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.