Abstract: In this talk, we show that on any tamed closed almost complex 4-manifold $(M,J)$ whose

dimension of $J$-anti-invariant cohomology is equal to $b^+(M)-1$, there exists a new symplectic

form compatible with the given almost complex structure $J$. In particular, if $b^+(M)=1$, we give

an affirmative answer to Donaldson’s question for tamed closed almost complex 4-manifolds. 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 symplectic version of Kodaira conjecture.