Timeline for 3-fold of general type homeomorphic to rational 3-fold
Current License: CC BY-SA 4.0
10 events
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| May 27, 2020 at 12:08 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 27, 2020 at 11:01 | answer | added | Nick L | timeline score: 1 | |
| Dec 19, 2018 at 22:28 | comment | added | Nick L | That would be great! I think the paper "cubic forms and compex 3-folds" by Okonek and Van de Ven is also a nice source for these things. If by chance one of these 3-folds had $b_{2}=1$ then there would be only finite number of possibilities to check since the rational 3-fold would be Fano. | |
| Dec 19, 2018 at 22:02 | comment | added | Jason Starr | Presumably topologists are well-aware of the following, but I just learned of the Manifold Atlas Project. There is a wonderful page about simply connected 6-manifolds and a complete list of homeomorphism invariants: map.mpim-bonn.mpg.de/6-manifolds:_1-connected I will try to see if the simply connected examples from the following answer might be homeomorphic to rational 3folds: mathoverflow.net/questions/318990/… | |
| Dec 18, 2018 at 23:43 | comment | added | Enrico | For sure all $h^{p,0}$ are birational invariants. I am not how to prove the full statement though | |
| Dec 18, 2018 at 16:02 | comment | added | Nick L | @ Enrico, thanks, how do you show that the homeomorphism preserves Hodge numbers? It will be true that $h^{1,0}=0$ since rational $3$-folds are simply connected, but I am not sure about $h^{2,0},h^{3,0}$. | |
| Dec 18, 2018 at 14:29 | comment | added | Dmitri Panov | Maybe it's worth to ask the authors of arxiv.org/abs/1606.09237 ... | |
| Dec 18, 2018 at 10:42 | comment | added | Enrico | To the best of my knowledge, I do not remember any smooth threefold of general type with $h^{1,0}=h^{2,0}=h^{3,0}=0$. However, if you allow basically harmless (e.g. quotient) singularities you can find examples of canonically polarised threefolds where $|-K|$ is empty (and the other hodge number vanishes as well by Lefschetz). Go to grdb.co.uk/search/gt3 and simply pick the threefold where no ambient weight is equal to 1. (of course this is just a necessary condition, far from sufficient) | |
| Dec 17, 2018 at 19:44 | history | edited | Nick L | CC BY-SA 4.0 |
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| Dec 17, 2018 at 19:33 | history | asked | Nick L | CC BY-SA 4.0 |