Timeline for Lefschetz hyper-plane theorem for singular projective varieties?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 18, 2022 at 11:08 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed, cf. https://math.meta.stackexchange.com/a/34713/228959
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| Mar 10, 2011 at 16:42 | comment | added | roy smith | Is the weak Lefschetz theorem for intersection cohomology of interest? igitur-archive.library.uu.nl/math/2007-0224-201418/… | |
| Mar 9, 2011 at 5:53 | comment | added | Sándor Kovács | Mohammad, I did not claim that there was nothing more known. I just said I know about these. You must see the difference. In your very good example, I don't see why you would expect an isomorphism for $H_4$. | |
| Mar 8, 2011 at 20:13 | comment | added | Mohammad Farajzadeh-Tehrani | A very good eaxmple to keep in mind is $X:(x_1^2+\cdots+x_4^2=0) \subset \mathbb{P}^4$ which has a node and has $b_2=1$, $b_4=2$ . A generic hyperplane section is just quadratic 3-fold with $b_2=1$, $b_4=1$. So there is a isomorphism on $H_2$ but not on $H_4$ and note that $2< $ min{ real codim of singualrity, dim $X$-1} | |
| Mar 8, 2011 at 20:08 | comment | added | Mohammad Farajzadeh-Tehrani | The reason for condition on codim $X^{sing}$ is this: if for exmple codim $^{sing}=2$ then first of all we have an inclusion map $H_2(Y) \rightarrow H_2(X)$. We want to show this is an isomorphsim. since codim $Y^{sing}=2$ any 2nd homology class can be represented by some thing which lies in smooth locus and so I expect to get an isomorphism in this case from what we know for smooth varieties. | |
| Mar 8, 2011 at 19:46 | comment | added | Mohammad Farajzadeh-Tehrani | can you say more details on the first paragraph of your answer. or can you mention one of those several versions which is most relevant to this case. The case I have in mind is when $X$ is a projective 4-fold with canonical singularities and $Y$ is a hyperplane section. You can even assume $X$ is toric. | |
| Mar 8, 2011 at 19:44 | comment | added | Mohammad Farajzadeh-Tehrani | so you mean there is nothing none yet when $Y$ is a generic hyperplane section and not necessarily one which has the singular locus? | |
| Mar 8, 2011 at 15:10 | comment | added | Sándor Kovács | If you downvote (Roy, I know it wasn't you), would you mind telling me why? | |
| Mar 8, 2011 at 4:50 | comment | added | roy smith | the version where the hyperplane contains the singular points is already in the standard reference of milnor, morse theory. p.41. at least over the complex numbers. | |
| Mar 8, 2011 at 4:22 | history | answered | Sándor Kovács | CC BY-SA 2.5 |