In this [question](https://mathoverflow.net/questions/22111/extending-vector-bundles-on-a-given-open-subscheme), Ariyan asks about the question of uniqueness of extensions of vector bundles when they exist.


Sasha's answer suggests that extensions of vector bundles don't always exist.


More precisely, if $F$ is a vector bundle on an open subscheme $U$, there does not always exist a vector bundle $F'$ on the ambient space $X$ such that $F'|\_U \cong F$.


Can anyone give me a simple example of such an $F$?


I am mainly interested in the case when $X$ is a variety (over $\mathbb{C}$), and $U$ is an open subvariety. Probably I want $X$ to be smooth.



Extending vector bundles on a given open subscheme, reprise
The simplest example is the following. Take $X = A^3$ with coordinates $(x,y,z)$, and let $E = Ker(O\_X \oplus O\_X \oplus O\_X \stackrel{(x,y,z)}\to O\_X)$. Let $U$ be the complement of the point $(0,0,0) \in X$. Then $E\_{|U}$ is a vector bundle. On the other hand, $E$ is not a vector bundle, but $E^{\*\*} \cong E$, hence $E$ is the reflexive envelope of $i\_\*i^\*E$, and thus there is no vector bundle on $X$ extending $E\_{|U}$.




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[Edit by Anton: I just spent some time digesting some pieces of the above answer, so figured I'd include the results for future readers similar to me.]



**("$E$ is not a vector bundle")** The sequence $O\_X\xrightarrow{\pmatrix{z\\ y \\ x}}O\_X^3\xrightarrow{\pmatrix{y & -z & 0\\ -x & 0 & z\\ 0 &x&-y}}O\_X^3\xrightarrow{\pmatrix{x& y& z}}O\_X$ is exact, so $E$ is the cokernel of the first map. Since taking fibers commutes with taking cokernels, we compute that $E$ has 2-dimensional fibers away from the origin, and 3-dimensional fiber at the origin.




**("$E^{\*\*}\cong E$")** Note that $E$ is $S\_2$ (i.e. sections defined away from codimension 2 extend uniquely) since it is the kernel of a map from an $S\_2$ sheaf to a torsion-free sheaf (the section of $O\_X^3$ extends uniquely, and its image is zero away from codimension 2, so must be zero, so the extended section is in $E$). Note also that the dual of any sheaf is $S\_2$ (if $\phi\colon F\to O\_X$ is defined on an open set $V$ with codimension 2 complement and $s$ is a section, $\phi(s)$ must be the unique extension of $\phi(s|\_V)$ as a section of $O\_X$), so $E^{\*\*}$ is $S\_2$. The canonical map $E\to E^{\*\*}$ is then a map of $S\_2$ sheaves which is an isomorphism away from codimension 2, so it must be an isomorphism.




**("and thus there is no vector bundle on $X$ extending $E|\_U$")** If $F$ is an $S\_2$ extension of $E|\_U$ (i.e. $i^\*F=i^\*E$), then there is a map $F\to i\_\*i^\*E\to (i\_\*i^\*E)^{\*\*}=E$ which is an isomorphism over $U$, so is an isomorphism by the argument in the previous paragraph. A vector bundle extension would be a different $S\_2$ extension.



