Higher arithmetic degrees of dominant rational self-maps

TL;DR

This paper extends the Kawaguchi-Silverman conjecture from points to higher-dimensional subvarieties, defining and exploring higher arithmetic degrees and their relation to dynamical degrees in algebraic dynamics.

Contribution

It introduces a new framework for higher arithmetic degrees of subvarieties and formulates conjectures linking these to dynamical degrees, expanding the scope of the original conjecture.

Findings

Defined arithmetic degrees for higher-dimensional subvarieties

Developed theory paralleling existing results for dynamical degrees

Formulated conjectures relating higher arithmetic degrees to dynamical degrees

Abstract

Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $\lambda_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $X$. We begin by defining a set of arithmetic degrees of $f$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.