Fixed Points and Entropy of Morphisms on Varieties of Kodaira Dimension Zero

大量的不动点公式证明了映射不动点的重要性。采用渐近的观点，不动点的研究可以对簇上的态射的分类做出重要贡献-后者是代数几何的一个重要的总体目标。此外，这种渐近方法在上同调群上的作用上展示了新的见解。这是与复杂动力系统的一个重要联系，因为态射的熵在这个阶段得到了准确的研究和计算。熵是这些系统的一个不变量，它衡量了它们的混沌程度，最近已成为代数曲面和阿贝尔簇研究的焦点。本课题的主要目的是确定正熵的态射，进而确定正熵的精确值。在这个项目中，目的是对零维Kodaira簇的态射的两个方面进行分类，一方面是渐近不动点行为的可能类型，另一方面是熵的出现值。由于它们在结构上的联系，第一个问题中的见解应该对第二个问题富有成效，反之亦然。更具体地说，它的目的是找到关于态射的条件，以确定确切的不动点行为和熵的具体值。本课题包括三个部分：(1)简单交换簇上的不动点和不动点的几何分布：具有完全不定四元数乘法的交换簇和具有第二类自同态代数的交换簇上的不动点、不动点和熵的几何分布；(2)Kodaira零维簇(K3曲面、Enrique曲面、超椭圆曲面)上的不动点和熵；(3)双态态射的熵。

The big importance of fixed points of maps is demonstrated by a large number of fixed-point formulas. Adopting an asymptotic perspective the investigation of fixed points can contribute essentially to the classification of morphisms on varieties - the latter is an important general aim in Algebraic Geometry. Additionally, this asymptotic approach exhibits new insights in the actions on the cohomology groups. Therein lies an important connection to complex Dynamical Systems, as the entropy of morphisms is studied and computed exactly on this stage. The entropy, an invariant of these systems, measures their level of chaos and has recently been brought into the focus of research on algebraic surfaces and abelian varieties. The main topic is to determine the morphisms of positive entropy and further the exact values of entropy.In this project it is the aim to classify two aspects of morphisms of varieties of Kodaira dimension zero, on the one hand the possible types of asymptotic fixed-point behaviour and on the other hand the occurring values of entropy. Because of their structural link insights in the first issue should be made fruitful for the second and vice versa. More specifically, it is the aim to find conditions on the morphisms to determine the exact fixed-point behaviour and concrete value of entropy. The project has three parts: (1) Fixed points and entropy on simple abelian varieties: Geometric distribution of fixed points, fixed points and entropy on abelian varieties with totally indefinite quaternion multiplication and on abelian varieties with endomorphism algebra of the second kind (2) Fixed points and entropy on varieties of Kodaira dimension zero (K3 surfaces, Enriques surfaces, hyperelliptc surfaces) (3) Entropy of birational morphisms.