Almost complex structures, transverse complex structures, and transverse Dolbeault cohomology (Q6620657)

An almost complex structure \(j\) is a smooth field of endomorphisms of the tangent bundle whose square is minus the identity \((j^2=-\mathrm{Id})\). Dolbeault cohomology is defined for a manifold endowed with an integrable almost complex structure. In the present paper, the authors suggest another Dolbeault cohomology defined only in terms of the almost complex structure \(j\), which they call transverse Dolbeault cohomology. It is defined as the usual Dolbeault cohomology of a natural transverse complex structure induced by the given almost complex structure. They define a transverse Dolbeault cohomology associated to any almost complex structure \(j\) on a smooth manifold \(M\). This they do by extending the notion of transverse complex structure and by introducing a natural \(j\)-stable involutive limit distribution with such a transverse complex structure. They relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of \((M,j)\) introduced by \textit{J. Cirici} and \textit{S. O. Wilson} [Adv. Math. 391, Article ID 107970, 52 p. (2021; Zbl 1478.32085)] showing that the \((p,0)\) cohomology spaces coincide. This study of transversality leads them to suggest a notion of minimally non-integrable almost complex structure. \N\NThe paper consists of the following sections. It starts with an introduction to the subject and statement of results. Section 1 deals with derived distributions associated to an almost complex structure. Section 2 is devoted to \(\mathcal{D}\)-transverse structures where \(\mathcal{D}\) can be the space of smooth sections of a real smooth involutive distribution \(D\) on a manifold \(M\) (not necessarily of constant rank), stable under \(j\), or can be the involutive limit of an increasing sequence of nested spaces of sections. In Section 3, the authors define the transverse Dolbeault cohomology associated to an almost complex structure \(j\). Section 4 deals with a notion of minimal non-integrability. Here, the authors suggest a minimality condition for a non-integrable almost complex structure, they write this condition in a homogeneous framework, and they study the example of the Kodaira-Thurston 4-dimensional manifold.