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There are three types of Dolbeault complexes arising from representations ofholonomy 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 algebraiccondition. As applications, we give all Dolbeault complexes on a compactRiemannian manifold with exceptional holonomy. Each cohomology can be describedby harmonic forms.

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