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

TL;DR

This paper investigates invariant Dirac operators on CR manifolds, establishing vanishing theorems for harmonic spinors and cohomology groups, and deriving obstructions related to Webster curvature, advancing understanding in CR geometry.

Contribution

It introduces CR invariant Kohn-Dirac and twistor operators, links harmonic spinors to cohomology, and proves new vanishing theorems and curvature obstructions in CR geometry.

Findings

Vanishing theorems for harmonic spinors and Kohn-Rossi groups.

CR invariant twistor operators of weight ll.

Obstructions to positive Webster curvature.

Abstract

We study Kohn-Dirac operators $D_\theta$ on strictly pseudoconvex CR manifolds with ${\rm spin}^{\mathbb C}$ structure of weight $\ell\in{\mathbb Z}$. Certain components of $D_\theta$ are CR invariants. We also derive CR invariant twistor operators of weight $\ell$. Harmonic spinors correspond to cohomology classes of some twisted Kohn-Rossi complex. Applying a Schr\"odinger-Lichnerowicz-type formula, we prove vanishing theorems for harmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructions to positive Webster curvature.