I will discuss joint work with Yalong Cao and Yukinobu Toda where we use reduced Gromov-Witten theory to define new invariants of holomorphic-symplectic 4-folds. The invariants are (a) conjecturally integers, and (b) in an ideal geometry should be enumerative for the counts of curves in primitive curve classes. This leads to explicit predictions for the number of genus 2 curves of minimal degree on very general polarized HK 4-folds of K3[2]-type. For example there should be precisely 3465 genus 2 curves of degree 11 on a very general Debarre-Voisin 4-fold. For the case K3xK3 it motivates a closed evaluation of Fujiki constants of Chern classes of the tangent bundle of Hilb(K3) in terms of quasi-modular forms. Speaker(s): Georg Oberdieck (University of Bonn)