A brief note concerning hard Lefschetz for Chow groups (Q2788798)

Let \(X\) be a smooth projective variety over \(\mathbb{C}\) of dimension \(n\), equipped with an ample line bundle \(L\). Let \(A^jX_{\mathbb{Q}}\) denote the Chow group of codimension \(j\) algebraic cycles with \(\mathbb{Q}\) coefficients, and \(A_{AJ}^jX_{\mathbb{Q}}\) the subgroup of Abel-Jacobi trivial cycles. The main result of this paper shows that the ``hard Lefschetz property'' can be proved in some special cases. More precisely, the author proves that the map \(L^r:A_{AJ}^jX_{\mathbb{Q}}\rightarrow A_{AJ}^{j+r}X_{\mathbb{Q}}\) is injective for \(j\leq r+1\), and \(L^r:A^jX_{\mathbb{Q}}\rightarrow A^{j+r}X_{\mathbb{Q}}\) is surjective for \(j>n-2r\), under the assumption that (0) the Voisin standard conjecture holds, (1) either the motive of \(X\) is finite-dimensional, or the Griffith group \(\mathrm{Griff}^n(X\times X)_{\mathbb{Q}}\) vanishes, (2) the Lefschetz standard conjecture \(B(X)\) holds, and (3) \(H^i(X,\mathbb{Q})=N^rH^i(X,\mathbb{Q})\) for all \(i\in [n-r+1,n]\).