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

Abstract

Let X be a projective variety and σ a wild automorphism on X, i.e., whenever σ(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 σ of zero entropy (and proved to be so when dim⁡X≤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≤3 and X is not a Calabi-Yau threefold, and also show that σ is of zero entropy when dim⁡X≤4 and the Kodaira dimension κ(X)≥0.