On isotropic and numerical equivalence of cycles

Abstract

We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with F p -coefficients). This conjecture is essential for understanding the structure of the isotropic motivic category and that of the tensor triangulated spectrum of Voevodsky category of motives. We prove the conjecture for the new range of cases. In particular, we show that, for a given variety X, it holds for sufficiently large primes p. We also prove the p-adic analogue. This permits to interpret integral numerically trivial classes in CH(X) as p ∞ -anisotropic ones. Isotropic Chow groupsEverywhere below k will be a field of characteristic zero.Let n ∈ N and Ch * = CH * /n be Chow groups modulo n.Definition 2.1 Let n ∈ N and X be a scheme of finite type over k. We say that X is n-anisotropic, if degrees of all closed points on X are divisible by n.Definition 2.2 Let n ∈ N, X be a scheme over k and x ∈ CH r (X). Then x is n-anisotropic, if there exists a proper map f : Y → X from an n-anisotropic scheme Y and y ∈ CH r (Y ) such that f * (y) = x. Now we can introduce isotropic Chow groups:Ch k/k := Ch /( n-anisotropic classes ).