The effective Bogomolov conjecture for tori and abelian varieties

Let $G$ be a torus or an abelian variety, and let $\hat h$ be a canonical height associated with an ample line bundle on a compactification of $G$.

Effective Bogomolov conjecture. There is a real number $c>0$ such that every proper closed irreducible subvariety $X$ of $G$ that is not contained in a translate of a proper algebraic subgroup of $G$ satisfies

$$\hat\mu(X)\geq c\,(\deg(X))^{-1/\operatorname{codim}(X)}.$$

This is the expected optimal degree dependence for an effective lower bound on the essential minimum. The source records the bound for tori up to an arbitrarily small loss in the exponent and presents the displayed statement as the natural conjectural extension to abelian, and possibly semi-abelian, varieties.