On the Kawaguchi--Silverman Conjecture for birational automorphisms of irregular varieties

TL;DR

This paper investigates the Kawaguchi--Silverman Conjecture for birational automorphisms on irregular varieties, proving it in specific cases and exploring the existence of Zariski dense orbits with explicit examples.

Contribution

It proves the conjecture for varieties with Kodaira dimension zero and high irregularity, and for irregular threefolds, advancing understanding of dynamical degrees and orbits.

Findings

Conjecture holds for varieties with Kodaira dimension zero and irregularity q(X) ≥ dim X - 1.

Confirmed the conjecture for irregular threefolds, with one possible exception.

Provided explicit examples of Zariski dense orbits.

Abstract

We study the main open parts of the Kawaguchi--Silverman Conjecture, asserting that for a birational self-map $f$ of a smooth projective variety $X$ defined over $\overline{\mathbb Q}$, the arithmetic degree $\alpha_f(x)$ exists and coincides with the first dynamical degree $\delta_f$ for any $\overline{\mathbb Q}$-point $x$ of $X$ with a Zariski dense orbit. Among other results, we show that this holds when $X$ has Kodaira dimension zero and irregularity $q(X) \ge \dim X -1$ or $X$ is an irregular threefold (modulo one possible exception). We also study the existence of Zariski dense orbits, with explicit examples.