The generalized Bogomolov–Gieseker stability conjecture for twisted varieties

Let $X$ be a smooth projective variety equipped with a Brauer class $\alpha\in\operatorname{Br}(X)$, and let $\omega$ and $D$ be real divisors on $X$, with $\omega$ ample. Define

$$Z_{\omega,D}(E)=-\int_X e^{-i\omega}\cdot\operatorname{ch}^{D}(E).$$

Twisted stability-condition conjecture. There exists a bounded t-structure on $\mathrm{D}_{\mathrm{perf}}(X,\alpha)$ with heart $\mathcal{A}_{\omega,D}$ such that $(Z_{\omega,D},\mathcal{A}_{\omega,D})$ is a full numerical stability condition. This is the twisted extension of the conjectural existence of stability conditions proposed by Bayer, Macrì, and Toda. The supplied status evidence indicates that the statement is resolved in the relevant setting, with a theorem giving a solution when $\dim X=2$ over a field of characteristic zero; the general conjecture is not established by that theorem.