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

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 10 2 n ≥ 10 (respectively 6 ≥ 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).