  Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin^C
structure induced by an almost-complex structure \omega. Each connection A on
the determinant line bundle induces a unique connection \nabla^A, and Dirac
operator \D^A on spinor fields. Let \sigma: W^+ --> \Lambda^+ be the natural
squaring map, taking self-dual (= positive) spinors to self-dual 2-forms.
  In this paper, we characterize the self-dual 2-forms that are images of
self-dual spinor fields through \sigma. They are those \alpha for which (off
zeros) c_1(\alpha) = c_1(\omega), where c_1(\alpha) is a suitably defined Chern
class. We also obtain the formula: || \phi ||^2 D^A \phi = i (2 d^*
\sigma(\phi) + < \nabla^A \phi, i \phi >)* \phi.
  Using these, we establish a bijective correspondence between: {Kahler forms
\alpha compatible with a metric scalar-multiple of g, and with c_1(\alpha) =
c_1(\omega)} and {gauge classes of pairs (\phi, A), with \nabla^A \phi = 0}, as
well as a bijective correspondence between: {Symplectic forms \alpha compatible
with a metric conformal to g, and with c_1(\alpha) = c_1(\omega)} and {gauge
classes of pairs (\phi, A), with D^ A \phi = 0, and < \nabla^A \phi, i \phi > =
0, and \phi nowhere-zero}.
