Complete curves in the moduli space of polarized K3 surfaces and hyper-Kähler manifolds

Abstract

Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who treated the case $e=14$), we prove that the moduli space of polarized K3 surfaces of degree $2e$ contains complete curves for all $e\ge 62$ and for some sporadic lower values of $e$ (starting at $14$). We also construct complete curves in the moduli spaces of polarized hyper-Kähler manifolds of ${K3}^{[n]}$-type or ${\mathrm{Kum}}_n$-type for all $n\ge 1$ and polarizations of various degrees and divisibilities.

20 pages. v2: fixed and clarified the computations for lower degrees; improved the exposition