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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!

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    $\begingroup$ This is false. Let $X = Y$ be a non-projective Kähler manifold and $\phi = 0$. $\endgroup$
    – Spenser
    Commented May 19, 2022 at 15:45
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    $\begingroup$ The version where $\phi\neq 0$ is also false, taking $Y$ non-projective Kähler and $X = Y \times S$ where $S$ is a K3 surface, abelian surface, etc. and $\phi$ is pulled-back from $S$. $\endgroup$
    – Will Sawin
    Commented May 19, 2022 at 15:48
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    $\begingroup$ It is not a good idea to ask here for a solution of your exam problem! $\endgroup$
    – Sasha
    Commented May 19, 2022 at 15:52
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    $\begingroup$ @Sasha I'm not studying at university and I study kahler geometry for myself. Advises and solutions of certain problem help me to understand subject. $\endgroup$
    – UserIn
    Commented May 19, 2022 at 15:58
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    $\begingroup$ @UserIn: I just happen to know that this problem was given yesterday for a home examination, so this looks exactly as if you are trying to use people here to solve this problem for you. $\endgroup$
    – Sasha
    Commented May 19, 2022 at 18:52

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