Timeline for cohomology classes of complex submanifolds
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 4, 2020 at 19:40 | comment | added | abx | No, of course. What I meant is that this is a much studied problem, with proven methods to decide. Look at "divisors and line bundles" in any algebraic geometry book. | |
| May 4, 2020 at 19:14 | comment | added | Stella Dubois | @abx I just meant cohomology classes of complex subvarieties of $(\mathbb{C}/\mathbb{Z}[i])^2$. Thanks for letting me know about the condition that it has to be in $H^{1,1}(T^4,\mathbb{C}) \cap H^2(T^4,\mathbb{Z})$. I'm sure I'm being very stupid, though, but is it really true that any element in the subgroup generated by complex subvarieties can itself be represented by a complex subvariety? I can see that if C_1 and C_2 are subvarieties, then [C_1] + [C_2] is represented by an actual subvariety, but is the same true of [C_1] - [C_2]? | |
| Apr 27, 2020 at 12:26 | review | Close votes | |||
| Apr 29, 2020 at 16:13 | |||||
| Apr 26, 2020 at 18:53 | comment | added | Arun Debray | This is related to the Steenrod realizability problem, though that's for the laxer question of representing classes by oriented manifolds. | |
| Apr 26, 2020 at 6:52 | comment | added | Denis Nardin | en.wikipedia.org/wiki/Hodge_conjecture | |
| Apr 26, 2020 at 6:33 | comment | added | abx | Your question is ambiguous. Are you asking about properties of cohomology classes of complex subvarieties of $(\mathbb{C}/\mathbb{Z}[i])^2$, in which case the answer is well-known (Lefschetz theorem), or of classes of submanifolds which are complex subvarieties of $T^4$ for some complex structure of $T^4$? (also well-known, but might be more interesting in general). | |
| Apr 26, 2020 at 6:26 | review | First posts | |||
| Apr 26, 2020 at 7:53 | |||||
| Apr 26, 2020 at 6:18 | history | asked | Stella Dubois | CC BY-SA 4.0 |