  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.
