On a spectral sequence for twisted cohomologies

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Let (Ω(M), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there are a twisted de Rham cochain complex (Ω(M), d+H) and its associated twisted de Rham cohomology H*(M,H). We show that there exists a spectral sequence {Eᵖ, qᵣ, dᵣ} derived from the filtration Fₚ(Ω(M))=ᵢ≥ ₚΩⁱ(M) of Ω(M), which converges to the twisted de Rham cohomology H*(M,H). We also show that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.