Dolbeault cohomology for almost complex manifolds

TL;DR

This paper generalizes Dolbeault cohomology to almost complex manifolds, introduces a spectral sequence and harmonic theory, and explores applications to non-integrable structures like nearly Kähler manifolds.

Contribution

It extends Dolbeault cohomology to almost complex manifolds, develops a spectral sequence and harmonic theory, and applies these to study non-integrable structures.

Findings

Spectral sequence converges to ordinary cohomology with Dolbeault as first page.

Harmonic theory injects into Dolbeault cohomology.

Dolbeault cohomology can obstruct nearly Kähler metrics.

Abstract

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 nilmanifolds. Finally, we study applications to maximally non-integrable manifolds, including nearly K\"ahler $6$-manifolds, and show Dolbeault cohomology can be used to prohibit the existence of nearly K\"ahler metrics.