Almost complex structures, transverse complex structures, and transverse Dolbeault cohomology

TL;DR

This paper introduces a new transverse Dolbeault cohomology for almost complex structures on manifolds, relating it to existing generalized cohomology theories and proposing a concept of minimal non-integrability.

Contribution

It extends the notion of transverse complex structures to almost complex structures and connects the new cohomology to prior generalized Dolbeault cohomology.

Findings

The (p,0) cohomology spaces coincide with those of Cirici and Wilson.

A new framework for studying almost complex structures via transversality.

Proposal of a notion of minimally non-integrable almost complex structures.

Abstract

We define a transverse Dolbeault cohomology associated to any almost complex structure $j$ on a smooth manifold $M$. This we 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. We relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of (M,j) introduced by Cirici and Wilson, showing that the (p,0) cohomology spaces coincide. This study of transversality leads us to suggest a notion of minimally non-integrable almost complex structure.