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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 $dy^j$.

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  • $\begingroup$ I don't understand the second version of your question. If you meant was originally posted, it became very unclear. I rolled back to the original version. $\endgroup$ – user98602 Apr 3 '15 at 2:29
  • $\begingroup$ Given a symplectic structure, there's no invariant way to distinguish the "$x$-directions" and the "$y$-directions." An easy way to see this is to note that for any Darboux coordinates $(x^1,\dots,x^n,y^1,\dots,y^n)$, the new coordinates $(\tilde x^1,\dots,\tilde x^n,\tilde y^1,\dots,\tilde y^n)$ given by $\tilde x^i = y^i$, $\tilde y^i = - x^i$, are also Darboux coordinates, and they interchange the $x$-directions and the $y$-directions. Some extra structure is needed, such as a pair of complementary Lagrangian foliations. $\endgroup$ – Jack Lee Apr 3 '15 at 18:53

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