On the Beilinson-Hodge conjecture for \(H^2\) and rational varieties (Q2391592)

The Beilinson-Hodge conjecture, in its current incarnation by the author and others, says that the Betti cycle class map, \[ \text{cl}_{m,m} : H^m_{\mathcal M}(X,{\mathbb Q}(m)) \to \Gamma\big(H^m(X,{\mathbb Q}(m))\big) := \hom_{\text{MHS}} ({\mathbb Q}(0),H^m(X,{\mathbb Q}(m)), \] is surjective, for any smooth complex algebraic variety \(X\), where \(H^m_{\mathcal M}(X,{\mathbb Q}(m))\) is motivic cohomology. The author proves the following main result (paraphrased): Theorem. Let \(X\) be smooth and connected, and let \(\overline{X}\) be a smooth compactification of \(X\). If the degree map from the Chow group of \(0\)-cycles, \(\deg : \text{CH}_0(\overline{X})\otimes {\mathbb Q} \to {\mathbb Q}\) is an isomorphism, then \[ \text{cl}_{2,2} : H^2_{\mathcal M}(X,{\mathbb Q}(2)) \to \Gamma\big(H^2(X,{\mathbb Q}(2))\big), \] is surjective. The proof is broken down into two steps: \(\bullet\) The fact that \(\deg : \text{CH}_0(\overline{X})\otimes {\mathbb Q} \to {\mathbb Q}\) is an isomorphism, implies that the motive of \(\overline{X}\) ``degenerates'', by a argument going back to Bloch and Colliot-Thélène. Using this, one can easily show that at the generic point \(\eta\) of \(X\), \[ \text{cl}_{2,2} : H^2_{\mathcal M}(\eta,{\mathbb Q}(2)) \to \Gamma\big(H^2(\eta,{\mathbb Q}(2))\big), \] is surjective. \(\bullet\) Using a Bloch-Ogus type spectral sequence construction, the author reduces the surjectivity of \[ \text{cl}_{2,2} : H^2_{\mathcal M}(X,{\mathbb Q}(2)) \to \Gamma\big(H^2(X,{\mathbb Q}(2))\big), \] to the case where \(X\) is replaced by \(\eta\), which follows from the assumptions of the main theorem.