Timeline for Mixed structures on Hom spaces induced by mixed sheaves
Current License: CC BY-SA 3.0
10 events
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| Oct 8, 2012 at 17:01 | comment | added | Reladenine Vakalwe | I really like the elliptic curve example (I haven't fully thought it through, but it's simple)! Do you have any thoughts on the general pointwise purity situation though (for instance when the strata are not cells)? Does the local global spectral sequence still degenerate in this situation? I am almost convinced that it doesn't necessarily, and Homs being pure is a very very special situation, but I'd be happy to be proved wrong. | |
| Oct 8, 2012 at 16:49 | comment | added | Geordie Williamson | Search for "What's an example of whose stalks are pure but not pointwise pure?" for an example of non pointwise purity. I learnt a nice example from Luca Migliorini: take a family of elliptic curves with smooth total space and some singular fibres. Then the decomposition theorem says that the direct image of the constant sheaf on the total space breaks into its cohomology sheaves (on a curve IC's are shifts of sheaves). Now it is not difficult to see that the "middle" summand (coming from the $H^1$ of the elliptic curves in the family) cannot be pointwise pure. | |
| Oct 8, 2012 at 16:24 | comment | added | Reladenine Vakalwe | Ok, the flag variety example is fine, the degeneration does follow from pointwise purity, since the strata are cells. So pointwise purity gives purity for Homs on restriction to each strata and this can be inductively built up. | |
| Oct 8, 2012 at 16:06 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Oct 8, 2012 at 15:58 | comment | added | Reladenine Vakalwe | @Geordie Williamson (contd.): However, you are absolutely right that if the stalk wasn't pure at some point, then Hom from/to skyscraper wouldn't be pure. | |
| Oct 8, 2012 at 15:39 | comment | added | Reladenine Vakalwe | @Geordie Williamson: I am slightly worried that the local to global degeneration for weight reasons that you mention uses that Hom between the ICs is pure. No? Let $X=X_0 \supset ⋯\supset X_1$ be the filtration by closed subspaces corresponding to the stratification. Let $v_k:X_k \to X$ be inclusion. The degeneration is obtained by looking $Hom(v^∗_k M,−)$ applied to $i_∗i^!v^!_k N \to v^!_k N \to j_∗j^!v^!_k N$ (I hope what my $i$ and $j$ are is clear). Now for degeneration we want the connecting map in the long exact to be zero. Without weights (or parity vanishing) I dont see how to get it | |
| Oct 8, 2012 at 12:30 | comment | added | Geordie Williamson | In the cases I know best (IC's on flag varieties) the statement is true by pointwise purity. (The local global spectral sequence degenerates for weight reasons, eg. in the BGG argument mentioned in your other question "About an argument in Koszul duality..."). Hence to have a counterexample one needs to consider morphisms between non pointwise pure sheaves. However this is easy: take a non pointwise pure IC and consider hom to or from a skyscraper sheaf. | |
| Oct 6, 2012 at 17:14 | comment | added | Reladenine Vakalwe | @Donu Arapura: Do you know whether the corresponding statement for $\ell$-adic sheaves is true? | |
| Oct 6, 2012 at 12:44 | comment | added | Donu Arapura | It's not clear to me either right now. As for a sufficient condition, if $X$ is smooth and $M,N$ are locally constant, then it should be fine, because the internal Hom can be constructed more naively for variations of pure Hodge structures. | |
| Oct 6, 2012 at 6:38 | history | asked | Reladenine Vakalwe | CC BY-SA 3.0 |