On a spectral sequence for twisted cohomologies

Abstract

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 {E , d r } derived from the filtration $F_p (\Omega ^ * (M)) = \mathop \oplus \limits_{i \geqslant p} \Omega ^i (M)$ 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.