TITLE: What is the Poincare dual of a symplectic form?
QUESTION [12 upvotes]: Every symplectic form on a manifold $M^n$ determines a De Rham cohomology class in $H^2(M)$ (often a nontrivial class), and this in turn determines a class in $H_{n-2}(M)$.  What in general can be said about this class?  For example, over the rationals this class is represented by a submanifold of $M$; is it possible to explicitly describe such a submanifold in terms of the symplectic structure?
If there is a nice answer to this question, does it also shed light on the Poincare duals of $\omega^2$, $\omega^3$, etc?

REPLY [3 votes]: Paul Biran has also studied this situation where $(M, \omega, J)$ is Kahler, $[\omega] \in H^2(M, \mathbb{Z})$ and $\Sigma$ is a complex hypersurface Poincare dual to $k[\omega]$.
He has proved that in this setting, $M$ can be decomposed symplectically as the unit normal disk bundle of $\Sigma$ and an isotropic CW-complex.
This decomposition result is then used to investigate various embedding questions into $M$.
Biran: "Lagrangian barriers and symplectic embeddings.  Geom. Funct. Anal.  11  (2001),  no. 3, 407–464"
Biran and Cieliebak: "Symplectic topology on subcritical manifolds.  Comment. Math. Helv.  76  (2001),  no. 4, 712–753"
Biran: "Lagrangian non-intersections.  Geom. Funct. Anal.  16  (2006),  no. 2, 279–326"