Picard group of isotropic realizations of twisted Poisson manifolds

Let $B$ be a twisted Poisson manifold with a fixed tropical affine structure given by a period bundle $P$. In this paper, we study the classification of almost symplectically complete isotropic realizations (ASCIRs) over $B$ in the spirit of \cite{DD}. We construct a product among ASCIRs in analogy with tensor product of line bundles, thereby introducing the notion of the Picard group of $B$. We give descriptions of the Picard group in terms of exact sequences involving certain sheaf cohomology groups, and find that the `N\'eron-Severi group' is isomorphic to $H^2(B, \underline{P})$. An example of an ASCIR over a certain open subset of a compact Lie group is discussed.


Introduction
A completely integrable Hamiltonian system is a symplectic 2n-dimensional manifold with n independent smooth functions being pairwise in involution, i.e. the Poisson brackets (induced by the symplectic structure) of any pair of them vanish. On the one hand, the Liouville-Arnold theorem, a well-known basic result on the local structure of completely integrable Hamiltonian systems, asserts the existence of local action-angle coordinates. Duistermaat, on the other hand, first explored the global structure in [Dui] in the early Date: September 1, 2015. 1 80s, where he worked in the slightly more general setting of Lagrangian fiber bundles. He introduced two topological invariants, namely monodromy and Chern classes. The triviality of the former quantity gives a necessary (but not sufficient) condition for the existence of global action coordinates, while the vanishing of the latter, together with a certain condition on the symplectic form, amount to the existence of global angle coordinates. Duistermaat's foundational work has since elicited generalizations in mainly two contexts, superintegrable systems (mechanical systems where the number of independent smooth functions in involution is less than half of the dimensions of the systems) and almost symplectic systems, to be explained below.
Dazord and Delzant in [DD] studied, in the spirit of [Dui], the global structure of symplectically complete isotropic realizations (SCIR) which correspond to superintegrable systems. In particular, they defined the Lagrangian class, an invariant finer than the Chern class and the vanishing of which is equivalent to the existence of global angle coordinates. Moreover, they introduced what is now called Dazord-Delzant homomorphism of certain sheaf cohomology groups and used it to give a sufficient and necessary topological condition for a torus bundle over a Poisson manifold to have a compatible symplectic form so as to be an SCIR. More details on their results can also be found in [Va].
In recent years there has been a growing interest in the study of nonholonomic systems, and as a preliminary step of investigation in this context almost symplectic systems are considered. For instance, in [FS] Fassò and Sansonetto studied almost symplectic integrable Hamiltonian systems (IASHS, see Definition 2.1), which are basically SCIRs except that they are almost symplectic. They obtained a generalization of Liouville-Arnold Theorem under the condition of the existence of strongly Hamiltonian vector fields (See Theorem 2.7). The global theory in this direction in the spirit of Duistermaat/Dazord-Delzant was first outlined by Sjamaar in [Sj], where the special case of almost Lagrangian fiber bundles were considered. Sansonetto and Sepe, in the recent paper [SS], generalized the Dazord-Delzant theory in both contexts of superintegrable systems and almost symplectic systems. They studied what we call in this paper almost symplectically complete isotropic realizations (ASCIR, see Definition 2.8), which are equivalent to IASHSs. They describe the tropical affine structure (called transversally integral affine structure in [SS]. See Definition 2.6) coming from ASCIRs and Chern class, and use the sheaf cohomology groups (which are exactly the same as those in [DD]) to describe Dazord-Delzant homomorphism in the case of ASCIRs. They brought twisted Poisson structures into play, seeing as the base space of an ASCIR inherits a twisted Poisson structure (cf. Theorem 2.7). Motivated by string theory, twisted Poisson manifolds were first introduced in [SW], and provide the framework for the study of such nonholonomic systems as the Veselova systems and the Chaplygin sphere (see [BG-N]).
In this paper, following the ideas outlined in [Sj], we continue the study of ASCIRs. In particular, we show, by a symplectic reduction argument, that there is a 'tensor product' of ASCIRs over a fixed twisted Poisson manifold B with a given tropical affine structure, analogous to the tensor product of line bundles. The resulting group of isomorphism classes of ASCIRs, which we call the Picard group of B, admits descriptions in terms of exact sequences of certain sheaf cohomology groups of B as in Theorem 5.10. We also consider the twisted Poisson structure on an open subset of a compact Lie group described in [SW], and exhibit an example of an ASCIR over it.
The organization of this paper is as follows. Section 2 reviews the analogue of Liouville-Arnold Theorem for IASHSs as in [FS] and the notion of twisted Poisson manifolds. AS-CIRs are defined and shown to be equivalent to IASHSs. In Sections 3 and 4 we define certain topological structures ASCIRs possess, namely torsors and tropical affine structures. We also review Chern classes and introduce certain sheaf cohomology groups to set the scene for classifying ASCIRs over a fixed twisted Poisson manifold with a given tropical affine structure. In Section 5, we explain in detail the construction of the tensor product of ASCIRs and define the Picard group. We restate the main result of [SS], which gives, by means of the Dazord-Delzant homomorphism, the necessary and sufficient condition for a torsor to be an ASCIR with a certain twisting 3-form. Our main results on the classification of ASCIRs and the description of the Picard group are also given and proved. In Section 6 we discuss the example of an ASCIR over an open subset of a compact Lie group with the twisted Poisson structure as in [SW]. Section 7 points out how the present work fits into the more general framework of integration of twisted Poisson and Dirac structures, and Picard groups in Poisson geometry. patient guidance and generously sharing his ideas which constitute a major part of this paper. He would also like to thank the referees for their detailed and critical comments and suggestions for improvements.

Integrable almost symplectic Hamiltonian systems
In [FS], the notion of strong Hamiltonianicity is introduced and exploited to generalize the Liouville-Arnold theorem on local action-angle coordinates in the context of integrable almost symplectic Hamiltonian systems (cf. Definition 2.1), which are mechanical systems that, in terms of generality, lie between Hamiltonian ones and nonholonomic ones (cf. [SS]). In what follows we shall first recall the relevant notions in this general setting before stating their results.
Definition 2.1. An almost symplectic manifold is a pair (M, ω), where M is a smooth manifold and ω ∈ Ω 2 (M ) is a non-degenerate 2-form. Any ω as above is referred to as an almost symplectic form.
(3) Fibers of π are isotropic. (4) For any point b ∈ B, there exists a π-saturated neighborhood U such that there exist linearly independent, strongly Hamiltonian vector fields Y 1 , · · · , Y n (here n := 2d − k) in U which are tangent to the fibers of π.
Definition 2.4 (cf. [SW]). A twisted Poisson manifold is a pair (B, Π) where Π is a bivector field such that . If Π is of constant rank then we say the twisted Poisson structure is regular. We also define X f := −Π ♯ (df ).
Remark 2.5. The term 3 Π ♯ (η) measures how Π fails to satisfy the Jacobi identity. The condition [Π, Π] = 3 Π ♯ (η) amounts to requiring that for any smooth functions f , g and h, Similar to the Poisson case, a twisted Poisson structure induces a (in general singular) foliation in which each leaf is equipped with an almost symplectic form, whose exterior differential is the restriction of the twisting form η to the leaf (cf. [SW]). In particular, if two twisting forms η 1 and η 2 correspond to the same twisted Poisson bivector field Π, then η 1 − η 2 vanishes on restriction to each almost symplectic leaf. Conversely, if η 1 is a twisting 3-form corresponding to Π, and η 2 another 3-form such that η 1 − η 2 vanishes on restriction to each leaf, then η 2 is also a twisting 3-form corresponding to Π. That is because the vector field X f is tangent to the leaves, and as a result (η 1 − η 2 )(X f , X g , X h ) = 0 and η 2 satisfies Equation (1) as well.
Throughout this paper, (B, Π) is always a regular twisted Poisson manifold with almost symplectic foliation denoted by F, unless other specified.
Definition 2.6. Let (B, F) be a manifold endowed with a regular foliation. A tropical affine structure transversal to the leaves of B is a system of foliated atlas A = {(U α , p α )} for F consisting of submersions p α which locally define the foliation of the leaf space such that on U αβ , the change of coordinates is given by p β = A αβ p α + z αβ for A αβ ∈ GL(n, Z) and a constant vector z αβ ∈ R n . In other words, the transition functions take values in the tropical affine group Trop(n) := GL(n, Z) ⋉ R n .
(2) Local action coordinates a 1 , · · · , a n , local angle coordinates α 1 , · · · , α n and b 1 , · · · , b 2d−2n exist. ω in these coordinates has the following representation where A ij , B ij and C ij are independent of the angle coordinates, and C ij is nonsingular. In particular, dω = π * η for some closed 3-form η of B k . (3) B is an η-twisted regular Poisson manifold of rank 2d − 2n, where the twisted Poisson structure Π is coinduced by ω, i.e. the twisted Poisson bracket {·, ·} B satisfies π * {f, g} B = ω(X π * f , X π * g ) (4) The almost symplectic leaves are locally the level sets of the action coordinates a 1 , · · · , a n on B, which are locally the Casimirs of the twisted Poisson structure of B.
(5) The local action coordinates give rise to a tropical affine structure transversal to the almost symplectic leaves of B.
In Theorem 2.7, we start with the almost symplectic and strongly Hamiltonian structures of the integrable system M , and deduce the various compatible structures of B, i.e. the regular twisted Poisson structure and the tropical affine structure. This is the geometric mechanical approach taken in [FS]. In the rest of this paper, we shall take the theoretical approach as in [DD] in analogy to the integration of Lie algebroids (for more explanation see Section 7) by starting with the regular twisted Poisson and tropical affine structures on B and investigate what integrable almost symplectic Hamiltonian systems B can support (or what torus bundles over B support compatible integrable almost symplectic Hamiltonian structures). This question was first answered in [DD] in the genuine symplectic case and later by [SS] in the almost symplectic case by generalizing the arguments in [DD].
Definition 2.8. An almost symplectically complete isotropic realization (ASCIR) (M, ω) over a regular twisted Poisson manifold (B, Π) is a surjective submersion π : M → B whose fibers are compact and connected and such that π is an almost Poisson map (here the almost Poisson structure on M is induced by ω), and dω = π * η for some closed 3-form η, which we call the twisting form of the ASCIR.
We note that the almost Poisson bi-vector induced by the almost symplectic structure of M satisfies [Π M , Π M ] = 3 Π ♯ M (dω). π being an almost Poisson map, the Poisson bivector Π of B satisfies [Π, Π] = 3 Π ♯ (η). So B is η-twisted if dω = π * η. The following Proposition justifies that there is nothing lost in taking the aforementioned 'theoretical' approach.

Lie group bundles, torsors, tropical affine structures and period bundles
Definition 3.1.
(1) A bundle of Lie groups G over B is a locally trivial fiber bundle with fibers being a Lie group G such that the transition functions take values in the automorphism group of G.
(2) An action of the bundle of Lie groups G → B on a fiber bundle π : M is a G-torsor if each fiber of G acts on the corresponding fiber of M freely and An example of bundles of Lie groups is any Euclidean vector bundle, whose fibers are isomorphic to R n . A principal G-bundle over B furnishes an example of a torsor, where the bundle of Lie groups acting on it is the trivial bundle B × G. On the other hand, if M is a G-torsor, then it is locally a principal G-bundle. Note that a bundle of Lie groups is an example of a Lie groupoid, where the source and target maps are identical.
is the leaf passing through b is a bundle of abelian Lie groups (in fact, it is a bundle of abelian Lie algebras and a Lie algebroid. See [SS,P2]). It gives a natural action of bundle of Lie groups in the following way. A form α ∈ ν * b F gives rise to a vector field ω ♯ (π * α) ∈ Γ(T M | π −1 (b) ). Since α = d b f for some function f defined in a neighborhood of b and constant on the almost symplectic leaves of B, ω ♯ (π * α) = X π * f . On the other hand, π * (X π * f ) = X f (see the proof of Proposition 2.9), and f is a local Casimir of the twisted Poisson structure, so X f = 0 and X π * f is tangent to the fibers. In particular, ω ♯ (π * α) is a vector field of π −1 (b), which is compact by assumption. Therefore ω ♯ (π * α) integrates to a time 1 flow, which we denote by ϕ b (α) : π −1 (b) → π −1 (b). Letting b vary smoothly yields a smooth action of ν * F on M . Let P be the stabilizer subbundle of this action. This is a smooth subbundle, and the quotient ν * F/P is a bundle of abelian Lie groups (cf. [SS,Theorem 2 (1) and (3)], which is proved using the same arguments in the symplectic case in [Va,Theorem 8.15]).
Definition 3.2. We call P the period bundle of the ASCIR (M, ω).
We can say something more about the action ϕ and the period bundle P . Note that the is an isomorphism. This, together with the assumption that π −1 (b) is connected, implies that the action ϕ b is transitive and locally free. It follows that the stabilizer P b is a discrete subgroup of ν is an ASCIR over a regular twisted Poisson manifold (B, Π, η), then M is a ν * F/P -torsor, where P , the period bundle of M , is a Z n -bundle.
Note that ω ν * F is closed, but not nondegenerate unless the dimension of the leaves is 0, i.e. ν * F = T * B. Moreover, ω ν * F vanishes on restriction to any local section of P by Corollary 3.5 and ξ * ω ν * F = dξ. Hence ω ν * F descends to a 2-form on ν * F/P . Definition 3.7. Let T be ν * F/P , and ω T the 2-form obtained by descending ω ν * F to T .
By Theorems 2.7 and 2.9, an ASCIR admits local action coordinates (a 1 , · · · , a n ) which give rise to a tropical affine structure of B transversal to F. One can glue the locally trivial Z n -bundles U × (Zda 1 ⊕ · · · ⊕ Zda n ) with transition functions being the linear part of the change of the action coordinate functions to get the period bundle P . Conversely, the tropical affine structure of B can be recovered from the period bundle P of any ASCIR: we may take a local frame (s 1 , · · · , s n ) of P over a small enough neighborhood U of B so that by Poincaré Lemma and Corollary 3.5, s i is locally exact. Any choice of the primitives of the local frame serves as local action coordinates. The linear part of the transition functions of these primitives is exactly the transition functions of P , while the translational part is accounted for by the ambiguities of different choices of the primitives, which differ by a constant vector. So the primitives give a tropical affine structure of B. To summarize,

Chern classes and some sheaf cohomology groups
In this section, we present a topological invariant, the Chern class, and certain sheaf cohomology groups to classify a subcategory of ASCIRs over a fixed regular twisted Poisson manifold B with a given tropical affine structure following [Dui] and [DD].
Definition 4.1. Let (B, Π) be a regular twisted Poisson manifold with a tropical affine structure transversal to F induced by the period bundle P . We define, on B, (1) Ω k F :=the sheaf of differential k-forms which vanish on each almost symplectic leaf, if k > 1. Ω 0 F :=the sheaf of smooth functions which are constant on each almost symplectic leaf.
(2) Z k F :=the sheaf of closed k-forms in Ω k F . (3) T :=the sheaf of smooth sections of T . (4) P :=the sheaf of locally constant sections of P .
(5) O P :=the structure sheaf of B with tropical affine structure P , defined by It is easy to see that the various sheaves in Definition 4.1 are linked in the following web of short exact sequences, which has been known since the work [DD].
As shown in Proposition 3.3, any ASCIR is a T -torsor. Therefore, as a first step to addressing the classification problem, we may forget the almost symplectic structure for the time being and classify T -torsors over B. Let M be a T -torsor over B, and U = {U α } be an open cover of B such that U α admits a smooth local section s α . Because of the transitivity and the freeness of the T -action on M , we can define t αβ ∈ T (U αβ ) by ϕ(t αβ )s β = s α {t αβ } defines a Cech 1-cocyle and a class [t] in the sheaf cohomology H 1 (B, T ), which gives the obstruction for M to possess a global section. The short exact sequence 0 −→ P −→ Ω 1 F −→ T −→ 0 induces the long exact sequence of sheaf cohomology groups F is a fine sheaf, δ is an isomorphism. Definition 4.2 (cf. [Dui]). The class c(M ) := δ([t]) ∈ H 2 (B, P) is called the Chern class of M as a T -torsor.
A T -torsor has vanishing Chern class if and only if it possesses a global section or, equivalently, is isomorphic to T as a T -torsor. The Chern classes measure the obstruction of the existence of a global section of a T -torsor, so they are purely topological in nature and independent of the almost symplectic structure, if any, of the T -torsor. They provide the complete invariant of the classification of T -torsors on B. In category theoretic language, if Tor (B,T ) denotes the category of T -torsors over B, then H 2 (B, P) ∼ = π 0 (Tor (B,T ) ).
Definition 4.3. A 2-form Θ is called a characteristic form of a regular twisted Poisson manifold B if its restriction to any leaf is its almost symplectic form.
For the existence of characteristic forms, see the proof of [BG-N, Proposition 3.1], which shows that forms satisfying the conditions in Definition 4.3 are not unique. As we are going to take the almost symplectic structure into account in solving the classification problem, we shall first consider a subcategory of ASCIRs over B.
Proof. T -equivariance of ϕ s follows from its definition. Let ξ ∈ Γ(T ). We observe that (dξ is well-defined because any representatives of ξ in Ω 1 F differ from each other by a closed form represented by a global section of P ) =s * (ω + π * dξ) − dξ − s * ω =0 To show that ϕ * s ω − ω T − π * T s * ω is 0, it remains to show that it is basic. Let a 1 , · · · , a n be the local action coordinates in an open neighborhood U of B. Then {X π * T a i := ω ♯ T (π * T da i )} 1≤i≤n spans the tangent spaces of the fibers of T over U . So it suffices to show that ι X π * T a i (ϕ * s ω − ω T − π * T s * ω) = 0. On the one hand, we have ι X π * T a i π * T s * ω = 0 and ι X π * T a i ω T = π * T da i . On the other hand, Differentiating with respect to t at t = 0 yields So ι X π * T a i ϕ * s ω = ι ω ♯ (π * da i ) ω = π * da i , and this completes the proof.
Proof. Take any local section s ′ : U → M . Then ϕ * s ′ ω = ω T | U + π * T | U s ′ * ω by Lemma 4.5. On the other hand, π * T | U (Θ − s ′ * ω) is obviously closed. π T | U being a submersion, Θ − s ′ * ω is closed as well. So by Poincaré lemma and restricting U to a smaller neighborhood if necessary, Θ − s ′ * ω = dt ss ′ for some 1-form t ss ′ on U . Let s := ϕ(t ss ′ )s ′ . We then have and thus independent of M . We shall drop the superscript M from the notation for the sheaf of automorphisms from now on. As will be clear later, Aut (B,Π,P,Θ) should be thought of as the 'structure sheaf' of B.

Proof.
(1) Obviously the kernel of ϕ•d is O P . It remains to show that ϕ•d is surjective and the sequence is exact in the middle. For any point b ∈ B, we can find a neighborhood U such that (π −1 , where a i are local action coordinates with da i ∈ P(U ), and α i are angle coordinates. By the above remark, we may consider In local coordinates, f (a 1 , · · · , a n , b 1 , · · · , b 2d−2n , α 1 , · · · , α n ) = (a 1 , · · · , a n , b 1 , · · · , b 2d−2n , f 1 , · · · , f n ) which in turn forces f i −α i to be functions independent of the coordinates b 1 , · · · , b 2d−2n , α 1 , · · · , α n (though both α i and f i are R/Z-valued functions, their difference f i − α i can be lifted to a well-defined function, as f is a local diffeomorphism). It follows that n By Poincaré lemma and restricting U further if necessary, n i=1 (f i − α i )da i is exact. This proves the surjectivity of ϕ • d. If f is identity, then n i=1 (f i − α i )da i ∈ P(U ) and any of its primitives is in O P (U ) by definition. This shows the exactness in the middle of the sequence.
Remark 4.8. By (2) of Proposition 4.7 the isomorphism class of Aut (B,Π,P,Θ) is actually independent of the characteristic form.
(1) The ASCIRs in ASCIR 0 (B, Π, P, Θ) are classified, up to isomorphism which descends to identity on B, by the sheaf cohomology group H 1 (B, Aut (B,Π,P,Θ) ), which is isomorphic to H 1 (B, K F ). When the leaves of B are points, then H 1 (B, K F ) ∼ = H 2 (B, O P ).
(2) (cf. [DD,Proposition 4.1])We have the long exact sequence which is induced by the short exact sequence Identifying H 1 (B, P) with H 0 (B, T ) and H 2 (B, F) with H 0 (B, Z 2 F ), the map d P, * is induced by the exterior differential map d P : T → Z 2 F .

Proof.
(1) That ASCIRs in ASCIR 0 (B, Π, P, Θ) are classified by H 1 (B, Aut (B,Π,P,Θ) ) follows from Lemma 4.6 and that Aut (B,Π,P,Θ) is the sheaf of transition functions of those ASCIRs. That H 1 (B, Aut (B,Π,P,Θ) ) is isomorphic to H 1 (B, K F ) follows from (2) of Proposition 4.7. Now suppose B is foliated by points. The short exact sequence in (1) of the same Proposition induces the long exact sequence of sheaf cohomology groups F), we get the desired long exact sequence and the interpretation of the Dazord-Delzant homomorphism ∂ 2 and d P, * .
Remark 4.11. Identifying H 1 (B, K F ) with π 0 (ASCIR 0 (B, Π, P, Θ)), we have that the map H 2 (B, F) → H 1 (B, K F ) in the long exact sequence sends the class [γ] to the isomorphism class of the ASCIR (T, ω T + π * T (γ + Θ)). The map H 1 (B, K F ) → H 2 (B, P) just forgets the symplectic structure and gives the Chern class of the ASCIR as a T -torsor.

Picard group and classification of ASCIRs
In the last Section we obtain the identification of π 0 (ASCIR 0 (B, Π, P, Θ)), the set of isomorphism classes of ASCIRs over B with period bundle P and twisting form dΘ, with sheaf cohomology group H 2 (B, O P ). This begs the question of whether there exists a natural group law among ASCIRs that makes the identification a group isomorphism. The answer is affirmative. In this Section we shall construct a 'tensor product' analogous to that of line bundles by means of a symplectic reduction construction (cf. [Xu]), and hence define the notion of Picard group (see Definition 5.5). We also restate the main result of [SS], Theorem 5.9, which gives the necessary and sufficient condition for a torsor to be an ASCIR with a certain twisting 3-form. Based on this and results in Section 4 we give our main results on the classification of ASCIRs and the description of Picard groups (see Theorem 5.10).
Definition 5.1. Let ASCIR(B, Π, P ) be the category of ASCIRs over B with P as the period bundle.
Let (M 1 , ω 1 ), (M 2 , ω 2 ) ∈ ASCIR(B, Π, P ) and dω i = π * η i , i = 1, 2. Fix a choice of the characteristic form Θ. The fiber product M 1 × B M 2 is a T × B T -torsor. By Theorems 2.7 and 2.9, ω i can be written in the local normal form n j=1 da j ∧ dα B (i) and C are skew-symmetric and C is nondegenerate. The same is true of Θ, which can be written as 1 2 j,l D jl da j ∧ da l + j<l E jl da j ∧ db l + 1 2 j,l C jl db j ∧ db l . Let π i : M 1 × B M 2 → M i and π ′ : M 1 × B M 2 → B be natural projections. The (n + 2d) × (n + 2d) matrix associated to the local normal form Let T − be the Lie group bundle over B whose fibers are anti-diagonal of the fibers of j , j = 1, · · · , n, and again by means of local normal forms, ι ∂ (2) j (π * 1 ω 1 + π * 2 ω 2 ) = 0. So π * 1 ω 1 + π * 2 ω 2 can be pushed down to a 2-form, which we denote by ω 1 ⊕ ω 2 , of the orbit space M 1 × B M 2 /T − , which is a T -torsor (there is a natural residual T -action on the orbit space). π ′ * Θ is obviously pushed down to π * Θ. The local normal form of ω 1 ⊕ ω 2 − π * Θ then is the 2d × 2d matrix  Definition 5.2. We define a binary operation ⊗ on ASCIR(B, Π, P ) such that ( Proposition 5.3. The 'tensor product' on ASCIR(B, Π, P ) as in Definition 5.2 descends to π 0 (ASCIR(B, Π, P )), the isomorphism classes of ASCIRs on (B, Π) with a fixed period bundle P .
Definition 5.5. We call the abelian group π 0 (ASCIR(B, Π, P )) as in Proposition 5.4 the Picard group of the twisted Poisson manifold (B, Π) with a fixed characteristic form Θ and period bundle P , and is denoted by Pic(B, Π, P, Θ).
Proposition 5.8. (cf. [SS,Proposition 5]) The following square is commutative where the left vertical map is the Chern class map and the right one takes a 3-form to its relative cohomology class.
In fact, the topological condition in Proposition 5.8 is also sufficient for a T -torsor to be an ASCIR. This is basically the content of the main result in [SS], which is a generalization of a similar result in [DD] where the special case of symplectically complete isotropic realization is treated. Proof. We shall show the last claim that any T -torsor can be made an ASCIR, which was not shown in [SS]. Suppose M is a T -torsor over (B, Π, P, Θ). Let η 0 be a twisting 3-form of (B, Π), and α = [dΘ−η 0 ]−∂ 2 c(M ) ∈ H 3 (B, F). Choose a representative 3-form α ∈ Z 3 F of the class α. Then ∂ 2 c(M ) = [dΘ − (η 0 + α)], and η 0 + α is a twisting 3-form of (B, Π) as well (see Remark 2.5). Hence M can be equipped with a compatible almost symplectic form which makes it an ASCIR with the twisting form η 0 + α.
We are now in a position to present two descriptions of Pic(B, Π, P, Θ), derived from the commutative square in Proposition 5.8, thereby giving some answers to the problem of classifying ASCIRs. In these two descriptions, we fit Pic(B, Π, P, Θ) into the short exact sequences built from the top and left maps in that commutative square.
(1) We have the exact sequence where the last map is the one in Corollary 5.7.
Plugging this to f * ω T = ω T + π * T γ, we get which shows that f i are independent of α i , the angle coordinates. In other words, f = ϕ(s) for some s ∈ Ω 1 F . Now ϕ(s) * ω T = ω T + π * T ds and therefore γ = ds. Definition 5.11. We denote the image of Ω 2 F /dΩ 1 F in the exact sequence in (2) of Theorem 5.10 by Pic 0 (B, Π, P, Θ).
The exact sequence in (2) of Theorem 5.10 is a generalization of the one in Corollary 4.10. As H 2 (B, P) is a discrete group, Pic 0 (B, Π, P, Θ) is the identity component of Pic(B, Π, P, Θ). Pic 0 (B, Π, P, Θ) is to the subgroup of Picard group consisting of isomorphism classes of degree 0 invertible sheaves in algebraic geometry what H 2 (B, P) is to the Néron-Severi group (see [Ha,Definitions p.109] and [Ha,Remark 6.10.3] and the references therein for definitions of invertible sheaves and Néron-Severi groups). This analogy first appeared in [Sj].
There is an action of Ω 2 F on ASCIR(B, Π, P ), namely, γ · (M, ω) = (M, ω + π * γ). We call this action the coarse gauge transformation. If two ASCIRs are in the same orbit of coarse gauge transformation, we say they are coarse gauge equivalent. Obviously this equivalence relation descends to the Picard group. Part (2) of Theorem 5.10 just says that there is a bijection between the coarse gauge equivalence classes and the isomorphism classes of T -torsors, and that any coarse gauge equivalence class is a principal homogeneous space of Ω 2 F /dΩ 1 F .

An example
This section is devoted to an example inspired by an SCIR over a certain open subset in g * with the Lie-Poisson structure studied in [DD] (cf. Remark 6.9). The twisted Poisson manifold in this example was discussed in [SW].
6.1. Twisted Poisson structure of an open subset of a compact Lie group. Let G be a compact connected Lie group of rank n and ·, · be a bi-invariant inner product on g. A G-invariant closed 3-form on G is given by η = θ L , [θ L , θ L ] , where θ L is the left-invariant Maurer-Cartan connection 1-form. In [SW], a twisted Dirac structure of G with twisting form η (called the Cartan-Dirac structure) is given by the following maximal isotropic subbundle L of the Courant algebroid T G ⊕ T * G ∼ = T G ⊕ T G (we identify T G with T * G by the inner product, so the bilinear form on T G ⊕ T G is given by ((X 1 , Y 1 ), (X 2 , Y 2 )) = X 1 , Y 2 + Y 1 , X 2 ).
Here X L (resp. X R ) is the left-(resp. right-)invariant vector field generated by X ∈ g. Let G 0 = {g ∈ G|Ad g + 1 is invertible}. Noting that {(X R + X L ) g |X ∈ g} = T g G if and only if g ∈ G 0 , we can define the almost Poisson structure whose graph is exactly L restricted to G 0 . In fact, Π is an η-twisted Poisson structure. The vector field X R − X L being the fundamental vector field of the conjugation action, the almost symplectic leaves are precisely the conjugacy classes of G in G 0 . For example, if G = SU (2), G \ G 0 is the conjugacy class containing i 0 0 −i . If we identify G with S 3 , then G \ G 0 can be identified with the equatorial 2-sphere, and G 0 consists of two connected components of hemispheres of S 3 . The almost symplectic leaves are regular conjugacy classes (except G \ G 0 ), which are all diffeomorphic to S 2 , and the two oneelement conjugacy classes {I 2 } and {−I 2 }.
Proposition 6.1. (cf. [ABM,Equation (40)], [AMM,Proposition 3.1] and [GHJW]) The conjugacy classes of G in G 0 together with the following data (1) the G-action by conjugation, (2) the almost symplectic forms induced by the η-twisted Poisson structure Π ♯ , and (3) the inclusion map ι : C → G as the G-valued moment map are quasi-Hamiltonian manifolds as in [AMM]. The almost symplectic form of a conjugacy class C in G 0 is given by where X ♯ is the fundamental vector field induced by the infinitesimal conjugation action by X ∈ g.
Let G reg 0 be the set of regular elements in G 0 . We fix a maximal torus T of G and a Weyl alcove ∆, which parametrizes the set of conjugacy classes in G. Let µ : G → ∆ be the point-orbit projection map, and X ∈ ∆ such that exp(X) ∈ G \ G 0 . Then −1 must be one of the eigenvalues of Ad exp(X) , which are e α(X) for α in the set of roots R. It follows that the image of G \ G 0 under µ is the intersection of ∆ with the union of finitely many hyperplanes α∈R α −1 (πi) in t. Hence Im(µ| G reg 0 ) = int(∆) \ α∈R α −1 (πi), which is a disjoint union of finitely many open polytopes. For definiteness we choose the open polytope Q with one vertex being 0, and let B = µ −1 (Q), which is the twisted Poisson manifold we consider in the example. B → Q is a fiber bundle whose fibers are all regular conjugacy classes, which are diffeomorphic to G/T . As Q is contractible, this bundle is trivial.
Using κ to identify B and G/T × Q, we have that ν * F is isomorphic to the trivial bundle π * 2 T * Q = G/T × T * Q. Furthermore, Proposition 6.6.
Proof. P being a trivial Z n -bundle, we have a canonical isomorphism H 2 (B, P) ∼ = H 2 (B, Z)⊗ H 0 (B, P), where H 0 (B, P) is spanned by the basis {d(ξ i • µ)} n i=1 . The Dazord-Delzant homomorphism then amounts to the composition Using this interpretation the Proposition follows easily.
Definition 6.7. Let M be the ν * F/P -torsor G × Q over B where the projection map is (g, X) → gexp(X)g −1 and the torsor structure is induced by the G/T × T * Q-action given by (gT, X, ζ) · (g, X) = (gexp(− ζ, · ), X) It is not hard to see that M is isormphic to the torsor M X 1 ξ 1 × B M X 2 ξ 2 × B · · · × B M Xn ξn and hence its Chern class is represented by the map (X → X, · ) ∈ Hom Z (Λ, Λ * ).
Proposition 6.8. There exists an almost symplectic form on M such that it is an ASCIR over B with the twisting form η.
Proof. It suffices to check that ∂ 2 c(M ) = [dΘ B − η] by Theorem 5.9. Let D be the open subset in g * which is diffeomorphic to B through the exponential map. By [AMM], there exists a 2-form ̟ ∈ Ω 2 (g * ) satisfying exp * η = d̟ and ω O = exp * ω C − i * O ̟, where O is a coadjoint orbit, ω O its Kostant-Kirillov-Soriau 2-form and C = exp(O). We have that On the other hand, by Proposition 6.6 Now it boils down to showing that [exp * Θ B − ̟] = n i=1 exp * c 1 (M ξ i ) · (ξ i • µ • exp) ∈ H 2 (D, R) ∼ = H 2 (G/T × Q, R). Restricting to the coadjoint orbit O ζ containing ζ ∈ Q, we have )) = 0. By applying Künneth formula to H 2 (G/T × Q, R) and the acyclicity of H * (Q, R), we have that [exp * Θ B − ̟] = n i=1 exp * c 1 (M ξ i ) · (ξ i • µ • exp), and the desired conclusion follows. Remark 6.9. The example of ASCIR discussed in this Section should be viewed as the 'exponentiated version' of the example of SCIR over the union of regular coadjoint orbits in g * equipped with the Lie-Poisson structure in [DD]. The difference between the Lie-Poisson structure of D and the twisted Poisson structure of B is accounted for by the introduction of the 2-form ̟, whose exterior differential is the twisted 3-form η. This example is also inspired by the AMM action groupoid G ⋊ G, where G acts on itself by conjugation and which integrates the Cartan-Dirac structure (cf. [BCWZ,Section 7] and [BXZ]).

Some general remarks
All results in this paper fits into the framework of integration of twisted Poisson or, even more generally, twisted Dirac structures (cf. [BCWZ], [CF1], [CF2] and [CX]). Viewing B more generally as a twisted Dirac manifold, we see that the ASCIRs over B are nothing but special examples of presymplectic realizations (cf. [BCWZ,Definition 7.1]). If we regard B with Π = 0 and an exact twisting 3-form dξ as a Lie algebroid, then (T * B, ω can + π * T * B ξ) is an almost symplectic groupoid with source and target maps being identical and fibers of the source map being simply-connected, which integrates B, in the sense of Definition 2.1 of [CX]. If in addition B possesses a tropical affine structure P , then (T, ω T + π * T ξ) is a proper almost symplectic groupoid.
The notions of tensor product and Picard groups are inspired by the paper [BW] where they are explored in the more general framework of symplectic dual pairs. Using their terminology Pic(B, Π, P, ξ) is the static Picard group of the groupoid (T, ω T + π * T ξ) if Π = 0.