Almost complex manifolds with small Nijenhuis tensor

TL;DR

This paper constructs explicit examples of compact manifolds with almost complex structures that have arbitrarily small Nijenhuis tensors, highlighting differences in complex structure existence across dimensions.

Contribution

It provides explicit 4- and 6-dimensional examples of manifolds with small Nijenhuis tensor, illustrating the subtlety of complex structure existence.

Findings

4D examples lack complex structures

6D example may lack a complex structure

Small Nijenhuis tensor does not guarantee integrability

Abstract

We give several explicit examples of compact manifolds with a $1$-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the $C^0$-norm. The $4$-dimensional examples possess no complex structure, whereas the $6$-dimensional example does not possesses a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.