Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties

Abstract:

Let $f:X\to X$ be an endomorphism of a normal projective variety defined over a global field $K$. We prove that for every $x\in X(\bar {K})$, the arithmetic degree $\alpha _f(x)=\lim _{n\to \infty }h_X(f^n(x))^{1/n}$ of $x$ exists, is an algebraic integer, and takes on only finitely many values as $x$ varies over $X(\bar {K})$. Further, if $X$ is an abelian variety defined over a number field, $f$ is an isogeny, and $x\in X(\bar {K})$ is a point whose $f$-orbit is Zariski dense in $X$, then $\alpha _f(x)$ is equal to the dynamical degree of $f$. The proofs rely on two results of independent interest. First, if $D_0,D_1,\ldots \in \mathrm {Div}(X)\otimes \mathbb {C}$ form a Jordan block with eigenvalue $\lambda$ for the action of $f^*$ on $\mathrm {Pic}(X)\otimes \mathbb {C}$, then we construct associated canonical height functions $\hat {h}_{D_k}$ satisfying Jordan transformation formulas $\hat {h}_{D_k}\circ f = \lambda \hat {h}_{D_k} + \hat {h}_{D_{k-1}}$. Second, if $A/\bar {\mathbb {Q}}$ is an abelian variety and $\hat {h}_D$ is the canonical height on $A$ associated to a nonzero nef divisor $D$, then there is a unique abelian subvariety $B_D\subsetneq A$ such that $\hat {h}_D(P)=0$ if and only if $P\in B_D(\bar {\mathbb {Q}})+A(\bar {\mathbb {Q}} )_{\mathrm {tors}}$.