I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this does not fit the scope of this site.
Let $M$ a complex surface and $\omega\in H^0(\Omega_M^2,M)$ a non degenerate holomomorphic form.
I've read somewhere (without proof), that then the first chern class of the symplecitc manifold $(M,Re~ \omega)$ vanishes.
Why is this true? As far as I know, the Chern class of $(M, Re~ \omega)$ is the chern class of any complex vector bundle with almost complex structure compatible with $Re ~\omega$. My first guess was that the original complex structure is compatible with $Re ~\omega$. But this is not true, as then $Re ~\omega$ would be of type $(1,1)$ (with repect to the original almost complex structure).