Fixpunkte von Endomorphismen komplexer Tori

Jede holomorphe Selbstabbildung auf einem komplexen Torus entspricht der Translation eines eindeutigen Endomorphismus, der dieselbe Fixpunktanzahl wie die Ausgangsfunktion hat. In dieser Dissertation wird nach dem asymptotischen Verhalten der Fixpunktanzahl gefragt, d.h., es wird das Verhalten der F...

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Bibliographic Details
Main Author: Herrig, Thorsten
Contributors: Bauer, Thomas (Prof. Dr.) (Thesis advisor)
Format: Doctoral Thesis
Published: Philipps-Universität Marburg 2017
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Table of Contents: Every holomorphic self-map of a complex torus corresponds to a translation of a unique endomorphism whose number of fixed-points equals the one of the original function. In this dissertation one asks for the asymptotic behaviour of the number of fixed-points, i.e., one studies how the number of fixed-points behaves when an endomorphism is iterated. Using the Holomorphic Lefschetz Fixed-Point Formula, three possible and in fact occurring types of the fixed-point behaviour on two-dimensional complex tori are determined. The endomorphism algebras of simple abelian varieties of arbitrary dimension are entirely classified. In terms of the concrete structure of an endomorphism on a simple abelian variety of arbitrary dimension and by use of a second version of the Holomorphic Lefschetz Fixed-Point Formula, criteria are developed in this thesis to determine the exact behaviour of the number of fixed-points for the iterated endomorphism. The eigenvalues of the endomorphisms are important for these results. Further, they also determine the spectral radius of the action on the cohomology groups induced by the endomorphism which defines the entropy of the endomorphism. Therefore, criteria are given in this paper to decide whether the entropy of an endomorphism is zero or positive. Afterwards, the questions about the asymptotic behaviour of the number of fixed-points are answered for endomorphisms and automorphisms on K3 surfaces and Kummer varieties.