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

TL;DR

This paper investigates the deformations of almost complex structures on compact four-manifolds, specifically calculating the dimension of a certain cohomology subgroup on the 4-torus and exploring generic properties on symplectic 4-manifolds.

Contribution

It provides explicit calculations of the $J$-anti-invariant cohomology dimension on $bT^4$ and shows that for generic symplectic structures, this dimension vanishes.

Findings

The dimension of $H_J^-$ on $bT^4$ is explicitly computed.

For generic symplectic structures on 4-manifolds, the $J$-anti-invariant cohomology subgroup is trivial.

The results suggest stability of certain cohomological properties under deformation.

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.