On log motives

TL;DR

This paper introduces categories of log motives and log mixed motives, establishing their properties, relations to existing categories, and discussing conjectures with verification in the case of curves.

Contribution

It defines new categories of log motives, proves their semisimplicity and Tannakian properties under certain conditions, and explores their realizations and conjectures.

Findings

Log motives form a semisimple abelian category if numerical and homological equivalences coincide.

Log mixed motives provide a new formulation for mixed motives.

Tate and Hodge conjectures are verified in the case of curves.

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 that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.