Piyaratne–Toda refined Bogomolov–Gieseker inequality conjecture

Fix real numbers $\alpha_0>0$ and $\beta_0$, and let $E\in\operatorname{Coh}^{\alpha_0\omega,B+\beta_0\omega}(X)$ be a $\nu_{\alpha_0\omega,B+\beta_0\omega}$-semistable object. Define

$$\bar\beta(E):=\frac{2v^B_2(E)}{v^B_1(E)+\sqrt{\overline{\Delta}_{\omega,B}(E)}}.$$

Call $E$ $\bar\beta$-stable if it remains $\nu_{\alpha\omega,B+\beta\omega}$-stable for all $(\alpha,\beta)$ in an appropriate open neighborhood of $(0,\bar\beta(E))$ with $\alpha>0$. Piyaratne–Toda refined Bogomolov–Gieseker inequality conjecture. If $E$ is a $\bar\beta$-stable object, then

$$\operatorname{ch}^{B+\bar\beta(E)\omega}_3(E)\leq 0.$$

This is a reduction form of the BG-type inequality: the source introduces it as the first reduction, and it is intended to establish the preceding conjectural inequality by reducing to objects stable near the boundary point $(0,\bar\beta(E))$.