K3曲面的自同构和Salem数

K3 surfaces are important surfaces, and have been used in the study of many problems in algebraic geometry. Salem numbers are a special kind of algebraic integers, and have appeared in many areas in mathematics. In 2002, McMullen observed that automorphisms of K3 surfaces and Salem numbers are closely related. This relation has been studied from many different aspects, but there are still many famous unsolved problems. This project studies several problems which are very closely related to this relation, and the themes are the followings: 1. to determine the minimum of positive topological entropies of automorphisms of Enriques surfaces; 2. to study the relation between automorphisms of supersingular K3 surfaces and automorphisms of non-projective K3 surfaces via Salem numbers; 3. to study the relation between singular K3 surfaces and Salem numbers with the help of elliptic fibrations. The study of this project will not only help to deepen the understanding of the relation between K3 surfaces and Salem numbers, but also help to enhance the interactions among algebraic geometry, number theory, complex dynamics, group theory and so on.

K3曲面是一类重要的曲面, 被应用于研究代数几何里的许多问题. Salem数是一类特别的代数整数, 出现在多个数学领域中. 在2002年, McMullen观察到K3曲面的自同构和Salem数有一种密切的关系. 这种关系已经被广泛地研究, 但是仍然有诸多著名的问题待解决. 本项目将围绕这种关系研究几个问题, 主要研究内容有：1. 确定Enriques曲面的自同构的最小正拓扑熵; 2. 以Salem数为切入点，研究超奇异K3曲面的自同构和非射影K3曲面的自同构之间的关系; 3. 借助于椭圆纤维化，研究奇异K3曲面的自同构和Salem数的关系. 本项目的研究不仅有助于加深人们对K3曲面和Salem数的关系的理解, 而且有助于促进代数几何、数论、复动力系统、群论等多个领域的交叉融合.

通过研究K3曲面的自同构、Enriques曲面的自同构、Salem数三者之间的关系，我们确定了Enriques曲面自同构的最小正拓扑熵。该成果加上McMullen之前的系列工作就彻底解决Enriques-Kodaira复曲面分类中各类复曲面自同构的最小正拓扑熵问题。通过研究2-初等K3曲面的自同构群，我们取得了Coble在1919年提出的一个关于经典Coble曲面自同构群的问题的新进展。我们证明了Hassett除子两两相交非空, 并且每一个Hassett除子都包含了由有理三次四流形构成的维数为18的子簇。我们对光滑三次三流形的自同构群进行了完整的分类。我们得到了正特征上光滑完备代数簇的向量丛的Hodge上同调blow-up公式，进而证明了Hochschild-Kostant-Rosenberg谱序列的E2退化性的blow-up不变性。我们构造出了第一个满足如下两条性质的复射影有理流形的例子:(1)该复射影有理流形的自同构群是离散的且不能被有限个元素生成;(2)该复射影有理流形拥有无限多个互不同构的实形式。