# Known norm varieties and the Bloch-Kato conjecture

The Bloch-Kato conjecture states that

$K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$.

A important part in the proof of the Bloch-Kato conjecture is to proof the existence of norm varieties. In

M. Rost, Norm varieties and algebraic cobordism, Proc. of the Int. Congress of Math., vol. 2 (Beijing, 2002), 77{85, Higher Ed. Press, Beijing, 2002

there are examples for norm varieties for special cases of $n,l$. For example if $l=2$ and any $n$,we write $BK(\infty,2)$, we get the Milnor conjecture and can use Pfister-Forms or better said their varieties.

1.-What newer examples of norm varieties are known?

2.-Is there hope to find a whole class of varieties,like in the case of quadratic forms,that fulfill a certain $BK(n,l)$? Fulfilling $BK(n,l)$ is also classifying feature though.

3.In the theory of quadratic forms we have the Witt-ring and the graded Witt-ring,which is also isomorphic to the galois cohomology ring. Knowing that,the Arason-Pfister Hauptsatz gives a lower bound for the rank of forms isomorphic to symbols contained in $H^n(k,\mu^{\otimes n}_l)$. Therefore it would be plausible for a general $l$ that norm varieties represented by symbols in $H^n(k,\mu^{\otimes n}_l)$ have bigger dimension the bigger $n$ is.Wouldnt it?

That whole proof of BK is hard to grasp for me at the moment.So it might be that my questions are to vage and im overrating the possibilities of current motivic research by far.

In the paper of Rost you mentioned, there are not just examples of norm varieties - there is an outline of how to construct norm varieties in general. The preprint version of Rost's article can be found following this link. Note that in that very same article, on page 2, you find Voevodsky's characterization of norm varieties. Part of this characterization is $\dim X=p^{n-1}-1$, answering your third question. In Section 3 of the paper, Rost gives a sketch of a proof that norm varieties exist for all symbols in $K^M_n/p$, for any $p,n$. Therefore, the positive answer to your second question is also in Rost's paper. A detailed proof of the existence of norm varieties can be found in A. Suslin, J. Joukhovitski, JPAA 206, (2006), 245--276. (Note that the proof that the varieties constructed are norm varieties in weight $n$ depends on the Bloch-Kato conjecture in weight $\leq n-1$.) All the above can be found in the article on norm varieties on Wikipedia.