Effective Eigendivisors and the Kawaguchi-Silverman Conjecture

TL;DR

This paper investigates the properties of eigendivisors associated with surjective endomorphisms of projective varieties, exploring their base loci and spectral characteristics to shed light on the Kawaguchi-Silverman conjecture.

Contribution

It introduces a linear algebraic condition on morphisms and analyzes eigendivisors with zero Iitaka dimension to advance understanding of the conjecture.

Findings

Characterization of eigendivisors with eigenvalue equal to spectral radius

Analysis of the base locus of such eigendivisors

Identification of a linear algebraic condition relevant to the conjecture

Abstract

Let $f\colon X\rightarrow X$ be a surjective endomorphism of a normal projective variety defined over a number field. The dynamics of $f$ may be studied through the dynamics of the linear action $f^*\colon Pic(X)_\mathbb{R}\rightarrow Pic(X)_\mathbb{R}$, which are governed by the spectral theory of $f^*$. Let $\lambda_1(f)$ be the spectral radius of $f^*$. We study $\mathbb{Q}$-divisors $D$ with $f^*D=\lambda_1(f) D$ and $\kappa(D)=0$ where $\kappa(D)$ is the Iitaka dimension of the divisor $D$. 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. Finally we identify a linear algebraic condition on surjective morphisms that may be useful in proving instances of the Kawaguchi-Silverman conjecture.