Timeline for Compact Kaehler submanifolds of projectivized Hilbert space
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 20, 2019 at 22:44 | comment | added | John Baez | What are the possible problems? | |
| Jan 19, 2019 at 3:44 | comment | added | Francois Ziegler | Note: That argument is in Remark 2.2.9 of Loi-Zedda (2018; arXiv). But I’m not (yet quite) convinced it can be turned into a watertight proof of the present conjecture — which they don’t claim. | |
| Jan 19, 2019 at 1:18 | comment | added | John Baez | That's what I've been thinking - especially after I posted my question. This argument seems sort of magical to me because it leverages some finiteness of the intrinsic geometry of $M$ (the finite-dimensionality of $H^0(M,L)$ to get finiteness of its extrinsic geometry (the finite-dimensional of the smallest projective space in which $M$ sits.) | |
| Jan 18, 2019 at 0:08 | comment | added | YangMills | I think you can argue like this: $PH$ has a hyperplane bundle, which pulls back to a line bundle $L$ on $M$. The inclusion $M\subset PH$ can be written in homogeneous coordinates as $z\mapsto [s_0(z):s_1(z):...]$ where the $s_j$'s are global holomorphic sections of $L$. If the image of $M$ is not contained in any finite-dimensional linear subspace then the sections $s_j$'s would be linearly independent (and there are countably many of them). But the dimension of the space of global sections $H^0(M,L)$ is finite since $M$ is compact, contradiction. | |
| Jan 17, 2019 at 7:34 | comment | added | John Baez | Yes; I'd like this to be true for some work I'm doing on geometric quantization. | |
| Jan 17, 2019 at 2:29 | comment | added | Konstantinos Kanakoglou | Is this somehow related to the concept of geometric quantization? To be more specific, does this have to do with the fact that to any compact Kähler manifold one can associate a finite dimensional quantum Hilbert space reflecting the size of the classical phase space ? | |
| Jan 17, 2019 at 0:47 | history | asked | John Baez | CC BY-SA 4.0 |