Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Archive for the ‘Harada’s proof of the standard conjectures’ Category

Updated version of Harada’s preprints

Posted by Andreas Holmstrom on March 16, 2009

New versions of both preprints, here and here, with some minor corrections.

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Update on Harada’s proof – no error after all

Posted by Andreas Holmstrom on February 4, 2009

In a previous post I wrote that there is a mistake in Masana Harada’s proof of the standard conjectures. Now it seems that I was wrong about this. As James Milne kindly points out in a comment,  his paper is indeed misquoted, but the argument of Harada is still valid, because, and I quote, “the Tate conjecture (including num=hom) implies that the category of motives over finite fields is generated by abelian varieties, and so the standard conjectures for abelian varieties over finite fields then implies it for all varieties over finite fields”.

Also, Harada posted a revised version of his second preprint a few days ago, fixing a mistake in the proof of Theorem 6.1. 

Apparently the proof of Harada builds on an unsuccessful attempt by Thomason to prove the Tate conjecture. Is there anyone who knows where to find a copy of the original preprint of Thomason?

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Error in the article of Harada

Posted by Andreas Holmstrom on January 10, 2009

I wrote earlier about the recent preprints of Harada, in which he claims to prove the standard conjectures of Grothendieck. Having asked some experts what they think about this, most seem sceptical, and my supervisor pointed out to me that there is at least one serious mistake in The Tate-Thomason conjecture. After Theorem 6.3 on page 19, he states that the Hodge conjecture for CM abelian varieties implies the standard conjecture of Hodge type for varieties of positive characteristics, and he gives a reference to Milne: Polarizations and Grothendieck’s Standard Conjectures. However, Milne only proves that the Hodge conjecture for CM abelian varieties implies the standard conjecture of Hodge type for abelian varieties, so the argument of Harada is not valid. Of course, if this is the only mistake in the article, his results would still be extremely interesting, but in any case it seems like the standard conjectures are certainly not proven in full.

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A proof of Grothendieck’s standard conjectures?!?

Posted by Andreas Holmstrom on December 11, 2008

Amazing news: Seven days ago, Masana Harada of Kyoto University posted a preprint on the K-theory archive claiming to prove Grothendieck’s standard conjectures on algebraic cycles. If this is true, it will no doubt be one of the major mathematical achievements of our time.

Who is Masana Harada? Well, he is assistant professor at Kyoto University. He was a student of Robert Thomason, getting his PhD from John Hopkins University in 1987. His list of reviewed publications on MathSciNet is very short (only four items), but presumable this list does not include articles written in Japanese?

The proof presented is contained in “The Tate-Thomason conjecture”, a 21-page preprint (number 919 on the K-theory archive), using results from another recent preprint by the same author (“Higher K-theory of algebraic curves”; number 915). The list of proof ingredients includes Deligne’s result on the Riemann hypothesis over finite fields, the results of Voevodsky-Rost (-Weibel-Suslin-Friedlander???) on the Milnor-Bloch-Kato conjecture, and some work of Kahn and Levine on Azumaya algebras. The whole article is full of homotopical machinery invented by people like Bousfield, Quillen and Thomason.

Although it is of course far too early to celebrate, I find it amazing that anyone dares to approach the standard conjectures! It will be exciting to see what comes out of all this.

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