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.