Wild Automorphisms and Abelian Varieties

Abstract

An automorphism $\sigma$ of a projective variety $X$ is said to be \textit{wild} if $\sigma(Y)\neq Y$ for every non-empty subvariety $Y\subsetneq X$. In MR2227726 Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if $X$ is an irreducible projective variety admitting a wild automorphism then $X$ is an abelian variety, and proved this conjecture for $\dim(X)\leq2$. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension $0$ admitting wild automorphisms. This essentially reduces the Kodaira dimension $0$ case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.