Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 011, 25 pages      arXiv:2102.02477

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

Felipe Leitner
Universität Greifswald, Institut für Mathematik und Informatik,Walter-Rathenau-Str. 47, D-17489 Greifswald, Germany

Received July 23, 2020, in final form January 22, 2021; Published online February 04, 2021

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 Schrödinger-Lichnerowicz-type formula, we prove vanishing theorems for harmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructions to positive Webster curvature.

Key words: CR geometry; spin geometry; Kohn-Dirac operator; harmonic spinors; Kohn-Rossi cohomology; vanishing theorems.

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