Abstract
An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety \({Y \subsetneq X}\) . In [1] 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) ≤ 2. 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.
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Barth, W., Peters, C., de Ven, A.V.: Compact complex surfaces. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, vol. 4, pp. 1–288. Springer, Berlin (1984)
Beauville, A.: Some remarks on Kähler manifolds with c 1 = 0. Classification of algebraic and analytic manifolds (Katata, 1982). Progr. Math., vol. 39, pp. 1–2. Birkhäuser Boston, Boston, MA (1983). 86c:32031
Birkenhake C., Lange H.: Complex Abelian Varieties. Springer, Berlin (2004)
Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York (1977). MR0463157 (57#116)
Miyaoka Y.: The Chern classes and Kodaira dimension of a minimal variety. Algebraic Geometry Sendai 10, 449–476 (1985)
Oguiso K.: On the finiteness of fiber-space structures on a Calabi–Yau 3-fold. J. Math. Sci. 106(5), 3320–3335 (2001)
Reichstein, Z., Rogalski, D., Zhang, J.J.: Projectively simple rings. Adv. Math. 203(2), 365–407 (2006). MR2227726 (2007i:16071)
Wilson P.: The Kähler cone on Calabi-Yau threefolds. Inventiones mathematicae 107(1), 561–583 (1992)
Wilson, P.: The Role of c 2 in Calabi-Yau Classification—a Preliminary Survey. Mirror Symmetry II, AMS/IP 1, 381–392 (1997)
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Kirson, A. Wild automorphisms of varieties with Kodaira dimension 0. Ann. Univ. Ferrara 56, 327–333 (2010). https://doi.org/10.1007/s11565-010-0104-2
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DOI: https://doi.org/10.1007/s11565-010-0104-2