The Bigraded Rumin Complex via Differential Forms

We give a new CR invariant treatment of the bigraded Rumin complex and related cohomology groups via differential forms. A key benefit is the identification of balanced A ∞ A_ -structures on the Rumin and bigraded Rumin complexes. We also prove related Hodge decomposition theorems. Among many applications, we give a sharp upper bound on the dimension of the Kohn–Rossi groups H 0, q (M 2 n + 1) H^0, q (M^2n+1), 1 ≤ q ≤ n − 1 1 q n-1, of a closed strictly pseudoconvex manifold with a contact form of nonnegative pseudohermitian Ricci curvature; we prove a sharp CR analogue of the Frölicher inequalities in terms of the second page of a natural spectral sequence; we give new proofs of selected topological properties of closed Sasakian manifolds; and we generalize the Lee class L ∈ H 1 (M ; P) L H¹ (M; P) — whose vanishing is necessary and sufficient for the existence of a pseudo-Einstein contact form — to all nondegenerate orientable CR manifolds.