The modified Bogomolov–Gieseker inequality for families of stability parameters

Let $X$ be a smooth projective threefold, let $H\in\operatorname{NS}(X)$ be ample, and let $B\in\operatorname{NS}_{\mathbb{R}}(X)$. For $\beta\in\mathbb{R}$ and $\alpha\in\mathbb{R}_{>0}$, let $\mathcal{B}_{H,B+\beta H}$ be the tilted heart and let $\nu_{H,B+\beta H,\alpha}$ denote its tilt slope. For $E\in D^b(X)$ define

$$D^{B,\xi}_{\alpha,\beta}(E)=\operatorname{ch}_3^{B+\beta H}(E)+\left(\Lambda-\left(\xi+\frac{1}{6}\alpha^2\right)H^2\right)\operatorname{ch}_1^{B+\beta H}(E).$$

Let $A:B+\mathbb{R}\langle H\rangle\to\mathbb{R}_{\geq0}$ be continuous. The modified Bogomolov–Gieseker inequality conjecture. There exist $\Lambda\in H^4(X,\mathbb{Q})$ with $\Lambda\cdot H=0$ and $\xi(A)\in\mathbb{R}_{\geq0}$ such that, whenever $\alpha\geq A(B+\beta H)$, every $\nu_{H,B+\beta H,\alpha}$-tilt-slope-stable object $E\in\mathcal{B}_{H,B+\beta H}$ satisfying $\nu_{H,B+\beta H,\alpha}(E)=0$ obeys

$$D^{B,\xi(A)}_{\alpha,\beta}(E)\leq0.$$

Consequently, the same inequality holds for every $\xi\geq\xi(A)$. This modification is intended to restore a Bogomolov–Gieseker-type inequality after counterexamples to the original threefold conjecture; it would yield families of Bridgeland stability conditions, and remains open in the stated generality.