc(\unicode[STIX]{x1D719})(f,g)=\widetilde{\{\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)\}}-\unicode[STIX]{x1D719}(\{f,g\})+t^{2}\mathbb{A}\in t\mathbb{A}/t^{2}\mathbb{A}.\end{eqnarray}$$

Lemma 6.12. Let $\unicode[STIX]{x1D6F7}:\mathbb{A}_{0}\rightarrow \mathbb{A}_{1}$ be a morphism of DQ-algebras. Then

$$\begin{eqnarray}c(\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})=c(\unicode[STIX]{x1D719}).\end{eqnarray}$$

Proof. We have

$$\begin{eqnarray}\displaystyle tc(\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719})(f,g) & = & \displaystyle \widetilde{\{\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719}(g)\}}-\unicode[STIX]{x1D6F7}\circ \unicode[STIX]{x1D719}(\{f,g\})+t^{2}\mathbb{A}_{1}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6F7}(\widetilde{\{\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)\}}-\unicode[STIX]{x1D719}(\{f,g\})+t^{2}\mathbb{A}_{0})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6F7}(tc(\unicode[STIX]{x1D719})(f,g)).\nonumber\end{eqnarray}$$

The statement follows from the commutativity of the following diagram.

Proof. (i) The skew-symmetry of $c(\unicode[STIX]{x1D719})$ is clear. Applying $(\widetilde{\unicode[STIX]{x1D719}})^{-1}$ to the identity

$$\begin{eqnarray}[\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)\unicode[STIX]{x1D719}(h)]=\unicode[STIX]{x1D719}(g)[\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(h)]+[\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)]\unicode[STIX]{x1D719}(h),\end{eqnarray}$$

expanding the result in powers of $t$ and comparing the coefficients of $t^{2}$ we obtain

$$\begin{eqnarray}c(\unicode[STIX]{x1D719})(f,gh)+{\textstyle \frac{1}{2}}\{f,\{g,h\}\}=gc(\unicode[STIX]{x1D719})(f,h)+{\textstyle \frac{1}{2}}\{g,\{f,h\}\}+c(\unicode[STIX]{x1D719})(f,g)h+{\textstyle \frac{1}{2}}\{\{f,g\},h\},\end{eqnarray}$$

which implies that

$$\begin{eqnarray}c(\unicode[STIX]{x1D719})(f,gh)=gc(\unicode[STIX]{x1D719})(f,h)+c(\unicode[STIX]{x1D719})(f,g)h\end{eqnarray}$$

i.e.  $c(\unicode[STIX]{x1D719})$ is a derivation in the second variable. Skew-symmetry implies that it is a biderivation.

(ii) Applying $(\widetilde{\unicode[STIX]{x1D719}})^{-1}$ to the identity

$$\begin{eqnarray}[\unicode[STIX]{x1D719}(f),[\unicode[STIX]{x1D719}(g),\unicode[STIX]{x1D719}(h)]]=[\unicode[STIX]{x1D719}(g),[\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(h)]]+[[\unicode[STIX]{x1D719}(f),\unicode[STIX]{x1D719}(g)],\unicode[STIX]{x1D719}(h)],\end{eqnarray}$$

expanding the result in powers of $t$ and comparing the coefficients of $t^{3}$ we obtain

$$\begin{eqnarray}c(\unicode[STIX]{x1D719})(f,\{g,h\})+\{f,c(\unicode[STIX]{x1D719})(g,h)\}=c(\unicode[STIX]{x1D719})(g,\{f,h\})+\{g,c(\unicode[STIX]{x1D719})(f,h)\}+c(\unicode[STIX]{x1D719})(\{f,g\},h)+\{c(\unicode[STIX]{x1D719})(f,g),h\}\end{eqnarray}$$

which is equivalent to $d_{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D719})=0$ .

(iii) If $\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}^{\prime }\in \unicode[STIX]{x1D6F4}(\mathbb{A})$ , then there exists $R=1+\sum _{i=1}^{\infty }R_{i}t^{i}$ , where $R_{i}$ are differential operators and $R_{1}=\unicode[STIX]{x1D709}$ is a vector field such that $\unicode[STIX]{x1D719}^{\prime }=\unicode[STIX]{x1D719}\circ R$ . Direct calculation shows that

(14) $$\begin{eqnarray}[\unicode[STIX]{x1D719}^{\prime }(f),\unicode[STIX]{x1D719}^{\prime }(g)]=\unicode[STIX]{x1D719}(\{f,g\}t+(\{\unicode[STIX]{x1D709}(f),g\}+\{f,\unicode[STIX]{x1D709}(g)\}+c(\unicode[STIX]{x1D719})(f,g))t^{2}+\cdots \,).\end{eqnarray}$$

Hence,

$$\begin{eqnarray}\displaystyle \frac{1}{t}[\unicode[STIX]{x1D719}^{\prime }(f),\unicode[STIX]{x1D719}^{\prime }(g)]-\unicode[STIX]{x1D719}^{\prime }(\{f,g\}) & = & \displaystyle \unicode[STIX]{x1D719}(\{f,g\})+\unicode[STIX]{x1D719}(\{\unicode[STIX]{x1D709}(f),g\}+\{f,\unicode[STIX]{x1D709}(g)\}+c(\unicode[STIX]{x1D719})(f,g))t-\unicode[STIX]{x1D719}^{\prime }(\{f,g\})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D719}(\{\unicode[STIX]{x1D709}(f),g\}+\{f,\unicode[STIX]{x1D709}(g)\}-\unicode[STIX]{x1D709}(\{f,g\})+c(\unicode[STIX]{x1D719})(f,g))t+t^{2}\mathbb{A}\nonumber\end{eqnarray}$$

and

(15) $$\begin{eqnarray}c(\unicode[STIX]{x1D719}^{\prime })(f,g)=c(\unicode[STIX]{x1D719})(f,g)+\{\unicode[STIX]{x1D709}(f),g\}+\{f,\unicode[STIX]{x1D709}(g)\}-\unicode[STIX]{x1D709}(\{f,g\}).\end{eqnarray}$$

The sections $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D719}^{\prime }$ are equivalent modulo $t^{2}$ if and only if $\unicode[STIX]{x1D709}=0$ in which case (15) implies $c(\unicode[STIX]{x1D719}^{\prime })=c(\unicode[STIX]{x1D719})$ .

(iv) Note that (15) is equivalent to $c(\unicode[STIX]{x1D719}^{\prime })(f,g)=c(\unicode[STIX]{x1D719})(f,g)+d_{\unicode[STIX]{x1D6F1}}\unicode[STIX]{x1D709}(f,g)$ .◻

According to parts (i)–(iii) of Proposition 6.13 the assignment $c:\unicode[STIX]{x1D719}\mapsto c(\unicode[STIX]{x1D719})$ descends to a map

(16) $$\begin{eqnarray}c:\unicode[STIX]{x1D6F4}_{1}(\mathbb{A})\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}}.\end{eqnarray}$$

which, according to (iv) of Proposition 6.13, is a morphism of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}$ -torsors compatible with the map of sheaves of groups $d_{\unicode[STIX]{x1D6F1}}:\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}}$ . In other words, the map (16) endows $\unicode[STIX]{x1D6F4}_{1}(\mathbb{A})$ with a canonical structure of a $(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})$ -torsor.

Lemma 6.12 implies that the assignment $\mathbb{A}\mapsto \unicode[STIX]{x1D6F4}_{1}(\mathbb{A})$ , $\unicode[STIX]{x1D6F7}\mapsto \unicode[STIX]{x1D6F4}_{1}(\unicode[STIX]{x1D6F7})$ defines a morphism of stacks

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{1}:\text{DQ}\text{-}\text{alg}_{X/{\mathcal{P}}}\rightarrow (\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\displaystyle \xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})[1].\end{eqnarray}$$

6.7 Bi-invertible bi-modules

Suppose that $\mathbb{A}_{0}$ and $\mathbb{A}_{1}$ are DQ-algebras.

In what follows we suppose that $\mathbb{A}_{01}$ is a bi-invertible $\mathbb{A}_{1}\otimes _{\mathbb{C}[[t]]}\mathbb{A}_{0}^{\mathtt{op}}$ -module. Let $\operatorname{gr}\mathbb{A}_{01}=\mathbb{A}_{01}/t\mathbb{A}_{01}$ and let $\operatorname{gr}:\mathbb{A}_{01}\rightarrow \operatorname{gr}\mathbb{A}_{01}$ denote the canonical projection. Let $\unicode[STIX]{x1D70B}$ denote the common Poisson bi-vector of $\mathbb{A}_{0}$ and $\mathbb{A}_{1}$ and let $\unicode[STIX]{x1D6F1}$ denote the corresponding Lie algebroid (structure on $\unicode[STIX]{x1D6FA}_{X/{\mathcal{P}}}^{1}$ , see § 4.1.2).

Let $\mathbb{A}_{01}^{\times }$ denote the subsheaf of $\mathbb{A}_{01}$ consisting of sections $b\in \mathbb{A}_{01}$ such that the maps $\mathbb{A}_{1}\ni a\mapsto (a\otimes 1)b\in \mathbb{A}_{01}$ and $\mathbb{A}_{0}\ni a\mapsto (1\otimes a)b\in \mathbb{A}_{01}$ are isomorphisms of $\mathbb{A}_{1}$ -modules (respectively, $\mathbb{A}_{0}^{\mathtt{op}}$ -modules). For $b\in \mathbb{A}_{01}^{\times }$ let

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{b}:\mathbb{A}_{0}\rightarrow \mathbb{A}_{1}\end{eqnarray}$$

denote the morphism of DQ-algebras uniquely determined by

$$\begin{eqnarray}(\unicode[STIX]{x1D6F7}_{b}(a)\otimes 1)\cdot b=(1\otimes a)b.\end{eqnarray}$$

The map $\operatorname{gr}:\mathbb{A}_{01}\rightarrow \operatorname{gr}\mathbb{A}_{01}$ restricts to $\operatorname{gr}:\mathbb{A}_{01}^{\times }\rightarrow (\operatorname{gr}\mathbb{A}_{01})^{\times }$ .

The assignment $b\mapsto \unicode[STIX]{x1D6F7}_{b}$ defines a map

(17) $$\begin{eqnarray}\mathbb{A}_{01}^{\times }\rightarrow \text{}\underline{\operatorname{Hom}}_{\text{DQ}\text{-}\text{alg}}(\mathbb{A}_{0},\mathbb{A}_{1}),\end{eqnarray}$$

while the functor $\unicode[STIX]{x1D6F4}_{1}$ gives rise to the map

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{1}:\text{}\underline{\operatorname{Hom}}_{\text{DQ}\text{-}\text{alg}}(\mathbb{A}_{0},\mathbb{A}_{1})\rightarrow \text{}\underline{\operatorname{Hom}}_{(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})[1]}(\unicode[STIX]{x1D6F4}_{1}(\mathbb{A}_{0}),\unicode[STIX]{x1D6F4}_{1}(\mathbb{A}_{1})).\end{eqnarray}$$

Proof. A section $b\in \mathbb{A}_{01}^{\times }$ determines the isomorphism $\unicode[STIX]{x1D6F7}_{b}:\mathbb{A}_{0}\rightarrow \mathbb{A}_{1}$ and a compatible isomorphism of $\mathbb{A}_{01}$ with $\mathbb{A}_{1}$ viewed as $\mathbb{A}_{1}\otimes _{\mathbb{C}[[t]]}\mathbb{A}_{1}^{\mathtt{op}}$ -module. Functoriality of $\unicode[STIX]{x1D6F4}_{1}$ shows that it is sufficient to prove the statement in the case when $\mathbb{A}_{0}=\mathbb{A}_{1}=\mathbb{A}$ , and $\mathbb{A}_{01}=\mathbb{A}$ as an $\mathbb{A}$ -bi-module. In this case the morphism

$$\begin{eqnarray}\mathbb{A}^{\times }\rightarrow \text{}\underline{\operatorname{Hom}}_{\text{DQ}\text{-}\text{alg}}(\mathbb{A},\mathbb{A})\end{eqnarray}$$

is given by $a\mapsto \operatorname{Ad}a$ , and both parts of the lemma follow from the formula for action of inner automorphisms on $\unicode[STIX]{x1D6F4}_{1}(\mathbb{A})$ (Lemma 6.10 (iii)):

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{1}(\operatorname{Ad}a)(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D719}+d_{\unicode[STIX]{x1D6F1}}\log (\unicode[STIX]{x1D70E}(a)),\quad \unicode[STIX]{x1D719}\in \unicode[STIX]{x1D6F4}_{1}(\mathbb{A}).\square\end{eqnarray}$$

The proof of the following lemma is left to the reader.

6.8 Quasi-classical limits of bi-modules

The canonical contravariant connection on the classical limit of a bi-module deformation was constructed in [Reference BursztynBur02] (see also [Reference Bursztyn and WaldmannBW04]) in the setting of star-products. The construction presented below is a generalization of that of [Reference BursztynBur02] to the case of DQ-algebras.

Suppose that $\mathbb{A}_{01}$ is a bi-invertible $\mathbb{A}_{1}\otimes _{\mathbb{C}[[t]]}\mathbb{A}_{0}^{\mathtt{op}}$ -module. By Lemma 6.16  $\mathbb{A}_{0}$ and $\mathbb{A}_{1}$ give rise to the same Poisson bi-vector $\unicode[STIX]{x1D70B}$ . Let $\unicode[STIX]{x1D6F1}$ denote the corresponding Lie algebroid (see § 4.1.2).

Then, $\operatorname{gr}\mathbb{A}_{01}$ has a canonical structure of an ${\mathcal{O}}_{X/{\mathcal{P}}}$ -bi-module which is central, i.e. the two ${\mathcal{O}}_{X/{\mathcal{P}}}$ -module structures coincide. Moreover, $\operatorname{gr}\mathbb{A}_{01}$ is locally free of rank one over ${\mathcal{O}}_{X/{\mathcal{P}}}$ .

Therefore, the composition

$$\begin{eqnarray}\mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}\otimes \mathbb{A}_{01}\rightarrow \mathbb{A}_{01}\xrightarrow[]{\operatorname{gr}}\operatorname{gr}\mathbb{A}_{01}\end{eqnarray}$$

defined by $(a_{1},a_{0})\otimes b\mapsto (a_{1}b-ba_{0})+t\mathbb{A}_{01}$ is trivial. Thus, the first map factors through the inclusion $t\mathbb{A}_{01}{\hookrightarrow}\mathbb{A}_{01}$ . The composition

$$\begin{eqnarray}\mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}\otimes \mathbb{A}_{01}\rightarrow t\mathbb{A}_{01}\xrightarrow[]{t^{-1}}\mathbb{A}_{01}\rightarrow \operatorname{gr}\mathbb{A}_{01}\end{eqnarray}$$

factors through a unique map

$$\begin{eqnarray}\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}/t^{2}\mathbb{A}_{0}\otimes \operatorname{gr}\mathbb{A}_{01}\rightarrow \operatorname{gr}\mathbb{A}_{01}\end{eqnarray}$$

which gives rise to the map

(19) $$\begin{eqnarray}\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}/t^{2}\mathbb{A}_{0}\rightarrow \text{}\underline{\operatorname{End}}_{\mathbb{C}}(\operatorname{gr}\mathbb{A}_{01}).\end{eqnarray}$$

The sheaf $\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\ominus \mathbb{A}_{0}/t^{2}\mathbb{A}_{0}$ (the Baer difference of extensions of ${\mathcal{O}}_{X/{\mathcal{P}}}$ by ${\mathcal{O}}_{X/{\mathcal{P}}}$ ) is defined by the push-out square

where the left vertical map is defined by $(f_{1},f_{0})\mapsto f_{1}-f_{0}$ . Here, for a DQ-algebra $\mathbb{A}$ we use the identification ${\mathcal{O}}_{X/{\mathcal{P}}}\cong t\mathbb{A}/t^{2}\mathbb{A}$ given by the multiplication by $t$ .

The sheaf $\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}/t^{2}\mathbb{A}_{0}$ has a canonical structure of a Lie algebra as a Lie subalgebra of $\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\times \mathbb{A}_{0}/t^{2}\mathbb{A}_{0}$ with the bracket $\widetilde{\{~,~\}}$ (see § 6.6) on each factor. The bracket on $\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}/t^{2}\mathbb{A}_{0}$ descends to a Lie bracket on $\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\ominus \mathbb{A}_{0}/t^{2}\mathbb{A}_{0}$ .

Proof. For $(a,a^{\prime })\in \mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}$ and $b\in \mathbb{A}_{01}$ let $[(a,a^{\prime }),b]:=ab-ba^{\prime }$ .

(i) For $(a_{i},a_{i}^{\prime })\in \mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}$ , $i=1,2$ , and $b\in \mathbb{A}_{01}$

$$\begin{eqnarray}\displaystyle & & \displaystyle [(a_{1},a_{1}^{\prime }),[(a_{2},a_{2}^{\prime }),b]]-[(a_{2},a_{2}^{\prime }),[(a_{1},a_{1}^{\prime }),b]]\nonumber\\ \displaystyle & & \displaystyle \quad =[(a_{1},a_{1}^{\prime }),a_{2}b-ba_{2}^{\prime }]-[(a_{2},a_{2}^{\prime }),a_{1}b-ba_{1}^{\prime }]\nonumber\\ \displaystyle & & \displaystyle \quad =a_{1}a_{2}b-a_{2}ba_{1}^{\prime }-a_{1}ba_{2}^{\prime }+ba_{2}^{\prime }a_{1}^{\prime }-a_{2}a_{1}b+a_{1}ba_{2}^{\prime }+a_{2}ba_{1}^{\prime }-ba_{1}^{\prime }a_{2}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad =[a_{1},a_{2}]b-b[a_{1}^{\prime },a_{2}^{\prime }]=[([a_{1},a_{2}],[a_{1}^{\prime },a_{2}^{\prime }]),b],\nonumber\end{eqnarray}$$

which implies the claim.

(ii) The composition

$$\begin{eqnarray}({\mathcal{O}}_{X/{\mathcal{P}}}\times {\mathcal{O}}_{X/{\mathcal{P}}})\otimes \operatorname{gr}\mathbb{A}_{01}\rightarrow \mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\times _{{\mathcal{O}}_{X/{\mathcal{P}}}}\mathbb{A}_{0}/t^{2}\mathbb{A}_{0}\otimes \operatorname{gr}\mathbb{A}_{01}\rightarrow \operatorname{gr}\mathbb{A}_{01}\end{eqnarray}$$

is given by $(f_{1},f_{0})\otimes b\mapsto (f_{1}-f_{0})\cdot b$ . This means that the map (19) factors through $\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\ominus \mathbb{A}_{0}/t^{2}\mathbb{A}_{0}$ .

(iii) We have

$$\begin{eqnarray}[(a,a^{\prime }),fb]-f[(a,a^{\prime }),fb]=[a,f]b=t\{\unicode[STIX]{x1D70E}(a),f\}b+t^{2}\mathbb{A}_{01}\end{eqnarray}$$

which shows that $\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\ominus \mathbb{A}_{0}/t^{2}\mathbb{A}_{0}$ acts by differential operators of order one.

(iv) The same calculation shows that the principal symbol of the operator $b\mapsto [(a,a^{\prime }),b]$ is given by $d_{\unicode[STIX]{x1D6F1}}\unicode[STIX]{x1D70E}(a)$ .◻

Let $\widetilde{\unicode[STIX]{x1D6F1}}_{\operatorname{gr}\mathbb{A}_{01}}=\unicode[STIX]{x1D6F1}\times _{{\mathcal{T}}_{X/{\mathcal{P}}}}{\mathcal{A}}_{\operatorname{gr}\mathbb{A}_{01}}$ denote the ${\mathcal{O}}_{X/{\mathcal{P}}}$ -extension of $\unicode[STIX]{x1D6F1}$ obtained by pull-back by the anchor map $\widetilde{\unicode[STIX]{x1D70B}}$ . Thus, we have the following commutative diagram.

For $\unicode[STIX]{x1D719}_{i}\in \unicode[STIX]{x1D6F4}_{1}(\mathbb{A}_{i})$ , $i=0,1$ , the composition

$$\begin{eqnarray}{\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{(\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{0})}\mathbb{A}_{1}/t^{2}\mathbb{A}_{1}\ominus \mathbb{A}_{0}/t^{2}\mathbb{A}_{0}\rightarrow \widetilde{\unicode[STIX]{x1D6F1}}_{\operatorname{gr}\mathbb{A}_{01}}\end{eqnarray}$$

is a derivation, hence factors uniquely as

$$\begin{eqnarray}{\mathcal{O}}_{X/{\mathcal{P}}}\xrightarrow[]{d}\unicode[STIX]{x1D6F1}\xrightarrow[]{\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{0}}}\widetilde{\unicode[STIX]{x1D6F1}}_{\operatorname{gr}\mathbb{A}_{01}}\end{eqnarray}$$

with $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{0}}\in \operatorname{Conn}_{\unicode[STIX]{x1D6F1}}(\widetilde{\unicode[STIX]{x1D6F1}}_{\operatorname{gr}\mathbb{A}_{01}})$ .

The assignment $(\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{0})\mapsto \unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{0}}$ defines the map

(20) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}(\mathbb{A}_{01}):\unicode[STIX]{x1D6F4}_{1}(\mathbb{A}_{1})\ominus \unicode[STIX]{x1D6F4}_{1}(\mathbb{A}_{0})\rightarrow \operatorname{Conn}_{\unicode[STIX]{x1D6F1}}(\widetilde{\unicode[STIX]{x1D6F1}}_{\operatorname{gr}\mathbb{A}_{01}}).\end{eqnarray}$$

The proof of the following lemma is left to the reader.

Suppose that $\mathbb{A}_{i}$ , $i=0,1,2$ , is a DQ-algebra and $\mathbb{A}_{i,i+1}$ , $i=0,1$ is a bi-invertible $\mathbb{A}_{i+1}\otimes _{\mathbb{C}[[t]]}\mathbb{A}_{i}^{\mathtt{op}}$ -module. Thus, all DQ-algebras $\mathbb{A}_{i}$ give rise to the same associated Poisson bi-vector $\unicode[STIX]{x1D70B}$ , hence the same Lie algebroid $\unicode[STIX]{x1D6F1}$ .

Let $\mathbb{A}_{02}:=\mathbb{A}_{01}\otimes _{\mathbb{A}_{1}}\mathbb{A}_{12}$ . Then, $\mathbb{A}_{02}$ is a bi-invertible $\mathbb{A}_{2}\otimes _{\mathbb{C}[[t]]}\mathbb{A}_{0}^{\mathtt{op}}$ -module with $\operatorname{gr}\mathbb{A}_{02}=\operatorname{gr}\mathbb{A}_{01}\otimes _{{\mathcal{O}}_{X/{\mathcal{P}}}}\operatorname{gr}\mathbb{A}_{12}$ .

The bi-linear paring $\operatorname{gr}\mathbb{A}_{01}\times \operatorname{gr}\mathbb{A}_{12}\rightarrow \operatorname{gr}\mathbb{A}_{02}$ gives rise to the map $(\operatorname{gr}\mathbb{A}_{01})^{\times }\times (\operatorname{gr}\mathbb{A}_{12})^{\times }\rightarrow (\operatorname{gr}\mathbb{A}_{02})^{\times }$ .

7.1 DQ-algebroids

Recall the definition of DQ-algebroids from [Reference Kashiwara and SchapiraKS12].

In other words, a DQ-algebroid is a $\mathbb{C}[[t]]$ -algebroid locally equivalent to a star-product.

For a DQ-algebroid ${\mathcal{C}}$ we denote by ${\mathcal{C}}/t\cdot {\mathcal{C}}$ the separated prestack with the same objects as ${\mathcal{C}}$ and $\operatorname{Hom}_{{\mathcal{C}}/t\cdot {\mathcal{C}}}(L_{1},L_{2})=\unicode[STIX]{x1D6E4}(U;\operatorname{gr}\text{}\underline{\operatorname{Hom}}_{{\mathcal{C}}}(L_{1},L_{2}))$ for $L_{1},L_{2}\in {\mathcal{C}}(U)$ and by $\operatorname{gr}{\mathcal{C}}$ the associated stack. We denote by $\operatorname{gr}:{\mathcal{C}}\rightarrow \operatorname{gr}{\mathcal{C}}$ the ‘principal symbol’ functor.

For $U\subseteq X$ let $\mathfrak{DQ}_{X/{\mathcal{P}}}(U)$ denote the 2-category of DQ-algebroids. The assignment $U\mapsto \mathfrak{DQ}_{X/{\mathcal{P}}}(U)$ extends to a 2-stack which we denote by $\mathfrak{DQ}_{X/{\mathcal{P}}}$ .

The assignment ${\mathcal{C}}\mapsto \operatorname{gr}^{\times }{\mathcal{C}}:=i\operatorname{gr}{\mathcal{C}}$ gives rise to the morphism of 2-stacks

(21) $$\begin{eqnarray}\operatorname{gr}^{\times }:\mathfrak{DQ}_{X/{\mathcal{P}}}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}^{\times }[2].\end{eqnarray}$$

7.2 The associated Poisson structure

The following proposition is an immediate consequence of Lemmas 7.2 and 6.16.

Proposition 7.3. There exists a unique Poisson bracket

$$\begin{eqnarray}\{.\,,.\}^{{\mathcal{C}}}:{\mathcal{O}}_{X/{\mathcal{P}}}\otimes {\mathcal{O}}_{X/{\mathcal{P}}}\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}\end{eqnarray}$$

such that for any $U\subset X$ with ${\mathcal{C}}(U)\neq \varnothing$ and any $L\in {\mathcal{C}}(U)$ the restriction of $\{.\,,.\}^{{\mathcal{C}}}$ to $U$ coincides with the Poisson bracket associated to the DQ-algebra $\text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L)$ .

The assignment ${\mathcal{C}}\mapsto \unicode[STIX]{x1D70B}^{{\mathcal{C}}}$ give rise to the canonical morphism

(22) $$\begin{eqnarray}\mathfrak{DQ}_{X/{\mathcal{P}}}\rightarrow \unicode[STIX]{x1D6EC}^{2}{\mathcal{T}}_{X/{\mathcal{P}}}.\end{eqnarray}$$

For $\unicode[STIX]{x1D70B}\in \unicode[STIX]{x1D6E4}(X;\unicode[STIX]{x1D6EC}^{2}{\mathcal{T}}_{X/{\mathcal{P}}})$ we denote by $\mathfrak{DQ}_{X/{\mathcal{P}}}^{\unicode[STIX]{x1D70B}}$ the fiber of (22) over $\unicode[STIX]{x1D70B}$ .

7.3 The associated $\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}$ -connective structure

Suppose that ${\mathcal{C}}$ is a DQ-algebroid.

For $U\subseteq X$ an open subset such that ${\mathcal{C}}(U)\neq \varnothing$ , $L,L^{\prime }\in {\mathcal{C}}(U)$ the $\text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L^{\prime })\otimes \text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L)^{\mathtt{op}}$ -module $\text{}\underline{\operatorname{Hom}}_{{\mathcal{C}}}(L,L^{\prime })$ is bi-invertible by Lemma 7.2. An isomorphism $\unicode[STIX]{x1D719}:L\rightarrow L^{\prime }$ induces the morphism of DQ-algebras $\operatorname{Ad}(\unicode[STIX]{x1D719}):\text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L)\rightarrow \text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L^{\prime })$ defined by $\unicode[STIX]{x1D713}\mapsto \unicode[STIX]{x1D719}\circ \unicode[STIX]{x1D713}\circ \unicode[STIX]{x1D719}^{-1}$ .

The proof of the following lemma is left to the reader.

For $U\subseteq X$ an open subset such that ${\mathcal{C}}(U)\neq \varnothing$ , $L\in {\mathcal{C}}(U)$ the $\mathbb{C}[[t]]$ -algebra $\text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L)$ is a DQ-algebra and we set

$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{{\mathcal{C}}}(L):=\unicode[STIX]{x1D6F4}_{1}(\text{}\underline{\operatorname{End}}_{{\mathcal{C}}}(L))\in (\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{1}\xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{2,\text{cl}})[1].\end{eqnarray}$$

For $U\subseteq X$ an open subset such that ${\mathcal{C}}(U)\neq \varnothing$ , $L,L^{\prime }\in {\mathcal{C}}(U)$ , let

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}^{{\mathcal{C}}}(L,L^{\prime }) & := & \displaystyle \unicode[STIX]{x1D6F4}_{1}(\text{}\underline{\operatorname{Hom}}_{{\mathcal{C}}}(L,L^{\prime })):\text{}\underline{\operatorname{Hom}}_{{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }[1]}(\operatorname{gr}(L)^{\times },\operatorname{gr}(L^{\prime })^{\times })\nonumber\\ \displaystyle & = & \displaystyle (\operatorname{gr}\text{}\underline{\operatorname{Hom}}_{{\mathcal{C}}}(L,L^{\prime }))^{\times }\rightarrow \text{}\underline{\operatorname{Hom}}_{(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{1}\xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{2,\text{cl}})[1]}(\unicode[STIX]{x1D6FB}^{{\mathcal{C}}}(L),\unicode[STIX]{x1D6FB}^{{\mathcal{C}}}(L^{\prime })).\nonumber\end{eqnarray}$$

By Proposition 6.22 the above assignments define a functor, denoted by

$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{{\mathcal{C}}}:i({\mathcal{C}}/t{\mathcal{C}})\rightarrow (\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{1}\xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{2,\text{cl}})[1].\end{eqnarray}$$

As the target is a stack, this functor induces the functor

$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{{\mathcal{C}}}:i\operatorname{gr}{\mathcal{C}}\rightarrow (\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{1}\xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}}^{2,\text{cl}})[1],\end{eqnarray}$$

i.e. a $\unicode[STIX]{x1D6F1}^{{\mathcal{C}}}$ -connective structure with flat curving on $i\operatorname{gr}{\mathcal{C}}$ .

The assignment ${\mathcal{C}}\mapsto (\operatorname{gr}{\mathcal{C}},\unicode[STIX]{x1D6FB}^{{\mathcal{C}}})$ defines the morphism of 2-stacks

(23) $$\begin{eqnarray}\widetilde{\operatorname{gr}^{\times }}:\mathfrak{DQ}_{X/{\mathcal{P}}}^{\unicode[STIX]{x1D70B}}\rightarrow ({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})[2]\end{eqnarray}$$

(see §§7.2 and 5.2 for notation) lifting the morphism (21). We refer to (23) as the morphism of quasi-classical limit.

7.4 Obstruction to quantization

The construction of the canonical Poisson connective structure with flat curving on the classical limit of a DQ-algebroid imposes restrictions on the class of the classical limit. We will say that a twisted form ${\mathcal{S}}$ of ${\mathcal{O}}_{X/{\mathcal{P}}}$ admits a deformation along a Poisson bi-vector $\unicode[STIX]{x1D70B}\in \unicode[STIX]{x1D6E4}(X;\bigwedge ^{2}{\mathcal{T}}_{X/{\mathcal{P}}})$ if there exists a DQ-algebroid ${\mathcal{C}}$ with $\unicode[STIX]{x1D70B}^{{\mathcal{C}}}=\unicode[STIX]{x1D70B}$ and $\operatorname{gr}{\mathcal{C}}\cong {\mathcal{S}}$ .

Theorem 7.5. A twisted form ${\mathcal{S}}$ of ${\mathcal{O}}_{X/{\mathcal{P}}}$ admits a deformation along a Poisson bi-vector $\unicode[STIX]{x1D70B}\in \unicode[STIX]{x1D6E4}(X;\bigwedge ^{2}{\mathcal{T}}_{X/{\mathcal{P}}})$ only if the class of ${\mathcal{S}}$ in $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times })$ is in the image of the map

(24) $$\begin{eqnarray}H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})\rightarrow H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}$ denotes the Lie algebroid associated with $\unicode[STIX]{x1D70B}$ .

Proof. The map (24) is induced by the map of complexes of sheaves

$$\begin{eqnarray}({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}^{\times }.\end{eqnarray}$$

The induced morphism of 2-stacks

$$\begin{eqnarray}({\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})[2]\rightarrow {\mathcal{O}}_{X/{\mathcal{P}}}^{\times }[2]\end{eqnarray}$$

is the functor of forgetting the $\unicode[STIX]{x1D6F1}$ -connective structure given by the $({\mathcal{S}},\unicode[STIX]{x1D705})\mapsto {\mathcal{S}}$ .

If there exists a DQ-algebroid ${\mathcal{C}}$ with the associated Poisson bi-vector $\unicode[STIX]{x1D70B}$ and $\operatorname{gr}^{\times }{\mathcal{C}}$ equivalent to ${\mathcal{S}}$ , then the class of $\widetilde{\operatorname{gr}^{\times }}{\mathcal{C}}$ in $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}})$ is a lift of the class of ${\mathcal{S}}$ in $H^{2}(X;{\mathcal{O}}_{X/{\mathcal{P}}}^{\times })$ .◻

The following corollary of Theorem 7.5 is well-known (see for example [Reference Bressler, Gorokhovsky, Nest and TsyganBGNT07]).

8.1 The de Rham class of an ${\mathcal{O}}_{X}^{\times }$ -gerbe

We recall the construction of a closed differential 3-form representing the class of an ${\mathcal{O}}_{X}^{\times }$ -gerbe. See [Reference BrylinskiBry93] for further details.

Suppose that ${\mathcal{S}}$ is a twisted form of ${\mathcal{O}}_{X}$ . Under the map $H^{2}(X;{\mathcal{O}}_{X}^{\times })\xrightarrow[]{d\log }H^{2}(X;\unicode[STIX]{x1D6FA}_{X}^{1,\text{cl}})\cong H_{dR}^{3}(X)$ the equivalence class $[{\mathcal{S}}]\in H^{2}(X;{\mathcal{O}}_{X}^{\times })$ is mapped to the class $[{\mathcal{S}}]_{dR}\in H_{dR}^{3}(X)$ . We briefly recall the construction of a representative $H\in \unicode[STIX]{x1D6E4}(X;\unicode[STIX]{x1D6FA}_{X}^{3,\text{cl}})$ of the class $[{\mathcal{S}}]_{dR}$ .

The map $H^{2}(X;{\mathcal{O}}_{X}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{X}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{X}^{2})\rightarrow H^{2}(X;{\mathcal{O}}_{X}^{\times })$ is surjective, in other words, every ${\mathcal{O}}_{X}^{\times }$ -gerbe admits a connective structure with curving (also known as a classical $B$ -field). The de Rham differential gives rise to the map of complexes of sheaves $d:({\mathcal{O}}_{X}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{X}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{X}^{2})[2]\rightarrow \unicode[STIX]{x1D6FA}_{X}^{3,\text{cl}}$ . The induced map on cohomology $d:H^{2}(X;{\mathcal{O}}_{X}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{X}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{X}^{2})\rightarrow \unicode[STIX]{x1D6E4}(X;\unicode[STIX]{x1D6FA}_{X}^{3,\text{cl}})$ corresponds to the map which associates to a gerbe with connective structure with curving $({\mathcal{S}},\unicode[STIX]{x1D705})$ a closed 3-form $c(\unicode[STIX]{x1D705})$ called the curvature of the curving $\unicode[STIX]{x1D705}$ . The class of $c(\unicode[STIX]{x1D705})$ in $H_{dR}^{3}(X)$ depends only on the equivalence class of ${\mathcal{S}}$ (and does not depend on the choice of the connective structure) and is denoted by $[{\mathcal{S}}]_{dR}\in H_{dR}^{3}(X)$ .

The map $H^{2}(X;{\mathcal{O}}_{X}^{\times })\rightarrow H_{dR}^{3}(X)$ given by $[{\mathcal{S}}]\mapsto [{\mathcal{S}}]_{dR}$ coincides with the composition $H^{2}(X;{\mathcal{O}}_{X}^{\times })\rightarrow H^{2}(X;\unicode[STIX]{x1D6FA}_{X}^{1,\text{cl}})\cong H_{dR}^{3}(X)$ induced by the map of sheaves $d\log :{\mathcal{O}}_{X}^{\times }\rightarrow \unicode[STIX]{x1D6FA}_{X}^{1,\text{cl}}$ .

8.2 Deformations of twisted forms of ${\mathcal{O}}_{X}$  [Reference Bressler, Gorokhovsky, Nest and TsyganBGNT15]

Suppose that ${\mathcal{S}}$ is a twisted form of ${\mathcal{O}}_{X}$ . Lemma 6.5 in conjunction with [Reference Bressler, Gorokhovsky, Nest and TsyganBGNT15, Remark 6.6] imply that equivalence classes of DQ-algebroids with associated Poisson bi-vector $\unicode[STIX]{x1D70B}$ and classical limit (equivalent to) ${\mathcal{S}}$ are in bijective correspondence with equivalence classes of pairs $(\unicode[STIX]{x1D70B}_{t},H)$ , where:

1. (a) $\unicode[STIX]{x1D70B}_{t}=0+\unicode[STIX]{x1D70B}_{1}t+\unicode[STIX]{x1D70B}_{2}t^{2}+\cdots \in \unicode[STIX]{x1D6E4}(X;\bigwedge ^{2}{\mathcal{T}}_{X}\widehat{\otimes }t\mathbb{C}[[t]])$ ;

2. (b) $H\in \unicode[STIX]{x1D6E4}(X;\unicode[STIX]{x1D6FA}_{X}^{3,\text{cl}})$

satisfying:

1. (i) $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}$ ;

2. (ii) $H$ is a representative of $[{\mathcal{S}}]_{dR}\in H_{dR}^{3}(X)$ ;

3. (iii) $[\unicode[STIX]{x1D70B}_{t},\unicode[STIX]{x1D70B}_{t}]=\widetilde{\unicode[STIX]{x1D70B}_{t}}^{\wedge 3}(H)$ ;

where $\widetilde{\unicode[STIX]{x1D70B}_{t}}:\unicode[STIX]{x1D6FA}_{X}^{1}\rightarrow {\mathcal{T}}_{X}\widehat{\otimes }t\mathbb{C}[[t]]$ is the adjoint of $\unicode[STIX]{x1D70B}_{t}$ .

8.3 $\unicode[STIX]{x1D6F1}$ -connective structures from quasi-classical data

Suppose that $\unicode[STIX]{x1D70B}$ is a Poisson bi-vector on $X$ with associated Lie algebroid denoted by $\unicode[STIX]{x1D6F1}$ and $({\mathcal{S}},\unicode[STIX]{x1D6FB})$ is gerbe equipped with a connective structure $\unicode[STIX]{x1D6FB}$ with curving whose curvature is equal to $H\in \unicode[STIX]{x1D6E4}(X;\unicode[STIX]{x1D6FA}_{X}^{3,\text{cl}})$ .

Given $\unicode[STIX]{x1D70B}_{t}\in \unicode[STIX]{x1D6E4}(X;\bigwedge ^{2}{\mathcal{T}}_{X}\widehat{\otimes }t\mathbb{C}[[t]])$ such that the pair $(\unicode[STIX]{x1D70B}_{t},H)$ satisfies the conditions in § 8.2 one obtains a canonical $\unicode[STIX]{x1D6F1}$ -connective structure with flat curving on ${\mathcal{S}}$ as follows.

To a locally defined object $L\in {\mathcal{S}}$ the connective structure $\unicode[STIX]{x1D6FB}$ associates a $\unicode[STIX]{x1D6FA}_{X}^{1}$ -torsor $\unicode[STIX]{x1D6FB}(L)$ equipped with the curvature map $c_{L}:\unicode[STIX]{x1D6FB}(L)\rightarrow \unicode[STIX]{x1D6FA}_{X}^{2}$ . As $H$ is the curvature of the curving of $\unicode[STIX]{x1D6FB}$ , the composition $\unicode[STIX]{x1D6FB}(L)\xrightarrow[]{c_{L}}\unicode[STIX]{x1D6FA}_{X}^{2}\xrightarrow[]{d}\unicode[STIX]{x1D6FA}_{X}^{3,\text{cl}}$ is constant and equal to $H$ .

The $\unicode[STIX]{x1D6FA}_{X}^{1}$ -torsor $\unicode[STIX]{x1D6FB}(L)$ gives rise to the $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}$ -torsor $\widetilde{\unicode[STIX]{x1D70B}}\unicode[STIX]{x1D6FB}(L)$ . The map $(\widetilde{\unicode[STIX]{x1D70B}}(c_{L})-\unicode[STIX]{x1D70B}_{2}):\widetilde{\unicode[STIX]{x1D70B}}\unicode[STIX]{x1D6FB}(L)\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2}$ (where $\unicode[STIX]{x1D70B}_{2}\in \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2}$ is viewed as a constant map) is a morphism of torsors compatible with the map of groups $d_{\unicode[STIX]{x1D6F1}}:\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{1}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2}$ .

The condition $[\unicode[STIX]{x1D70B}_{t},\unicode[STIX]{x1D70B}_{t}]=\widetilde{\unicode[STIX]{x1D70B}_{t}}^{\wedge 3}(H)$ implies that $[\unicode[STIX]{x1D70B},\unicode[STIX]{x1D70B}_{2}]=\widetilde{\unicode[STIX]{x1D70B}}^{\wedge 3}(H)$ . Therefore, the composition $\widetilde{\unicode[STIX]{x1D70B}}\unicode[STIX]{x1D6FB}(L)\xrightarrow[]{\widetilde{\unicode[STIX]{x1D70B}}(c_{L})-\unicode[STIX]{x1D70B}_{2}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2}\xrightarrow[]{d_{\unicode[STIX]{x1D6F1}}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{3}$ is equal to zero. Consequently, the map $\widetilde{\unicode[STIX]{x1D70B}}(c_{L})-\unicode[STIX]{x1D70B}_{2}$ takes values in $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F1}}^{2,\text{cl}}$ and, hence, the assignment ${\mathcal{S}}\ni L\mapsto (\widetilde{\unicode[STIX]{x1D70B}}\unicode[STIX]{x1D6FB}(L),\widetilde{\unicode[STIX]{x1D70B}}(c_{L})-\unicode[STIX]{x1D70B}_{2})$ defines a $\unicode[STIX]{x1D6F1}$ -connective structure with flat curving on ${\mathcal{S}}$ .

8.4 Quasi-classical limit and formality

Suppose that ${\mathcal{C}}$ is a DQ-algebroid. Let $(\unicode[STIX]{x1D70B}_{t},H)$ be a pair representing the equivalence class of ${\mathcal{C}}$ under the bijection (see § 8.2) induced by a choice of a formality isomorphism.