Let $(M,\omega)$ be a symplectic manifold. Define $J(M)=\{\text{smooth almost complex structures compatible with the orientation of M}\}$.
That specific definition gives me a few things to think. How does an almost complex structure induce an orientation? If it doesn't naturally induce a orientation how do i know it is compatible? Are there other definitions of compatibility in that sense?
I know that $[\omega^n]$ induces a volume form and that if there exists a almost complex structure on $M$, $M$ is orientable.