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This is a little perverse, but rather than answering the question, I want to explain what can go wrong when attempting to construct an example. This is the sort of thing one never does normally so I think it's kind of interesting.

1. If $X$ is a smooth curve, then any vector bundle $E$ on an open set $U$ extends. To see this, we can assume after shrinking $X$, that $E$ is trivial. Then it can be extended to a trivial bundle (the extension is not unique).

2. If $X$ is smooth surface, then any vector bundle $E$ on an open set $U$ extends. (I think that Olivier Benoist's answer contains a very nice idea, but I don't think the conclusion is OK.) To simplify the argument, assume that $X-U=\{p_1,p_2\ldots \}$ is zero dimensional. We can find finitely sections in a neigbourhood $V$ of $p_i$ which generate $E^*$. This yields an inclusion $E|_V\hookrightarrow \oplus \mathcal{O}_V^n$, and therefore $j_*E|_V \hookrightarrow\mathcal{O}_X^n$, where $j:U\hookrightarrow X$ is the inclusion. It follows easily, that $j_*E$ is coherent. Therefore $F=(j_*E)^{**}$ is a reflexive extension of $E$. However, reflexive sheaves have depth 2. Since by Auslander-Buchsbaum-Serre depth+proj.dim=2 in $\mathcal{O}_{p_i}$, we can conclude that $F$ is in fact locally free.

3. In view of jvp's answer, we see that 2 does not hold in the analytic category.

4. One might seek a topological obstruction involving Chern classes as in David Treumann's comment, however: Claim: Any Chern class on $U$ extends to $X$, where $X$ is a smooth partial compactification. Proof: With a bit of fiddling one can see that $c_p(E)$ would lie in $W_{2p}H^{2p}(U)=im H^{2p}(X)$ W_{2p}H^{2p}(U,\mathbb{Q})=im H^{2p}(X,\mathbb{Q})$by Deligne, Theorie de Hodge II, III 4 added 40 characters in body This is a little perverse, but rather than answering the question, I want to explain what can go wrong when attempting to construct an example. This is the sort of thing one never does normally so I think it's kind of interesting. 1. If$X$is a smooth curve, then any vector bundle$E$on an open set$U$extends. To see this, we can assume after shrinking$X$, that$E$is trivial. Then it can be extended to a trivial bundle (the extension is not unique). 2. If$X$is smooth surface, then any vector bundle$E$on an open set$U$extends. (I think that Olivier Benoist's answer contains a very nice idea, but I don't think the conclusion is OK.) To simplify the argument, assume that $X-U=\{p_1,p_2\ldots \}$ is zero dimensional. We can find finitely sections in a neigbourhood$V$of$p_i$which generate$E^*$. This yields an inclusion $E|_V\hookrightarrow \oplus \mathcal{O}_V^n$, and therefore $j_*E|_V \hookrightarrow\mathcal{O}_X^n$, where $j:U\hookrightarrow X$ is the inclusion. It follows easily, that $j_*E$ is coherent. Therefore $F=(j_*E)^{**}$ is a reflexive extension of$E$. However, reflexive sheaves have depth 2. Since by Auslander-Buchsbaum-Serre depth+proj.dim=2 in$\mathcal{O}_{p_i}$, we can conclude that$F$is in fact locally free. 3. In view of jvp's answer, we see that 2 does not hold in the analytic category. 4. One might seek a topological obstruction involving Chern classes as in David Treumann's comment, however: Claim: Any Chern class on$U$extends to$X$. X$, where $X$ is a smooth compactification. Proof: With a bit of fiddling one can see that $c_p(E)$ would lie in $W_{2p}H^{2p}(U)=im H^{2p}(X)$ by Deligne, Theorie de Hodge II, III

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This is a little perverse, but rather than answering the question, I want to explain what can go wrong when attempting to construct an example. This is the sort of thing one never does normally so I think it's kind of interesting.

1. If $X$ is a smooth curve, then any vector bundle $E$ on an open set $U$ extends. To see this, we can assume after shrinking $X$, that $E$ is trivial. Then it can be extended to a trivial bundle (the extension is not unique).

2. If $X$ is smooth surface, then any vector bundle $E$ on an open set $U$ extends. (I think that Olivier Benoist's answer contains a very nice idea, but I don't think the conclusion is OK.) To simplify the argument, assume that $X-U=\{p_1,p_2\ldots \}$ is zero dimensional. We can find finitely sections in a neigbourhood $V$ of $p_i$ which generate $E^*$. This yields an inclusion $E|_V\hookrightarrow \oplus \mathcal{O}_V^n$, and therefore $j_*E|_V \hookrightarrow\mathcal{O}_X^n$, where $j:U\hookrightarrow X$ is the inclusion. It follows easily, that $j_*E$ is coherent. Therefore $F=(j_*E)^{**}$ is a reflexive extension of $E$. However, reflexive sheaves have depth 2. Since by Auslander-Buchsbaum-Serre depth+proj.dim=2 in $\mathcal{O}_{p_i}$, we can conclude that $F$ is in fact locally free.

3. In view of jvp's answer, we see that 2 does not hold in the analytic category.

4. One might seek a topological obstruction involving Chern classes as in David Treumann's comment, however: Claim: Any Chern class on $U$ extends to $X$. Proof: With a bit of fiddling one can see that $c_p(E)$ would lie in $W_{2p}H^2p(U)=im W_{2p}H^{2p}(U)=im H^{2p}(X)$ by Deligne, Theorie de Hodge II, III

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