# On the Anti-Invariant Cohomology of Almost Complex Manifolds

## TL;DR

This paper investigates the space of anti-invariant forms on almost complex manifolds, constructing examples with infinite and large finite dimensions of anti-invariant cohomology, revealing complex geometric structures.

## Contribution

It introduces new constructions of almost complex structures with prescribed anti-invariant cohomology dimensions, including non-compact and compact cases.

## Key findings

- Infinite-dimensional anti-invariant cohomology on non-compact manifolds.
- Large anti-invariant cohomology in compact 6-manifolds.
- Explicit families with maximum anti-invariant cohomology on the Kodaira-Thurston manifold.

## Abstract

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non compact. We construct families of (non integrable) almost complex structures on $\R^4$, such that the space of closed $J$-anti-invariant forms is infinite dimensional, and also $0$- or $1$-dimensional. In the compact case, we construct $6$-dimensional almost complex manifolds with arbitrary large anti-invariant cohomology and a $2$-parameter family of almost complex structures on the Kodaira-Thurston manifold whose anti-invariant cohomology group has maximum dimension.