Donaldson Question: “Tamed to Compatible"
In this talk, we show that on any tamed closed almost complex fourmanifold (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 a_rmative 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 uni_ed proof of the Kodaira conjecture. Thus, our main result gives an a_rmative answer to the Kodaira conjecture in symplectic version.