For every smooth complete threefold X and every integer n≥ 0, let pi_n:operatornameHilb^n(X)→ S^nX be the Hilbert--Chow morphism. The refined MNOP conjecture asks for the characteristic-class formula, relating pi_n[operatornameHilb^n(X)]^vir∈ A_0(S^nX) to the Chern--Schwartz--MacPherson classes c_SM(S^nX) and the associated virtual curve-counting data. The conjecture is asserted integrally for arbitrary smooth complete threefolds; the cited work establishes the proposed equality only modulo n!-torsion when X is a Calabi--Yau threefold.

Refined MNOP conjecture

For every smooth complete threefold $X$ and every integer $n\ge 0$ , let $\pi_n:\operatorname{Hilb}^n(X)\to S^nX$ be the Hilbert--Chow morphism. The refined MNOP conjecture asks for the characteristic-class formula, relating $\pi_{n*}[\operatorname{Hilb}^n(X)]^{\mathrm{vir}}\in A_0(S^nX)$ to the Chern--Schwartz--MacPherson classes $c_{\mathrm{SM}}(S^nX)$ and the associated virtual curve-counting data. The conjecture is asserted integrally for arbitrary smooth complete threefolds; the cited work establishes the proposed equality only modulo $n!$ -torsion when $X$ is a Calabi--Yau threefold.

References

Chern-Schwartz-MacPherson classes of symmetric powers and applications — arXiv — Barbara Fantechi, Lothar Goettsche

Source: arXiv

Posted 3 hours ago · edited 3 hours ago

Progress summary

Refreshed 3 hours ago

Open · progress reported

A new paper proves the proposed formula only after ignoring certain factorial torsion for Calabi–Yau threefolds, so the full conjecture remains open.

The refined MNOP conjecture links characteristic classes of symmetric powers with virtual curve-counting data on threefolds. Fantechi and Goettsche introduce this refined formulation and establish a substantial torsion-limited case.

October 2026 torsion-limited result

Fantechi and Goettsche compute Chern–Schwartz–MacPherson classes of symmetric powers and claim the refined virtual-class formula for Calabi–Yau threefolds after quotienting by $n!$ -torsion. The result does not prove the conjecture integrally or for arbitrary smooth projective threefolds, and remains unverified as a complete resolution.

Current status (as of October 2026): The formula is claimed modulo $n!$ -torsion for Calabi–Yau threefolds, while the integral conjecture and the case of arbitrary smooth projective threefolds remain open.

Sources

arxiv.org

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