  Reformulations of Donaldson's "tamed to compatible" question are obtained in
terms of spaces of exact forms on a compact almost complex manifold
$(M^{2n},J)$. 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.
