Timeline for Białynicki-Birula theory for non-complete varieties
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 17, 2014 at 14:42 | comment | added | Allen Knutson | Another family of examples: if $\mathbb G_m$ acts on $M$ (of finite type, I guess) with this limit-existing property, then some circle in $(\mathbb G_m)^2$ acts on $T^* M$ similarly. | |
| Nov 14, 2014 at 14:59 | comment | added | Qfwfq | @Dave Anderson: I found the second isomorphism on page 19 of this set of slides: mysite.science.uottawa.ca/jlema072/Borel-Moore.pdf Probably, yes, what is denoted $H^{i}_{BM}$ should be compactly supported cohomology. | |
| Nov 14, 2014 at 0:01 | comment | added | Dave Anderson | @Qfwfq, I think you've got it, but I'm not aware of any "Borel-Moore cohomology"... (That second "Poincare duality" isomorphism should be $H^i_c(X) = H_{n-i}(X)$, where $c$ denotes cohomology with compact support.) | |
| Nov 13, 2014 at 20:51 | comment | added | Dori Bejleri | Yes it should be BM homology and I believe what you said about Poincare duality is right. | |
| Nov 13, 2014 at 20:36 | comment | added | Qfwfq | My previous comment is confusing. I've just looked up in some references that for $X$ a (real) smooth manifold we have the following version of Poincaré duality: $H_{i}^{BM}(X)\cong H^{n-i}(X)$ and $H_{BM}^i(X)\cong H_{n-i}(X)$ where $n = \dim X$. So there's just a 'reflection' in the degree. Checking on the example of $\mathbb{C}^{*}$ acting on $\mathbb{C}$ I would say that, yes, BB computes BM (co)homology (with the cells of complex dimension $i$ generating degree $2i$ BM (co)homology). | |
| Nov 13, 2014 at 20:11 | comment | added | Qfwfq | @Dave Anderson: Ok, so I'm probably confusing BM homology with BM cohomology? Say we work with rational coefficients. You claim $H^{i}_{BM}(X)\cong H^{i}(X)$ whether $X$ is compact or not. On the other hand, according to the Wikipedia entry on BM homology, we would have $H_{i}^{BM}(\mathbb{C}^n)=\mathbb{Q}$ for $i=2n$ and trivial otherwise; while for ordinary homology $H_{i}(\mathbb{C}^n)=0$ (in positive degree) for the contractible space $\mathbb{C}^n$. This would suggest that in general $H_{i}^{BM}(X)\ncong H_{BM}^{i}$ as rational vector spaces? | |
| Nov 13, 2014 at 19:15 | comment | added | Dave Anderson | @Qfwfq, this may be what you mean, but to clarify: The cell decomposition coming from this answer computes Borel-Moore homology, which for a (compact or non-compact) manifold is isomorphic to singular cohomology. | |
| Nov 13, 2014 at 15:08 | comment | added | Qfwfq | A question: when you talk about BB cells computing cohomology, do you mean singular cohomology or Borel-Moore homology (since we're on a non-compact manifold)? | |
| Nov 13, 2014 at 14:44 | vote | accept | Qfwfq | ||
| Nov 13, 2014 at 14:34 | history | edited | Qfwfq | CC BY-SA 3.0 |
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| Nov 13, 2014 at 12:32 | history | edited | Dori Bejleri | CC BY-SA 3.0 |
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| Nov 12, 2014 at 23:14 | history | answered | Dori Bejleri | CC BY-SA 3.0 |