Bayer–Macrì–Toda generalized Bogomolov–Gieseker inequality for tilt-stable objects

Let $(X,H)$ be a polarized smooth projective $3$-fold. For $b\bitin\mathbb{R}$ and $a>0$, let $\sigma_{a,b}=(\mathcal{A}^b,Z_{a,b})$ be the tilt-stability condition, and let $\Delta_H(E)$ denote the discriminant

$$\Delta_H(E)=\big(H^2\operatorname{ch}_1(E)\big)^2-2H^3\operatorname{ch}_0(E)H\operatorname{ch}_2(E).$$

For an object $E$, write $\operatorname{ch}^{bH}(E)=e^{-bH}\operatorname{ch}(E)$. Bayer–Macrì–Toda generalized Bogomolov–Gieseker inequality. If $E$ is $\sigma_{a,b}$-semistable, then

$$Q_{a,b}(E):=a^2\Delta_H(E)+4\big(H\operatorname{ch}_2^{bH}(E)\big)^2-6\big(H^2\operatorname{ch}_1^{bH}(E)\big)\operatorname{ch}_3^{bH}(E)\geq 0.$$

This inequality was proposed as a higher-dimensional analogue of the classical Bogomolov–Gieseker inequality and is intended to support the construction of Bridgeland stability conditions on derived categories of threefolds. Its status is recorded here as open.