A high-codimensional Yuan's inequality and its application to higher arithmetic degrees

In this article, we consider a dominant rational self-map $f:X⇢X$ of a normal projective variety defined over a number field. We study the arithmetic degree ${\alpha }_k(f)$ for $f$ and ${\alpha }_k(f,V)$ of a subvariety $V$, which generalize the classical arithmetic degree ${\alpha }_1(f,P)$ of a point $P$. We generalize Yuan’s arithmetic version of Siu’s inequality to higher codimensions and utilize it to demonstrate the existence of the arithmetic degree ${\alpha }_k(f)$. Furthermore, we establish the relative degree formula ${\alpha }_k(f)=\max \{{\lambda }_k(f),{\lambda }_{k-1}(f)\}$. In addition, we prove several basic properties of the arithmetic degree ${\alpha }_k(f,V)$ and establish the upper bound ${\bar{\alpha }}_{k+1}(f,V)\le \max \{{\lambda }_{k+1}(f),{\lambda }_k(f)\}$, which generalizes the classical result ${\bar{\alpha }}_f(P)\le {\lambda }_1(f)$. Finally, we discuss a generalized version of the Kawaguchi-Silverman conjecture that was proposed by Dang et al, and we provide a counterexample to this conjecture.