On tamed almost complex four manifolds

TL;DR

This paper demonstrates that under specific cohomological conditions, tamed almost complex four-manifolds admit compatible symplectic forms, providing an affirmative answer to Donaldson's question in certain cases.

Contribution

It establishes the existence of compatible symplectic forms on tamed four-manifolds with particular cohomological properties, extending previous results and using techniques similar to Buchdahl's approach.

Findings

Existence of compatible symplectic forms under cohomological conditions

Affirmative answer to Donaldson's question when Betti number is one

Unified proof approach similar to Buchdahl's for the Kodaira conjecture

Abstract

This paper proves that on any tamed closed almost complex four-manifold $(M,J)$ whose dimension of $J$-anti-invariant cohomology is equal to the 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 affirmative answer to a question of Donaldson for tamed closed almost complex four-manifolds. Our approach is along the lines used by Buchdahl to give a unified proof of the Kodaira conjecture.