Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds

Yanghyun Byun, Y. Lee, J. Park, J. S. Ryu

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spinc structure implies that the underlying smooth manifold admits a Kahler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the spinc structure in which the non-zero parallel spinor lives is equivalent to the canonical spinc structure associated to the Kahler structure.

Original languageEnglish
Pages (from-to)1161-1168
Number of pages8
JournalProceedings of the American Mathematical Society
Volume129
Issue number4
StatePublished - 2001 Dec 1

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Spinors
Symplectic Structure
4-manifold
Spinor
Smooth Manifold
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Keywords

  • Kahler manifold
  • Parallel positive spinor
  • Spin structure
  • Symplectic manifold

Cite this

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Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds. / Byun, Yanghyun; Lee, Y.; Park, J.; Ryu, J. S.

In: Proceedings of the American Mathematical Society, Vol. 129, No. 4, 01.12.2001, p. 1161-1168.

Research output: Contribution to journalArticle

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N2 - We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spinc structure implies that the underlying smooth manifold admits a Kahler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the spinc structure in which the non-zero parallel spinor lives is equivalent to the canonical spinc structure associated to the Kahler structure.

AB - We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spinc structure implies that the underlying smooth manifold admits a Kahler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the spinc structure in which the non-zero parallel spinor lives is equivalent to the canonical spinc structure associated to the Kahler structure.

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