Lagrangian fibrations on Hilbert schemes of points on K3 surfaces

Abstract

Let ${\mathrm{Hilb}}^{g}S$ be the Hilbert scheme of $g$ points on a K3 surface $S$. Suppose that $\mathrm{Pic}S\cong \mathbb{Z}C$ where $C$ is a smooth curve with $C^2=2(g-1)n^2$. We prove that ${\mathrm{Hilb}}^{g}S$ is a Lagrangian fibration.

21 pages, original (stronger) version of Theorem 2 proved