Why is it true, that complex submanifolds of Kähler manifolds are Kähler?
A Kähler manifold is $(M,J, \omega)$, where $(M, \omega)$ is symplectic, $(M,J)$ is a complex manifold.
Now let $W \subset M$ be a complex submanifold, so $(W, J|_{W})$ is a complex manifold. Then I need to show, that $\omega|_{W \times W}$ is non-degenerate.
Now assume that for $v \in W$, $\omega(v,u)=0$ for all $u \in W$.
I need now to show that this is somehow a contradiction to the complex structure of $W$.
Can somebody give me a hint on how to do that?
Hint: Don't think about the restriction of the Kähler form, but about the restriction of the metric.