### 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 Kahler 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 Kahler structure.

Original language | English |
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Pages (from-to) | 1161-1168 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 129 |

Issue number | 4 |

State | Published - 2001 Dec 1 |

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### Keywords

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

### Cite this

*Proceedings of the American Mathematical Society*,

*129*(4), 1161-1168.

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*Proceedings of the American Mathematical Society*, vol. 129, no. 4, pp. 1161-1168.

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds

AU - Byun, Yanghyun

AU - Lee, Y.

AU - Park, J.

AU - Ryu, J. S.

PY - 2001/12/1

Y1 - 2001/12/1

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.

KW - Kahler manifold

KW - Parallel positive spinor

KW - Spin structure

KW - Symplectic manifold

UR - http://www.scopus.com/inward/record.url?scp=23044522308&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:23044522308

VL - 129

SP - 1161

EP - 1168

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 4

ER -