Timeline for Hyperbolicity for algebraic varieties and relation to curves on them
Current License: CC BY-SA 2.5
16 events
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| Nov 18, 2010 at 18:24 | comment | added | diverietti | That's exactly what I meant. A subvariety of a complex torus is Kobayashi hyperbolic if and only if it does not contain any translate of a subtorus | |
| Nov 18, 2010 at 11:05 | comment | added | Francesco Polizzi | If $A$ is a complex torus and $X$ an irreducible subvariety of $A$ which is non-degenerated, locally complete intersection and such that $2 \dim X > \dim A$, then $\pi_1(X)=\pi_1(A)$. So the fundamental group of $X$ is Abelian, hence amenable, in particular $X$ is not Kahler hyperbolic. If you know that $X$ is Kobayashi hyperbolic, this actually gives an example. | |
| Nov 18, 2010 at 9:20 | comment | added | diverietti | Recently it has been proved by Damian Brotbek that generic projective complete intersections of high multi-degree are hyperbolic. These should give other examples. Otherwise, what about subvarieties of complex tori which do not contain any translate of a sub-tours ? | |
| Nov 17, 2010 at 17:19 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Nov 17, 2010 at 16:05 | comment | added | sfilip | Thanks so much! I guess I need to learn how to do computations, not only the statements of the theorems :) | |
| Nov 17, 2010 at 16:01 | vote | accept | sfilip | ||
| Nov 17, 2010 at 10:57 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Nov 17, 2010 at 10:54 | comment | added | Francesco Polizzi | Fixed. By the way Diverietti, do you know any other examples? | |
| Nov 17, 2010 at 10:47 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Nov 17, 2010 at 10:26 | comment | added | diverietti | Just a last comment, to be precise. This is not Siu's conjecture, this is the Kobayashi conjecture. The strategy to prove it is by Siu. | |
| Nov 17, 2010 at 10:11 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Nov 17, 2010 at 10:04 | comment | added | Francesco Polizzi | Ok, thank you for pointing this out. By the way, Lefschetz is true also for $n=3$, so surfaces in $\mathbb{P}^3$ are examples too. However, the computations on $H^{2,0}$ works only for $n \geq 4$. I have edited the answer. | |
| Nov 17, 2010 at 9:49 | comment | added | diverietti | Let's say that Siu indicated a quite precise strategy to prove that. Up to now, we have a full proof only for surfaces in $\mathbb P^3$ and threefolds in $\mathbb P^4$. So your exemple works for the moment just for $n=4$. | |
| Nov 17, 2010 at 9:35 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Nov 17, 2010 at 9:29 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Nov 17, 2010 at 9:11 | history | answered | Francesco Polizzi | CC BY-SA 2.5 |