Good time of day.

I have the following question.

$X$- is a compact Kahler manifold(it may be projective or not). And $Y\subset X$ a complex submanifold. Also There is a holomorphic two-form $\phi \in \Gamma(X,\Omega_{X}^{2})$ such that $\phi|_{Y}=0$. And I try to understand why $Y$ is projective.

My thoughts about this are the following: It is known that complex submanifolds of Kähler manifolds are Kähler manifolds. It follows that $Y$ is a Kahler manifold too. After this I'm trying to use Kodaira embedding theorem for proving that $Y$ is projective. I don't know how to prove that we have positive line bundle over $Y$.

Possibly there are other more convenient attempts to this problem. Please if you don't mind please explain it in more details. Thank you!

Good time of day.

I have the following question.

$X$- is a compact Kähler manifold  (it may be projective or not). And $Y\subset X$ a complex submanifold. Also there is a holomorphic two-form $\phi \in \Gamma(X,\Omega_{X}^{2})$ such that $\phi|_{Y}=0$. And I try to understand why $Y$ is projective.

My thoughts about this are the following: It is known that complex submanifolds of Kähler manifolds are Kähler manifolds. It follows that $Y$ is a Kähler manifold too. After this I'm trying to use Kodaira embedding theorem for proving that $Y$ is projective. I don't know how to prove that we have positive line bundle over $Y$.

Possibly there are other more convenient attempts to this problem. Please if you don't mind please explain it in more details. Thank you!