  We use Noether-Lefschetz theory to study the reduced Gromov--Witten
invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields
strong evidence for a new conjectural formula that expresses Gromov-Witten
invariants of this geometry for arbitrary classes in terms of primitive
classes. The formula generalizes an earlier conjecture by Pandharipande and the
author for K3 surfaces. Using Gromov-Witten techniques we also determine the
generating series of Noether-Lefschetz numbers of a general pencil of
Debarre-Voisin varieties. This reproves and extends a result of Debarre, Han,
O'Grady and Voisin on HLS divisors on the moduli space of Debarre-Voisin
fourfolds.
