Timeline for Growth of norm of curvature under direct sum or existence of universal connection
Current License: CC BY-SA 3.0
14 events
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| Dec 11, 2015 at 19:00 | comment | added | Yasha | Actually tracefree does not really help since at least in the formulation above one could restrict wlog to SU(n) connections. So at least for the formulation above the answer looks to be: no there is no growth. | |
| Dec 11, 2015 at 15:16 | comment | added | Yasha | As I posted my last comment I realized you were probably right, but it was too late to change. Anyway as the question as stated now there does not seem to be a problem? | |
| Dec 11, 2015 at 15:12 | comment | added | Sebastian Goette | Operator norm with respect to which norm on $E^n$, then? If you take the Hermitian norm induced by the direct product metric, the operator norm stays constant - if not, feel free to send me a proof. On the other hand, Chern numbers $(c_{k_1}(E^n)\cdots c_{k_\ell}(E^n))[X]$ can be estimated using the product of the operator norm and a polynomial in $n\mathrm{rk} E$ depending on the numbers $k_1,\dots, k_\ell$. Hence, they may grow even though the operator norm does not. | |
| Dec 11, 2015 at 15:00 | comment | added | Yasha | I think this is just a difference of what we mean by operator norm, for me this is: $\mathfrac{||Av}||}{||v||}$, for a direct sum operator the operator norm adds. | |
| Dec 10, 2015 at 14:30 | comment | added | Yasha | Sebastian the original version of the question was also ok, since the operator norm if the direct sum connection is exactly double. Adding tracefree does not substantially change the question, but I keep this change since ultimately this is what I want. | |
| Dec 9, 2015 at 16:47 | history | edited | Yasha | CC BY-SA 3.0 |
edited body
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| Dec 9, 2015 at 13:33 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 8, 2015 at 16:38 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 8, 2015 at 14:00 | comment | added | Sebastian Goette | Btw, have you read Gromov's article? Somewhere in section 4-6 he does something looking very similar to your question. | |
| Dec 8, 2015 at 12:51 | history | edited | Yasha | CC BY-SA 3.0 |
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| Dec 7, 2015 at 18:53 | comment | added | Sebastian Goette | There is a paper by Narasimhan and Ramanan that answers your second question - every connection can be pulled back from the standard connection on the tautological bundle over the Grassmann manifold $G_{r,N}(\mathbb C)$, provided $N$ is large enough ($N$ is determined by $r$ and $\dim X$ alone, if I remember correctly). | |
| S Dec 7, 2015 at 18:09 | history | suggested | Silvia Ghinassi | CC BY-SA 3.0 |
fixed mathjax
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| Dec 7, 2015 at 17:45 | review | Suggested edits | |||
| S Dec 7, 2015 at 18:09 | |||||
| Dec 7, 2015 at 17:04 | history | asked | Yasha | CC BY-SA 3.0 |