Abstract
Abstract In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ 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 $\mathbb{P}^3$.
Highlights
Pn(S, β) ∼= Hilbm(C/|β|), where Hilbm(C/|β|) denotes the relative Hilbert scheme of m points on the fibres of C → |β| and m = n + g(β) − 1 = n + 21 β(β + KS). This isomorphism was exploited in order to determine the surface contribution to stable pair invariants of local surfaces TotS(KS) in [KT2]
We remark that most stable pair invariants of local surfaces calculated in this paper are small
[Pn(S, β)]vir is the virtual class induced from the relative Hilbert scheme (Section 4.1), IS = {O → F} denotes the universal stable pair on S × Pn(S, β), and πPS : S × Pn(S, β) → Pn(S, β) is the projection
Summary
For Calabi-Yau 4-folds, KlemmPandharipande [KP] defined Gopakumar-Vafa type invariants using Gromov-Witten theory and conjectured their integrality. Gromov-Witten invariants vanish for genus g 2 for dimensional reasons and one only needs to consider the genus zero and one cases. The genus zero Gromov-Witten invariants of X for class β ∈ H2(X, Z) are defined using an insertion. For the genus one case, the virtual dimension of M 1,0(X, β) is zero and one defines. −1 mβ ,β log(1 − qβ1+β2 ), β1,β2 where σ(d) = i|d i and mβ1,β2 ∈ Z are called meeting invariants, which can be inductively determined by the genus zero Gromov-Witten invariants of X. The genus zero integrality conjecture has been proved by Ionel-Parker using symplectic geometry [IP, Thm. 9.2]
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