Let $(K, G, \Omega, J)$ be a Kähler manifold and $(S, \omega)$ be a symplectic manifold. Let $i : S \to K$ be a symplectic embedding. Is it possible to endow $S$ with a Kähler manifold structure, compatible with $\omega$, through $i$?
Clearly, I can define $g = i^* G$ a Riemann metric on $S$, but what about its complex structure $j$? I thought about defining it through $g(X,Y) = \omega (X, jY)$, but this doesn't seem to produce $j^2 = -\textrm {id}$. I also thought about defining it as $j(X) = i_* ^{-1} \textrm{pr} \ J (i_* X)$, where $\textrm {pr}$ is the orthogonal projection onto $i_* (TS)$ obtained with the aid of $G$, $\Omega$ or $H$ (the Hermitic structure on $K$), but again this does not seem to produce $j^2 = -\textrm {id}$.
Can my question be answered in the affirmative, then? If so, how?