Symplectic forms and cohomology decomposition of almost complex 4-manifolds

TL;DR

This paper demonstrates a cohomology decomposition for 4-dimensional compact almost complex manifolds using J-invariant and J-anti-invariant forms, with estimates and implications for symplectic structures and Donaldson's question.

Contribution

It establishes a cohomology decomposition in dimension 4 based on J-invariant and J-anti-invariant subgroups, extending previous definitions and connecting to symplectic and Donaldson's questions.

Findings

Cohomology decomposition in 4D almost complex manifolds.

Estimates for dimensions of cohomology subgroups under symplectic taming.

Reformulation of Donaldson's question in this context.

Abstract

For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by $J$-invariant, respectively, $J$-anti-invariant real $2-$forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of $H^2(M, \mathbb{R})$. This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.