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 $M$?
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3$\begingroup$ One of the main goals of symplectic geometry is to answer this question in the special case where $M$ is the unique leaf... $\endgroup$– Tim PerutzCommented Jan 22, 2013 at 18:34
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$\begingroup$ I agree with Tim that, as currently phrased, this question is an extremely hard open problem, way beyond current techniques. Perhaps though with different hypotheses, it might become a hard, open, but not completely impossible question. For example, what if the leaves are 2-dimensional? (Lefschetz fibrations play nicely with symplectic topology, after all.) What if we're allowed foliations and Poisson structures with singularities or degeneracies? (Perhaps along the lines of broken pencils on 4-manifolds?) $\endgroup$– Joel FineCommented Jan 22, 2013 at 21:53
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3$\begingroup$ If the leaves are $2$-dimensional, then the only obstruction is whether the leaves have a consistent orientation, i.e., whether there is a nonvanishing bi-vector $\pi$ on $M$ that is tangent to the leaves of $\mathcal{F}$. In this case, $[\pi,\pi]=0$ is an identity since $\pi$ is tangent to a foliation, so any such $\pi$ will be a Poisson structure. $\endgroup$– Robert BryantCommented Jan 22, 2013 at 22:38
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$\begingroup$ Thanks Robert, thats very clear. (And not so hard after all!) $\endgroup$– Joel FineCommented Jan 23, 2013 at 12:14
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1 Answer
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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.
