On the Kodaira dimension of orthogonal modular varieties

• 成果类型: Article
• 署名作者: Ma, Shouhei
• 署名单位: Institute of Science Tokyo; Tokyo Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0781-x
• 发表日期: 2018
• 页码: 859-911
• 关键词: kac-moody algebras; k3 surfaces; automorphic products; singular weight; forms; SPACE; proportionality; CLASSIFICATION; MANIFOLDS; lattices

摘要

We prove that up to scaling there are only finitely many integral lattices L of signature (2, n) with or such that the modular variety defined by the orthogonal group of L is not of general type. In particular, when , 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 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.