With a choice of (almost) complex structure on a symplectic (Kahler) manifold, has anyone thought of how or what it means to globally define "real" versions of the Dolbeault operators?
i.e. Something like
$\partial_x : \Omega_x^k(M) \oplus \Omega_y^l(M) \to \Omega_x^{k+1}(M)\oplus \Omega_y^l(M)$
and
$\partial_y : \Omega_x^k(M) \oplus \Omega_y^l(M) \to \Omega_x^k(M)\oplus \Omega_y^{l+1}(M)$
with $d=\partial_x + \partial_y$ where, in Darboux coordinates for example, $\Omega^1_x(M)$ is spanned by the $dx^i$ and $\Omega^1_x(M)$ by the $dy^j$.