Applications of the minimal modelprogram in arithmetic dynamics

Abstract

Let F be a surjective endomorphism of a normal projective variety X defined over a number field. The dynamics of F may be studied through the dynamics of the linear action of an associated linear pull-back action on divisors. This linear action is governed by the spectral theory of pull-back.

We first study eigen-divisors (that is eigenvectors of the pullback action) that have Iitaka dimension 0. We analyze the base locus of such divisors and interpret the set of small eigenvalues in terms of the canonical heights of Jordan blocks described by Kawaguchi and Silverman. We identify a linear algebraic condition on surjective morphisms that may be useful in proving instances of the Kawaguchi-Silverman conjecture. We prove the Kawaguchi-Silverman conjecture and verify the aforementioned linear algebraic condition holds for projective bundles over an elliptic curve that are direct sums of Atiyah bundles. This represents new progress in the last remaining case of the Kawaguchi-Silverman conjecture for projective bundles over curves.

By a result of Silverman and Kawaguchi, the arithmetic degree of a point P on X is an eigenvalue of the associated linear-pull back mapping. We give examples of endomorphisms of abelian varieties that possess an eigenvalue which is not an arithmetic degree. On the other hand, we show using the minimal model program that if X is a simplicial toric variety then every eigenvalue of the linear pullback action is an arithmetic degree.

Finally, we give a program to study the arithmetic dynamics of higher-dimensional projective varieties using the minimal model program. In particular, we describe how one might use the minimal model program to determine if certain surjective morphisms have a dense set of pre-periodic points, and how to study the Medvedev-Scanlon conjecture for certain surjective endomorphisms using the minimal model program.