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

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 in 2001, 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.