K3曲面及其相关问题的研究

K3曲面是一类重要的曲面, 有着丰富的几何和算术性质, 与代数几何以及其它数学领域的很多课题密切相关. 本项目将围绕与K3曲面密切相关的几个未解决问题展开研究, 主要研究内容有: 1. K3曲面的自同构和Salem数的关系, 包括实现Lehmer数的K3曲面自同构的具体几何描述; 2. Coble在1919年提出的关于经典Coble曲面自同构群的一个问题; 3. 三次四流形的有理性. 本项目的研究不仅有助于加深人们对K3曲面以及相关课题的认识, 并且对代数几何、数论、复动力系统、群论等多个领域的交叉融合会起到很好的促进作用.

K3 surfaces are important surfaces, have rich geometric and arithmetic properties, and are closely related to many topics in algebraic geometry and other mathematical fields. This project studies several unsolved problems closely related to K3 surfaces, the themes are the followings: 1. The close relations between automorphisms of K3 surfaces and Salem numbers, including explicit geometric descriptions of K3 surface automorphisms which realize the Lehmer number; 2. A question raised by Coble in 1919 on the automorphism groups of classical Coble surfaces; 3. The rationality of cubic fourfolds. The study of this project will not only help to deepen the understanding of K3 surfaces and related topics, but also help to enhance the interactions among algebraic geometry, number theory, complex dynamics, group theory and so on.