Symplectic forms and cohomology decomposition of almost complex 4-manifolds

For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups HJ⁺(M), HJ⁻(M) of the degree 2 real de Rham cohomology group H²(M, R) in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by J-invariant, respectively, J-anti-invariant real $2-$forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of H²(M, R). This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.