Almost Kahler forms on rational 4-manifolds

TL;DR

This paper investigates conditions under which almost complex structures on rational four-manifolds admit compatible almost Kahler forms, extending techniques to broader classes of manifolds and answering key geometric questions.

Contribution

It extends Taubes' subvarieties technique to $J$-nef classes, providing affirmative answers to Nakai-Moishezon and Donaldson's questions for various rational four-manifolds.

Findings

Affirmative answers for all tamed structures on $S^2$ bundles over $S^2$

Extension of techniques to del Pezzo and other rational four-manifolds

Broader understanding of almost Kahler forms on rational 4-manifolds

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.