A math journal has retracted a 2015 paper after three outside experts informed the editors that “the paper contains errors which invalidate its main results.”

According to the retraction notice, published in the July 2017 issue of *Manuscripta Mathematica*, the author Ilya Karzhemanov “has not admitted to the alleged errors and disagrees with the retraction.”

It’s unclear when exactly the paper was retracted, but Karzhemanov, now associate professor at the Moscow Institute of Physics and Technology, posted the now-retracted paper on arXiv in June 2017 and explained his “strong disagreement” with the retraction:

The present situation around my paper is a result of personal conflict and is a part of continuous manhunt on me (initiated unfortunately by my former advisor Cheltsov).

We asked Karzhemanov about the personal conflict, but he said he did not want to discuss “personal matters.”

Ivan Cheltsov, who supervised Karzhemanov’s PhD from 2007 to 2010, told us “there is no conflict.” Cheltsov, a professor of mathematics at the University of Edinburgh, explained that he did not write to *Manuscripta Mathematica* about Karzhemanov’s now-retracted paper but said “it was absolutely clear that there is a problem with this paper:”

I emailed people working on these types of problem and informed them about this paper… I asked them to check the proof. And I emailed Ilya as well so that he knows what I am doing.

Karzhemanov argued that aside from “one or two small misprints,” which he says he corrected in the arXiv version, his paper is correct. He told us that he sent the journal explanations defending his work (1, 2), but “the evidence in its favor were completely ignored.”

Here’s the retraction notice for “Some pathologies of Fano manifolds in positive characteristic:”

The article Some pathologies of Fano manifolds in positive characteristic by Ilya Karzhemanov published under DOI:10.1007/s00229-015-0796-9 has been retracted by the journal’s editorial board.

The retraction has been made because in a post-publication peer review process it was reported by three independent reviewers that the paper contains errors which invalidate its main results.

All three reviewers presented examples which show that Proposition 2.9 and Lemma 2.12 are wrong. Furthermore it was pointed out that the way the Fano variety X in the Main Theorem 1.4 is constructed is in contradiction with the Theorem of Campana and Kollar-Miyaoka-Mori that every smooth Fano variety over an algebraically closed field is chain connected.

The author of the paper has not admitted to the alleged errors and disagrees with the retraction.

The online version of this article contains the full text of the retracted article as electronic supplementary material.

The retraction notice, though not the paper, has been indexed by Clarivate Analytics’ Web of Science, which we typically use to determine how often a paper has been cited. We asked the editor for more details but he referred us to the notice.

**Other problematic papers?**

Cheltsov elaborated on the personal situation Karzhemanov referenced in his arXiv post. Cheltsov told us that a few years go several mathematicians informed him that Karzhemanov had written some problematic papers. Cheltsov says he checked a few of Karzhemanov’s papers and discussed them with him to help resolve any issues.

Karzhemanov, however, says that the discussion about the quality of his work occurred behind his back and has affected his career. “It was impossible for me to argue” with the criticism, he says.

Cheltsov explained that he believes Karzhemanov considered these efforts to discuss the papers and correct any inaccuracies “a personal offense.”

It’s not the first time authors have protested a retraction on arXiv. In one recent case, two physicists protested the retraction of their paper and defended their work on arXiv.

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In mathematics, you’re either right or you’re wrong. There are no gray areas in mathematics. If counter-examples are presented and verified, that is it. The emotional appeals are unnecessary. It is more productive to move forward and correct the proofs/issue corrigenda and errata/retract those proofs that are unworkable.

I thought exactly that before I got a BS. Mathematics. In fact, at the bleeding edges, math is indeed subject to the interpretation of other mathematicians. Don’t worry though, there is practically nothing that most of us will ever do in math that is subject to such interpretation.

John’s suggestion that “math is subject to the interpretation of other mathematicians” is rather generous. Few papers are carefully read before being published, as an article

by a Fields medalist linked from

a recent weekend reads suggests. It is mostly one’s charisma and likeability which are judged; I would guess this is not completely different from other non-applied fields.

I think your interpretation of the article by Gowers (the Fields medalist) is mistaken, insofar as you equivocate (not necessarily deliberately!) between two distinct meanings of “publish”—two meanings which correspond to the two distinct meanings of “peer review” that Gowers distinguishes, namely, “formal” and “informal” peer review.

It is indeed true, as you say, that “Few [mathematics] papers are carefully read before being published”

when “published” is understood to mean “informally published”(nowadays meaning, in mathematics, placed on the arXiv, as were the papers by Perelman and by Blum that Gowers discusses; formerly meaning, in mathematics, distributed in preprint form, whether by the author or recursively by someone who has already received a preprint). The one example I know about, of a (mathematical) paper that was “carefully read before being published” even in preprint form, is the case of Andrew Wiles’s manuscript proving (modulo eventual corrections that even postdated its preprintandits publication in theAnnals, and which were eventually published in a collaborative paper) “Fermat’s Last Theorem”: his Princeton colleague Nick Katz was, by agreement, itsonlypre-preprint reader, and he read itverycarefully (and even so there was a subtle error left to be corrected!). That example (and maybe a few other such examples every decade, that don’t get as much publicity) is the rare case of someone following (which is not to say, taking) the advice some serious old editor used to give (I think it may have been Paul Halmos), that a mathematical author shouldalwaysmake sure a few friends or colleagues have read (and commented on!) a new paper before sending it out into the world, on the grounds that if even a friend won’t read all the way through a paper you give them, why should a journal subscriber?It is also (sadly) probably still true (as it was during the many years when the median number of papers that a Ph.D. mathematician published in an entire career was one) that few papers are carefully read

afterbeing published, even when “published” is understood to mean “formally published”: many papers are just of no interest at all to anyone but (maybe) their authors (although even those may turn out to be interesting years later…). But many other papers, once they have been “informally published”, and especially now that the arXiv exists,arecarefully readbefore being “formally published”; some, even, which for one reason or another never end up being formally published. And it is those careful readings of preprints that Gowers calls “informal peer review”.…Everything I’m saying, particularly the quantifiers “few”, “many”, “most”, is of course based on my own experience filtered through my own preconceptions and rigorously kept separate from any statistical analysis. However, I have been observing the culture of mathematicians, their preprints, and their formal publications and “informal peer reviews”

avant la lettre, for (exactly!) 50 years, and actively participating in them for the past 46. Indeed, my introduction to what has ended up inspiring most of my life’s work consisted in being around in 1971-2 while one of Lefschetz’s “correct theorems” from the 1920s—the eponymous “Hard Lefschetz Theorem—had what turned out to be its “incorrect proof” informally peer-reviewed to within an inch of its life by a couple of great topologists (one of whom is widely believed only to have failed to get a Fields Medal because he then distributed everything in pre-print form but was very slow to get stuff off to journals, and the other of whom was, years later, at the heart of the informal but WILDLY rigorous peer-review of Perelman’s work).By “published”, I meant “published in a journal”. Daniel Biss’s papers in the Annals and Inventiones, which include his thesis is one example. Errata retracting the main claims, but not the papers themselves, were published about 5 years later. He is now running for the governor of Illinois and bragging about his mathematical prowess. I am aware of other papers in these journals that claim to establish previously expected, but very useful, results. They are often cited for the claimed results, but I have not been able to find anyone willing to discuss their proofs publicly (which have been available on arXiv almost 20 years for now); the authors have refused to do so.

I think arXiv is fantastic and would be even more so if they had added some kind of discussion overlay in the spirit of MathOverflow. This would make it more possible for the community to discuss the articles and for the authors to improve them. The quality of the articles, could then be more carefully judged on these discussion, instead of based on which journals they were published based on confidential advice of anonymous referees. My interpretation of Gower’s article is that this is precisely what he is advocating and that he is expressing concern about the overreliance on where the papers are published in hiring and funding decisions. In my experience, many mathematicians quietly agree with this, but go with the flow.

This is a nice ideal, but it has very little resemblance to the reality of mathematical practice and publishing. Many great papers have mistakes, ranging from extremely minor to very important. Most papers don’t get retracted for it.

Sometimes what happens is that other people realize how a flawed proof can be fixed, and let the original stand, and even cite it for the result (even if the final, correct proof is elsewhere). Even if there’s a counterexample found, it’s not necessarily the end: maybe the paper has presented something interesting, and the theorem simply needs a hypothesis added. In that case it would be rare for the journal to run a correction, and instead the correct statement would just be generally known to “those in the know”.

There is some movement toward computer-verified proofs, but it is very slow, and at present a tiny minority of papers in a narrow set of areas.

It was said (somewhat hyperbolically…) that Solomon Lefschetz never stated a false theorem, and never gave a correct proof.

I am happy that posts like this exist publicly online. I recently came across the paper “One construction of a K3 surface with a dense set of rational points,” authored by Karzhemanov. The result surprised me, so much so that I suspected an error.

The proof of potential density used an elliptic fibration, which cannot exist on a Picard rank 1 K3 surface. But the paper had been cited by another author, and I am not an expert in the number-theoretic aspect of K3 surfaces. Without a post like this, I wouldn’t have understood the situation, and may have fruitlessly spent time trying to understand the proof or use the result.

Mathematics definitely needs a “error arXiv” of sorts, which publicly compiles lists of known incorrect papers, perhaps with short comments on the main errors involved. Otherwise, only people “in the know” of that subfield have access to the knowledge of which papers are wrong.