The generalized Bogomolov–Gieseker conjecture for tilt-stable objects on threefolds

Let $X$ be a smooth projective threefold. Let $u_{\omega,B}$ be the tilt slope on the tilted heart $\operatorname{Coh}^{\omega,B}(X)$, where $\omega$ is ample and $B$ is a real divisor class. An object $E\in\operatorname{Coh}^{\omega,B}(X)$ is $\nu_{\omega,B}$-semistable if it satisfies the tilt-stability condition with respect to this slope. The generalized Bogomolov–Gieseker conjecture. For any $\nu_{\omega,B}$-semistable object $E\in\operatorname{Coh}^{\omega,B}(X)$ satisfying $\nu_{\omega,B}(E)=0$, one has

$$\operatorname{ch}_3^B(E)\leq \frac{\omega^2}{18}\operatorname{ch}_1^B(E).$$

This conjecture is the main inequality needed to construct Bridgeland stability conditions on threefolds. It is known in several important cases, including the abelian and Calabi–Yau threefolds of abelian type treated in the paper, but is not known for arbitrary smooth projective threefolds.