Complete curves in the moduli space of polarized K3 surfaces and hyper-K\"ahler manifolds

TL;DR

This paper proves the existence of complete curves in the moduli spaces of polarized K3 surfaces and hyper-K"ahler manifolds for a wide range of degrees and types, extending previous results and constructing new examples.

Contribution

It establishes the presence of complete curves in moduli spaces for all degrees above a certain threshold and for various hyper-K"ahler types, broadening the understanding of their geometric structure.

Findings

Complete curves exist in moduli spaces of polarized K3 surfaces for all degrees ≥62.

Constructs complete curves in moduli spaces of hyper-K"ahler manifolds of various types.

Extends previous results to a wider range of degrees and manifold types.

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 $e\geq 62$ and for some sporadic lower values of $e$ (starting at $14$). We also construct complete curves in the moduli spaces of polarized hyper-K\"ahler manifolds of $\mathrm{K3}^{[n]}$-type or $\mathrm{Kum}_n$-type for all $n\ge 1$ and polarizations of various degrees and divisibilities.