Timeline for Comparison between $E_2$-terms of Leray and "second hypercohomology" spectral sequences
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 12, 2013 at 13:00 | vote | accept | Nick Switala | ||
| Jul 3, 2013 at 19:44 | comment | added | Nick Switala | Here $Y$ began as some projective scheme embedded both in $\mathbb{P}^n_k$ and $\mathbb{P}^m_k$, and to help compare the embeddings I'm viewing $Y$ as a closed subscheme of the product of those projective spaces via the diagonal. $Y$ need not be smooth. | |
| Jul 3, 2013 at 19:41 | comment | added | Nick Switala | @Karl: The fibers are all projective spaces, so they are Fano; in the particular case I care about, $f$ is the projection $\mathbb{P}^n_k \times \mathbb{P}^m_k \rightarrow \mathbb{P}^n_k$. The complex does have a sheaf of differentials as its starting point: it is the complex obtained by applying the (sheafified) "supports at $Y$" functor to the Cousin complex $E^{\bullet}(\omega_{X})$ of the canonical sheaf on $X = \mathbb{P}^n_k \times \mathbb{P}^m_k$, so its cohomology sheaves are the local cohomology sheaves $\mathcal{H}_Y^{\bullet}(\omega_X)$. | |
| Jul 3, 2013 at 15:56 | comment | added | Karl Schwede | Ok, as Dan Peterson points out below, there probably isn't much hope unless one knows more. Is there any chance that the complex $I^{\bullet}$ is made up of sheaves of differentials, or pushed down from some other scheme made up of sheaves of differentials from some other scheme (ie, a dualizing complex, or something similar even from a simplicial scheme). That's one general case where there are lots of vanishing theorems like the kind you need. Alternately, if you know a lot more about the fibers of your morphism $f$ (are they all Fano?), there might be other things you can try. | |
| Jul 2, 2013 at 20:35 | comment | added | Nick Switala | @Karl: thanks, I've edited the question. | |
| Jul 2, 2013 at 20:34 | comment | added | Nick Switala | @Mariano: indeed, appropriately restrictive hypotheses on $f$ making these $E_2$-terms isomorphic are exactly what I'm trying to find. | |
| Jul 2, 2013 at 20:32 | history | edited | Nick Switala | CC BY-SA 3.0 |
added 45 characters in body
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| Jul 2, 2013 at 5:31 | answer | added | Dan Petersen | timeline score: 5 | |
| Jul 2, 2013 at 1:58 | comment | added | Karl Schwede | Even if they both degenerate at the E2 page (meaning the maps of the spectral sequence are zero), I don't see why two spectral sequences have the same terms. | |
| Jul 2, 2013 at 0:00 | comment | added | Mariano Suárez-Álvarez | Shouldn't some significant hypothesis be put on the dynamics of $f$?: One s.s. does not depend on $f$ while the other does. | |
| Jul 1, 2013 at 23:41 | history | asked | Nick Switala | CC BY-SA 3.0 |