Incompressibility of Products of Pseudo-homogeneous Varieties

We show that the conjectural criterion of p -incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties which includes the norm varieties associated to symbols in Galois cohomology of arbitrary degree. Let F be a field. A smooth complete irreducible F -variety X is incompressible, if every rational self-map X 99K X is dominant. This means that cdimX = dimX, where the canonical dimension cdimX is defined as the minimum of dimY for Y running over closed irreducible subvarieties of X admitting a rational map X 99K Y . The notion of canonical dimension has been originally introduced in [1]. For the whole exposition, let p be a fixed prime number. Canonical p-dimension cdimp X – the p -local version of cdimX – is defined as the minimum of dimY for Y running over closed irreducible subvarieties of X admitting a correspondence X p′ Y of degree 0 and of p -prime multiplicity. For arbitrary F -varieties X, Y with irreducible X, a correspondence X Y of degree 0 is an element of the Chow group CHd(X × Y ), where d := dimX (cf. [3, §62]); its multiplicity is its image under the push-forward homomorphism CHd(X × Y ) → CHd X = Z with respect to the projection X × Y → X. We refer to [3, §75] for basic properties of the multiplicity of a correspondence. The notion of canonical p -dimension has been originally introduced in [12]. We refer to [7] and [18, §4] for motivation, history, and general discussion of canonical (p -)dimension. The variety X is p-incompressible if every self-correspondence X p′ X of degree 0 and of p -prime multiplicity is “dominant”, i.e., if cdimpX = dimX. The rational equivalence class of the closure of the graph of a rational map is a correspondence of degree 0 and of multiplicity 1; therefore we always have cdimpX ≤ cdimX ≤ dimX. In particular, a variety that is p -incompressible (for at least one p) is incompressible. Studying canonical p -dimension, it is more appropriate to use the Chow group Ch with coefficients in F := Fp := Z/pZ rather than the Chow group CH with integer coefficients. Multiplicities of correspondences as well as degrees of classes of 0-cycles take then values in F. We also consider the Chow motives with coefficients in F, see [3, Chapter XII]. Date: 11 November 2015. Revised: 20 April 2016.