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Stable pair invariants of local Calabi-Yau 4-folds

Yalong Cao, Martijn Kool, Sergej Monavari

TL;DR

The paper establishes a concrete link between Gopakumar–Vafa type invariants for Calabi–Yau 4-folds and stable-pair invariants in local Calabi–Yau geometries by proving a formula for $P_{n,eta}([ ext{pt}])$ in terms of intersection theory on Hilbert schemes of points on a surface $S$ when stable pairs are supported on the zero section. It develops the necessary framework comparing virtual classes of stable pairs on the local fourfold and the corresponding surface moduli, using a twisted tangent bundle and relative Hilbert schemes. It then verifies Cao–Maulik–Toda conjectures in numerous low-degree cases, combining the main theorem with genus-zero GV data from log-local results and with toric-geometry vertex calculations. The work also clarifies when the local moduli spaces reduce to surface data, and it provides practical computational tools via Atiyah–Bott localization to produce new verifications and tables of invariants. These results strengthen the sheaf-theoretic interpretation of GV invariants and illuminate their structure in noncompact Calabi–Yau fourfolds.

Abstract

In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao-Maulik-Toda conjectures for low degree curve classes and find connections to Carlsson-Okounkov numbers. Some of our verifications involve genus zero Gopakumar-Vafa type invariants recently determined in the context of the log-local principle by Bousseau-Brini-van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao-Maulik-Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$.

Stable pair invariants of local Calabi-Yau 4-folds

TL;DR

The paper establishes a concrete link between Gopakumar–Vafa type invariants for Calabi–Yau 4-folds and stable-pair invariants in local Calabi–Yau geometries by proving a formula for in terms of intersection theory on Hilbert schemes of points on a surface when stable pairs are supported on the zero section. It develops the necessary framework comparing virtual classes of stable pairs on the local fourfold and the corresponding surface moduli, using a twisted tangent bundle and relative Hilbert schemes. It then verifies Cao–Maulik–Toda conjectures in numerous low-degree cases, combining the main theorem with genus-zero GV data from log-local results and with toric-geometry vertex calculations. The work also clarifies when the local moduli spaces reduce to surface data, and it provides practical computational tools via Atiyah–Bott localization to produce new verifications and tables of invariants. These results strengthen the sheaf-theoretic interpretation of GV invariants and illuminate their structure in noncompact Calabi–Yau fourfolds.

Abstract

In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When is the total space of the sum of two line bundles over a surface , and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on . As an application, we obtain new verifications of the Cao-Maulik-Toda conjectures for low degree curve classes and find connections to Carlsson-Okounkov numbers. Some of our verifications involve genus zero Gopakumar-Vafa type invariants recently determined in the context of the log-local principle by Bousseau-Brini-van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao-Maulik-Toda conjectures when thickened curves contribute and also for the case of local .

Paper Structure

This paper contains 21 sections, 12 theorems, 138 equations, 6 tables.

Table of Contents

  1. Introduction
  2. GW/GV invariants of Calabi-Yau 4-folds
  3. Stable pair invariants of Calabi-Yau 4-folds
  4. Stable pair invariants of local surfaces
  5. Verifications
  6. Vertex calculations
  7. Acknowledgements
  8. Background
  9. DT invariants of Calabi-Yau 4-folds
  10. Stable pair invariants of Calabi-Yau 4-folds
  11. Moduli spaces
  12. Compactness I
  13. Compactness II
  14. Invariants
  15. Virtual classes of relative Hilbert schemes
  16. ...and 6 more sections

Key Result

Theorem 1.5

Let $S$ be a smooth projective surface with $b_1(S) = p_g(S) = 0$ and $L_1, L_2 \in \mathop{\rm Pic}\nolimits(S)$ such that $L_1 \otimes L_2 \cong K_S$. Suppose $\beta \in H_2(S,\mathbb{Z})$ and $n \geqslant 0$ are chosen such that $P_n(X,\beta) \cong P_n(S,\beta)$ for $X = \mathrm{Tot}_S(L_1 \oplus when $\beta^2 \geqslant 0$. Here $m := n + g(\beta) - 1$ and $h := c_1(\mathcal{O}(1))$. Moreover,

Theorems & Definitions (35)

  • Conjecture 1.1
  • Conjecture 1.2
  • Example 1.3
  • Example 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Corollary 1.8
  • Remark 1.9
  • Proposition 1.10
  • ...and 25 more