On the Kodaira dimension of orthogonal modular varieties

Abstract

We prove that up to scaling there are only finitely many integral lattices L of signature (2,n) with n>20 or n=17 such that the modular variety defined by the orthogonal group of L is not of general type. In particular, when n>107, every modular variety defined by an arithmetic group for a rational quadratic form of signature (2,n) is of general type. We also obtain similar finiteness in n>8 for the stable orthogonal groups. As a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order, which proves a conjecture of Gritsenko and Nikulin.

essentially superseding 1309.7121

References in corpus (2)

• The Kodaira dimension of the moduli of K3 surfaces
• Automorphic products of singular weight for simple lattices

Cited by in corpus (17)

• The classification of free algebras of orthogonal modular forms
• The classification of 2-reflective modular forms
• The Kodaira dimension of some moduli spaces of elliptic K3 surfaces
• On the Kodaira Dimension of the Moduli of Deformation Generalised Kummer Varieties
• The singularities and Kodaira dimension of unitary Shimura varieties
• Irregular cusps of orthogonal modular varieties
• Free algebras of modular forms on ball quotients
• Irregular cusps of ball quotients
• Uniruledness of some low-dimensional ball quotients
• On the irrationality of moduli spaces of projective hyperkähler manifolds
• Fano Shimura varieties with mostly branched cusps
• Reflective obstructions of unitary modular varieties
• Reflective modular forms on lattices of prime level
• Reflective modular forms: A Jacobi forms approach
• Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac--Moody algebras
• General type results for moduli of deformation generalised Kummer varieties
• Theta block conjecture for Siegel paramodular forms