On log motives

Abstract

We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to the latter being a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in the curve case.