Manifolds with parallel differential forms and Kähler identities for G2-manifolds

Let M be a compact Riemannian manifold equipped with a parallel differential form ω. We prove a version of Kähler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquotient (H∗ c (M), d), called the pseudocohomology of M . When M is compact and Kähler and ω is its Kähler form, (H∗ c (M), d) is isomorphic to the cohomology algebra of M . This gives another proof of homotopy formality for Kähler manifolds, originally shown by Deligne, Griffiths, Morgan and Sullivan. We compute H∗ c (M) for a compact G2manifold, showing that H c (M) ∼= H(M) unless i = 3, 4. For i = 3, 4, we compute H∗ c (M) explicitly in terms of the first order differential operator ∗d : Λ3(M)−→ Λ(M).