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 $2n\ge 10$ (respectively $\ge 6$) 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).

19 pages, misprint corrected in reference [5], to appear in Selecta Mathematica

Cited by in corpus (6)

• Dolbeault and $J$-invariant cohomologies on almost complex manifolds
• Rank of the Nijenhuis tensor on parallelizable almost complex manifolds
• Hodge-de Rham numbers of almost complex 4-manifolds
• On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms
• Bott-Chern Laplacian on almost Hermitian manifolds
• Invariant and non-invariant almost complex structures on compact quotients of Lie groups