Timeline for Proof of the holomorphic Frobenius theorem in Voisin's book on Hodge theory (Theorem 2.26)
Current License: CC BY-SA 3.0
10 events
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| Sep 10, 2017 at 6:36 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Apr 23, 2017 at 20:22 | comment | added | Saal Hardali | @nfdc23 I asked about what version since I didn't really understand the part of the answer that used it before the further elaboration graciously provided now. Everything is clear now. Thanks! | |
| Apr 23, 2017 at 20:20 | vote | accept | Saal Hardali | ||
| Apr 23, 2017 at 14:30 | comment | added | nfdc23 | @SaalHardali: there is only one version of the holomorphic implicit function theorem; its proof is identical to the one in the $C^{\infty}$-setting (via reduction to the inverse function theorem, whose proof is the same too). Perhaps your skepticism on various things could be cleared up if you state what you know about equivalences among various possible definitions of holomorphicity for functions of several complex variables (say assuming continuity, to avoid digressing into irrelevant measure-theoretic issues). Have you tried speaking with an expert in complex analysis at your university? | |
| Apr 23, 2017 at 11:12 | history | edited | Ben McKay | CC BY-SA 3.0 |
more details
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| Apr 23, 2017 at 11:05 | comment | added | Saal Hardali | Also It's not clear to me why the lifts of holomorphic vector fields via the projection are unique. | |
| Apr 23, 2017 at 10:59 | comment | added | Saal Hardali | Aha, you were talking about the fiber I see. Sorry again but, it's not clear to me what version of the holomorphic implicit function theorem you're using. | |
| Apr 23, 2017 at 10:55 | comment | added | Ben McKay | I only ask that $V_0$ is $dy=0$, at the origin. Since $V_0 \subset T_0 M$ is some linear subspace, a linear change of local coordinates will arrange that the various $dy$ 1-forms are linearly independent on $V_0$, and that is actually all we use here: $dy$ is a linear function of $dx$ on $V_0$. | |
| Apr 23, 2017 at 10:52 | comment | added | Saal Hardali | It's not clear to me how one can get that $V$ is $dy=0$. This seems to me like the bulk of the proof. That is showing that an involutive subbundle is locally the kernel of exact holomorphic 1-forms. I think linear algebra gives you that it is locally the kernel of closed holomorphic 1-forms see here: math.stackexchange.com/questions/1419718/… | |
| Apr 23, 2017 at 10:44 | history | answered | Ben McKay | CC BY-SA 3.0 |