Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamed
to compatible" question (Draghici T, Zhang W. A note on exact forms on almost
complex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011).
That is, for a compact almost complex 4-manifold whose almost complex structure is
tamed by a symplectic form, is there a symplectic form compatible with this almost
complex structure? They got several equivalent forms of this problem by studying the
space of exact forms on such a manifold. With these equivalent forms, they proved
a result which can be thought as a further partial answer to Donaldson's question in
dimension 4. In this note, we give another simpler proof of their result.
