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Motivic periods and Grothendieck arithmetic invariants. (English) Zbl 1433.14017
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 [J. Ayoub, Eur. Math. Soc. Newsl. 91, 12–18 (2014; Zbl 1306.14006)] and [J.-B. Bost and 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 \(\mathbf{DM}^{\text{eff}}_{\text{ét}}\) [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.
MSC:
14F42 Motivic cohomology; motivic homotopy theory
14F40 de Rham cohomology and algebraic geometry
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14L15 Group schemes
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