# On the Kodaira dimension of orthogonal modular varieties

## TL;DR

This paper establishes finiteness results for orthogonal modular varieties of certain signatures, showing that most such varieties are of general type, and confirms a conjecture regarding reflective modular forms.

## Contribution

It proves finiteness of lattices with non-general type modular varieties and confirms a conjecture on reflective modular forms, advancing understanding of orthogonal modular varieties.

## Key findings

- Finiteness of lattices with non-general type varieties for n>20 or n=17
- All varieties are of general type for n>107
- Finiteness of lattices with reflective modular forms of bounded vanishing order

## 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.