Timeline for Rational or elliptic curves on Calabi-Yau threefolds
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2014 at 11:01 | comment | added | diverietti | @LiYutong, this is because trivial canonical class implies existence of Ricci flat Kähler metrics (this is Yau), then Kobayashi-Lübke inequalities give you that $c_2(X)\ge 0$ and you have equality if and only if the Ricci flat metric is indeed flat itself. Now you conclude by the classical Bieberbach theorem. | |
| Feb 21, 2014 at 23:23 | comment | added | Li Yutong | @diverietti Why "a manifold with trivial canonical class and $c_2(X)=0$ is a finite étale quotient of a complex torus"? Could you explain more about this? | |
| Jul 7, 2011 at 22:49 | comment | added | diverietti | did you mean Picard? :p | |
| Jul 7, 2011 at 17:54 | comment | added | Sándor Kovács | ps: (of the right Picrad number) | |
| Jul 7, 2011 at 17:53 | comment | added | Sándor Kovács | Right. Probably Oguiso-Sakurai meant something like that: Is it true that all CY threefolds admit a rational curve or a non-trivial map from an abelian variety. Perhaps even asking that that map from the abelian variety be dominant. | |
| Jul 7, 2011 at 17:47 | comment | added | diverietti | On the other hand, coming back to my original question, these CY threefold with $c_2(X)=0$ certainly admit a non-constant map from a complex torus. | |
| Jul 7, 2011 at 17:44 | comment | added | Sándor Kovács | Actually, Oguiso and Sakurai say so themselves right before posing that question "Secondly, our statements (II) and (III) show that there certainly exist smooth Calabi-Yau threefolds containing no rational curves if $\rho(X) = 2$ and $3$, but, on the other hand, suggest some hope to ask the following:" then comes the question quoted by Francesco. Obviously they meant some variation of the question. | |
| Jul 7, 2011 at 17:35 | comment | added | diverietti | Thank you for your hints. Anyway, a manifold with trivial canonical class and $c_2(X)=0$ is a finite étale quotient of a complex torus, so that it cannot certainly contain rational curves... | |
| Jul 7, 2011 at 17:30 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |