Invariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry

Abstract

We study Kohn-Dirac operators D θ on strictly pseudoconvex CR manifolds with spin C structure of weight ∈ Z. Certain components of D θ are CR invariants. We also derive CR invariant twistor operators of weight. Harmonic spinors correspond to cohomology classes of some twisted Kohn-Rossi complex. Applying a Schrödinger-Lichnerowicz-type formula, we prove vanishing theorems for harmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructions to positive Webster curvature.