On the J-anti-invariant cohomology of almost complex 4-manifolds

TL;DR

This paper investigates the structure of certain cohomology subgroups in almost complex 4-manifolds, revealing generic triviality of the anti-invariant subgroup and establishing semi-continuity properties of their dimensions.

Contribution

It provides new results on the behavior of J-anti-invariant cohomology classes, including generic triviality and semi-continuity, for almost complex 4-manifolds.

Findings

H^-_J subgroup is trivial for generic structures when b^+ = 1

Computed dimensions h^{ ext{±}}_J for related structures

Established semi-continuity properties for h^{ ext{±}}_J

Abstract

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that for generic almost complex structures on $M$, the subgroup $H^-_J$ is trivial. Computations of the subgroups and their dimensions $h^{\pm}_J$ are obtained for almost complex structures related to integrable ones. We also prove semi-continuity properties for $h^{\pm}_J$.