On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a Poisson structure on $M$?



What foliations are symplectic foliations?
Even though each leaf can carry a symplectic structure still there may be no global Poisson structure. Some steps in understanding the problem (and as far as I know the only) were contained in a work by Bertelson <http://arxiv.org/abs/math/0010191>, also Commun. Contemp. Math. 3 441-456 (2001). This paper gives some indications on what may go wrong.


On the positive side some results are obtained by the same author at the end of her paper "A h-principle for open relations invariant under foliated isotopies",J. Symplectic Geom. Volume 1, Number 2 (2002), 369-425. 


The underlying idea is that if the leaves are open manifolds then the most easy obstructions vanish, but nevertheless the "openess" of the leaves should be somehow uniform,


All this, of course, taking for granted that "foliation" means "regular foliation". 


For the non regular case I doubt anything interesting, apart from examples, is known.


