Chow-valued GW/DT correspondence for nonsingular projective threefolds

Let $X$ be a nonsingular projective threefold, let $\beta\in H_2(X,\mathbb{Z})$, and let $\operatorname{Chow}(X,\beta)$ be the seminormalized Chow variety of curves of class $\beta$. The seminormalized moduli spaces of stable maps and ideal sheaves have natural maps

$$\overline{M}'_{g}(X,\beta)_{\mathrm{sn}}\to \operatorname{Chow}(X,\beta),\qquad I_n(X,\beta)_{\mathrm{sn}}\to \operatorname{Chow}(X,\beta).$$

Chow-valued GW/DT conjecture. The equivalence of Corollary TChow, namely the equality of the corresponding Gromov–Witten and Donaldson–Thomas homological push-forwards after $e^{iu}=-q$, holds for all nonsingular projective threefolds $X$. The proven corollary in the supplied context applies to nonsingular projective toric threefolds; the conjecture proposes this refinement for arbitrary nonsingular projective threefolds.