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

Abstract

We study Kohn-Dirac operators 𝐷θ on strictly pseudoconvex CR manifolds with spinℂ structure of weight ℓ ∈ ℤ. Certain components of 𝐷θ are CR invariants. We also derive CR invariant twistor operators of weight ℓ. 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.