  This paper proves that on any tamed closed almost complex four-manifold
$(M,J)$ whose dimension of $J$-anti-invariant cohomology is equal to the
self-dual second Betti number minus one, there exists a new symplectic form
compatible with the given almost complex structure $J$. In particular, if the
self-dual second Betti number is one, we give an affirmative answer to a
question of Donaldson for tamed closed almost complex four-manifolds. Our
approach is along the lines used by Buchdahl to give a unified proof of the
Kodaira conjecture.
