$\Gamma$-generalized Bayer–Macrì–Toda inequality

Let $(X,H)$ be a polarised smooth projective threefold. Let $\operatorname{CH}_1(X)_{\mathbb R}$ denote the real Chow group of $1$-cycles, and let $\nu_{b,w}$, $\Delta_H$, $\operatorname{ch}^{bH}$ and $Q^{\Gamma}_{b,w}$ be defined as in the paper. $\Gamma$-generalized BMT inequality. There exists a $1$-cycle $\Gamma\in \operatorname{CH}_1(X)_{\mathbb R}$ with $\Gamma.H\geq 0$ such that, for any

$$w>\tfrac12 b^2+\tfrac12\bigl(b-\lfloor b\rfloor\bigr)\bigl(1-b+\lfloor b\rfloor\bigr)$$

and any $\nu_{b,w}$-semistable object $E$, one has

$$0\leq Q^{\Gamma}_{b,w}(E)\coloneqq (2w-b^2)\!\left(\Delta_H(E)+3\,\frac{\Gamma.H}{H^3}\bigl(\operatorname{ch}_0^{bH}(E)H^3\bigr)^2\right)+2\bigl(\operatorname{ch}_2^{bH}(E).H\bigr)\!\left(2\,\operatorname{ch}_2^{bH}(E).H-3(\Gamma.H)\operatorname{ch}_0^{bH}(E)\right)-6\bigl(\operatorname{ch}_1^{bH}(E).H^2\bigr)\!\left(\operatorname{ch}_3^{bH}(E)-\Gamma.\operatorname{ch}_1^{bH}(E)\right).$$

This is a generalized Bogomolov–Gieseker inequality intended to control tilt-semistable objects on threefolds. Its general status is not resolved in the supplied text.