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Stream: deprecated: algebraic geometry

Topic: Holomorphic differential cohomology

Tim Hosgood (May 25 2020 at 00:11):

Is anybody familiar with differential cohomology (in the sense of Urs Schreiber's work on it)? It seems like all the things I can read about it deal only with the smooth case, and say that things should sort of work the same in the holomorphic case, but I don't think this is very obvious at all...

Tim Hosgood (May 25 2020 at 02:02):

yes, i’m interested in deligne cohomology, which is why i’m trying to understand differential cohomology!

Tim Hosgood (May 25 2020 at 02:11):

to be a bit more specific, here’s an example of something i’d like to know: the theory of $\infty$ -connections is studied in a lot of detail in the context of differential cohomology, but holomorphic connections don’t usually exist (globally), and we have no partitions of unity, and generally the complex world is much less well behaved than the smooth one

Tim Hosgood (May 25 2020 at 02:12):

so does this introduce problems when actually trying to apply this theory to the holomorphic case? or are there sneaky ways around it?

Tim Hosgood (May 25 2020 at 02:16):

David Michael Roberts (May 25 2020 at 09:02):

Yes, do get in touch with Finnur, he is a very nice guy.

David Michael Roberts (May 25 2020 at 09:05):

Also, if you are interested in specific examples, then I believe the basic gerbe on a complex reductive Lie group can be equipped with at least a connection, if not a curving (working from memory here), hence something corresponding to a cocycle in holomorphic Deligne cohomology. There's a preprint by Brylinski (basically the last before he left academia) and a more recent paper (in the last 6 years or so) on this. I was once trying to write down, with Raymond Vozzo, an explicit model for, and connection and curving on, the gerbe on $SL(2,\mathbb{C})$ , but we ended up doing something else.

Tim Hosgood (May 25 2020 at 09:36):

will get in touch with Finnur, thanks for the suggestions :)

Tim Hosgood (May 25 2020 at 09:37):

@David Michael Roberts do you remember the titles of these two papers?

David Michael Roberts (May 25 2020 at 09:54):

Here's the Brylinski one: https://arxiv.org/abs/math/0002158

David Michael Roberts (May 25 2020 at 09:54):

I'll have to hunt around and dig up the other one, my searching of obvious things isn't turning it up very quickly.

David Michael Roberts (May 25 2020 at 09:57):

Ah, here it is: http://www.tac.mta.ca/tac/volumes/32/30/32-30abs.html

David Michael Roberts (May 25 2020 at 09:57):

I hope these are helpful, @Tim Hosgood !

David Michael Roberts (May 25 2020 at 09:57):

@Rongmin Lu I don't recall the question, perhaps you can remind me (it was a long time ago!)

David Michael Roberts (May 25 2020 at 09:58):

@Tim Hosgood check this one out as well, it's also by Brylinski, published in the 90s: http://www.numdam.org/item/?id=AST_1994__226__145_0

Tim Hosgood (May 25 2020 at 12:07):

wonderful, thank you!