Timeline for Does the Hodge star operator respect complex structure?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2010 at 20:20 | vote | accept | Abtan Massini | ||
| Nov 21, 2010 at 20:01 | answer | added | Donu Arapura | timeline score: 6 | |
| Nov 21, 2010 at 19:49 | comment | added | Abtan Massini | Great. Thanks for your help. @Donu Please just copy and paste your comments into an answer so I can mark as the accepted answer. | |
| Nov 21, 2010 at 19:42 | comment | added | Donu Arapura | Our comments seemed to have crossed. Yes, p 82 not 66, and yes. (Be warned that some authors use a different convention, where $N-p,N-q$ get switched. It's a question of whether $*$ is linear or antilinear.) | |
| Nov 21, 2010 at 19:42 | comment | added | Spiro Karigiannis | Since $∗$ is a real operator, to be more precise, you should say that after complexifying the space of forms, and extending $∗$ to be complex linear, then indeed $∗$ maps $\Omega^{p,q}$ to $\Omega^{N-q,N-p}$. Frequently, it is preferable to use $\bar *$, which is the composition of $∗$ with complex conjugation, to map $\Omega{p,q}$ to $\Omega{N−p,N−q}$. This way, we have $\alpha \wedge \bar * \beta = g(\alpha, \beta) vol$, where $g$ is the Hermitian metric. | |
| Nov 21, 2010 at 19:38 | comment | added | Donu Arapura | I forgot to say that for the above work, you need to choose $*$ with respect to a Hermitean metric or equivalently a Riemannian metric $g$ satisfying $g(JX,JY)= g(X,Y)$, where $J$ is the complex structure. | |
| Nov 21, 2010 at 19:34 | comment | added | Abtan Massini | Thanks for the reference. In the edition I've laid my hands it's page 82, but no worries. So the answer to my question is that for a complex manifold of dimension $2N$, the Hodge star operator sends $\Omega^{p,q}$ to $\Omega^{N-p,N-q}$? | |
| Nov 21, 2010 at 19:24 | history | edited | Abtan Massini | CC BY-SA 2.5 |
deleted 65 characters in body
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| Nov 21, 2010 at 19:24 | comment | added | Abtan Massini | The last question is just a rewording of the previous line. Since it is non-essential and apparently confusing I'll delete it. | |
| Nov 21, 2010 at 19:18 | comment | added | Donu Arapura | So yes in the sense you mean. | |
| Nov 21, 2010 at 19:14 | comment | added | Donu Arapura | The last question is ambiguous, but I think the 2nd sentence is also a question, so you should put a question mark after it. You can find the answer in many places, e.g. Griffiths-Harris page 66. | |
| Nov 21, 2010 at 18:50 | history | asked | Abtan Massini | CC BY-SA 2.5 |