A note on the deformations of almost complex structures on compact four-manifolds

Abstract

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.