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 (respectively ≥ 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).

Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

“…is a compatible symplectic structure, namely (J, ω) is an almost-Kähler structure on T 6 . Notice now that by a direct computation…”

“…Let Γ be the subgroup of matrices with integral entries, let X := Γ\H(1, 2) and define M := X × T 3 . Denoting with u, v, w coordinates on T 3 we consider the following left-invariant 1-forms e 1 := dx 2 , e 2 := dx 1 , e 3 := dy, e 4 := du, e 5 := dz 1 − x 1 dy, e 6 := dz 2 − x 2 dy, e 7 := dv, e 8 := dw, and the structure equations become de 1 = de 2 = de 3 = de 4 = de 7 = de 8 = 0, de 5 = −e 23 , de 6 = −e 13 .…”

“…Recently, Cirici and Wilson in [4] gave a definition for the Dolbeault cohomology in the non-integrable setting considering also the operator μ together with ∂. Such cohomology groups might be infinite-dimensional on compact almost-complex manifolds as shown by [6]. On the other side, fixing an almost-Hermitian metric g on (X, J) one can develop an Hodge theory for harmonic forms on (X, J, g) without a cohomological counterpart.…”

“…−e 13 + e 24 de6 = −e 14 − e 23 .Let us consider the non integrable left-invariant almost-complex structure J given by φ 1 = e 1 + ie 6 , φ 2 = e 2 + ie 5 , φ 3 = e 3 + ie 4being a global coframe of (1, 0)-forms. By a direct computation the structure equations become (cf.…”

Bott-Chern harmonic forms and primitive decompositions on compact almost Kähler manifolds

“…If X is compact these spaces are finite-dimensional but they do not have a cohomological counterpart. In fact, the almost complex Dolbeault, Bott-Chern and Aeppli cohomology groups might be infinite dimensional (see [2], [4]).…”