Cones of Noether-Lefschetz divisors and moduli spaces of 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 $OG6$ and ${\mathrm{Kum}}_n$-type. Finally, we show that any family of polarized hyperkähler fourfolds of ${\mathrm{Kum}}_2$-type with polarization of degree $2$ and divisibility $2$ over a projective base is isotrivial.

v3: Minor changes. Final version. To appear in Math. Ann