Virasoro conjecture for stable pairs on simply-connected 3-folds: the full descendent algebra beyond $(p,p)$-cohomology and the vanishing of $(p,0)$-descendants

The Virasoro conjecture for the stable pairs (Pandharipande-Thomas) descendent theory of a smooth projective 3-fold predicts that certain Virasoro operators $\mathcal{L}_k$ annihilate all descendent invariants. Moreira (J. Lond. Math. Soc. 2022) extended the conjecture from toric 3-folds, whose cohomology has only $(p,p)$ Hodge summands, to simply-connected 3-folds with non-$(p,p)$ cohomology, proving it conditionally for the subalgebra $\mathbb{D}_{\mathrm{PT}}^{X,p>0}$ generated by descendents of classes with positive first Hodge index (assuming the stationary Gromov-Witten Virasoro constraints and the stationary GW/PT descendent correspondence for $X$), and unconditionally in two specializations: the Hilbert scheme of points on any surface with $H^1=0$, and the cubic 3-fold in the line class. As of August 2026, the remaining core is open: proving the constraints for the full descendent algebra $\mathbb{D}_{\mathrm{PT}}^{X}$ — above all for insertions of Hodge type $(0,q)$, for which the stationary GW/PT intertwining route has no input — together with the source's Conjecture 3, the predicted total vanishing of descendents of $(p,0)$-classes, which would immediately settle the $(0,2)$- and $(0,3)$-cases. Adjacent progress (vertex-algebra wall-crossing proofs of Virasoro constraints for moduli of sheaves on surfaces and curves, K3 monodromy arguments, new GW/PT correspondence cases for Fano complete intersections and conifold transitions, and 2026 proofs of the companion rationality conjecture for Fano 3-folds and superpositive curve classes) leaves this 3-fold core untouched.

Background

A stable pair on a smooth projective 3-fold $X$ is a two-term complex $[\mathcal{O}_X\to F]$ whose sheaf part $F$ is pure of dimension one; stable pairs were introduced by Pandharipande and Thomas as an alternative to stable maps for counting curves. The moduli space $P_n(X,\beta)$ of stable pairs with Euler characteristic $n$ and support in the curve class $\beta\in H_2(X;\mathbb{Z})$ carries a virtual fundamental class $[P_n(X,\beta)]^{\mathrm{vir}}$ (constructed by the standard perfect-obstruction-theory method), and PT invariants are integrals of cohomology classes against it. Besides primary insertions one integrates descendents: for a class $\gamma\in H^*(X)$ and $k\ge0$, $\mathrm{ch}_k(\gamma)$ is defined from the Chern character of the universal stable pair pulled back from $X\times P_n(X,\beta)$, and the descendent algebra $\mathbb{D}_{\mathrm{PT}}^{X}$ is the polynomial algebra on these generators. The partition function $Z_{\mathrm{PT}}^{X}(q\mid D)_\beta=\sum_n q^n\langle D\rangle_{n,\beta}^{X,\mathrm{PT}}$ records all Euler characteristics at fixed $D$ and $\beta$.

The Virasoro conjecture for stable pairs, proposed by Moreira-Oblomkov-Okounkov-Pandharipande for 3-folds whose cohomology has only $(p,p)$ Hodge summands (as for toric 3-folds), predicts that for every $k\ge-1$ the operator $\mathcal{L}_k=R_k+T_k+S_k$ on $\mathbb{D}_{\mathrm{PT}}^{X}$ annihilates all invariants: $\langle\mathcal{L}_k(D)\rangle_{n,\beta}^{X,\mathrm{PT}}=0$. Here $R_k$ raises descendent degrees with factorial coefficients depending on the first Hodge index $p$ of the insertion class (the algebraic counterpart of the Virasoro raising operators acting on fermions in physics), $T_k$ is multiplication by a fixed element of the descendent algebra built from descendents of the Chern classes $c_1$ and $c_1c_2$ of $X$ (the descendent-theoretic counterpart of the Todd-class insertion in the Gromov-Witten Virasoro operators, reflecting $\mathrm{td}_3(X)=\frac{1}{24}c_1c_2$), and $S_k=(k+1)!\sum_{p_i^L=0}R_{-1}[\gamma_i^L]\,\mathrm{ch}_{k+1}(\gamma_i^R)$ is a correction summing over the Kunneth components $\gamma_i^L\otimes\gamma_i^R$ of the diagonal class of $X\times X$ whose left factor has Hodge type $(0,q)$; when $h^{0,q}(X)=0$ for all $q>0$ (the $(p,p)$-world of toric 3-folds) this sum collapses to the single point-class term $(k+1)!\,R_{-1}\,\mathrm{ch}_{k+1}(\mathsf{p})$, with $\mathsf{p}$ the class of a point on $X$, so the genuinely new contributions to $S_k$ arise only from non-$(p,p)$ cohomology. The toric case was proved (in the stationary sector) by intertwining these operators, through the explicit stationary GW/PT descendent correspondence of Oblomkov-Okounkov-Pandharipande (itself proven for toric 3-folds via vertex operators), with the Virasoro constraints of Gromov-Witten theory, which are known for toric targets.

Moreira extended the conjecture to simply-connected 3-folds with non-$(p,p)$ cohomology and proved two things. First, Theorem 10: if $X$ satisfies (i) the stationary Gromov-Witten Virasoro constraints and (ii) the stationary GW/PT descendent correspondence, then the stable pairs Virasoro constraints hold on the subalgebra $\mathbb{D}_{\mathrm{PT}}^{X,p>0}$ generated by descendents of classes with positive first Hodge index. Second, two unconditional verifications: the specialization $X=S\times\mathbb{P}^1$, $\beta=n[\mathbb{P}^1]$ at minimal Euler characteristic, where $P_n(X,\beta)\cong S^{[n]}$ (the Hilbert scheme of $n$ points on the surface $S$) for any $S$ with $H^1(S)=0$ — giving universal Virasoro constraints on descendent integrals over Hilbert schemes of points — and the cubic 3-fold in the line class, via a full computation of its stable pairs theory using the Fano variety of lines.

What fails beyond $\mathbb{D}_{\mathrm{PT}}^{X,p>0}$ is exactly the Hodge periphery. The stationary GW/PT correspondence controls only insertions built from classes with positive first Hodge index; for descendents of $(0,q)$-classes the intertwining argument has no input, and the paper states explicitly that no adequate intertwining statement was obtained. For $(p,0)$-classes the conjecture predicts instead a total vanishing (Conjecture 3 of the source): for $\gamma\in H^{p,0}(X)$ with $p=2$ or $3$, $\langle\mathrm{ch}_k(\gamma)D\rangle_{n,\beta}^{X,\mathrm{PT}}=0$ for all $k\ge0$ and all $D$ — speculated to hold already at the cycle level, $\mathrm{ch}_k(\gamma)\cap[P_n(X,\beta)]^{\mathrm{vir}}=0$ — and this vanishing would immediately imply the $(0,2)$- and $(0,3)$-cases of the constraints, since every term of the expanded $\langle\mathcal{L}_k(D)\rangle$ would then contain such a vanishing descendent. The Hilbert scheme connection makes the problem consequential beyond PT theory itself: the surface specializations are the seed cases for the now-active Virasoro program for moduli spaces of sheaves.

Problem Statement

Let $X$ be a smooth projective simply-connected 3-fold with arbitrary Hodge-type cohomology. Prove the Virasoro conjecture for the stable pairs descendent theory of $X$ in its full form (Conjecture 2 of the source): for every $k\ge-1$, $n\in\mathbb{Z}$, $\beta\in H_2(X;\mathbb{Z})$ and every descendent $D\in\mathbb{D}_{\mathrm{PT}}^{X}$ — including products containing descendents $\mathrm{ch}_j(\gamma)$ with $\gamma$ of Hodge type $(0,q)$ or $(p,0)$ — the constraint

$\langle\mathcal{L}_k(D)\rangle_{n,\beta}^{X,\mathrm{PT}}=0$

holds unconditionally, i.e. without assuming the stationary Gromov-Witten Virasoro constraints or the stationary GW/PT descendent correspondence for $X$. Two coupled sub-questions, identified in the source as the heart of the obstruction, lie inside this single objective:

(1) $(0,q)$-insertions. Prove the constraint for descendents of $(0,q)$-classes, thereby extending the constraints from the subalgebra $\mathbb{D}_{\mathrm{PT}}^{X,p>0}$ covered by Theorem 10 to the full descendent algebra $\mathbb{D}_{\mathrm{PT}}^{X}$.

(2) Vanishing of $(p,0)$-descendants (Conjecture 3 of the source). Prove or refute that for $\gamma\in H^{p,0}(X)$ with $p=2$ or $3$, one has $\langle\mathrm{ch}_k(\gamma)D\rangle_{n,\beta}^{X,\mathrm{PT}}=0$ for all $k\ge0$, all $n$, $\beta$ and all $D\in\mathbb{D}_{\mathrm{PT}}^{X}$. The source further speculates the stronger cycle-level vanishing $\mathrm{ch}_k(\gamma)\cap[P_n(X,\beta)]^{\mathrm{vir}}=0$; deciding this stronger form is not required for the main objective. A positive resolution of (2) yields the $(0,2)$- and $(0,3)$-cases of (1) immediately.

References

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