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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-Kähler manifolds of \(\mathrm {K3}^{[n]}\) -type or \(\mathrm {Kum}_n\) -type for all \(n\ge 1\) and polarizations of various degrees and divisibilities.

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Complete Curves in the Moduli Space of Polarized K3 Surfaces and Hyper-Kähler Manifolds

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  • 发表日期：

    2025年

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  • Olivier Debarre,
  • Emanuele Macrì

摘要

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-Kähler manifolds of \(\mathrm {K3}^{[n]}\) -type or \(\mathrm {Kum}_n\) -type for all \(n\ge 1\) and polarizations of various degrees and divisibilities.