Modified Bloch–Beilinson prediction via the niveau filtration

Let $X$ be an abelian variety of dimension $d$ over $\mathbf{C}$, let $\widetilde{N}^1H^d(X)$ be the first niveau-filtration step, and let $G^dCH_0(X)$ be the deepest piece of the divisor-intersection filtration. Suppose $f:X\to X$ is an automorphism such that the induced map

$$f_*:\frac{H^d(X)}{\widetilde{N}^1H^d(X)}\to\frac{H^d(X)}{\widetilde{N}^1H^d(X)}$$

is the identity. Modified Bloch–Beilinson prediction. The restriction of $f_*:CH_0(X)\to CH_0(X)$ induces the identity

$$f_*:G^dCH_0(X)\to G^dCH_0(X).$$

The modification replaces the top holomorphic-form quotient by the quotient through the niveau filtration; its validity is open in the stated generality.