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

The Kawaguchi-Silverman conjecture predicts that if f: X → X is a dominant rational-self map of a projective variety over ρ, and P is a ρ-point of X with a Zariski dense orbit, then the dynamical and arithmetic degrees of f coincide: λ 1(f) = α 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 ρ. We additionally obtain results on the periodic subvarieties of automorphisms for which the dynamical degrees are as large as possible subject to log concavity.