Motivic decomposition of a compactification of a Merkurjev-Suslin variety

Abstract

We provide a motivic decomposition of a twisted form of a smooth hyperplane section of Gr(3,6). This variety is a norm variety corresponding to a symbol in K M 3 /3. As an application we construct a torsion element in the Chow group of this variety.

Motivic decomposition of compactifications of certain group varieties

“…Any smooth compactification of E is a norm variety of the element s := [D] ∪(c) ∈ H 3 et (F, Z/pZ (2)). It has been shown by N. Semenov in [26] for p = 3 (and char F = 0) that the motive of a certain smooth equivariant compactification of E decomposes in a direct sum, where one of the summands is the Rost motive of s, another summand is a motive ε vanishing over any field extension of F splitting D, and each of the remaining summands is a shift of the motive of the Severi-Brauer variety of D. All these summands (but ε) are indecomposable and ε was expected to be 0.…”

“…We note that the compactification in [26] (for p = 3) has the property required in Theorem 1.1 (see Example 10.6).…”

“…Example 10.6. Let X be the (non-toroidal) equivariant compactification of SL 1 (D) with p = 3 considered in [26]. Since P X (t) = t 8 + t 7 + 2t 6 + 3t 5 + 4t 4 + 3t 3 + 2t 2 + t + 1, we have The Chow group CH >0 (R) has been computed in [19, Appendix RM] (the characteristic assumption is needed here).…”

Integrality of the Chern character in small codimension

“…This can also be used to obtain informations on gr K ′ 0 (X). For example, a direct consequence of Proposition 8.3 is that gr 2 K ′ 0 (D) is contains non-zero 3-torsion, where D is the variety of [Sem08] (over a field of characteristic zero).…”

Special correspondences and Chow traces of Landweber-Novikov operations

“…1.5 Example. An example of a non-homogeneous variety which possesses a special correspondence for p = 3 was provided recently by N. Semenov (see [Se08]). By Theorem 1.3 it also provides an example of a 4-splitting variety.…”