Maximally non-integrable almost complex structures: an $h$-principle and cohomological properties

TL;DR

This paper demonstrates that almost complex structures with maximal Nijenhuis tensor rank satisfy an $h$-principle, allowing broad existence results and revealing complex cohomological properties, especially on high-dimensional and certain 4-manifolds.

Contribution

It establishes an $h$-principle for almost complex structures with lower bounds on Nijenhuis tensor rank and characterizes their existence on specific manifolds.

Findings

All parallelizable manifolds admit such structures.

High-dimensional manifolds (dimension ≥ 10) admit structures with maximal Nijenhuis tensor rank.

The Dolbeault cohomology can be infinite dimensional for non-integrable structures.

Abstract

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an $h$-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension $2n\geq 10$ (respectively $\geq 6$) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed $4$-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds).