Timeline for Extending vector bundles on a given open subscheme, reprise
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2020 at 21:05 | comment | added | myzhang24 | Remark to point 2: as soon as dimension $X$ is at least $3$, one can find examples. See my answer. | |
| Aug 12, 2012 at 14:40 | comment | added | Federigo | In order to say that there exists $W$ coherent on $X$ such that $W_{|U}\simeq E$ can't we use Hartshorne's Algebraic Geometry, II, ex. 5.15 p. 126? There is no assumption on $X\U$. Of course $W$ is not isomorphic to $j_*(E)$ but I don't think we need it in point 2. Let me point out that point 2 can be applied to the following situation, if I am not wrong: if $X$ is a relative smooth connected curve over a Dedekind scheme (i.e. $dim(X)=2$) a vector bundle over its generic fiber can be extended to $X$. Sorry for my 2 years delay but what is important is that it's still August! :-) | |
| Aug 20, 2010 at 15:00 | comment | added | David Treumann | I am also wrong about torsion, the example in my previous comment is bogus. I've asked another question about this. | |
| Aug 18, 2010 at 18:37 | comment | added | Donu Arapura | Right. I made the appropriate changes. | |
| Aug 18, 2010 at 18:35 | history | edited | Donu Arapura | CC BY-SA 2.5 |
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| Aug 18, 2010 at 15:59 | comment | added | David Treumann | I am wrong about X having to be compact, since U and X will have a common compactification. Torsion Chern classes still give obstructions: e.g. if X is the total space of O(n), n < -1, on P^1 and U is the complement of the zero section, there are line bundles on U that do not extend across X or a compactification of X. | |
| Aug 17, 2010 at 15:21 | history | edited | Donu Arapura | CC BY-SA 2.5 |
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| Aug 17, 2010 at 15:18 | comment | added | Donu Arapura | Yes, sorry, let me add that. | |
| Aug 17, 2010 at 15:16 | comment | added | David Treumann | 4. is nice! It requires X smooth and compact, so Chern classes can still give obstructions e.g. to extending vector bundles across the special fiber of a degeneration. | |
| Aug 17, 2010 at 15:04 | history | edited | Donu Arapura | CC BY-SA 2.5 |
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| Aug 17, 2010 at 14:47 | history | edited | Donu Arapura | CC BY-SA 2.5 |
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| Aug 17, 2010 at 14:37 | history | answered | Donu Arapura | CC BY-SA 2.5 |