In this talk, our objective is tamed and compatible almost complex
structures. There are two interesting questions.
The classical Nakai-Moishezon theorem (for surfaces) states the duality
between ample divisor cone and curve cone for projective surfaces.
Demailly-Paun, Buchdahl and Lamari generalized this duality to Kahler
surfaces. It is natural to ask for such a duality for almost Kahler
surfaces.
Another interesting questin is raised by Donaldson. He asked that if there
is a J-tamed symplectic form, do we have a J-compatible symplectic form as
well?
We answered these two questions affirmatively for all tamed almost complex
structures on spheres bundles over sphere. We also answer them in many
interesting cases for other rational surfaces.