Some remarks concerning the Grothendieck period conjecture

Abstract

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. Let ℚ ¯ ${\overline{\mathbb {Q}}}$ be the algebraic closure of ℚ in ℂ, let X be a smooth projective variety over ℚ ¯ ${\overline{\mathbb {Q}}}$ and let X ℂ an $X^{\rm an}_\mathbb {C}$ denote the compact complex analytic manifold that it defines. The Grothendieck period conjecture in codimension k on X, denoted GPC k ( X ) $\mathrm {GPC}^k(X)$ , asserts that any class α in the algebraic de Rham cohomology group H dR 2 k ( X / ℚ ¯ ) $H^{2k}_{{\rm dR}}(X/{\overline{\mathbb {Q}}})$ of X over ℚ ¯ ${\overline{\mathbb {Q}}}$ such that 1 ( 2 π i ) k ∫ γ α ∈ ℚ $ \frac{1}{(2\pi i)^k} \int _\gamma \alpha \in \mathbb {Q}$ for every rational homology class γ in H 2 k ( X ℂ an , ℚ ) $H_{2k}(X^{\rm an}_\mathbb {C}, \mathbb {Q})$ is the class in algebraic de Rham cohomology of some algebraic cycle of codimension k in X, with rational coefficients. We notably establish that GPC 1 ( X ) $\mathrm {GPC}^1(X)$ holds when X is a product of curves, of abelian varieties, and of K3 surfaces, and that GPC 2 ( X ) $\mathrm {GPC}^2(X)$ holds for a smooth cubic hypersurface X in ℙ ℚ ¯ 5 $\mathbb {P}^5_{\overline{\mathbb {Q}}}$ . We also discuss the conjectural relationship of Grothendieck classes with the weight filtration on cohomology.