Almost Kähler Forms on Rational 4-Manifolds

Source abstract

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-current-form 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.