On a Spectral Sequence for Twisted Cohomologies

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 is a twisted de Rham cochain complex （Ω＊ （M）, d ＋ H∧） and its associated twisted de Rham cohomology H＊ （M, H）. The authors show that there exists a spectral sequence {Ep/r.q, dr } derived from the filtration Fp（Ω＊ （M）） = （¤i〉p Ωi（M） of Ω＊ （M）, which converges to the twisted de Rham cohomology H＊（M, H）. It is also shown 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.