Motivic periods and Grothendieck arithmetic invariants

TL;DR

This paper introduces a period regulator for motivic cohomology, formulates a conjecture relating it to algebraic cycles, and verifies it in specific cases, extending classical conjectures in algebraic geometry.

Contribution

It constructs a new period regulator for motivic cohomology and formulates a conjecture on its surjectivity, verified in several cases, generalizing Grothendieck's period conjecture.

Findings

Verification of the period conjecture in certain cases.

Establishment of a fully faithful Betti--de Rham realization for 1-motives.

Extension of divisibility properties of motivic cohomology.

Abstract

We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Th\'el\`ene--Raskind).