The Noether-Lefschetz conjecture and generalizations

• 成果类型: Article
• 署名作者: Bergeron, Nicolas; Li, Zhiyuan; Millson, John; Moeglin, Colette
• 署名单位: Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Fudan University; University System of Maryland; University of Maryland College Park; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0695-z
• 发表日期: 2017
• 页码: 501-552
• 关键词: k3 surfaces; arithmetic quotients; fundamental lemma; automorphic-forms; picard-groups; k-3 surfaces; moduli space; VARIETIES; REPRESENTATIONS; PRODUCTS

摘要

We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles, i.e. sub-arithmetic manifolds of the same type. For compact manifolds this was proved in [3], here we extend the results of [3] to non-compact manifolds. This allows us to apply our results to the moduli spaces of quasi-polarized K3 surfaces.