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

TL;DR

The article advances the Kawaguchi--Silverman conjecture for birational self-maps on irregular varieties by establishing the existence and value of arithmetic degrees for points with dense orbits in new irregular contexts. It proves KSC (2) when $\kappa(X)=0$ and $q(X)\ge \dim X-1$, with sharper statements depending on $q(X)$; it extends these results to irregular threefolds, identifying an exception when $X$ is covered by rational surfaces, and it analyzes the existence of Zariski dense $\overline{\mathbb Q}$-orbits, providing a complete classification framework tied to the Kodaira dimension and irregularity and constructing explicit examples realizing the predicted phenomena. The methods combine dynamical degrees, height theory, and birational geometry, including Albanese fibrations and minimal-model techniques, to transfer dynamical questions to more tractable bases or fibers. Overall, the work broadens the scope of KSC (2) in the irregular setting and clarifies when Zariski dense orbits can occur, offering concrete examples across a range of Kodaira dimensions and irregularities. These results have significant implications for understanding the arithmetic dynamics of birational maps beyond the realm of morphisms and abelian varieties.

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 $α_f(x)$ exists and coincides with the first dynamical degree $δ_f$ 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.

Theorems & Definitions (63)

• Conjecture 1.1: Kawaguchi--Silverman Conjecture (KSC)
• Remark 1.2
• Conjecture 1.4: See e.g.JSXZ21
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 3.1
• proof
• Lemma 3.2
• proof