Tuesday, November 18, 2008

Virtualization Forum 2008

After the morning seminar of vertex operator algebras in Komaba, I went to the Virtualization Forum 2008 held by VMWare.

I heard the talks of HP, NEC, and Hitachi. They distributed some handouts of the presentations as well as some "commercial bag" of VMWare. At the coffee break, I went to the panel session, to see a blade server of NEC as well as the desktop virtualization by DELL or HP. They did not explain how they were involved with the Xen virtualization or which one was the smartest virtualization machine.

There was also the annoucement of Xen Summit 2009 North America. http://www.xen.org/community/xensummit.html

(See, also 2008 Tokyo

Wednesday, November 05, 2008

IPMU Seminar: Conjectures on mixed motives

Sergey Gorchinsky had a 2-hour seminar from 10:30 - 12:30. It started from the Chow groups of smooth irreducible variety $X$ over a field $k$. This was a story on the algebraic cycles modulo rational equivalence on $Z^p (X)$. Then it was on the conjecture by Bloch in algebraic K-theory; if $h^{(0,2)} > 0$, then the kernel of the Albanese map $T (X) = 0$. In the 70's, it is known for $X$ NOT of general type (with $h^{0,2}=0$). He also explained that (by Beilinson?) for $X$-smooth projective surface over $\mathbb{C}$, suppose $\alpha \in CH^2 (X \times X)$ the Hodge realization $h (\alpha) = 0$. Then $\alpha_{\ast}|_{T (X)} = 0$. The remark was on the preservation of the filtration.

After a break, $X$ will be more generally an arbitrary variety over a field $k$. The corresponding categories are the de Rham, e'tale, and Betti (mixed motives like onions (by Bloch)). He then worked on the mixed motive version $\mathcal{MM}_k$ as a right tensor abelian $\mathbb{Q}$-linear category from $Vect_k$. Of course we assumed the vanishing conjecture of Beilinson-Soule' for the existence of the category of mixed motives. Well, he then functored $Vect_k$ to a triangulated category $DM_k$, which has a non-degenerated $t$-structure s.t. its heart is the image of the inclusion of $\mathcal{MM}_k$ to $DM_k$. The story came to the Chow motive $CHM_k$ to pure motive $\mathcal{PM}_k$. And there was a "believable" conjecture of Beilinson. Then Sergei showed some examples of smooth projective $X$ of Murre. Finally he explained the theorem of he and Guletskii of when $X$ is a smooth projective 3-folds over $\mathbb{C}$, then there exists a split (or decomposition?) of $M(X)$.

After the talk, I asked him whether there exists a relation between Kontsevich's motivic integration and the conjectural category of mixed (Tate) motives after Bloch-Kriz. He informed me of the paper by Lunts, which showed that there is not a homemorphism from a motivic measure category $M_k$ to the $K_0$ group of smooth scheme $X$. I am not sure about the case of "refined" motivic integration, which has the value not on the Grothendieck group, but rather a trianglated category.