Timeline for Differntial of the Torelli morphism and the multiplication map
Current License: CC BY-SA 3.0
8 events
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| Aug 31, 2011 at 2:00 | comment | added | roy smith | if you are really interested in this question as stated, and not just in the easier one of the induced map of deformations spaces, i.e. the differential of the map of functors, you might look at: Algebraic geometry: Angers 1979 ; variétés de petite dimension: the article of oort and steenbrink, on the local torelli problem. | |
| Aug 10, 2011 at 8:33 | vote | accept | Cyrus | ||
| Aug 10, 2011 at 6:00 | comment | added | Torsten Ekedahl | Dan is completely right, the only thing you have to do (as Dan indicates) in positive characteristic is to note that first order deformations deformations of principally polarised abelian varieties is a subspace of the first order deformation space of abelian varieties. Hence we get a subspace of $\mathrm{Hom}(H^0(C,\Omega^1),H^1(C,\mathcal O_C))=H^1(C,\mathcal O_C)^{\oplus2}$ consisting of the symmetric tensors. This is $\Gamma^2H^1(C,\mathcal O_C)$ rather than $S^2H^1(C,\mathcal O_C)$. | |
| Aug 10, 2011 at 4:46 | comment | added | roy smith | Nowadays I suppose those letters refer to "stacks", which I understand to mean you don't care that the tangent space at a singular point of Mg is not the same as at a smooth point hence neither is the differential. ignorant comment by Old guy. | |
| Aug 9, 2011 at 22:06 | history | edited | Dan Petersen | CC BY-SA 3.0 |
added 5 characters in body
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| Aug 9, 2011 at 22:04 | comment | added | Dan Petersen | Although I don't know a reference, I can not imagine that the map $H^1(T_C) \otimes H^0(\Omega_C) \to H^1(O_C)$ is not given by cup product in positive characteristic. That would be madness. The only issue I can see is that you will not be able to identify symmetric tensors with the symmetric square in characteristic two. | |
| Aug 9, 2011 at 21:07 | comment | added | Cyrus | Thank you very much Dan for your complete answer. Is your argument only valid over C or it works in any characteristic? I think the proof in ACG book only applies over C. On the other hand I have some reasons to believe that this identification can not be made in characteristic p. Could you clarify this? Thank you in advance. | |
| Aug 9, 2011 at 20:51 | history | answered | Dan Petersen | CC BY-SA 3.0 |