Symplectic and Poisson geometry on b-manifolds

Abstract

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

Examples and counter-examples of log-symplectic manifolds

“…While Proposition 2.6 gives a list of simple invariants associated to a log-symplectic structure in [10], Guillemin, Miranda and Pires showed that these are in fact all invariants associated to a neighbourhood of the singular locus. Indeed, the following is a direct consequence of the results in [10]:…”

Integrable systems and group actions

“…As it can be seen in [26], this example is indeed a canonical model for adapted integrable systems because we have Moser normal forms for these manifolds (see [25]). …”

“…As it is seen in [25], the Poisson geometry of the manifold can be reconstructed semilocally from the critical hypersurface Z (a codimension one symplectic foliation admitting a transverse Poisson vector field). We can use a similar strategy to give natural global examples of integrable systems on -Poisson manifolds (including the compact ones).…”

“…We have recently studied the symplectic and Poisson geometry of these manifolds in [24,25] together with Guillemin and Pires.…”

A characterization of diagonal Poisson structures

“…We avoid Goto's terminology in this paper because in the context of C ∞ Poisson geometry, a log-symplectic manifold is usually defined as a generically symplectic Poisson manifold with reduced and smooth degeneracy divisor, see [4,12]. Log-symplectic manifolds in this strict sense are also labeled topologically stable Poisson manifolds [22], b-Poisson manifolds [14] and b-log-symplectic manifolds [18].…”