The Bigraded Rumin Complex via Differential Forms

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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_\infty -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\leq q\leq 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 ) \mathcal {L}\in H^1(M;\mathscr {P}) — whose vanishing is necessary and sufficient for the existence of a pseudo-Einstein contact form — to all nondegenerate orientable CR manifolds.