Talk:K-stability

Tian and resolution of YTD conjecture

Tian's role in the solution of YTD conjecture is disputed, to say the least. The community seems not to have assigned him credit for this result. 73.149.246.232 (talk) 22:00, 16 March 2020 (UTC)

Currently the page makes no mention of the controversy (and was originally written according to the mainstream consensus, that Tian made significant contributions to K-stability and the development of the field, but that ultimately the credit for proving YTD goes to Chen-Donaldson-Sun, but this has since been changed). In fact I would say in its current state the article probably underplays Tian's contributions to the field in terms of the original development of K-stability, but that the resolution of YTD should not be credited to him. Perhaps some discussion of the controversy is warranted. Tazerenix (talk) 01:00, 17 March 2020 (UTC)

I don't have an opinion on whether the controversy is notable for this page or the biography pages of the people involved. However, this article currently adds "and Tian" to the list of solvers of the problem, which would be appropriate if he had an independent approach to the problem, even if it appeared later. My non-expert understanding is that Tian's solution is essentially the same as D-S-C's solution, but in each instance published later and with fewer key details (and without crediting papers of DSC that he probably had seen), which led to controversy over whether his works were independent or constituted a solution. If this is also the general understanding of the expert community then "and Tian" should be removed. 73.149.246.232 (talk) 17:40, 17 March 2020 (UTC)

I removed the reference to Tian per WP:BOLD. If anyone has references showing community acceptance (outside of Tian and his students, coauthors or other affiliates) of Tian's work as an independent solution of YTD then they should revert. 73.149.246.232 (talk) 01:40, 21 March 2020 (UTC)

Tian's solution is not essentially the same as CDS's solution. Moreover, there is plenty of evidence showing community acceptance. For example, the following paper by Blum and Jonsson clearly pointed out that Tian's proof and CDS's proof are different. Both authors are outside of Tian and his students, coauthors, or other affiliates. Undermining Tian's work could only be a disgrace of the CDS school itself.

References

This is incorrect. Firstly, the paper you have cited does not claim that the proofs by CDS and Tian are essentially different. It claims that the proof of the YTD conjecture for singular Fano's given by BBJ is different to the proof in the smooth case, for which both CDS and Tian's paper are cited. In fact, this implies that the two proofs can be considered essentially the same, directly contrary to your comments. This is one of only a small number of papers in the literature which cite the two proofs on the same footing. As previously discussed, the consensus of the community and therefore the position on the wikipedia article is that key components of Tian's proof were plagiarized from CDS, and the credit to the proof of the YTD conjecture for Fanos goes to CDS.

Any edits removing the credit from CDS (as you did) or adding credit to Tian will be removed as per the previous discussion in this thread.Tazerenix (talk) 12:46, 2 October 2020 (UTC)

Your argument is incorrect. Tian's paper was posted on arXiv earlier than two of the three papers of CDS, and there is no evidence that "only a small number of papers in the literature which cite the two proofs on the same footing". The citation number for both Tian's proof and CDS's proof is almost equal. As previously discussed, the consensus of the community and therefore the position on the wikipedia article is that key components of CDS's proof were plagiarized from Tian, and the credit to the proof of the YTD conjecture for Fanos goes to CDS.

Any edits removing the credit from Tian (as you did) or adding credit to CDS will be removed as per the previous discussion in this thread. — Preceding unsigned comment added by 148.75.196.19 (talk) 07:12, 6 October 2020 (UTC)

Firstly, you cannot make an edit based on a justification you have given by citing a paper, and when that justification is immediately rebutted due to a reading comprehension error, simply pretend that you didn't say it and invent a new justification (or in this case, invent no new justification and simply revert to your previous edit again).

Secondly, there is no one, not even Tian, who claims that the proof of YTD was plagiarized from him by CDS, and if you think this is the case you are mistaken. The timeline of the proof is clearly laid out by CDS in the document on Donaldson's website. Tian's responses made no mention of the critical lemma which had exactly the same proof as in the CDS paper that Tian's paper quickly followed, and instead consisted of autohagiographic comments and unsubstantiated quotes from emails which had little relevance to the actual mathematics, instead intended to demonstrate that Tian had been working in the field for some time. This was in an attempt to make it appear as though Tian was a lone wolf (a la Perelman), who had miraculously produced the exact same proof of critical parts of the argument as CDS in isolation just days after the details of those results became public in CDS's work. The consensus within the community is indicated by the fact that Chen-Donaldson-Sun were awarded the Veblen prize for their resolution of the YTD conjecture, with Tian conspicuously absent. To claim that the proof of the conjecture should be credited solely to Tian is absolutely ludicrous, given that the proof relies critically on Donaldson-Sun's work on Gromov-Hausdorff limits of Kahler-Einstein manifolds at the very least. The fact that you have edited the page twice now to remove all reference to CDS for the credit of the conjecture indicates that you are more concerned with scrubbing the history books in Tian's favour than faithfully representing the consensus of the mathematical community.

I have reverted your edits again and will seek further input from WT:WPM before making any further changes. Please do not change the page again until further input has been received.Tazerenix (talk) 11:31, 7 October 2020 (UTC)

Nonsense. — Preceding unsigned comment added by 155.97.233.178 (talk) 14:03, 7 October 2020 (UTC)

It is not the role of Wikipedia to participate to any controversy. Here we have an attribution that is supported by reliable WP:secondary sources (at least Veblen Prize). On the other hand, it seems to not exist any source independent from Tian that attributes to him the priority of the proof. So, Tian's claim of priority is original research in the sense of Wikipedia (see WP:OR), and the policy of Wikipedia is to challenge such original research; it does not matter whether Tian's assertion is true of not. In particular, any attribution change without the support of independent secondary source has to be reverted. D.Lazard (talk) 15:30, 7 October 2020 (UTC)

Tian's paper is as widely cited in the literature as Chen-Donaldson-Sun's, so I think it must also be mentioned. However it is very clear that Tian's paper should not be mentioned instead of Chen-Donaldson-Sun's, as the IP user seems to want. I believe it would be fair and neutral to attribute the theorem to both Chen-Donaldson-Sun and to Tian, and to also say (at the least) that Chen, Donaldson, and Sun allege that Tian's paper contains gaps or inaccuracies. This seems to me to be a reliable source. I think it would also be fair to say that Tian has rebutted their charges. I think it'd be inappropriate for the article to choose a side or to judge whether or not Tian's defense is good. Gumshoe2 (talk) 23:48, 8 October 2020 (UTC)

I have added credit of the proof of the theorem to Tian, citing his paper. I also added notes linking to both CDS's rejection of his claimed proof as well as Tian's rebuttals to CDS. I hope this resolves the issue for now.Tazerenix (talk) 13:10, 9 October 2020 (UTC)

I made a small edit, just trying to make it sound as neutral as possible. Gumshoe2 (talk) 17:12, 9 October 2020 (UTC)

Tian's proof was not completed until 2015. See Remark that this corrigendum has gone through the peer-review process, so it should be viewed as a more reliable resource than people's claims in their homepages in order to show that Tian's proof was not completed until 2015. 144.92.38.236 (talk) 01:19, 11 November 2020 (UTC)

Ved Datar and Gábor Székelyhidi's paper and Chen-Sun-Wang's paper have been cited by almost the same amount of people as Chen-Donaldson-Sun's paper or Tian's paper, so they should also be mentioned if you cite Tian's. I would like to remark that their results were all completed before Tian's completion in 2015. 144.92.38.236 (talk) 01:26, 11 November 2020 (UTC)

I don't think it's appropriate to judge the significance of Tian's correction using my own knowledge in Kaehler geometry because Wikipedia relies on peer-reviewed papers rather than editors' own knowledge. However, as you can imagine, no one would submit a correction for a minor issue if there was a priority disputation, so the published correction should be considered by default as evidence that Tian's first version contained a gap. The peer-reviewed Veblen prize provided another evidence because they only mentioned CDS. 144.92.38.236 (talk) 01:55, 11 November 2020 (UTC)

There seems to be no metric by which Datar-Székelyhidi or Chen-Sun-Wang's papers have been cited as much as Chen-Donaldson-Sun or Tian's (according to google scholar, the difference is between the numbers 89 and 43 and the numbers 328 and 282.) More importantly, I think your use of errata or corrigendums as proof of guilt would set a very bad standard, considering at the least that many (possibly most?) papers should have them but do not. Having said that, the lack of citation to Tian's corrigendum was a notable oversight in the previous version of this article. Also, I have never heard of anyone crediting Datar-Székelyhidi or Chen-Sun-Wang with resolving the conjecture, and as far as I know none of those authors have claimed to do so, so I have changed the language to refer to them as "other proofs". Gumshoe2 (talk) 05:23, 11 November 2020 (UTC)

When you read Datar-Székelyhidi or Chen-Sun-Wang's paper carefully, you will see that the reason for them not being considered as the first proof was because they all claimed themselves as "a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun" without mentioning Tian. If you think that the corrigendum is not proof of incompleteness, then there must be another reason for both of them to consider Tian's paper as an incomplete paper. 144.92.38.236 (talk) 07:16, 11 November 2020 (UTC)

It must be noted that Chen and Sun from Chen-Sun-Wang are also the Chen and Sun from Chen-Donaldson-Sun, and Wang was Chen's student. (Additionally, Szekelyhidi was Donaldson's student.) That isn't to say that I personally disagree with their choice; although I haven't studied the YTD papers carefully, I know that many of Tian's papers have errors or omissions. Gumshoe2 (talk) 07:35, 11 November 2020 (UTC)

For your information, Datar was Song's student and Song was known for his collaboration with Tian. 144.92.38.236 (talk) 07:45, 11 November 2020 (UTC)

I also disagree with your counting of citations because not all citations carry the same weight. For example, in another well-cited preprint by Berman, Boucksom, and Jonsson in September 2015, they said "it was settled a few years ago by Chen–Donaldson–Sun [CDS15] (see also [Tia15])". Such a sentence provides the same amount of citation but clearly, the importance between Chen–Donaldson–Sun and Tian were NOT considered the same. 144.92.38.236 (talk) 07:27, 11 November 2020 (UTC)

I think the relevant question is what you presently think is on the page which should be improved, and why. Gumshoe2 (talk) 07:36, 11 November 2020 (UTC)

I have changed the sentences according to Berman, Boucksom, and Jonsson's JAMS paper. 144.92.38.236 (talk) 07:42, 11 November 2020 (UTC)

As far as I know, Berman, Boucksom, and Jonsson are neither Donaldson's students nor Tian's collaborators' students. In some sense, their opinions should be considered as opinions without conflict of interest. 144.92.38.236 (talk) 07:52, 11 November 2020 (UTC)

You should not edit the page by copying directly the text of BBJ's article, as you've done. Anyway, this has all been discussed before here and on wikiproject:math, and I don't think that one particular paper's choices change anything. But I will wait for others to weigh in before I edit again Gumshoe2 (talk) 08:14, 11 November 2020 (UTC)

BBJ's paper is not "one particular paper", but another paper solving the YTD conjecture. The choices in their paper, as well as Datar-Székelyhidi and Chen-Sun-Wang's papers carry more weight than other papers. 144.92.38.236 (talk) 08:18, 11 November 2020 (UTC)

I think User:144.92.38.236's citations help lend citable credibility to the statement that Tian's proof did not have primacy over CDS's. I think there are two separate discussions happening here, so we should probably split them. One is whether CDS or Tian get the "primary" credit for the resolution of the conjecture, which we had mostly settled before the recent edits, and I don't think anything has shifted on this front.

The second is to what extent the page should discuss the alternative proofs of the YTD conjecture (by alternative I of course mean those that came after CDS, which now includes Tian, BBJ, DG, CSW). This is certainly worthwhile to have in that section (truthfully this sort of thing should probably be saved for a dedicated page on the YTD conjecture, which does not yet exist), but I think as currently written it appears quite clumsily in the statement of the theorem. It may be best to avoid putting the credit for the resolution of the conjecture in the text of the theorem statement, and instead summarise clearly the primacy of proofs below it.Tazerenix (talk) 12:08, 11 November 2020 (UTC)

To be clear, do any of you disagree with the following statement?: in the literature, for the resolution of the YTD conjecture, some authors credit Chen-Donaldson-Sun, some authors credit Chen-Donaldson-Sun and Tian equally, and nobody credits Datar-Szekelyhidi, Chen-Sun-Wang, or Berman-Boucksom-Jonsson. Gumshoe2 (talk) 17:52, 11 November 2020 (UTC)

That seems to be fairly accurate to me. I have seen off-hand credit given to BBJ insofar as the novelty of their approach is that it works for singular varieties, but also resolves the smooth case. No one gives them equal credit to the smooth case as CDS or Tian however. If we interpret "resolution" in your statement to mean "first completed and reviewed proof" then I agree completely.Tazerenix (talk) 20:55, 11 November 2020 (UTC)

As far as I know, some authors such as Datar-Szekelyhidi, Chen-Sun-Wang credit Chen-Donaldson-Sun only, some authors such as Berman-Boucksom-Jonsson credit Chen-Sun-Wang strongly and credit Tian weakly. Could you find any notable paper which credits Chen-Donaldson-Sun and Tian equally? As for the credit of Datar-Szekelyhidi, Chen-Sun-Wang, or Berman-Boucksom-Jonsson, many people credit them as a different proof of the YTD conjecture but in the smooth case, no one views them as the "first completed and reviewed proof". 144.92.38.236 (talk) 07:12, 12 November 2020 (UTC)

Apologies for leaving this hanging for so long. Of course there are papers that credit CDS and Tian equally, and they are easy to find by looking through the hundreds of citations of Tian's paper, which continues to be as cited as CDS. . Again, although I personally would credit CDS with the theorem, I think that for this website it is totally clear that CDS and Tian should be credited equally, and other authors can be mentioned with their other proofs. The fact that CDS have been recognized by Veblen and breakthrough prizes should be a striking fact for any readers of the page, and there is no need to further editorialize it. I have rewritten the section in an attempt to be totally objective and clear. I think neither supporters of CDS nor of Tian could dispute what is written there now. Gumshoe2 (talk) 00:06, 16 January 2022 (UTC)

Looks good thank you. The previous setting with about 10 names listed after the theorem was a little ridiculous. Tazerenix (talk) 06:16, 16 January 2022 (UTC)

2026 cscK YTD preprints

I am Jihao Liu, the author of arXiv:2608.19301, and therefore have a conflict of interest concerning this material. I will not make further direct changes while this discussion is open and request review by uninvolved editors.

An uninvolved editor rewrote the material in explicitly attributed language under WP:PREPRINT and WP:SCHOLARSHIP in Special:Diff/1370434641. The Liu paragraph was subsequently removed in Special:Diff/1370450012 and again in Special:Diff/1370469728, with edit summaries stating that an AI-assisted paper had not been verified by humans.

That factual claim is incorrect: Appendix A of the preprint describes final review and checking by human experts: Appendix A. This is separate from the legitimate question whether a recent arXiv preprint has sufficient reliability and due weight under WP:RS, WP:PRIMARY, WP:EXCEPTIONAL, and WP:DUE.

I request that uninvolved editors determine whether and in what attributed form this result should be included. Please discuss the issue here rather than continuing to revert it. ~2026-43164-95 (talk) 08:39, 21 August 2026 (UTC)

To be more clear. An uninvolved editor appropriately revised the material in Special:Diff/1370434641 to reflect its status as a preprint, using attributed and conditional wording such as “According to a preprint” and “this would give”. I agree that an arXiv preprint should be described in this conditional form.

The later editor ~2026-45719-54 nevertheless removed the Liu paragraph in Special:Diff/1370450012 and again in Special:Diff/1370469728, stating that the AI-assisted paper had not been verified by humans. That factual reason is incorrect: Appendix A expressly records subsequent review and checking by human experts: Appendix A.

I request that the independently revised, attributed and conditional wording from Special:Diff/1370434641 be used as the basis for consensus. Please discuss any further sourcing or due-weight objections here rather than continuing to revert. ~2026-43164-95 (talk) 08:43, 21 August 2026 (UTC)

RfC on reporting Liu's 2026 cscK YTD counterexample preprint

This LLM-generated text has been collapsed and should be excluded from assessments of consensus.

The following discussion has been closed. Please do not modify it.

Should the “Extension to constant scalar curvature Kähler metrics” section include the attributed and conditional summary introduced by an uninvolved editor in Special:Diff/1370434641, stating that Jihao Liu’s preprint arXiv:2608.19301 reports a K-polystable smooth polarised projective fivefold with no cscK metric, and that this would give a counterexample to the K-polystability formulation of the cscK Yau–Tian–Donaldson conjecture? ~2026-43164-95 (talk) 10:19, 21 August 2026 (UTC)

COI disclosure: I am Jihao Liu, the author of arXiv:2608.19301.

My position: The attributed and conditional summary should be included.

First, Wikipedia’s preprint guideline does not prohibit the use of preprints. WP:PREPRINT states that preprints are “seldom reliable”, not that they are never usable, and expressly recognizes that they may sometimes satisfy the criteria for acceptable use of self-published sources. Reliability is contextual and depends on the precise claim being supported.

Second, the proposed text does not ask Wikipedia to certify the mathematical proof. Under WP:PRIMARY, a primary source may be used carefully for a straightforward descriptive statement. The proposed statement is that the preprint reports a particular theorem and that, if correct, the theorem would give a counterexample. These claims are stated explicitly in the title, abstract, and main theorem of the cited preprint; no Wikipedia editor is deriving, interpreting, or independently validating the proof.

This is precisely why the wording introduced by an uninvolved editor in Special:Diff/1370434641 begins “According to a preprint” and uses “would give a counterexample”. That editor expressly cited WP:PREPRINT and WP:SCHOLARSHIP, but did not treat those guidelines as requiring deletion. Instead, the editor retained both the Liu and Trusiani results while changing them to attributed language appropriate for preprints.

Third, the result is directly relevant to the central subject of the subsection. The subsection states the cscK Yau–Tian–Donaldson conjecture and currently describes the implication from K-polystability to the existence of a cscK metric as the significant unresolved direction. The Liu preprint reports a counterexample to exactly that implication. A short, expressly attributed paragraph is therefore proportionate to its relevance and does not require Wikipedia to present the result as an established consensus.

The subsection also continues to report Antonio Trusiani’s 2026 arXiv preprint using the same “According to a preprint” formulation. The two results should be assessed consistently as recent mathematical preprints: Trusiani’s preprint reports the uniform version, while Liu’s preprint reports a counterexample to the K-polystability version. The Liu preprint expressly states that the Fano case and the uniform and completed versions are unaffected.

Finally, the later deletion was justified by edit summaries asserting that the AI-assisted paper had not been verified by humans. That is not an accurate description of the source. Appendix A records review and checking by human experts: Appendix A. The cited source is a mathematical preprint issued under a named human author’s responsibility, not an unattributed chatbot response. Human checking does not turn a preprint into a peer-reviewed publication, which is why conditional attribution remains appropriate; but disclosed AI assistance is not itself a policy-based reason to erase an otherwise properly attributed report of the preprint’s result.

Proposed article wording:

According to a preprint first submitted to arXiv on 19 August 2026 by Jihao Liu, there is a smooth polarised projective fivefold that is K-polystable with respect to every normal ample algebraic test configuration of every positive exponent, but whose polarisation class contains no extremal Kähler metric and hence no cscK metric; this would give a counterexample to the conjecture stated above, as formulated in Donaldson's 2002 paper and interpreted using the standard modern convention of normal ample algebraic test configurations. The paper also treats arbitrary ample test configurations, with equality in the Donaldson–Futaki invariant precisely for configurations that are products in codimension two. The reported counterexample would not affect the Fano case or the uniform and completed versions of the cscK correspondence.

References

~2026-43164-95 (talk) 10:19, 21 August 2026 (UTC)

To be honest the page is due for a bit of a touch up since I originally wrote it, as there have been quite a few developments which were only in their infancy in 2019-2020. The NA story and the subsequent Chi Li model story has matured into the primary framing of YTD *in practice* and is now arguably more correct than the original definition of K-stability that Donaldson gave. In particular the article should have an expanded discussion of stability with respect to models, uniform K-stability, the results of Darvas-Zhang and Boucksom-Jonnson and then the marvelous result of Trusiani characterising the *positive* version of YTD, and then the counterweight of the recent preprint characterising the *negative* version of YTD.

A balanced presentation of the subject on this page should include both points of view that, generally speaking, the *literal* statement of YTD was considered "morally true, literally false" and that the combined results of recent years now characterize this general sentiment precisely in both directions (i.e. results which appear to almost saturate the "morally true" part and results which now precisely characterise the "literally false" part).

The specific remarkable circumstances around the disproof of the standard YTD statement (i.e. that it used AI) is probably not worth, *on this article*, more than a cursory mention. I suspect any extended discussion of that will quickly become outdated in a few years time when the use of LLMs as tools in proof becomes more commonplace (though this stands out as a particularly remarkable usage of them).

My general sense is that most of the content on this page is quite "long held" and old, and the recent work preprints are quite bleeding edge. I would make an exception in this case though since this page is the primary YTD page on Wikipedia and the results (both positive and negative) are clearly significant enough to be mentioned even as preprints. I will review the wording and try put some effort into making the broader context up to date soon. Tazerenix (talk) 11:54, 21 August 2026 (UTC)

Sounds good to me. This preprint is a pretty stunning result, although normally I would want to wait for secondary sources. Sławomir Biały (talk) 12:43, 21 August 2026 (UTC)

There have been previous arxiv preprints claiming to prove or disprove major conjectures — sometimes correctly and sometimes incorrectly. In the past, Wikipedia doesn’t report on them until, at the least, they’ve received some major academic attention. I don’t see why this case should be different … shouldn’t we wait until there’s at least one secondary source? Gumshoe2 (talk) 14:28, 21 August 2026 (UTC)

My view is that applying notability and verifiability standards, whether you should or shouldn't include a preprint depends on how notable the result is for the main subject of the page and whether the coverage of the topic would be considered "balanced" if you exclude it. In this case since this is basically the primary page for YTD, and since this paper (and Trusiani's paper) directly and notably address that problem, its defensible to reference them more quickly than you would otherwise. In any case I suspect there will be some further sources referencing this which clearly tip it from "notable but unverified" to just "notable" quite quickly. Tazerenix (talk) 22:10, 21 August 2026 (UTC)

I think if what you’re saying is true, there’s very little cost in waiting until there are such further sources, especially if it will be a quick wait. All the moreso given that this is an AI result, which I believe makes it a priori less trustworthy. There needs to be evidence that somebody (not just the ‘author’/communicator) has absorbed the counterexample and believes it’s legit. Gumshoe2 (talk) 22:50, 21 August 2026 (UTC)

The proof is also largely AI generated. I don't think the mathematical community here, or at large, has really settled on its treatment of AI generated results. In his 2026 ICM keynote Terrance Tao, lays out a proposal for how AI generated result can become accepted by the mathematical community, with transitions from Open Problems -> Unverified Solutions -> Verified Solutions (eg Lean) -> Well Written Solution (requiring a good proof explanation) -> Accepted Solution (Journal publication) -> Definitive Solution (Happens after a number of years when the community accepts a reworked version of the proof into the mathematical cannon). This paper has no formal verification in Lean or similar, so we cannot be sure the proof does not have has some hallucinated steps. This paper does have an attempt at a proof explanation, Tao discusses how some AI explanations tend to be unbalanced, spending many pages on simple lemmas and only a few lines on the most interesting and novel parts of the argument. I can't judge how good the proof explanation is, and we should really wait for journal publication. --Salix alba (talk): 07:07, 22 August 2026 (UTC)

• Now, I am not a mathematician, but WP:PREPRINT notes that preprints are rarely WP:Reliable sources. I'd wait until journal publication or any news outlet/magazine/etc reporting on the potential discovery. Shocksingularity (talk) 02:31, 29 August 2026 (UTC)
• Note: I have collapsed and closed the RFC per WP:LLMTALK. To those who commented, I would strongly urge you to not engage and just do the same when you happen upon something like this. –Deacon Vorbis (carbon • videos) 05:50, 8 September 2026 (UTC)