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 P2 . 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 hyperkähler 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.