Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

TL;DR

This paper introduces new Gopakumar-Vafa type invariants for holomorphic symplectic 4-folds, providing conjectural integrality, sheaf-theoretic interpretations, and explicit computations that extend enumerative geometry to higher-dimensional Calabi-Yau analogues.

Contribution

The authors define novel invariants for holomorphic symplectic 4-folds, conjecture their integrality, and relate them to reduced Donaldson-Thomas invariants, extending enumerative theories to fourfolds.

Findings

Conjecture that the invariants are integers.

Computed invariants for specific examples like K3 surfaces and cotangent bundles.

Derived a new formula for Fujiki constants of tangent bundle Chern classes.

Abstract

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds, Klemm and Pandharipande for Calabi-Yau 4-folds, Pandharipande and Zinger for Calabi-Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced $4$-dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two $K3$ surfaces and for the cotangent bundle of $\mathbb{P}^2$. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a $K3$ surface. This yields a conjectural formula for the number of isolated genus $2$ curves of minimal degree on a very general hyperk\"ahler $4$-fold of $K3^{[2]}$-type. The formula may be viewed as a $4$-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on $K3$ surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on $K3$ surfaces and generalized Kummer varieties.