Dolbeault-type complexes on G2- and Spin(7)-manifolds

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. We list all Dolbeault complexes on compact G(2)-and Spin(7)-manifolds. Each cohomology group can be described by harmonic forms.