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

• Applicant: Dr. Thorsten Herrig
• Subject Area: Mathematics
• Term: from 2018 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415052336

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.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Robert Lazarsfeld, Ph.D.