Wild automorphisms of projective varieties, the maps which have no invariant proper subsets

Abstract

Let $X$ be a projective variety and $\sigma$ a wild automorphism on $X$, i.e., whenever $\sigma (Z)=Z$ for a non-empty Zariski-closed subset $Z$ of $X$, we have $Z=X$. Then $X$ is conjectured to be an abelian variety with $\sigma$ of zero entropy (and proved to be so when $\dim X\le 2$) by Z. Reichstein, D. Rogalski and J. J. Zhang in their study of projectively simple rings. This conjecture has been generally open for more than a decade. In this note, we confirm this original conjecture when $\dim X\le 3$ and $X$ is not a Calabi-Yau threefold, and also show that $\sigma$ is of zero entropy when $\dim X\le 4$ and the Kodaira dimension $\kappa (X)\ge 0$.

arXiv commentAdvances in Mathematics, Volume 396, 12 February 2022, 108173

References in corpus (6)

• Criteria for σ-ampleness
• Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy
• Polarized endomorphisms of uniruled varieties (with Appendix by Y. Fujimoto and N. Nakayama)
• Compact Kähler manifolds admitting large solvable groups of automorphisms
• Automorphism groups and anti-pluricanonical curves
• Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic -- A remark to a paper of Dinh-Oguiso-Zhang

Cited by in corpus (1)

• Wild automorphisms of compact complex spaces of lower dimensions