Torsion motives

TL;DR

This paper investigates torsion Chow motives, introduces invariants like the p-level, and explores their implications on the motives' structure and cohomology, with detailed analysis of surface motives such as Godeaux torsion motives.

Contribution

It introduces the p-level invariant for torsion motives and establishes its bounds and restrictions, providing a new perspective on their structure and cohomological properties.

Findings

p-level bounds the dimension of the variety

Restrictions on motivic and singular cohomology

Classification of p-torsion motives of surfaces

Abstract

In this paper we study Chow motives whose identity map is killed by a natural number. Examples of such objects were constructed by Gorchinskiy-Orlov. We introduce various invariants of torsion motives, in particular, the $p$-level. We show that this invariant bounds from below the dimension of the variety a torsion motive $M$ is a direct summand of and imposes restrictions on motivic and singular cohomology of $M$. We study in more details the $p$-torsion motives of surfaces, in particular, the Godeaux torsion motive. We show that such motives are in 1-to-1 correspondence with certain Rost cycle submodules of free modules over $H^*_{et}$. This description is parallel to that of mod-$p$ reduced motives of curves.