For any compact almost complex manifold (M, J), the last two authors  defined two subgroups H+J(M), H-J(M) of the degree 2 real de Rham cohomology group. 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. This is a specifically four-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.
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We appreciate V. Apostolov for his very useful comments, R. Hind and T. Perutz for their interest, A. Fino and A. Tomassini for sending us their paper , and National Science Foundation for the partial support. We also thank the referees for their careful reading of the manuscript and useful remarks.