  In this paper, we calculate the dimension of the $J$-anti-invariant
cohomology subgroup $H_J^-$ on $\mathbb{T}^4$. Inspired by the concrete
example, $\mathbb{T}^4$, we get that:
  On a closed symplectic $4$-dimensional manifold $(M, \omega)$, $h^-_J=0$ for
generic $\omega$-compatible almost complex structures.
