Calabi–Yau property conjecture for the moduli stack of one-dimensional sheaves

Let $X$ be a smooth projective Calabi–Yau threefold, let $\beta\in N_1(X)$, and let $\mathcal{M}_X(\beta)$ be the moduli stack of one-dimensional sheaves with class $\beta$. Let $\operatorname{Chow}_X(\beta)$ be the Chow variety and let the Hilbert–Chow map send a sheaf to its fundamental one-cycle. The stack $\mathcal{M}_X(\beta)$ is Calabi–Yau at every point $\gamma\in\operatorname{Chow}_X(\beta)$ for the Hilbert–Chow map.

This conjectural Calabi–Yau condition is used to construct the relevant perverse sheaves and invariants. The paper proves the condition for the local Calabi–Yau threefold $X=\operatorname{Tot}_S(K_S)$ when $S$ is a smooth projective surface, but the general projective Calabi–Yau case remains open.