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Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-mapof a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchiand Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a pointwith well-defined forward orbit, then the growth rate of the height along theorbit 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 oforbits of higher-dimensional subvarieties of $X$. We begin by defining a set ofarithmetic degrees of $f$, independent of the choice of cycle, and we thendevelop the theory of arithmetic degrees in parallel to existing results fordynamical degrees. We formulate several conjectures governing these higherarithmetic degrees, relating them to dynamical degrees.

Higher arithmetic degrees of dominant rational self-maps