Timeline for Kähler metric on projective bundle
Current License: CC BY-SA 4.0
5 events
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| Jul 8, 2022 at 3:51 | comment | added | Mjr | @JasonStarr, thanks, your answer seems promising. But I am not an expert in algebraic geometry and I don't understand your construction well. Could you explain a little about what is the relative canonical sheaf and why there is a closed holomorphic embedding as you mentioned? | |
| Jul 7, 2022 at 13:58 | comment | added | Jason Starr | @NickL Yes, indeed! I did interpret the OP's question to mean the case when the total space is a complex manifold and the projection to the base is a holomorphic submersion that (analytically locally on the base) is just a product of the base and a fixed projective space. | |
| Jul 7, 2022 at 12:53 | comment | added | Nick L | I guess it depends what you mean precisely (i.e. what category this bundle with fibres equal to projective space is in). For example if it is just a topological bundle then there are $\mathbb{CP}^1$-bundles over $T^2$ which are not orientable. I am guess Jason Starr's answer assumes that the bundle is a complex manifold and the bundle map is holomorphic . | |
| Jul 7, 2022 at 10:58 | comment | added | Jason Starr | Yes. The pushforward of the dual of the relative canonical (invertible) sheaf is a holomorphic vector bundle on the base Kaehler manifold. The projectivization is a Kaehler manifold. There is a closed holomorphic embedding of the total space of the original bundle in this new Kaehler manifold. | |
| Jul 7, 2022 at 9:45 | history | asked | Mjr | CC BY-SA 4.0 |