Dolbeault cohomology for almost complex manifolds

This paper extends Dolbeault cohomology and its surrounding theory to arbitrary almost complex manifolds. We define a spectral sequence converging to ordinary cohomology, whose first page is the Dolbeault cohomology, and develop a harmonic theory which injects into Dolbeault cohomology. Lie-theoretic analogues of the theory are developed which yield important calculational tools for Lie groups and nil manifolds. Finally, we develop applications to maximally non integrable manifolds, including nearly Kahler 6-manifolds, and show Dolbeault cohomology can be used to prohibit the existence of nearly Kahler metrics. (c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).