Can anyone explain to me what the Hodge decomposition form of a symplectic form in a special symplectic manifold looks like?



Hodge decomposition of a symplectic form.
Using the additional information that the OP provided in the comments to Yael Fregier's answer, I can elaborate as follows:


I still don't know what "special complex manifold" means, but in any case, I will assume the following. If $(M, J, \nabla)$ is a complex manifold with a connection $\nabla$ coming from a metric $g$, (that is, $\nabla$ is the Levi-Civita connection of $g$), then we get an associated symplectic form $\omega(X,Y) = g(JX, Y)$, and $\omega$ is parallel with respect to $\nabla$ if and only if $J$ is parallel with respect to $\nabla$, if and only if $J$ is integrable and $\omega$ is closed. That is, $(M, g, J, \omega)$ is Kaehler. In this case, $\omega$ is harmonic, so its Hodge decomposition is $\omega = \omega \in \Delta\_2$, where $\Delta\_2$ is the space of harmonic $2$-forms on $M$, using the notation of the OP.


If $\nabla$ does not come from a metric, you still need some metric to define the co-derivative $d^\* = \delta$ of $d$, and to define the Laplacian $\Delta$. One can indeed do this with a different connection $\nabla$, as long as you have a metric. But in this case it is not clear to me what the symplectic form $\omega$ is, and how it is related to $J$ and $\nabla$.


**Added later:** I think I just realized that the OP is not asking about the Hodge decomposition of the form $\omega$ in particular, just the "Hodge decomposition" for a "special symplectic manifold." There is a version of "symplectic Hodge theory." See, for example, these notes by Victor Guillemin: [<http://www-math.mit.edu/~vwg/shlomo-notes.pdf>](http://www-math.mit.edu/~vwg/shlomo-notes.pdf) --- I don't know if this is the same thing mentioned in Yael Fregier's answer. Otherwise, I remain confused by the question.


