Wild automorphisms of projective varieties, the maps which have no invariant proper subsets. (English) Zbl 1484.14082
Authors’ 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 \(\dim X \leq 2\)) by Z. Reichstein et al. in their study of projectively simple rings [Adv. Math. 203, No. 2, 365–407 (2006; Zbl 1120.16040)]. This conjecture has been generally open for more than a decade. In this note, we confirm this original conjecture when \(\dim X \leq 3\) and \(X\) is not a Calabi-Yau threefold, and also show that \(\sigma\) is of zero entropy when \(\dim X\leq 4\) and the Kodaira dimension \(\kappa(X) \geq 0\).
Reviewer: Jin-Xing Cai (Beijing)
MSC:
| 14J50 | Automorphisms of surfaces and higher-dimensional varieties |
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37F80 | Higher-dimensional holomorphic and meromorphic dynamics |
Citations:
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