On variation of dynamical canonical heights, and Intersection numbers

Abstract

We study families of varieties endowed with polarized canonical eigensystems of several maps, inducing canonical heights on the dominating variety as well as on the "good" fibers of the family. We show explicitely the dependence on the parameter for global and local canonical heights defined by Kawaguchi when the fibers change, extending previous works of J. Silverman and others. Finally, fixing an absolute value $v\in K$ and a variety $V/K$, we descript the Kawaguchi`s canonical local height ${\hat{\lambda }}_{V,E,\mathcal{Q},}(.,v)$ as an intersection number, provided that the polarized system $(V,\mathcal{Q})$ has a certain weak Néron model over Spec$({\mathcal{O}}_v)$ to be defined and under some conditions depending on the special fiber. With this we extend Néron's work strengthening Silverman's results, which were for systems having only one map.

arXiv commentKeywords: Canonical Heights, Local heights, Intersection numbers, Families of varieties, Dynamical systems

References in corpus (6)

• Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space
• Jordan Normal and Rational Normal Form Algorithms
• Examples of dynamical degree equals arithmetic degree
• Canonical metrics of commuting maps
• Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint
• Canonical Height Functions For Monomial Maps