Title: On tamed almost complex four manifolds,
Abstract: This paper proves that on any tamed closed almost complex four-manifold
$(M,J)$ whose dimension of $J$-anti-invariant cohomology is equal to 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 Donaldson question
for tamed closed almost complex four-manifolds that is a conjecture in joint
paper of Tosatti, Weinkove and Yau. Our approach is along the lines used by
Buchdahl to give a unified proof of the Kodaira conjecture. Thus, our main
result gives an affirmative answer to the Kodaira conjecture in symplectic
version.