Bayer–Macrì–Toda Bogomolov–Gieseker inequality

Let $X$ be the smooth projective threefold considered in the paper, let $H$ be its ample divisor, let $U$ be the domain of parameters $(b,w)$, and let $\nu_{b,w}$ be the associated tilt slope on the heart $\mathcal D(X)$. For $E\in\mathcal D(X)$, write $\operatorname{ch}_i^{bH}(E)$ for the $bH$-twisted Chern characters and $\Delta_H(E)$ for the discriminant.

Bayer–Macrì–Toda Bogomolov–Gieseker inequality. For any $(b,w)\in U$ and $\nu_{b,w}$-semistable $E\in\mathcal D(X)$, one has

$$B_{b,w}(E):=(2w-b^2)\Delta_H(E)+4\bigl(\operatorname{ch}_2^{bH}(E).H\bigr)^2-6\bigl(\operatorname{ch}_1^{bH}(E).H^2\bigr)\operatorname{ch}_3^{bH}(E)\geq 0.$$

This is the conjectural Bogomolov–Gieseker inequality of Bayer–Macrì–Toda, used throughout the paper as an assumption. Its validity supplies the support property for the relevant stability conditions and underlies the rank-reduction arguments for Donaldson–Thomas theory; the supplied text does not state that it has been proved or disproved in this setting.