Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture. (2nd April 2019) - Lesieutre, John; Satriano, Matthew
Abstract: The Kawaguchi–Silverman conjecture predicts that if f: X X is a dominant rational-self map of a projective variety over Q, and P is a Q -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 → X of a hyper-Kähler manifold defined over 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.