A note on exact forms on almost complex manifolds

Abstract

Reformulations of Donaldson’s “tamed to compatible” question are obtained in terms of spaces of exact forms on a compact almost complex manifold (M, J). In dimension 4, we show that J admits a compatible symplectic form if and only if J admits tamed symplectic forms with arbitrarily given J-anti-invariant parts. Some observations about the cohomology of J-modified de Rham complexes are also made.