Incompressibility of Products

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. We identify the properties of projective homogeneous varieties actually needed for the proof to go through. For instance, generically split (nonhomogeneous) varieties also satisfy these properties. 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 . For the whole exposition, let p be a fixed prime number. Canonical p-dimension cdimpX is defined as the minimum of dimY for Y running over closed irreducible subvarieties of X admitting a degree 0 correspondence X p Y of p-prime multiplicity. The variety X is p-incompressible, if every degree 0 self-correspondence X p X of p-prime multiplicity is dominant, i.e., if cdimpX = dimX . The closure of the graph of a rational map is a degree 0 correspondence of multiplicity 1; therefore a p-incompressible (for at least one p) variety is incompressible. Studying canonical p-dimension, instead of the integral Chow group CH, it is more appropriate to use the Chow group Ch with coefficients in Fp := Z/pZ. Multiplicities of correspondences as well as degrees of 0-cycles take then values in Fp. We also consider the Chow motives with coefficients in Fp, see [2, Chapter XII]. Now we are going to introduce a class of varieties, called nice here, for which we can prove that the following criterion holds (see Theorem 9): the product X × Y of F varieties X and Y is p-incompressible if and only if the varieties XF (Y ) and YF (X) are p-incompressible. A smooth complete variety is split, if its motive decomposes into a finite direct sum of Tate motives. By Tate motive, we mean an arbitrary shift of the motive of the point SpecF . For instance, an (absolutely) cellular variety is split, [2, Corollary 66.4]. A smooth complete variety X is nice, if it has the following three properties: (i) The variety X is geometrically split, that is, there exists a field extension L/F such that the L-variety XL is split. Date: January 31, 2015.