Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture. (2nd April 2019)

• Record Type: Journal Article
• Authors: Lesieutre, John; Satriano, Matthew
• Is Part Of: International mathematics research notices. Volume 2021:Number 10(2021)
• Page Start: 7677
• Page End: 7714
• Publication Date: 2019-04-02
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz067
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: 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.