Deformations of symplectic groupoids

Cristian Camilo Cárdenas
Universidade Federal Fluminense, Instituto de Matemática e Estatística,Rua Prof. Marcos Waldemar de Freitas Reis, S/n, 24210-201, Niterói, RJ, Brazil

João Nuno Mestre
University of Coimbra, CMUC, Department of Mathematics, Apartado 3008,
EC Santa Cruz,
3001-501 Coimbra, Portugal

Ivan Struchiner
Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão 1010, 05508-090, São Paulo, SP, Brazil

We describe the deformation cohomology of a symplectic groupoid, and use it to study deformations via Moser path methods, proving a symplectic groupoid version of the Moser Theorem.

Our construction uses the deformation cohomologies of Lie groupoids and of multiplicative forms, and we prove that in the symplectic case, deformation cohomology of both the underlying groupoid and of the symplectic groupoid have de Rham models in terms of differential forms.

We use the de Rham model, which is intimately connected to the Bott-Shulman-Stasheff double complex, to compute deformation cohomology in several examples. We compute it for proper symplectic groupoids using vanishing results; alternatively, for groupoids satisfying homological 2-connectedness conditions we compute it using a simple spectral sequence.

Finally, without making assumptions on the topology, we constructed a map relating differentiable and deformation cohomology of the underlying Lie groupoid of a symplectic groupoid, and related it to its Lie algebroid counterpart via van Est maps.

§ INTRODUCTION

In this paper we study deformations of a symplectic groupoid $(\G,\omega)$, constructing the deformation cohomology $H_\mathrm{def}^\bullet(\G,\omega)$ controlling them. Any deformation of $(\G,\omega)$ gives rise to a deformation class $[\eta]$ in the cohomology, which can be used in Moser path methods. We carry out computations for the cohomology in examples, comparing it with cohomologies associated to related objects.

Symplectic groupoids were introduced by Karasëv [47], Weinstein [72] and Zakrzewski [75], motivated by the search for a quantization procedure for Poisson structures. They are first of all Lie groupoids, objects introduced by Ehresmann in the 1950's [36], which generalize Lie groups and which, among other uses, permit the unified study of differential geometric objects such as foliations or Lie group actions. A Lie groupoid equipped with a compatible, i.e., multiplicative symplectic structure, becomes well adapted to the study of Poisson structures. Poisson structures are Lie algebra structures on the algebra of smooth functions on a manifold, which in addition are biderivations of the associative algebra structure. Although these Lie algebras are infinite dimensional, they are on the infinitesimal side of a rich Lie theory, with symplectic groupoids (which are finite dimensional!) playing the part of the corresponding global object.

In analogy with the relation between Lie groups and Lie algebras, any symplectic groupoid $\G\rightrightarrows M$ induces a Poisson structure $\pi$ on its space of units $M$, such that the Lie algebroid of $\G$ is naturally isomorphic to the cotangent algebroid $(T^*M)_\pi$ associated to the Poisson manifold $(M,\pi)$.  Not all Poisson structures arise in this way, but for the ones which do, called integrable, it was shown in in [62] that there is always a natural symplectic groupoid structure on the source $1$-connected integration $\G$ of $(T^*M)_\pi$. The symplectic groupoid structure on $\G$ was also obtained by an infinite dimensional reduction procedure in [15], and the characterization of which Poisson manifolds are integrable was finally settled in [20].

The survey [73] gives an account of Poisson geometry at the early stages of development of symplectic groupoids, and already then they feature prominently in the results and the problems presented, some of them now settled, some still open.

First and foremost (from our specific viewpoint) symplectic groupoids are useful in the study of the Poisson geometry of the base, both in its local and global properties.
But we should mention that symplectic groupoids have also feature in a variety of other treatments, for example in geometric quantization [43], deformation quantization (cf. e.g. [9, 15]), the study of the geometry of varieties appearing in Poisson-Lie theory (for example [53]) and of moduli spaces of flat connections (e.g. [51, 71]).

Particulary motivating for this work, was the role played by symplectic groupoids, both in the study of local and of global Poisson geometry,  via Moser path arguments, also known as “the Moser trick”. We mention how it came to influence our work. Starting with the famous Moser Theorem, the Moser trick has been used to prove many rigidity and normal form results in symplectic geometry, and related topics (see e.g. [11]), by reducing deformation problems to cohomological conditions. It was first used in the context of Lie groupoids, to our knowledge, by Weinstein [74], in the pursuit of linearization theorems for proper Lie groupoids. Next, in the geometric proof of Conn's Theorem of [21], a Poisson version of the Moser Theorem was used, together with symplectic groupoids that solved the cohomological conditions. Further, similar techniques were used around leaves of a Poisson manifold [27]. These techniques eventually led to the geometric proof of the linearization of proper groupoids around orbits [29], and to the development of the deformation cohomology $H^\bullet_\mathrm{def}(\G)$ of an arbitrary Lie groupoid in [25].

Recent developments related to deformation theory for Lie groupoids include a rigidity result for foliations [32], the study of deformations of vector bundles over Lie groupoids [49], and of morphisms of Lie groupoids [12]. Specific classes of deformations of Lie groupoids (deformations to the normal cone) are also of importance in noncommutative geometry (see e.g. [30, 31]).

Our work brings this new understanding on deformations of Lie groupoids full circle back to Poisson geometry, as exemplified by a symplectic groupoid version of the classical Moser Theorem (detailed further in Sections 7 and 8).

Let $(\widetilde{\calG},\omega)$ be a deformation of a compact symplectic groupoid $(\calG,\omega_\calG)$ and let $[\eta]$ be the deformation class in $H^2_\mathrm{def}(\G, \omega)$ associated to it. Then, the deformation $(\widetilde{\calG},\omega)$ is trivial if and only if $[\eta]$ is exact.

This result also leads to local descriptions of moduli spaces of multiplicative symplectic forms on a compact Lie groupoid $\G$, given by an analogue of a Kodaira-Spencer map (See Theorem <ref> for a more precise and detailed version).

Let $(\G,\omega)$ be a compact symplectic groupoid, and denote by $\mathcal{S}({\G})$ the space of multiplicative symplectic structures on $\G$.

Then there is a neighbourhood $\mathcal{U}$ of $\omega$ in $\mathcal{S}({\G})$ and a neighbourhood $\mathcal{V}$ of $0$ in $H^2_\mathrm{def}(\G,\omega)$, such that there is an explicit 1:1 correspondence
\[\kappa:\ \mathcal{U}\big/\sim\ \  \stackrel{\mathrm{1:1}}{\longrightarrow} \  \mathcal{V}\subset H^2_\mathrm{def}(\G,\omega),\]
sending the equivalence class of $\omega$ to $0$. The equivalence relation is such that $\omega_1 \sim \omega_2$ if $\omega_1+\e(\omega_2-\omega_1)$ is a trivial family of multiplicative forms for all $\e$.

The key insight in order to construct the deformation cohomology $H^2_\mathrm{def}(\G, \omega)$, where the cohomological equations of the Moser Theorem naturally live, was that the pair of a Lie groupoid and an extra structure on it is more manageable when interpreted as a diagram. Then the new tools on deformations of morphisms of Lie groupoids from [12] could be used.

Moreover, in the specific case where the extra structure is symplectic, there is a description in terms of differential forms on the nerve $\G^{(\bullet)}$ of $\G$, of $H^\bullet_\mathrm{def}(\G)$, and of $H^\bullet_\mathrm{def}(\G, \omega)$. We call the latter the de Rham model for deformation cohomology of the symplectic groupoid $(\G,\omega)$, and we summarize the results here (see Proposition <ref> and Theorem <ref> for more detailed versions).

Let $(\calG,\omega)$ be a symplectic groupoid. Then $\omega^{b}$ induces a quasi-isomorphism
\[C^{\bullet}_\mathrm{def}(\calG)\sim \Omega^{1}(\calG^{(\bullet)}).\]

The deformation complex $C^{\bullet}_\mathrm{def}(\G,\omega)$ of the symplectic groupoid $(\G,\omega)$ is isomorphic to the mapping cone of the de Rham differential
\[-d_{dR}:\Omega^{1}_\mathrm{def}(\G^{(\bullet)})\to\Omega_\mathrm{cl}^{2}(\G^{(\bullet)}).\]

In order to effectively use a symplectic groupoid to study the Poisson structure on the base, it is desirable to have a good description of the symplectic groupoid, or at least to know some of its properties. There are conditions to characterize whether a Poisson manifold is integrable [20] and if it is, there is an infinite-dimensional reduction procedure for the construction of the source $1$-connected integration [15]. Nonetheless, constructing a concrete integration is often still challenging.

Fortunately, symplectic groupoids for several classes of Poisson manifolds have been described, usually immediately bringing about new insight about the Poisson structures they integrate; we mention a few of them. Some of the simplest cases are symplectic manifolds, the zero Poisson structure and linear Poisson structures on vector spaces and vector bundles [17]. Even for the zero Poisson structure, symplectic groupoids may provide rich geometry in the form of integral affine structures, as shown recently in [23, 24]. Integrations for any Poisson-Lie group are described in [52, 54], and these and other symplectic groupoids also have interpretations in terms of moduli spaces of flat connections, for example in [51, 71]. Integrations of Poisson fibrations are studied in [4]; integrations of Log-symplectic manifolds are studied using blow-up and gluing techniques in [42]; a description of the integration of neighborhoods of Poisson transversal submanifolds is given in [38]; more examples appear in relation to the theory of Poisson-Lie groups - for example, symplectic leaves of double Bruhat cells are symplectic groupoids studied in [53]; and this list is far from complete.

We carried out computations for the deformation cohomology in a few of these examples. Most computable examples $(\G,\omega)$ are typically of one of two kinds: either we assumed some sort of compactness condition, or vanishing of $H^1_\mathrm{dR}(\G)$ and $H^2_\mathrm{dR}(\G)$. Clearly we cannot ask for both  vanishing $H^2_\mathrm{dR}(\G)$ and compactness, because we deal with symplectic groupoids. In fact, even examples of compact symplectic groupoids with the more reasonable condition of having simply connected source fibres are very hard to find [24, 63, 77].

Asking for the symplectic groupoid to be proper (a compactness-type condition), we obtained the following result (See Section 6 for more a detailed version of it), and applied it for linear Poisson and zero Poisson structures of proper type.

Let $(\G, \omega)$ be a proper symplectic groupoid, and let $\nu$ denote the normal distribution to the symplectic leaves. Then there is a 7-term exact sequence

\begin{align*}
0  & \to \Omega^1_\mathrm{cl}(M)^{\G-\mathrm{inv}} \to \Gamma(\nu^*)^{\G-\mathrm{inv}} \to \Omega^2_\mathrm{cl}(M)^{\G-\mathrm{inv}} \to H^{1}_\mathrm{def}(\G, \omega) \to \\
&  \to \Gamma(\nu)^{\G-\mathrm{inv}}\to H_\mathrm{def}^{1}(\omega)\to H^{2}_\mathrm{def}(\G, \omega)\to 0
\end{align*}

and $H^{k}_\mathrm{def}(\G,\omega)\cong H^{k-1}_\mathrm{def}(\omega)$ for $k> 2$.

In particular, expressed in terms of the de Rham model,
\[H^{2}_\mathrm{def}(\G, \omega)\cong H_\mathrm{def}^{1}(\omega)/d(H^1_\mathrm{def}(\G)).\]

With the help of some simple spectral sequence arguments we also provide descriptions of the deformation cohomology of an arbitrary symplectic groupoid. Those descriptions, however, become much simpler in the case of proper symplectic groupoids, as detailed in Sections <ref> and <ref>.

If on the other hand we ask for moderately strong topological conditions - vanishing of first and second de Rham cohomologies, we see a very close relation between deformation cohomology of $(\G,\omega)$ and Poisson cohomology $H_\pi(M)$ of the base. Poisson cohomology of $(M,\pi)$ is computed by the complex $\mathfrak{X}^\bullet(M)$ of multivector fields on $M$, with differential $d_\pi=[\pi,\cdot]_{SN}$, i.e., given by the Schouten-Nijenhuis bracket with $\pi$. It is the cohomology controlling deformations of the Poisson structure $\pi$, and in general it is hard to compute. We refer to [66] for an account of techniques that can be used to compute it in different classes of Poisson manifolds, as well as for the very recent developments in loc. cit.

Call a Lie groupoid $\G$ homologically $2$-connected if all the spaces $\G^{(n)}$ of its nerve, as well as its source-fibres, are homologically $2$-connected.
In this situation we obtained the following result (Section 9 contains precise and improved versions).

Let $(\G,\omega) \rightrightarrows (M,\pi)$ be a homologically $2$-connected symplectic groupoid.
\[H_\mathrm{def}^0(\G,\omega)\cong H^0_\pi(M)/\mathbb{R}, \ \ \ H_\mathrm{def}^1(\G,\omega)\cong H^1_\pi(M)\ \ \ H_\mathrm{def}^2(\G,\omega)\cong H^2_\pi(M),\]
and there is an injective map
\[H_\mathrm{def}^{3}(\G,\omega)\to H^{3}_\pi(M).\]

Although the conditions do sound restrictive, we focus in Section 9 on some situations in which they are satisfied, for example in the context of linear Poisson structures, Poisson-Lie groups, and cotangent VB-groupoids.

As a general result without further assumptions on topology of $(\G,\omega)$ other than connectivity of the source fibres, we construct a map $i_\G: H^\bullet(\G)\to H^\bullet_\mathrm{def}(\G)$ as a global counterpart to the map $i: H^\bullet_\pi(M)\to H^\bullet_\mathrm{def}(A)$ of [26] and prove the following (more details, and the statement at the level of cochains in Theorem <ref>).

Let $(\calG,\omega)$ be a $s$-connected symplectic groupoid. Then  $i_{\calG}$ together with $i$ and the van Est maps for differentiable and deformation cohomology form the commutative diagram.

\begin{equation*}
\begin{tikzcd}[column sep=4em, row sep=10ex]
H^{k}(\calG) \MySymb[\circlearrowright]{dr} \arrow{d}[swap]{VE} \arrow{r}{i_{\calG}} & H^{k}_\mathrm{def}(\calG) \arrow{d}{VE_\mathrm{def}}\\
H^{k}(T^{*}M) \arrow{r}{i} & H^{k}_\mathrm{def}(T^{*}M).
\end{tikzcd}
\end{equation*}

We would like to make some final remarks.
We believe that the description of the deformation complex of a symplectic groupoid is important by itself, because this work should serve as prototype. It opens the road to similar studies for other structures on groupoids, which should shed light on deformation theoretic properties of objects described by them. We expect that similar techniques can be used, but revealing different phenomena, for multiplicative contact, Poisson, complex structures, foliations, etc.

The theory developed in this paper should have a version for local and formal symplectic groupoids, likely using the version of the Bott-Shulman-Stasheff for local Lie groupoids [50]. In this direction, it would be interesting to study the classes of deformations of formal symplectic groupoids [14] induced by star products arising from quantization, or pairs of star products, as in the works of [9] and [45, 46]. We expect that local and formal versions automatically satisfy many of the topological assumptions we used in computations, bringing the theory closer to the deformations of the underlying Poisson structures.

Finally, all the work developed in this paper is intimately connected with the Bott-Shulman-Stasheff double complex [2], therefore lending itself to generalizations towards twisted presymplectic, and higher (shifted) (pre)symplectic groupoids.

§.§ Outline of the paper

In Section 2 we recall needed background material about Lie groupoid cohomology and deformations, VB-groupoids, multiplicative forms and symplectic groupoids, and Poisson manifolds.

In Section 3 we introduce deformations and equivalence of deformations of symplectic groupoids, and provide examples of trivial and non-trivial deformations.

In Section 4 we explain how the symplectic form of a symplectic groupoid $(\G,\omega)$ permits a description of deformation cohomology of $\G$ in terms of differential forms.

In Section 5 we come to the central object of this paper, the deformation cohomology $H_\mathrm{def}(\G, \omega)$ of a symplectic groupoid. We explain how to construct it out of the deformation cohomologies of the $\G$ and of $\omega$. We also prove that there is a de Rham model for it, in which cochains are differential forms, used for the computations in the rest of the paper.

In Section 6 we carry out the first computations of deformation cohomology, for symplectic manifolds, and proper symplectic groupoids; we then specialize to zero Poisson structures of proper type and linear Poisson structures of proper type. Using simple spectral sequence arguments we present some explicit computations for general symplectic groupoids.

In sections 7 and 8 we use the deformation cohomology to prove symplectic groupoid versions of the Moser theorem, first for specific (source-constant), and then for general deformations. We also study consequences moduli spaces of multiplicative symplectic forms.

In Section 9 we use another spectral sequence to carry out more computations of deformation cohomology, this time for symplectic groupoids for which the nerve has vanishing first and second de Rham cohomology. The relation with Poisson cohomology is very close in these cases; we include computations for linear Poisson structures, the zero Poisson structure, Poisson-Lie groups, and cotangent groupoids.

in Section 10 we study, now without topological assumptions on $(\G, \omega)$, relations between the differentiable and deformation cohomology of $\G$, the Poisson cohomology of the base, and the deformation cohomology of the corresponding algebroid. We do so by providing a map $i_\G: H^\bullet(\G)\to H^\bullet_\mathrm{def}(\G)$ and proving that it is related by van Est maps to the map $i: H^\bullet_\pi(M)\to H^\bullet_\mathrm{def}(A)$ of [26].

Finally, we collect in an appendix the needed material on double structures, used to prove the Theorem of Section 10.

§.§ Notation and conventions

We denote a Lie groupoid $\G$ over a base $M$ by $\G\rightrightarrows M$, usually having source and target maps denoted by $s$ and $t$, multiplication denoted by $m$, inversion map $i$ and unit map $u$.

We say that a Lie groupoid $\G$ is $s$-connected (respectively $s$-$k$-connected) if all fibres of its source map are connected (respectively $k$-connected).

We denote the tangent map of a differentiable map $f:M\to N$ either by $df$ or $Tf$.
To avoid confusion with other cohomologies, we denote de Rham cohomology of a manifold $M$ by $H^\bullet_{dR}(M)$.

§.§ Acknowledgements

The authors would like to thank Alejandro Cabrera, Rui Loja Fernandes, Ioan Mărcuţ, Cristián Ortiz, and Luca Vitagliano for useful discussions related to the work of this paper.

C.C.C. was partially supported by a PNPD postdoctoral fellowship at UFF, and would also like to thank the University of São Paulo, where this work was initiated.
J.N.M. was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES, and by grant 2019/20789-4, São Paulo Research Foundation (FAPESP). I.S. was partially supported by FAPESP 2015/22059-2 and CNPq 307131/2016-5.

§ BACKGROUND

We start by recalling some background on Lie groupoids: their cohomology and deformations, VB-groupoids, and symplectic groupoids. For a general introduction to Lie groupoids we refer to [22, 59, 65].

§.§ Deformations of Lie groupoids

More details on the notions in this section can be found in [25], where  they are discussed in detail.
A family of Lie groupoids over a manifold $B$ consists of a Lie groupoid $\widetilde{\calG}\arrows\tilde{M}$, together with a submersion $\textbf{p}:\tilde{M}\to B$, such that $\textbf{p}\circ \tilde{s}=\textbf{p}\circ \tilde{t}$.

In other words, a family of Lie groupoids over $B$ consists of a submersive Lie groupoid map $\textbf{p}:(\widetilde{\calG}\arrows \tilde{M})\to(B \arrows B)$, from $\widetilde{\calG}$ to the unit groupoid of $B$. We denote by $\calG_b\arrows M_b$ the fibre of $\textbf{p}$ over $b\in B$, i.e. $(\textbf{p}\circ \tilde{s})^{-1}(b)\arrows \textbf{p}^{-1}(b)$.

An equivalence between families $\widetilde{\calG}$ and $\widetilde{\calG}'$ over $B$ is given by a Lie groupoid isomorphism $F:\widetilde{\calG}\to\tilde{\calG'}$ such that $\textbf{p}'\circ F=\textbf{p}$.
A family is said to be trivial if it is equivalent to the product family given by
\[(\widetilde{\calG}\arrows \tilde{M})\times(B \arrows B)\stackrel{\textbf{p}=pr_B}{\longrightarrow}(B \arrows B).\]

A deformation of a Lie groupoid $\calG$ is a family of Lie groupoids $\widetilde{\calG}\arrows\tilde{M}$ over an open interval $I$ containing zero, such that $\calG_0$ is equal to $\calG$.

An equivalence between deformations $\widetilde{\calG}$ and $\widetilde{\calG}'$ of $\calG$ is given by an equivalence of families $F:\widetilde{\calG}_{|_J}\to\widetilde{\calG}_{|_J}'$ such that $F_0:=F_{|_{\calG_0}}=id_{\calG_0}$, where $J\subset I\cap I'$ is an open interval containing zero, and $\widetilde{\calG}_{|_J}$ denotes the restriction of the family $\widetilde{\calG}$ to the interval $J$.

A deformation $\widetilde{\G}$ of a Lie groupoid $\calG$ for which $\widetilde{\G}=\G\times I$ as smooth manifolds is called a strict deformation.

It is called an $s$-constant deformation if additionally the source map is left constant, $i.e.$, $s_\varepsilon=s_0$ for all $\e\in I$.

A deformation of a Lie groupoid $\G$ determines a collection $\{\G_\epsilon\}$ of Lie groupoids parametrized by $\epsilon\in I$ (the fibres of the family), varying smoothly with respect to $\epsilon$ in the sense that they fit as fibres of a submersion. If the deformation is strict then the $\G_\e$ are equal as manifolds, and only the structural maps change.

Accordingly, we often use denote a deformation of $\G$ simply by $\{\G_\epsilon\}$, keeping in mind that there is a specified family that the members of this family fit into.

Similarly, we often denote an equivalence $F:\widetilde{\G}\to\widetilde{\G}'$ of deformations by the associated family of maps $F_\epsilon:\G_\epsilon\to\G_\epsilon '$.

§.§ The nerve of a Lie groupoid and differentiable cohomology

Let $\G\arrows M$ be a Lie groupoid and denote its space of strings of $k$ composable arrows by

\[\calG^{(k)} =\{(g_1,\ldots,g_k) : s(g_i) = t(g_{i+1}) \text{ for all } 1\leq i\leq k-1\},\]

and let $\G^{(0)}=M$. The nerve of $\G$ is the simplicial manifold $\G^{(\bullet)}$ for which the space of $k$-simplices is $\calG^{(k)}$, the face maps are $d_i:\G^{(n)}\to\G^{(n-1)}$ given by

\[d_i(g_1,\ldots,g_n) =
\begin{cases}
(g_2,\ldots,g_n) & \text{ if }  i=0\\
(g_1,\ldots,g_i g_{i+1}, \ldots,g_n)& \text{ if }   1\leq i\leq n-1\\
(g_1,\ldots,g_{n-1})& \text{ if } i=n,\end{cases}
\]
and degeneracies $s_i:\G^{(n)}\to\G^{(n+1)}$ are given by inserting the appropriate unit in the $i$-th position, $s_i(g_1,\ldots,g_n)=(g_1,\ldots,g_{i-1},1_,g_i,\ldots, g_n)$, for $1\leq i\leq n+1$.

Smooth functions on the nerve form a cosimplicial space, out of which we can construct the cochain complex $C^\bullet_\mathrm{diff}(\G)$ (also denoted just by $C^\bullet(\G)$) computing the differentiable cohomology of $\G$, denoted by $H^\bullet_\mathrm{diff}(\G)$. Explicitly, its spaces of cochains are $C^k_\mathrm{diff}(\G)=C^\infty(\G^{(k)})$, and the differentials are given by
\[\delta=\sum_{i=0}^{k+1} (-1)^i d_i^*: C^k_\mathrm{diff}(\G)\to C^{k+1}_\mathrm{diff}(\G).\]
The subcomplex of normalized cochains consists of those cochains $c\in C^\bullet_\mathrm{diff}(\G)$ which vanish on all degeneracies, i.e., $s_i^*c=0$ for all $i$; it is quasi-isomorphic to $C^\bullet_\mathrm{diff}(\G)$.

More generally, $k$-forms on the nerve of $\G$ together with a differential $\delta$ defined by the same formula form the cochain complex $\Omega^k(\G^{\bullet})$. These complexes form the lines of the Bott-Shulman-Stasheff double complex $\left(\Omega^{\bullet}(\calG^{(\bullet)}),\delta, d\right)$ [2]; the horizontal differentials $\delta:\Omega^{q}(\calG^{(\bullet)})\longrightarrow\Omega^{q}(\calG^{(\bullet+1)})$ are determined by the simplicial structure of the nerve of $\calG$, as described above, and vertical differentials $d:\Omega^{\bullet}(\calG^{(p)})\longrightarrow\Omega^{\bullet+1}(\calG^{(p)})$ are given by the de Rham differential of forms.

§.§ The deformation complex of a Lie groupoid

We recall the definition from [25] of the deformation complex of a Lie groupoid $\calG$.
To that end, let $\bar{m}$ denote the division map  of $\calG$, defined as

\[\bar{m}(p,q) = pq^{-1}, \text{ for all } p,q \in \calG\ \text{ such that }\ s(p) = s(q).\]

The deformation complex $(C_\mathrm{def}^\bullet(\mathcal{G}),\delta)$ of a Lie groupoid $\calG$, whose cohomology is denoted $H_\mathrm{def}^\bullet(\calG)$, is defined as follows. For $k\geq 1$, the $k$-cochains $c\in C_\mathrm{def}^k(\mathcal{G})$ are the smooth maps

\[ c:\calG^{(k)}\longrightarrow T\calG, \ \ (g_1,\ldots,g_k)\mapsto c(g_1,\ldots,g_k)\in T_{g_1}\calG,\]

which are $s$-projectable in the sense that

\[ ds\circ c(g_1,g_2,\ldots,g_k)\]

does not depend on $g_1$.

The differential of $c\in C_\mathrm{def}^k(\mathcal{G})$ is defined by

\begin{align*}(\delta c)(g_1,\ldots,g_{k+1})  = &- d\bar{m}(c(g_1g_2,\ldots,g_{k+1}),c(g_2,\ldots,g_{k+1}))  \\
&+\sum_{i=2}^k(-1)^{i} c(g_1,\ldots, g_ig_{i+1},\ldots, g_{k+1}) +(-1)^{k+1}c(g_1,\ldots, g_k).
\end{align*}

For $k= 0$, $C_\mathrm{def}^0(\mathcal{G}): = \Gamma(A)$ and the differential of $\alpha\in \Gamma(A)$ is defined by

\[ \delta(\alpha)=\overrightarrow{\alpha}+\overleftarrow{\alpha}\in C_\mathrm{def}^1(\mathcal{G}),\]

where $\overrightarrow{\alpha}$ and $\overleftarrow{\alpha}$ are the right-invariant and the left-invariant vector fields on $\G$ induced by $\alpha$.

Note that, to interpret the case $k= 0$, one way to think of a section of $A$ is as a map $c:\calG^{(0)}=M\longrightarrow T\calG$, with $c(1_x)=c_x \in T_{1_x}\calG$, such that $ds\circ c_x=0_x$.

§.§ The deformation class of a deformation of Lie groupoids

We recall the definition from [25] of the deformation class of a deformation $\widetilde{\G}\arrows M\stackrel{\textbf{p}}{\to} I$ of a Lie groupoid $\G$.

If $\widetilde{\G}$ is an $s$-constant deformation, then the expression

\begin{equation}\label{def-s-constant-cocycle}
\xi(g,h):=\left.\frac{d}{d\e}\right|_{\e=0}\bar{m}_\e(gh,h)
\end{equation}

defines a $2$-cochain in deformation cohomology. It is a cocycle and its cohomology class only depends on the equivalence class of the deformation <cit.>. Moreover, If we let $\tilde{\xi}=-\delta(\frac{\partial}{\partial\e})\in C_\mathrm{def}^1(\widetilde{\G})$, then $\xi=\tilde{\xi}_{|\G_0}$ <cit.>.

The expression (<ref>) no longer makes sense if the deformation $\widetilde{\G}$ is not $s$-constant, but defining $\xi$ in terms of $\tilde{\xi}$ is still possible. For that, we need an appropriate analogue for the vector field $\frac{\partial}{\partial\e}$ on $\widetilde{\G}$.

Let $\widetilde{\G}\arrows M\stackrel{\textbf{p}}{\to} I$ be a deformation of a Lie groupoid $\G$. A transverse vector field for $\widetilde{\G}$ is a vector field $\tilde{X}\in\mathfrak{X}(\widetilde{\G})$ which is $s$-projectable to a vector field $\tilde{V}\in\mathfrak{X}(\tilde{M})$, which is in turn $\textbf{p}$-projectable to $\frac{\partial}{\partial\e}\in\mathfrak{X}(I)$.

Let $\widetilde{\G}$ be a deformation of $\G$. Then

* There exist transverse vector fields for $\widetilde{\G}$;

* For  $\tilde{X}\in\mathfrak{X}(\widetilde{\G})$ transverse, the restriction of $\delta\tilde{X}$ to $\G_0$ defines a cocycle $\xi_0\in C^2_\mathrm{def}(\G)$;

* The cohomology class of $\xi_0$ does not depend on the choice of $\tilde{X}$.

The resulting cohomology class $[\xi_0]\in H^2_\mathrm{def}(\G)$ is called the deformation class associated to the deformation $\widetilde{\G}$ of $\G$.

Again, the deformation class is invariant under equivalence of deformations, and as remarked in [49], this follows already from the independence of the chosen transverse vector field.

Note that $\widetilde{\G}$ can also be seen as a deformation of its other fibres $\G_\e$, and for such, the deformation class will be $[\xi_\e]=[\delta\tilde{X}_{|\G_\e}]$.

§.§ Deformations of Lie groupoid morphisms and diagrams

We recall some background on deformations of Lie groupoid morphisms, which are studied in detail in [12].

Let $F:\G\to\mathcal{H}$ be a morphism of Lie groupoids. A deformation of the morphism $F$ is a smooth map $\tilde{F}: \G\times I \to \mathcal{H}$ such that $\tilde{F}(\cdot, 0)=F$ and for each $\epsilon\in I$, the map $F_\epsilon: \mathcal{G}\to \mathcal{H}$ is a morphism.

More generally, and this will be useful in our study, one can at once deform the whole diagram $F: \G \to \mathcal{H}$, which for simplicity we denote by $F$ as well.

A deformation of the diagram $F$ consists of a triple $(\widetilde{\G},\tilde{F},\tilde{\mathcal{H}})$, where $\widetilde{\G}$ and $\tilde{\mathcal{H}}$ are deformations of $\G$ and $\mathcal{H}$, and the map $\tilde{F}:\widetilde{\G}\to \tilde{\mathcal{H}}$ is a Lie groupoid morphism sending $\G_\epsilon$ to $\mathcal{H}_\epsilon$, such that $F_0:=\tilde{F}_{|_{\G_0}} = F$.

Note that the requirement that $\tilde{F}$ is a morphism implies in particular that each $F_\epsilon:=\tilde{F}_{|_{\G_\epsilon}}:\G_\epsilon\to \mathcal{H}_\epsilon$ is a Lie groupoid morphism as well.

For the purposes of this paper we will be interested in simultaneous deformations of the Lie groupoid $\G$ and the morphism $F$, or in other words, deformations of the diagram $F$ where $\mathcal{H}$ is kept fixed.

§.§ The deformation complex of a Lie groupoid morphism

The deformation complex $C^\bullet_\mathrm{def}(F)$ of a Lie groupoid morphism $F:\G\to \mathcal{H}$ was briefly described in [25] and further explored in [12]. It has the following description, similar to that of $C^\bullet_\mathrm{def}(\G)$.
For $k\geq 1$,
\[C_\mathrm{def}^k(F)=\left\{c:\G^{(k)}\to T\mathcal{H}\ \middle|\ c(g_1,\ldots,c_k)\in T_{F(g_1)}\mathcal{H}\ \text{and}\ c \ \text{is}\ s_\mathcal{H}-\text{projectable} \right\},\]
where $c$ being
$s_\mathcal{H}-$projectable means that $ds_\mathcal{H}\circ c(g_1, \ldots, g_k)$ is independent of $g_1$.

The differential of $c$ is defined by the same formula as the differential for $C^*_\mathrm{def}(\G)$ (see Definition <ref>), except using the division $d\bar{m}_\mathcal{H}$ of $T\mathcal{H}$ instead of the division $d\bar{m}$ of $T\G$.

In degree 0, we have $C_\mathrm{def}^0(F)=\Gamma({F^*A_\mathcal{H}})$, and $\delta \alpha = \overrightarrow{\alpha}+\overleftarrow{\alpha}\in C_\mathrm{def}^1(F)$, where $\overrightarrow{\alpha}(g)=dR_{F(g)}(\alpha_{t(g)})$ and $\overleftarrow{\alpha}(g)=dL_{F(g)}(di (\alpha_{s(g)}))$.

There is a natural cochain complex map $F_*:C^\bullet_\mathrm{def}(\G)\to C^\bullet_\mathrm{def}(F)$, given by $F_*c(g_1,\ldots,g_k):=dF_{g_1}(c(g_1,\ldots,g_k))$.

Analogously, for example, to the cases of deformations of  morphisms of associative algebras [39], of Lie algebras [37], and of deformations of holomorphic maps (see [44] and references therein), it is shown in [12] that simultaneous deformations of $\G$ and $F$ are controlled by the mapping cone complex of $F_*$,
\[\text{Cone}(F_*)^n=C^{n}_\mathrm{def}(\G)\oplus C^{n-1}_\mathrm{def}(F),\ \ \delta(c,Y)=(\delta c, F_*c-\delta Y).\]

§.§ VB-groupoids

VB-groupoids can be thought of as groupoid objects in the category of vector bundles. They provide alternative ways to look at the representation theory and the deformation theory of Lie groupoids (See [25], [41] and Proposition <ref> below).
We review some basics on VB-groupoids which will be useful in the following sections. For more details we refer to [59], [41] and [6].

A VB-groupoid $(\Gamma, E, \calG, M)$ consists of two Lie groupoids together with two vector bundle structures as in the diagram

\begin{equation}\label{VB-grpd}
\begin{tikzcd}[column sep=4em, row sep=10ex]
\Gamma \arrow[yshift=2pt]{r}{\tilde{s}} \arrow[yshift=-2pt]{r}[swap]{\tilde{t}} \arrow{d}[swap]{\tilde{q}} & E \arrow{d}{q}\\
\calG \arrow[yshift=2pt]{r}{s} \arrow[yshift=-2pt]{r}[swap]{t}    & M,
\end{tikzcd}
\end{equation}

where $\tilde{q}$ and $q$ are vector bundles and $(\Gamma\rightrightarrows E)$, $(\calG\rightrightarrows M)$ are Lie groupoids such that the structure maps of the groupoid $\Gamma$
are vector bundle morphisms over the corresponding structure maps of the groupoid $\calG$.

A morphism of VB-groupoids
\[(\phi_\Gamma,\phi_E,\phi_\calG,\phi_M):(\Gamma, E, \calG, M)\longrightarrow (\Gamma', E', \calG', M')\]
is a Lie groupoid map $(\phi_\Gamma,\phi_E):\Gamma\rightrightarrows E\to\Gamma'\rightrightarrows E'$, such that $\phi_\Gamma$ and $\phi_E$ are vector bundle morphisms covering $\phi_\calG:\calG\longrightarrow \calG'$ and $\phi_M:M\longrightarrow M'$, respectively. Restricting $\phi_\Gamma$ to the zero section, $\phi_\calG$ turns out to be a Lie groupoid morphism.

The requirement that $m_\Gamma:\Gamma^{(2)}\longrightarrow\Gamma$ is a vector bundle morphism, is taken with respect to the vector bundle structure of $\Gamma^{(2)}$ over $\calG^{(2)}$.

(Tangent VB-groupoid)

Given a Lie groupoid $\calG\rightrightarrows M$ with source, target and multiplication maps $s$, $t$ and $m$, by applying the tangent functor one gets the tangent groupoid $T\calG\rightrightarrows TM$ with structure maps $ds$, $dt$, $dm$ and so on. This tangent groupoid is further a VB-groupoid over $\calG\rightrightarrows M$ (with respect to the tangent projections).

Note that in the previous example one has the following short exact sequences of vector bundles over $\calG$,

\begin{equation}\label{left}
s^{*}(A_\calG)\stackrel{-l\circ di}{\longrightarrow} T\calG\stackrel{(dt)^{!}}{\longrightarrow}t^{*}(TM)
\end{equation}

\begin{equation}\label{right}
t^{*}(A_\calG)\stackrel{r}{\longrightarrow} T\calG\stackrel{(ds)^{!}}{\longrightarrow}s^{*}(TM)
\end{equation}

where $r$ and $l$ are the right and left multiplication on vectors tangent to the $s$-fibres and $t$-fibres of $\calG$, respectively; $(ds)^{!}$ and $(dt)^{!}$ are the maps induced by $ds$ and $dt$ with image in the corresponding pullback bundles.

(Cotangent groupoid)
As noticed in [17], the cotangent bundle of a Lie groupoid $\calG$ inherits a groupoid structure over the dual of the Lie algebroid of $\calG$,

$$T^{*}\calG\rightrightarrows A_\calG^{*},$$

with source and target maps induced, respectively, from the dual of the exact sequences (<ref>) and (<ref>). Explicitly, for $\alpha_g\in T^{*}_{g}\calG$ and $a\in\Gamma(A_\calG)$,

$$\left\langle \tilde{s}(\alpha_g),a_{s(g)}\right\rangle=-\left\langle \alpha_g, l_g\circ di(a)\right\rangle$$

$$\left\langle \tilde{t}(\alpha_g), a_{t(g)}\right\rangle=\left\langle \alpha_g, r_g(a)\right\rangle.$$

With multiplication determined by
$$\left\langle \tilde{m}(\alpha_g,\beta_h),Tm(v_g,w_h)\right\rangle=\left\langle \alpha_g,v_g\right\rangle+\left\langle \beta_h,w_h\right\rangle,$$

for $(v_g.w_h)\in (T\calG)^{(2)}$.

VB-groupoid cohomology
Let $\Gamma$ be a VB-groupoid. The differentiable complex of $\Gamma$ (as Lie groupoid) has a natural subcomplex $C^\bullet_\mathrm{lin}(\Gamma)$ given by the fibrewise linear cochains of $\Gamma$.

The VB-groupoid complex  $C^\bullet_\mathrm{VB}(\Gamma)$ of $\Gamma$ ([41]), is a subcomplex of $C^\bullet_\mathrm{lin}(\Gamma)$ which takes into account simultaneously the Lie groupoid and vector bundle structures of $\Gamma$. It consists of the left-projectable elements, that is, those $c\in C^\bullet_\mathrm{lin}(\Gamma)$ for which

* $c(0_{g_1},\gamma_{g_2},...,\gamma_{g_k})=0$,

* $c(0_g\cdot\gamma_{g_1},\gamma_{g_2},...,\gamma_{g_k})=c(\gamma_{g_1},\gamma_{g_2},...,\gamma_{g_k})$.

This complex turns out to compute the cohomology of the base groupoid with coefficients in a representation up to homotopy. For the case of the tangent VB-groupoid it  yields an interpretation of the adjoint representation ([41]). We will make use of the following fact.

The deformation complex of a Lie groupoid $\calG$ is isomorphic to the VB-groupoid complex of its cotangent VB-groupoid, $C^\bullet_\mathrm{VB}(T^{*}\calG)$. The isomorphism is given by $C^\bullet_\mathrm{def}(\calG)\longrightarrow C^\bullet_\mathrm{VB}(T^{*}\calG)$, $c\mapsto c'$ with

$$c'(\eta_{g_1},...,\eta_{g_k})=\left\langle\eta_{g_1}, c(g_1,...,g_k)\right\rangle.$$

We recall also the following result of Cabrera and Drummond, that will let us substitute the VB-complex by the complex of linear cochains for most purposes.

Let $(\Gamma, E, \calG, M)$ be a VB-groupoid. The inclusion $\iota:C^\bullet_\mathrm{VB}(\Gamma) \hookrightarrow C^\bullet_\mathrm{lin}(\Gamma)$ induces an isomorphism of right $H^\bullet(\calG)$-modules in cohomology.

§.§ Multiplicative forms, symplectic groupoids, and Poisson manifolds

Given a Lie groupoid $\G$, a differential $k$-form $\alpha\in\Omega^k(\G)$ is called multiplicative if it satisfies
\[m^*\alpha=pr_1^*\alpha + pr_2^*\alpha,\]
where $m$ is the multiplication of $\G$ and $pr_1,pr_2:\G^{(2)}\to\G$ are the two projections. We denote by $\Omega^\bullet_\mathrm{mult}(\G)\subset \Omega^\bullet(\G)$ the subcomplex of multiplicative forms on $\G$.

Multiplicativity can be expressed in yet a few other equivalent ways, some of which will be useful in this study. By definition, multiplicativity of $\alpha$ amounts to $\delta\alpha=0$, where $\delta=pr_1^* - m^* + pr_2^*$ is the horizontal differential of the first column in the Bott-Shulman-Stasheff double complex (cf. Section <ref>).

Moreover, note that to a $k$-form $\alpha\in \Omega^k(\G)$ one can associate the map
\[\hat\alpha:\oplus^k_\G T\G\to\mathbb{R},\]
given by contraction of $\alpha$ with tangent vectors. The map $\hat{\alpha}$ is skewsymmetric and multilinear (with respect to the linear structure of $\oplus^k_\G T\G$ over $\G$). We can now determine multiplicativity of $\alpha$ in terms of $\hat{\alpha}$.

<cit.> The form $\alpha$ is multiplicative if and only if the map $\hat{\alpha}$ is a groupoid morphism (where the groupoid structure on $\oplus^k_\G T\G$ is the natural extension of the structure of $T\G$, and $\mathbb{R}$ is seen as an additive group).

A symplectic groupoid is a pair $(\G,\omega)$, where $\G$ is a Lie groupoid and $\omega$ is a multiplicative symplectic form on $\G$.
As seen above this can be interpreted as having a pair of a Lie groupoid $\G$ and a symplectic form $\omega$ on $\G$ for which $\hat\omega:\oplus^2_\G T\G\to \mathbb{R}$ is a groupoid morphism. This point of view will be useful, because it lets us draw from the study of deformations of Lie groupoid morphisms [12] in order to study deformations of symplectic groupoids.

Symplectic groupoids are very closely related to Poisson manifolds. Recall that a Poisson structure on a manifold $M$ is a Lie bracket

\[\{\cdot,\cdot\}:C^\infty(M)\times C^\infty(M)\to C^\infty(M)\]

which is a derivation in each entry, i.e. it satisfies the Leibniz identity
\[\{f,gh\}=\{f,g\}h + g\{f,h\}.\]

Being a biderivation, it can be seen as a bivector field $\pi\in \Gamma(\wedge^2(TM))$.
The pair $(M,\pi)$ is called a Poisson manifold.

Any Poisson manifold $(M,\pi)$ gives rise to a Lie algebroid structure on the bundle $T^*M\to M$, called the cotangent Lie algebroid. The anchor $\pi^\sharp: T^*M\to TM$ is given by contraction with the Poisson bivector, $\pi^\sharp(\alpha)=\pi(\alpha,\cdot)$, while the Lie bracket on $\Omega^1(M)$ is defined by
\[[\alpha, \beta]_\pi=\mathcal{L}_{\pi^\sharp(\alpha)}\beta-\mathcal{L}_{\pi^\sharp(\beta)}\alpha-d(\pi(\alpha,\beta)).\]

Returning to symplectic groupoids, let us recall a few of the many notable properties they satisfy (see for example [17] for more).
Let $(\G, \omega)$ be a symplectic groupoid over $M$. Then

* The dimension of $\G$ is twice the dimension of $M$.

* $M$ (embedded via the unit map) is a Lagrangian submanifold of $\G$. This is a consequence of the multiplicativity of $\omega$ and of the relation between the dimensions.

* Let $A$ be the Lie algebroid of $\G$. There is a natural splitting at the units
\[T\G_{|M}=A\oplus TM,\]

and since $(\G, \omega)$ is symplectic, it follows from the dimension relation that $A$ and $TM$ have the same rank. Moreover, the fact that $M$ is Lagrangian, together with the splitting $T\G_{|M}=A\oplus TM$, implies that $\omega:\bigoplus^2 T\G\to \mathbb{R}$ induces a non-degenerate pairing $TM\oplus A\to \mathbb{R}$, and hence an isomorphism $\omega^\flat_{|TM}:TM\to A^*$. Its dual map $-\omega^\flat_{|A}$ induces an isomorphism $A\cong T^*M$.

* The anchor of $A$, viewed via the identification above as a map $T^*M\to TM$, is anti-selfadjoint, giving us a bivector $\pi\in \mathfrak{X}^2(M)$. Moreover, $\pi$ is in fact Poisson, and the induced algebroid structure on $T^*M$ is that of the cotangent algebroid of $(M, \pi)$.

A Poisson manifold $(M, \pi)$ is called integrable if there is a symplectic groupoid $(\G, \omega)$ over $M$ inducing the Poisson structure on the base. In that case, as mentioned above,  $\G$ integrates the cotangent Lie algebroid $T^*M$. There is a converse to this statement.

Let $(M,\pi)$ be a Poisson manifold such that the cotangent Lie algebroid $T^*M$ is integrable. Then the source $1$-connected integration of $T^*M$ admits a natural multiplicative symplectic structure, which induces the Poisson structure $\pi$ on $M$.

Let us mention a few examples of integrable Poisson manifolds, and of symplectic groupoids integrating them.

Symplectic manifolds
Any  symplectic manifold $(M, \omega)$ can be seen as a Poisson manifold,  with the Poisson bivector being given by the inverse of the symplectic form, $\pi=\omega^{-1}$. The pair groupoid $M\times M \rightrightarrows M$, equipped with the symplectic form $pr_2^*\omega-pr_1^*\omega$ is a symplectic groupoid integrating $\pi$. The source $1$-connected symplectic groupoid integrating $(M,\omega)$ is the symplectic fundamental groupoid $(\Pi_1(M), s^*\omega-t^*\omega)$.

Any other $s$-connected Lie groupoid $\G$ integrating $T^*M$ lies between these two extremes, as there are groupoid submersions $\Pi_1(M)\to \G \to M\times M$. Moreover, $\G$ is a symplectic groupoid when equipped with the form $s^*\omega-t^*\omega$.

Zero Poisson structure

Let $(M,0)$ be a manifold equipped with the zero Poisson structure, meaning that the Poisson bracket of any two functions is zero. The source $1$-connected symplectic groupoid integrating $(M,0)$ is the cotangent bundle of $M$, with the canonical symplectic structure $(T^*M, \omega_\mathrm{can})$. The source and target maps of $T^*M$ are equal to the vector bundle projection, and the groupoid multiplication is the fibrewise addition.

Linear Poisson manifolds

A linear Poisson structure on a vector space $V$ is one for which the Poisson bracket of linear functions is again linear. This happens if and only if there is a Lie algebra $\mathfrak{g}$ such that $V=\mathfrak{g}^*$, and the Poisson bracket is the Kirillov-Konstant-Souriau bracket, which extends the Lie bracket on $\mathfrak{g}$ (seen as the space of linear functions on $\mathfrak{g}^*$) to all smooth functions on $\mathfrak{g}^*$. We denote the corresponding Poisson bivector by $\pi_\mathfrak{g}$. Explicitly,
\[\{f,g\}(\eta):=\langle\eta, [d_\eta f,d_\eta g]_\mathfrak{g}\rangle, \ \ \forall \eta\in \mathfrak{g}^*, \ f,g\in C^\infty(\mathfrak{g}^*).\]

The $s$-connected symplectic groupoids integrating the linear Poisson manifold $\mathfrak{g}^*$ are of the form $(T^*G, \omega_\mathrm{can})$, for any connected Lie group $G$ integrating $\mathfrak{g}$. The groupoid structure of $T^*G$ is that of the cotangent groupoid of $G$, as explained in Example <ref>.

The source 1-connected symplectic groupoid integrating $(\mathfrak{g}^*, \pi_\mathfrak{g})$ is the cotangent groupoid $(T^*G, \omega_\mathrm{can})$ of the $1$-connected group $G$ integrating $\mathfrak{g}$.

The cotangent groupoid $T^*G$ is also isomorphic to the action groupoid $G\ltimes \mathfrak{g}^*$ of the coadjoint action of $G$ on $\mathfrak{g}^*$. An isomorphism is given by the trivialization of the cotangent bundle via right translations, $T^*G\cong G\times \mathfrak{g}^*$.

Cotangent VB-groupoids

Generalizing the previous case, Lie algebroid structures on a vector bundle $A$ are in $1:1$ correspondence with Poisson structures on its dual $A^*$ which are linear along the fibres [18]. By that we mean a Poisson structure such that the Poisson bracket of two fibrewise linear functions on $A^*$ is linear, and the bracket of a fibrewise linear function with a basic function is basic, and the bracket of two basic functions is zero.

Given any Lie groupoid $\G \rightrightarrows M$ with Lie algebroid $A=\mathrm{Lie}(\G)$, the cotangent Lie groupoid $T^*\G\rightrightarrows A^*$ (Example <ref>) is a symplectic groupoid integrating $A^*$. The source 1-connected integration of $(A^*, \pi_\mathrm{lin})$ is the cotangent groupoid $(T^*\G, \omega_\mathrm{can})$ of the source $1$-connected groupoid $\G$ integrating $A$.

§ DEFORMATIONS OF SYMPLECTIC GROUPOIDS

We now come to the notion of a deformation of a symplectic groupoid $(\mathcal{G},\omega)$. It naturally consists of an underlying deformation of Lie groupoids of $\mathcal{G}$ (Definition <ref>), with a compatible deformation of the multiplicative symplectic form $\omega$. For simplicity let us start with strict deformations.

(Strict deformations of symplectic groupoids)

Let $(\calG,\omega)$ be a symplectic groupoid, let $\calG_\e=(\calG,\bar{m}_{\e})$ be a strict deformation of $\calG$ and let $\omega_\e\in\Omega^{2}_\mathrm{cl}(\calG)$ be a deformation of $\omega$ by symplectic forms. The family $(\calG_\e,\omega_\e)$ is called a strict deformation of $(\calG,\omega)$ if each $(\calG_{\e},\omega_\e)$ is a symplectic groupoid.

Similarly, we define an  $s$-constant deformation of symplectic groupoids as one for which the underlying Lie groupoid deformation is $s$-constant.

An $s$-constant deformation of $(\G, \omega)$ such that $(\bar{m}_{\e},\omega_{\e})=(\bar{m},\omega)$ for all $\e$ is called a constant deformation of $(\calG,\omega)$.

Examples of deformations can be easily constructed (at least) in two simple ways.

Consider $(\calG,\omega)$ a symplectic groupoid with $(M,\pi)$ the Poisson structure induced on the base $M$.

* (Diffeomorphisms) Let $\phi_\e:\calG\longrightarrow\calG$ be a smooth family of diffeomorphisms with $\phi_0=id_{\calG}$. For every $\e$, induce the groupoid structure $\calG_\e:=(\calG,\bar{m}_\e)$ in such a way that $\phi_\e:\calG_\e\longrightarrow\calG$ is an isomorphism of Lie groupoids. Then $(\calG_\e,\phi^{*}_\e\omega)$ turns out to be a (strict) deformation of $(\calG,\omega)$.

* (Basic gauge transformations) Let $\alpha_\e\in\Omega^{2}_\mathrm{cl}(M)$ be a smooth family of closed 2-forms, with $\alpha_0=0$. Denote by $\Omega_{\alpha_\e}=\omega+s^{*}\alpha_\e-t^{*}\alpha_\e$ the family of multiplicative 2-forms on $\G$ induced by $\alpha_\e$. Then $(\calG,\Omega_{\alpha_\e})$ is an $s$-constant deformation of $(\G,\omega)$ as long as $\Omega_{\alpha_\e}$ stays non-degenerate for small enough $\e$ (for example, this happens if $\G$ is compact).
In particular, $(\calG,\Omega_{\alpha_\e})$ is the symplectic groupoid integrating the gauge transformation $\pi^{\alpha_\e}$ of $\pi$ by $\alpha_\e$.

These classes of deformations will be considered trivial, and lead us to the notion of equivalence of deformations.

We say that a deformation $(\calG'_{\e},\omega'_{\e})$ of $(\G,\omega)$ is equivalent to another deformation $(\calG_{\e},\omega_{\e})$ of $(\G,\omega)$ if there exists a smooth family of isomorphisms $F_\e:\calG_\e\rightarrow\calG'_{\e}$ and a smooth family of closed 2-forms $\alpha_{\e}$ on $M$, such that $F_0=id_{\calG}$, $\alpha_0=0$, and

$$F_\e^{*}\omega'_\e=\omega_\e+\delta_\e(\alpha_\e), \text{ for every }\e\in J,$$

where $\delta_\e(\alpha_\e)=s^{*}_{\e}(\alpha_\e)-t^{*}_{\e}(\alpha_\e)$, and $J\subset I\cap I'$ is an open interval containing zero. A deformation is said to be trivial if it is equivalent to a constant deformation.

General deformations of symplectic groupoids, and their equivalences,  are defined similarly, allowing for non-strict deformations of the underlying Lie groupoid.

A deformation of the symplectic groupoid $(\calG,\omega_\calG)$ consists of a deformation $\widetilde{\calG}\rightrightarrows\tilde{M}\stackrel{\textbf{p}}{\rightarrow} I$ of $\calG$, together with a multiplicative 2-form $\omega$ on $\widetilde{\calG}$ such that, for every $\e$, the restriction $\omega_\e$ of $\omega$ to $\calG_\e$ makes $(\G_\e, \omega_\e)$ into a symplectic groupoid, and $(\calG_0,\omega_0)=(\calG,\omega_\calG)$.

We recall that an equivalence between deformations $\widetilde{\G}$ and $\widetilde{\G}'$ of a Lie groupoid $\G$ is a Lie groupoid isomorphism $F:\widetilde{\G}_{|_J}\to\widetilde{\G}_{|_J}'$ such that $\textbf{p}'\circ F=\textbf{p}$, and $F_{|\G_0}=\mathrm{id}_{\G_0}$, where $\widetilde{\G}_{|_J}\rightrightarrows \widetilde{M}_{|_J}$ denotes the restriction of the family $\widetilde{\G}\rightrightarrows \widetilde{M}$ to some open interval $J\subset I\cap I'$ containing zero.

Let $\mathcal{F}$ and $\mathcal{F}_M$ denote the foliations of $\widetilde{\G}$ and $\widetilde{M}$ by the fibres of $\textbf{p}$. Given a form $\omega\in\Omega^k(\widetilde{\G})$ denote by $\omega_\mathcal{F}\in \Omega^k_\mathcal{F}(\widetilde{\G})$ its corresponding foliated form.

An equivalence between deformations $(\widetilde{\calG},\omega)$ and $(\widetilde{\calG}',\omega')$ of $(\calG,\omega_\calG)$ is given by an equivalence of deformations of Lie groupoids $F:\widetilde{\G}_{|_J}\to\widetilde{\G}_{|_J}'$, and a closed foliated $2$-form $\alpha_\mathcal{F}\in\Omega^{2}_{\mathcal{F}_M}(\widetilde{M}_{|_J})$ such that $\alpha_\mathcal{F}$ vanishes along $\G_0$, and

\[(F^{*}\omega')_\mathcal{F}=\omega_\mathcal{F}+\delta(\alpha_\mathcal{F}),\]
for some open interval $J\subset I\cap I'$ containing zero.

Given a strict deformation $\calG_\e$ of a Lie groupoid $\calG$ there is an induced deformation of Lie algebroids $A_\e=\mathrm{Lie}(\calG_\e)$. It in turn induces a deformation of Poisson manifolds $(A_\e^*, \pi_\e)$ such that each $\pi_\e$ is the fibrewise linear Poisson structure on $A^*_\e$ corresponding to the algebroid $A_\e$. Finally, each of these Poisson manifolds is integrated by the symplectic groupoid $(T^*\calG_\e, \omega_\mathrm{can})$. Since the deformation of Lie groupoids $\calG_\e$ is strict (in fact, a strict deformation of $\mathrm{VB}$-groupoids, cf. [49]), $(T^*\calG_\e, \omega_\mathrm{can})=(T^*\calG,\omega_\mathrm{can})$ as symplectic manifolds, but the Lie groupoid structure of $T^*\calG_\e$ can vary.

In short, any strict deformation of Lie groupoids induces deformations of Lie algebroids, fibrewise linear Poisson manifolds, and symplectic groupoids.

\[\G_\e \rightsquigarrow A_\e \rightsquigarrow (A_\e^*, \pi_\e)\rightsquigarrow (T^*\calG_\e, \omega_\mathrm{can})\]

§ THE DEFORMATION COMPLEX OF A (SYMPLECTIC) GROUPOID, REVISITED

We revisit the deformation complex of a Lie groupoid $\calG$ in the case that $(\calG,\omega)$ is a symplectic groupoid. We will use the symplectic form $\omega$ to identify $C^\bullet_\mathrm{def}(\calG)$ with a subcomplex $\Omega^{1}_\mathrm{def}(\calG^{(\bullet)})$ of $\Omega^{1}(\calG^{(\bullet)})$, the complex of 1-forms on the nerve of $\calG$.

Step 1 Recall that the deformation complex $C^\bullet_\mathrm{def}(\calG)$ can be identified with the VB-groupoid complex $C_\mathrm{VB}^\bullet(T^{*}\calG)$ of $T^{*}\calG$ (Prop. <ref>), which is a subcomplex of $C^\bullet_\mathrm{lin}(T^{*}\calG)$.

Step 2We use the fact that the vector bundle isomorphism $\omega^{b}:T\calG\rightarrow T^{*}\calG$ is actually a VB-groupoid isomorphism, due to the multiplicativity of $\omega$ (Lemma 3.6 in [7]). Therefore it determines an isomorphism between $C^\bullet_\mathrm{lin}(T^{*}\calG)$ and  $C^\bullet_\mathrm{lin}(T\calG)$. We use it to further identify the deformation complex of $\calG$ with $C_\mathrm{VB}^\bullet(T\calG)\subset C^\bullet_\mathrm{lin}(T\calG)$.

Step 3 The inclusion $C_\mathrm{VB}^\bullet(T\calG) \subset C^\bullet_\mathrm{lin}(T\calG)$ is a quasi-isomorphism (Lemma 3.1 in [10], Lemma <ref>).

Step 4 For each integer $k$, there is a natural isomorphism $(T\calG)^{(k)}\cong T\calG^{(k)}$. In this way, the elements of $C^{k}_\mathrm{lin}(T\calG)$ can be viewed as the fibrewise linear functions on $T\calG^{(k)}$, i.e., as elements of $C^{\infty}_\mathrm{lin}(T\calG^{(k)})\cong\Omega^{1}(\calG^{(k)})$.

Thus, at the level of cochains, we have the following chain of identifications and quasi-isomorphisms:

\[C^{k}_\mathrm{def}(\calG)\cong C_\mathrm{VB}^{k}(T^{*}\calG) \stackrel{\omega^{b}}{\cong} C_\mathrm{VB}^{k}(T\calG) \stackrel{\mathrm{q. i.}}{\hookrightarrow} C^{k}_\mathrm{lin}(T\calG):=C^{\infty}_\mathrm{lin}(T\calG^{(k)})\cong\Omega^{1}(\calG^{(k)})\,\text{.}\]

The last identification of $C^\bullet_\mathrm{lin}(T\calG)$ with $\Omega^\bullet(\calG)$ is compatible with the differentials of the complexes involved. In fact, the two differentials correspond to the two different ways of considering the alternated sum of pullbacks of 1-forms $\alpha$ by the face maps $d_i:\calG^{(n)}\to\calG^{(n-1)}$ of the nerve of $\calG$: as the usual pullback of forms $d_i^{*}\alpha$, or as the pullback of functions $(Td_i)^{*}:C^{\infty}_\mathrm{lin}(T\calG^{(n+1)})\longrightarrow C^{\infty}_\mathrm{lin}(T\calG^{(n)})$ by $Td_i$.

We summarize now the description of the deformation complex of $\calG$ obtained above, in the symplectic case.

Let $(\calG,\omega)$ be a symplectic groupoid. Then $\omega^{b}$ induces an isomorphism between the deformation complex $C^{*}_\mathrm{def}(\calG)$ of $\calG$ and a subcomplex $\Omega^{1}_\mathrm{def}(\calG^{(\bullet)})$ of $\Omega^{1}(\calG^{(\bullet)})$. Moreover, $\Omega^{1}_\mathrm{def}(\calG^{(\bullet)})\hookrightarrow\Omega^{1}(\calG^{(\bullet)})$ is a quasi-isomorphism.

The complex $\Omega^{1}_\mathrm{def}(\calG^{(\bullet)})$ obtained under the identification above can be explicitly described by translating the conditions of the VB-groupoid complex to conditions on 1-forms on the nerve of $\calG$. Namely,
a 1-form $\alpha$ on $\calG^{(k)}$, with $k>0$ will belong to $\Omega^{1}_\mathrm{def}(\calG^{(k)})$ if the following two conditions are satisfied (when $k=0$ there are no conditions imposed).

* $\alpha\in\Gamma_{\calG^{(k)}}(T^{*}\calG^{(k)})$ comes from a section $\hat{\alpha}$ of the bundle $pr_{1}^{*}T^{*}\calG\rightarrow\calG^{(k)}$, where $pr_{1}:\calG^{(k)}\rightarrow\calG$ is the projection on the first component. That is, there exists a necessarily unique section $\hat{\alpha}$ such that the left triangle below commutes;

\begin{equation}\label{triangledefiningsigma}
\begin{tikzcd}[column sep=3em, row sep=10ex]
T^{*}\calG^{(k)} \arrow{d}[swap]{c_{\calG^{(k)}}} & pr_{1}^{*}(T^{*}\calG) \arrow{l}[swap]{(d\,pr_1)^{*}} \arrow[xshift=-3pt]{ld}{ } \arrow{r}{ } & T^{*}\calG \arrow{d}{c_{\calG}}\\
\calG^{(k)}	\arrow[xshift=3pt]{ur}[swap]{\hat{\alpha}} \arrow[xshift=3pt]{u}[swap]{\alpha} \arrow{rr}{pr_1} &  & \calG
%\calG \arrow[xshift=2pt]{d}{t} \arrow[xshift=-2pt]{d}[swap]{s}    & M,
\end{tikzcd}
\end{equation}

* $\alpha$ is left-invariant for the action $g\cdot(g_1,\ldots,g_k)\mapsto (gg_1,\ldots,g_k)$ of $\calG$ on $\calG^{(k)}$.

In terms of cochains, the identification of Proposition <ref> is the composition of the following maps

\[
\begin{tikzcd}[column sep=4em, row sep=0.1em]
\parbox[0]{4em}{$C^{k}_\mathrm{def}(\calG)$} \arrow{r}{} & \parbox[2em]{4em}{$C^{k}_\mathrm{lin}(T^{*}\calG)$} \arrow{r}{(\omega^{b})^{*}} & \parbox[0]{4em}{$C^{k}_\mathrm{lin}(T\calG)$} \arrow[<->]{r}{} & \parbox[0]{4em}{$\Omega^{1}_\mathrm{def}(\calG^{(k)})$}\\
\parbox[t]{4em}{\hspace{1em}$c$} \arrow[|->]{r}{} & \parbox[t]{4em}{\hspace{1.5em}$c'$} \arrow[|->]{r}{} & \parbox[t]{4em}{$(\omega^{b})^{*}c'$} \arrow[|->]{r}{} & \parbox[t]{4em}{\hspace{1em}$\hat{c}$,}
\end{tikzcd}
\]

where $c'(\eta_{g_{1}},\ldots,\eta_{g_{k}}):=\left\langle \eta_{g_{1}}, c(g_{1},\ldots, g_{k})\right\rangle$ and, by construction,

\begin{equation}\label{identification}
\begin{split}
\hat{c}(v_{g_{1}},\ldots,v_{g_{k}})&=(\omega^{b})^{*}c'(v_{g_{1}},\ldots,v_{g_{k}})\\
&=-\left\langle\omega^{b}(c(g_{1},\ldots, g_{k})),v_{g_{1}}\right\rangle\\
&=-\left[(dpr_{1})^{*}(\omega^{b}(c(g_{1},\ldots, g_{k})))\right](v_{g_{1}},\ldots,v_{g_{k}}).
\end{split}
\end{equation}

In other words, $c\in C^{k}_\mathrm{def}(\calG)$ corresponds to the section $-\omega^{b}\circ c:\calG^{(k)}\rightarrow pr_{1}^{*}T^{*}\calG$ of condition $(1)$ above. Condition $(2)$ holds for $\hat{c}$ as it amounts to the $s$-projectability condition of $c$. In degree zero, this correspondence $C^0_\mathrm{def}(\G)=\Gamma(A)\cong \Omega^1(M)$ amounts to the identification $-\omega^\flat_{|A}:A\cong T^*M$, at the level of sections.

Note that, in the identification above, we only use the multiplicativity and non-degeneracy conditions of $\omega$. Then, in particular, such an identification also holds for twisted symplectic groupoids (cf. [8]).

Denote by $\sigma$ the vector bundle maps $(d\,pr_1)^{*}$ given by the diagram (<ref>) above, for any $k$. If $(\calG_\e,\omega_\e)$ is an $s$-constant deformation of $(\calG,\omega)$, then under the identification above the deformation cocycle $\xi_0$ of $\calG_\e$ corresponds to the 1-form $\zeta_0=-\sigma(\omega(\xi_0,\cdot))\in\Omega^{1}_\mathrm{def}(\calG^{(2)})$. And, since $\sigma=Id$ for $k=1$, it follows that for $X\in C^{1}_\mathrm{def}(\calG)$ the corresponding 1-form on $\calG$ will be $-i_{X}\omega=-\omega^{b}(X)$.

§ THE DEFORMATION COMPLEX OF A SYMPLECTIC GROUPOID

Our approach to find the deformation complex of a symplectic groupoid is to construct it out of the deformation complex of a Lie groupoid [25] (taking also into account the discussion of Section <ref>), and the deformation complex for multiplicative forms, introduced in [12].

In section <ref> we will study the resulting complex, looking at cocycles and coboundaries related to deformations and equivalences, confirming that this approach leads indeed to the correct complex.

§.§ Deformations of symplectic groupoids in terms of morphisms

Let $(\calG_\e,\omega_\e)$ be a deformation of the symplectic groupoid $(\calG,\omega)$. The first step is to consider the associated morphism $\hat\omega:\bigoplus^2T\G\to\mathbb{R}$, described in Remark <ref>. Doing so,
$(\calG_\e,\omega_\e)$ can be replaced by the simultaneous deformation $(\bigoplus^{2}T\calG_\e,\hat{\omega}_\e)$ of the Lie groupoid $\bigoplus^{2}T\calG$ and of the morphism $\hat{\omega}$.

Note that this is a very particular type of simultaneous deformation of a Lie groupoid and a morphism having that groupoid as domain. The deformation of $\bigoplus^{2}T\G$ is of the form $\bigoplus^{2}T(\calG_\e)$ and the deformation of $\hat\omega$ is  by (closed, non-degenerate) bilinear and skewsymmetric morphisms $\hat{\omega}_\e$. Let us then first examine these separate pieces.

§.§ Deformations of multiplicative 2-forms

Let us first describe the deformation complex $\hat{C}_\mathrm{def}^\bullet(\omega)$ of a multiplicative 2-form $\omega$, defined in [12]. It is the subcomplex of bilinear and skewsymmetric cochains of the deformation complex $C^\bullet_\mathrm{def}(\hat{\omega})$ of the associated Lie groupoid morphism $\hat{\omega}$. We remark that our notation differs from the one in [12], where the deformation complex $\hat{C}_\mathrm{def}^\bullet(\omega)$ was denoted simply by $C_\mathrm{def}^\bullet(\omega)$. We reserve the latter notation for the deformation complex of a closed multiplicative form $\omega$.

As shown in [12], it is isomorphic to the complex $\Omega^{2}(\G^{(\bullet)})$ of 2-forms on the nerve of $\G$, and it can also be identified with the subcomplex $C^{\bullet}(\bigoplus^{2}_{p_\G}T\G)_\mathrm{2-lin,\, sk}$ of 2-linear skewsymmetric cochains of the differentiable complex of the groupoid $\bigoplus^{2}_{p_\G}T\G$.
As remarked in [12], a deformation $\omega_\e$ of $\omega$ induces a family of cocycles $\frac{d}{d\e}\hat{\omega}_\e$ in $\hat{C}^{1}_\mathrm{def}(\omega_\e)$.

In our setting, $\omega$ is closed and we are interested in deformations of it via closed multiplicative forms. The deformation complex to consider is then the subcomplex $C_\mathrm{def}^\bullet(\omega)\subset \hat{C}_\mathrm{def}^\bullet(\omega)$, corresponding to closed $2$-forms on the nerve of $\G$.

§.§ Tangent lifts of deformation cochains

For any Lie groupoid $\G$, the tangent lift of deformation cochains, described in [49], is the inclusion of cochain complexes   $T_\mathrm{def}:C^{\bullet}_\mathrm{def}(\calG)\to C^{\bullet}_\mathrm{def}(T\calG)$  determined by composing the canonical involution (also called canonical flip, cf. <cit.>) $J_\calG:TT\calG\to TT\calG$  and the differential $d$ of a smooth function:

$$T_\mathrm{def}c:=J_\calG\circ dc.$$

We also call tangent lift to its natural extension $\oplus^{2}T_\mathrm{def}: C^\bullet_\mathrm{def}(\calG)\to C^\bullet_\mathrm{def}(\oplus^{2}T\calG)$ defined as $\oplus^{2}T_\mathrm{def}(c):=\oplus^{2}(T_\mathrm{def}c),$ of which we will make use.

As remarked in [12, 49], the tangent lift includes the deformation cocycles of $\G$ into the deformation complex of $\oplus^2 T\G$, as cocycles associated to deformations of the form $\bigoplus^{2}T(\G_\e)$.

§.§ The deformation complex

We come back to the idea that deformations of symplectic groupoids can be seen as particular simultaneous deformations of $\bigoplus^{2}T\calG$, of the form $\bigoplus^{2}T\calG_\e$, and of a morphism $\hat{\omega}$, via morphisms associated to (closed, non-degenerate) multiplicative forms.

In this situation, we can use the idea from [12] of using the mapping cone construction to control simultaneous deformations.
As described in Section <ref>, $\hat{\omega}$ induces a map $\hat{\omega}_{*}:C_\mathrm{def}^{\bullet}(\bigoplus^{2}T\G)\to C_\mathrm{def}^{\bullet}(\hat\omega)$ between deformation complexes. In fact, the bilinearity and skewsymmetry of $\omega$ imply, by direct computation, that its image is contained in the subcomplex $C_\mathrm{def}^{\bullet}(\omega)$.

The deformation complex of the symplectic groupoid $(\calG, \omega)$, denoted by $C^{\bullet}_\mathrm{def}(\G,\omega)$, is the mapping cone complex of the map

$$\hat{\omega}_{*}\circ\oplus^{2}T_\mathrm{def}:C^{\bullet}_\mathrm{def}(\G)\to C_\mathrm{def}^{\bullet}(\omega).$$

Considering deformations of symplectic groupoids as simultaneous deformations of a morphism of Lie groupoids and of its domain, the work of [12], already indicates that the element

$$\xi_\e-\frac{d}{d\e}\hat{\omega}_\e\in C^{2}_\mathrm{def}(\G_\e)\bigoplus C^{1}_\mathrm{def}(\omega_\e)$$

should be the deformation cocycle associated to the deformation $(\G_\e,\omega_\e)$. Here $\xi_\e$ is the deformation class of the underlying deformation of Lie groupoids, described in Section <ref>. We will reach this same construction explicitly (see Section <ref>) using the alternative description of the deformation complex given by the Theorem <ref> below.

The next Theorem shows that using identifications given by the symplectic form $\omega$, the mapping cone complex defined above has a more familiar presentation, given in terms of differential forms.

Let $(\G,\omega)$ be a symplectic groupoid. Under the correspondence $C_\mathrm{def}^\bullet(\calG)\cong \Omega_\mathrm{def}^{1}(\calG^{(\bullet)})$, induced by the isomorphism $\omega^{b}$ (see Proposition <ref>), the map $\hat{\omega}_{*}\circ\oplus^{2}T_\mathrm{def}$ agrees with the de Rham differential $d_{dR}:\Omega_\mathrm{def}^{1}(\calG^{(\bullet)})\to\Omega^{2}_\mathrm{cl}(\calG^{(\bullet)})$ up to a sign. That is, the following diagram is commutative.

\begin{equation}\label{diagrama:deRham-mapping}
\xymatrix@C+1pc{C_\mathrm{def}^\bullet(\calG) \ar[r]^{\hat{\omega}_{*}\circ\oplus^{2}T_\mathrm{def}} \ar[d]_{-\sigma\circ\omega^{b}} & C_\mathrm{def}^\bullet(\omega) \ar[d]^{\cong}\\
\Omega^{1}_\mathrm{def}(\G^{(\bullet)}) \ar[r]^{-d_{dR}} & \Omega_\mathrm{cl}^{2}(\G^{(\bullet)})}
\end{equation}

Let us first review the correspondence $C_\mathrm{def}^{\bullet}(\calG)\cong \Omega_\mathrm{def}^{1}(\calG^{(\bullet)})$ of the previous section in terms of vector fields. Given any splitting $\tau:t^{*}TM\to T\G$ of the target map $dt:T\G\to t^{*}TM$ of $\G$, any deformation cochain $c$ of $C^{k}_\mathrm{def}(\calG)$ induces a vector field $X_c\in\mathfrak{X}(\calG^{(k)})$ on $\calG^{(k)}$. For example, if $c\in C_\mathrm{def}^{2}(\G)$ is a degree 2 deformation cochain, the vector field $X_{c}$ is determined by

Analogously, for a higher degree cochain $c$, the components of the vector field $X_c$ are determined by successively $s$-projecting and $\tau$-lifting the deformation cochain $c$. The correspondence $C_\mathrm{def}^\bullet(\calG)\cong \Omega_\mathrm{def}^{1}(\calG^{(\bullet)})$ induced by the isomorphism $\omega^{b}$ can be written as

$$c\mapsto -i_{X_c}(pr_1^{*}\omega) \in \Omega^{1}_\mathrm{def}(\calG^{(k)});$$

where $pr_1:\G^{(k)}\to\G$ is the 1st-component projection map. Note that even though $X_c$ depends on $\tau$,  the form $-i_{X_c}(pr_1^{*}\omega)$ does not.

Therefore, when doing the same for the tangent groupoid $T\G$, we are free to choose a convenient splitting $\tilde{\tau}$ of the target map $d\tilde{t}:TT\G\to\tilde{t}^{*}TTM$, where $\tilde{t}:T\G\to TM$ is the target map of $T\G$.
We choose $\tilde{\tau}$ to be induced by the tangent lift $T\tau$ of the previous splitting $\tau$. Explicitly, under the canonical isomorphism of bundles $\tilde{t}^{*}T(TM)\cong T(t^{*}TM)$, the splitting $\tilde{\tau}$ is defined by

$$\tilde{\tau}:=J_\G\circ T\tau,$$

where $J_\G:TT\G\to TT\G$ is the canonical involution of the double tangent bundle.

We denote by  $\tilde{X}_{\tilde{c}}$ the vector field induced by the splitting $\tilde{\tau}$ and the deformation cochain $\tilde{c}\in C_\mathrm{def}^{k}(T\G)$.

A direct computation now shows that there is a compatibility between $\tilde{X}_{\tilde{c}}$ and the tangent lift $X_c^{T}\in\mathfrak{X}(T\G^{(k)})$ of the vector field $X_c$. Precisely, one obtains

Finally, we are ready to check the commutativity of the diagram (<ref>) above. Let $c\in C^{k}_\mathrm{def}(\G)$ and $\tilde{pr}_1:(T\G)^{(k)}\to T\G$ be the projection on the first component. Then

\begin{align*}
\hat{\omega}_{*}\circ\oplus^{2}T_\mathrm{def}(c)&=d\hat{\omega}(T_\mathrm{def}c, T_\mathrm{def}c)\\
&=d\hat{\omega}(d\tilde{pr}_1\tilde{X}_{T_\mathrm{def}c}, d\tilde{pr}_1\tilde{X}_{T_\mathrm{def}c})\\
&=d\hat{\omega}(d\tilde{pr}_1 X^{T}_{c}, d\tilde{pr}_1 X^{T}_{c}).
\end{align*}

Let $F_\e$ be the flow of the vector field $X_c$. Then $TF_\e$ will be the flow of the tangent lift $X^{T}_{c}$ and the expression above becomes

\begin{align*}
d\hat{\omega}(d\tilde{pr}_1 X^{T}_{c}, d\tilde{pr}_1 X^{T}_{c})&=\frac{d}{d\e}\Bigr|_{\e=0}\ \hat{\omega}(\tilde{pr}_1TF_\e, \tilde{pr}_1TF_\e)\\
&=\frac{d}{d\e}\Bigr|_{\e=0}\ F_\e^{*}(pr_1)^{*}\omega\\
\end{align*}

which proves the commutativity of the diagram.

The deformation complex $C^{\bullet}_\mathrm{def}(\G,\omega)$ of the symplectic groupoid $(\G,\omega)$ is isomorphic to the mapping cone of the de Rham differential
\[-d_{dR}:\Omega^{1}_\mathrm{def}(\G^{(\bullet)})\to\Omega_\mathrm{cl}^{2}(\G^{(\bullet)}).\]

Explicitly, according to our convention for the mapping cone complex, the differential $D:\Omega_\mathrm{def}^{1}(\calG^{(\bullet)})\oplus\Omega_\mathrm{cl}^{2}(\calG^{(\bullet-1)})\to\Omega_\mathrm{def}^{1}(\calG^{(\bullet+1)})\oplus\Omega_\mathrm{cl}^{2}(\calG^{(\bullet)})$ is given by

$$(\zeta,\omega)\mapsto (\delta\zeta,-d_{dR}\zeta-\delta\omega).$$
We call this complex the de Rham model for deformation cohomology of the symplectic groupoid $(\G,\omega)$.

Notice that $d_{dR}:\Omega^{1}(\G^{(\bullet)})\to\Omega^{2}(\G^{(\bullet)})$ is a part of the vertical differential of the Bott-Shulman-Stasheff double complex $(\Omega^{\bullet}(\G^{(\bullet)}),\delta,d_{dR})$. This allows us to identify the deformation complex $C^{\bullet}_\mathrm{def}(\G,\omega)$ with the total complex of a sub-double complex $B^{\bullet,\bullet}(\G)\subset\Omega^{\bullet}(\G^{(\bullet)})$. This sub-double complex consists of the subset $\Omega^{1}_\mathrm{def}(\G^{(\bullet)})$ of the second line $\Omega^{1}(\G^{(\bullet)})$ and of the subset $\Omega^{2}_\mathrm{cl}(\G^{(\bullet)})$ of the third line $\Omega^{2}(\G^{(\bullet)})$ of the Bott-Shulman-Stasheff double complex (and it is zero in all other bidegrees). In fact, the map

$$\varphi:\Omega^{1}_\mathrm{def}(\calG^{(n)})\oplus\Omega_\mathrm{cl}^{2}(\calG^{(n-1)})\to Tot^{n+1}(B^{\bullet,\bullet}(\G))$$

defined by

$$(\zeta,\omega)\mapsto (\zeta,(-1)^n\omega)$$

gives us the isomorphism of the complexes. The shift in degree appears naturally since multiplicative symplectic forms, while being degree two forms, give rise to degree three cocycles in the Bott-Shulman-Stasheff cohomology. More generally, a groupoid $\G$ equipped with a pair $(\phi, \omega)\in \Omega^2(\G)\oplus \Omega^3(M)$ such that $0 \oplus 0 \oplus \omega \oplus\phi \in Tot^3(\Omega^\bullet(\G^{(\bullet)}))$ is a cocycle is called a $\phi$-twisted presymplectic groupoid [8].

Another interpretation for Bott-Shulman-Stasheff cohomology is that it is the de Rham cohomology of the differentiable stack $M//\G$ presented by $\G$. We note also that degree three cocycles belonging to $B^{\bullet,\bullet}$ i.e. corresponding to deformation cocycles of $(\G,\omega)$, and satisfying an integrality condition are related to $S^1$-central extensions of $\G$ <cit.>.

§ EXAMPLES AND COMPUTATIONS

§.§ Deformation cohomology in small degrees

We start by interpreting cohomology classes in small degrees in a geometric way via direct computations; but see also the computations of small degrees in terms of a spectral sequence, in Section <ref>. Using the de Rham model for the deformation cohomology, a simple computation shows that
\[H_\mathrm{def}^0(\G, \omega)=\{ \alpha\in \Omega^1_\mathrm{def}(M)\mid  d\alpha=0, \ \delta\alpha=0\}=\Omega^1_\mathrm{cl}(M)^{\G-\mathrm{inv}},\]
i.e. the space of closed invariant $1$-forms on $M$.

In degree 1, again using the de Rham model, a deformation cocycle on $(\G,\omega)$ is given by a pair $(\zeta, \alpha)\in \Omega^1_\mathrm{def}(\G)\oplus \Omega^2_\mathrm{cl}(M)$ such that $\delta{\zeta}=0$ and $d\zeta = \delta \alpha$. Using the terminology of [8, 16], this means that $\zeta$ is a relatively $\alpha$-closed multiplicative $1$-form.

Under the isomorphism $\omega^\flat: T\G\to T^*\G$, the form $\zeta$ corresponds to a vector field $X$ on $\G$ such that $-\omega^\flat(X)=\zeta$.
Multiplicativity of $\zeta$ and of $\omega$ mean that $X$ is  multiplicative as well, meaning that $X$ is a Lie groupoid morphism $X:\G\to T\G$. Therefore, $X$ is an infinitesimal automorphism of $\G$, its flow preserving the Lie groupoid structure. However, since $\zeta$ is not closed but only relatively $\alpha$-closed, the flow $\phi_\e$ of $X$ does not preserve the symplectic form $\omega$. Instead, it satisfies $\phi_\e^*\omega=\omega+\delta(\e\alpha)$, so it  applies a gauge transformation by $\delta(\e \alpha)$ to $\zeta$. Because of this we call 1-cocycles twisted infinitesimal automorphisms of $(\G,\omega)$.

A degree 1 coboundary is a pair of the form $(\delta\beta, d\beta)\in \Omega^1_\mathrm{def}(\G)\oplus \Omega^2_\mathrm{cl}(M)$. Adding it to a relatively $\alpha$-closed multiplicative $1$-form $\zeta$ we obtain a relatively $(\alpha+d\beta)$-closed multiplicative $1$-form $\zeta+\delta\beta$. We call these transformations of the form $(\zeta,\alpha)\mapsto (\zeta + \delta\beta, \alpha+d\beta)$ trivial gauge transformations. We conclude that

\[H^1_\mathrm{def}(\G,\omega)=\frac{ \text{Twisted infinitesimal automorphisms of }(\G,\omega)}{\text{Trivial gauge transformations}}.\]

In degree 2, as mentioned in Remark <ref>, deformations of symplectic groupoids give rise, after taking a first-order approximation, to cocycles in $C^2_\mathrm{def}(\G,\omega)$; equivalences between deformations provide 1-cochains transgressing the difference between the corresponding deformation cocycles. Thus

\[H^2_\mathrm{def}(\G,\omega)=\frac{ \text{Infinitesimal deformations of }(\G,\omega)}{\text{Infinitesimal equivalences}}.\]

This is the main use of the deformation cohomology, and we will study it in detail in Sections <ref> and <ref>.

§.§ The cone exact sequence and proper Lie groupoids

Before we start analysing the deformation cohomology of a symplectic groupoid in particular examples, we recall a simple consequence of the definition of the deformation complex. As the mapping cone of $\hat{\omega}_{*}\circ\oplus^{2}T_\mathrm{def}:C^{\bullet}_\mathrm{def}(\G)\to C_\mathrm{def}^{\bullet}(\omega)$, it fits in a short exact sequence of complexes

\[0\to C_\mathrm{def}^{\bullet}(\omega)[-1] \to C^{\bullet}_\mathrm{def}(\G, \omega) \to C^{\bullet}_\mathrm{def}(\G) \to 0.\]
This induces the long exact sequence in cohomology

\begin{equation}\label{eq-long-ex-seq-cone}
\cdots \to H_\mathrm{def}^{k-1}(\omega) \to H^{k}_\mathrm{def}(\G, \omega) \to H^{k}_\mathrm{def}(\G)\stackrel{\partial}{\to} H_\mathrm{def}^{k}(\omega)\to \cdots
\end{equation}
where the connecting homomorphism $\partial=H(\hat{\omega}_{*}\circ\oplus^{2}T_\mathrm{def})$ is the map induced in cohomology by $\hat{\omega}_{*}\circ\oplus^{2}T_\mathrm{def}$.

[Symplectic manifolds]

A Lie groupoid $\G$ integrating a connected non-degenerate Poisson manifold $(M,\omega^{-1})$ is transitive. For a transitive Lie groupoid it holds <cit.> that $H^\bullet_\mathrm{def}(\G)\cong H^\bullet(G,\mathfrak{g})$, where $G$ is the isotropy Lie group at any point of the base, acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation. By simple dimension counting we know that the isotropy groups are discrete, so $\mathfrak{g}=0$.

Thus, for a symplectic groupoid $(\G, s^*\omega-t^*\omega)\rightrightarrows(M,\omega^{-1})$, the long exact sequence (<ref>) implies that
\[H_\mathrm{def}^{\bullet}(\G, s^*\omega-t^*\omega)\cong H_\mathrm{def}^{\bullet-1}(s^*\omega-t^*\omega).\]

For the pair groupoid of $M$, any multiplicative $2$-form is cohomologically trivial, so $H_\mathrm{def}^2(M\times M, pr_1^*\omega-pr_2^*\omega)=0$. This translates to the fact that deformations of $(M\times M, pr_1^*\omega-pr_2^*\omega)$ are given by gauge transformation of $\delta\omega=pr_1^*\omega-pr_2^*\omega$ by $\delta \omega_\e$, where $\omega + \omega_\e$ is a deformation of the symplectic structure on the base.

Another situation in which we can make further use of this long exact sequence is for proper symplectic groupoids.
A Lie groupoid is called proper if it is Hausdorff and
\[(s,t):\G\to M\times M\]
is a proper map. Examples of proper groupoids include compact Lie groupoids, pair groupoids, and the cotangent Lie groupoid $T^*G$ of a compact Lie group $G$.

Proper symplectic groupoids are the subject of recent extensive work [23, 24, 76], and satisfy many remarkable properties not shared by a general symplectic groupoid, and much less by non-integrable Poisson manifolds. They generalize many aspects of the geometry of compact symplectic manifolds and of Lie algebras of compact type.

Proper symplectic groupoids are also very special among proper Lie groupoids, in a way which is analogous to how the adjoint action of a compact Lie group $G$ on its Lie algebra $\mathfrak{g}$ is special among actions of $G$ (compare for example [23, 24] with Chapters 2 and 3 of [35]). For example, for an arbitrary Lie groupoid $\G \rightrightarrows M$ the isotropy Lie algebra $\mathfrak{i}_x$ and the normal space $\nu_x$ to the orbit of $\G$ through $x\in M$ are the kernel and cokernel of the anchor map of $A=\mathrm{Lie}(\G)$ at $x$. In general there is no further relation between these two objects, which feature prominently in the study of the geometry of proper Lie groupoids; when $\G$ is a symplectic groupoid, however, $\nu_x$ inherits a linear Poisson structure from $M$ by linearization at $x$, such that its dual Lie algebra $\nu_x^*$ is naturally isomorphic to $\mathfrak{i}_x$.

We recall the following vanishing result for the deformation cohomology of proper Lie groupoids.

If $\G$ is a proper Lie groupoid, then
\[H^0_\mathrm{def}(\G)\cong\Gamma(\mathfrak{i})^{\G-\mathrm{inv}},\ \ \ H^1_\mathrm{def}(\G)\cong\Gamma(\nu)^{\G-\mathrm{inv}},\ \text{ and }\  H^{k}_\mathrm{def}(\G)=0 \text{ for } k\geq 2.\]

Let us give a brief explanation of the objects in the statement of this result. The isotropy bundle $\mathfrak{i}$ is the kernel of the anchor map of the Lie algebroid of $\G$,
\[\mathfrak{i}=\mathrm{Ker}(\rho:A\to TM).\]
Differentiating at the units the conjugation by elements $g:x\to y$ induces a map $\mathrm{ad}_g:\mathfrak{i}_x\to \mathfrak{i}_y$.

Strictly speaking, $\mathfrak{i}$ is a distribution with possibly varying rank, so it is only a bundle if $\G$ is a regular groupoid. Nonetheless, it makes sense to consider $\Gamma(\mathfrak{i})$ as the space of smooth sections of $A$ which land in $\mathfrak{i}\subset A$. Among these, the space of invariant ones is
\[\Gamma(\mathfrak{i})^{\G-\mathrm{inv}}=\{\alpha\in \Gamma(\mathfrak{i})\ |\ \mathrm{ad}_g(\alpha_x)=\alpha_y\}.\]
It was shown in [25] that in fact $H^0_\mathrm{def}(\G)\cong\Gamma(\mathfrak{i})^{\G-\mathrm{inv}}$ for any Lie groupoid, not only proper ones.

The normal bundle of $\G$ is
\[\nu=\mathrm{Coker(\rho)}=TM/\rho(A),\]
and once again it is only a smooth vector bundle for regular $\G$. But once more, there is a way to make sense of its space of smooth invariant sections $\Gamma(\nu)^{\G-\mathrm{inv}}$. First of all, the space of smooth sections is defined as $\Gamma(\nu):=\mathfrak{X}(M)/\rho(A)$. A section $[V]\in \Gamma(\nu)$ is called invariant if there is a vector field $X\in \mathfrak{X}(\G)$ which is both $s$ and $t$-projectable to $V$. This condition is independent of the choice of representative $V$. In the regular case, this notion recovers the usual notion of invariant sections of the normal representation of $\G$. We refer to Section $4.4$ of [25] for more detail.

For an arbitrary Lie groupoid $\G$, deformation cocycles of degree one are multiplicative vector fields on $\G$, i.e., vector fields which are Lie groupoid morphisms $X:\G\to T\G$. Any such vector field $X$ is in particular $s$ and $t$-projectable to a vector field $V$ on the base. Mapping $X$ to the class of $V$ modulo $\mathrm{Im}(\rho)$ induces a linear map $H^1_\mathrm{def}(\G)\to\Gamma(\nu)^{\G-\mathrm{inv}}$ <cit.>. Part of Theorem <ref>  is that for proper Lie groupoids this map becomes an isomorphism.

Let $(\G, \omega)$ be a proper symplectic groupoid. Then there is a 7-term exact sequence

\begin{align*}
0  & \to \Omega^1_\mathrm{cl}(M)^{\G-\mathrm{inv}} \to \Gamma(\nu^*)^{\G-\mathrm{inv}} \to \Omega^2_\mathrm{cl}(M)^{\G-\mathrm{inv}} \to H^{1}_\mathrm{def}(\G, \omega) \to \\
&  \to \Gamma(\nu)^{\G-\mathrm{inv}}\to H_\mathrm{def}^{1}(\omega)\to H^{2}_\mathrm{def}(\G, \omega)\to 0
\end{align*}

and $H^{k}_\mathrm{def}(\G,\omega)\cong H^{k-1}_\mathrm{def}(\omega)$ for $k> 2$.

In particular, in terms of the de Rham model for deformation cohomology,
\[H^{2}_\mathrm{def}(\G, \omega)\cong \frac{H_\mathrm{def}^{1}(\omega)}{d(H^1_\mathrm{def}(\G))}\cong \frac{H_\mathrm{def}^{1}(\omega)}{d[\Omega^1_\mathrm{mult}(\G)/\delta(\Omega^1(M))]}.\]

Simply use Theorem <ref> together with the long exact sequence (<ref>), the discussion of Section <ref> for deformation cohomology in small degrees, and recall that $\mathfrak{i}_x\cong  \nu_x^*$ for symplectic groupoids.

§.§ Zero Poisson structures of proper type

The source $1$-connected integration of the zero Poisson structure on a manifold $M$ is given by the symplectic groupoid $(T^*M, \omega_\mathrm{can})$, viewed as a bundle of abelian Lie groups over $M$, and so it is never proper. However, $(M,0)$ may still be integrated by a non simply connected proper symplectic groupoid, having interesting consequences for the geometry of $M$. Proper $s$-connected symplectic integrations of $(M,0)$ are in 1:1 correspondence with integral affine structures on $M$ (<cit.>, see also <cit.>).

An integral affine structure on $M$ can be described by a Lagrangian smooth full rank lattice $\Lambda$ in $T^*M$ i.e., a subbundle $\Lambda\subset T^*M$, whose fibres are full rank lattices in the fibres of $T^*M$, and which is a Lagrangian submanifold of $T^*M$. The proper symplectic integration corresponding to $\Lambda$ is $(T^*M, \omega_\mathrm{can})/\Lambda$. The Lie groupoid $\mathcal{T}_\Lambda:=T^*M/\Lambda$ is a torus bundle and the form $\omega_\mathrm{can}$ descends to a symplectic form $\omega_\mathcal{T}$ on it, so $(\mathcal{T}_\Lambda,\omega_\mathcal{T})$ is a symplectic torus bundle over $M$.

Since $(\mathcal{T}_\Lambda,\omega_\mathcal{T})$ is a bundle of Lie groups, its source and target maps coincide, so using the de Rham model for deformation cohomology we see that
\[H_\mathrm{def}^0(\mathcal{T}_\Lambda, \omega_\mathcal{T})=\Omega^1_\mathrm{cl}(M)^{\mathcal{T}_\Lambda-\mathrm{inv}}=\Omega^1_\mathrm{cl}(M).\]
\[H_\mathrm{def}^1(\mathcal{T}_\Lambda)\cong \Omega^1_\mathrm{mult}(\mathcal{T}_\Lambda)/\delta(\Omega^1(M))=\Omega^1_\mathrm{mult}(\mathcal{T}_\Lambda)\]
\[H_\mathrm{def}^1(\omega_\mathcal{T})= \Omega^2_\mathrm{cl,mult}(\mathcal{T}_\Lambda)/\delta(\Omega_\mathrm{cl}^2(M))=\Omega^2_\mathrm{cl, mult}(\mathcal{T}_\Lambda).\]

Therefore, in this case the 7-term sequence from Proposition <ref> becomes

\begin{align*}
0  & \to \Omega^1_\mathrm{cl}(M) \to \Omega^1(M) \stackrel{d}{\to} \Omega^2_\mathrm{cl}(M) \to H^{1}_\mathrm{def}(\mathcal{T}_\Lambda, \omega_\mathcal{T}) \to \\
&  \to \Omega^1_\mathrm{mult}(\mathcal{T}_\Lambda)\stackrel{d}{\to} \Omega^2_\mathrm{cl, mult}(\mathcal{T}_\Lambda)\to H^{2}_\mathrm{def}(\mathcal{T}_\Lambda,\omega_\mathcal{T})\to 0
\end{align*}

and $H^{k}_\mathrm{def}(\mathcal{T}_\Lambda,\omega_\mathcal{T})\cong H^{k-1}_\mathrm{def}(\omega_\mathcal{T})$ for $k> 2$.

In particular,
\[H^{2}_\mathrm{def}(\mathcal{T}_\Lambda,\omega_\mathcal{T})\cong H^2(\Omega^\bullet_\mathrm{mult}(\mathcal{T}_\Lambda), d_{dR}),\]

the second de Rham cohomology of the subcomplex of multiplicative forms on $\mathcal{T}_\Lambda$.

§.§ Linear Poisson manifolds of proper type

The Lie groupoid $T^*G\cong G\ltimes \mathfrak{g}^*$ integrating the linear Poisson manifold $(\mathfrak{g}^*,\pi_\mathfrak{g})$ is proper if and only if the Lie group $G$ is compact. In this case, we know from Proposition <ref> that
\[H^{2}_\mathrm{def}(T^*G, \omega_\mathrm{can})\cong \frac{H_\mathrm{def}^1(\omega_\mathrm{can})}{d[\Omega^1_\mathrm{mult}(T^*G)/\delta(\Omega^1(\mathfrak{g}^*))]}.\]

But since $\mathfrak{g}^*$ is contractible, $H_\mathrm{dR}^2(\mathfrak{g}^*)=0$, so $\delta \Omega_\mathrm{cl}^2(\mathfrak{g}^*)=\delta d\Omega^1(\mathfrak{g}^*)=d \delta\Omega^1(\mathfrak{g}^*)$ and therefore

\[H^{2}_\mathrm{def}(T^*G, \omega_\mathrm{can})\cong \frac{\Omega^2_\mathrm{mult, cl}(T^*G)/\delta \Omega^2_\mathrm{cl}(\mathfrak{g}^*)}{d[\Omega^1_\mathrm{mult}(T^*G)/\delta(\Omega^1(\mathfrak{g}^*))]}=\frac{\Omega^2_\mathrm{mult, cl}(T^*G)/d \delta\Omega^1(\mathfrak{g}^*)}{d[\Omega^1_\mathrm{mult}(T^*G)/\delta(\Omega^1(\mathfrak{g}^*))]},\]

which is the same as $H^2(\Omega^\bullet_\mathrm{mult}(T^*G), d_{dR})$.

§.§ Spectral sequence computations for deformation cohomology I

For computational purposes, it is useful to use the identification of the deformation complex of $(\G,\omega)$ with the total complex of the very simple double complex

\[\begin{tikzcd}
0    & 0     & 0    & 0    &        \\
\Omega^{2}_\mathrm{cl}(M) \arrow[u] \arrow[r, "\delta"]                               & \Omega^{2}_\mathrm{cl}(\G) \arrow[u] \arrow[r, "\delta"]                                         & \Omega^{2}_\mathrm{cl}(\mathcal{G}^{(2)}) \arrow[u] \arrow[r, "\delta"]                                               & \Omega^{2}_\mathrm{cl}(\mathcal{G}^{(3)}) \arrow[u] \arrow[r]                                               & \cdots \\
\Omega^{1}_\mathrm{def}(M) \arrow[u, "d"] \arrow[r, "\delta"] & \Omega^{1}_\mathrm{def}(\mathcal{G}) \arrow[u, "d"] \arrow[r, "\delta"] & \Omega^{1}_\mathrm{def}(\mathcal{G}^{(2)}) \arrow[u, "d"] \arrow[r, "\delta"] & \Omega^{1}_\mathrm{def}(\mathcal{G}^{(3)}) \arrow[u, "d"] \arrow[r] & \cdots
\end{tikzcd}\]

as per definition of the mapping cone, and detailed in Remark <ref>. We recall again that the inclusion $\Omega^{1}_\mathrm{def}(\G^{(\bullet)})\subset \Omega^{1}(\G^{(\bullet)})$ is a quasi-isomorphism. As such, we can also compute deformation cohomology of a symplectic groupoid using the cone of $-d_{dR}:\Omega^{1}(\G^{(\bullet)})\to\Omega_\mathrm{cl}^{2}(\G^{(\bullet)})$ instead of $-d_{dR}:\Omega^{1}_\mathrm{def}(\G^{(\bullet)})\to\Omega_\mathrm{cl}^{2}(\G^{(\bullet)})$. For the “rows first” spectral sequence associated to the corresponding double complex  <cit.>, the second page $E_2$ is

(m) [matrix of math nodes,
nodes in empty cells,nodes=minimum width=5ex,
minimum height=5ex,outer sep=-5pt,
column sep=1ex,row sep=1ex]

1          H_dR(Ω^2_cl(M)^-inv)      H_dRH_δ(Ω^2_cl())      H_dRH_δ(Ω^2_cl(^(2)))     ⋯

0          Ω^1_cl(M)^-inv      (H^1_def())_cl      (H^2_def())_cl       ⋯

0       1       2

[thick] (m-1-1.east) – (m-4-1.east) ;
[thick] (m-4-1.north) – (m-4-5.north) ;

where $(H^n_\mathrm{def}(\G))_\mathrm{cl}$ denotes $\mathrm{Ker}(d)\subset H_\delta(\Omega^{1}(\G^{(n)}))$. Let us also use the notation $H_{dR}^2(M)^{\G-\mathrm{inv}}=H^2(\Omega^\bullet(M)^{\G-\mathrm{inv}},d_{dR})$. Since the double complex was concentrated in the first two rows, the spectral sequence stabilizes at this page. Therefore $H_\mathrm{def}^k(\G,\omega)\cong E_2^{k,0}\oplus E_2^{k-1,1}$ as vector spaces, and more explicitly
\begin{equation}\label{eq:E_2}
\end{equation}
When we know more about $\G$, it is possible to have a better description of these pieces. For example, either equation (<ref>) for $k=2$ or the same argument used for linear Poisson manifolds of proper type (Section <ref>) shows the following.

If $\G\rightrightarrows M$ is a symplectic groupoid and $H^2_\mathrm{dR}(M)=0$, then
\[H^2_\mathrm{def}(\G,\omega)\cong H^2_\mathrm{mult}(\G)\oplus (H^2_\mathrm{def}(\G))_\mathrm{cl}.\]
If additionally $\G$ is proper, then
\[H^2_\mathrm{def}(\G,\omega)\cong H^2_\mathrm{mult}(\G).\]

By the previous discussion of this section and of Section <ref> we can also compute the cohomology in degrees zero and one, so for the sake of completeness, we recall that $H_\mathrm{def}^0(\G,\omega)=\Omega^1_\mathrm{cl}(M)^{\G-\mathrm{inv}}$ and $H^1_\mathrm{def}(\G,\omega)\cong H_{dR}^2(M)^{\G-\mathrm{inv}}\oplus (H^1_\mathrm{def}(\G))_\mathrm{cl}$.

§.§ Spectral sequence computations for deformation cohomology II

Let us now use the “columns first” spectral sequence for the double complex associated to the mapping cone of $-d_{dR}:\Omega_\mathrm{def}^{1}(\G^{(\bullet)})\to\Omega_\mathrm{cl}^{2}(\G^{(\bullet)})$. Its first page is

(m) [matrix of math nodes,
nodes in empty cells,nodes=minimum width=5ex,
minimum height=5ex,outer sep=2pt,
column sep=3ex,row sep=1ex]

1          H^2_dR(M)      Ω^2_cl()/d_dR(Ω^1_def())      Ω^2_cl(^(2))/d_dR(Ω^1_def(^(2)))      ⋯

0          Ω^1_def(M)_cl       Ω^1_def()_cl      Ω^1_def(^(2))_cl      ⋯

0       1       2      3
[-stealth] (m-2-2.east) – (m-2-3.west);
[-stealth] (m-2-3.east) – (m-2-4.west);

[-stealth] (m-3-2.east) – (m-3-3.west);
[-stealth] (m-3-3.east) – (m-3-4.west);

[thick] (m-1-1.east) – (m-4-1.east) ;
[thick] (m-4-1.north) – (m-4-5.north) ;

where $\Omega^1_\mathrm{def}(\G^{(k)})_\mathrm{cl}$ denotes the kernel of $d_{dR}:\Omega_\mathrm{def}^{1}(\G^{(k)})\to\Omega_\mathrm{cl}^{2}(\G^{(k)})$.
This spectral sequence stabilizes at the third page, and so $H_\mathrm{def}^k(\G,\omega)\cong E^{k,0}_3\oplus E^{k-1,1}_3$, where

\[E^{k,0}_3=H^{k}_{\delta_{\mathrm{def}}}(\Omega^{1}_{\mathrm{def}}(\G^{\bullet})_\mathrm{cl})\]

\[E^{k-1,1}_3=\mathrm{Ker}\left(d^{k-1,1}_2:H^{k-1}_{\delta}\left(\frac{\Omega^{2}_\mathrm{cl}(\G^{(k-1)})}{d_{dR}(\Omega^1_\mathrm{def}(\G^{(k-1)}))}\right)\to H^{k+1}_{\delta_{\mathrm{def}}}(\Omega^{1}_{\mathrm{def}}(\G^{\bullet})_\mathrm{cl})\right).\]

Again, if we know more about $\G$, these formulas become simpler. If $\G$ is proper, we can make use of the following variation of Theorem <ref>.

If $(\G,\omega)$ is a proper Lie groupoid, then
\[H_\delta^k(\Omega_\mathrm{def}(\G^{(\bullet)})_\mathrm{cl})=0\]
for all $k\geq 2$.

We will first rewrite the homotopy operator of $C^\bullet_\mathrm{def}(\G)$ in terms of the de Rham model $\Omega^{1}_\mathrm{def}(\G^{(\bullet)})$; we then check that it restricts to a homotopy operator for the subcomplex $\Omega^{1}_\mathrm{def}(\G^{(\bullet)})_\mathrm{cl}$. We recall that the homotopy operator is described in terms of a Haar system and a cut-off function, which exist for any proper Lie groupoid, and permit integration along the target fibres. We refer to [25] for details and for the proof of Theorem <ref> using this operator.

Let $k\geq 2$ and let $\beta\in\Omega^{1}_\mathrm{def}(\G^{(k+1)})_{\delta-\mathrm{cl}}$. The homotopy operator on the de Rham model $\Omega^{1}_\mathrm{def}(\G^{(\bullet)})$ assigns to $\beta$ the transgression $\alpha\in \Omega^{1}_\mathrm{def}(\G^{(k)})$ defined by

where $\tau:t^{*}TM\to T\G$ is a splitting of the target map of $\G$. Notice that, since $\beta\in\Omega^{1}_\mathrm{def}(\G^{(k+1)})$, the expression in the integral actually does not depend on the value of the tangent vectors at $h\in\G$ (cf. Remark <ref>). We remark that this expression can also be obtained by using the homotopy operator for VB-cohomology as in [10]. Equivalently, if $X\in\mathfrak{X}(\G^{(k)})$ the homotopy operator has the form

where $X^{(k+1)}\in\mathfrak{X}(\G^{(k+1)})$ is the vector field

induced by $X$.

We now prove that the transgression $\alpha$ of a closed 1-form $\beta\in \Omega^{1}_\mathrm{def}(\G^{(\bullet)})$ is a closed 1-form.
Recall that the de Rham differential of 1-forms can be written as

\begin{equation*}
\begin{split}
\end{split}
\end{equation*}

Let $\beta\in\Omega^{1}_\mathrm{def}(\G^{(k+1)})_{\delta-\mathrm{cl}}$ be such that $d\beta=0$, and let $X,Y\in\mathfrak{X}(\G^{(k)})$. Then,

\begin{align*}
&=\int_{t\text{-fibres}}\frac{d}{d\e}\Bigr|_{\e=0}\beta((\varphi^{X^{(k+1)}}_\e)_{*}Y^{(k+1)})\circ\varphi^{X^{(k+1)}}_\e d\mu_\G\ \ \ \\ %(\text{because } \beta\in\Omega^{1}_\mathrm{def}(\G^{(k+1)}))\\
\end{align*}

using that $\beta\in\Omega^{1}_\mathrm{def}(\G^{(k+1)})$ to pass from the second to the third lines. Similarly we can compute $(L_Y\alpha)(X)$. And finally,

\begin{align*}
\alpha([X,Y])&=\int_{t\text{-fibres}}\beta([X,Y]^{(k+1)})d\mu_\G\\
&=\int_{t\text{-fibres}}\beta(L_{X^{(k+1)}}Y^{(k+1)})d\mu_\G\ \ (\text{because }\beta\in\Omega^{1}_\mathrm{def}(\G^{(k+1)})).
\end{align*}

Therefore, with these computations we get that

which proves the claim.

This result, together with the formulas for $E^{k,0}_3$ and for $E^{k-1,1}_3$ from the beginning of this section, lead to a simpler description of $H_\mathrm{def}^k(\G,\omega)\cong E^{k,0}_3\oplus E^{k-1,1}_3$.

Let $\G$ be a proper Lie groupoid. Then for $k\geq 2$
\[H_\mathrm{def}^k(\G,\omega)\cong H^{k}_{\delta}\left(\frac{\Omega^{2}_\mathrm{cl}(\G^{(k-1)})}{d_{dR}(\Omega^1_\mathrm{def}(\G^{(k-1)}))}\right).\]

§ THE DEFORMATION COCYCLE OF $S$-CONSTANT DEFORMATIONS, AND MOSER TRICKS

In this section we describe the deformation cocycle mentioned in remark <ref> by using the simpler presentation of the deformation complex given by corollary <ref>.
We then use such a cocycle to state a result analogous to the classical Moser Theorem in the context of symplectic groupoids. We first carry out this work for $s$-constant deformations, where the deformation cocycles involved are canonically determined. In the case of  general deformations it is only the deformation cohomology class that will be well-defined. We consider that case in Section <ref>.

§.§ The deformation class

Let $(\calG_\e,\omega_\e)$ be a deformation of $(\calG,\omega)$. In remark <ref> we associate the cocycle $\xi_\e-\frac{d}{d\e}\omega_\e\in C^{2}_\mathrm{def}(\G,\omega)$ to the deformation $(\calG_\e,\omega_\e)$. Alternatively, using corollary <ref>, we can express such a cocycle as an element $\eta_\e$ of the mapping cone complex of the de Rham differential $-d_{dR}:\Omega^{1}(\G^{(\bullet)})\to\Omega^{2}(\G^{(\bullet)})$. Explicitly,

As we will show in proposition <ref> and remark <ref> below, every $\eta_\e$ is indeed a 2-cocycle of the deformation complex $C^{\bullet}_\mathrm{def}(\G_\e,\omega_\e)$ when viewed in terms of differential forms as in corollary <ref>. The corresponding cohomology class $[\eta_\e]$ will be called the deformation class.

We now prove that $\eta_0\in C^{2}_\mathrm{def}(\G,\omega)$ is in fact a cocycle, which will be called the infinitesimal deformation cocycle associated to the deformation $(\calG_\e,\omega_\e)$.

Let $(\G_\e,\omega_\e)$ be an $s$-constant deformation of $(\G,\omega)$. The 2-cochain $\eta_0=\zeta_0-\dot{\omega}_0\in\Omega^{1}(\G^{(2)})\bigoplus\Omega^{2}(\G)$, defined above, is a 2-cocycle. The corresponding cohomology class in $H^{2}_\mathrm{def}(\calG,\omega)$ depends only on the equivalence class of the deformation.

To prove this Proposition we first enunciate two technical lemmas. In what follows, denote by $\calG^{[2]}$ the domain of the division map $\overline{m}$ of $\calG$, i.e., the space of pairs of arrows with the same source; denote by $p_i:\calG^{[2]}\longrightarrow\calG$ the projections on $\calG$, and denote by $pr_i:\calG^{(2)}\longrightarrow\calG$ the projections from the space of composable arrows to $\calG$, for $i=1,2$.

A $k$-form $\omega\in\Omega^{k}(\calG)$ is multiplicative if, and only if

\begin{equation}\label{multipl.1}
\overline{m}^{*}\omega=p_1^{*}\omega-p_2^{*}\omega.
\end{equation}

We just need to show that equation (<ref>) is equivalent to the equation

\begin{equation}\label{multipl.2}
\end{equation}

This follows from the diffeomorphism $\psi:\calG^{[2]}\longrightarrow\calG^{(2)},$ $(p,q)\longmapsto(p,i(q))$, where $i:\calG\longrightarrow\calG$ is the inversion map of $\calG$. Indeed, by applying $\psi^{*}$ to equation (<ref>) and since $i^{*}\omega=-\omega$ ([8], Lemma 3.1), we obtain the equation (<ref>).

Let $(\calG,\overline{m}_\e)$ be an $s$-constant deformation of $\calG$. Define for every $\e\in I$ the diffeomorphism $\overline{M}_\e:\calG^{[2]}\longrightarrow\calG^{[2]},\ (p,q)\longmapsto(\overline{m}_\e(p,q),i_\e(q))$ and the map $\psi_\e:\calG^{[2]}\longrightarrow\calG^{(2)}_\e,\ (p,q)\longmapsto (p,i_\e(q))$. These families of maps satisfy the following properties:

*   $\overline{m}_\e\circ\overline{M}_\e^{-1}=p_1=(pr_1)_\e\circ\psi_\e$;

*   $p_1\circ\overline{M}_\e^{-1}=m_\e\circ\psi_\e$;

*   $p_2\circ\overline{M}_\e^{-1}=i_\e\circ p_2=(pr_2)_\e\circ\psi_\e$.

* If $V_\e$ denotes the time-dependent vector field on $\calG^{[2]}$ associated to the smooth family of diffeomorphisms $\overline{M}_\e$ (i.e., $\left.\frac{d}{d\t}\right|_{\t=\e}\overline{M}_\t(p,q)=V_\e(\overline{M}_\e(p,q))$), then

Items (1)-(3) are straightforward once that $\overline{M}_\e^{-1}(p,q)=(m_\e(p,i_\e(q)),i_\e(q))$.
For item (4) note that $V_\e(p,q)=\left.\frac{d}{d\t}\right|_{\t=\e}(\overline{M}_\t\circ\overline{M}_\e^{-1})(p,q)$, so its proof follows from applying $\left.\frac{d}{d\t}\right|_{\t=\e}$ to $(\overline{M}_\t\circ\overline{M}_\e^{-1})(p,q)=(\overline{m}_\t(m_\e(p,i_\e(q)),i_\e(q)),i_\t(i_\e(q)))$.

of proposition <ref>

The proof of the first part follows from differentiating the multiplicativity condition (<ref>) of the family $\omega_\e$. In fact, by taking derivative with respect to $\e$ and reordering the terms, we get

\begin{equation}\label{derivative}
\left.\frac{d}{d\e}\right|_{\e=\l}\bar{m}_{\e}^{*}\omega_\e=\left(-\bar{m}_\l^{*}(\left.\frac{d}{d\e}\right|_{\e=\l}\omega_\e)+p_1^{*}(\left.\frac{d}{d\e}\right|_{\e=\l}\omega_\e)-p_2^{*}(\left.\frac{d}{d\e}\right|_{\e=\l}\omega_\e)\right).
\end{equation}

Now, we use Lemma <ref> and the family of diffeomorphisms $\bar{M}_\e:\calG^{[2]}\longrightarrow\calG^{[2]}$ defined there. Since $\bar{M}_{\e}(p,q)=(\bar{m}_\e(p,q),i_\e(q))$, then $\left.\frac{d}{d\e}\right|_{\e=\l}\bar{m}_{\e}^{*}\omega_\l$ is equal to $\left.\frac{d}{d\e}\right|_{\e=\l}\bar{M}_{\e}^{*}pr_1^{*}\omega_\l$, which is equivalent to $\left.\frac{d}{d\e}\right|_{\e=\l}\bar{M}_{\l}^{*}(\bar{M}_\e\circ\bar{M}_{\l}^{-1})^{*}pr_1^{*}\omega_\l$. In this way, letting $\varphi^{\e,\l}:=\bar{M}_\e\circ\bar{M}_{\l}^{-1}$ (which, by construction of $V_\e$, is its time-dependent flow), then we have

\begin{equation*}
\begin{split}
\left.\frac{d}{d\e}\right|_{\e=\l}\bar{M}_{\l}^{*}(\bar{M}_\e\circ\bar{M}_{\l}^{-1})^{*}p_1^{*}\omega_\l&=\bar{M}_{\l}^{*}\left.\frac{d}{d\e}\right|_{\e=\l}(\varphi^{\e,\l})^{*}p_1^{*}\omega_\l\\
%&=\bar{M}_{\l}^{*}d(\i_{V_\l}p_1^{*}\omega_\l). ...........
\end{split}
\end{equation*}

That is, by item (1) of Lemma <ref>, equation (<ref>) becomes

\begin{equation*}
\begin{split}
\end{split}
\end{equation*}

Also, since $\omega_\l$ is a closed form, by Cartan's formula we have

\begin{equation*}
\begin{split}
(\psi_\l^{-1})^{*}L_{V_\l}(p_1^{*}\omega_\l)&=d\left((\psi_\l^{-1})^{*}(V_\l\intprod p_1^{*}\omega_\l)\right)\\%.............\\
\end{split}
\end{equation*}

where the last equality follows from identification (<ref>) in Section <ref>. Namely,

\begin{equation*}
\begin{split}
((\psi_\l)_{*}V_\l\intprod &(pr_1)_\l^{*}\omega_\l)_{(g,h)}=\\ &=\left.(dpr_1)_\l^{*}\right|_{(g,h)}\left[\left.\omega_\l^{b}\right|_{g}\left(d(pr_1)_\l\circ(d\psi_\l)_{(g,i_\l(h))}(V_\l(g,i_\l(h)))\right)\right]\\
&=-\zeta_\l(g,h)\ \ \ \ (\text{by identification } \eqref{identification}).
\end{split}
\end{equation*}

Therefore, we conclude that

That is,
This completes the first part of the proof by taking $\l=0$.

Take now $(\calG'_{\e},\omega'_\e)$ an equivalent deformation of $(\calG,\omega)$, and denote by $\eta'_0$ its infinitesimal cocycle. We will prove that $\eta_0-\eta'_0$ is exact. This follows from taking derivatives at $\e=0$ of the equivalence condition $F_\e^{*}\omega'_\e=\omega_\e+\delta_\e(\alpha_\e)$ (recall that $\alpha_0=0$). Doing so, we obtain

where $Z_\e$ is the time-dependent vector field associated to the family $F_\e$. Hence,

On the other hand, since $F_\e:\calG_\e\rightarrow\calG'_\e$ is an isomorphism for every $\e$, then the groupoid deformation cocycles $\xi_0$ and $\xi'_0$ satisfy that $\xi'_0-\xi_0=\delta^\textrm{def}_0(Z_0)$. In terms of differential forms, this amounts to $\zeta'_0-\zeta_0=\delta_0(-i_{Z_0}\omega)$ (see identification (<ref>)). Therefore, the equation above becomes

\begin{align*}
\eta_0-\eta'_0&=-\dot{\omega}_0+ \zeta_0-(-\dot{\omega}'_0+\zeta'_0)\\
&=\zeta_0-\zeta'_0 + (\dot{\omega}'_0-\dot{\omega}_0)\\
\end{align*}

The 2-cocycle associated to the trivial deformation vanishes in cohomology.

Since a deformation $(\calG_{\e},\omega_\e)$ of $(\calG_0,\omega_0)$ can be seen as the deformation $(\calG_{\e+\l},\omega_{\e+\l})$ of $(\calG_{\l},\omega_\l)$ for any $\l\in I$, then the corresponding cochain $\eta_{\l}\in C^{2}_\mathrm{def}(\calG_\l,\omega_\l)$ is also a cocycle. Furthermore, in the case of a trivial deformation we can say even more: every $\eta_\l$ is exact. This follows from noting that the deformation $(\calG_{\e},\omega_\e)$ of $(\calG_0,\omega_0)$ being trivial implies that $(\calG_{\e+\l},\omega_{\e+\l})$ is also a trivial deformation of $(\calG_{\l},\omega_\l)$, for any $\l\in I.$

In the next section we show that, under appropriate regularity and compactness conditions, the converse statement of the previous remark is also true.

§.§ Moser arguments and moduli spaces

(Moser Theorem for symplectic groupoids)

Let $(\calG_\e,\omega_\e)$ be an $s$-constant deformation of a compact symplectic groupoid $(\calG,\omega)$. Then, the deformation $(\calG_\e,\omega_\e)$ is trivial if, and only if, the deformation class
\[\eta_\e:=\zeta_\e-\dot{\omega}_\e \in C^{2}_\mathrm{def}(\calG_\e,\omega_\e)\]
is smoothly exact.

We will show that the smooth exactness of the deformation class $\eta_\e$ implies the triviality of the deformation $(\calG_\e,\omega_\e)$. The remark <ref> above shows the converse statement. Thus, let us assume that $\eta_\e$ is a smoothly exact family of cocycles. Then, we get

\begin{equation*}
\delta_{\t}(-\chi_\t)=\zeta_\t \text{ and } -\dot{\omega}_\t=-d(-\chi_\t)-\delta_\t(\tilde{\alpha}_\t);
\end{equation*}

where $\chi_\t\in\Omega^{1}(\calG_\t)$ and $\tilde{\alpha}_\t\in\Omega^{2}(M)$ are smooth families of forms. The first equation amounts to $\delta^{\t}_\mathrm{def}(X_\t)=\xi_\t$, where $X_\t$ is the smooth family of vector fields such that $\chi_\t=\iota_{X_\t}\omega_\t$ for every $\t$. Let $(\phi_\e,\varphi_\e)$ be the flow of the time-dependent vector field $X_\e$ (starting at time zero). Then, by letting  $\bar{\alpha}_\t:=\int_{0}^{\t}(\phi_\e^{*}\tilde{\alpha}_\e) d\e$ be the  primitive of the curve $\t\mapsto\phi_\t^{*}\tilde{\alpha}_\t$, we obtain

which is,

That is,

$$\phi_\e^{*}\omega_\e-\delta_0(\alpha_\e)=\omega, \text{ for } \alpha_\e=\bar{\alpha}_0-\bar{\alpha}_\e.$$

Hence, one concludes that the deformation $(\G_\e,\omega_\e)$ is trivial.

As a consequence of this version of the Moser Theorem, then for a compact symplectic groupoid $(\G,\omega)$ we can describe a neighbourhood of $\omega$ in a moduli space of multiplicative symplectic structures on $\G$. We say a moduli space and not the moduli space, because there are a few options of which equivalence relation to consider. We will use one that is suggested by the equivalence relation on deformations of symplectic groupoids.

The space of closed multiplicative $2$-forms on a Lie groupoid $\G$ is the subspace $\Omega_\mathrm{cl,mult}^2(\G)\subset \Omega^2(\G)$. It is a linear subspace, being the intersection of the kernel of the two linear operators $d_{dR}$ and $\delta$. Non-degeneracy is an open condition, so the space of multiplicative symplectic structures on $\G$ is an open neighbourhood of $\omega$ in  $\Omega_\mathrm{cl,mult}^2(\G)$, which we denote by $\mathcal{S(\G)}$.

Two multiplicative symplectic forms $\omega_1,\omega_2\in\mathcal{S(\mathcal{\G})}$ are said to be convex gauge-isotopic if the deformation $(\G,\omega_\e)$ given by $\omega_1 +\e(\omega_2-\omega_1)$ is a trivial deformation of symplectic groupoids (in particular $\omega_\e$ is required to be non-degenerate for all $\e\in[0,1]$).

Denote by $\sim_\mathrm{cgi}$ the equivalence relation on $\mathcal{S}(\G)$ generated by convex gauge-isotopies.

Let $(\G,\omega)$ be a compact symplectic groupoid. Then there is a neighbourhood $\mathcal{U}$ of $\omega$ in $\mathcal{S}({\G})$ and a neighbourhood $\mathcal{V}$ of $0$ in $H^2_\mathrm{def}(\G,\omega)$, such that the map $\kappa :\mathcal{U}\to H^2_\mathrm{def}(\G,\omega)$, defined by $\kappa(\omega_1) = [0\oplus (\omega_1-\omega)]$ induces a 1:1 correspondence
\[\mathcal{U}\big/\sim_\mathrm{cgi}\ \  \longleftrightarrow\ \  \mathcal{V}\subset H^2_\mathrm{def}(\G,\omega),\]
sending the equivalence class of $\omega$ to $0$.

The map $\kappa$ descends to the quotient: if $\omega_1$ and $\omega_2$ are convex gauge-isotopic, then the deformation $(\G,\omega_1 +\e(\omega_2-\omega_1))$ of $(\G,\omega_1)$ is trivial, so its deformation class $[0\oplus(\omega_2-\omega_1)]\in H^2_\mathrm{def}(\G,\omega_1)$ vanishes. Therefore $[0\oplus(\omega_2-\omega_1)]\in H^2_\mathrm{def}(\G,\omega)$ also vanishes. Thus $\kappa(\omega_2)-\kappa(\omega_1)=[0\oplus(\omega_2-\omega)]-[0\oplus(\omega_1-\omega)]=[0\oplus(\omega_2-\omega_1)]=0$.

Let $\mathcal{U}''$ be a convex neighbourhood of $\omega$ in $\mathcal{S}(\G)$, and let $\omega_1,\omega_2\in \mathcal{U}''$ be such that $\kappa(\omega_1)-\kappa(\omega_2)=0$. By definition, $[0\oplus(\omega_1-\omega_2)]=0\in H^2_\mathrm{def}(\G,\omega)$, meaning that $(\omega_1-\omega_2)$ is exact in deformation cohomology. We can consider the $s$-constant deformation $(\G,\omega_2+\e(\omega_1-\omega_2))$ of $(\G,\omega_2)$ (it is a deformation of symplectic groupoids because $\mathcal{U}''$ is convex). This deformation has a constant deformation class $[0\oplus(\omega_1-\omega_2)]=0\in H^2_\mathrm{def}(\G,\omega_2)$ for all $\e$, which we know to be exact. Therefore the deformation classes are smoothly exact and by Theorem <ref> we conclude the triviality of the deformation. This implies that $(\G,\omega_1)$ and $(\G,\omega_2)$ are convex gauge-isotopic. Hence $\kappa$ descends to an injective map.

Neighbourhoods $\mathcal{U}'$ of $\omega$ and $\mathcal{V}'$ of $0$ can be chosen such that $\kappa$ maps $\mathcal{U}'$ onto $\mathcal{V}'$. The reason for this is that by Proposition <ref> any class in $H^2_\mathrm{def}(\G,\omega)$ has a representative of the form $0\oplus \omega_1$ with $\omega_1\in \Omega_\mathrm{mult}^2(\G)$. Let $\mathcal{V}'$ be small enough that for every class in $\mathcal{V}'$ there is such a representative $0\oplus \omega_1$ for which $\omega+\omega_1$ is non-degenerate, and $\omega_1$ belongs to $\mathcal{U}''$. By definition $\kappa(\omega+\omega_1)=[0\oplus \omega_1]$, so $\kappa:\mathcal{U}'=\kappa^{-1}(\mathcal{V}')\to \mathcal{V'}$ becomes surjective. Finally, let $\mathcal{U}\subset \mathcal{U}'$ be a convex neighbourhood of $\omega$, and let $\mathcal{V}=\kappa(\mathcal{U})$.

The map $\kappa$ plays a similar role to a Kodaira-Spencer map in the theory of deformations of complex structures <cit.>. Allowing for some heuristics, and if we disregard issues with spaces of infinite dimension, there is a tautological (infinite-dimensional) family of symplectic groupoids
\[pr_{\mathcal{S}(\G)}: \mathfrak{G}=(\G\times \mathcal{S}(\G),\widetilde{\omega})\longrightarrow \mathcal{S}(\G),\]

where $\tilde{\omega}$ is the tautological $pr_{S(\G)}$-foliated $2$-form on $\mathfrak{G}=\G \times S(\G)$ with values $\tilde{\omega}(g,\omega)=\omega(g)$. The fibre $pr_{\mathcal{S}(\G)}^{-1}(\omega)$ of this family over $\omega\in \mathcal{S}(\G)$ is equal to $(\G,\omega)$.

By analogy with the case of complex manifolds, the Kodaira-Spencer map of this family is a map $KS:T_\omega \mathcal{S}(\G)\to H^2_\mathrm{def}(\G,\omega)$; given a smooth curve $\gamma:I\to \mathcal{S}(\G)$, define $KS(\dot{\gamma}(0))$ to be the deformation class of the deformation $\textbf{p}:\gamma^*\mathfrak{G}\to I$.
Since $\mathcal{S}(\G)$ is an open subset of the vector space $\Omega^2_\mathrm{mult,cl}(\G)$, there is a natural identification $T_\omega \mathcal{S}(\G)\cong \Omega^2_\mathrm{mult,cl}(\G)$. Under this identification, the Kodaira-Spencer map $KS$ corresponds to $\kappa$.

For a compact symplectic groupoid $(\G,\omega)$, taking the analogy further, a germ of a neighbourhood of $0$ in $H^2_\mathrm{def}(\G,\omega)$ plays the role of what is called a Kuranishi space (cf. <cit.>, see also Corollary 9 in loc. cit.) because infinitesimal deformations of $(\G,\omega)$ are unobstructed, as the proof of Theorem <ref> shows.

It also makes sense to study deformations of multiplicative symplectic forms on a fixed Lie groupoid $\G$, up to a stricter equivalence relation where triviality is given by paths of symplectic groupoid isomorphisms (i.e., not allowing for cohomologically trivial gauge transformations as equivalences). In this case, the correct deformation complex to consider would be $(\Omega^\bullet_\mathrm{mult}(\G), d_{dR})$, and a Moser theorem in that setting is detailed in [12]. It is similar to Theorem <ref> in the sense that when $\G$ is compact, deformations will be trivial if and only if their deformation classes in $H^2_\mathrm{mult}(\G)$ are smoothly exact.
This in turn will lead to a similar correspondence to that of Theorem <ref>, of the form

\[\mathcal{U}/\sim\ \ \longleftrightarrow\ \  \mathcal{V}\subset H^2_\mathrm{mult}(\G),\]
for neighbourhoods $\mathcal{U}$ of $\omega$ in $\mathcal{S}(\G)$ and $\mathcal{V}$ of $0$ in $H^2_\mathrm{mult}(\G)$. Here $\sim$ denotes the equivalence relation generated by convex isotopies, where $\omega_0$ and $\omega_1$ are convex-isotopic if the form $\omega_\e=\omega_0+\e(\omega_1-\omega_0)$ is symplectic for all $\e\in [0,1]$, and there is an isotopy $F_\e$ of $\G$ such that each $F_\e$ is a Lie groupoid isomorphism satisfying $F_\e^*\omega_\e=\omega_0$.

Any deformation $\textbf{p}:(\widetilde{\G}\rightrightarrows \tilde{M})\to (I\rightrightarrows I)$ of a compact Lie groupoid $\G$ is a trivial deformation <cit.>. If moreover $\textbf{p}$ is proper, then the entire family $\widetilde{\G}$ is trivial, without the need to restrict to a smaller interval $J\subset I$.
This means that any deformation $(\G_\e,\omega_\e)$ of a compact symplectic groupoid is equivalent to one of the form $(\G,\phi_\e^*\omega_\e)$, and in particular $s$-constant, for $\e$ small enough. Thus, by Proposition <ref>, any deformation class in $H^2_\mathrm{def}(\G,\omega)$ is represented by a cocycle of the form $0\oplus \dot{\omega}_0\in C^2_\mathrm{def}(\G)\oplus C^1_\mathrm{def}(\omega)$, and has also a corresponding cocycle in $H^2_\mathrm{mult}(\G)$.

Therefore, if the space of Lie groupoid structures on a compact manifold $\G$ (or perhaps having an underlying submersion $s:\G\to M$ as source) were locally path-connected, then the correspondences of Theorem <ref> and of Remark <ref> can be upgraded. They would give local descriptions not only of moduli spaces of multiplicative symplectic forms on $\G$, but of symplectic groupoids structures on the manifold $\G$ (respectively, with underlying source $s:\G\to M$). Although we do not know if the space of Lie groupoid structures on $\G$ is locally path-connected, in this direction we note that the space of actions of a compact Lie group on a compact manifold is, indeed, locally path-connected [68].

§ THE DEFORMATION CLASS OF GENERAL DEFORMATIONS

In this section we show how general deformations of symplectic groupoids can also be handled in an infinitesimal way.

§.§ Defining the deformation class

In order to describe the deformation class of a deformation $(\widetilde{\G},\omega)$ we first reinterpret the definition of the deformation cocycle in the case of $s$-constant deformations $(\widetilde{\G}=\calG\times I,\omega_\e)$. The deformation cocycle $\eta_\e=\zeta_\e-\dot{\omega}_\e$ can be alternatively expressed as

\begin{equation}\label{eq:cocycle2}
\eta_\e=\sigma(\omega(\delta_\mathrm{def}(\partial/\partial\e),\cdot))_{|_{\G^{(2)}_\e}}-(\mathcal{L}_{\partial/\partial\e}\omega)_{|_{\G_\e}},
\end{equation}

where $\partial/\partial\e$ is the vector field on $\widetilde{\G}$ whose flow $F_\e$ is $(g,\l)\mapsto (g,\l+\e)$, and
\[\sigma:pr_1^{*}T^{*}\widetilde{\calG}\to T^{*}\widetilde{\calG}^{(2)}\]
is the map of vector bundles over $\G^{(2)}$ defined on Remark <ref> by $\sigma=(dpr_1)^*$.

Note also that $\omega$ is a 2-form on $\widetilde{\calG}$ which restricts to $\omega_\e$ over the fibre $\calG\times \{\e\}$.
This alternative description of the deformation cocycle
can be extended in such a way that the deformation class still makes sense for a general deformation.

Let $(\widetilde{\calG},\omega)$ be a deformation of $(\G,\omega_\G)$, and let $X\in\mathfrak{X}(\widetilde{\G})$ be a transverse vector field; that is, a vector field on $\widetilde{\G}$ which projects by $\textbf{p}\circ\tilde{s}$ to the vector field $\partial/\partial\e$ on $I$. Then,

\begin{equation}\label{eq:generalcocycle}
\eta_\e:=\sigma(\omega(\delta_\mathrm{def}X,\cdot))|_{\G^{(2)}_\e}-(\mathcal{L}_{X}\omega)|_{\G_\e}\  \in\ \Omega^{1}_\mathrm{def}(\calG_\e^{(2)})\oplus\Omega^{2}(\calG_\e).
\end{equation}

defines a family of cocycles $\eta_\e\in C^{2}_\mathrm{def}(\calG_\e,\omega_\e)$.
Moreover, the cohomology class $[\eta_0]$ at time zero does not depend on the choice of the transverse vector field $X$.

The resulting cohomology class $[\eta_0]\in H^2_\mathrm{def}(\G,\omega_\G)$ is called the deformation class associated to the deformation $(\widetilde{\G},\omega)$.

Equivalently, the family of cocycles $\eta_\e$ can be viewed as a single cocycle $\eta$ on the “foliated version” $C^\bullet_\mathrm{def}(\widetilde{\calG},\omega)_{\mathcal{F}}$ of $C^\bullet_\mathrm{def}(\widetilde{\calG},\omega)$. That is, on the mapping cone complex of the leafwise de Rham differential $d_{\mathcal{F}}:\Omega^{1}_\mathrm{def}(\widetilde{\calG}^{(\bullet)})_{\mathcal{F}}\to\Omega^{2}_{\mathcal{F}}(\widetilde{\calG}^{(\bullet)})$. Here, to shorten the notation we let $\mathcal{F}$ refer to the foliation $\calF^{(n)}$ on each $\widetilde{\G}^{(n)}$ with leaves $\G_\e^{(n)}$, induced by the foliation on $\widetilde{\G}$ previously denoted by $\calF$, having the fibres $\G_\e$ as leaves. The space $\Omega^{1}_\mathrm{def}(\widetilde{\calG}^{(\bullet)})_{\mathcal{F}}$ is the space of the foliated forms corresponding to forms in $\Omega^{1}_\mathrm{def}(\widetilde{\calG}^{(\bullet)})$. Explicitly, we have

\begin{equation}
\eta=\sigma(\omega(\delta_\mathrm{def}X,\cdot))|_{\mathcal{F}^{(2)}}-(\mathcal{L}_{X}\omega)|_{\mathcal{F}}\ \ \in\ \Omega^{1}_\mathrm{def}(\widetilde{\calG}^{(2)})_\mathcal{F}\oplus\Omega^{2}_\mathcal{F}(\widetilde{\calG}).
\end{equation}

of Proposition <ref>

Let $\calM:\widetilde{\calG}^{(2)}\rightarrow\widetilde{\calG}^{(2)}$ be the diffeomorphism defined by $\calM(g,h)=(M(g,h),i(h))$, where $M$ is the multiplication of $\widetilde{\calG}$.

On the one hand, notice that

\begin{align*}
&=\calM^{*}\calL_{X^{pr_1}}pr_1^{*}\omega,\ \ \ \ \text{where }\ \ X^{pr_1}_{(g,h)}:=(X_g,(i_{*}X)_h)\in T_{(g,h)}\widetilde{\calG}^{(2)}\\
&=\calL_{X^{M}}\calM^{*}pr_1^{*}\omega,\ \ \ \ \text{where }\ \ X^{M}_{(g,h)}\ :=(\calM_{*}^{-1}X^{pr_1})_{(g,h)}\\ & \hspace{5.5cm} =(X_{gh}\cdot(i_{*}X)_{h^{-1}},X_h)\in T_{(g,h)}\widetilde{\calG}^{(2)}\\
\end{align*}

That is,

\begin{align*}
&=(d\i_{X^{M}}pr_1^{*}\omega)_{(g,h)}+(d_{(g,h)}pr_1)^{*}\big(d\omega(X_{gh}\cdot di(X_h),-,-)\big)\\ &+ (\calL_{X^{M}}p_2^{*}\omega)_{(g,h)}.
\end{align*}

On the other hand,

\begin{align*}
\end{align*}

\[p_2^{*}\calL_X\omega=\calL_{X^{M}}p_2^{*}\omega.\]

\begin{align*}
(pr_1^{*}\calL_X\omega &-M^{*}\calL_X\omega +p_2^{*}\calL_X\omega)_{(g,h)} = \\ &=(d_{(g,h)}pr_1)^{*}d\omega(X_g,-,-)+(d\;pr_1^{*}\i_X\omega)_{(g,h)}-(d\i_{X^{M}}pr_1^{*}\omega)_{(g,h)}\\ &-(d_{(g,h)}pr_1)^{*}\big(d\omega(X_{gh}\cdot di(X_h),-,-)\big)\\
&=(d_{(g,h)}pr_1)^{*}d\omega(X_g-X_{gh}\cdot di(X_h),-,-)+ \big(d(pr_1^{*}\i_X\omega-\i_{X^{M}}pr_1^{*}\omega)\big)_{(g,h)}\\
&=\big(\i_{(\delta_\mathrm{def}X,0)}pr_1^{*}d\omega\big)_{(g,h)} + \big(d\i_{(\delta_\mathrm{def}X,0)}pr_1^{*}\omega\big)_{(g,h)}.
\end{align*}

Thus, since $(\delta_\mathrm{def}X)_{(g,h)}$ is tangent to the fibres of $\G_\e$ of $\widetilde{\calG}$ and each restriction $\omega_\e\equiv\omega|_{\calG_\e}$ is a closed 2-form, then

for all $\e$, which proves the proposition.

Alternatively, since the tangent lift map $\oplus^{2}T_\mathrm{def}$ takes transversal vector fields to transversal vector fields, Proposition <ref> can also be proved in terms of deformations of Lie groupoid morphisms by using the results of [12].

§.§ Triviality of deformations with exact class

In this section we generalize Theorem <ref> to general deformations of compact symplectic groupoids.

Let $(\widetilde{\calG},\omega)$ be a deformation of a compact symplectic groupoid $(\calG,\omega_\calG)$ and let $\eta=\sigma(\omega(\delta_\mathrm{def}X,\cdot))|_{\mathcal{F}^{(2)}}-(\mathcal{L}_{X}\omega)|_{\mathcal{F}}$ be the deformation cocycle in the foliated cohomology $C^{\bullet}_\mathrm{def}(\G,\omega)_{\mathcal{F}}$, associated to the deformation via a transverse vector field $X$ on $\widetilde{\G}$. Then, $(\widetilde{\calG},\omega)$ is trivial if and only if $\eta$ is exact.

The proof of this result will be based on the following lemma.

Let $(\widetilde{\G},\omega)$ be a deformation of $(\G,\omega_\G)$ as in the theorem above. Then, the associated foliated form $\omega_\calF$ induces the isomorphism

$$C^{\bullet}_\mathrm{def}(\widetilde{\calG})_{\calF}\to \Omega^{1}_\mathrm{def}(\widetilde{\calG}^{(\bullet)})_\calF,\ \xi\mapsto -\sigma(\omega(\xi,\cdot))_{\calF}=-\sigma_\calF(\omega_\calF(\xi,\cdot)).$$

Under this isomorphism, the image $\mathcal{Y}=-i_Y\omega_\calF$ of an element $Y\in C^{1}_\mathrm{def}(\widetilde{\calG})_{\calF}$ satisfies

$$d_\calF\mathcal{Y}=-(\mathcal{L}_{Y}\omega)_{\calF},\ \text{and }\  \delta_\calF\mathcal{Y}=-\sigma(\omega(\delta_\mathrm{def}Y,\cdot))_\calF.$$

Checking that $\omega_\calF$ induces an isomorphism is done by essentially the same arguments as those for Proposition <ref>.

\begin{align*}
\end{align*}

where the second equality is due to the fact that $\omega$ is $d_\calF$-closed. Moreover,

\begin{align*}
\delta_\calF\mathcal{Y}&=-\delta_\calF(\i_Y\omega_\calF) =-\delta_\calF(\i_Y\omega)_\calF =-(\delta\i_Y\omega)_\calF\\
& =(\delta(\omega^{b})^{*}(Y))_\calF\\
& =((\omega^{b})^{*}(\delta_\mathrm{def}Y))_\calF \\
& =-\sigma(\omega(\delta_\mathrm{def}Y,\cdot))_\calF. \qedhere
\end{align*}

of Theorem <ref>

If $\eta$ is exact, then there exists $\alpha\oplus\mathcal{Y}\in\Omega^{2}_{\calF}(\widetilde{M})\oplus\Omega^{1}_\mathrm{def}(\widetilde{\calG})_{\calF}$ such that

\begin{equation}\label{eq:2*}
\end{equation}

Since the foliated 2-form $\omega_\calF$ induces the isomorphism $C^{*}_\mathrm{def}(\calG)_{\calF}\to \Omega^{1}_\mathrm{def}(\calG^{(*)})_\calF$ (see previous Lemma), it follows that $\mathcal{Y}=-i_{Y}\omega_\calF$ for some vector field $Y$ on $\widetilde{\G}$ which is tangent to the fibres $\G_\e$ of $\widetilde{\G}$. Therefore, the first equation is equivalent to

\begin{equation}\label{eq:1*}
\sigma(\omega(\delta_\mathrm{def}X,\cdot))|_{\mathcal{F}^{(2)}}=-\sigma(\omega(\delta_\mathrm{def}Y,\cdot))|_{\mathcal{F}^{(2)}},
\end{equation}

and equation (<ref>) becomes

\begin{equation}\label{eq:2**}
\end{equation}

Thus, equation (<ref>) says that $X+Y$ is a multiplicative and transverse vector field on $\widetilde{\G}$. Let us denote by $F_\e$ the flow of $X+Y$ on $\widetilde{\G}$. Then the equation (<ref>) is equivalent to

Denote by $(\Phi_\e,\phi_\e)$ the family of restricted morphisms $F_\e|_{{\G}_0}:\G_0\to{\G}_\e$. Then, by restricting the previous equation to the fibre ${\G}_0$ we get

\begin{align*}
\frac{d}{d\e}\Phi_\e^{*}\omega_\e&=\Phi_\e^{*}(\delta_{{\G}_\e}(\alpha_\e))\\
\end{align*}

where $\bar{\alpha}_\t:=\int_{0}^{\t}(\phi_\e^{*}\alpha_\e) d\e$ is the primitive of the curve $\e\mapsto\phi_\e^{*}\alpha_\e$. Hence, the sequence of equalities tells us that the expression $\Phi_\e^{*}\omega_\e-\delta_{\widetilde{\G}_0}(\bar{\alpha}_\e)$ is constant in $\e$. Thus,

Therefore, if $\Phi:\G\times I\to\widetilde{\G}$ is the isomorphism determined by the family $\Phi_\e$ and $\bar{\alpha}\in\Omega^{2}_\mathcal{F}(M\times I)_\mathrm{cl}$ is the  closed foliated 2-form which equals $\bar{\alpha}_\e$ when restricted to the fibre $\G_\e$, then

where $pr_\G:\G\times I\to \G$ is the obvious projection to $\G$. This proves the triviality of the deformation $(\widetilde{\calG},\omega)$.

Alternatively, Theorem <ref> could be  deduced from Theorem <ref> by first using the Reeb stability theorem for the foliation of $\widetilde{\G}$ by the fibres $\G_\e$ (at least for $\widetilde{\G}$ Hausdorff) and using the triviality of proper deformations of compact Lie groupoids <cit.>. Nonetheless, it seems to us that it is useful  having a direct proof of the general version of the theorem at hand, without the need to pass through a specific type of deformation. This proof should also leave more room for generalizations to relative or semi-local versions for groupoids admitting well adapted tubular neighborhoods, or normal forms, around submanifolds as for example in <cit.>, or [38].

§ MORE EXAMPLES AND RELATION WITH POISSON COHOMOLOGY

We once more use a spectral sequence in order to compute deformation cohomology, as done in sections <ref> and <ref>. This time we turn to the “columns first” spectral sequence for the double complex associated to the mapping cone of $-d_{dR}:\Omega^{1}(\G^{(\bullet)})\to\Omega_\mathrm{cl}^{2}(\G^{(\bullet)})$. Its first page is

(m) [matrix of math nodes,
nodes in empty cells,nodes=minimum width=5ex,
minimum height=5ex,outer sep=2pt,
column sep=3ex,row sep=1ex]

1          H_dR^2(M)      H_dR^2()      H_dR^2(^(2))     H_dR^2(^(3))

0          Ω^1_cl(M)      Ω^1_cl()      Ω^1_cl(^(2))      Ω^1_cl(^(3))

0       1       2      3
[-stealth] (m-2-2.east) – (m-2-3.west);
[-stealth] (m-2-3.east) – (m-2-4.west);
[-stealth] (m-2-4.east) – (m-2-5.west);
[-stealth] (m-3-2.east) – (m-3-3.west);
[-stealth] (m-3-3.east) – (m-3-4.west);
[-stealth] (m-3-4.east) – (m-3-5.west);
[thick] (m-1-1.east) – (m-4-1.east) ;
[thick] (m-4-1.north) – (m-4-6.north) ;

If we assume that $M$, $\G$ and $\G^{(2)}$ have vanishing first and second de Rham cohomology, then this page becomes

(m) [matrix of math nodes,
nodes in empty cells,nodes=minimum width=5ex,
minimum height=5ex,outer sep=2pt,
column sep=3ex,row sep=1ex]

1          0      0     0     H_dR^2(^(3))

0          d(C^∞(M))      d(C^∞())     d(C^∞(^(2)))      Ω^1_cl(^(3))

0       1       2      3

[-stealth] (m-3-2.east) – (m-3-3.west);
[-stealth] (m-3-3.east) – (m-3-4.west);
[-stealth] (m-3-4.east) – (m-3-5.west);
[thick] (m-1-1.east) – (m-4-1.east) ;
[thick] (m-4-1.north) – (m-4-6.north) ;

In  this case the second page is

(m) [matrix of math nodes,
nodes in empty cells,nodes=minimum width=5ex,
minimum height=5ex,outer sep=2pt,
column sep=2ex,row sep=1ex]

1          0      0     0      H_δ(H_dR^2(^(3)))

0          H_δ(d(C^∞(M)))      H_δ(d(C^∞()))     H_δ(d(C^∞(^(2))))      H_δ(Ω^1_cl(^(3)))

0       1       2      3
[thick] (m-1-1.east) – (m-4-1.east) ;
[thick] (m-4-1.north) – (m-4-6.north) ;

and therefore the deformation cohomology of the symplectic groupoid $(\G,\omega)$ is very closely related in degrees up to $3$ (the most important ones for deformation theory) to the differentiable cohomology of $\G$, which is $H_\delta(C^\infty(\G^{(\bullet)}))$.

Recall also that there is a van Est map relating differentiable cohomology of $\G$ with the algebroid cohomology of its Lie algebroid [19], which is naturally isomorphic to the Poisson cohomology of the base, in the case of a symplectic groupoid.

We now take a closer look at some (families of) symplectic groupoids satisfying these conditions. We start by illustrating a general strategy with the following case.

§.§ Linear Poisson structures - Part 2

Let $(\mathfrak{g}^*, \pi_\mathfrak{g})$ be a linear Poisson structure and let $G$ be the $1$-connected Lie group integrating the Lie algebra $\mathfrak{g}$. We recall that the cotangent groupoid $(T^*G, \omega_\mathrm{can})$ is a source 1-connected integration of $(\mathfrak{g}^*, \pi_\mathfrak{g})$, and that $T^*G\cong G\ltimes \mathfrak{g}^*$.

Let $\mathfrak{g}$ be a Lie algebra and let $G$ be the $1$-connected Lie group integrating it. Then there are isomorphisms

\[H_\mathrm{def}^0(T^*G,\omega_\mathrm{can})\cong H^0_{\pi_\mathfrak{g}}(\mathfrak{g}^*)/\mathbb{R}, \ \ \ H_\mathrm{def}^i(T^*G,\omega_\mathrm{can})\cong H^i_{\pi_\mathfrak{g}}(\mathfrak{g}^*),\]
for $i=1,2$, and an injective map
\[H_\mathrm{def}^3(T^*G,\omega_\mathrm{can})\to H^3_{\pi_\mathfrak{g}}(\mathfrak{g}^*).\]

Using that $T^*G\cong G\ltimes \mathfrak{g}^*$, we compute the deformation cohomology of $(T^*G,\omega_\mathrm{can})$ using the “columns first” spectral sequence for the double complex associated to the mapping cone of $d:\Omega^1((G\ltimes \mathfrak{g}^*)^{(\bullet)})\to\Omega^2_\mathrm{cl}((G\times \mathfrak{g}^*)^{(\bullet)})$. Its first page is

(m) [matrix of math nodes,
nodes in empty cells,nodes=minimum width=5ex,
minimum height=5ex,outer sep=2pt,
column sep=3ex,row sep=1ex]

1          H_dR^2(𝔤^*)      H_dR^2(G⋉𝔤^*)      H_dR^2((G⋉𝔤^*)^(2))     H_dR^2((G⋉𝔤^*)^(3))

0          Ω^1_cl(𝔤^*)      Ω^1_cl(G⋉𝔤^*)      Ω^1_cl((G⋉𝔤^*)^(2))      Ω^1_cl((G⋉𝔤^*)^(3))

0       1       2      3
[-stealth] (m-2-2.east) – (m-2-3.west);
[-stealth] (m-2-3.east) – (m-2-4.west);
[-stealth] (m-2-4.east) – (m-2-5.west);
[-stealth] (m-3-2.east) – (m-3-3.west);
[-stealth] (m-3-3.east) – (m-3-4.west);
[-stealth] (m-3-4.east) – (m-3-5.west);
[thick] (m-1-1.east) – (m-4-1.east) ;
[thick] (m-4-1.north) – (m-4-6.north) ;

For the action groupoid $G\ltimes \mathfrak{g}^*$, the nerve is $(G\ltimes \mathfrak{g}^*)^{(n)}\cong G^n\times \mathfrak{g}^*$. Because $G$ is a $1$-connected Lie group, and since $\pi_2(G)=0$ for any Lie group, we conclude by Hurewicz's Theorem that $(G\ltimes \mathfrak{g}^*)^{(n)}$ has vanishing first and second de Rham cohomology for any $n$.

Thus, the first line of the page above vanishes, and the de Rham differential induces a surjective cochain map onto the $0$-th line, $d: C^\infty(G^\bullet\times \mathfrak{g}^*)\to \Omega_\mathrm{cl}^1(G^\bullet\times \mathfrak{g}^*)$. The first page becomes

(m) [matrix of math nodes,
nodes in empty cells,nodes=minimum width=5ex,
minimum height=5ex,outer sep=2pt,
column sep=3ex,row sep=1ex]

1          0      0      0     0

0          dC^∞(𝔤^*)      dC^∞(G⋉𝔤^*)      dC^∞((G⋉𝔤^*)^(2))      dC^∞((G⋉𝔤^*)^(3))

0       1       2      3
[-stealth] (m-2-2.east) – (m-2-3.west);
[-stealth] (m-2-3.east) – (m-2-4.west);
[-stealth] (m-2-4.east) – (m-2-5.west);
[-stealth] (m-3-2.east) – (m-3-3.west);
[-stealth] (m-3-3.east) – (m-3-4.west);
[-stealth] (m-3-4.east) – (m-3-5.west);
[thick] (m-1-1.east) – (m-4-1.east) ;
[thick] (m-4-1.north) – (m-4-6.north) ;

Since both $G$ and $\mathfrak{g}^*$ are connected, the kernel of $d$ consists of constant functions, forming the complex $\mathbb{R}\stackrel{0}{\to}\mathbb{R}\stackrel{id}{\to}\mathbb{R}\stackrel{0}{\to}\mathbb{R}\stackrel{id}{\to}\cdots$. Its cohomology vanishes in all degrees except $0$, where it is isomorphic to $\mathbb{R}$. Therefore, the terms of this spectral sequence stabilize at the second page, where we have on the $0$-th line that $H_\delta(\Omega_\mathrm{cl}^1(\mathfrak{g}^*)))\cong H_\mathrm{diff}^0(G\ltimes \mathfrak{g}^*)/\mathbb{R}$ and for $i>0$, $H_\delta(\Omega_\mathrm{cl}^1(G^i\times \mathfrak{g}^*))\cong H_\mathrm{diff}^i(G\ltimes \mathfrak{g}^*)$.

Finally, since the $s$-fibres of $G\ltimes \mathfrak{g}^*$ are $2$-connected, the van Est map relating differentiable and algebroid cohomology is an isomorphism up to degree 2, and injective in degree 3. The statement follows, as the algebroid cohomology of the algebroid of $G\ltimes \mathfrak{g}^*$ is naturally isomorphic to the Poisson cohomology of $\mathfrak{g}^*$.

§.§ Groupoids with homologically 2-connected nerve