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The Noether-Lefschetz conjecture and generalizations

2016; Springer Science+Business Media; Volume: 208; Issue: 2 Linguagem: Inglês

10.1007/s00222-016-0695-z

1432-1297

Nicolas Bergeron, Zhiyuan Li, John J. Millson, Colette Mœglin,

Geometry and complex manifolds

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.