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 $X$?