The modified generalized Bogomolov–Gieseker conjecture

Let $(X,H)$ be a smooth complex projective polarized variety. Let $\gamma=\gamma_{X,H}$ be a class as in the source's definition of $\gamma$, and let $f=f_{X,H}^\gamma\colon\mathbb{R}\to\mathbb{R}$ be an upper semi-continuous function. The $\gamma$-generalized Bogomolov–Gieseker inequality is the condition that the quadratic inequality

$$\overline{Q}_{\alpha,\beta}(v_H^\gamma(E))\geq 0$$

holds for every $\nu^\gamma_{H,\alpha,\beta}$-semistable object $E$ in $\operatorname{Coh}^\gamma_{H,\beta}(X)$. The modified generalized Bogomolov–Gieseker conjecture. There exist such $\gamma$ and $f$ for which $(X,H)$ satisfies the $\gamma$-generalized BG inequality for every $(\alpha,\beta)\in\mathbb{R}^2$ with $\alpha>f(\beta)$. This modification is proposed after the original conjecture was disproved and is intended to recover a sufficiently broad region in which the inequality yields Bridgeland stability conditions.