Symplectic Forms and Cohomology Decomposition of almost Complex Four-Manifolds

For any compact almost complex manifold (M, J), the last two authors [8] defined two subgroups H-J(+)(M), H-J(-)(M) of the degree 2 real de Rham cohomology group H-2(M,R). These are the sets of cohomology classes which can be represented by J-invariant, respectively, J-antiinvariant real 2-forms. In this paper, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of H-2(M,R). This is a specifically four-dimensional result, as it follows from a recent work of Fino and Tomassini [6]. 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.