On the injectivity of cycle maps (Q1318882)

For \(X\) a smooth complex projective variety and \(D^{go}\) the bounded derived category of mixed Hodge modules of geometric origin, as previously defined by the author [in: Complex geometry and Lie theory, Proc. Symp., Sundance 1989, Proc. Symp. Pure Math. 53, 283-303 (1991; Zbl 0761.14002) and in: Algebraic Geometry, Proc. US-USSSR Sympos., Chicago 1989, Lect. Notes Math. 1479, 196-215 (1991; Zbl 0761.14003)] let \(H^{2p}_{MH} (X, \mathbb{Q} (p))\) be canonically defined by \(\text{Ext}^{2p}_{D^{go}}\) of \(\mathbb{Q}_X\) and Tate twist. There is a cycle map CH\(^p (X)_\mathbb{Q} \to H^{2p}_{MH} (X, \mathbb{Q} (p))\). In this paper the injectivity is detected by a factorisation of the cycle map via mixed Hodge modules of geometric level \(\leq \text{dim} X\) and reduced to the surjectivity of some natural morphisms associated to smooth subvarieties. The surjectivity is a consequence of the Hodge conjectures and the surjectivity of a cycle map from Bloch's higher Chow groups. The author claims that the injectivity of this cycle map would imply Bloch's conjecture.