I have a question regarding Kähler manifolds and proposition 2.63 of the book: "Introduction to symplectic topology" of McDuff and Salamon.
According to the wikipedia page, a Kähler manifold is a symplectic manifold with a compatible almost-integrable structure, but according to proposition 2.63 of McDuff (see image), every symplectic manifold admits an almost complex compatible structure.
I do not understand how this statements are compatible, since there are well known examples of non-Kähler symplectic manifolds. In particular, symplectic manifolds which have topological obstructions to being Kähler, which means, the problem is not a confusion about the choice of the metric.
So my question is, what am I getting wrong? Any help is appreciated.
