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, and 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？hler 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. 

Publication: 

Commun. Math. Phys. 405, 26 (2024). 

https://doi.org/10.1007/s00220-023-04882-8 

Author: 

Yalong Cao 

Morningside Center of Mathematics & Institute of Mathematics, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Beijing, 100190, China 

Email: yalongcao@amss.ac.cn 

Georg Oberdieck 

Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden 

Yukinobu Toda 

Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan