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

小平零维变体的不动点和态射熵

• 批准号: 415052336
• 负责人: Dr. Thorsten Herrig
• 金额: --
• 依托单位: Arbeitsgruppe Mathematik und ihre Didaktik
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/415052336?language=en
• 关键词: Fixed; Points; Entropy; Morphisms; Varieties

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.

大量的不动点公式证明了映射不动点的重要性。采用渐近的观点调查的不动点可以基本上有助于分类的态射的品种-后者是一个重要的一般目标代数几何。此外，这种渐近方法在上同调群上的作用方面表现出新的见解。这与复杂动力系统有着重要的联系，因为态射的熵是在这个阶段被研究和精确计算的。熵，这些系统的不变量，衡量他们的混乱程度，最近已被纳入研究的重点代数曲面和阿贝尔品种。本课题的主要目的是确定正熵的态射并进一步确定熵的精确值，本课题的目的是将科代拉零维簇的态射分为两个方面，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。由于它们之间的结构联系，对第一个问题的见解应该对第二个问题产生成果，反之亦然。更具体地说，它的目的是找到态射的条件，以确定确切的不动点行为和具体的熵值。该项目包括三个部分：（1）单阿贝尔簇上的不动点和熵：全不定四元数乘法阿贝尔簇和第二类自同态代数阿贝尔簇上的不动点、不动点和熵的几何分布;（2）零维科代拉簇（K3曲面、Enriques曲面、超椭圆曲面）上的不动点和熵;（3）双有理态射的熵。