Timeline for Pushing forward a complex structure by submersion
Current License: CC BY-SA 4.0
8 events
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| Dec 7, 2018 at 17:21 | comment | added | Robert Bryant | @lolo: By the way, I forgot to respond to your worry about the apparent conflict of such examples with Voisin's text. The reason that there isn't a contradiction is that she is assuming that $E$ is a holomorphic subbundle of $TX$, whereas you have only assumed that $\mathrm{ker}\,\mathrm{d}\phi$ is a complex subbundle of $TX$ (one that is also, of course, Frobenius as a (real) subbundle of $TX$, but that doesn't make it holomorphic). | |
| Dec 7, 2018 at 13:07 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added an example and some explanations
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| Dec 7, 2018 at 12:40 | comment | added | Robert Bryant | @lolo: Yes, I'll edit my answer to include the example and the argument for the second question. I'm sorry that I didn't do it first, but I had to stop after I put in my answer and tend to some other business. | |
| Dec 7, 2018 at 10:39 | vote | accept | lolo | ||
| Dec 7, 2018 at 10:04 | comment | added | lolo | and yes I made a stupid mistake in the question, of course $J$ can not fix anything but zero vector | |
| Dec 7, 2018 at 10:04 | comment | added | lolo | Then she claims that there exists an almost complex structure on $V$ in which the differential becomes $\mathbb{C}$-linear. | |
| Dec 7, 2018 at 10:04 | comment | added | lolo | Do you have a reference or an explicit example at hand? I am asking because theorem 2.26 in volume 1 of Voisin's text says "Let X be a complex manifold of dimension n, and let E be a holomorphic distribution of rank k over X, i.e. a holomorphic vector subbundle of rank k of the holomorphic tangent bundle TX . Then E is integrable in the holomorphic sense if and only if we have the integrability condition $[E, E]=E$." To prove it, she considers the real part of $E$, applies Frobenius theorem to it so she gets a submersion $U\rightarrow V$ as in the question... | |
| Dec 7, 2018 at 9:56 | history | answered | Robert Bryant | CC BY-SA 4.0 |