Timeline for How much of a variety can be reconstructed from codimension-zero data?
Current License: CC BY-SA 3.0
6 events
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| Mar 29, 2013 at 6:49 | comment | added | Dmitry Vaintrob | Thanks Sasha! Neat. I agree that when coherent cohomology is trivial, we get no information. However are you sure that there's no data if only $H^n(O)=0$? For example what if we compose your first map with the multiplication map $Hom(O_X,O_S)\otimes Hom(O_X, O_S)\to Hom(O_X, O_S)$ to get a map $Hom(O_X,O_S)\otimes Hom(O_X,O_S)\otimes Ext^n(O_S,O_X)\to Ext^n(O_X,O_X)$. Then if $H^n(O_X)=0$ then this product will of course be zero, but there will be a well-defined and possibly nonzero Massey product. Is it clear that this Massey product will be zero? | |
| Mar 29, 2013 at 4:36 | vote | accept | Dmitry Vaintrob | ||
| Mar 27, 2013 at 4:19 | comment | added | Sasha | Dmitry, you are completely right. I edited the answer, now you see that in some cases you can reconstruct a lot of geometry! | |
| Mar 27, 2013 at 4:18 | history | edited | Sasha | CC BY-SA 3.0 |
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| Mar 26, 2013 at 2:54 | comment | added | Dmitry Vaintrob | Hm... I agree that it seems like something is lost in dimension $>1$, but not sure your argument is correct, or else I'm not following it. On a curve, if you take the extension closed abelian subcategory of $(D^b)Coh(X)$ generated by $O_X$ and all structure sheaves of points, you get all of $Coh$, which certainly lets you recover $X$. So despite the fact that Homs and Exts between objects in the subcategory <i>consisting</i> of $O_X$ and skyscraper sheaves depend only on the coherent cohomology, once you start taking extensions things can become more subtle. How does Hartog change this? | |
| Mar 25, 2013 at 19:15 | history | answered | Sasha | CC BY-SA 3.0 |