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Formality theorem for gerbes

We extend the formality theorem of Maxim Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes on smooth and complex manifolds.


Introduction
This is the second of two articles devoted to extension of M. Kontsevich formality theorem [16] to deformations of gerbes. In the previous paper [3] we discuss extension of the notion of Deligne 2-groupoid to L ∞ algebras. In particular, due to the work of J. Duskin [8], E. Getzler [9], and V. Hinich [11] one has a natural notion of Deligne 2-groupoid for an L ∞ algebra g which we denote MC 2 (g). In the present paper we use these results to construct L ∞ algebra controlling the deformation theory of a gerbe.
In the fundamental paper [16] M. Kontsevich showed that the set of equivalence classes of formal deformations of the algebra of functions on a manifold is in oneto-one correspondence with the set of equivalence classes of formal Poisson structures on the manifold. This result was obtained as a corollary of the formality of the Hochschild complex of the algebra of functions on the manifold conjectured by M. Kontsevich (cf. [15]) and proven in [16]. Later proofs by a different method were given in [22] and in [6].
In this paper we extend the formality theorem of M. Kontsevich to deformations of gerbes on smooth manifolds or complex analytic manifolds, using the method of [6].
Consider first the case of a C ∞ manifold X. We denote by O X the sheaf of complex valued C ∞ functions on X. For a twisted form S of O X regarded as an algebroid stack (see Section 6.2) we denote by [S] dR ∈ H 3 dR (X) the de Rham class of S. The main result of this paper establishes an equivalence of 2-groupoid valued functors on Artin C-algebras between Def(S) (the formal deformation theory of S, see [2]) and the Deligne 2-groupoid of the L ∞ -algebra of multivector fields on X twisted by a closed three-form representing [S] dR : Theorem 1.1. Let X be a smooth manifold. Suppose that S is a twisted form of O X . Let H be a closed 3-form on X which represents [S] dR ∈ H 3 dR (X). For any Artin algebra R with maximal ideal m R there is an equivalence of 2-groupoids MC 2 (s(O X ) H ⊗ m R ) ∼ = Def(S) (R) natural in R.
Here, s(O X ) H denotes the L ∞ -algebra of multivector fields with the trivial differential, the binary operation given by Schouten bracket, the ternary operation given by H (see 5.2) and all other operations equal to zero. As a corollary of this result we obtain that the isomorphism classes of formal deformations of S are in a bijective correspondence with equivalence classes of the formal twisted Poisson structures defined by P. Severa and A. Weinstein in [21]. More precisely, equivalence classes of R-deformations of a gerbe are in a bijection with equivalence classes of π ∈ Γ(X; 2 T X ) ⊗ m R , satisfying the equation [π, π] = Φ(H)(π, π, π).
A construction of an algebroid stack associated to a twisted formal Poisson structure was proposed by P.Ševera in [20].
To give a treatment which also covers a complex analytic case and to incorporate other examples we work in the framework of the manifold with two transversal integrable polarizations. Let as before X be a smooth manifold. We shall assume that the manifold X is equipped with two complementary integrable complex distributions P and Q. We do suppose that P Q = 0 and P ⊕ Q = T C X . We denote by O X/P the sheaf of complex valued C ∞ functions on X constant along P; see the section 3 for the details. In the case when P = 0 we recover the case of the smooth manifold X with the sheaf O X/P = O X of complex valued C ∞ functions. In the case when P is a complex structure, i.e. P = Q we recover the case of complex analytic manifold X with the sheaf O X/P of holomorphic functions. The decomposition P ⊕ Q = T C X induces bigrading on forms Ω n X = p+q=n Ω p,q X with Ω p,q X = p P ⊥ ⊗ q Q ⊥ and filtration: dR (X) can be represented by a form H ∈ Γ(X; F −1 Ω 3 X ). Such a form H defines a structure of L ∞ -algebra on the P-Dolbeault resolution of P-holomorphic multivector fields, see Section 5.4. We denote this L ∞ -algebra by s(O X/P ). Then more general version of the Theorem 1.1 is the following: Suppose that X is a C ∞ -manifold equipped with a pair of complementary complex integrable distributions P and Q, and S is a twisted form of O X/P (6.2). Let H ∈ Γ(X; F −1 Ω 3 X ) be a representative of [S] (6.2). Then, for any Artin algebra R with maximal ideal m R there is an equivalence of bi-groupoids where the left-hand side is defined as in (6.1), natural in R.
The proof of the Theorem proceeds along the following lines. As a starting point we use the construction of the Differential Graded Lie Algebra (DGLA) controlling the deformations of S, obtained in [1,2]. Next we construct a chain of L ∞ -quasiisomorphisms between this DGLA and s(O X ) H , using the techniques of [6]. Then we use the results of [3] to deduce the equivalence of bigroupoids.
The paper is organized as follows.
Section 2 contains a short exposition of the proof of Kontsevich formality theorem given in [6].
Section 3 contains a short review of differential calculus and differential geometry of jets in the presence of an integrable distribution.
In section 4, the proof from [6] is modified to the twisted case.
The section 5 is devoted to constructions of L ∞ -structures on the algebra of multivectors and related algebras.
Finally, in the Section 6 we prove the main results on the deformations of algebroid stacks.

Formality
We give a synopsis of the results of [6] in the notation of loc. cit. Let k be a field of characteristic zero. For a k-cooperad C and a complex of k-vector spaces V we denote by F C (V ) the cofree C-coalgebra on V .
We denote by e 2 the operad governing Gerstenhaber algebras. The latter is Koszul, and we denote by e 2 ∨ the dual cooperad.
2.1. Hochschild cochains. For a k-algebra A, p = 0, 1, 2, . . . the space There is a canonical isomorphism of graded k-modules g(A) = Der k (coAss(A[1])), where coAss(V ) denotes the co-free co-associative co-algebra on the graded vector space V . In particular, g(A) = C(A) [1] has a canonical structure of a graded Lie algebra under the induced Gerstenhaber bracket. Let m ∈ C 2 (A) and let δ denote the adjoint action of m with respect to the Gerstenhaber bracket. Thus, δ is an endomorphism of degree one and a derivation of the Gerstenhaber bracket. The fact that δ • δ = 0 is equivalent to the associativity of m. The graded space C • (A) equipped with the Hochschild differential δ is called the Hochschild complex of A.
The graded space g(A) = C • (A) [1] equipped with the Gerstenhaber bracket and the Hochschild differential is a DGLA which controls the deformation theory of A.

2.2.
Outline of the proof of formality for cochains. For an associative kalgebra A the Hochschild complex C • (A) has a canonical structure of a brace algebra, hence a structure of homotopy e 2 -algebra. The latter structure is encoded in a differential (i.e. a coderivation of degree one and square zero) M : Suppose from now on that A is regular commutative algebra over a field of characteristic zero (the regularity assumption is not needed for the constructions). Let viewed as a complex with trivial differential. In this capacity V • (A) has a canonical structure of an e 2 -algebra which gives rise to the differential d V • (A) on F e 2 ∨ (V • (A)); we have: B e 2 ∨ (V • (A)) = (F e 2 ∨ (V • (A)), d V • (A) ) (see [6], Theorem 1 for notation).
Additionally, concerning the DGLA structures relevant to applications to deformation theory, deduced from respective e 2 -algebra structures we have the following result.
2.3. Some (super-)symmetries. For applications to deformation theory of algebroid stacks we will need certain equivariance properties of the maps described in 2.2.
For a ∈ A let i a : denote the adjoint action (in the sense of the Gerstenhaber bracket and the identification A = C 0 (A)). It is given by the formula i a D(a 1 , . . . , a n ) = n i=0 (−1) k D(a 1 , . . . , a i , a, a k+1 , . . . , a n ).
The operation i a extends uniquely to a coderivation of F e 2 ∨ (C • (A)); we denote this extension by i a as well. Furthermore, the subcoalgebra Ξ(A) is preserved by i a .
Since the operation i a is a derivation of the cup product as well as of all of the brace operations on C • (A) and the homotopy-e 2 -algebra structure on C • (A) is given in terms of the cup product and the brace operations it follows that i a anticommutes with the differential M . Hence, the coderivation i a induces a derivation of the differential graded e 2 -algebra Ω e 2 (F e 2 ∨ (C • (A)), M ) which will be denoted by i a as well. For the same reason the DGLA Ω e 2 (F e 2 ∨ (C • (A)), M ) [1] and C • (A) [1] are quasi-isomorphic in a way which commutes with the respective operations i a .
On the other hand, let i a : V • (A) → V • (A)[−1] denote the adjoint action in the sense of the Schouten bracket and the identification A = V 0 (A). The operation i a extends uniquely to a coderivation of F e 2 ∨ (V • (A)) which anticommutes with the differential d V • (A) because i a is a derivation of the e 2 -algebra structure on V • (A). We denote this coderivation as well as its unique extension to a derivation of the differential graded e 2 -algebra Ω e 2 (B e 2 ∨ (V • (A))) by i a . The counit map η e 2 : Ω e 2 (B e 2 ∨ (V • (A))) → V • (A) commutes with respective operations i a .
The subcoalgebra Ξ(A) of F e 2 ∨ (C • (A)) and F e 2 ∨ (V • (A)) is preserved by the respective operations i a . Moreover, the restrictions of the two operations to Ξ(A) coincide, i.e. the maps in (2.1) commute with i a and, therefore, so do the maps in (2.2) and (2.3).

Extensions and generalizations.
The constructions and results of [6] apply in a variety of situations. First of all, observe that the constructions of all objects and morphisms involved can be carried out in any closed symmetric monoidal category such as for example the category of sheaves of k-modules, k a sheaf of commutative algebras (over a field of characteristic zero). As is pointed out in [6], Section 4, the proof of Theorem 2.1 is based on the flatness of the module Der(A) and the Hochschild-Kostant-Rosenberg theorem.
The considerations above apply to a sheaf of k-algebras K yielding the sheaf of Hochschild cochains C p (K),the Hochschild complex (of sheaves) C • (K) and the (sheaf of) DGLA g(K).
Suppose that X is a C ∞ -manifold. The sheaf C p (O X ) coincides with the sheaf of multilinear differential operators O ×p X → O X by the celebrated theorem of J. Peetre [18], [19].

Deformations of O and Kontsevich formality.
Suppose that X is a manifold. Let O X (respectively, T X ) denote the structure sheaf (respectively, the sheaf of vector fields). The construction of 2.2 yield the diagram of sheaves of differential graded e 2 -algebras ) denotes the sheaf of multivector fields. Theorem 2.1 says that the morphisms Ω e 2 (σ) and ν in (2.4) are quasi-isomorphisms of sheaves of differential graded e 2 -algebras.

Calculus in the presence of distribution
In this section we briefly review basic facts regarding differential calculus in the presence of an integrable complex distribution. We refer the reader to [Ko], [R] and [FW] for details and proofs.
For a C ∞ manifold X we denote by O X (respectively, Ω i X ) the sheaf of complex valued C ∞ functions (respectively, differential forms of degree i) on X. Throughout this section we denote by T X the sheaf of real valued vector fields on X. Let T C X := T X ⊗ R C.
For a distribution P on X we denote by P ⊥ ⊆ Ω 1 X the annihilator of P (with respect to the canonical duality pairing).
A distribution P of rank r on X is called integrable if, locally on X, there exist functions f 1 , . . . , f r ∈ O X such that df 1 , . . . , df r form a local frame for P ⊥ .
It is easy to see that an integrable distribution is involutive. The converse is true when P is real, i.e. P = P (Frobenius) and when P is a complex structure, i.e. P ∩ P = 0 and P ⊕ P = T C X (Newlander-Nirenberg). See [R] for generalizations.
3.2. The Hodge filtration. We assume from now on that P is an integrable distribution on X. (Involutivity suffices for basic constructions.) Let F • Ω • X denote the filtration by the powers of the differential ideal generated by The wedge product of differential forms induces a structure of a commutative DGA on (Gr F • Ω • X , ∂). In particular, Suppose that E is a vector bundle on X, i.e. a locally free O Xmodule of finite rank. A connection along P on E is, by definition, a map ∇ P : The two avatars of a connection along P are related by ∇ P ξ (e) = ι ξ ∇ P (e). A connection along P on E is called flat if the corresponding map ∇ P (•) : P → End C (E) is a morphism of Lie algebras. We will refer to a flat connection along P on E as a ∂-operator on E.
A connection on E along P extends uniquely to a derivation of the graded The vanishing of higher ∂-cohomology of O X (3.1) generalizes easily to vector bundles.
Remark 3.2. Suppose that F is a locally free O X/P -module of finite rank. Then, O X ⊗ O X/P F is a locally free O X -module of rank rk O X/P F and is endowed in a canonical way with a ∂-operator, namely, ∂ ⊗ Id. The assignments F → (O X ⊗ O X/P F, ∂ ⊗ Id) and (E, ∂ E ) → ker(∂ E ) are mutually inverse equivalences of suitably defined categories.

3.4.
Calculus. The adjoint action of P on T C X preserves P, hence descends to an action on T C X /P. The latter action defines a connection along P, i.e. a canonical ∂-operator on T C X /P which is easily seen to coincide with the one induced via the duality pairing between the latter and P ⊥ . 2 Let T X/P := (T C X /P) P (the subsheaf of P invariant section, equivalently, the kernel of the ∂-operator on T C X /P. The Lie bracket on T C X (respectively, the action of T C X on O X ) induces a Lie bracket on T X/P (respectively, an action of T X/P on O X/P ). Together the two endow T X/P with a structure of an O X/P -Lie algebroid.
The action of P on Ω 1 X by Lie derivative restricts to a flat connection along P, i.e. a canonical ∂-operator on P ⊥ and, therefore, on i P ⊥ for all i. It is easy to see that the multiplication map The wedge product of differential forms induces a structure of a graded-commutative algebra on Ω • X/P : The multiplication induces an isomorphism i O X/P Ω 1 X/P → Ω i X/P . The de Rham differential d restricts to the map d : Ω i X/P → Ω i+1 X/P and the complex Ω • X/P : The action of T C X /P on O X/P the pairing and the de Rham differential are related by the usual formula ξ(f ) = ι ξ df , for ξ ∈ T X/P and f ∈ O X/P .

3.5.
Jets. Let pr i : X × X → X, i = 1, 2, denote the projection on the i th factor. The restriction of the canonical map The sheaf I X/P plays the role of the defining ideal of the "diagonal embedding X/P → X/P × X/P": there is a short exact sequence of sheaves on X × X For a locally-free O X/P -module of finite rank E let . It is clear from the above definition that J k X/P is, in a natural way, a commutative algebra and J k X/P (E) is a J k X/P -module. In what follows, unless stated explicitly otherwise, we regard J k X/P (E) as a O X/Pmodule via the map 1 (k) . Let For 0 ≤ k ≤ l the inclusion I l+1 X/P → I k+1 X/P induces the surjective map π l,k : J l X/P (E) → J k X/P (E). The sheaves J k X/P (E), k = 0, 1, . . . together with the maps π l,k , k ≤ l form an inverse system. Let J X/P (E) = J ∞ X/P (E) := lim ←− J k X/P (E). Thus, J X/P (E) carries a natural topology.
The maps 1 (k) (respectively, j k ), k = 0, 1, 2, . . . are compatible with the projections π l,k , i.e. π l,k • 1 (l) = 1 (k) (respectively, π l,k • j l = j k ). Let 1 := lim the exterior derivative along the first factor. It satisfies for each k and, therefore, induces the map d (k) The maps d X ⊗ O X F is a flat connection. The flat connection ∇ extends uniquely to a differential ∇ on Ω • X ⊗ O X F subject to the Leibniz rule with respect to the Ω • X -module structure. We will make use of the following notation: is a complex of O X -modules with a flat connection ∇ = (∇ i ) i∈Z , i.e. for each i ∈ Z, ∇ i is a flat connection on F i and [d, ∇] = 0. Then, is a double complex. We denote by DR(F) the associated simple complex.
O X -modules equipped with flat connections form a closed monoidal category. In particular, the sheaf of O X -multilinear maps J ×p X → J X is endowed with a canonical flat connection induced by ∇ can . It follows directly from the definitions that ∇ can preserves C p (J X ).

Formality for the algebroid Hochschild complex
Until further notice we work with a fixed C ∞ manifold X equipped with an integrable complex distribution P and denote J X,P (respectively, J X,P , J X,P (E)) by J (respectively, J , J (E)).

Hochschild cochains in formal geometry.
In what follows we will be interested in the Hochschild complex in the context of formal geometry. To this end we define C p (J ) to be the sheaf of continuous (with respect to the adic topology) O X -multi-linear Hochschild cochains on J . The Gerstenhaber bracket endows g(J ) = C • (J ) [1] with a structure of a graded Lie algebra. The product on J is a global section of C 2 (J ), hence the Hochschild differential preserves C • (J ).
The complex Γ(X; DR(C • (J ))) = (Γ(X; Ω • X ⊗ C • (J X )), ∇ can + δ) is a differential graded brace algebra in a canonical way. The abelian Lie algebra J = C 0 (J ) acts on the brace algebra C • (J X ) by derivations of degree −1 via the restriction of the adjoint action with respect to the Gerstenhaber bracket. The above action factors through an action of J . Therefore, the abelian Lie algebra Γ(X; Ω 2 X ⊗ J ) acts on the brace algebra Ω • X ⊗ C • (J ) by derivations of degree +1; the action of an element a is denoted by i a .
Let The canonical flat connection ∇ can on J induces a flat connection which we denote ∇ can as well on each of the objects in the diagram (4.1). Moreover, the maps Ω e 2 (σ) and ν are flat with respect to ∇ can hence induce the maps of respective de Rham complexes All objects in the diagram (4.2) have canonical structures of differential graded e 2 -algebras and the maps are morphisms of such.
For each of the objects in (4.2) we denote by i ω the operation which is induced by the one described in 2.3 and the wedge product on Ω • X . Thus, for each differential graded e 2 -algebra (N • , d) in (4.2) we have a derivation of degree one and square zero i ω which anticommutes with d and we denote by (N • , d) ω the ω-twist of (N • , d), i.e. the differential graded e 2 -algebra (N • , d + i ω ). Since the morphisms in (4.2) commute with the respective operations i ω , they give rise to morphisms of respective ω-twists Let G • Ω • X denote the filtration given by As is easy to see, the associated graded complex is given by . It is clear that the morphisms DR(Ω e 2 (σ)) and DR(ν) are filtered with respect to G • .

L ∞ -structures on multivectors.
The canonical pairing , : Ω 1 X/P ⊗T X/P → O X extends to the pairing denote the map given by the formula In what follows we consider the (shifted) de Rham complex Ω • X/P [2] as a differential graded Lie algebra with the trivial bracket.
Lemma 5.1. The map ω → Φ(ω) defines a morphism of sheaves of differential graded Lie algebras [1]. Proof. Recall the explicit formulas for the Schouten bracket. If f and g are functions and X i , Y j are vector fields, then Note that for a one-form ω and for vector fields X and Y From the two formulas above we deduce by an explicit computation that Note that Lie algebra cochains are invariant under the symmetric group acting by permutations multiplied by signs that are computed by the following rule: a permutation of π i and π j contributes a factor (−1) |π i ||π j |+|π i |+|π j | . We use the explicit formula for the bracket on the Lie algebra complex.
Notation. For a section B ∈ Γ(X; Ω 2 X ⊗J ) we denote by B its image in Γ(X; Ω 2 X ⊗J ).
Proof. For the first claim it suffices to show that ∇ can d d R B = 0. This follows from the assumption that ∇ can B = 0 and the fact that d d R : Remark 5.3. Lemma 5.2 shows that this notation is unambiguous with reference to the previously introduced notation for the twist. In the notation introduced above, and "allows" us to "identify" B with d d R B.

Dolbeault complexes.
We shall assume that the manifold X admits two complementary integrable complex distributions P and Q. We do suppose that P Q = 0 and P ⊕ Q = T C X . The latter decomposition induces a bi-grading on differential forms: Ω n X = p+q=n Ω p,q X with Ω p,q X = p P ⊥ ⊗ q Q ⊥ . The bi-grading splits the Hodge filtration: F −i Ω n = p≥i Ω p,n−p X . Two cases of particular interest in applications are • P = 0 • P is a complex structure, i.e. P = Q. The map (5.1) extends to the morphism of sheaves of DGLA We leave the verification of the following claim to the reader. Thus, the image under (5.6) of a closed 3-form H ∈ Γ(X; F −1 Ω 3 X ), dH = 0, gives rise to a structure of an L ∞ -algebra on Ω 0,• X ⊗ O X/P V • (O X/P ) [1] (whereas general closed 3-forms give rise to curved L ∞ -structures). Moreover, cohomologous closed 3forms give rise to gauge equivalent Maurer-Cartan elements, hence to L ∞ -isomorphic L ∞ -structures.
Notation. For H as above we denote by s(O X/P ) H the P-Dolbeault complex of the sheaf of multi-vector fields equipped with the corresponding L ∞ -algebra structure: Remark 5.5. In the case when P = 0, in other words, X is a plain C ∞ manifold, the map (5.6) simplifies to [1], a DGLA with the Schouten bracket and the trivial differential. These are the unary and the binary operations in the L ∞ -structure on s(O X ) H , H a closed 3-form on X; the ternary operation is induced by H and all operations of higher valency are equal to zero. The L ∞ -structure on multi-vector fields induced by a closed three-form appeared earlier in [20] and [21]. 5.5. Formal geometry vs. Dolbeault. Compatibility of the two constructions, one using formal geometry, the other using Dolbeault resolutions, is the subject of the next theorem.
Before embarking upon a proof of Theorem 5.6 we introduce some notations. Let Ω p,q X = J X (Ω p,q X ), Ω n X = p+q=n Ω p,q X = J X (Ω n X ). The differentials ∂ and ∂ induce, respectively, the differentials ∂ and ∂ in Ω •,• X which are horizontal with respect to the canonical flat connection. The complex Ω p,• X with the differential ∂ is a resolution of The map (5.3) extends to the map of DGLA [1] which gives rise to the map of DGLA [1]) and cohomologous cocycles determine L ∞ -quasi-isomorphic structures.
Passing to global sections we conclude that s DR (J ) j ∞ (H) and g DR (J ) B are L ∞quasi-isomorphic. Together with (5.9) this implies the claim.
6. Deformations of algebroid stacks 6.1. Algebroid stacks. Here we give a very brief overview, referring the reader to [4,12] for the details. Let k be a field of characteristic zero, and let R be a commutative k-algebra. Definition 6.1. A stack in R-linear categories C on X is an R-algebroid stack if it is locally nonempty and locally connected, i.e. satisfies (1) any point x ∈ X has a neighborhood U such that C(U ) is nonempty; (2) for any U ⊆ X, x ∈ U , A, B ∈ C(U ) there exits a neighborhood V ⊆ U of x and an isomorphism A| V ∼ = B| V .
For a prestack C we denote by C the associated stack.
For a category C denote by iC the subcategory of isomorphisms in C; equivalently, iC is the maximal subgroupoid in C. If C is an algebroid stack then the stack associated to the substack of isomorphisms iC is a gerbe.
For an algebra K we denote by K + the linear category with a single object whose endomorphism algebra is K. For a sheaf of algebras K on X we denote by K + the prestack in linear categories given by U → K(U ) + . Let K + denote the associated stack. Then, K + is an algebroid stack equivalent to the stack of locally free K opmodules of rank one.
By a twisted form of K we mean an algebroid stack locally equivalent to K + . The equivalence classes of twisted forms of K are in bijective correspondence with H 2 (X; Z(K) × ), where Z(K) denotes the center of K. To see this note that there is a canonical monoidal equivalence of stacks in monoidal categories α : i Z(K) + → Aut( K + ). Here, i Z(K) + is the stack of locally free modules of rank one over the commutative algebra Z(K) and isomorphisms thereof with the monoidal structure given by the tensor product; Aut( K + ) is the stack of auto-equivalences of K + . The functor α is given by α(a)(L) = a ⊗ Z(K) L for a ∈ Z(K) + and L ∈ K + . The inverse associates to an auto-equivalence F the Z(K)-module Hom(Id, F ). 6.2. Twisted forms of O. Twisted forms of O X/P are in bijective correspondence with O × X/P -gerbes: if S is a twisted form of O X/P , the corresponding gerbe is the substack iS of isomorphisms in S. We shall not make a distinction between the two notions. The equivalence classes of twisted forms of O X/P are in bijective correspondence with H 2 (X; O × X/P ). The composition . We denote by [S] the image in the latter space of the class of S. Let B ∈ Γ(X; Ω 2 X ⊗ J ) denote a representative of [S]. Since the map Γ(X; Ω 2 Let H ∈ Γ(X; F −1 Ω 3 X ) denote the closed form which represents the class of d d R B.

Deformations of linear stacks.
Here we describe the notion of 2-groupoid of deformations of an algebroid stack. We follow [2] and refer the reader to that paper for all the proofs and additional details.
For an R-linear category C and homomorphism of algebras R → S we denote by C ⊗ R S the category with the same objects as C and morphisms defined by For a prestack C in R-linear categories we denote by C ⊗ R S the prestack associated to the fibered category U → C(U ) ⊗ R S. Lemma 4.13). Suppose that A is a sheaf of R-algebras and C is an R-algebroid stack. Then C ⊗ R S is an algebroid stack.
Suppose now that C is a stack in k-linear categories on X and R is a commutative Artin k-algebra. We denote by Def(C)(R) the 2-category with • objects: pairs (B, ̟), where B is a stack in R-linear categories flat over R and ̟ : B ⊗ R k → C is an equivalence of stacks in k-linear categories • 1-morphisms: a 1-morphism (B (1) , ̟ (1) ) → (B (2) , ̟ (2) ) is a pair (F, θ) where F : B (1) → B (2) is a R-linear functor and θ : The 2-category Def(C)(R) is a 2-groupoid. Let B be a prestack on X in R-linear categories. We say that B is flat if for any U ⊆ X, A, B ∈ B(U ) the sheaf Hom B (A, B) is flat (as a sheaf of R-modules). Lemma 6.3 ([2], Lemma 6.2). Suppose that B is a flat R-linear stack on X such that B ⊗ R k is an algebroid stack. Then B is an algebroid stack.

Deformations of twisted forms of O.
Suppose that S is a twisted form of O X . We will now describe the DGLA controlling the deformations of S.
The main result of the present paper (Theorem 6.5 below) is a description of Def(S) (R) in terms of the L ∞ -algebra s(O X/P ) H defined in 5.4 in the situation when X is a C ∞ -manifold which admits a pair of complementary integrable complex distributions P and P. The statement of the result, which is analogous to that of Theorem 6.4, necessitates an extension of the domain of the Deligne-Getzler 2groupoid functor to nilpotent L ∞ -algebras satisfying the same vanishing condition.
To this end we set, for a nilpotent L ∞ -algebra g which satisfies g i = 0 for i ≤ −2, (6.1) MC 2 (g) := Bic Π 2 (Σ(g)), where Σ(g) is the Kan simplicial set defined for any nilpotent L ∞ -algebra (see [3] 3.2 for the definition and properties) and Π 2 is the projector on Kan simplicial sets of Duskin ([8]) which is supplied with a natural transformation Id → Π 2 . The latter transformation induces isomorphisms on sets of connected components as well as homotopy groups in degrees one and two (component by component), while higher homotopy groups of a simplicial set in the image of Π 2 vanish (component by component).
In [8] the image of Π 2 is characterized as the simplicial sets arising as simplicial nerves of bi-groupoids (see [3] 2.1.3 and 2.2) and Bic denotes the functor which "reads the bi-groupoid off" the combinatorics of its simplicial nerve, i.e., for example, the simplicial set Π 2 (Σ(g)) is the simplicial nerve of Bic Π 2 (Σ(g)).
Theorem 6.5. Suppose that X is a C ∞ -manifold equipped with a pair of complementary complex integrable distributions P and Q, and S is a twisted form of O X/P (6.2). Let H ∈ Γ(X; F −1 Ω 3 X ) be a representative of [S] (6.2). Then, for any Artin algebra R with maximal ideal m R there is an equivalence of bi-groupoids where the left-hand side is defined as in (6.1), natural in R.
Proof. We refer the reader to [3] for notations. By Theorem 5.6, s(O X/P ) H is L ∞ -quasi-isomorphic to g DR (J ) B . Proposition 3.4 of [3] implies that Σ(s(O X/P ) H ⊗ m R ) is weakly equivalent to Σ(g DR (J ) B ⊗ m R ). In particular, Σ(s(O X/P ) H ⊗ m R ) is a Kan simplicial set with homotopy groups vanishing in dimensions larger then two. By Duskin (cf. [8]), the natural transformation Id → Π 2 induces a homotopy equivalence between Σ(s(O X/P ) H ⊗ m R ) and N MC 2 (s(O X/P ) H ⊗ m R ), the nerve of the two-groupoid MC 2 (s(O X/P ) H ⊗ m R ).
On the other hand, by Theorem 3.7 (alternatively, Theorem 6.6) of [3], Σ(g DR (J ) B ⊗ m R ) is equivalent to N MC 2 (g DR (J ) B ⊗m R ). Combining all of the above equivalences we obtain an equivalence of 2-groupoids The result now follows from Theorem 6.4. Remark 6.6. In the case when P = 0, i.e. X is a plain C ∞ -manifold isomorphism classes of formal deformations of S are in bijective correspondence with equivalence classes of Maurer-Cartan elements of the L ∞ -algebra s DR (O X ) H ⊗m R . These are the formal twisted Poisson structures in the terminology of [21], i.e. elements π ∈ Γ(X; 2 T X ) ⊗ m R , satisfying the equation [π, π] = Φ(H)(π, π, π).