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

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.