Timeline for Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?
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13 events
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| Apr 20, 2014 at 9:20 | comment | added | Dan Petersen | By the way, the fact that $\overline M_{g,n}$ is a quotient of a smooth projective variety by a finite group action was first proven (over the complex numbers) by Looijenga (the paper on "Prym level structures"). | |
| Apr 20, 2014 at 9:14 | comment | added | Dan Petersen | @Gina Yes the whole theory of weights works also for stacks. But a remark is that you can bypass stacks completely in this case, since (as you mention) the spaces $\overline M_{g,n}$ are quotients of a smooth projective variety by a finite group. In general one has for rational cohomology $H^\bullet(X/G) = H^\bullet(X)^G$ ($G$-invariants); when $X$ is an algebraic variety, $H^\bullet(X)^G$ is a sub-Hodge structure of $H^\bullet(X)$ and in particular it is pure if $H^\bullet(X)$ is pure. | |
| Apr 19, 2014 at 23:40 | vote | accept | Gina | ||
| Apr 19, 2014 at 23:27 | answer | added | Francesco Polizzi | timeline score: 10 | |
| Apr 19, 2014 at 22:16 | comment | added | Donu Arapura | Gina, yes, the purity follows from Deligne, Theorie de Hodge III, Thm 8.2.4 (iv) + plus the fact that orbifolds are rational homology manifolds. I'm sure it's Peters-Steenbrink also if that is preferable. | |
| Apr 19, 2014 at 21:50 | comment | added | Gina | @FrancescoPolizzi : You are correct that it only has quotient singularities. Does this imply that $H^k$ is pure of weight $k$? I can't find a statement like that in any of the surveys I've consulted, but as I said I am not an expert in this area. | |
| Apr 19, 2014 at 21:49 | comment | added | Gina | @Ari : So the whole theory of weights works for stacks? Is there a down-to-earth reference for that? I was reading the paper just thinking of the coarse moduli spaces. Is there a way to see this using from the orbifold perspective, i.e. from the fact that there is a finite orbifold cover that is smooth (such a thing was constructed by de Jong-Pikaart).? | |
| Apr 19, 2014 at 21:34 | comment | added | Ariyan Javanpeykar | @Gina The DM-stack $\overline { \mathcal M_{g,p}}$ is smooth; see Theorem 2.1 in math.jussieu.fr/~freixas/Site/Recherche_files/SingARR_arxiv.pdf . | |
| Apr 19, 2014 at 21:29 | comment | added | Francesco Polizzi | But it has only quotient singularities, if I remember correctly. | |
| Apr 19, 2014 at 21:25 | comment | added | Gina | @Ari : Yes, but $\overline{\mathcal{M}}_{g,p}$ is not smooth; it has singularities along its boundary. | |
| Apr 19, 2014 at 21:19 | review | First posts | |||
| Apr 19, 2014 at 21:25 | |||||
| Apr 19, 2014 at 21:13 | comment | added | Ariyan Javanpeykar | See en.wikipedia.org/wiki/Hodge_structure and especially the second example in "Examples" on that page. | |
| Apr 19, 2014 at 21:01 | history | asked | Gina | CC BY-SA 3.0 |