Higher Chow groups and the Hodge-\({\mathcal D}\)-conjecture (Q1816475)

Let \(X\) be a complex quasi-projective variety, and let \(\text{CH}(X,m)\) denote Bloch's higher Chow groups of \(X\). In this paper, the author investigates a possible relationship between mixed Hodge structure of \(X\) and Chow groups. As a basic invariant of the latter, he introduces the notion of ``level'' analogous to that for the former. Then, under some assumptions including ``the hard Lefschetz conjecture'', he proves that \[ \text{Level} (N^{k-\ell} H^{2k- \ell-m} (X,\mathbb{Q}))= \ell-m\quad \Rightarrow\quad\text{Level(CH}^k (X,m)) \geq\ell-m. \] He points out that the inequality \[ \text{Level} (H^*(X)) \leq\text{Level(CH}^* (X,m)_\mathbb{Q}) \] holds under the same hypothesis plus the general Hodge conjecture, and conjectures \[ \text{Level} (H^*(X)) =\text{Level(CH}^* (X,m)_\mathbb{Q}) \] holds for all \(m\geq 0\). Moreover, he shows that this conjecture holds for any general hypersurfaces of degree \(d\) in \(\mathbb{P}^{n+1}\) satisfying the numerical condition \(k(n+2-k) +1- {k+d \choose d} \geq 0\) where \(k= [(n+1)/d]\).