It is well-known that the curvature forms of the (complexified) Levi-Civita connection can be used to provide explicit representatives for Chern classes of compact Kähler manifolds. This is not true for symplectic manifolds. So my question is, for a general compact symplectic manifold, does there exist any method to produce explicit closed forms to represent Chern classes.


Thanks in advances!



Representatives of Chern classes for compact symplectic manifolds
The symplectic manifolds equipped with an adapted almost complex structure are also called almost-Kahler manifolds. There are many connections on the tangent bundle compatible with the metric induced by the symplectic form and the almost complex structure. Each one of them is uniquely determined by its torsion.


According to Paul Gauduchon, the Chern connection on an almost Kahler manifold is the connection compatible with the metric and whose torsion is equal to the Nijenhuis tensor $N$. It is the one with the smallest torsion amongst all the connections compatible with both the metric and the almost complex structure and it is precisely the connection used by Taubes in his work on Seiberg-Witten equations on symplectic $4$-manifolds. For more details and references [see Section 1.4.1 of these notes.](http://www3.nd.edu/~lnicolae/swnotes.pdf)


Obviously in the case, in the case of Kahler manifolds, the Nijenhuis tensor is zero and the Chern connection coincides with the Levi--Civita connection.


