Kimura — Chow Groups Are Finite Dimensional, In Some Sense (2005)

When S is a surface with pg(S){\(>\)}0, Mumford proved that its Chow group A*S is not ``finite dimensional’’ in some sense. In this paper, we propose another definition of ``finite dimensionality’’ for the Chow groups. Using this new definition, at least the Chow group of some surface S with pg(S){\(>\)}0 (for example, the product of two curves) becomes finite dimensional. The finite dimensionality of the Chow groups follows from the finite dimensionality of the Chow motives. It turns out that the finite dimensionality of the Chow motives is a very strong property. For example, we can prove Bloch’s conjecture (representability of the Chow groups of surfaces with pg(S)=0) under the assumption that the Chow motive of S is finite dimensional.