Functorial Bogomolov-type lower bound for transverse subvarieties

Let $(A,\mathcal{L})$ be a polarized abelian variety of dimension $g$ defined over a number field $\mathbb{K}$, and let $X\subset A$ be a transverse subvariety with finite stabilizer. Let $\psi:A\to A$ be an isogeny. Write $\mu_{\psi^*\mathcal{L}}(X)$ for the essential minimum associated with the pulled-back polarization, and let $\operatorname{cod}X$ denote the codimension of $X$. Then Functorial Bogomolov-type bound.

$$\mu_{\psi^*\mathcal{L}}(X)>c\bigl(g,[\mathbb{K}:\mathbb{Q}],h_{\mathcal{L}}(A)\bigr)\min_{\eta'=\pm\eta}\left(\frac{\deg_{\psi^*\mathcal{L}}A}{\deg_{\psi^*\mathcal{L}}X}\right)^{\frac{1}{2\operatorname{cod}X}+\eta'}.$$

Here $c(g,[\mathbb{K}:\mathbb{Q}],h_{\mathcal{L}}(A))$ depends only on $g$, $[\mathbb{K}:\mathbb{Q}]$, and $h_{\mathcal{L}}(A)$. The bound is proposed as a functorial effective lower bound for the essential minimum, uniform under isogenies and depending on the polarization through the displayed degrees.