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

大量的不动点公式证明了地图不动点的重要性。采用渐近的观点来研究不动点的问题，本质上有助于变体上态射的分类——后者是代数几何中的一个重要的一般目标。此外，这种渐近方法在上同调群上的作用方面也有了新的认识。这是与复杂动力系统的重要联系，因为在这个阶段对态射熵进行了精确的研究和计算。熵是这些系统的一个不变量，用来衡量它们的混沌程度，最近成为代数曲面和阿贝尔变体研究的焦点。主要的课题是确定正熵的态射，并进一步确定熵的确切值。在这个项目中，目的是对Kodaira维零的变体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，另一方面是熵的发生值。由于它们的结构联系，在第一个问题上的见解应该在第二个问题上取得成果，反之亦然。更具体地说，它的目的是找到态射的条件，以确定精确的不动点行为和熵的具体值。本课题包括三个部分：(1)简单阿贝尔变体上的不动点和熵：完全不定四元数乘法阿贝尔变体上的不动点和熵的几何分布、完全不定四元数乘法阿贝尔变体上的不动点和熵、第二类自同态代数阿贝尔变体上的不动点和熵；(2)Kodaira维零（K3曲面、Enriques曲面、超椭圆曲面）变体上的不动点和熵；

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.