1. Background

Recently, arithmetic dynamics of dominant self-maps and their iterations of a normal projective variety defined over an algebraic closure of a number field (hence all defined over a finite extension of the rational number field) caught much attention by several researchers, especially in connection with Kawaguchi-Silverman Conjecture (KSC).  KSC concerns relations between arithmetic and geometric complexities of a dominant self-map of a variety, and among other things, it predicts the arithmetical complexity of a point under iteration (arithmetic dynamical degree) is measured by the complex dynamical (or complex algebro-geometric) complexity of the dominant selfmaps (dynamical degree), if the forwarded orbit of the point is Zariski dense. There are now fairly ample affirmative answers for KSC when the self-map is a morphism due mainly by Kawaguchi and Silverman. However, there seems no known series of examples of strict Calabi-Yau manifolds of both dimension and Picard number are greater than 2 for which KSC for a morphism is fully proved, while KSC for an automorphism of a projective hyperkaehler manifold is proved by Leiseutre and Satriano. So, there still remain many interesting open problems for KSC for higher dimensional varieties and morphisms even in concrete cases. On the other hand, very little seems known and KSC is not yet much studied for dominant self-maps or even for birational self-maps, except basic works of Amerik and Matszawa. For instance, it is unknown if the “limsup” in the definition of arithmetic degree is really “limsup” or actually “limit” , nor algebraicity of the arithmetic dynamical degree; one could even expect a negative answer from the current work in complex dynamics. In a more concrete level, it is also unknown KSC for a birational automorphism of a projective hyperkaehler manifold. This course is an introduction to this current research area also for me.
 

2. Outline

In this course, in a first few lectures, I would like to introduce Weil height and the so called Weil Height Machine following M. Hindry and J.H. Silverman. Then I would like to introduce the notion of arithmetic degree for a self-morphism of a normal projective variety and its basic properties following Kawaguchi-Silverman. Then as concrete important cases, I would like to explain the full solution of KSC for an endomorphism of a surface and an endomorphism of an abelian variety (Kawaguchi-Silverman-Sano-Matsuzawa-Shibata) and a self-morphism for hyperkaehler manifold by Leiseutre and Satriano. Then (probably the last one third of lectures) we start to generalize for dominant self-maps or birational self-maps. This part is under progress by many researchers, and, except some basics, the choice of topics will be fixed later.

3. Credit: 1

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