Dolbeault-type complexes on $G_2$- and $\mathrm{Spin}(7)$-manifolds

TL;DR

This paper classifies Dolbeault-type complexes on manifolds with exceptional holonomy groups G_2 and Spin(7), showing their ellipticity conditions and describing cohomology via harmonic forms.

Contribution

It provides a complete classification of Dolbeault complexes on G_2 and Spin(7) manifolds and characterizes their cohomology in terms of harmonic forms.

Findings

All Dolbeault complexes on G_2 and Spin(7) manifolds are classified.

Ellipticity of these complexes depends on algebraic conditions of generators.

Cohomology groups are described explicitly by harmonic forms.

Abstract

There are three types of Dolbeault complexes arising from representations of holonomy group on a Riemannian manifold, two of which are dual to each other. Such a complex is elliptic if and only if its generator satisfies an algebraic condition. As applications, we give all Dolbeault complexes on a compact Riemannian manifold with exceptional holonomy. Each cohomology can be described by harmonic forms.