The strengthened Bogomolov–Gieseker conjecture for tilt-semistable objects

Let $X$ be a smooth projective threefold and let $H\in\operatorname{NS}(X)$ be an ample class. For a real number $\beta$, write $\operatorname{ch}_i^\beta$ for the $\beta H$-twisted Chern character, and let $\overline{\Delta}_H(E)$ denote the $H$-discriminant. Let $\nu^H_{\alpha,\beta}$ be the rescaled tilt slope

$$\nu_{\alpha,\beta}^H(E)=\frac{H\operatorname{ch}_2^\beta(E)-\frac{1}{2}\alpha^2H^3\operatorname{ch}_0^\beta(E)}{H^2\operatorname{ch}_1^\beta(E)}.$$

The strengthened Bogomolov–Gieseker conjecture. If $E$ is $\nu^H_{\alpha,\beta}$-semistable, then

$$\alpha^2\overline{\Delta}_H(E)+4\left(H\operatorname{ch}_2^\beta(E)\right)^2-6H^2\operatorname{ch}_1^\beta(E)\operatorname{ch}_3^\beta(E)\geq 0.$$

This strengthens the zero-slope generalized Bogomolov–Gieseker inequality by imposing a bound for arbitrary tilt-semistable objects. It is formulated as a tool for constructing stability conditions with larger numerical parameter spaces and remains open in general.