Let $M^{2n}$ be a Poisson manifold with Poisson bivector field $\Pi$. We say that $M$ is b-Poisson if the map $\Pi^n:M\to\Lambda^{2n}(TM)$ intersects the zero section transversally on a codimension one submanifold $Z\subset M$. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of $(M,\Pi)$ in the neighbourhood of $Z$ and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.Comment: 34 pages. Some changes have been implemented mainly in Sections 2 and 6. Minor changes in exposition. References have been adde

Dedicated to the memory of Paulette Libermann whose cosymplectic manifoldsplay a fundamental role in this paper.Abstract. In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a Poisson b-manifold as we will see in [GMP].

Abstract. We study Hamiltonian actions on b-symplectic manifolds with a focus on the effective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classifies these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus.

We construct a distance on the moduli space of symplectic toric manifolds of dimension four. Then we study some basic topological properties of this space, in particular, path-connectedness, compactness, and completeness. The construction of the distance is related to the Duistermaat-Heckman measure and the Hausdorff metric. While the moduli space, its topology and metric, may be constructed in any dimension, the tools we use in the proofs are four-dimensional, and hence so is our main result.Comment: To appear in Geometriae Dedicata, minor changes to previous version, 19 pages, 6 figure

We consider the ellipsoid embedding functions cH b (z) for symplectic embeddings of ellipsoids of eccentricity z into the family of nontrivial rational Hirzebruch surfaces H b with symplectic form parametrized by b ∈ [0, 1). This function was known to have an infinite staircase in the monotone cases (b = 0 and b = 1/3). It was also known that for each b there is at most one value of z that can be the accumulation point of such a staircase. In this manuscript, we identify three sequences of open, disjoint, blocked b-intervals, consisting of b-parameters where the ellipsoid embedding function for H b does not contain an infinite staircase. There is one sequence in each of the intervals (0, 1/5), (1/5, 1/3), and (1/3, 1). We then establish six sequences of associated infinite staircases, one occurring at each endpoint of the blocked b-intervals. The staircase numerics are variants of those in the Fibonacci staircase for the projective plane (the case b = 0). We also show that there is no staircase at the point b = 1/5, even though this value is not blocked. The focus of this paper is to develop techniques, both graphical and numeric, that allow identification of potential staircases, and then to understand the obstructions well enough to prove that the purported staircases really do have the required properties. A subsequent paper will explore in more depth the set of b that admit infinite staircases.