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Cones of Noether-Lefschetz divisors and moduli spaces of hyperkähler manifolds

Ignacio Barros, Pietro Beri, Laure Flapan, Brandon Williams

TL;DR

This work advances the birational study of orthogonal modular varieties by deriving a general formula for generators of the NL-cone and then explicitly computing the NL-cone and its extremal rays for key moduli spaces of polarized K3 surfaces and hyperkähler manifolds of known deformation type at low degrees. It leverages the correspondence between the rational Picard group and vector-valued modular forms to obtain concrete, polyhedral generators and provides a Sage package, \texttt{heegner_cones}, to implement these computations, including explicit boundary divisors for arbitrarily large polarizations. By analyzing the NL-cone together with Eisenstein-series bounds, the paper proves uniruledness results for moduli spaces of ${\rm OG6}$-type and ${\rm Kum}_n$-type hyperkähler manifolds, and establishes isotriviality for families of Kum$_2$-type fourfolds with polarization degree $2$ and divisibility $2$ over projective bases. Overall, the results connect modular-form techniques to the birational geometry of moduli spaces, producing explicit, computable descriptions of NL-cones and enabling new geometric conclusions about the moduli of K3s and hyperkähler manifolds.

Abstract

We give a general formula for generators of the NL-cone, the cone of effective linear combinations of irreducible components of Noether-Lefschetz divisors, on an orthogonal modular variety. We then fully describe the NL-cone and its extremal rays in the cases of moduli spaces of polarized K3 surfaces and hyperkähler manifolds of known deformation type for low degree polarizations. Moreover, we exhibit explicit divisors in the boundary of NL-cones for polarizations of arbitrarily large degrees. Additionally, we study the NL-positivity of the canonical class for these modular varieties. As a consequence, we obtain uniruledness results for moduli spaces of primitively polarized hyperkähler manifolds of ${\rm{OG6}}$ and ${\rm{Kum}}_n$-type. Finally, we show that any family of polarized hyperkähler fourfolds of ${\rm{Kum}}_2$-type with polarization of degree $2$ and divisibility $2$ over a projective base is isotrivial.

Cones of Noether-Lefschetz divisors and moduli spaces of hyperkähler manifolds

TL;DR

This work advances the birational study of orthogonal modular varieties by deriving a general formula for generators of the NL-cone and then explicitly computing the NL-cone and its extremal rays for key moduli spaces of polarized K3 surfaces and hyperkähler manifolds of known deformation type at low degrees. It leverages the correspondence between the rational Picard group and vector-valued modular forms to obtain concrete, polyhedral generators and provides a Sage package, \texttt{heegner_cones}, to implement these computations, including explicit boundary divisors for arbitrarily large polarizations. By analyzing the NL-cone together with Eisenstein-series bounds, the paper proves uniruledness results for moduli spaces of -type and -type hyperkähler manifolds, and establishes isotriviality for families of Kum-type fourfolds with polarization degree and divisibility over projective bases. Overall, the results connect modular-form techniques to the birational geometry of moduli spaces, producing explicit, computable descriptions of NL-cones and enabling new geometric conclusions about the moduli of K3s and hyperkähler manifolds.

Abstract

We give a general formula for generators of the NL-cone, the cone of effective linear combinations of irreducible components of Noether-Lefschetz divisors, on an orthogonal modular variety. We then fully describe the NL-cone and its extremal rays in the cases of moduli spaces of polarized K3 surfaces and hyperkähler manifolds of known deformation type for low degree polarizations. Moreover, we exhibit explicit divisors in the boundary of NL-cones for polarizations of arbitrarily large degrees. Additionally, we study the NL-positivity of the canonical class for these modular varieties. As a consequence, we obtain uniruledness results for moduli spaces of primitively polarized hyperkähler manifolds of and -type. Finally, we show that any family of polarized hyperkähler fourfolds of -type with polarization of degree and divisibility over a projective base is isotrivial.
Paper Structure (19 sections, 20 theorems, 127 equations, 2 tables)

This paper contains 19 sections, 20 theorems, 127 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Applications to moduli
  3. Uniruledness results
  4. Preliminaries
  5. Heegner and NL divisors
  6. Rational Picard group of orthogonal modular varieties
  7. Effective and NL cones
  8. Generators of the NL cone
  9. Moduli of K3 surfaces
  10. Hyperkähler fourfolds of ${\rm{K3}}^{[2]}$-type
  11. Cubic Fourfolds
  12. Uniruledness
  13. Moduli of ${\rm{OG6}}$-type hyperkähler manifolds
  14. Moduli of ${\rm{Kum}}_n$-type hyperkähler manifolds
  15. Declarations
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Lambda$ be an even lattice of signature $(2,n)$ with $n\ge 4$ splitting off two copies of the hyperbolic plane and $X=\mathcal{D}/\widetilde{\rm{O}}^+(\Lambda)$ its modular variety. Fixing $b$ as above, there are explicit bounds $\Xi$ and $\Omega$, depending on $k$, the discriminant of $\Lambd

Theorems & Definitions (44)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Theorem 2.2: Bor99McG03Bru02Bru14BLMM17BZ24
  • Remark 2.3
  • Lemma 3.1
  • proof
  • ...and 34 more