Dolbeault cohomology for G2-manifolds

Cocalibrated G(2)-manifolds are seven-dimensional Riemannian manifolds with a distinguished 3-form which is coclosed. For such a manifold M, S. Salamon in Riemannian Geometry and Holonomy Groups (Longman, 1989) defined a differential complex (A(q)(M),D-q(V)) related with the G(2)-structure of M. In this paper we study the cohomology H*(V)(M) of this complex; it is treated as an analogue of a Dolbeault cohomology of complex manifolds. For compact G(2)-manifolds whose holonomy group is a subgroup of G(2) special properties are proved. The cohomology H*(V)(Gamma/K) of any cocalibrated G(2)-nilmanifold Gamma\K is also studied.