The bigraded Rumin complex via differential forms

2021/08/31 by Case, Jeffrey S.
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.13911

Abstract

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_∞-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(M²ⁿ⁺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∈ H¹(M;\mathscrP) -- whose vanishing is necessary and sufficient for the existence of a pseudo-Einstein contact form -- to all nondegenerate orientable CR manifolds.

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