ℓ-independent Lefschetz standard conjecture in degree i

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, let $h\in \operatorname{CH}^1(X)$ be the first Chern class of an ample line bundle, let $0\leq i\leq n$, and for every prime $\ell\neq\operatorname{char}(k)$ define

$$L_{\ell}^{n-i}=(-)\cup h^{n-i}:\mathrm{H}^i_{\mathrm{\acute{e}t}}(X,\mathbf{Q}_{\ell})\to\mathrm{H}^{2n-i}_{\mathrm{\acute{e}t}}(X,\mathbf{Q}_{\ell}).$$

ℓ-independent Lefschetz standard conjecture in degree $i$. For $X$, $h$, and $i$ as above, there exists a codimension $i$ cycle $\lambda\in \operatorname{CH}^i(X\times X)_{\mathbf{Q}}$ such that for every prime $\ell\neq\operatorname{char}(k)$ the induced map

$$\lambda^*:\mathrm{H}_{\mathrm{\acute{e}t}}^{2n-i}(X,\mathbf{Q}_{\ell})\to\mathrm{H}_{\mathrm{\acute{e}t}}^{i}(X,\mathbf{Q}_{\ell})$$

is the inverse of $L_{\ell}^{n-i}$.

This is the étale-cohomological analogue of the Lefschetz standard conjecture and supplies the motivic input for the paper's results over arbitrary algebraically closed fields. Its general status is open.