Timeline for Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?
Current License: CC BY-SA 4.0
12 events
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| Jan 1, 2022 at 20:00 | history | edited | Ben Webster♦ | CC BY-SA 4.0 |
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| Jul 31, 2021 at 5:46 | comment | added | random123 | @Z.M Thanks! I found the result referred to in the above comments. | |
| Jul 30, 2021 at 22:03 | comment | added | Z. M | @random123 Seemingly they were talking about "Equations différentielles à points singuliers réguliers". | |
| Apr 1, 2018 at 10:54 | comment | added | random123 | @BCnrd Which book of Deligne were you talking about here? Thanks! | |
| Mar 27, 2010 at 4:04 | comment | added | BCnrd | See 2.23 in Deligne's book for a brilliant inductive proof in the smooth case over an arbitrary analytic base (allowed non-smooth); taking base to be point is smooth case which was the focus of interest in the question. I wrote up the smooth case with base a point in notes on Riemann-Hilbert correspondence on my webpage (see Theorem 2.6, Lemma 1.6 there). I think my memory got confused about Deligne working in relative case over any (possibly non-smooth) base; most likely smoothness of structure map to the base cannot be dropped. Feel free to delete comments about non-smoothness; mea culpa. | |
| Mar 25, 2010 at 18:25 | comment | added | Ketil Tveiten | I was thinking of the smooth case (Riemann surfaces, actually), but the general case would be interesting too. Which book by Deligne are you talking about? I have 'Equations Differentielles', but there it was 'well known'. | |
| Mar 25, 2010 at 17:42 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Mar 25, 2010 at 17:40 | comment | added | Ben Webster♦ | Brian- Obviously it doesn't. In my interpretation of the question, that was not what the OP was confused about (checking the rank is the difficult part of the question if you understand how the bijection should work). It would great to see an answer which did cover this point. I would certainly vote it up, and I wouldn't blame the OP for unaccepting my answer and accepting a more complete one. | |
| Mar 25, 2010 at 3:02 | comment | added | BCnrd | How does this answer address the real difficulty in the question, which is the proof that the kernel of the flat connection is locally constant of the "expected" rank when the base space is an arbitrary (not necessarily smooth) complex-analytic space? Going from the local system to the bundle with flat connection is the easy direction; the other one requires new work when the base is not assumed to be smooth. I don't think this is at all obvious. Deligne's proof in his thin SLN book is very beautiful, and requires a real idea. | |
| Nov 4, 2009 at 20:16 | vote | accept | Ketil Tveiten | ||
| Nov 4, 2009 at 20:16 | comment | added | Ketil Tveiten | I'd suspected this, but I wasn't able to make it explicit. Now that you mention it, it ought to have been obvious. Thanks! | |
| Nov 4, 2009 at 19:40 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |