Is a symplectic submanifold of a Kähler manifold Kähler?


That is, if $X$ is a Kähler manifold with symplectic form $\omega$ and $i:Y\hookrightarrow X$ is an embedded submanifold such that $i^\*\omega$ is symplectic, is $Y$ a Kähler submanifold of $X$?



Is a symplectic submanifold of a Kähler manifold Kähler?
No. In $\mathbf C^2$ with standard 2-form and complex structure, the real span of $U=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix}i\\1\end{smallmatrix}\right)$ is a symplectic but not complex (hence not Kähler) subspace, since
$$
\omega(U,V)=1,
\qquad
JU=\left(\begin{smallmatrix}i\\0\end{smallmatrix}\right)\notin\operatorname{span}\_{\mathbf R}(U,V).
$$


