The Gromov-Witten/Donaldson-Thomas (GW/DT) correspondence predicts that two a priori different ways of counting algebraic curves in a nonsingular complex projective 3-fold — integration over moduli spaces of stable maps and integration over Hilbert schemes of ideal sheaves — yield the same generating series after a change of variables. Maulik, Oblomkov, Okounkov, and Pandharipande proved it for toric 3-folds and, for projective toric X, refined it to an identity of classes pushed forward to the Chow variety of curves, conjecturing the same Chow-level equivalence for all nonsingular projective 3-folds and noting that descendent insertions escape that formulation entirely. The current status is as follows. The capped-vertex polynomiality conjectures that motivated the toric proof were resolved in 2026 by Maulik-Ranganathan (in a preprint), and the primary numerical correspondence is now known for all threefolds with nef anticanonical bundle (Pardon), for Calabi-Yau complete intersections (Pandharipande-Pixton), and for Fano complete intersections (Oberdieck). What remains open is the source question in its two surviving forms: the Chow-level primary equivalence for arbitrary nonsingular projective 3-folds — open for every non-toric 3-fold, including all Calabi-Yaus — and the descendent correspondence, which remains open beyond toric targets and a small set of special geometries.

Background

Counting algebraic curves on a fixed variety is one of the oldest problems in enumerative geometry. In complex dimension three, it acquires a rich quantum-field-theoretic structure, and two sheaf- and map-theoretic counting formalisms turned out to be mysteriously equivalent.

Gromov-Witten theory. Let XX be a nonsingular complex projective 3-fold and β∈H2(X,Z)\beta \in H_2(X,\mathbb{Z}) a curve class. The Gromov-Witten (GW) invariants of XX are defined as integrals over moduli spaces of stable maps — maps f:C→Xf : C \to X from (possibly disconnected, prestable, marked) algebraic curves to XX satisfying f∗[C]=βf_*[C] = \beta, modulo isomorphism — against the virtual fundamental class [M‾]vir[\overline{M}]^{\mathrm{vir}}, a homology class of the 'expected' dimension that replaces the classical fundamental class because these moduli spaces are typically singular and of excess dimension. Integrating cohomology classes on XX pulled back by the evaluation maps at the markings (primary insertions), or tautological classes on the moduli space of curves (descendent insertions, obtained by intersecting Chern classes of tautological vector bundles such as the Hodge bundle of relative differentials), produces numerical invariants that are assembled into generating series.

Donaldson-Thomas theory and the MNOP conjecture. Donaldson-Thomas (DT) invariants count curves sheaf-theoretically: the Hilbert scheme In(X,β)I_n(X,\beta) of ideal sheaves (rank-1 torsion-free sheaves with trivial determinant, equivalently subschemes of dimension at most one with Euler characteristic nn and fundamental cycle β\beta) carries its own virtual class, and its degrees are the DT invariants. Maulik, Nekrasov, Okounkov, and Pandharipande conjectured (MNOP, 2006) what is now called the GW/DT correspondence: for every nonsingular projective 3-fold, the reduced DT partition function (reduced meaning: with the contribution of embedded points, which carry no curve class, divided out) is a rational function of qq, and after the change of variables eiu=−qe^{iu} = -q linking the genus parameter uu to the Euler-characteristic parameter qq (which tracks the DT variable nn; the curve class β\beta is kept fixed), it equals the reduced GW series in primary insertions, once the degree-0 DT contribution ZDT(X,q)0=M(−q)∫Xc3(TX⊗KX)Z_{DT}(X,q)_0 = M(-q)^{\int_X c_3(T_X \otimes K_X)} (a MacMahon-function evaluation, M(q)=∏n≥1(1−qn)−nM(q) = \prod_{n \geq 1}(1-q^n)^{-n} counting 3-dimensional partitions) is divided out.

The source paper and its toric theorem. The source paper (Maulik-Oblomkov-Okounkov-Pandharipande, arXiv:0809.3976, published in Inventiones Mathematicae in 2011) proved this correspondence for all nonsingular toric 3-folds — varieties carrying a dense algebraic torus (C∗)3(\mathbb{C}^*)^3-action — equivariantly, that is, for the torus-equivariant versions of both theories in which the torus action enters through formal weight parameters. The method is virtual localization: the torus action isolates finitely many fixed loci, and the invariants become explicit sums over combinatorial data on trivalent graphs, with partitions (Young diagrams) carried by edges and vertices. For projective toric XX, the paper then obtained a refinement (its Corollary 1): both the stable-map moduli spaces and the Hilbert schemes admit natural maps to the Chow variety Chow(X,β)Chow(X,\beta) — the projective variety parametrizing effective 1-cycles in class β\beta, defined as a seminormalization of a variety of Chow forms — and the virtual classes push forward to an identity there. This is strictly stronger than numerical agreement of invariants: it equates entire cycle-valued classes, forcing agreement of every pairing. The paper's Conjecture 1 predicts that this Chow-level equivalence holds for all nonsingular projective 3-folds. The same paper notes explicitly that the Chow statement 'does not capture the full GW/DT correspondence as descendent insertions are not pulled-back from the Chow variety,' since descendents see the geometry of the domain curve rather than only the image cycle.

The capped vertex. The technical heart of the toric proof is the capped vertex CDT(λ,μ,ν,t1,t2,t3)C_{DT}(\lambda,\mu,\nu,t_1,t_2,t_3): a residue invariant on (P1)3(\mathbb{P}^1)^3 (minus the toric boundary) associated to each torus-fixed point, depending on three partitions (λ,μ,ν)(\lambda,\mu,\nu) encoding relative conditions along the three invariant directions, and three equivariant parameters (t1,t2,t3)(t_1,t_2,t_3). The paper proved the capped vertex is rational in Q(q,t1,t2,t3)\mathbb{Q}(q,t_1,t_2,t_3), announced (its Theorem 3, proof deferred by footnote) that its poles in qq occur only at roots of unity and 0, and conjectured (its Conjecture 2) the stronger Laurent polynomiality CDT∈Q(t1,t2,t3)[q,1/q]C_{DT} \in \mathbb{Q}(t_1,t_2,t_3)[q,1/q]. It also recorded that its proof yields an algorithm computing capped vertices, together with small-partition evaluations of a remainder factor R(λ,μ,ν)R(\lambda,\mu,\nu) appearing in an explicit factorization of the connected capped vertex, suggesting RR is itself a Laurent polynomial in qq.

Stable pairs and the modern formulation. The sheaf side was subsequently recast via stable pairs (Pandharipande-Thomas): two-term complexes [OX→F][\mathcal{O}_X \to F] with FF a pure one-dimensional sheaf, whose moduli carries a virtual class and behaves better under degeneration; Toda's wall-crossing theorem identifies the DT and stable-pairs (PT) generating series, so the GW/DT and GW/PT correspondences are equivalent targets. In this language the Chow-level statement was reformulated by Okounkov as an equality of series of Chow-pushed classes with an explicit virtual-dimension power prefactor.

Where the boundary lies (2026). The correspondence is currently proven: for toric 3-folds including descendents (Pandharipande-Pixton; Oblomkov-Okounkov-Pandharipande); for Calabi-Yau (trivial canonical class) complete intersections in products of projective spaces, including the quintic hypersurface (Pandharipande-Pixton); for Fano (ample anticanonical) complete intersections and the reduced theories of products S×CS \times C of a K3 surface (a simply-connected complex surface with trivial canonical class) with a curve, for all curve classes of divisibility at most 2 over the surface (Oberdieck); across conifold and del Pezzo transitions (operations contracting rational curves to ordinary double points, and del Pezzo surfaces — surfaces with ample anticanonical class — to points, respectively, followed by smoothing; Lin-Wang; Lee-Wang-Wang); and most generally for all complex threefolds with nef anticanonical bundle (nef: the anticanonical divisor intersects every curve nonnegatively; this class comprises all Calabi-Yau and all Fano threefolds) by Pardon's universal curve-counting theorem, which reduces to local curves. Maulik-Ranganathan (2026) proved the capped-vertex Laurent polynomiality for toric pairs. Beyond these classes the primary correspondence for arbitrary projective 3-folds — in particular for 3-folds of general type (maximal Kodaira dimension, the opposite extreme from Fano), whose anticanonical bundle is not nef — is open, no non-toric instance of the Chow-level refinement is known, and the descendent correspondence is open outside the listed geometries.

Problem Statement

The problem concerns the source paper's own program for extending its toric theorem to arbitrary nonsingular projective 3-folds: its Chow-level Conjecture 1, together with the stationary descendent extension prompted by the paper's explicit remark that descendent insertions escape its Chow formulation.

Target 1 (Chow-level primary equivalence; Conjecture 1 of the source paper). Prove that for every nonsingular projective 3-fold XX and every curve class β∈H2(X,Z)\beta \in H_2(X,\mathbb{Z}), the identity proved for projective toric XX as Corollary 1 holds:

∑gu2g−2 ρGW∗([M‾g′(X,β)]vir)  =  1ZDT(X,q)0∑nqn ρDT∗([In(X,β)]vir)\sum_g u^{2g-2}\,\rho_{GW*}\left(\left[\overline{M}'_g(X,\beta)\right]^{\mathrm{vir}}\right) \;=\; \frac{1}{Z_{DT}(X,q)_0}\sum_n q^{n}\,\rho_{DT*}\left(\left[I_n(X,\beta)\right]^{\mathrm{vir}}\right)

in H∗(Chow(X,β),Q)⊗Q(q)H_*(Chow(X,\beta),\mathbb{Q}) \otimes \mathbb{Q}(q), after the change of variables eiu=−qe^{iu} = -q. Here M‾g′(X,β)\overline{M}'_g(X,\beta) is the moduli space of stable maps to XX from possibly disconnected genus-gg curves representing β\beta and not collapsing any connected component (this is what makes the left side the reduced series), In(X,β)I_n(X,\beta) is the Hilbert scheme of ideal sheaves with Euler characteristic nn and fundamental cycle β\beta, the maps ρGW\rho_{GW} and ρDT\rho_{DT} are the natural morphisms from the associated seminormalized moduli spaces to the Chow variety Chow(X,β)Chow(X,\beta) of effective 1-cycles in class β\beta (seminormalization of a variety of Chow forms), the subscript ∗* denotes pushforward of virtual fundamental classes, and ZDT(X,q)0=M(−q)∫Xc3(TX⊗KX)Z_{DT}(X,q)_0 = M(-q)^{\int_X c_3(T_X \otimes K_X)} is the degree-0 DT partition function with MM the MacMahon function.

Target 2 (stationary descendent correspondence). Prove the stationary (descendents paired only with cohomology classes γ∈H≥2(X)\gamma \in H^{\geq 2}(X) of degree at least 2, following the convention of the cited vertex-operator conjecture) descendent GW/PT correspondence for arbitrary nonsingular projective 3-folds: the explicit identification of the generating series of descendent GW invariants with the generating series of descendent stable-pairs invariants, in the vertex-operator form conjectured by Oblomkov-Okounkov-Pandharipande — an explicit formula built from vertex operators, the creation and annihilation operators on the Fock space of symmetric functions — or an equally explicit identification of the two series. The identification includes, as a well-posedness prerequisite, the rationality, in the Euler-characteristic variable qq (for each fixed curve class), of the descendent stable-pairs series.

Numerical shadow. Since Target 1 implies the primary numerical MNOP correspondence for all nonsingular projective 3-folds, a solution must in particular cover 3-folds whose anticanonical bundle is not nef (for example 3-folds of general type), for which no instance is currently known, and must establish the Chow-level refinement even for Calabi-Yau 3-folds, where the numerical statement is known by other methods but no non-toric Chow-level instance is.

The two targets are the two forms of the beyond-toric question arising from the source paper — the Chow-level refinement of its main theorem, conjectured there as Conjecture 1, and the stationary descendent extension prompted by the paper's explicit observation that its Chow formulation cannot express descendent insertions, formulated as a conjecture by Oblomkov-Okounkov-Pandharipande — and together they cover the primary GW/DT correspondence for all nonsingular projective 3-folds and its stationary descendent sector; non-stationary descendent insertions lie beyond both targets.

References

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