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

SIGMA 17 (2021), 011, 25 pages      arXiv:2102.02477      https://doi.org/10.3842/SIGMA.2021.011

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

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 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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References

1. Blair D.E., Contact manifolds in Riemannian geometry, Lecture Notes in Math., Vol. 509, Springer-Verlag, Berlin - New York, 1976.
2. Čap A., Gover A.R., CR-tractors and the Fefferman space, Indiana Univ. Math. J. 57 (2008), 2519-2570 arXiv:math.DG/0611938.
3. Case J.S., Yang P., A Paneitz-type operator for CR pluriharmonic functions, Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013), 285-322, arXiv:1309.2528.
4. Folland G.B., Kohn J.J., The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, Vol. 75, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1972.
5. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
6. Jerison D., Lee J.M., The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167-197.
7. Kohn J.J., Boundaries of complex manifolds, in Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, Berlin, 1965, 81-94.
8. Kohn J.J., Rossi H., On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. 81 (1965), 451-472.
9. Kordyukov Y.A., Vanishing theorem for transverse Dirac operators on Riemannian foliations, Ann. Global Anal. Geom. 34 (2008), 195-211, arXiv:0708.1698.
10. Lawson Jr. H.B., Michelsohn M.L., Spin geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, Princeton, NJ, 1989.
11. Lee J.M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), 157-178.
12. Leitner F., The first eigenvalue of the Kohn-Dirac operator on CR manifolds, Differential Geom. Appl. 61 (2018), 97-132.
13. Leitner F., Parallel spinors and basic holonomy in pseudo-Hermitian geometry, Ann. Global Anal. Geom. 55 (2019), 181-196.
14. Lichnerowicz A., Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9.
15. Michelsohn M.L., Clifford and spinor cohomology of Kähler manifolds, Amer. J. Math. 102 (1980), 1083-1146.
16. Morgan J.W., The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, Vol. 44, Princeton University Press, Princeton, NJ, 1996.
17. Moroianu A., On Kirchberg's inequality for compact Kähler manifolds of even complex dimension, Ann. Global Anal. Geom. 15 (1997), 235-242.
18. Petit R., ${\rm Spin}^c$-structures and Dirac operators on contact manifolds, Differential Geom. Appl. 22 (2005), 229-252.
19. Pilca M., Kählerian twistor spinors, Math. Z. 268 (2011), 223-255, arXiv:0812.3315.
20. Stadtmüller C., Horizontal Dirac operators in CR geometry, Ph.D. Thesis, Humboldt University Berlin, 2017, available at https://edoc.hu-berlin.de/handle/18452/18801.
21. Tanaka N., A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, Vol. 9, Kinokuniya Book-Store Co., Ltd., Tokyo, 1975.
22. Webster S.M., Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), 25-41.