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I have two questions. For the first, we consider a projective morphism between two smooth, projective varieties over $k$, $f:X\rightarrow Y$. Let $\mathcal{L}$ be a line bundle and $f^*\mathcal{L}$ the pullback. Under what assumptions is $f*\mathcal{L}$ again a line bundle? (more generally what happens for a vector bundle $\mathcal{E}$ ?) For the second, we consider a closed embedding of a smooth variety $X$ into a smooth and projective variety $Y$ , $j:X\rightarrow Y$ and $\mathcal{L}$ a line bundle on $X$. When is $j_*\mathcal{L}$ again a line bundle? (more generally what for a vector bundle $\mathcal{E}$ ?)

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    $\begingroup$ Pullback of a vector bundle is always a vector bundle. Pushforward of a nontrivial vector bundle by a nontrivial embedding is never a vector bundle --- it is trivial outside the image of the embedding. $\endgroup$ – user64687 Feb 3 '14 at 17:20
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(Making my comment into an answer.)

The pullback of a vector bundle is always a vector bundle. The pushforward of a nontrivial vector bundle by a nontrivial embedding is never a vector bundle — it is trivial outside the image of the embedding.

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