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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 7th 2012

started Narasimhan–Seshadri theorem, for the moment just to collect references.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 21st 2014
• (edited May 21st 2014)

I have added a bit more of an actual statement to Narasimhan–Seshadri theorem. I followed Jonathan Evans’s good idea (linked to in the entry) to highlight that for line bundles on surfaces this is what ought to be called the Hodge-Maxwell theorem; and I created a brief entry for that.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeMay 21st 2014

Some of the business about numerical invariants like rank and degree inducing corresponding filtrations needing to do the strata of moduli spaces are invented by Grothendieck in FGA when introducing moduli spaces like Quot, Pic, Hilbert schemes. See the modern era remake FGA explained, n particular the chapter

Then Mumford introduced the stability in his works on geometric invarant theory. Urs, a small question – I do not understand why the later Takemoto Nagoya references from 1972/1973 is claimed to have the original discovery as well ? I do not know the details, but probably you have some reason why you consider this an original reference about the notion ?

I thought that we already had the related entry Harder-Narasimhan filtration but it does not look so under the inspection. Maybe I had it in my mind when in a spree of writing the circle of (in one chain of correspondences) somewhat related stubs like stability, Bridgeland stability, Stokes phenomenon, slope

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 21st 2014
• (edited May 21st 2014)

Thanks, Zoran. Right, so I guess you saw me start to create stubs for

etc. This is not done yet, clearly. Please add references as you happen to have them readily available.

Regarding your question, I am not sure which entry the question refers to, sorry. But I am pretty sure that you know more about which author was first in this case, so please just go and fix it if you see the need.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeMay 21st 2014
• (edited May 21st 2014)

Well, Takemoto says in his first part

We give the definition of an H-stable vector bundle on a variety X. This definition is a generalization of Mumford’s definition on a curve.

what is in favor of your wording about the origins of Mumford stability in the references at stable vector bundle.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeMay 21st 2014
• (edited May 21st 2014)

OK, I added a couple of lines at rank of a coherent sheaf.

We do have a stub for Quot scheme. The entry stability is a disambiguation page and we do not have yet a Mumford stability page.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 21st 2014

okay, thanks. Meanwhile I have started a stub for Harder-Narasimhan theorem with a pointer to Evans’s lectures.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 21st 2014
• (edited May 21st 2014)

And I have added to holomorphic vector bundle a brief paragraph on the characterisation over Riemann surfaces by $U(n)$-principal connections, due to Atiyah-Bott.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMay 21st 2014

Generally,I am adding pointers to sections 7 and 8 of

• Michael Atiyah, Raoul Bott, section 8 of The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)

to various of these entries.