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

Byun, Y.; Lee, Y.; Park, J.; Ryu, J.

doi:10.1090/S0002-9939-00-05587-8

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

Proc. Amer. Math. Soc. 129 (2001), 1161-1168 Request permission

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.

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