Bayer–Macrì–Toda generalized Bogomolov–Gieseker conjecture

Let $X$ be a smooth projective threefold, let $H$ be an ample divisor, and let $\alpha>0$ and $\beta\in\mathbb{R}$. Let $\mathcal{B}_{H}^{\beta}$ denote the tilted heart, and let $Z_{\alpha,\beta}^{2}$ be the second-stage weak stability function. For an object $E\in\mathcal{B}_{H}^{\beta}$, write

$$\operatorname{ch}^{\beta}(E)=e^{-\beta H}\operatorname{ch}(E).$$

Bayer–Macrì–Toda generalized Bogomolov–Gieseker conjecture. For any $Z_{\alpha,\beta}^{2}$-semistable object $E\in\mathcal{B}_{H}^{\beta}$ satisfying

$$\operatorname{Im} Z_{\alpha,\beta}^{3}(E)=0,$$

we have

$$\operatorname{ch}_{3}^{\beta}(E)<\frac{\alpha^{2}}{2}H^{2}\operatorname{ch}_{1}^{\beta}(E).$$

This inequality is equivalent to the assertion that the twice-tilted heart with central charge $Z^{3}_{\alpha,\beta}$ defines a stability condition. It is a central open case of the generalized Bogomolov–Gieseker problem for threefolds.