The refined Gopakumar–Vafa and perverse Hodge number conjecture

Let $X$ be a projective Calabi–Yau threefold and let $M_{\beta,1}$ be the moduli space of stable one-dimensional sheaves of class $\beta$ and Euler characteristic $1$. Let $\rho:M_{\beta,1}\to\operatorname{Chow}_{\beta}$ be the Hilbert–Chow map, let $\Phi$ be the associated perverse sheaf, and define

$${^ph}^{i,j}_{\beta}:=h^j\bigl({^p\mathcal H}^{i}(R\rho_*\Phi)\bigr).$$

Let $\mathsf{GV}_{\beta}(u,p)$ be the refined Gopakumar–Vafa polynomial.

Refined Gopakumar–Vafa conjecture. One has

$$\mathsf{GV}_{\beta}(u,p)=\sum_{i,j}{^ph}^{i,j}_{\beta}(-1)^{i+j}p^i u^j.$$

This is the motivic refinement of the Maulik–Toda relationship, identifying refined curve-counting invariants with perverse Hodge data. The source attributes the conjecture to discussions with J. Shen; no resolution evidence is supplied.