Spinors as automorphisms of the tangent bundle

We show that, on a 4-manifold M endowed with a spin^c structure induced by an almost-complex structure, a self-dual (= positive) spinor field \phi \in \Gamma(W^+) is the same as a bundle morphism \phi: TM \to TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of \phi on tangent vectors, and that the squaring map \sigma: W^+ \to \Lambda^+ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kahler and symplectic structures.