The Bogomolov-type lower bound for essential minima

Let $X\subset G$ be irreducible and not a translate, and let $H$ be the irreducible translate of minimal dimension containing $X$. Write $\operatorname{codim}_H X$ for the codimension of $X$ in $H$. Bogomolov-type bound. For every real $\eta>0$, there exists an effective positive constant $c(G,\eta)$ such that

$$\mu(X)\geq c(G,\eta)\frac{(\deg H)^{\frac{1}{\operatorname{codim}_H X}-\eta}}{(\deg X)^{\frac{1}{\operatorname{codim}_H X}+\eta}}.$$

This is an effective version of the Bogomolov conjecture and is intended to provide height control needed for effective intersection results; its full generality remains open.