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

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.