Timeline for Why Calabi-Yau manifolds should be complex?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 2, 2017 at 9:52 | comment | added | diverietti | People use it! But this is the group of a "generic" orientable riemannian manifold, it does not give much more restrictions than this. On the complex side, $U(n)$ is the holonomy group of a "generic" Kähler manifold. While $SU(n)$ is the "first" holonomy group in Berger's list which gives some restrictive condition on the manifold. | |
| Nov 1, 2017 at 20:36 | comment | added | Xige Yang | Oh I forgot to add... Compare to $SO(n)$, $SU(n)$is "complex" in the sense that it's a subgroup a $GL(n,\mathbb{C})$. So why don't people use $SO(n)$ for holonomy groups? | |
| Oct 29, 2017 at 9:49 | comment | added | diverietti | I am sorry Xige, not sure to understand correctly your question. The Lie group $SU(n)$ is indeed a real Lie group! | |
| Oct 28, 2017 at 13:15 | comment | added | Xige Yang | Thank you for the answer! Is there any intuitive reason for a $SU(n)$ holonomy group instead of a real one? | |
| Oct 27, 2017 at 10:38 | history | edited | diverietti | CC BY-SA 3.0 |
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| Oct 27, 2017 at 10:37 | comment | added | diverietti | Yes, it should! :) I'll fix it. | |
| Oct 27, 2017 at 10:23 | comment | added | Vincent | Shouldn't that be $n = \frac{1}{2} \dim_{\mathbb{R}} M$? | |
| Oct 27, 2017 at 9:52 | history | answered | diverietti | CC BY-SA 3.0 |