Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture.

The Kawaguchi–Silverman conjecture predicts that if |$f: X \dashrightarrow X$| is a dominant rational-self map of a projective variety over |$\overline{{\mathbb{Q}}}$|⁠ , and |$P$| is a |$\overline{{\mathbb{Q}}}$| -point of |$X$| with a Zariski dense orbit, then the dynamical and arithmetic degrees of |$f$| coincide: |$\lambda _1(f) = \alpha _f(P)$|⁠. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than |$1$|⁠ , and all endomorphisms of hyper-Kähler manifolds in any dimension. In the latter case, we construct a canonical height function associated with any automorphism |$f: X \to X$| of a hyper-Kähler manifold defined over |$\overline{{\mathbb{Q}}}$|⁠. We additionally obtain results on the periodic subvarieties of automorphisms for which the dynamical degrees are as large as possible subject to log concavity. [ABSTRACT FROM AUTHOR]