Strong Bogomolov–Gieseker inequality for stable sheaves at minimal first Chern class

Let $X$ be a smooth projective threefold, let $B,\omega\in\operatorname{NS}_{\mathbb Q}(X)$ with $\omega$ ample, and let $c$ be the minimum positive value of $\omega^2\operatorname{ch}_1^B(F)$ for objects $F\in\mathcal B_{\omega,B}$. Let $E$ be a $\mu_{\omega,B}$-stable sheaf satisfying $\omega^2\operatorname{ch}_1^B(E)=c$ and

$$\frac{\omega\operatorname{ch}_2^B(E)}{\omega^3\operatorname{ch}_0^B(E)}=\frac16.$$

Stable-sheaf inequality conjecture. Then

$$\frac{\operatorname{ch}_3^B(E)}{\omega^2\operatorname{ch}_1^B(E)}\leq\frac1{18}.$$

This is a special case of the strong Bogomolov–Gieseker conjecture for tilt-stable objects. The paper presents it as a consequence of that conjecture rather than proving it.