We study Nakai-Moishezon type question and Donaldson’s “tamed to compatible” question
for almost complex structures on rational four manifolds. By extending Taubes’ subvarieties-currentform
technique to J-nef genus 0 classes, we give affirmative answers of these two questions for all
tamed almost complex structures on S^2 bundles over S^2 as well as for many geometrically interesting
tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.
Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds:
a sequence of A-infinity algebras built of differential forms on the symplectic manifold.
We show that these symplectic A-infinity algebras have a simple topological interpretation.
Namely, when the cohomology class of the symplectic form is integral, these A-infinity algebras
are equivalent to the standard de Rham differential graded algebra on certain odd dimensional
sphere bundles over the symplectic manifold. From this equivalence, we
deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic A-infinity algebras
satisfy the Calabi-Yau property, and importantly, that they can be used to define an
intersection theory for coisotropic/isotropic chains. We further demonstrate that these
symplectic A-infinity algebras satisfy several functorial properties and lay the groundwork for
addressing Weinstein functoriality and invariance in the smooth category.