Timeline for On Grothendieck's period relations
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2012 at 15:38 | vote | accept | Hugo Chapdelaine | ||
| Jun 17, 2012 at 10:17 | answer | added | François Brunault | timeline score: 13 | |
| Jun 17, 2012 at 3:59 | answer | added | Keerthi Madapusi | timeline score: 14 | |
| Jun 17, 2012 at 3:57 | comment | added | Keerthi Madapusi | Sorry, I initially meant it to be a comment and then it sort of expanded. | |
| Jun 16, 2012 at 15:55 | history | edited | David Zureick-Brown | CC BY-SA 3.0 |
fixed title
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| Jun 16, 2012 at 14:42 | comment | added | Martin Brandenburg | @Keerthi: MO has an answer box for answers and a comment box for comments. :-) | |
| Jun 16, 2012 at 14:38 | comment | added | Keerthi Madapusi | But the fact that cycle classes have to be preserved means that the isomorphism is actually a trivialization of a torsor under a much smaller group (the 'motivic galois group' for $V$), which means that you should have non-trivial relations between the co-ordinates. Grothendieck conjectured that these are all the relations. Combined with the Hodge conjecture this would imply that the transcendence degree of the extension generated by the co-ordinates is the dimension of the Mumford-Tate group for $V$. | |
| Jun 16, 2012 at 14:33 | comment | added | Keerthi Madapusi | Suppose the cycle class of an algebraic cycle has co-ordinates $(a_1,\ldots,a_n)$ on the de Rham side and $(b_1,\ldots,b_n)$ on the Betti side. Since the comparison isomorphism has to preserve cycle classes, we get the relation $\sum_j\omega_{ij}a_j=b_i$. One can also think about it as follows: If there were no constraints on $\omega$, it's just a random isomorphism between two complex vector spaces. The space of such isomorphisms is a torsor under $GL(n,\mathbb{C})$, and so the co-ordinates of a generic point of the space will generate an extension of $\mathbb{Q}$ of transcendence degree $n^2 | |
| Jun 16, 2012 at 14:01 | history | asked | Hugo Chapdelaine | CC BY-SA 3.0 |