A note on exact forms on almost complex manifolds

TL;DR

This paper explores conditions for the existence of compatible symplectic forms on almost complex manifolds, reformulating Donaldson's question through exact forms and analyzing cohomological properties in dimension four.

Contribution

It provides new reformulations of Donaldson's question using spaces of exact forms and characterizes compatible symplectic forms in dimension four via tamed forms with specific anti-invariant parts.

Findings

J admits a compatible symplectic form iff it admits tamed forms with arbitrary J-anti-invariant parts in dimension 4

Observations on the cohomology of J-modified de Rham complexes

Reformulations of Donaldson's 'tamed to compatible' question

Abstract

Reformulations of Donaldson's "tamed to compatible" question are obtained in terms of spaces of exact forms on a compact almost complex manifold $(M^{2n},J)$. In dimension 4, we show that $J$ admits a compatible symplectic form if and only if $J$ admits tamed symplectic forms with arbitrarily given $J$-anti-invariant parts. Some observations about the cohomology of $J$-modified de Rham complexes are also made.