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 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.