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Say I have an embedding: $f: X\to Y$ and a non-vanishing form $\omega$ on $Y$. Are there conditions on $f$ so that the pull-back $f^* \omega$ vanishes everywhere?

Or in a specific case, say $\omega$ is the non-trivial curvature form of some line bundle on $Y$, when does $f^* \omega$ vanish?

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