Degree Formulas

Abstract

This chapter uses algebraic cobordism to establish some degree formulas. It presents δ‎ as a function from a class of smooth projective varieties over a field 𝑘 to some abelian group. Here, a degree formula for δ‎ is a formula relating δ‎(𝑋), δ‎(𝑌), and deg(𝑓) for any generically finite map 𝑓 : 𝑌 → 𝑋 in this class. The formula is usually δ‎(𝑌)=deg(𝑓)δ‎(𝑋). These degree formulas are used to prove that any norm variety over 𝑘 is a ν‎ n−1-variety. Using a standard result for the complex bordism ring 𝑀𝑈*, which uses a gluing argument of equivariant bordism theory, this chapter establishes Rost's DN (Degree and Norm Principle) Theorem for degrees, and defines the invariant η‎(𝑋/𝑆) of a pseudo-Galois cover.