Fully supersingular Bloch--Beilinson conjecture

Let $X$ be a smooth projective variety over $k$ which is fully Shioda supersingular. For $0\leq i\leq \dim(X)$, write ${\rm CH}^{i}(X)_{\operatorname{alg}}$ for algebraically trivial codimension-$i$ cycles modulo rational equivalence, and let $Ab^{i}(X)$ denote an abelian variety.

Fully supersingular Bloch--Beilinson conjecture. Numerical and algebraic equivalence coincide on ${\rm CH}^{i}(X)_{\mathbb{Q}}$, so $\operatorname{Griff}^{i}(X)_{\mathbb{Q}}=0$; there is a regular surjective homomorphism

$$\nu_{i}:{\rm CH}^{i}(X)_{\operatorname{alg}}\to Ab^{i}(X)$$

to the algebraic representative, with finite kernel, $\dim Ab^{i}(X)=\frac{1}{2}b_{2i-1}(X)$, and $Ab^{i}(X)$ supersingular; and the intersection product on ${\rm CH}^{*}(X)_{\operatorname{alg}}$ is zero. Consequently, the kernel of

$${\rm CH}^{*}(X)_{\mathbb{Q}}\twoheadrightarrow \overline{\rm CH}^{*}(X)_{\mathbb{Q}}$$

is the square-zero graded ideal $Ab^{*}(X)_{\mathbb{Q}}:=\bigoplus_i Ab^{i}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$.

This conjecture predicts a precise Bloch--Beilinson-type description of Chow groups for fully supersingular varieties. The paper derives it for symplectic varieties from the supersingular abelian motive conjecture, but does not establish that motivic conjecture in general.