Beilinson--Bloch positivity conjectures

Let $X$ be a smooth projective variety of dimension $d$ over a global field, and let $L$ be the Lefschetz operator induced by an ample line bundle. Write $\operatorname{CH}^p_{\mathrm{hom}}(X)_{\mathbb{Q}}$ for the rational homologically trivial codimension-$p$ Chow group, and let $\langle\,\ \rangle_X$ denote the Beilinson--Bloch height pairing.

Beilinson--Bloch positivity conjectures. For $p\leq \frac{d+1}{2}$, the map

$$L^{d+1-2p}\colon \operatorname{CH}^p_{\mathrm{hom}}(X)_{\mathbb{Q}}\to \operatorname{CH}^{d+1-p}_{\mathrm{hom}}(X)_{\mathbb{Q}}$$

is an isomorphism. If $z\in \operatorname{CH}^p_{\mathrm{hom}}(X)_{\mathbb{Q}}\setminus\{0\}$ is primitive, meaning that $L^{d+2-2p}z=0$, then

$$(-1)^p\langle z,L^{d+1-2p}z\rangle_X>0.$$

These conjectures generalize the positivity of the Néron--Tate pairing on abelian varieties to Beilinson--Bloch height pairings on arbitrary smooth projective varieties. Their status is not resolved in the supplied text.