Some remarks concerning the Grothendieck Period Conjecture

TL;DR

This paper explores the Grothendieck period conjecture, providing new evidence and confirming its validity in specific cases such as products of curves, abelian varieties, K3 surfaces, and smooth cubic fourfolds.

Contribution

It establishes the conjecture in degree 1 for certain varieties and in degree 2 for smooth cubic fourfolds, advancing understanding of transcendence properties.

Findings

Grothendieck period conjecture holds in degree 1 for products of curves, abelian varieties, and K3 surfaces.

The conjecture holds in degree 2 for smooth cubic fourfolds.

Provides new evidence supporting the conjecture's validity in specific geometric cases.

Abstract

We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich-Zagier concerning transcendence properties of the torsors of periods of varieties over number fields. We notably establish that the Grothendieck period conjecture holds in degree 1 for products of curves, of abelian varieties, and of K3 surfaces, and that it holds in degree 2 for smooth cubic fourfolds.