Timeline for Compactifications of varieties with small complement
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 17, 2018 at 0:24 | comment | added | Dan Petersen | If the codimension is $c$, then $H^i(\overline X) \to H^i(X)$ is an isomorphism for $i<2c-1$ and injective for $i=2c-1$ (use the long exact sequence of a pair). This gives cohomological obstructions to the existence of such a compactification; for example, the Hodge structure on $H^i(X)$ must be pure of weight $i$ in low degrees. | |
| Oct 11, 2010 at 6:49 | comment | added | Lars | Ah, yes, of course! | |
| Oct 10, 2010 at 18:08 | answer | added | Torsten Ekedahl | timeline score: 6 | |
| Oct 10, 2010 at 12:52 | comment | added | Torsten Ekedahl | Vector bundles extend to vector bundles in codimension $2$ but not necessarily beyond that so your equivalence of categories is OK for surfaces but not for higher dimension. | |
| Oct 10, 2010 at 12:38 | comment | added | Lars | Yes, I should have mentioned this, there are stronger restrictions. For example, at least if $\overline{X}$ is smooth, then we must have $\Gamma(X,\mathcal{O}_X)=k$, and the category of vector bundles on is equivalent to the category of vector bundles on $X$. | |
| Oct 10, 2010 at 11:57 | comment | added | Daniel Loughran | Just a small remark to get things going, perhaps you knew this already, but the complement of an affine variety is always a divisor. So if you want larger codimension $X$ must not be affine. | |
| Oct 10, 2010 at 10:06 | history | asked | Lars | CC BY-SA 2.5 |