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For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $Pic(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $Pic(M)$ and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead, in particular, to the proof of a conjecture from [BW04] stating that for any compact simple Lie algebra $mathfrak{g}$ the group $Pic(mathfrak{g}^*)$ concides with the group of outer automorphisms of $mathfrak{g}$.
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This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties of PMCTs, which already show that they are the analogues of compact symplectic manifolds, thus placing them in a prominent position among all Poisson manifolds. For instance, their Poisson cohomology behaves very much like the de Rham cohomology of compact symplectic manifolds (Hodge decomposition, non-degenerate Poincare duality pairing, etc.) and the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and Symplectic Topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a nontrivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian $G$-spaces, foliation theory, Lie theory and symplectic gerbes.
In this thesis, we study the deformation problem of coisotropic submanifolds in Jacobi manifolds. In particular we attach two algebraic invariants to any coisotropic submanifold $S$ in a Jacobi manifold, namely the $L_infty[1]$-algebra and the BFV-complex of $S$. Our construction generalizes and unifies analogous constructions in symplectic, Poisson, and locally conformal symplectic geometry. As a new special case we also attach an $L_infty[1]$-algebra and a BFV-complex to any coisotropic submanifold in a contact manifold. The $L_infty[1]$-algebra of $S$ controls the formal coisotropic deformation problem of $S$, even under Hamiltonian equivalence. The BFV-complex of $S$ controls the non-formal coisotropic deformation problem of $S$, even under both Hamiltonian and Jacobi equivalence. In view of these results, we exhibit, in the contact setting, two examples of coisotropic submanifolds whose coisotropic deformation problem is obstructed.
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine orbifolds. We prove that the cohomology class of the leafwise symplectic form varies linearly and that there is a distinguished polynomial function describing the leafwise sympletic volume. The leaf space of a PMCT carries a natural Duistermaat-Heckman measure and a Weyl type integration formula holds. We introduce the notion of a symplectic gerbe, and we show that they obstruct realizing PMCTs as the base of a symplectic complete isotropic fibration (a.k.a. a non-commutative integrable system).
We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with $b_1=0$ and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale-Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estimates than the examples constructed previously. The manifold can be chosen spin or non-spin.
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