Torsion refined Gopakumar–Vafa stability conjecture for non-Kähler small resolutions

Let $X$ be a Calabi–Yau threefold with only terminal conifold singularities and no Kähler small resolution. Let $\mathcal{S}$ be the set of all small resolutions $\widehat{X}$ of $X$, let $\alpha\in\operatorname{Br}(\widehat{X})$, and let $\mathcal{K}_\alpha$ be the corresponding component of the Kähler moduli space. For each $\widehat{X}\in\mathcal{S}$, write $\operatorname{Coh}_{\leq 1}(\widehat{X})$ for the category of coherent sheaves on $\widehat{X}$ whose support has dimension at most $1$, viewed as a subcategory of $D^b(\widehat{X},\alpha)$. The torsion refined Gopakumar–Vafa stability conjecture. The category $D^b(\widehat{X},\alpha)$ supports Bridgeland stability conditions parametrized by $\mathcal{K}_\alpha$; among these are stability conditions whose hearts contain

$$\bigcup_{\widehat{X}\in\mathcal{S}}\operatorname{Coh}_{\leq 1}(\widehat{X})\,.$$

On each $\operatorname{Coh}_{\leq 1}(\widehat{X})$, the Bridgeland stability condition coincides with Gieseker stability. This is proposed as a mathematical definition of torsion refined Gopakumar–Vafa invariants for Calabi–Yau threefolds without Kähler small resolutions. The claim concerns the compatibility of stability conditions across the different small resolutions, using their derived equivalences and the common Brauer group.