Motivic periods and Grothendieck arithmetic invariants (Q2007742)

Let $X$ be a separated scheme of finite type over a field $K\subseteq \mathbb{C}$. If $X$ is smooth projective, the classical period conjecture of Grothendieck asserts that if $K=\overline{\mathbb{Q}}$, the cycle map \[ Z^k(X)_{\mathbb{Q}}\to H^{2k}_{dR}(X) \] is surjective over $H^{2k}(X_{\mathrm{an}},\mathbb{Q}(k))\cap H^{2k}_{dR}(X)$. In other words, that a cohomology class is algebraic if and only if it comes from an algebraic cycle. A detailed history of the conjecture can be found in [\textit{J. Ayoub}, Eur. Math. Soc. Newsl. 91, 12--18 (2014; Zbl 1306.14006)] and [\textit{J.-B. Bost} and \textit{F. Charles}, J. Reine Angew. Math. 714, 175--208 (2016; Zbl 1337.14009)]. In the paper under review, the authors formulate an analogue period conjecture for the étale motivic cohomology, removing the assumptions on $X$. Ayoub's period isomorphism in Voevodsky's motivic category $\mathbf{DM}^{\text{eff}}_{\text{ét}}$ [\textit{J. Ayoub}, J. Reine Angew. Math. 693, 1--149 (2014; Zbl 1299.14020)] induces, for any scheme $X$, the isomorphism \[ \varpi^{p,q}_X : H^p(X_{\mathrm{an}},\mathbb{Z}_{\mathrm{an}}(q)) \otimes_\mathbb{Z} \mathbb{C} \to H^p_{dR}(X) \otimes_K \mathbb{C}, \] where $\mathbb{Z}_{\mathrm{an}}(\bullet)$ is the motivic complex of the analytic category $\mathbf{DM}^{\text{eff}}_{\mathrm{an}}$, which, following Ayoub [loc. cit.], computes Betti cohomology. The authors consider the following arithmetic invariant \[ H^{p,q}_\varpi(X):=H^p_{dR}(X) \cap H^p(X_{\mathrm{an}},\mathbb{Z}_{\mathrm{an}}(q)) \subseteq H^p(X_{\mathrm{an}},\mathbb{Z}_{\mathrm{an}}(q)) \] and construct a regulator map from the étale motivic cohomology \[ r^{pq}:H^{p,q}(X):=H^p_{\text{éh}}(X,\mathbb{Z}(q))\to H^{p,q}_\varpi(X). \] In this context the analogue of Grothendieck's period conjecture asserts that if $K=\overline{\mathbb{Q}}$, the regulator $r^{p,q}$ is surjective. The main result of this paper is the proof of the latter conjecture in the case $p=1$ and all $q$. In order to attack it, the authors rivisit the definitions in terms of 1-motives and make use of the description of $H^1$ via the motivic Albanese map. The main step becomes showing that a realizaion of 1-motives in a period category, the Betti-de Rham realization, is fully faithful. In the appendix, some divisibility properties of motivic cohomology are proved and used to link this conjecture with the classical period conjecture of Grothendieck for $X$ smooth and projective.