Existence of Norm Varieties

Abstract

This chapter constructs norm varieties for symbols ª = {𝑎1, ...,𝑎𝑛} over a field 𝑘 of characteristic 0, and starts the proof that norm varieties are Rost varieties. It first recalls the definition of a norm variety for a symbol ª in 𝐾𝑀 𝑛(𝑘)/𝓁; if 𝑛 ≥ 2 and 𝑘 is 𝓁-special, norm varieties are geometrically irreducible. Next, the chapter uses the Chain Lemma to produce a specific ν‎ n−1-variety ℙ(𝒜), and a pencil Q of splitting varieties over 𝔸1—{0} whose fibers 𝑄𝑊 are fixed point equivalent to ℙ (𝒜). Using a bordism result, this chapter shows that any equivariant resolution 𝑄(ª) of 𝑄𝑊 is a ν‎ n−1-variety. Next, one of Rost's degree formulas is used to show that any norm variety for ª is ν‎ n−1 because 𝑄(ª) is. Finally, a norm variety for ª is constructed by induction on 𝑛, making use of the global inductive assumption that BL(n − 1) holds.