The arithmetic degree conjectures of Kawaguchi and Silverman

Let $K$) be an algebraically closed field with a suitable theory of height functions, let $X$ be a smooth projective variety over $K$, let $\varphi:X\dashrightarrow X$ be a dominant rational map, and let $x\in X(K)$ have a well-defined forward orbit

$$\mathcal{O}_{\varphi}(x)=\{\varphi^n(x):n\ge 0\}.$$

For an ample divisor $H$ on $X$, write

$$\delta(\varphi)=\lim_{n\to\infty}\left(((\varphi^n)^*H)\cdot H^{d-1}\right)^{1/n}$$

and

$$\alpha(\varphi,x)=\lim_{n\to\infty}h_{X,H}(\varphi^n(x))^{1/n},$$

when the latter limit exists. Kawaguchi--Silverman conjectures. (a) The limit defining $\alpha(\varphi,x)$ exists. (b) $\alpha(\varphi,x)$ is an algebraic integer. (c) The set

$$\{\alpha(\varphi,x):x\in X\text{ such that }\mathcal{O}_{\varphi}(x)\text{ exists}\}$$

is finite. (d) If $\mathcal{O}_{\varphi}(x)$ is Zariski dense in $X$, then $\alpha(\varphi,x)=\delta(\varphi)$. These conjectures relate arithmetic complexity of orbits to the geometric complexity measured by the dynamical degree; in the paper's setting, the inequality $\alpha(\varphi,x)\leq\delta(\varphi)$ is known, and the conjectures are proved for abelian varieties.