## 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.

## James Milne said

Actually, I’m not sure it is an error. It is true that I only prove that the Hodge conjecture for CM abelian varieties implies the standard conjectures for abelian varieties over finite fields. However, 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.

J.S. Milne

## Update on Harada’s proof « Motivic stuff said

[…] by homotopical 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 […]