Timeline for complex deformations of abelian varieties
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 16, 2012 at 14:00 | comment | added | Donu Arapura | Hugo: (1) yes, Dolbeault. (2) That's an interesting thought. It seems entirely plausible to me that there should be a deformation theory for varieties with endomorphisms, and that CM abelian vars may be rigid. But I haven't seen such theory/computation worked out. | |
| Apr 16, 2012 at 13:25 | comment | added | Hugo Chapdelaine | Donu, Is there a way using (only) deformation theory to see that an abelian variety with complex multiplication can be defined over number field? When $X$ is a smooth projective variety over $\amthbf{C}$ such that $H^1(X,\Theta_X)=0$ then since the Kodaira-Spencer map is trivial we see readily that $X$ can be defined over a number field. I know about the classical proof for elliptic curves which uses the $j$-invariant but I'm wondering if there exists some refinement of the deformation theory argument that I have just explained. | |
| Apr 16, 2012 at 13:19 | comment | added | Hugo Chapdelaine | Thanks Donu, so you are using the Dolbeault's isomorphism $H^1(A,\mathcal{O}_A)≃H_{\overline{\partial}}^{0,1}(A)$. | |
| Apr 16, 2012 at 13:09 | vote | accept | Hugo Chapdelaine | ||
| Apr 16, 2012 at 12:44 | history | answered | Donu Arapura | CC BY-SA 3.0 |