Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds
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Abstract:
We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spin$^c$ structure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the $spin^c$ structure in which the non-zero parallel spinor lives is equivalent to the canonical spin$^c$ structure associated to the Kähler structure.References
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Additional Information
- Y. Byun
- Affiliation: Department of Mathematics, College of Natural Science, Hanyang University, Sungdong-gu, Seoul 133-791, Korea
- Email: yhbyun@fermat.hanyang.ac.kr
- Y. Lee
- Affiliation: Department of Mathematics, College of Natural Science, Inha University, Incheon-si 402-751, Korea
- Email: ywlee@math.inha.ac.kr
- J. Park
- Affiliation: Department of Mathematics, College of Natural Science, Dongguk University, Joong-gu, Seoul 100-715, Korea
- Email: jpark@cakra.dongguk.ac.kr
- J. S. Ryu
- Affiliation: Department of Mathematics Education, College of Education, Hongik University, Mapo-gu, Seoul 121-791, Korea
- Email: jsryu@math.hongik.ac.kr
- Received by editor(s): April 30, 1999
- Received by editor(s) in revised form: June 21, 1999
- Published electronically: September 20, 2000
- Additional Notes: The first author was partially supported by the Hanyang University Research Fund. The second author was supported in part by 1998-015-D00044. The third author was partially supported by the Dongguk University Research Fund. The fourth author was supported in part by GARC
- Communicated by: Ronald A. Fintushel
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1161-1168
- MSC (1991): Primary 53C05, 53C07
- DOI: https://doi.org/10.1090/S0002-9939-00-05587-8
- MathSciNet review: 1707139