I’ve heard reports from Harvard that yesterday and today Heisuke Hironaka has been giving talks in the math department there, claiming to have a proof that singularities of algebraic varieties can be resolved for any dimension in characteristic p. This would be a major advance in the field of algebraic geometry. I don’t know any details of the proof, but Hironaka is, at 77, an extremely well-respected mathematician, not known for making claims unless they are very solid.

Hironaka won the Fields Medal in 1970, largely based on his 1964 proof of the resolution of singularities, in the characteristic zero case. For an introduction to that proof, see this article from the Bulletin of the AMS.

**Update**: From comments here it seems that the source of my information about Hironaka’s talk was most likely overly optimistic about exactly what Hironaka was claiming. The current situation seems to be that several groups are working on this, with promising ideas of how to get to a proof, but with no definitive proof yet done. There will be a workshop at RIMS in December, with the goal of sorting out the current situation:

The aim of the workshop is to review recent advances in the resolution of singularities of algebraic varieties with special emphasis on the positive characteristic case. After many years of slow progress, this is now a rapidly developing area with several promising new approaches. Our aim is to keep the program flexible, in order to give the maximum opportunity to discuss and explore new developments. We expect a joint effort to understand characteristic p, and that the purpose is not that everybody exposes his/her own results.

“singularities of algebraic varieties can be resolved for any dimension in characteristic p. ”

Can you explain, with an example, why is that a very important result?

Well, there are a lot of problems in characteristic zero (where resolution is known) which are solved by taking a resolution of singularities, proving the result (or defining the object, or the like) on the resolution, and then pushing forward to the singular object. The point of the resolution is that it’s smooth, but as close as possible to the singular variety. So presumably the reason is that there are some conjectures that hold if resolution does, but we can’t prove otherwise. I know that some stuff on alterations by deJong at Columbia has been used to chip away at some of them (positive char stuff isn’t my subject, so I’m working from heresay mostly) and stronger statements can be obtained with full resolution.

Of course, there’s also the fact that it’s been an open problem for decades, and so a proof would most likely involve powerful new techniques that could be used to solve problems.

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Peter,

It is a vast overstatement to say that Hironaka is claiming to have a proof that singularities of algebraic varieties can be resolved for any dimension in characteristic p. He outlined a new approach at Harvard, but made no claims that he solved the problem.

In Argentina Orlando Villamayor of the University of Madrid recently gave a very nice talk in which he claimed he had reduced the problem to some very special cases. He will also speak at the Harvard meeting.

And there are some others who are claiming progress.

Thanks Terry,

Were you at Hironaka’s talk and do you know exactly what he claimed? The story some people at Columbia were told was that there had been a claimed proof, but our informant from Harvard was not an expert on the subject, and may have missed something about exactly what Hironaka was claiming.

Peter,

I didn’t attend Hironaka’s talk. My information comes directly from a Harvard senior mathematics professor who did attend. I don’t know any of the details of Hironaka’s new approach.

Terry,

Thanks. My info also comes from a Harvard senior mathematics professor….

If any of my readers did attend the talk or know someone who did, I hope they can help us resolve this disagreement amongst the Harvard faculty…

Recently people in that subject have started to think that the proof of characteristic p resolution is imminent, so I think people are sort of jumping the gun staking their claims, and their exact wording might affect the impression they give. I suggest waiting to see what happens before making these speculations.

Peter -

That’s hilarious. I sent my Harvard math department informant an email asking about the inconsistency, but I haven’t heard back yet. My informant has made considerable contributions to algebraic geometry, so he very much knows what he’s talking about in that field. But he is not a singularity expert as such.

I’ll keep you apprised!

O, and my informant did attend the Hironaka talk.

I don’t think it makes much difference until there’s a paper. About a decade ago one of Hironaka’s students gave a lecture series at Harvard on his proof of resolution in positive characteristic. (or approach-to-proof or however it was described). It sounded promising. But the details didn’t hold up and the claim was eventually withdrawn.

Hironaka may well have a proof, a usable strategy, or neither. Resolution is a very technical problem and, unlike with Wiles or Perelman’s proofs, I don’t think a quick provisional determination of correctness would necessarily be easy to reach.

Hi Peter,

My Harvard math department informant has responded, and he’s pretty certain:

“Actually, I cannot imagine any of my colleagues spreading the word that Hironaka had solved the problem. Movement on it is in the air, but it’s not all by Hironaka, nor is the program complete.”

My informant has been to the talks and he definitely walks the algebraic geometry walk! I’m quite confident in him, for whatever that is worth.