Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau \(4\)-folds. (English) Zbl 1491.14077

Cao, Yalong; Maulik, Davesh; Toda, Yukinobu

J. Eur. Math. Soc. (JEMS) 24, No. 2, 527-581 (2022).

This paper studies the relationship between two types of enumerative invariants of Calabi-Yau \(4\)-folds \(X\). Pandharipande-Thomas (PT) invariants count pairs \(s\colon \mathcal{O}_X \to \mathcal{E}\), consisting of a pure \(1\)-dimensional coherent sheaf \(\mathcal{E}\) on \(X\) and a section \(s\), satisfying a certain stability condition. On the other hand, Gopakumar-Vafa (GV) invariants are conjecturally-integer quantities underlying the Gromov-Witten (GW) theory of \(X\), defined formally from its multiple-cover formulas. Earlier work [R. Pandharipande and R. P. Thomas, J. Am. Math. Soc. 23, No. 1, 267–297 (2010; Zbl 1250.14035)] proposed a relationship between PT and GV invariants for CY3s. The recent construction in [D. Borisov and D. Joyce, Geom. Topol. 21, No. 6, 3231–3311 (2017; Zbl 1390.14008)] of virtual fundamental classes \([M]^{\mathrm{vir}}\) for moduli spaces \(M\) of sheaves on CY4s enables the generalization of such a PT/GV correspondence to CY4s, modulo the (new) problem of how to consistently pick certain required orientation data for \(M\).
The main conjecture of this paper is that, for a suitable choice of orientation data, \begin{align*} P_{1,\beta}(\gamma_1, \ldots, \gamma_m) &= \sum_{\substack{\beta_1+\beta_2=\beta\\ \beta_1,\beta_2\ge 0}} n_{0,\beta_1}(\gamma_1, \ldots, \gamma_m) \cdot P_{0,\beta_2} \\ \sum_{\beta \ge 0} P_{0,\beta} q^\beta &= \prod_{\beta > 0} M(q^\beta)^{n_{1,\beta}} \end{align*} where the \(P_{n,\beta}\) (resp. \(n_{g,\beta}\)) are PT invariants (resp. GV invariants) of \(X\) in class \(\beta\), and \(M(q) = \prod_{k > 0} (1-q^k)^{-k}\) is the MacMahon function. Note that when \(n= \chi(\mathcal{E})=1\), non-trivial descendent insertions \(\gamma_i \in H^*(X; \mathbb{Z})\) are required for \(P_{1,\beta}\) to be non-zero, and when genus \(g>1\), GW and therefore the GV invariants \(n_{g,\beta}\) vanish on \(4\)-folds.
Many independent pieces of evidence are given to support the conjecture. A general proof is given assuming all relevant families of curves in \(X\) deform with expected properties. Explicit computational checks, for certain curve classes \(\beta\), are also given for: (compact case) sextic hypersurfaces, elliptic fibrations over \(\mathbb{P}^3\), and products of elliptic curves and CY3s; (non-compact case) local Fano \(3\)-folds, and local \(\mathbb{P}^1\) and elliptic curves. Many of these computations proceed by reduction to known conjectures/results for \(3\)-folds (e.g. [Y. Cao, Commun. Contemp. Math. 22, No. 7, Article ID 1950060, 25 p. (2020; Zbl 1452.14053)]) or genus-\(0\) GV for CY4s defined using \(4\)-fold Donaldson-Thomas invariants (e.g. [Y. Cao et al., Adv. Math. 338, 41–92 (2018; Zbl 1408.14177)]). Such reductions, especially to moduli spaces of sheaves on \(X\) of lower dimension, often give non-trivial insights on the correct choice of orientation in the \(4\)-fold setting. Finally, invariants for local curves are computed in low degrees via equivariant localization.

Reviewer: Huaxin Liu (Oxford)

MSC:

14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J32	Calabi-Yau manifolds (algebro-geometric aspects)

Keywords:

stable pairs; Gopakumar-Vafa type invariants; Calabi-Yau \(4\)-folds

Citations:

Zbl 1250.14035; Zbl 1390.14008; Zbl 1452.14053; Zbl 1408.14177

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