A Note on Donaldson's "Tamed to Compatible" Question (Communications in Mathematical Research, 2021)
Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamedto compatible" question (Draghici T, Zhang W. A note on exact forms on almostcomplex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011).That is, for a compact almost complex 4-manifold whose almost complex structure istamed by a symplectic form, is there a symplectic form compatible with this almostcomplex structure? They got several equivalent forms of this problem by studying thespace of exact forms on such a manifold. With these equivalent forms, they proveda result which can be thought as a further partial answer to Donaldson's question indimension 4. In this note, we give another simpler proof of their result.