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.
All Science Journal Classification (ASJC) codes
- General Mathematics